LMFDB - Embedded newform 1152.3.m.f.415.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (of order $4$, degree $2$, not 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:	$31.3897264543$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			415.5
Root			$1.78012 - 0.911682i$ of defining polynomial
Character	$\chi$	$=$	1152.415
Dual form			1152.3.m.f.991.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00772 + 1.00772i) q^{5} +10.0236 q^{7} +O(q^{10})$	$q+(1.00772 + 1.00772i) q^{5} +10.0236 q^{7} +(2.26517 - 2.26517i) q^{11} +(6.88229 - 6.88229i) q^{13} +22.3801 q^{17} +(16.8918 + 16.8918i) q^{19} -33.2007 q^{23} -22.9690i q^{25} +(-24.6412 + 24.6412i) q^{29} -41.3761i q^{31} +(10.1010 + 10.1010i) q^{35} +(6.60031 + 6.60031i) q^{37} +47.1477i q^{41} +(48.8218 - 48.8218i) q^{43} -45.6048i q^{47} +51.4717 q^{49} +(25.1401 + 25.1401i) q^{53} +4.56532 q^{55} +(6.23974 - 6.23974i) q^{59} +(-35.9513 + 35.9513i) q^{61} +13.8709 q^{65} +(-10.2045 - 10.2045i) q^{67} -11.9529 q^{71} +111.332i q^{73} +(22.7051 - 22.7051i) q^{77} +4.46031i q^{79} +(10.1751 + 10.1751i) q^{83} +(22.5530 + 22.5530i) q^{85} +21.9364i q^{89} +(68.9850 - 68.9850i) q^{91} +34.0444i q^{95} +107.309 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) $ 16 * q + 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 + 96 * q^35 + 96 * q^37 - 160 * q^43 + 112 * q^49 - 160 * q^53 - 256 * q^55 - 128 * q^59 + 32 * q^61 + 32 * q^65 - 320 * q^67 - 512 * q^71 + 224 * q^77 - 160 * q^83 - 160 * q^85 + 480 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00772	+	1.00772i	0.201544	+	0.201544i	0.800661	−	0.599117i	$-0.204482\pi$
$5$	1.00772	+	1.00772i	0.201544	+	0.201544i	−0.599117	+	0.800661i	$0.704482\pi$
$6$		0			0
$7$	10.0236			1.43194			0.715969	−	0.698133i	$-0.245985\pi$
$7$	10.0236			1.43194			0.715969	+	0.698133i	$0.245985\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.26517	−	2.26517i	0.205925	−	0.205925i	−0.596608	−	0.802533i	$-0.703485\pi$
$11$	2.26517	−	2.26517i	0.205925	−	0.205925i	0.802533	+	0.596608i	$0.203485\pi$
$12$		0			0
$13$	6.88229	−	6.88229i	0.529407	−	0.529407i	−0.390989	−	0.920395i	$-0.627867\pi$
$13$	6.88229	−	6.88229i	0.529407	−	0.529407i	0.920395	+	0.390989i	$0.127867\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	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	$-0.0336092\pi$
$19$	16.8918	+	16.8918i	0.889041	+	0.889041i	−0.105390	+	0.994431i	$0.533609\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−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
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−24.6412	+	24.6412i	−0.849696	+	0.849696i	−0.990095	−	0.140399i	$-0.955161\pi$
$29$	−24.6412	+	24.6412i	−0.849696	+	0.849696i	0.140399	+	0.990095i	$0.455161\pi$
$30$		0			0
$31$		−	41.3761i		−	1.33471i	−0.744738	−	0.667357i	$-0.767426\pi$
$31$		−	41.3761i		−	1.33471i	0.744738	−	0.667357i	$-0.232574\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.1010	+	10.1010i	0.288599	+	0.288599i
$36$		0			0
$37$	6.60031	+	6.60031i	0.178387	+	0.178387i	0.790652	−	0.612266i	$-0.209742\pi$
$37$	6.60031	+	6.60031i	0.178387	+	0.178387i	−0.612266	+	0.790652i	$0.709742\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			47.1477i			1.14994i	0.818173	+	0.574972i	$0.194987\pi$
$41$			47.1477i			1.14994i	−0.818173	+	0.574972i	$0.805013\pi$
$42$		0			0
$43$	48.8218	−	48.8218i	1.13539	−	1.13539i	0.146124	−	0.989266i	$-0.453320\pi$
$43$	48.8218	−	48.8218i	1.13539	−	1.13539i	0.989266	−	0.146124i	$-0.0466799\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	45.6048i		−	0.970315i	−0.874427	−	0.485157i	$-0.838762\pi$
$47$		−	45.6048i		−	0.970315i	0.874427	−	0.485157i	$-0.161238\pi$
$48$		0			0
$49$	51.4717			1.05044
$50$		0			0
$51$		0			0
$52$		0			0
$53$	25.1401	+	25.1401i	0.474341	+	0.474341i	0.903316	−	0.428975i	$-0.141125\pi$
$53$	25.1401	+	25.1401i	0.474341	+	0.474341i	−0.428975	+	0.903316i	$0.641125\pi$
$54$		0			0
$55$	4.56532			0.0830059
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.23974	−	6.23974i	0.105758	−	0.105758i	−0.652248	−	0.758006i	$-0.726174\pi$
$59$	6.23974	−	6.23974i	0.105758	−	0.105758i	0.758006	+	0.652248i	$0.226174\pi$
$60$		0			0
$61$	−35.9513	+	35.9513i	−0.589366	+	0.589366i	−0.937460	−	0.348093i	$-0.886829\pi$
$61$	−35.9513	+	35.9513i	−0.589366	+	0.589366i	0.348093	+	0.937460i	$0.386829\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13.8709			0.213398
$66$		0			0
$67$	−10.2045	−	10.2045i	−0.152307	−	0.152307i	0.626841	−	0.779147i	$-0.284348\pi$
$67$	−10.2045	−	10.2045i	−0.152307	−	0.152307i	−0.779147	+	0.626841i	$0.784348\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−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	$0.276053\pi$
$73$			111.332i			1.52510i	−0.646929	+	0.762550i	$0.723947\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	22.7051	−	22.7051i	0.294871	−	0.294871i
$78$		0			0
$79$			4.46031i			0.0564596i	0.999601	+	0.0282298i	$0.00898702\pi$
$79$			4.46031i			0.0564596i	−0.999601	+	0.0282298i	$0.991013\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.1751	+	10.1751i	0.122592	+	0.122592i	0.765741	−	0.643149i	$-0.222373\pi$
$83$	10.1751	+	10.1751i	0.122592	+	0.122592i	−0.643149	+	0.765741i	$0.722373\pi$
$84$		0			0
$85$	22.5530	+	22.5530i	0.265329	+	0.265329i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			21.9364i			0.246476i	0.992377	+	0.123238i	$0.0393279\pi$
$89$			21.9364i			0.246476i	−0.992377	+	0.123238i	$0.960672\pi$
$90$		0			0
$91$	68.9850	−	68.9850i	0.758077	−	0.758077i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			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	$0.313429\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−100.780	−	100.780i	−0.997824	−	0.997824i	0.00217389	−	0.999998i	$-0.499308\pi$
$101$	−100.780	−	100.780i	−0.997824	−	0.997824i	−0.999998	+	0.00217389i	$0.999308\pi$
$102$		0			0
$103$	58.0562			0.563653			0.281826	−	0.959465i	$-0.409060\pi$
$103$	58.0562			0.563653			0.281826	+	0.959465i	$0.409060\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	112.747	−	112.747i	1.05371	−	1.05371i	0.0552381	−	0.998473i	$-0.482408\pi$
$107$	112.747	−	112.747i	1.05371	−	1.05371i	0.998473	−	0.0552381i	$-0.0175918\pi$
$108$		0			0
$109$	81.1384	−	81.1384i	0.744389	−	0.744389i	−0.229030	−	0.973419i	$-0.573555\pi$
$109$	81.1384	−	81.1384i	0.744389	−	0.744389i	0.973419	+	0.229030i	$0.0735554\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	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
$116$		0			0
$117$		0			0
$118$		0			0
$119$	224.329			1.88512
$120$		0			0
$121$			110.738i			0.915190i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	48.3394	−	48.3394i	0.386715	−	0.386715i
$126$		0			0
$127$		−	36.8333i		−	0.290026i	−0.989430	−	0.145013i	$-0.953678\pi$
$127$		−	36.8333i		−	0.290026i	0.989430	−	0.145013i	$-0.0463224\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.3686	−	12.3686i	−0.0944170	−	0.0944170i	0.658321	−	0.752738i	$-0.271267\pi$
$131$	−12.3686	−	12.3686i	−0.0944170	−	0.0944170i	−0.752738	+	0.658321i	$0.771267\pi$
$132$		0			0
$133$	169.316	+	169.316i	1.27305	+	1.27305i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	145.679i		−	1.06335i	−0.846949	−	0.531674i	$-0.821563\pi$
$137$		−	145.679i		−	1.06335i	0.846949	−	0.531674i	$-0.178437\pi$
$138$		0			0
$139$	−82.5709	+	82.5709i	−0.594035	+	0.594035i	−0.938719	−	0.344684i	$-0.887986\pi$
$139$	−82.5709	+	82.5709i	−0.594035	+	0.594035i	0.344684	+	0.938719i	$0.387986\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	31.1791i		−	0.218036i
$144$		0			0
$145$	−49.6629			−0.342503
$146$		0			0
$147$		0			0
$148$		0			0
$149$	196.248	+	196.248i	1.31710	+	1.31710i	0.916059	+	0.401043i	$0.131352\pi$
$149$	196.248	+	196.248i	1.31710	+	1.31710i	0.401043	+	0.916059i	$0.368648\pi$
$150$		0			0
$151$	64.5007			0.427157			0.213578	−	0.976926i	$-0.431488\pi$
$151$	64.5007			0.427157			0.213578	+	0.976926i	$0.431488\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	41.6956	−	41.6956i	0.269004	−	0.269004i
$156$		0			0
$157$	−54.4202	+	54.4202i	−0.346625	+	0.346625i	−0.858851	−	0.512226i	$-0.828821\pi$
$157$	−54.4202	+	54.4202i	−0.346625	+	0.346625i	0.512226	+	0.858851i	$0.328821\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−332.789			−2.06701
$162$		0			0
$163$	−104.803	−	104.803i	−0.642961	−	0.642961i	0.308321	−	0.951282i	$-0.400233\pi$
$163$	−104.803	−	104.803i	−0.642961	−	0.642961i	−0.951282	+	0.308321i	$0.900233\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−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
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−41.5780	+	41.5780i	−0.240335	+	0.240335i	−0.816989	−	0.576654i	$-0.804358\pi$
$173$	−41.5780	+	41.5780i	−0.240335	+	0.240335i	0.576654	+	0.816989i	$0.304358\pi$
$174$		0			0
$175$		−	230.231i		−	1.31561i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	53.0709	+	53.0709i	0.296486	+	0.296486i	0.839636	−	0.543150i	$-0.182769\pi$
$179$	53.0709	+	53.0709i	0.296486	+	0.296486i	−0.543150	+	0.839636i	$0.682769\pi$
$180$		0			0
$181$	66.6042	+	66.6042i	0.367979	+	0.367979i	0.866740	−	0.498761i	$-0.166211\pi$
$181$	66.6042	+	66.6042i	0.367979	+	0.367979i	−0.498761	+	0.866740i	$0.666211\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			13.3025i			0.0719056i
$186$		0			0
$187$	50.6949	−	50.6949i	0.271096	−	0.271096i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	113.753i		−	0.595567i	−0.954633	−	0.297784i	$-0.903753\pi$
$191$		−	113.753i		−	0.595567i	0.954633	−	0.297784i	$-0.0962474\pi$
$192$		0			0
$193$	−26.5596			−0.137615			−0.0688073	−	0.997630i	$-0.521919\pi$
$193$	−26.5596			−0.137615			−0.0688073	+	0.997630i	$0.521919\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	51.8935	+	51.8935i	0.263419	+	0.263419i	0.826442	−	0.563023i	$-0.190362\pi$
$197$	51.8935	+	51.8935i	0.263419	+	0.263419i	−0.563023	+	0.826442i	$0.690362\pi$
$198$		0			0
$199$	−136.741			−0.687140			−0.343570	−	0.939127i	$-0.611636\pi$
$199$	−136.741			−0.687140			−0.343570	+	0.939127i	$0.611636\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−246.992	+	246.992i	−1.21671	+	1.21671i
$204$		0			0
$205$	−47.5118	+	47.5118i	−0.231765	+	0.231765i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	76.5255			0.366151
$210$		0			0
$211$	141.171	+	141.171i	0.669057	+	0.669057i	0.957498	−	0.288441i	$-0.0931368\pi$
$211$	141.171	+	141.171i	0.669057	+	0.669057i	−0.288441	+	0.957498i	$0.593137\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	98.3975			0.457663
$216$		0			0
$217$		−	414.736i		−	1.91123i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	154.027	−	154.027i	0.696953	−	0.696953i
$222$		0			0
$223$			122.607i			0.549806i	0.961472	+	0.274903i	$0.0886457\pi$
$223$			122.607i			0.549806i	−0.961472	+	0.274903i	$0.911354\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−295.844	−	295.844i	−1.30328	−	1.30328i	−0.926168	−	0.377112i	$-0.876917\pi$
$227$	−295.844	−	295.844i	−1.30328	−	1.30328i	−0.377112	−	0.926168i	$-0.623083\pi$
$228$		0			0
$229$	−73.3817	−	73.3817i	−0.320444	−	0.320444i	0.528493	−	0.848937i	$-0.322757\pi$
$229$	−73.3817	−	73.3817i	−0.320444	−	0.320444i	−0.848937	+	0.528493i	$0.822757\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			156.229i			0.670509i	0.942128	+	0.335255i	$0.108822\pi$
$233$			156.229i			0.670509i	−0.942128	+	0.335255i	$0.891178\pi$
$234$		0			0
$235$	45.9569	−	45.9569i	0.195561	−	0.195561i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	13.1716i		−	0.0551113i	−0.999620	−	0.0275557i	$-0.991228\pi$
$239$		−	13.1716i		−	0.0551113i	0.999620	−	0.0275557i	$-0.00877235\pi$
$240$		0			0
$241$	−189.519			−0.786386			−0.393193	−	0.919456i	$-0.628630\pi$
$241$	−189.519			−0.786386			−0.393193	+	0.919456i	$0.628630\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	51.8692	+	51.8692i	0.211711	+	0.211711i
$246$		0			0
$247$	232.508			0.941328
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−27.4434	+	27.4434i	−0.109336	+	0.109336i	−0.759658	−	0.650322i	$-0.774634\pi$
$251$	−27.4434	+	27.4434i	−0.109336	+	0.109336i	0.650322	+	0.759658i	$0.274634\pi$
$252$		0			0
$253$	−75.2053	+	75.2053i	−0.297254	+	0.297254i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−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
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−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
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−194.213	+	194.213i	−0.721981	+	0.721981i	−0.969008	−	0.247028i	$-0.920546\pi$
$269$	−194.213	+	194.213i	−0.721981	+	0.721981i	0.247028	+	0.969008i	$0.420546\pi$
$270$		0			0
$271$			291.647i			1.07619i	0.842884	+	0.538095i	$0.180856\pi$
$271$			291.647i			1.07619i	−0.842884	+	0.538095i	$0.819144\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−52.0287	−	52.0287i	−0.189195	−	0.189195i
$276$		0			0
$277$	−305.166	−	305.166i	−1.10168	−	1.10168i	−0.994208	−	0.107475i	$-0.965723\pi$
$277$	−305.166	−	305.166i	−1.10168	−	1.10168i	−0.107475	−	0.994208i	$-0.534277\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			211.861i			0.753955i	0.926222	+	0.376978i	$0.123037\pi$
$281$			211.861i			0.753955i	−0.926222	+	0.376978i	$0.876963\pi$
$282$		0			0
$283$	−105.325	+	105.325i	−0.372175	+	0.372175i	−0.868269	−	0.496094i	$-0.834767\pi$
$283$	−105.325	+	105.325i	−0.372175	+	0.372175i	0.496094	+	0.868269i	$0.334767\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			472.588i			1.64665i
$288$		0			0
$289$	211.871			0.733117
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−171.289	−	171.289i	−0.584603	−	0.584603i	0.351562	−	0.936165i	$-0.385651\pi$
$293$	−171.289	−	171.289i	−0.584603	−	0.584603i	−0.936165	+	0.351562i	$0.885651\pi$
$294$		0			0
$295$	12.5758			0.0426300
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−228.497	+	228.497i	−0.764204	+	0.764204i
$300$		0			0
$301$	489.368	−	489.368i	1.62581	−	1.62581i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−72.4579			−0.237567
$306$		0			0
$307$	27.1124	+	27.1124i	0.0883140	+	0.0883140i	0.749884	−	0.661570i	$-0.230109\pi$
$307$	27.1124	+	27.1124i	0.0883140	+	0.0883140i	−0.661570	+	0.749884i	$0.730109\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−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	$-0.795865\pi$
$313$		−	374.501i		−	1.19649i	0.801313	−	0.598245i	$-0.204135\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−48.5840	+	48.5840i	−0.153262	+	0.153262i	−0.779573	−	0.626311i	$-0.784564\pi$
$317$	−48.5840	+	48.5840i	−0.153262	+	0.153262i	0.626311	+	0.779573i	$0.284564\pi$
$318$		0			0
$319$			111.633i			0.349947i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	378.040	+	378.040i	1.17040	+	1.17040i
$324$		0			0
$325$	−158.079	−	158.079i	−0.486398	−	0.486398i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	457.122i		−	1.38943i
$330$		0			0
$331$	1.88883	−	1.88883i	0.00570644	−	0.00570644i	−0.704248	−	0.709954i	$-0.748716\pi$
$331$	1.88883	−	1.88883i	0.00570644	−	0.00570644i	0.709954	+	0.704248i	$0.248716\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	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	$0.694669\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−93.7240	−	93.7240i	−0.274851	−	0.274851i
$342$		0			0
$343$	24.7757			0.0722325
$344$		0			0
$345$		0			0
$346$		0			0
$347$	441.887	−	441.887i	1.27345	−	1.27345i	0.329183	−	0.944266i	$-0.393227\pi$
$347$	441.887	−	441.887i	1.27345	−	1.27345i	0.944266	−	0.329183i	$-0.106773\pi$
$348$		0			0
$349$	−119.382	+	119.382i	−0.342068	+	0.342068i	−0.857144	−	0.515076i	$-0.827764\pi$
$349$	−119.382	+	119.382i	−0.342068	+	0.342068i	0.515076	+	0.857144i	$0.327764\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	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
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−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
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−112.192	+	112.192i	−0.307375	+	0.307375i
$366$		0			0
$367$			219.482i			0.598043i	0.954246	+	0.299021i	$0.0966602\pi$
$367$			219.482i			0.598043i	−0.954246	+	0.299021i	$0.903340\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	251.993	+	251.993i	0.679226	+	0.679226i
$372$		0			0
$373$	−425.005	−	425.005i	−1.13942	−	1.13942i	−0.988554	−	0.150870i	$-0.951793\pi$
$373$	−425.005	−	425.005i	−1.13942	−	1.13942i	−0.150870	−	0.988554i	$-0.548207\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			339.175i			0.899669i
$378$		0			0
$379$	−365.916	+	365.916i	−0.965476	+	0.965476i	−0.999424	−	0.0339473i	$-0.989192\pi$
$379$	−365.916	+	365.916i	−0.965476	+	0.965476i	0.0339473	+	0.999424i	$0.489192\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	213.276i		−	0.556857i	−0.960457	−	0.278428i	$-0.910187\pi$
$383$		−	213.276i		−	0.556857i	0.960457	−	0.278428i	$-0.0898135\pi$
$384$		0			0
$385$	45.7608			0.118859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−210.798	−	210.798i	−0.541898	−	0.541898i	0.382187	−	0.924085i	$-0.375171\pi$
$389$	−210.798	−	210.798i	−0.541898	−	0.541898i	−0.924085	+	0.382187i	$0.875171\pi$
$390$		0			0
$391$	−743.037			−1.90035
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.49475	+	4.49475i	−0.0113791	+	0.0113791i
$396$		0			0
$397$	−392.907	+	392.907i	−0.989690	+	0.989690i	−0.999947	−	0.0102579i	$-0.996735\pi$
$397$	−392.907	+	392.907i	−0.989690	+	0.989690i	0.0102579	+	0.999947i	$0.496735\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−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
$404$		0			0
$405$		0			0
$406$		0			0
$407$	29.9017			0.0734684
$408$		0			0
$409$			601.115i			1.46972i	0.678219	+	0.734860i	$0.262752\pi$
$409$			601.115i			1.46972i	−0.678219	+	0.734860i	$0.737248\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	62.5444	−	62.5444i	0.151439	−	0.151439i
$414$		0			0
$415$			20.5073i			0.0494153i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−518.885	−	518.885i	−1.23839	−	1.23839i	−0.960659	−	0.277729i	$-0.910418\pi$
$419$	−518.885	−	518.885i	−1.23839	−	1.23839i	−0.277729	−	0.960659i	$-0.589582\pi$
$420$		0			0
$421$	411.213	+	411.213i	0.976754	+	0.976754i	0.999736	−	0.0229817i	$-0.00731596\pi$
$421$	411.213	+	411.213i	0.976754	+	0.976754i	−0.0229817	+	0.999736i	$0.507316\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	514.049i		−	1.20953i
$426$		0			0
$427$	−360.360	+	360.360i	−0.843936	+	0.843936i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	41.1083i		−	0.0953789i	−0.998862	−	0.0476895i	$-0.984814\pi$
$431$		−	41.1083i		−	0.0953789i	0.998862	−	0.0476895i	$-0.0151858\pi$
$432$		0			0
$433$	−351.682			−0.812199			−0.406100	−	0.913829i	$-0.633111\pi$
$433$	−351.682			−0.812199			−0.406100	+	0.913829i	$0.633111\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−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	$0.844742\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−241.372	+	241.372i	−0.544858	+	0.544858i	−0.924949	−	0.380091i	$-0.875893\pi$
$443$	−241.372	+	241.372i	−0.544858	+	0.544858i	0.380091	+	0.924949i	$0.375893\pi$
$444$		0			0
$445$	−22.1058	+	22.1058i	−0.0496759	+	0.0496759i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−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
$452$		0			0
$453$		0			0
$454$		0			0
$455$	139.035			0.305572
$456$		0			0
$457$		−	515.244i		−	1.12745i	−0.825963	−	0.563725i	$-0.809368\pi$
$457$		−	515.244i		−	1.12745i	0.825963	−	0.563725i	$-0.190632\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.67717	−	5.67717i	0.0123149	−	0.0123149i	−0.700923	−	0.713237i	$-0.747228\pi$
$461$	5.67717	−	5.67717i	0.0123149	−	0.0123149i	0.713237	+	0.700923i	$0.247228\pi$
$462$		0			0
$463$		−	464.510i		−	1.00326i	−0.865082	−	0.501631i	$-0.832733\pi$
$463$		−	464.510i		−	1.00326i	0.865082	−	0.501631i	$-0.167267\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	495.985	+	495.985i	1.06207	+	1.06207i	0.997942	+	0.0641248i	$0.0204256\pi$
$467$	495.985	+	495.985i	1.06207	+	1.06207i	0.0641248	+	0.997942i	$0.479574\pi$
$468$		0			0
$469$	−102.286	−	102.286i	−0.218094	−	0.218094i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	221.180i		−	0.467610i
$474$		0			0
$475$	387.987	−	387.987i	0.816815	−	0.816815i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			378.802i			0.790818i	0.918505	+	0.395409i	$0.129397\pi$
$479$			378.802i			0.790818i	−0.918505	+	0.395409i	$0.870603\pi$
$480$		0			0
$481$	90.8504			0.188878
$482$		0			0
$483$		0			0
$484$		0			0
$485$	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	$0.451628\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	109.547	−	109.547i	0.223110	−	0.223110i	−0.586697	−	0.809807i	$-0.699572\pi$
$491$	109.547	−	109.547i	0.223110	−	0.223110i	0.809807	+	0.586697i	$0.199572\pi$
$492$		0			0
$493$	−551.473	+	551.473i	−1.11861	+	1.11861i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−119.810			−0.241067
$498$		0			0
$499$	360.523	+	360.523i	0.722491	+	0.722491i	0.969112	−	0.246621i	$-0.0793202\pi$
$499$	360.523	+	360.523i	0.722491	+	0.722491i	−0.246621	+	0.969112i	$0.579320\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	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
$506$		0			0
$507$		0			0
$508$		0			0
$509$	677.931	−	677.931i	1.33189	−	1.33189i	0.428208	−	0.903680i	$-0.359145\pi$
$509$	677.931	−	677.931i	1.33189	−	1.33189i	0.903680	−	0.428208i	$-0.140855\pi$
$510$		0			0
$511$			1115.95i			2.18385i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	58.5045	+	58.5045i	0.113601	+	0.113601i
$516$		0			0
$517$	−103.303	−	103.303i	−0.199812	−	0.199812i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			143.173i			0.274804i	0.990515	+	0.137402i	$0.0438753\pi$
$521$			143.173i			0.274804i	−0.990515	+	0.137402i	$0.956125\pi$
$522$		0			0
$523$	−226.187	+	226.187i	−0.432481	+	0.432481i	−0.889471	−	0.456991i	$-0.848927\pi$
$523$	−226.187	+	226.187i	−0.432481	+	0.432481i	0.456991	+	0.889471i	$0.348927\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	926.004i		−	1.75712i
$528$		0			0
$529$	573.288			1.08372
$530$		0			0
$531$		0			0
$532$		0			0
$533$	324.484	+	324.484i	0.608788	+	0.608788i
$534$		0			0
$535$	227.235			0.424739
$536$		0			0
$537$		0			0
$538$		0			0
$539$	116.592	−	116.592i	0.216312	−	0.216312i
$540$		0			0
$541$	156.708	−	156.708i	0.289663	−	0.289663i	−0.547284	−	0.836947i	$-0.684338\pi$
$541$	156.708	−	156.708i	0.289663	−	0.289663i	0.836947	+	0.547284i	$0.184338\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	163.530			0.300055
$546$		0			0
$547$	−247.357	−	247.357i	−0.452207	−	0.452207i	0.443880	−	0.896086i	$-0.353602\pi$
$547$	−247.357	−	247.357i	−0.452207	−	0.452207i	−0.896086	+	0.443880i	$0.853602\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−832.466			−1.51083
$552$		0			0
$553$			44.7082i			0.0808466i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−661.193	+	661.193i	−1.18706	+	1.18706i	−0.209184	+	0.977876i	$0.567081\pi$
$557$	−661.193	+	661.193i	−1.18706	+	1.18706i	−0.977876	+	0.209184i	$0.932919\pi$
$558$		0			0
$559$		−	672.011i		−	1.20217i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	246.685	+	246.685i	0.438162	+	0.438162i	0.891393	−	0.453231i	$-0.149729\pi$
$563$	246.685	+	246.685i	0.438162	+	0.438162i	−0.453231	+	0.891393i	$0.649729\pi$
$564$		0			0
$565$	173.171	+	173.171i	0.306497	+	0.306497i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			243.567i			0.428061i	0.976827	+	0.214030i	$0.0686592\pi$
$569$			243.567i			0.428061i	−0.976827	+	0.214030i	$0.931341\pi$
$570$		0			0
$571$	−59.9229	+	59.9229i	−0.104944	+	0.104944i	−0.757629	−	0.652685i	$-0.773642\pi$
$571$	−59.9229	+	59.9229i	−0.104944	+	0.104944i	0.652685	+	0.757629i	$0.273642\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			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	$0.462230\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	101.991	+	101.991i	0.175543	+	0.175543i
$582$		0			0
$583$	113.893			0.195357
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−331.817	+	331.817i	−0.565276	+	0.565276i	−0.930801	−	0.365525i	$-0.880889\pi$
$587$	−331.817	+	331.817i	−0.565276	+	0.565276i	0.365525	+	0.930801i	$0.380889\pi$
$588$		0			0
$589$	698.916	−	698.916i	1.18661	−	1.18661i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−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
$596$		0			0
$597$		0			0
$598$		0			0
$599$	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	$-0.863858\pi$
$601$		−	498.566i		−	0.829561i	0.909922	−	0.414780i	$-0.136142\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−111.593	+	111.593i	−0.184451	+	0.184451i
$606$		0			0
$607$		−	568.740i		−	0.936969i	−0.883472	−	0.468484i	$-0.844800\pi$
$607$		−	568.740i		−	0.936969i	0.883472	−	0.468484i	$-0.155200\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−313.865	−	313.865i	−0.513691	−	0.513691i
$612$		0			0
$613$	168.441	+	168.441i	0.274782	+	0.274782i	0.831022	−	0.556240i	$-0.187756\pi$
$613$	168.441	+	168.441i	0.274782	+	0.274782i	−0.556240	+	0.831022i	$0.687756\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	599.157i		−	0.971081i	−0.874214	−	0.485541i	$-0.838623\pi$
$617$		−	599.157i		−	0.971081i	0.874214	−	0.485541i	$-0.161377\pi$
$618$		0			0
$619$	−126.719	+	126.719i	−0.204715	+	0.204715i	−0.802017	−	0.597301i	$-0.796240\pi$
$619$	−126.719	+	126.719i	−0.204715	+	0.204715i	0.597301	+	0.802017i	$0.296240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			219.881i			0.352939i
$624$		0			0
$625$	−476.800			−0.762879
$626$		0			0
$627$		0			0
$628$		0			0
$629$	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	$0.322363\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	37.1177	−	37.1177i	0.0584531	−	0.0584531i
$636$		0			0
$637$	354.243	−	354.243i	0.556112	−	0.556112i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−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	$-0.276388\pi$
$643$	−75.2980	−	75.2980i	−0.117104	−	0.117104i	−0.763230	+	0.646126i	$0.776388\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	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
$650$		0			0
$651$		0			0
$652$		0			0
$653$	457.453	−	457.453i	0.700541	−	0.700541i	−0.263986	−	0.964527i	$-0.585037\pi$
$653$	457.453	−	457.453i	0.700541	−	0.700541i	0.964527	+	0.263986i	$0.0850371\pi$
$654$		0			0
$655$		−	24.9283i		−	0.0380584i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−430.079	−	430.079i	−0.652623	−	0.652623i	0.301001	−	0.953624i	$-0.402679\pi$
$659$	−430.079	−	430.079i	−0.652623	−	0.652623i	−0.953624	+	0.301001i	$0.902679\pi$
$660$		0			0
$661$	513.622	+	513.622i	0.777038	+	0.777038i	0.979326	−	0.202288i	$-0.0648376\pi$
$661$	513.622	+	513.622i	0.777038	+	0.777038i	−0.202288	+	0.979326i	$0.564838\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			341.246i			0.513152i
$666$		0			0
$667$	818.105	−	818.105i	1.22654	−	1.22654i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			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	$0.809761\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	633.271	+	633.271i	0.935408	+	0.935408i	0.998037	−	0.0626291i	$-0.0199485\pi$
$677$	633.271	+	633.271i	0.935408	+	0.935408i	−0.0626291	+	0.998037i	$0.519949\pi$
$678$		0			0
$679$	1075.62			1.58412
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−429.651	+	429.651i	−0.629065	+	0.629065i	−0.947833	−	0.318768i	$-0.896731\pi$
$683$	−429.651	+	429.651i	−0.629065	+	0.629065i	0.318768	+	0.947833i	$0.396731\pi$
$684$		0			0
$685$	146.803	−	146.803i	0.214312	−	0.214312i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	346.042			0.502239
$690$		0			0
$691$	151.617	+	151.617i	0.219417	+	0.219417i	0.808253	−	0.588836i	$-0.200414\pi$
$691$	151.617	+	151.617i	0.219417	+	0.219417i	−0.588836	+	0.808253i	$0.700414\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−166.417			−0.239449
$696$		0			0
$697$			1055.17i			1.51388i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−920.704	+	920.704i	−1.31341	+	1.31341i	−0.394533	+	0.918882i	$0.629094\pi$
$701$	−920.704	+	920.704i	−1.31341	+	1.31341i	−0.918882	+	0.394533i	$0.870906\pi$
$702$		0			0
$703$			222.982i			0.317186i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1010.18	−	1010.18i	−1.42882	−	1.42882i
$708$		0			0
$709$	−405.348	−	405.348i	−0.571718	−	0.571718i	0.360890	−	0.932608i	$-0.382473\pi$
$709$	−405.348	−	405.348i	−0.571718	−	0.571718i	−0.932608	+	0.360890i	$0.882473\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1373.72i			1.92667i
$714$		0			0
$715$	31.4199	−	31.4199i	0.0439439	−	0.0439439i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	880.704i		−	1.22490i	−0.790509	−	0.612450i	$-0.790184\pi$
$719$		−	880.704i		−	1.22490i	0.790509	−	0.612450i	$-0.209816\pi$
$720$		0			0
$721$	581.930			0.807115
$722$		0			0
$723$		0			0
$724$		0			0
$725$	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	$0.258455\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1092.64	−	1092.64i	1.49472	−	1.49472i
$732$		0			0
$733$	−540.306	+	540.306i	−0.737116	+	0.737116i	−0.972019	−	0.234903i	$-0.924523\pi$
$733$	−540.306	+	540.306i	−0.737116	+	0.737116i	0.234903	+	0.972019i	$0.424523\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−46.2301			−0.0627274
$738$		0			0
$739$	−893.726	−	893.726i	−1.20937	−	1.20937i	−0.971230	−	0.238142i	$-0.923462\pi$
$739$	−893.726	−	893.726i	−1.20937	−	1.20937i	−0.238142	−	0.971230i	$-0.576538\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−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
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1130.13	−	1130.13i	1.50885	−	1.50885i
$750$		0			0
$751$		−	229.818i		−	0.306016i	−0.988225	−	0.153008i	$-0.951104\pi$
$751$		−	229.818i		−	0.306016i	0.988225	−	0.153008i	$-0.0488961\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	64.9987	+	64.9987i	0.0860910	+	0.0860910i
$756$		0			0
$757$	373.678	+	373.678i	0.493630	+	0.493630i	0.909448	−	0.415818i	$-0.136505\pi$
$757$	373.678	+	373.678i	0.493630	+	0.493630i	−0.415818	+	0.909448i	$0.636505\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	384.012i		−	0.504615i	−0.967647	−	0.252307i	$-0.918811\pi$
$761$		−	384.012i		−	0.504615i	0.967647	−	0.252307i	$-0.0811894\pi$
$762$		0			0
$763$	813.296	−	813.296i	1.06592	−	1.06592i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	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	$0.309864\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.78859	−	1.78859i	−0.00231383	−	0.00231383i	0.705949	−	0.708263i	$-0.250521\pi$
$773$	−1.78859	−	1.78859i	−0.00231383	−	0.00231383i	−0.708263	+	0.705949i	$0.750521\pi$
$774$		0			0
$775$	−950.368			−1.22628
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−796.409	+	796.409i	−1.02235	+	1.02235i
$780$		0			0
$781$	−27.0753	+	27.0753i	−0.0346675	+	0.0346675i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−109.681			−0.139721
$786$		0			0
$787$	−143.702	−	143.702i	−0.182595	−	0.182595i	0.609891	−	0.792485i	$-0.291213\pi$
$787$	−143.702	−	143.702i	−0.182595	−	0.182595i	−0.792485	+	0.609891i	$0.791213\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1722.49			2.17761
$792$		0			0
$793$			494.855i			0.624029i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−477.929	+	477.929i	−0.599660	+	0.599660i	−0.940222	−	0.340562i	$-0.889383\pi$
$797$	−477.929	+	477.929i	−0.599660	+	0.599660i	0.340562	+	0.940222i	$0.389383\pi$
$798$		0			0
$799$		−	1020.64i		−	1.27740i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	252.187	+	252.187i	0.314056	+	0.314056i
$804$		0			0
$805$	−335.359	−	335.359i	−0.416595	−	0.416595i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1524.96i			1.88500i	0.334212	+	0.942498i	$0.391530\pi$
$809$			1524.96i			1.88500i	−0.334212	+	0.942498i	$0.608470\pi$
$810$		0			0
$811$	576.427	−	576.427i	0.710761	−	0.710761i	−0.255933	−	0.966694i	$-0.582383\pi$
$811$	576.427	−	576.427i	0.710761	−	0.710761i	0.966694	+	0.255933i	$0.0823828\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	211.224i		−	0.259170i
$816$		0			0
$817$	1649.37			2.01882
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−386.324	−	386.324i	−0.470552	−	0.470552i	0.431541	−	0.902093i	$-0.357970\pi$
$821$	−386.324	−	386.324i	−0.470552	−	0.470552i	−0.902093	+	0.431541i	$0.857970\pi$
$822$		0			0
$823$	377.870			0.459138			0.229569	−	0.973292i	$-0.426268\pi$
$823$	377.870			0.459138			0.229569	+	0.973292i	$0.426268\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	140.900	−	140.900i	0.170375	−	0.170375i	−0.616769	−	0.787144i	$-0.711559\pi$
$827$	140.900	−	140.900i	0.170375	−	0.170375i	0.787144	+	0.616769i	$0.211559\pi$
$828$		0			0
$829$	522.203	−	522.203i	0.629919	−	0.629919i	−0.318128	−	0.948048i	$-0.603054\pi$
$829$	522.203	−	522.203i	0.629919	−	0.629919i	0.948048	+	0.318128i	$0.103054\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1151.95			1.38289
$834$		0			0
$835$	−53.7226	−	53.7226i	−0.0643385	−	0.0643385i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	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
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−74.8417	+	74.8417i	−0.0885701	+	0.0885701i
$846$		0			0
$847$			1109.99i			1.31049i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−219.135	−	219.135i	−0.257503	−	0.257503i
$852$		0			0
$853$	494.617	+	494.617i	0.579856	+	0.579856i	0.934863	−	0.355007i	$-0.115522\pi$
$853$	494.617	+	494.617i	0.579856	+	0.579856i	−0.355007	+	0.934863i	$0.615522\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1676.75i		−	1.95654i	−0.207340	−	0.978269i	$-0.566481\pi$
$857$		−	1676.75i		−	1.95654i	0.207340	−	0.978269i	$-0.433519\pi$
$858$		0			0
$859$	−228.948	+	228.948i	−0.266529	+	0.266529i	−0.827700	−	0.561171i	$-0.810351\pi$
$859$	−228.948	+	228.948i	−0.266529	+	0.266529i	0.561171	+	0.827700i	$0.310351\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1603.23i			1.85774i	0.370401	+	0.928872i	$0.379220\pi$
$863$			1603.23i			1.85774i	−0.370401	+	0.928872i	$0.620780\pi$
$864$		0			0
$865$	−83.7981			−0.0968764
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.1034	+	10.1034i	0.0116264	+	0.0116264i
$870$		0			0
$871$	−140.461			−0.161264
$872$		0			0
$873$		0			0
$874$		0			0
$875$	484.533	−	484.533i	0.553752	−	0.553752i
$876$		0			0
$877$	−289.017	+	289.017i	−0.329552	+	0.329552i	−0.852416	−	0.522864i	$-0.824864\pi$
$877$	−289.017	+	289.017i	−0.329552	+	0.329552i	0.522864	+	0.852416i	$0.324864\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−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	$-0.122765\pi$
$883$	485.966	+	485.966i	0.550357	+	0.550357i	−0.376187	+	0.926544i	$0.622765\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	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
$890$		0			0
$891$		0			0
$892$		0			0
$893$	770.346	−	770.346i	0.862649	−	0.862649i
$894$		0			0
$895$			106.961i			0.119510i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1019.56	+	1019.56i	1.13410	+	1.13410i
$900$		0			0
$901$	562.638	+	562.638i	0.624460	+	0.624460i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			134.237i			0.148328i
$906$		0			0
$907$	−368.669	+	368.669i	−0.406471	+	0.406471i	−0.880506	−	0.474035i	$-0.842797\pi$
$907$	−368.669	+	368.669i	−0.406471	+	0.406471i	0.474035	+	0.880506i	$0.342797\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1592.43i		−	1.74800i	−0.485927	−	0.873999i	$-0.661518\pi$
$911$		−	1592.43i		−	1.74800i	0.485927	−	0.873999i	$-0.338482\pi$
$912$		0			0
$913$	46.0967			0.0504893
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−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	$0.570453\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−82.2632	+	82.2632i	−0.0891258	+	0.0891258i
$924$		0			0
$925$	151.602	−	151.602i	0.163894	−	0.163894i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−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
$932$		0			0
$933$		0			0
$934$		0			0
$935$	102.173			0.109276
$936$		0			0
$937$		−	248.875i		−	0.265609i	−0.991142	−	0.132804i	$-0.957602\pi$
$937$		−	248.875i		−	0.265609i	0.991142	−	0.132804i	$-0.0423982\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	884.188	−	884.188i	0.939626	−	0.939626i	−0.0586528	−	0.998278i	$-0.518681\pi$
$941$	884.188	−	884.188i	0.939626	−	0.939626i	0.998278	+	0.0586528i	$0.0186805\pi$
$942$		0			0
$943$		−	1565.34i		−	1.65996i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	462.279	+	462.279i	0.488151	+	0.488151i	0.907722	−	0.419571i	$-0.137820\pi$
$947$	462.279	+	462.279i	0.488151	+	0.488151i	−0.419571	+	0.907722i	$0.637820\pi$
$948$		0			0
$949$	766.221	+	766.221i	0.807398	+	0.807398i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	707.656i		−	0.742556i	−0.928522	−	0.371278i	$-0.878920\pi$
$953$		−	707.656i		−	0.742556i	0.928522	−	0.371278i	$-0.121080\pi$
$954$		0			0
$955$	114.632	−	114.632i	0.120033	−	0.120033i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1460.22i		−	1.52265i
$960$		0			0
$961$	−750.983			−0.781460
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−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	$0.643243\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	308.415	−	308.415i	0.317626	−	0.317626i	−0.530229	−	0.847855i	$-0.677894\pi$
$971$	308.415	−	308.415i	0.317626	−	0.317626i	0.847855	+	0.530229i	$0.177894\pi$
$972$		0			0
$973$	−827.654	+	827.654i	−0.850621	+	0.850621i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	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
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−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
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1620.92	+	1620.92i	−1.63895	+	1.63895i
$990$		0			0
$991$		−	436.650i		−	0.440616i	−0.975430	−	0.220308i	$-0.929294\pi$
$991$		−	436.650i		−	0.440616i	0.975430	−	0.220308i	$-0.0707062\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−137.797	−	137.797i	−0.138489	−	0.138489i
$996$		0			0
$997$	383.801	+	383.801i	0.384956	+	0.384956i	0.872884	−	0.487928i	$-0.162247\pi$
$997$	383.801	+	383.801i	0.384956	+	0.384956i	−0.487928	+	0.872884i	$0.662247\pi$
$998$		0			0
$999$		0			0

Twists

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

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

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00772 + 1.00772i) q^{5} +10.0236 q^{7} +O(q^{10})\)	\(q+(1.00772 + 1.00772i) q^{5} +10.0236 q^{7} +(2.26517 - 2.26517i) q^{11} +(6.88229 - 6.88229i) q^{13} +22.3801 q^{17} +(16.8918 + 16.8918i) q^{19} -33.2007 q^{23} -22.9690i q^{25} +(-24.6412 + 24.6412i) q^{29} -41.3761i q^{31} +(10.1010 + 10.1010i) q^{35} +(6.60031 + 6.60031i) q^{37} +47.1477i q^{41} +(48.8218 - 48.8218i) q^{43} -45.6048i q^{47} +51.4717 q^{49} +(25.1401 + 25.1401i) q^{53} +4.56532 q^{55} +(6.23974 - 6.23974i) q^{59} +(-35.9513 + 35.9513i) q^{61} +13.8709 q^{65} +(-10.2045 - 10.2045i) q^{67} -11.9529 q^{71} +111.332i q^{73} +(22.7051 - 22.7051i) q^{77} +4.46031i q^{79} +(10.1751 + 10.1751i) q^{83} +(22.5530 + 22.5530i) q^{85} +21.9364i q^{89} +(68.9850 - 68.9850i) q^{91} +34.0444i q^{95} +107.309 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) \) 16 * q + 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 + 96 * q^35 + 96 * q^37 - 160 * q^43 + 112 * q^49 - 160 * q^53 - 256 * q^55 - 128 * q^59 + 32 * q^61 + 32 * q^65 - 320 * q^67 - 512 * q^71 + 224 * q^77 - 160 * q^83 - 160 * q^85 + 480 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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