LMFDB - Embedded newform 144.3.m.c.19.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,3,Mod(19,144)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("144.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	144.m (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.92371580679$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			19.6
Root			$1.78012 + 0.911682i$ of defining polynomial
Character	$\chi$	$=$	144.19
Dual form			144.3.m.c.91.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.78012 - 0.911682i) q^{2} +(2.33767 - 3.24581i) q^{4} +(-1.00772 + 1.00772i) q^{5} +10.0236 q^{7} +(1.20220 - 7.90915i) q^{8} +O(q^{10})$	\(q+(1.78012 - 0.911682i) q^{2} +(2.33767 - 3.24581i) q^{4} +(-1.00772 + 1.00772i) q^{5} +10.0236 q^{7} +(1.20220 - 7.90915i) q^{8} +(-0.875146 + 2.71259i) q^{10} +(-2.26517 - 2.26517i) q^{11} +(-6.88229 - 6.88229i) q^{13} +(17.8432 - 9.13830i) q^{14} +(-5.07058 - 15.1753i) q^{16} +22.3801 q^{17} +(-16.8918 + 16.8918i) q^{19} +(0.915151 + 5.62660i) q^{20} +(-6.09740 - 1.96717i) q^{22} -33.2007 q^{23} +22.9690i q^{25} +(-18.5258 - 5.97686i) q^{26} +(23.4318 - 32.5346i) q^{28} +(24.6412 + 24.6412i) q^{29} +41.3761i q^{31} +(-22.8613 - 22.3911i) q^{32} +(39.8394 - 20.4036i) q^{34} +(-10.1010 + 10.1010i) q^{35} +(-6.60031 + 6.60031i) q^{37} +(-14.6695 + 45.4693i) q^{38} +(6.75875 + 9.18170i) q^{40} -47.1477i q^{41} +(-48.8218 - 48.8218i) q^{43} +(-12.6475 + 2.05709i) q^{44} +(-59.1013 + 30.2685i) q^{46} +45.6048i q^{47} +51.4717 q^{49} +(20.9404 + 40.8876i) q^{50} +(-38.4271 + 6.25007i) q^{52} +(-25.1401 + 25.1401i) q^{53} +4.56532 q^{55} +(12.0503 - 79.2779i) q^{56} +(66.3292 + 21.3994i) q^{58} +(-6.23974 - 6.23974i) q^{59} +(35.9513 + 35.9513i) q^{61} +(37.7219 + 73.6546i) q^{62} +(-61.1095 - 19.0167i) q^{64} +13.8709 q^{65} +(10.2045 - 10.2045i) q^{67} +(52.3174 - 72.6417i) q^{68} +(-8.77208 + 27.1898i) q^{70} -11.9529 q^{71} -111.332i q^{73} +(-5.73197 + 17.7667i) q^{74} +(15.3401 + 94.3149i) q^{76} +(-22.7051 - 22.7051i) q^{77} -4.46031i q^{79} +(20.4022 + 10.1827i) q^{80} +(-42.9837 - 83.9287i) q^{82} +(-10.1751 + 10.1751i) q^{83} +(-22.5530 + 22.5530i) q^{85} +(-131.419 - 42.3988i) q^{86} +(-20.6388 + 15.1924i) q^{88} -21.9364i q^{89} +(-68.9850 - 68.9850i) q^{91} +(-77.6124 + 107.763i) q^{92} +(41.5771 + 81.1821i) q^{94} -34.0444i q^{95} +107.309 q^{97} +(91.6260 - 46.9259i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} + 12 q^{8}+O(q^{10}) $ 16 * q + 12 * q^4 + 12 * q^8	$ 16 q + 12 q^{4} + 12 q^{8} - 56 q^{10} - 32 q^{11} + 44 q^{14} + 32 q^{16} - 32 q^{19} - 80 q^{20} + 32 q^{22} + 128 q^{23} + 100 q^{26} - 120 q^{28} - 32 q^{29} - 160 q^{32} + 96 q^{34} - 96 q^{35} - 96 q^{37} - 168 q^{38} + 48 q^{40} + 160 q^{43} - 88 q^{44} + 136 q^{46} + 112 q^{49} + 236 q^{50} - 48 q^{52} + 160 q^{53} - 256 q^{55} + 224 q^{56} + 144 q^{58} + 128 q^{59} - 32 q^{61} + 276 q^{62} - 408 q^{64} + 32 q^{65} + 320 q^{67} + 448 q^{68} - 384 q^{70} - 512 q^{71} - 348 q^{74} + 72 q^{76} - 224 q^{77} - 552 q^{80} - 40 q^{82} + 160 q^{83} + 160 q^{85} - 528 q^{86} + 480 q^{88} - 480 q^{91} - 496 q^{92} + 312 q^{94} + 440 q^{98}+O(q^{100}) $ 16 * q + 12 * q^4 + 12 * q^8 - 56 * q^10 - 32 * q^11 + 44 * q^14 + 32 * q^16 - 32 * q^19 - 80 * q^20 + 32 * q^22 + 128 * q^23 + 100 * q^26 - 120 * q^28 - 32 * q^29 - 160 * q^32 + 96 * q^34 - 96 * q^35 - 96 * q^37 - 168 * q^38 + 48 * q^40 + 160 * q^43 - 88 * q^44 + 136 * q^46 + 112 * q^49 + 236 * q^50 - 48 * q^52 + 160 * q^53 - 256 * q^55 + 224 * q^56 + 144 * q^58 + 128 * q^59 - 32 * q^61 + 276 * q^62 - 408 * q^64 + 32 * q^65 + 320 * q^67 + 448 * q^68 - 384 * q^70 - 512 * q^71 - 348 * q^74 + 72 * q^76 - 224 * q^77 - 552 * q^80 - 40 * q^82 + 160 * q^83 + 160 * q^85 - 528 * q^86 + 480 * q^88 - 480 * q^91 - 496 * q^92 + 312 * q^94 + 440 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.78012	−	0.911682i	0.890061	−	0.455841i
$2$	1.78012	−	0.911682i	0.890061	−	0.455841i
$3$		0			0
$3$		0			0
$4$	2.33767	−	3.24581i	0.584418	−	0.811453i
$5$	−1.00772	+	1.00772i	−0.201544	+	0.201544i	−0.800661	−	0.599117i	$-0.795518\pi$
$5$	−1.00772	+	1.00772i	−0.201544	+	0.201544i	0.599117	+	0.800661i	$0.295518\pi$
$6$		0			0
$7$	10.0236			1.43194			0.715969	−	0.698133i	$-0.245985\pi$
$7$	10.0236			1.43194			0.715969	+	0.698133i	$0.245985\pi$
$8$	1.20220	−	7.90915i	0.150274	−	0.988644i
$9$		0			0
$10$	−0.875146	+	2.71259i	−0.0875146	+	0.271259i
$11$	−2.26517	−	2.26517i	−0.205925	−	0.205925i	0.596608	−	0.802533i	$-0.296515\pi$
$11$	−2.26517	−	2.26517i	−0.205925	−	0.205925i	−0.802533	+	0.596608i	$0.796515\pi$
$12$		0			0
$13$	−6.88229	−	6.88229i	−0.529407	−	0.529407i	0.390989	−	0.920395i	$-0.372133\pi$
$13$	−6.88229	−	6.88229i	−0.529407	−	0.529407i	−0.920395	+	0.390989i	$0.872133\pi$
$14$	17.8432	−	9.13830i	1.27451	−	0.652736i
$15$		0			0
$16$	−5.07058	−	15.1753i	−0.316911	−	0.948455i
$17$	22.3801			1.31648			0.658240	−	0.752809i	$-0.271301\pi$
$17$	22.3801			1.31648			0.658240	+	0.752809i	$0.271301\pi$
$18$		0			0
$19$	−16.8918	+	16.8918i	−0.889041	+	0.889041i	−0.994431	−	0.105390i	$-0.966391\pi$
$19$	−16.8918	+	16.8918i	−0.889041	+	0.889041i	0.105390	+	0.994431i	$0.466391\pi$
$20$	0.915151	+	5.62660i	0.0457575	+	0.281330i
$21$		0			0
$22$	−6.09740	−	1.96717i	−0.277155	−	0.0894167i
$23$	−33.2007			−1.44351			−0.721755	−	0.692149i	$-0.756664\pi$
$23$	−33.2007			−1.44351			−0.721755	+	0.692149i	$0.756664\pi$
$24$		0			0
$25$			22.9690i			0.918760i
$26$	−18.5258	−	5.97686i	−0.712530	−	0.229879i
$27$		0			0
$28$	23.4318	−	32.5346i	0.836850	−	1.16195i
$29$	24.6412	+	24.6412i	0.849696	+	0.849696i	0.990095	−	0.140399i	$-0.0448385\pi$
$29$	24.6412	+	24.6412i	0.849696	+	0.849696i	−0.140399	+	0.990095i	$0.544839\pi$
$30$		0			0
$31$			41.3761i			1.33471i	0.744738	+	0.667357i	$0.232574\pi$
$31$			41.3761i			1.33471i	−0.744738	+	0.667357i	$0.767426\pi$
$32$	−22.8613	−	22.3911i	−0.714415	−	0.699722i
$33$		0			0
$34$	39.8394	−	20.4036i	1.17175	−	0.600105i
$35$	−10.1010	+	10.1010i	−0.288599	+	0.288599i
$36$		0			0
$37$	−6.60031	+	6.60031i	−0.178387	+	0.178387i	−0.790652	−	0.612266i	$-0.790258\pi$
$37$	−6.60031	+	6.60031i	−0.178387	+	0.178387i	0.612266	+	0.790652i	$0.290258\pi$
$38$	−14.6695	+	45.4693i	−0.386039	+	1.19656i
$39$		0			0
$40$	6.75875	+	9.18170i	0.168969	+	0.229543i
$41$		−	47.1477i		−	1.14994i	−0.818173	−	0.574972i	$-0.805013\pi$
$41$		−	47.1477i		−	1.14994i	0.818173	−	0.574972i	$-0.194987\pi$
$42$		0			0
$43$	−48.8218	−	48.8218i	−1.13539	−	1.13539i	−0.989266	−	0.146124i	$-0.953320\pi$
$43$	−48.8218	−	48.8218i	−1.13539	−	1.13539i	−0.146124	−	0.989266i	$-0.546680\pi$
$44$	−12.6475	+	2.05709i	−0.287444	+	0.0467521i
$45$		0			0
$46$	−59.1013	+	30.2685i	−1.28481	+	0.658011i
$47$			45.6048i			0.970315i	0.874427	+	0.485157i	$0.161238\pi$
$47$			45.6048i			0.970315i	−0.874427	+	0.485157i	$0.838762\pi$
$48$		0			0
$49$	51.4717			1.05044
$50$	20.9404	+	40.8876i	0.418808	+	0.817752i
$51$		0			0
$52$	−38.4271	+	6.25007i	−0.738983	+	0.120194i
$53$	−25.1401	+	25.1401i	−0.474341	+	0.474341i	−0.903316	−	0.428975i	$-0.858875\pi$
$53$	−25.1401	+	25.1401i	−0.474341	+	0.474341i	0.428975	+	0.903316i	$0.358875\pi$
$54$		0			0
$55$	4.56532			0.0830059
$56$	12.0503	−	79.2779i	0.215184	−	1.41568i
$57$		0			0
$58$	66.3292	+	21.3994i	1.14361	+	0.368955i
$59$	−6.23974	−	6.23974i	−0.105758	−	0.105758i	0.652248	−	0.758006i	$-0.273826\pi$
$59$	−6.23974	−	6.23974i	−0.105758	−	0.105758i	−0.758006	+	0.652248i	$0.773826\pi$
$60$		0			0
$61$	35.9513	+	35.9513i	0.589366	+	0.589366i	0.937460	−	0.348093i	$-0.113171\pi$
$61$	35.9513	+	35.9513i	0.589366	+	0.589366i	−0.348093	+	0.937460i	$0.613171\pi$
$62$	37.7219	+	73.6546i	0.608417	+	1.18798i
$63$		0			0
$64$	−61.1095	−	19.0167i	−0.954835	−	0.297136i
$65$	13.8709			0.213398
$66$		0			0
$67$	10.2045	−	10.2045i	0.152307	−	0.152307i	−0.626841	−	0.779147i	$-0.715652\pi$
$67$	10.2045	−	10.2045i	0.152307	−	0.152307i	0.779147	+	0.626841i	$0.215652\pi$
$68$	52.3174	−	72.6417i	0.769374	−	1.06826i
$69$		0			0
$70$	−8.77208	+	27.1898i	−0.125315	+	0.388426i
$71$	−11.9529			−0.168350			−0.0841752	−	0.996451i	$-0.526826\pi$
$71$	−11.9529			−0.168350			−0.0841752	+	0.996451i	$0.526826\pi$
$72$		0			0
$73$		−	111.332i		−	1.52510i	−0.646929	−	0.762550i	$-0.723947\pi$
$73$		−	111.332i		−	1.52510i	0.646929	−	0.762550i	$-0.276053\pi$
$74$	−5.73197	+	17.7667i	−0.0774591	+	0.240091i
$75$		0			0
$76$	15.3401	+	94.3149i	0.201843	+	1.24099i
$77$	−22.7051	−	22.7051i	−0.294871	−	0.294871i
$78$		0			0
$79$		−	4.46031i		−	0.0564596i	−0.999601	−	0.0282298i	$-0.991013\pi$
$79$		−	4.46031i		−	0.0564596i	0.999601	−	0.0282298i	$-0.00898702\pi$
$80$	20.4022	+	10.1827i	0.255027	+	0.127284i
$81$		0			0
$82$	−42.9837	−	83.9287i	−0.524192	−	1.02352i
$83$	−10.1751	+	10.1751i	−0.122592	+	0.122592i	−0.765741	−	0.643149i	$-0.777627\pi$
$83$	−10.1751	+	10.1751i	−0.122592	+	0.122592i	0.643149	+	0.765741i	$0.277627\pi$
$84$		0			0
$85$	−22.5530	+	22.5530i	−0.265329	+	0.265329i
$86$	−131.419	−	42.3988i	−1.52812	−	0.493010i
$87$		0			0
$88$	−20.6388	+	15.1924i	−0.234532	+	0.172641i
$89$		−	21.9364i		−	0.246476i	−0.992377	−	0.123238i	$-0.960672\pi$
$89$		−	21.9364i		−	0.246476i	0.992377	−	0.123238i	$-0.0393279\pi$
$90$		0			0
$91$	−68.9850	−	68.9850i	−0.758077	−	0.758077i
$92$	−77.6124	+	107.763i	−0.843613	+	1.17134i
$93$		0			0
$94$	41.5771	+	81.1821i	0.442309	+	0.863640i
$95$		−	34.0444i		−	0.358362i
$96$		0			0
$97$	107.309			1.10628			0.553140	−	0.833088i	$-0.313429\pi$
$97$	107.309			1.10628			0.553140	+	0.833088i	$0.313429\pi$
$98$	91.6260	−	46.9259i	0.934959	−	0.478835i
$99$		0			0
$100$	74.5530	+	53.6940i	0.745530	+	0.536940i
$101$	100.780	−	100.780i	0.997824	−	0.997824i	−0.00217389	−	0.999998i	$-0.500692\pi$
$101$	100.780	−	100.780i	0.997824	−	0.997824i	0.999998	+	0.00217389i	$0.000691973\pi$
$102$		0			0
$103$	58.0562			0.563653			0.281826	−	0.959465i	$-0.409060\pi$
$103$	58.0562			0.563653			0.281826	+	0.959465i	$0.409060\pi$
$104$	−62.7069	+	46.1592i	−0.602951	+	0.443839i
$105$		0			0
$106$	−21.8327	+	67.6722i	−0.205968	+	0.638417i
$107$	−112.747	−	112.747i	−1.05371	−	1.05371i	−0.998473	−	0.0552381i	$-0.982408\pi$
$107$	−112.747	−	112.747i	−1.05371	−	1.05371i	−0.0552381	−	0.998473i	$-0.517592\pi$
$108$		0			0
$109$	−81.1384	−	81.1384i	−0.744389	−	0.744389i	0.229030	−	0.973419i	$-0.426445\pi$
$109$	−81.1384	−	81.1384i	−0.744389	−	0.744389i	−0.973419	+	0.229030i	$0.926445\pi$
$110$	8.12684	−	4.16212i	0.0738803	−	0.0378375i
$111$		0			0
$112$	−50.8252	−	152.110i	−0.453797	−	1.35813i
$113$	171.844			1.52074			0.760371	−	0.649489i	$-0.225017\pi$
$113$	171.844			1.52074			0.760371	+	0.649489i	$0.225017\pi$
$114$		0			0
$115$	33.4571	−	33.4571i	0.290931	−	0.290931i
$116$	137.584	−	22.3776i	1.18607	−	0.192910i
$117$		0			0
$118$	−16.7962	−	5.41884i	−0.142340	−	0.0459224i
$119$	224.329			1.88512
$120$		0			0
$121$		−	110.738i		−	0.915190i
$122$	96.7740	+	31.2216i	0.793230	+	0.255915i
$123$		0			0
$124$	134.299	+	96.7238i	1.08306	+	0.780031i
$125$	−48.3394	−	48.3394i	−0.386715	−	0.386715i
$126$		0			0
$127$			36.8333i			0.290026i	0.989430	+	0.145013i	$0.0463224\pi$
$127$			36.8333i			0.290026i	−0.989430	+	0.145013i	$0.953678\pi$
$128$	−126.119	+	21.8603i	−0.985309	+	0.170784i
$129$		0			0
$130$	24.6918	−	12.6458i	0.189937	−	0.0972754i
$131$	12.3686	−	12.3686i	0.0944170	−	0.0944170i	−0.658321	−	0.752738i	$-0.728733\pi$
$131$	12.3686	−	12.3686i	0.0944170	−	0.0944170i	0.752738	+	0.658321i	$0.228733\pi$
$132$		0			0
$133$	−169.316	+	169.316i	−1.27305	+	1.27305i
$134$	8.86204	−	27.4686i	0.0661347	−	0.204990i
$135$		0			0
$136$	26.9053	−	177.008i	0.197833	−	1.30153i
$137$			145.679i			1.06335i	0.846949	+	0.531674i	$0.178437\pi$
$137$			145.679i			1.06335i	−0.846949	+	0.531674i	$0.821563\pi$
$138$		0			0
$139$	82.5709	+	82.5709i	0.594035	+	0.594035i	0.938719	−	0.344684i	$-0.112014\pi$
$139$	82.5709	+	82.5709i	0.594035	+	0.594035i	−0.344684	+	0.938719i	$0.612014\pi$
$140$	9.17307	+	56.3985i	0.0655219	+	0.402847i
$141$		0			0
$142$	−21.2776	+	10.8972i	−0.149842	+	0.0767410i
$143$			31.1791i			0.218036i
$144$		0			0
$145$	−49.6629			−0.342503
$146$	−101.500	−	198.185i	−0.695203	−	1.35743i
$147$		0			0
$148$	5.99399	+	36.8527i	0.0405000	+	0.249005i
$149$	−196.248	+	196.248i	−1.31710	+	1.31710i	−0.401043	+	0.916059i	$0.631352\pi$
$149$	−196.248	+	196.248i	−1.31710	+	1.31710i	−0.916059	+	0.401043i	$0.868648\pi$
$150$		0			0
$151$	64.5007			0.427157			0.213578	−	0.976926i	$-0.431488\pi$
$151$	64.5007			0.427157			0.213578	+	0.976926i	$0.431488\pi$
$152$	113.292	+	153.907i	0.745345	+	1.01254i
$153$		0			0
$154$	−61.1177	−	19.7180i	−0.396868	−	0.128039i
$155$	−41.6956	−	41.6956i	−0.269004	−	0.269004i
$156$		0			0
$157$	54.4202	+	54.4202i	0.346625	+	0.346625i	0.858851	−	0.512226i	$-0.171179\pi$
$157$	54.4202	+	54.4202i	0.346625	+	0.346625i	−0.512226	+	0.858851i	$0.671179\pi$
$158$	−4.06638	−	7.93990i	−0.0257366	−	0.0502525i
$159$		0			0
$160$	45.6018	−	0.473799i	0.285011	−	0.00296124i
$161$	−332.789			−2.06701
$162$		0			0
$163$	104.803	−	104.803i	0.642961	−	0.642961i	−0.308321	−	0.951282i	$-0.599767\pi$
$163$	104.803	−	104.803i	0.642961	−	0.642961i	0.951282	+	0.308321i	$0.0997671\pi$
$164$	−153.033	−	110.216i	−0.933126	−	0.672048i
$165$		0			0
$166$	−8.83647	+	27.3894i	−0.0532317	+	0.164996i
$167$	−53.3110			−0.319228			−0.159614	−	0.987180i	$-0.551025\pi$
$167$	−53.3110			−0.319228			−0.159614	+	0.987180i	$0.551025\pi$
$168$		0			0
$169$		−	74.2683i		−	0.439457i
$170$	−19.5859	+	60.7081i	−0.115211	+	0.357107i
$171$		0			0
$172$	−272.596	+	44.3370i	−1.58486	+	0.257773i
$173$	41.5780	+	41.5780i	0.240335	+	0.240335i	0.816989	−	0.576654i	$-0.195642\pi$
$173$	41.5780	+	41.5780i	0.240335	+	0.240335i	−0.576654	+	0.816989i	$0.695642\pi$
$174$		0			0
$175$			230.231i			1.31561i
$176$	−22.8889	+	45.8604i	−0.130051	+	0.260570i
$177$		0			0
$178$	−19.9990	−	39.0495i	−0.112354	−	0.219379i
$179$	−53.0709	+	53.0709i	−0.296486	+	0.296486i	−0.839636	−	0.543150i	$-0.817231\pi$
$179$	−53.0709	+	53.0709i	−0.296486	+	0.296486i	0.543150	+	0.839636i	$0.317231\pi$
$180$		0			0
$181$	−66.6042	+	66.6042i	−0.367979	+	0.367979i	−0.866740	−	0.498761i	$-0.833789\pi$
$181$	−66.6042	+	66.6042i	−0.367979	+	0.367979i	0.498761	+	0.866740i	$0.333789\pi$
$182$	−185.694	−	59.9094i	−1.02030	−	0.329172i
$183$		0			0
$184$	−39.9138	+	262.590i	−0.216923	+	1.42712i
$185$		−	13.3025i		−	0.0719056i
$186$		0			0
$187$	−50.6949	−	50.6949i	−0.271096	−	0.271096i
$188$	148.025	+	106.609i	0.787365	+	0.567070i
$189$		0			0
$190$	−31.0377	−	60.6032i	−0.163356	−	0.318964i
$191$			113.753i			0.595567i	0.954633	+	0.297784i	$0.0962474\pi$
$191$			113.753i			0.595567i	−0.954633	+	0.297784i	$0.903753\pi$
$192$		0			0
$193$	−26.5596			−0.137615			−0.0688073	−	0.997630i	$-0.521919\pi$
$193$	−26.5596			−0.137615			−0.0688073	+	0.997630i	$0.521919\pi$
$194$	191.023	−	97.8318i	0.984657	−	0.504288i
$195$		0			0
$196$	120.324	−	167.068i	0.613898	−	0.852386i
$197$	−51.8935	+	51.8935i	−0.263419	+	0.263419i	−0.826442	−	0.563023i	$-0.809638\pi$
$197$	−51.8935	+	51.8935i	−0.263419	+	0.263419i	0.563023	+	0.826442i	$0.309638\pi$
$198$		0			0
$199$	−136.741			−0.687140			−0.343570	−	0.939127i	$-0.611636\pi$
$199$	−136.741			−0.687140			−0.343570	+	0.939127i	$0.611636\pi$
$200$	181.665	+	27.6132i	0.908327	+	0.138066i
$201$		0			0
$202$	87.5216	−	271.281i	0.433275	−	1.34297i
$203$	246.992	+	246.992i	1.21671	+	1.21671i
$204$		0			0
$205$	47.5118	+	47.5118i	0.231765	+	0.231765i
$206$	103.347	−	52.9288i	0.501686	−	0.256936i
$207$		0			0
$208$	−69.5435	+	139.338i	−0.334344	+	0.669893i
$209$	76.5255			0.366151
$210$		0			0
$211$	−141.171	+	141.171i	−0.669057	+	0.669057i	−0.957498	−	0.288441i	$-0.906863\pi$
$211$	−141.171	+	141.171i	−0.669057	+	0.669057i	0.288441	+	0.957498i	$0.406863\pi$
$212$	22.8307	+	140.369i	0.107692	+	0.662119i
$213$		0			0
$214$	−303.493	−	97.9142i	−1.41819	−	0.457543i
$215$	98.3975			0.457663
$216$		0			0
$217$			414.736i			1.91123i
$218$	−218.409	−	70.4639i	−1.00188	−	0.323229i
$219$		0			0
$220$	10.6722	−	14.8182i	0.0485101	−	0.0673554i
$221$	−154.027	−	154.027i	−0.696953	−	0.696953i
$222$		0			0
$223$		−	122.607i		−	0.549806i	−0.961472	−	0.274903i	$-0.911354\pi$
$223$		−	122.607i		−	0.549806i	0.961472	−	0.274903i	$-0.0886457\pi$
$224$	−229.151	−	224.439i	−1.02300	−	1.00196i
$225$		0			0
$226$	305.903	−	156.667i	1.35355	−	0.693216i
$227$	295.844	−	295.844i	1.30328	−	1.30328i	0.377112	−	0.926168i	$-0.376917\pi$
$227$	295.844	−	295.844i	1.30328	−	1.30328i	0.926168	−	0.377112i	$-0.123083\pi$
$228$		0			0
$229$	73.3817	−	73.3817i	0.320444	−	0.320444i	−0.528493	−	0.848937i	$-0.677243\pi$
$229$	73.3817	−	73.3817i	0.320444	−	0.320444i	0.848937	+	0.528493i	$0.177243\pi$
$230$	29.0555	−	90.0599i	0.126328	−	0.391565i
$231$		0			0
$232$	224.514	−	165.267i	0.967735	−	0.712359i
$233$		−	156.229i		−	0.670509i	−0.942128	−	0.335255i	$-0.891178\pi$
$233$		−	156.229i		−	0.670509i	0.942128	−	0.335255i	$-0.108822\pi$
$234$		0			0
$235$	−45.9569	−	45.9569i	−0.195561	−	0.195561i
$236$	−34.8395	+	5.66655i	−0.147625	+	0.0240108i
$237$		0			0
$238$	399.333	−	204.516i	1.67787	−	0.859313i
$239$			13.1716i			0.0551113i	0.999620	+	0.0275557i	$0.00877235\pi$
$239$			13.1716i			0.0551113i	−0.999620	+	0.0275557i	$0.991228\pi$
$240$		0			0
$241$	−189.519			−0.786386			−0.393193	−	0.919456i	$-0.628630\pi$
$241$	−189.519			−0.786386			−0.393193	+	0.919456i	$0.628630\pi$
$242$	−100.958	−	197.127i	−0.417181	−	0.814575i
$243$		0			0
$244$	200.734	−	32.6488i	0.822679	−	0.133807i
$245$	−51.8692	+	51.8692i	−0.211711	+	0.211711i
$246$		0			0
$247$	232.508			0.941328
$248$	327.250	+	49.7422i	1.31956	+	0.200573i
$249$		0			0
$250$	−130.120	−	41.9799i	−0.520481	−	0.167920i
$251$	27.4434	+	27.4434i	0.109336	+	0.109336i	0.759658	−	0.650322i	$-0.225366\pi$
$251$	27.4434	+	27.4434i	0.109336	+	0.109336i	−0.650322	+	0.759658i	$0.725366\pi$
$252$		0			0
$253$	75.2053	+	75.2053i	0.297254	+	0.297254i
$254$	33.5802	+	65.5678i	0.132206	+	0.258141i
$255$		0			0
$256$	−204.578	+	153.895i	−0.799135	+	0.601152i
$257$	−135.375			−0.526752			−0.263376	−	0.964693i	$-0.584836\pi$
$257$	−135.375			−0.526752			−0.263376	+	0.964693i	$0.584836\pi$
$258$		0			0
$259$	−66.1586	+	66.1586i	−0.255438	+	0.255438i
$260$	32.4255	−	45.0222i	0.124714	−	0.173162i
$261$		0			0
$262$	10.7414	−	33.2939i	0.0409978	−	0.127076i
$263$	−31.6123			−0.120199			−0.0600994	−	0.998192i	$-0.519142\pi$
$263$	−31.6123			−0.120199			−0.0600994	+	0.998192i	$0.519142\pi$
$264$		0			0
$265$		−	50.6684i		−	0.191201i
$266$	−147.041	+	455.765i	−0.552784	+	1.71340i
$267$		0			0
$268$	−9.26715	−	56.9769i	−0.0345789	−	0.212600i
$269$	194.213	+	194.213i	0.721981	+	0.721981i	0.969008	−	0.247028i	$-0.0794538\pi$
$269$	194.213	+	194.213i	0.721981	+	0.721981i	−0.247028	+	0.969008i	$0.579454\pi$
$270$		0			0
$271$		−	291.647i		−	1.07619i	−0.842884	−	0.538095i	$-0.819144\pi$
$271$		−	291.647i		−	1.07619i	0.842884	−	0.538095i	$-0.180856\pi$
$272$	−113.480	−	339.625i	−0.417207	−	1.24862i
$273$		0			0
$274$	132.813	+	259.326i	0.484717	+	0.946444i
$275$	52.0287	−	52.0287i	0.189195	−	0.189195i
$276$		0			0
$277$	305.166	−	305.166i	1.10168	−	1.10168i	0.107475	−	0.994208i	$-0.465723\pi$
$277$	305.166	−	305.166i	1.10168	−	1.10168i	0.994208	−	0.107475i	$-0.0342765\pi$
$278$	222.265	+	71.7079i	0.799513	+	0.257942i
$279$		0			0
$280$	67.7467	+	92.0333i	0.241952	+	0.328691i
$281$		−	211.861i		−	0.753955i	−0.926222	−	0.376978i	$-0.876963\pi$
$281$		−	211.861i		−	0.753955i	0.926222	−	0.376978i	$-0.123037\pi$
$282$		0			0
$283$	105.325	+	105.325i	0.372175	+	0.372175i	0.868269	−	0.496094i	$-0.165233\pi$
$283$	105.325	+	105.325i	0.372175	+	0.372175i	−0.496094	+	0.868269i	$0.665233\pi$
$284$	−27.9419	+	38.7968i	−0.0983870	+	0.136608i
$285$		0			0
$286$	28.4254	+	55.5027i	0.0993897	+	0.194065i
$287$		−	472.588i		−	1.64665i
$288$		0			0
$289$	211.871			0.733117
$290$	−88.4060	+	45.2768i	−0.304848	+	0.156127i
$291$		0			0
$292$	−361.364	−	260.258i	−1.23755	−	0.891296i
$293$	171.289	−	171.289i	0.584603	−	0.584603i	−0.351562	−	0.936165i	$-0.614349\pi$
$293$	171.289	−	171.289i	0.584603	−	0.584603i	0.936165	+	0.351562i	$0.114349\pi$
$294$		0			0
$295$	12.5758			0.0426300
$296$	44.2680	+	60.1377i	0.149554	+	0.203168i
$297$		0			0
$298$	−170.430	+	528.262i	−0.571912	+	1.77269i
$299$	228.497	+	228.497i	0.764204	+	0.764204i
$300$		0			0
$301$	−489.368	−	489.368i	−1.62581	−	1.62581i
$302$	114.819	−	58.8041i	0.380196	−	0.194716i
$303$		0			0
$304$	341.988	+	170.686i	1.12496	+	0.561468i
$305$	−72.4579			−0.237567
$306$		0			0
$307$	−27.1124	+	27.1124i	−0.0883140	+	0.0883140i	−0.749884	−	0.661570i	$-0.769891\pi$
$307$	−27.1124	+	27.1124i	−0.0883140	+	0.0883140i	0.661570	+	0.749884i	$0.269891\pi$
$308$	−126.773	+	20.6194i	−0.411602	+	0.0669460i
$309$		0			0
$310$	−112.236	−	36.2102i	−0.362053	−	0.116807i
$311$	−371.124			−1.19333			−0.596663	−	0.802492i	$-0.703507\pi$
$311$	−371.124			−1.19333			−0.596663	+	0.802492i	$0.703507\pi$
$312$		0			0
$313$			374.501i			1.19649i	0.801313	+	0.598245i	$0.204135\pi$
$313$			374.501i			1.19649i	−0.801313	+	0.598245i	$0.795865\pi$
$314$	146.488	+	47.2607i	0.466524	+	0.150512i
$315$		0			0
$316$	−14.4773	−	10.4267i	−0.0458143	−	0.0329960i
$317$	48.5840	+	48.5840i	0.153262	+	0.153262i	0.779573	−	0.626311i	$-0.215436\pi$
$317$	48.5840	+	48.5840i	0.153262	+	0.153262i	−0.626311	+	0.779573i	$0.715436\pi$
$318$		0			0
$319$		−	111.633i		−	0.349947i
$320$	80.7448	−	42.4178i	0.252328	−	0.132555i
$321$		0			0
$322$	−592.406	+	303.398i	−1.83977	+	0.942230i
$323$	−378.040	+	378.040i	−1.17040	+	1.17040i
$324$		0			0
$325$	158.079	−	158.079i	0.486398	−	0.486398i
$326$	91.0149	−	282.108i	0.279187	−	0.865363i
$327$		0			0
$328$	−372.899	−	56.6808i	−1.13689	−	0.172807i
$329$			457.122i			1.38943i
$330$		0			0
$331$	−1.88883	−	1.88883i	−0.00570644	−	0.00570644i	0.704248	−	0.709954i	$-0.251284\pi$
$331$	−1.88883	−	1.88883i	−0.00570644	−	0.00570644i	−0.709954	+	0.704248i	$0.751284\pi$
$332$	9.24040	+	56.8125i	0.0278325	+	0.171122i
$333$		0			0
$334$	−94.9001	+	48.6027i	−0.284132	+	0.145517i
$335$			20.5667i			0.0613931i
$336$		0			0
$337$	−386.980			−1.14831			−0.574154	−	0.818747i	$-0.694669\pi$
$337$	−386.980			−1.14831			−0.574154	+	0.818747i	$0.694669\pi$
$338$	−67.7090	−	132.207i	−0.200323	−	0.391144i
$339$		0			0
$340$	20.4812	+	125.924i	0.0602388	+	0.370365i
$341$	93.7240	−	93.7240i	0.274851	−	0.274851i
$342$		0			0
$343$	24.7757			0.0722325
$344$	−444.832	+	327.446i	−1.29312	+	0.951877i
$345$		0			0
$346$	111.920	+	36.1080i	0.323468	+	0.104358i
$347$	−441.887	−	441.887i	−1.27345	−	1.27345i	−0.944266	−	0.329183i	$-0.893227\pi$
$347$	−441.887	−	441.887i	−1.27345	−	1.27345i	−0.329183	−	0.944266i	$-0.606773\pi$
$348$		0			0
$349$	119.382	+	119.382i	0.342068	+	0.342068i	0.857144	−	0.515076i	$-0.172236\pi$
$349$	119.382	+	119.382i	0.342068	+	0.342068i	−0.515076	+	0.857144i	$0.672236\pi$
$350$	209.898	+	409.840i	0.599707	+	1.17097i
$351$		0			0
$352$	1.06501	+	102.504i	0.00302560	+	0.291206i
$353$	515.642			1.46074			0.730371	−	0.683050i	$-0.239347\pi$
$353$	515.642			1.46074			0.730371	+	0.683050i	$0.239347\pi$
$354$		0			0
$355$	12.0452	−	12.0452i	0.0339301	−	0.0339301i
$356$	−71.2014	−	51.2801i	−0.200004	−	0.144045i
$357$		0			0
$358$	−46.0890	+	142.857i	−0.128740	+	0.399041i
$359$	−428.264			−1.19294			−0.596468	−	0.802637i	$-0.703430\pi$
$359$	−428.264			−1.19294			−0.596468	+	0.802637i	$0.703430\pi$
$360$		0			0
$361$		−	209.664i		−	0.580786i
$362$	−57.8418	+	179.286i	−0.159784	+	0.495264i
$363$		0			0
$364$	−385.177	+	62.6480i	−1.05818	+	0.172110i
$365$	112.192	+	112.192i	0.307375	+	0.307375i
$366$		0			0
$367$		−	219.482i		−	0.598043i	−0.954246	−	0.299021i	$-0.903340\pi$
$367$		−	219.482i		−	0.598043i	0.954246	−	0.299021i	$-0.0966602\pi$
$368$	168.347	+	503.830i	0.457464	+	1.36910i
$369$		0			0
$370$	−12.1277	−	23.6802i	−0.0327775	−	0.0640004i
$371$	−251.993	+	251.993i	−0.679226	+	0.679226i
$372$		0			0
$373$	425.005	−	425.005i	1.13942	−	1.13942i	0.150870	−	0.988554i	$-0.451793\pi$
$373$	425.005	−	425.005i	1.13942	−	1.13942i	0.988554	−	0.150870i	$-0.0482075\pi$
$374$	−136.461	−	44.0255i	−0.364868	−	0.117715i
$375$		0			0
$376$	360.695	+	54.8259i	0.959296	+	0.145814i
$377$		−	339.175i		−	0.899669i
$378$		0			0
$379$	365.916	+	365.916i	0.965476	+	0.965476i	0.999424	−	0.0339473i	$-0.0108078\pi$
$379$	365.916	+	365.916i	0.965476	+	0.965476i	−0.0339473	+	0.999424i	$0.510808\pi$
$380$	−110.502	−	79.5846i	−0.290794	−	0.209433i
$381$		0			0
$382$	103.707	+	202.495i	0.271484	+	0.530091i
$383$			213.276i			0.556857i	0.960457	+	0.278428i	$0.0898135\pi$
$383$			213.276i			0.556857i	−0.960457	+	0.278428i	$0.910187\pi$
$384$		0			0
$385$	45.7608			0.118859
$386$	−47.2793	+	24.2139i	−0.122485	+	0.0627303i
$387$		0			0
$388$	250.854	−	348.305i	0.646530	−	0.897694i
$389$	210.798	−	210.798i	0.541898	−	0.541898i	−0.382187	−	0.924085i	$-0.624829\pi$
$389$	210.798	−	210.798i	0.541898	−	0.541898i	0.924085	+	0.382187i	$0.124829\pi$
$390$		0			0
$391$	−743.037			−1.90035
$392$	61.8791	−	407.098i	0.157855	−	1.03852i
$393$		0			0
$394$	−45.0665	+	139.687i	−0.114382	+	0.354536i
$395$	4.49475	+	4.49475i	0.0113791	+	0.0113791i
$396$		0			0
$397$	392.907	+	392.907i	0.989690	+	0.989690i	0.999947	−	0.0102579i	$-0.00326524\pi$
$397$	392.907	+	392.907i	0.989690	+	0.989690i	−0.0102579	+	0.999947i	$0.503265\pi$
$398$	−243.415	+	124.664i	−0.611597	+	0.313226i
$399$		0			0
$400$	348.561	−	116.466i	0.871403	−	0.291165i
$401$	−29.3290			−0.0731396			−0.0365698	−	0.999331i	$-0.511643\pi$
$401$	−29.3290			−0.0731396			−0.0365698	+	0.999331i	$0.511643\pi$
$402$		0			0
$403$	284.762	−	284.762i	0.706606	−	0.706606i
$404$	−91.5224	−	562.705i	−0.226541	−	1.39283i
$405$		0			0
$406$	664.855	+	214.498i	1.63757	+	0.528321i
$407$	29.9017			0.0734684
$408$		0			0
$409$		−	601.115i		−	1.46972i	−0.678219	−	0.734860i	$-0.737248\pi$
$409$		−	601.115i		−	1.46972i	0.678219	−	0.734860i	$-0.262752\pi$
$410$	127.892	+	41.2612i	0.311933	+	0.100637i
$411$		0			0
$412$	135.716	−	188.440i	0.329409	−	0.457378i
$413$	−62.5444	−	62.5444i	−0.151439	−	0.151439i
$414$		0			0
$415$		−	20.5073i		−	0.0494153i
$416$	3.23584	+	311.440i	0.00777845	+	0.748654i
$417$		0			0
$418$	136.225	−	69.7669i	0.325897	−	0.166907i
$419$	518.885	−	518.885i	1.23839	−	1.23839i	0.277729	−	0.960659i	$-0.410418\pi$
$419$	518.885	−	518.885i	1.23839	−	1.23839i	0.960659	−	0.277729i	$-0.0895819\pi$
$420$		0			0
$421$	−411.213	+	411.213i	−0.976754	+	0.976754i	−0.999736	−	0.0229817i	$-0.992684\pi$
$421$	−411.213	+	411.213i	−0.976754	+	0.976754i	0.0229817	+	0.999736i	$0.492684\pi$
$422$	−122.599	+	380.005i	−0.290518	+	0.900485i
$423$		0			0
$424$	168.613	+	229.060i	0.397673	+	0.540236i
$425$			514.049i			1.20953i
$426$		0			0
$427$	360.360	+	360.360i	0.843936	+	0.843936i
$428$	−629.522	+	102.390i	−1.47084	+	0.239229i
$429$		0			0
$430$	175.160	−	89.7072i	0.407348	−	0.208622i
$431$			41.1083i			0.0953789i	0.998862	+	0.0476895i	$0.0151858\pi$
$431$			41.1083i			0.0953789i	−0.998862	+	0.0476895i	$0.984814\pi$
$432$		0			0
$433$	−351.682			−0.812199			−0.406100	−	0.913829i	$-0.633111\pi$
$433$	−351.682			−0.812199			−0.406100	+	0.913829i	$0.633111\pi$
$434$	378.107	+	738.281i	0.871215	+	1.70111i
$435$		0			0
$436$	−453.035	+	73.6850i	−1.03907	+	0.169002i
$437$	560.819	−	560.819i	1.28334	−	1.28334i
$438$		0			0
$439$	−775.613			−1.76677			−0.883386	−	0.468646i	$-0.844742\pi$
$439$	−775.613			−1.76677			−0.883386	+	0.468646i	$0.844742\pi$
$440$	5.48841	−	36.1079i	0.0124737	−	0.0820633i
$441$		0			0
$442$	−414.609	−	133.763i	−0.938030	−	0.302631i
$443$	241.372	+	241.372i	0.544858	+	0.544858i	0.924949	−	0.380091i	$-0.124107\pi$
$443$	241.372	+	241.372i	0.544858	+	0.544858i	−0.380091	+	0.924949i	$0.624107\pi$
$444$		0			0
$445$	22.1058	+	22.1058i	0.0496759	+	0.0496759i
$446$	−111.778	−	218.255i	−0.250624	−	0.489361i
$447$		0			0
$448$	−612.534	−	190.615i	−1.36726	−	0.425480i
$449$	−266.360			−0.593228			−0.296614	−	0.954997i	$-0.595858\pi$
$449$	−266.360			−0.593228			−0.296614	+	0.954997i	$0.595858\pi$
$450$		0			0
$451$	−106.798	+	106.798i	−0.236802	+	0.236802i
$452$	401.714	−	557.772i	0.888749	−	1.23401i
$453$		0			0
$454$	256.923	−	796.355i	0.565910	−	1.75409i
$455$	139.035			0.305572
$456$		0			0
$457$			515.244i			1.12745i	0.825963	+	0.563725i	$0.190632\pi$
$457$			515.244i			1.12745i	−0.825963	+	0.563725i	$0.809368\pi$
$458$	63.7277	−	197.529i	0.139143	−	0.431287i
$459$		0			0
$460$	−30.3837	−	186.807i	−0.0660514	−	0.406102i
$461$	−5.67717	−	5.67717i	−0.0123149	−	0.0123149i	0.700923	−	0.713237i	$-0.252772\pi$
$461$	−5.67717	−	5.67717i	−0.0123149	−	0.0123149i	−0.713237	+	0.700923i	$0.752772\pi$
$462$		0			0
$463$			464.510i			1.00326i	0.865082	+	0.501631i	$0.167267\pi$
$463$			464.510i			1.00326i	−0.865082	+	0.501631i	$0.832733\pi$
$464$	248.992	−	498.882i	0.536621	−	1.07518i
$465$		0			0
$466$	−142.431	−	278.106i	−0.305646	−	0.596794i
$467$	−495.985	+	495.985i	−1.06207	+	1.06207i	−0.0641248	+	0.997942i	$0.520426\pi$
$467$	−495.985	+	495.985i	−1.06207	+	1.06207i	−0.997942	+	0.0641248i	$0.979574\pi$
$468$		0			0
$469$	102.286	−	102.286i	0.218094	−	0.218094i
$470$	−123.707	−	39.9109i	−0.263207	−	0.0849167i
$471$		0			0
$472$	−56.8525	+	41.8497i	−0.120450	+	0.0886646i
$473$			221.180i			0.467610i
$474$		0			0
$475$	−387.987	−	387.987i	−0.816815	−	0.816815i
$476$	524.407	−	728.129i	1.10170	−	1.52968i
$477$		0			0
$478$	12.0083	+	23.4471i	0.0251220	+	0.0490525i
$479$		−	378.802i		−	0.790818i	−0.918505	−	0.395409i	$-0.870603\pi$
$479$		−	378.802i		−	0.790818i	0.918505	−	0.395409i	$-0.129397\pi$
$480$		0			0
$481$	90.8504			0.188878
$482$	−337.367	+	172.781i	−0.699932	+	0.358467i
$483$		0			0
$484$	−359.435	−	258.869i	−0.742633	−	0.534854i
$485$	−108.138	+	108.138i	−0.222964	+	0.222964i
$486$		0			0
$487$	147.446			0.302764			0.151382	−	0.988475i	$-0.451628\pi$
$487$	147.446			0.302764			0.151382	+	0.988475i	$0.451628\pi$
$488$	327.565	−	241.124i	0.671240	−	0.494107i
$489$		0			0
$490$	−45.0453	+	139.622i	−0.0919292	+	0.284942i
$491$	−109.547	−	109.547i	−0.223110	−	0.223110i	0.586697	−	0.809807i	$-0.300428\pi$
$491$	−109.547	−	109.547i	−0.223110	−	0.223110i	−0.809807	+	0.586697i	$0.800428\pi$
$492$		0			0
$493$	551.473	+	551.473i	1.11861	+	1.11861i
$494$	413.893	−	211.973i	0.837840	−	0.429096i
$495$		0			0
$496$	627.894	−	209.801i	1.26592	−	0.422985i
$497$	−119.810			−0.241067
$498$		0			0
$499$	−360.523	+	360.523i	−0.722491	+	0.722491i	−0.969112	−	0.246621i	$-0.920680\pi$
$499$	−360.523	+	360.523i	−0.722491	+	0.722491i	0.246621	+	0.969112i	$0.420680\pi$
$500$	−269.902	+	43.8989i	−0.539804	+	0.0877977i
$501$		0			0
$502$	73.8722	+	23.8329i	0.147156	+	0.0474760i
$503$	927.420			1.84378			0.921889	−	0.387454i	$-0.126645\pi$
$503$	927.420			1.84378			0.921889	+	0.387454i	$0.126645\pi$
$504$		0			0
$505$			203.117i			0.402211i
$506$	202.438	+	65.3114i	0.400075	+	0.129074i
$507$		0			0
$508$	119.554	+	86.1042i	0.235342	+	0.169496i
$509$	−677.931	−	677.931i	−1.33189	−	1.33189i	−0.903680	−	0.428208i	$-0.859145\pi$
$509$	−677.931	−	677.931i	−1.33189	−	1.33189i	−0.428208	−	0.903680i	$-0.640855\pi$
$510$		0			0
$511$		−	1115.95i		−	2.18385i
$512$	−223.872	+	460.462i	−0.437249	+	0.899340i
$513$		0			0
$514$	−240.984	+	123.419i	−0.468841	+	0.240115i
$515$	−58.5045	+	58.5045i	−0.113601	+	0.113601i
$516$		0			0
$517$	103.303	−	103.303i	0.199812	−	0.199812i
$518$	−57.4548	+	178.086i	−0.110917	+	0.343795i
$519$		0			0
$520$	16.6755	−	109.707i	0.0320682	−	0.210974i
$521$		−	143.173i		−	0.274804i	−0.990515	−	0.137402i	$-0.956125\pi$
$521$		−	143.173i		−	0.274804i	0.990515	−	0.137402i	$-0.0438753\pi$
$522$		0			0
$523$	226.187	+	226.187i	0.432481	+	0.432481i	0.889471	−	0.456991i	$-0.151073\pi$
$523$	226.187	+	226.187i	0.432481	+	0.432481i	−0.456991	+	0.889471i	$0.651073\pi$
$524$	−11.2324	−	69.0600i	−0.0214359	−	0.131794i
$525$		0			0
$526$	−56.2738	+	28.8204i	−0.106984	+	0.0547916i
$527$			926.004i			1.75712i
$528$		0			0
$529$	573.288			1.08372
$530$	−46.1934	−	90.1959i	−0.0871574	−	0.170181i
$531$		0			0
$532$	153.762	+	945.371i	0.289027	+	1.77701i
$533$	−324.484	+	324.484i	−0.608788	+	0.608788i
$534$		0			0
$535$	227.235			0.424739
$536$	−68.4415	−	92.9772i	−0.127689	−	0.173465i
$537$		0			0
$538$	522.783	+	168.662i	0.971716	+	0.313499i
$539$	−116.592	−	116.592i	−0.216312	−	0.216312i
$540$		0			0
$541$	−156.708	−	156.708i	−0.289663	−	0.289663i	0.547284	−	0.836947i	$-0.315662\pi$
$541$	−156.708	−	156.708i	−0.289663	−	0.289663i	−0.836947	+	0.547284i	$0.815662\pi$
$542$	−265.890	−	519.168i	−0.490571	−	0.957875i
$543$		0			0
$544$	−511.639	−	501.116i	−0.940512	−	0.921170i
$545$	163.530			0.300055
$546$		0			0
$547$	247.357	−	247.357i	0.452207	−	0.452207i	−0.443880	−	0.896086i	$-0.646398\pi$
$547$	247.357	−	247.357i	0.452207	−	0.452207i	0.896086	+	0.443880i	$0.146398\pi$
$548$	472.845	+	340.549i	0.862856	+	0.621440i
$549$		0			0
$550$	45.1839	−	140.051i	0.0821525	−	0.254638i
$551$	−832.466			−1.51083
$552$		0			0
$553$		−	44.7082i		−	0.0808466i
$554$	265.019	−	821.447i	0.478373	−	1.48276i
$555$		0			0
$556$	461.033	−	74.9858i	0.829196	−	0.134867i
$557$	661.193	+	661.193i	1.18706	+	1.18706i	0.977876	+	0.209184i	$0.0670808\pi$
$557$	661.193	+	661.193i	1.18706	+	1.18706i	0.209184	+	0.977876i	$0.432919\pi$
$558$		0			0
$559$			672.011i			1.20217i
$560$	204.503	+	102.067i	0.365183	+	0.182263i
$561$		0			0
$562$	−193.150	−	377.139i	−0.343684	−	0.671066i
$563$	−246.685	+	246.685i	−0.438162	+	0.438162i	−0.891393	−	0.453231i	$-0.850271\pi$
$563$	−246.685	+	246.685i	−0.438162	+	0.438162i	0.453231	+	0.891393i	$0.350271\pi$
$564$		0			0
$565$	−173.171	+	173.171i	−0.306497	+	0.306497i
$566$	283.516	+	91.4689i	0.500911	+	0.161606i
$567$		0			0
$568$	−14.3697	+	94.5372i	−0.0252988	+	0.166439i
$569$		−	243.567i		−	0.428061i	−0.976827	−	0.214030i	$-0.931341\pi$
$569$		−	243.567i		−	0.428061i	0.976827	−	0.214030i	$-0.0686592\pi$
$570$		0			0
$571$	59.9229	+	59.9229i	0.104944	+	0.104944i	0.757629	−	0.652685i	$-0.226358\pi$
$571$	59.9229	+	59.9229i	0.104944	+	0.104944i	−0.652685	+	0.757629i	$0.726358\pi$
$572$	101.202	+	72.8866i	0.176926	+	0.127424i
$573$		0			0
$574$	−430.850	−	841.265i	−0.750610	−	1.46562i
$575$		−	762.587i		−	1.32624i
$576$		0			0
$577$	136.609			0.236757			0.118378	−	0.992969i	$-0.462230\pi$
$577$	136.609			0.236757			0.118378	+	0.992969i	$0.462230\pi$
$578$	377.156	−	193.159i	0.652519	−	0.334185i
$579$		0			0
$580$	−116.096	+	161.196i	−0.200165	+	0.277925i
$581$	−101.991	+	101.991i	−0.175543	+	0.175543i
$582$		0			0
$583$	113.893			0.195357
$584$	−880.544	−	133.843i	−1.50778	−	0.229184i
$585$		0			0
$586$	148.754	−	461.076i	0.253846	−	0.786818i
$587$	331.817	+	331.817i	0.565276	+	0.565276i	0.930801	−	0.365525i	$-0.119111\pi$
$587$	331.817	+	331.817i	0.565276	+	0.565276i	−0.365525	+	0.930801i	$0.619111\pi$
$588$		0			0
$589$	−698.916	−	698.916i	−1.18661	−	1.18661i
$590$	22.3865	−	11.4652i	0.0379433	−	0.0194325i
$591$		0			0
$592$	133.629	+	66.6942i	0.225724	+	0.112659i
$593$	−131.285			−0.221391			−0.110695	−	0.993854i	$-0.535308\pi$
$593$	−131.285			−0.221391			−0.110695	+	0.993854i	$0.535308\pi$
$594$		0			0
$595$	−226.061	+	226.061i	−0.379934	+	0.379934i
$596$	178.221	+	1095.75i	0.299028	+	1.83850i
$597$		0			0
$598$	615.069	+	198.436i	1.02854	+	0.331833i
$599$	136.119			0.227243			0.113621	−	0.993524i	$-0.463755\pi$
$599$	136.119			0.227243			0.113621	+	0.993524i	$0.463755\pi$
$600$		0			0
$601$			498.566i			0.829561i	0.909922	+	0.414780i	$0.136142\pi$
$601$			498.566i			0.829561i	−0.909922	+	0.414780i	$0.863858\pi$
$602$	−1317.28	−	424.987i	−2.18818	−	0.705959i
$603$		0			0
$604$	150.781	−	209.357i	0.249638	−	0.346617i
$605$	111.593	+	111.593i	0.184451	+	0.184451i
$606$		0			0
$607$			568.740i			0.936969i	0.883472	+	0.468484i	$0.155200\pi$
$607$			568.740i			0.936969i	−0.883472	+	0.468484i	$0.844800\pi$
$608$	764.393	−	7.94198i	1.25723	−	0.0130625i
$609$		0			0
$610$	−128.984	+	66.0585i	−0.211449	+	0.108293i
$611$	313.865	−	313.865i	0.513691	−	0.513691i
$612$		0			0
$613$	−168.441	+	168.441i	−0.274782	+	0.274782i	−0.831022	−	0.556240i	$-0.812244\pi$
$613$	−168.441	+	168.441i	−0.274782	+	0.274782i	0.556240	+	0.831022i	$0.312244\pi$
$614$	−23.5455	+	72.9813i	−0.0383478	+	0.118862i
$615$		0			0
$616$	−206.874	+	152.282i	−0.335834	+	0.247211i
$617$			599.157i			0.971081i	0.874214	+	0.485541i	$0.161377\pi$
$617$			599.157i			0.971081i	−0.874214	+	0.485541i	$0.838623\pi$
$618$		0			0
$619$	126.719	+	126.719i	0.204715	+	0.204715i	0.802017	−	0.597301i	$-0.203760\pi$
$619$	126.719	+	126.719i	0.204715	+	0.204715i	−0.597301	+	0.802017i	$0.703760\pi$
$620$	−232.807	+	37.8654i	−0.375495	+	0.0610732i
$621$		0			0
$622$	−660.647	+	338.347i	−1.06213	+	0.543967i
$623$		−	219.881i		−	0.352939i
$624$		0			0
$625$	−476.800			−0.762879
$626$	341.426	+	666.658i	0.545409	+	1.06495i
$627$		0			0
$628$	303.854	−	49.4210i	0.483844	−	0.0786959i
$629$	−147.716	+	147.716i	−0.234842	+	0.234842i
$630$		0			0
$631$	668.283			1.05909			0.529543	−	0.848283i	$-0.322363\pi$
$631$	668.283			1.05909			0.529543	+	0.848283i	$0.322363\pi$
$632$	−35.2773	−	5.36216i	−0.0558185	−	0.00848444i
$633$		0			0
$634$	130.779	+	42.1923i	0.206275	+	0.0665494i
$635$	−37.1177	−	37.1177i	−0.0584531	−	0.0584531i
$636$		0			0
$637$	−354.243	−	354.243i	−0.556112	−	0.556112i
$638$	−101.774	−	198.720i	−0.159520	−	0.311474i
$639$		0			0
$640$	105.064	−	149.122i	0.164163	−	0.233004i
$641$	−484.574			−0.755966			−0.377983	−	0.925813i	$-0.623382\pi$
$641$	−484.574			−0.755966			−0.377983	+	0.925813i	$0.623382\pi$
$642$		0			0
$643$	75.2980	−	75.2980i	0.117104	−	0.117104i	−0.646126	−	0.763230i	$-0.723612\pi$
$643$	75.2980	−	75.2980i	0.117104	−	0.117104i	0.763230	+	0.646126i	$0.223612\pi$
$644$	−777.952	+	1080.17i	−1.20800	+	1.67728i
$645$		0			0
$646$	−328.306	+	1017.61i	−0.508213	+	1.57525i
$647$	582.307			0.900011			0.450006	−	0.893026i	$-0.351422\pi$
$647$	582.307			0.900011			0.450006	+	0.893026i	$0.351422\pi$
$648$		0			0
$649$			28.2682i			0.0435565i
$650$	137.282	−	425.518i	0.211204	−	0.654644i
$651$		0			0
$652$	−95.1754	−	585.164i	−0.145974	−	0.897491i
$653$	−457.453	−	457.453i	−0.700541	−	0.700541i	0.263986	−	0.964527i	$-0.414963\pi$
$653$	−457.453	−	457.453i	−0.700541	−	0.700541i	−0.964527	+	0.263986i	$0.914963\pi$
$654$		0			0
$655$			24.9283i			0.0380584i
$656$	−715.480	+	239.066i	−1.09067	+	0.364430i
$657$		0			0
$658$	416.750	+	813.734i	0.633359	+	1.23668i
$659$	430.079	−	430.079i	0.652623	−	0.652623i	−0.301001	−	0.953624i	$-0.597321\pi$
$659$	430.079	−	430.079i	0.652623	−	0.652623i	0.953624	+	0.301001i	$0.0973207\pi$
$660$		0			0
$661$	−513.622	+	513.622i	−0.777038	+	0.777038i	−0.979326	−	0.202288i	$-0.935162\pi$
$661$	−513.622	+	513.622i	−0.777038	+	0.777038i	0.202288	+	0.979326i	$0.435162\pi$
$662$	−5.08436	−	1.64034i	−0.00768031	−	0.00247785i
$663$		0			0
$664$	68.2440	+	92.7089i	0.102777	+	0.139622i
$665$		−	341.246i		−	0.513152i
$666$		0			0
$667$	−818.105	−	818.105i	−1.22654	−	1.22654i
$668$	−124.624	+	173.037i	−0.186562	+	0.259038i
$669$		0			0
$670$	18.7503	+	36.6112i	0.0279855	+	0.0546436i
$671$		−	162.872i		−	0.242730i
$672$		0			0
$673$	−1112.68			−1.65332			−0.826659	−	0.562703i	$-0.809761\pi$
$673$	−1112.68			−1.65332			−0.826659	+	0.562703i	$0.809761\pi$
$674$	−688.871	+	352.802i	−1.02206	+	0.523446i
$675$		0			0
$676$	−241.061	−	173.615i	−0.356599	−	0.256827i
$677$	−633.271	+	633.271i	−0.935408	+	0.935408i	−0.998037	−	0.0626291i	$-0.980051\pi$
$677$	−633.271	+	633.271i	−0.935408	+	0.935408i	0.0626291	+	0.998037i	$0.480051\pi$
$678$		0			0
$679$	1075.62			1.58412
$680$	151.262	+	205.488i	0.222444	+	0.302188i
$681$		0			0
$682$	81.3937	−	252.287i	0.119346	−	0.369922i
$683$	429.651	+	429.651i	0.629065	+	0.629065i	0.947833	−	0.318768i	$-0.103269\pi$
$683$	429.651	+	429.651i	0.629065	+	0.629065i	−0.318768	+	0.947833i	$0.603269\pi$
$684$		0			0
$685$	−146.803	−	146.803i	−0.214312	−	0.214312i
$686$	44.1038	−	22.5876i	0.0642913	−	0.0329265i
$687$		0			0
$688$	−493.330	+	988.439i	−0.717049	+	1.43668i
$689$	346.042			0.502239
$690$		0			0
$691$	−151.617	+	151.617i	−0.219417	+	0.219417i	−0.808253	−	0.588836i	$-0.799586\pi$
$691$	−151.617	+	151.617i	−0.219417	+	0.219417i	0.588836	+	0.808253i	$0.299586\pi$
$692$	232.150	−	37.7586i	0.335477	−	0.0545644i
$693$		0			0
$694$	−1189.47	−	383.753i	−1.71394	−	0.552958i
$695$	−166.417			−0.239449
$696$		0			0
$697$		−	1055.17i		−	1.51388i
$698$	321.353	+	103.676i	0.460391	+	0.148533i
$699$		0			0
$700$	747.287	+	538.205i	1.06755	+	0.768864i
$701$	920.704	+	920.704i	1.31341	+	1.31341i	0.918882	+	0.394533i	$0.129094\pi$
$701$	920.704	+	920.704i	1.31341	+	1.31341i	0.394533	+	0.918882i	$0.370906\pi$
$702$		0			0
$703$		−	222.982i		−	0.317186i
$704$	95.3473	+	181.500i	0.135437	+	0.257812i
$705$		0			0
$706$	917.906	−	470.102i	1.30015	−	0.665866i
$707$	1010.18	−	1010.18i	1.42882	−	1.42882i
$708$		0			0
$709$	405.348	−	405.348i	0.571718	−	0.571718i	−0.360890	−	0.932608i	$-0.617527\pi$
$709$	405.348	−	405.348i	0.571718	−	0.571718i	0.932608	+	0.360890i	$0.117527\pi$
$710$	10.4605	−	32.4233i	0.0147331	−	0.0456666i
$711$		0			0
$712$	−173.498	−	26.3718i	−0.243677	−	0.0370391i
$713$		−	1373.72i		−	1.92667i
$714$		0			0
$715$	−31.4199	−	31.4199i	−0.0439439	−	0.0439439i
$716$	48.1958	+	296.321i	0.0673125	+	0.413856i
$717$		0			0
$718$	−762.363	+	390.441i	−1.06179	+	0.543789i
$719$			880.704i			1.22490i	0.790509	+	0.612450i	$0.209816\pi$
$719$			880.704i			1.22490i	−0.790509	+	0.612450i	$0.790184\pi$
$720$		0			0
$721$	581.930			0.807115
$722$	−191.147	−	373.227i	−0.264746	−	0.516935i
$723$		0			0
$724$	60.4859	+	371.884i	0.0835440	+	0.513651i
$725$	−565.983	+	565.983i	−0.780667	+	0.780667i
$726$		0			0
$727$	1000.46			1.37615			0.688077	−	0.725637i	$-0.258455\pi$
$727$	1000.46			1.37615			0.688077	+	0.725637i	$0.258455\pi$
$728$	−628.547	+	462.680i	−0.863388	+	0.635549i
$729$		0			0
$730$	301.999	+	97.4320i	0.413697	+	0.133469i
$731$	−1092.64	−	1092.64i	−1.49472	−	1.49472i
$732$		0			0
$733$	540.306	+	540.306i	0.737116	+	0.737116i	0.972019	−	0.234903i	$-0.0754772\pi$
$733$	540.306	+	540.306i	0.737116	+	0.737116i	−0.234903	+	0.972019i	$0.575477\pi$
$734$	−200.097	−	390.704i	−0.272612	−	0.532295i
$735$		0			0
$736$	759.011	+	743.401i	1.03126	+	1.01006i
$737$	−46.2301			−0.0627274
$738$		0			0
$739$	893.726	−	893.726i	1.20937	−	1.20937i	0.238142	−	0.971230i	$-0.423462\pi$
$739$	893.726	−	893.726i	1.20937	−	1.20937i	0.971230	−	0.238142i	$-0.0765384\pi$
$740$	−43.1775	−	31.0970i	−0.0583480	−	0.0420229i
$741$		0			0
$742$	−218.841	+	678.316i	−0.294934	+	0.914172i
$743$	−1295.75			−1.74394			−0.871969	−	0.489561i	$-0.837157\pi$
$743$	−1295.75			−1.74394			−0.871969	+	0.489561i	$0.837157\pi$
$744$		0			0
$745$		−	395.527i		−	0.530909i
$746$	369.092	−	1144.03i	0.494761	−	1.53355i
$747$		0			0
$748$	−283.054	+	46.0380i	−0.378414	+	0.0615481i
$749$	−1130.13	−	1130.13i	−1.50885	−	1.50885i
$750$		0			0
$751$			229.818i			0.306016i	0.988225	+	0.153008i	$0.0488961\pi$
$751$			229.818i			0.306016i	−0.988225	+	0.153008i	$0.951104\pi$
$752$	692.066	−	231.243i	0.920300	−	0.307504i
$753$		0			0
$754$	−309.220	−	603.774i	−0.410106	−	0.800761i
$755$	−64.9987	+	64.9987i	−0.0860910	+	0.0860910i
$756$		0			0
$757$	−373.678	+	373.678i	−0.493630	+	0.493630i	−0.909448	−	0.415818i	$-0.863495\pi$
$757$	−373.678	+	373.678i	−0.493630	+	0.493630i	0.415818	+	0.909448i	$0.363495\pi$
$758$	984.973	+	317.776i	1.29944	+	0.419229i
$759$		0			0
$760$	−269.262	−	40.9280i	−0.354293	−	0.0538527i
$761$			384.012i			0.504615i	0.967647	+	0.252307i	$0.0811894\pi$
$761$			384.012i			0.504615i	−0.967647	+	0.252307i	$0.918811\pi$
$762$		0			0
$763$	−813.296	−	813.296i	−1.06592	−	1.06592i
$764$	369.222	+	265.918i	0.483275	+	0.348060i
$765$		0			0
$766$	194.440	+	379.658i	0.253838	+	0.495637i
$767$			85.8874i			0.111978i
$768$		0			0
$769$	865.026			1.12487			0.562436	−	0.826841i	$-0.309864\pi$
$769$	865.026			1.12487			0.562436	+	0.826841i	$0.309864\pi$
$770$	81.4598	−	41.7193i	0.105792	−	0.0541809i
$771$		0			0
$772$	−62.0876	+	86.2074i	−0.0804244	+	0.111668i
$773$	1.78859	−	1.78859i	0.00231383	−	0.00231383i	−0.705949	−	0.708263i	$-0.749479\pi$
$773$	1.78859	−	1.78859i	0.00231383	−	0.00231383i	0.708263	+	0.705949i	$0.249479\pi$
$774$		0			0
$775$	−950.368			−1.22628
$776$	129.007	−	848.725i	0.166246	−	1.09372i
$777$		0			0
$778$	183.066	−	567.428i	0.235303	−	0.729341i
$779$	796.409	+	796.409i	1.02235	+	1.02235i
$780$		0			0
$781$	27.0753	+	27.0753i	0.0346675	+	0.0346675i
$782$	−1322.70	+	677.413i	−1.69143	+	0.866257i
$783$		0			0
$784$	−260.991	−	781.098i	−0.332897	−	0.996299i
$785$	−109.681			−0.139721
$786$		0			0
$787$	143.702	−	143.702i	0.182595	−	0.182595i	−0.609891	−	0.792485i	$-0.708787\pi$
$787$	143.702	−	143.702i	0.182595	−	0.182595i	0.792485	+	0.609891i	$0.208787\pi$
$788$	47.1265	+	289.747i	0.0598052	+	0.367699i
$789$		0			0
$790$	12.0990	+	3.90342i	0.0153152	+	0.00494104i
$791$	1722.49			2.17761
$792$		0			0
$793$		−	494.855i		−	0.624029i
$794$	1057.63	+	341.216i	1.33203	+	0.429743i
$795$		0			0
$796$	−319.655	+	443.835i	−0.401577	+	0.557582i
$797$	477.929	+	477.929i	0.599660	+	0.599660i	0.940222	−	0.340562i	$-0.110617\pi$
$797$	477.929	+	477.929i	0.599660	+	0.599660i	−0.340562	+	0.940222i	$0.610617\pi$
$798$		0			0
$799$			1020.64i			1.27740i
$800$	514.301	−	525.101i	0.642877	−	0.656376i
$801$		0			0
$802$	−52.2091	+	26.7387i	−0.0650987	+	0.0333400i
$803$	−252.187	+	252.187i	−0.314056	+	0.314056i
$804$		0			0
$805$	335.359	−	335.359i	0.416595	−	0.416595i
$806$	247.299	−	766.524i	0.306823	−	0.951023i
$807$		0			0
$808$	−675.929	−	918.244i	−0.836545	−	1.13644i
$809$		−	1524.96i		−	1.88500i	−0.334212	−	0.942498i	$-0.608470\pi$
$809$		−	1524.96i		−	1.88500i	0.334212	−	0.942498i	$-0.391530\pi$
$810$		0			0
$811$	−576.427	−	576.427i	−0.710761	−	0.710761i	0.255933	−	0.966694i	$-0.417617\pi$
$811$	−576.427	−	576.427i	−0.710761	−	0.710761i	−0.966694	+	0.255933i	$0.917617\pi$
$812$	1379.08	−	224.303i	1.69837	−	0.276236i
$813$		0			0
$814$	53.2286	−	27.2608i	0.0653914	−	0.0334899i
$815$			211.224i			0.259170i
$816$		0			0
$817$	1649.37			2.01882
$818$	−548.026	−	1070.06i	−0.669958	−	1.30814i
$819$		0			0
$820$	265.281	−	43.1473i	0.323514	−	0.0526186i
$821$	386.324	−	386.324i	0.470552	−	0.470552i	−0.431541	−	0.902093i	$-0.642030\pi$
$821$	386.324	−	386.324i	0.470552	−	0.470552i	0.902093	+	0.431541i	$0.142030\pi$
$822$		0			0
$823$	377.870			0.459138			0.229569	−	0.973292i	$-0.426268\pi$
$823$	377.870			0.459138			0.229569	+	0.973292i	$0.426268\pi$
$824$	69.7950	−	459.176i	0.0847026	−	0.557252i
$825$		0			0
$826$	−168.357	−	54.3161i	−0.203822	−	0.0657580i
$827$	−140.900	−	140.900i	−0.170375	−	0.170375i	0.616769	−	0.787144i	$-0.288441\pi$
$827$	−140.900	−	140.900i	−0.170375	−	0.170375i	−0.787144	+	0.616769i	$0.788441\pi$
$828$		0			0
$829$	−522.203	−	522.203i	−0.629919	−	0.629919i	0.318128	−	0.948048i	$-0.396946\pi$
$829$	−522.203	−	522.203i	−0.629919	−	0.629919i	−0.948048	+	0.318128i	$0.896946\pi$
$830$	−18.6962	−	36.5056i	−0.0225255	−	0.0439826i
$831$		0			0
$832$	289.694	+	551.451i	0.348190	+	0.662802i
$833$	1151.95			1.38289
$834$		0			0
$835$	53.7226	−	53.7226i	0.0643385	−	0.0643385i
$836$	178.892	−	248.387i	0.213985	−	0.297114i
$837$		0			0
$838$	450.621	−	1396.74i	0.537734	−	1.66675i
$839$	442.133			0.526976			0.263488	−	0.964663i	$-0.415127\pi$
$839$	442.133			0.526976			0.263488	+	0.964663i	$0.415127\pi$
$840$		0			0
$841$			373.376i			0.443967i
$842$	−357.114	+	1106.91i	−0.424126	+	1.31462i
$843$		0			0
$844$	128.203	+	788.226i	0.151899	+	0.933917i
$845$	74.8417	+	74.8417i	0.0885701	+	0.0885701i
$846$		0			0
$847$		−	1109.99i		−	1.31049i
$848$	508.982	+	254.033i	0.600215	+	0.299567i
$849$		0			0
$850$	468.650	+	915.071i	0.551352	+	1.07655i
$851$	219.135	−	219.135i	0.257503	−	0.257503i
$852$		0			0
$853$	−494.617	+	494.617i	−0.579856	+	0.579856i	−0.934863	−	0.355007i	$-0.884478\pi$
$853$	−494.617	+	494.617i	−0.579856	+	0.579856i	0.355007	+	0.934863i	$0.384478\pi$
$854$	970.020	+	312.952i	1.13585	+	0.366454i
$855$		0			0
$856$	−1027.28	+	756.190i	−1.20009	+	0.883400i
$857$			1676.75i			1.95654i	0.207340	+	0.978269i	$0.433519\pi$
$857$			1676.75i			1.95654i	−0.207340	+	0.978269i	$0.566481\pi$
$858$		0			0
$859$	228.948	+	228.948i	0.266529	+	0.266529i	0.827700	−	0.561171i	$-0.189649\pi$
$859$	228.948	+	228.948i	0.266529	+	0.266529i	−0.561171	+	0.827700i	$0.689649\pi$
$860$	230.021	−	319.380i	0.267466	−	0.371372i
$861$		0			0
$862$	37.4777	+	73.1778i	0.0434776	+	0.0848931i
$863$		−	1603.23i		−	1.85774i	−0.370401	−	0.928872i	$-0.620780\pi$
$863$		−	1603.23i		−	1.85774i	0.370401	−	0.928872i	$-0.379220\pi$
$864$		0			0
$865$	−83.7981			−0.0968764
$866$	−626.038	+	320.622i	−0.722907	+	0.370234i
$867$		0			0
$868$	1346.15	+	969.517i	1.55087	+	1.11695i
$869$	−10.1034	+	10.1034i	−0.0116264	+	0.0116264i
$870$		0			0
$871$	−140.461			−0.161264
$872$	−739.281	+	544.192i	−0.847799	+	0.624074i
$873$		0			0
$874$	487.038	−	1509.61i	0.557252	−	1.72725i
$875$	−484.533	−	484.533i	−0.553752	−	0.553752i
$876$		0			0
$877$	289.017	+	289.017i	0.329552	+	0.329552i	0.852416	−	0.522864i	$-0.175136\pi$
$877$	289.017	+	289.017i	0.329552	+	0.329552i	−0.522864	+	0.852416i	$0.675136\pi$
$878$	−1380.69	+	707.113i	−1.57254	+	0.805367i
$879$		0			0
$880$	−23.1488	−	69.2801i	−0.0263055	−	0.0787274i
$881$	−937.450			−1.06408			−0.532038	−	0.846721i	$-0.678574\pi$
$881$	−937.450			−1.06408			−0.532038	+	0.846721i	$0.678574\pi$
$882$		0			0
$883$	−485.966	+	485.966i	−0.550357	+	0.550357i	−0.926544	−	0.376187i	$-0.877235\pi$
$883$	−485.966	+	485.966i	−0.550357	+	0.550357i	0.376187	+	0.926544i	$0.377235\pi$
$884$	−860.005	+	139.877i	−0.972856	+	0.158232i
$885$		0			0
$886$	649.726	+	209.617i	0.733326	+	0.236588i
$887$	534.193			0.602247			0.301123	−	0.953585i	$-0.402638\pi$
$887$	534.193			0.602247			0.301123	+	0.953585i	$0.402638\pi$
$888$		0			0
$889$			369.201i			0.415299i
$890$	59.5044	+	19.1976i	0.0668589	+	0.0215703i
$891$		0			0
$892$	−397.958	−	286.614i	−0.446142	−	0.321316i
$893$	−770.346	−	770.346i	−0.862649	−	0.862649i
$894$		0			0
$895$		−	106.961i		−	0.119510i
$896$	−1264.17	+	219.118i	−1.41090	+	0.244552i
$897$		0			0
$898$	−474.153	+	242.835i	−0.528010	+	0.270418i
$899$	−1019.56	+	1019.56i	−1.13410	+	1.13410i
$900$		0			0
$901$	−562.638	+	562.638i	−0.624460	+	0.624460i
$902$	−92.7475	+	287.479i	−0.102824	+	0.318712i
$903$		0			0
$904$	206.590	−	1359.14i	0.228529	−	1.50347i
$905$		−	134.237i		−	0.148328i
$906$		0			0
$907$	368.669	+	368.669i	0.406471	+	0.406471i	0.880506	−	0.474035i	$-0.157203\pi$
$907$	368.669	+	368.669i	0.406471	+	0.406471i	−0.474035	+	0.880506i	$0.657203\pi$
$908$	−268.668	−	1651.84i	−0.295890	−	1.81921i
$909$		0			0
$910$	247.500	−	126.756i	0.271978	−	0.139292i
$911$			1592.43i			1.74800i	0.485927	+	0.873999i	$0.338482\pi$
$911$			1592.43i			1.74800i	−0.485927	+	0.873999i	$0.661518\pi$
$912$		0			0
$913$	46.0967			0.0504893
$914$	469.739	+	917.198i	0.513938	+	1.00350i
$915$		0			0
$916$	−66.6408	−	409.726i	−0.0727519	−	0.447299i
$917$	123.978	−	123.978i	0.135199	−	0.135199i
$918$		0			0
$919$	−403.500			−0.439064			−0.219532	−	0.975605i	$-0.570453\pi$
$919$	−403.500			−0.439064			−0.219532	+	0.975605i	$0.570453\pi$
$920$	−224.395	−	304.839i	−0.243908	−	0.331347i
$921$		0			0
$922$	−15.2818	−	4.93029i	−0.0165747	−	0.00534738i
$923$	82.2632	+	82.2632i	0.0891258	+	0.0891258i
$924$		0			0
$925$	−151.602	−	151.602i	−0.163894	−	0.163894i
$926$	423.486	+	826.886i	0.457328	+	0.892965i
$927$		0			0
$928$	−11.5855	−	1115.07i	−0.0124844	−	1.20159i
$929$	−348.546			−0.375184			−0.187592	−	0.982247i	$-0.560068\pi$
$929$	−348.546			−0.375184			−0.187592	+	0.982247i	$0.560068\pi$
$930$		0			0
$931$	−869.449	+	869.449i	−0.933887	+	0.933887i
$932$	−507.089	−	365.211i	−0.544087	−	0.391858i
$933$		0			0
$934$	−430.734	+	1335.10i	−0.461171	+	1.42944i
$935$	102.173			0.109276
$936$		0			0
$937$			248.875i			0.265609i	0.991142	+	0.132804i	$0.0423982\pi$
$937$			248.875i			0.265609i	−0.991142	+	0.132804i	$0.957602\pi$
$938$	88.8292	−	275.334i	0.0947007	−	0.293533i
$939$		0			0
$940$	−256.600	+	41.7353i	−0.272978	+	0.0443992i
$941$	−884.188	−	884.188i	−0.939626	−	0.939626i	0.0586528	−	0.998278i	$-0.481319\pi$
$941$	−884.188	−	884.188i	−0.939626	−	0.939626i	−0.998278	+	0.0586528i	$0.981319\pi$
$942$		0			0
$943$			1565.34i			1.65996i
$944$	−63.0508	+	126.329i	−0.0667911	+	0.133823i
$945$		0			0
$946$	201.645	+	393.727i	0.213156	+	0.416202i
$947$	−462.279	+	462.279i	−0.488151	+	0.488151i	−0.907722	−	0.419571i	$-0.862180\pi$
$947$	−462.279	+	462.279i	−0.488151	+	0.488151i	0.419571	+	0.907722i	$0.362180\pi$
$948$		0			0
$949$	−766.221	+	766.221i	−0.807398	+	0.807398i
$950$	−1044.39	−	336.944i	−1.09935	−	0.354678i
$951$		0			0
$952$	269.687	−	1774.25i	0.283285	−	1.86371i
$953$			707.656i			0.742556i	0.928522	+	0.371278i	$0.121080\pi$
$953$			707.656i			0.742556i	−0.928522	+	0.371278i	$0.878920\pi$
$954$		0			0
$955$	−114.632	−	114.632i	−0.120033	−	0.120033i
$956$	42.7525	+	30.7909i	0.0447202	+	0.0322081i
$957$		0			0
$958$	−345.347	−	674.314i	−0.360487	−	0.703877i
$959$			1460.22i			1.52265i
$960$		0			0
$961$	−750.983			−0.781460
$962$	161.725	−	82.8267i	0.168113	−	0.0860984i
$963$		0			0
$964$	−443.033	+	615.143i	−0.459578	+	0.638115i
$965$	26.7647	−	26.7647i	0.0277354	−	0.0277354i
$966$		0			0
$967$	−841.240			−0.869949			−0.434974	−	0.900443i	$-0.643243\pi$
$967$	−841.240			−0.869949			−0.434974	+	0.900443i	$0.643243\pi$
$968$	−875.844	−	133.129i	−0.904797	−	0.137530i
$969$		0			0
$970$	−93.9112	+	291.086i	−0.0968157	+	0.300088i
$971$	−308.415	−	308.415i	−0.317626	−	0.317626i	0.530229	−	0.847855i	$-0.322106\pi$
$971$	−308.415	−	308.415i	−0.317626	−	0.317626i	−0.847855	+	0.530229i	$0.822106\pi$
$972$		0			0
$973$	827.654	+	827.654i	0.850621	+	0.850621i
$974$	262.472	−	134.424i	0.269478	−	0.138012i
$975$		0			0
$976$	363.278	−	727.866i	0.372211	−	0.745764i
$977$	1316.85			1.34786			0.673928	−	0.738797i	$-0.264606\pi$
$977$	1316.85			1.34786			0.673928	+	0.738797i	$0.264606\pi$
$978$		0			0
$979$	−49.6897	+	49.6897i	−0.0507556	+	0.0507556i
$980$	47.1044	+	289.611i	0.0480657	+	0.295521i
$981$		0			0
$982$	−294.880	−	95.1352i	−0.300285	−	0.0968791i
$983$	−504.538			−0.513263			−0.256632	−	0.966509i	$-0.582613\pi$
$983$	−504.538			−0.513263			−0.256632	+	0.966509i	$0.582613\pi$
$984$		0			0
$985$		−	104.588i		−	0.106181i
$986$	1484.46	+	478.922i	1.50554	+	0.485722i
$987$		0			0
$988$	543.528	−	754.677i	0.550129	−	0.763843i
$989$	1620.92	+	1620.92i	1.63895	+	1.63895i
$990$		0			0
$991$			436.650i			0.440616i	0.975430	+	0.220308i	$0.0707062\pi$
$991$			436.650i			0.440616i	−0.975430	+	0.220308i	$0.929294\pi$
$992$	926.457	−	945.911i	0.933929	−	0.953539i
$993$		0			0
$994$	−213.277	+	109.229i	−0.214565	+	0.109888i
$995$	137.797	−	137.797i	0.138489	−	0.138489i
$996$		0			0
$997$	−383.801	+	383.801i	−0.384956	+	0.384956i	−0.872884	−	0.487928i	$-0.837753\pi$
$997$	−383.801	+	383.801i	−0.384956	+	0.384956i	0.487928	+	0.872884i	$0.337753\pi$
$998$	−313.093	+	970.458i	−0.313720	+	0.972403i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.3.m.c.19.6		16
3.2	odd	2		48.3.l.a.19.3	✓	16
4.3	odd	2		576.3.m.c.559.4		16
8.3	odd	2		1152.3.m.c.991.5		16
8.5	even	2		1152.3.m.f.991.5		16
12.11	even	2		192.3.l.a.175.3		16
16.3	odd	4		1152.3.m.f.415.5		16
16.5	even	4		576.3.m.c.271.4		16
16.11	odd	4	inner	144.3.m.c.91.6		16
16.13	even	4		1152.3.m.c.415.5		16
24.5	odd	2		384.3.l.a.223.2		16
24.11	even	2		384.3.l.b.223.6		16
48.5	odd	4		192.3.l.a.79.3		16
48.11	even	4		48.3.l.a.43.3	yes	16
48.29	odd	4		384.3.l.b.31.6		16
48.35	even	4		384.3.l.a.31.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.3	✓	16	3.2	odd	2
48.3.l.a.43.3	yes	16	48.11	even	4
144.3.m.c.19.6		16	1.1	even	1	trivial
144.3.m.c.91.6		16	16.11	odd	4	inner
192.3.l.a.79.3		16	48.5	odd	4
192.3.l.a.175.3		16	12.11	even	2
384.3.l.a.31.2		16	48.35	even	4
384.3.l.a.223.2		16	24.5	odd	2
384.3.l.b.31.6		16	48.29	odd	4
384.3.l.b.223.6		16	24.11	even	2
576.3.m.c.271.4		16	16.5	even	4
576.3.m.c.559.4		16	4.3	odd	2
1152.3.m.c.415.5		16	16.13	even	4
1152.3.m.c.991.5		16	8.3	odd	2
1152.3.m.f.415.5		16	16.3	odd	4
1152.3.m.f.991.5		16	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	144.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.78012 - 0.911682i) q^{2} +(2.33767 - 3.24581i) q^{4} +(-1.00772 + 1.00772i) q^{5} +10.0236 q^{7} +(1.20220 - 7.90915i) q^{8} +O(q^{10})\)	\(q+(1.78012 - 0.911682i) q^{2} +(2.33767 - 3.24581i) q^{4} +(-1.00772 + 1.00772i) q^{5} +10.0236 q^{7} +(1.20220 - 7.90915i) q^{8} +(-0.875146 + 2.71259i) q^{10} +(-2.26517 - 2.26517i) q^{11} +(-6.88229 - 6.88229i) q^{13} +(17.8432 - 9.13830i) q^{14} +(-5.07058 - 15.1753i) q^{16} +22.3801 q^{17} +(-16.8918 + 16.8918i) q^{19} +(0.915151 + 5.62660i) q^{20} +(-6.09740 - 1.96717i) q^{22} -33.2007 q^{23} +22.9690i q^{25} +(-18.5258 - 5.97686i) q^{26} +(23.4318 - 32.5346i) q^{28} +(24.6412 + 24.6412i) q^{29} +41.3761i q^{31} +(-22.8613 - 22.3911i) q^{32} +(39.8394 - 20.4036i) q^{34} +(-10.1010 + 10.1010i) q^{35} +(-6.60031 + 6.60031i) q^{37} +(-14.6695 + 45.4693i) q^{38} +(6.75875 + 9.18170i) q^{40} -47.1477i q^{41} +(-48.8218 - 48.8218i) q^{43} +(-12.6475 + 2.05709i) q^{44} +(-59.1013 + 30.2685i) q^{46} +45.6048i q^{47} +51.4717 q^{49} +(20.9404 + 40.8876i) q^{50} +(-38.4271 + 6.25007i) q^{52} +(-25.1401 + 25.1401i) q^{53} +4.56532 q^{55} +(12.0503 - 79.2779i) q^{56} +(66.3292 + 21.3994i) q^{58} +(-6.23974 - 6.23974i) q^{59} +(35.9513 + 35.9513i) q^{61} +(37.7219 + 73.6546i) q^{62} +(-61.1095 - 19.0167i) q^{64} +13.8709 q^{65} +(10.2045 - 10.2045i) q^{67} +(52.3174 - 72.6417i) q^{68} +(-8.77208 + 27.1898i) q^{70} -11.9529 q^{71} -111.332i q^{73} +(-5.73197 + 17.7667i) q^{74} +(15.3401 + 94.3149i) q^{76} +(-22.7051 - 22.7051i) q^{77} -4.46031i q^{79} +(20.4022 + 10.1827i) q^{80} +(-42.9837 - 83.9287i) q^{82} +(-10.1751 + 10.1751i) q^{83} +(-22.5530 + 22.5530i) q^{85} +(-131.419 - 42.3988i) q^{86} +(-20.6388 + 15.1924i) q^{88} -21.9364i q^{89} +(-68.9850 - 68.9850i) q^{91} +(-77.6124 + 107.763i) q^{92} +(41.5771 + 81.1821i) q^{94} -34.0444i q^{95} +107.309 q^{97} +(91.6260 - 46.9259i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} + 12 q^{8}+O(q^{10}) \) 16 * q + 12 * q^4 + 12 * q^8	\( 16 q + 12 q^{4} + 12 q^{8} - 56 q^{10} - 32 q^{11} + 44 q^{14} + 32 q^{16} - 32 q^{19} - 80 q^{20} + 32 q^{22} + 128 q^{23} + 100 q^{26} - 120 q^{28} - 32 q^{29} - 160 q^{32} + 96 q^{34} - 96 q^{35} - 96 q^{37} - 168 q^{38} + 48 q^{40} + 160 q^{43} - 88 q^{44} + 136 q^{46} + 112 q^{49} + 236 q^{50} - 48 q^{52} + 160 q^{53} - 256 q^{55} + 224 q^{56} + 144 q^{58} + 128 q^{59} - 32 q^{61} + 276 q^{62} - 408 q^{64} + 32 q^{65} + 320 q^{67} + 448 q^{68} - 384 q^{70} - 512 q^{71} - 348 q^{74} + 72 q^{76} - 224 q^{77} - 552 q^{80} - 40 q^{82} + 160 q^{83} + 160 q^{85} - 528 q^{86} + 480 q^{88} - 480 q^{91} - 496 q^{92} + 312 q^{94} + 440 q^{98}+O(q^{100}) \) 16 * q + 12 * q^4 + 12 * q^8 - 56 * q^10 - 32 * q^11 + 44 * q^14 + 32 * q^16 - 32 * q^19 - 80 * q^20 + 32 * q^22 + 128 * q^23 + 100 * q^26 - 120 * q^28 - 32 * q^29 - 160 * q^32 + 96 * q^34 - 96 * q^35 - 96 * q^37 - 168 * q^38 + 48 * q^40 + 160 * q^43 - 88 * q^44 + 136 * q^46 + 112 * q^49 + 236 * q^50 - 48 * q^52 + 160 * q^53 - 256 * q^55 + 224 * q^56 + 144 * q^58 + 128 * q^59 - 32 * q^61 + 276 * q^62 - 408 * q^64 + 32 * q^65 + 320 * q^67 + 448 * q^68 - 384 * q^70 - 512 * q^71 - 348 * q^74 + 72 * q^76 - 224 * q^77 - 552 * q^80 - 40 * q^82 + 160 * q^83 + 160 * q^85 - 528 * q^86 + 480 * q^88 - 480 * q^91 - 496 * q^92 + 312 * q^94 + 440 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more