LMFDB - Embedded newform 450.6.f.d.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(107,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("450.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.f (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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.6.f.d.143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(52.0000 + 52.0000i) q^{7} +(45.2548 + 45.2548i) q^{8} +O(q^{10})$	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(52.0000 + 52.0000i) q^{7} +(45.2548 + 45.2548i) q^{8} -124.451i q^{11} +(-183.000 + 183.000i) q^{13} -294.156 q^{14} -256.000 q^{16} +(350.725 - 350.725i) q^{17} -2108.00i q^{19} +(352.000 + 352.000i) q^{22} +(1813.02 + 1813.02i) q^{23} -1035.20i q^{26} +(832.000 - 832.000i) q^{28} -6998.94 q^{29} +6784.00 q^{31} +(724.077 - 724.077i) q^{32} +1984.00i q^{34} +(-3723.00 - 3723.00i) q^{37} +(5962.32 + 5962.32i) q^{38} +12208.9i q^{41} +(-3180.00 + 3180.00i) q^{43} -1991.21 q^{44} -10256.0 q^{46} +(-7764.03 + 7764.03i) q^{47} -11399.0i q^{49} +(2928.00 + 2928.00i) q^{52} +(-22856.5 - 22856.5i) q^{53} +4706.50i q^{56} +(19796.0 - 19796.0i) q^{58} +35847.5 q^{59} +13740.0 q^{61} +(-19188.0 + 19188.0i) q^{62} +4096.00i q^{64} +(-34064.0 - 34064.0i) q^{67} +(-5611.60 - 5611.60i) q^{68} +48032.3i q^{71} +(29891.0 - 29891.0i) q^{73} +21060.5 q^{74} -33728.0 q^{76} +(6471.44 - 6471.44i) q^{77} -59328.0i q^{79} +(-34532.0 - 34532.0i) q^{82} +(-38132.9 - 38132.9i) q^{83} -17988.8i q^{86} +(5632.00 - 5632.00i) q^{88} +17045.5 q^{89} -19032.0 q^{91} +(29008.3 - 29008.3i) q^{92} -43920.0i q^{94} +(-96093.0 - 96093.0i) q^{97} +(32241.2 + 32241.2i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 208 q^{7}+O(q^{10}) $ 4 * q + 208 * q^7	$ 4 q + 208 q^{7} - 732 q^{13} - 1024 q^{16} + 1408 q^{22} + 3328 q^{28} + 27136 q^{31} - 14892 q^{37} - 12720 q^{43} - 41024 q^{46} + 11712 q^{52} + 79184 q^{58} + 54960 q^{61} - 136256 q^{67} + 119564 q^{73} - 134912 q^{76} - 138128 q^{82} + 22528 q^{88} - 76128 q^{91} - 384372 q^{97}+O(q^{100}) $ 4 * q + 208 * q^7 - 732 * q^13 - 1024 * q^16 + 1408 * q^22 + 3328 * q^28 + 27136 * q^31 - 14892 * q^37 - 12720 * q^43 - 41024 * q^46 + 11712 * q^52 + 79184 * q^58 + 54960 * q^61 - 136256 * q^67 + 119564 * q^73 - 134912 * q^76 - 138128 * q^82 + 22528 * q^88 - 76128 * q^91 - 384372 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−2.82843	+	2.82843i	−0.500000	+	0.500000i
$2$	−2.82843	+	2.82843i	−0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$		−	16.0000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	52.0000	+	52.0000i	0.401105	+	0.401105i	0.878622	−	0.477517i	$-0.158463\pi$
$7$	52.0000	+	52.0000i	0.401105	+	0.401105i	−0.477517	+	0.878622i	$0.658463\pi$
$8$	45.2548	+	45.2548i	0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$		−	124.451i		−	0.310110i	−0.987906	−	0.155055i	$-0.950445\pi$
$11$		−	124.451i		−	0.310110i	0.987906	−	0.155055i	$-0.0495555\pi$
$12$		0			0
$13$	−183.000	+	183.000i	−0.300326	+	0.300326i	−0.841141	−	0.540815i	$-0.818116\pi$
$13$	−183.000	+	183.000i	−0.300326	+	0.300326i	0.540815	+	0.841141i	$0.318116\pi$
$14$	−294.156			−0.401105
$15$		0			0
$16$	−256.000			−0.250000
$17$	350.725	−	350.725i	0.294337	−	0.294337i	−0.544454	−	0.838791i	$-0.683263\pi$
$17$	350.725	−	350.725i	0.294337	−	0.294337i	0.838791	+	0.544454i	$0.183263\pi$
$18$		0			0
$19$		−	2108.00i		−	1.33964i	−0.742526	−	0.669818i	$-0.766372\pi$
$19$		−	2108.00i		−	1.33964i	0.742526	−	0.669818i	$-0.233628\pi$
$20$		0			0
$21$		0			0
$22$	352.000	+	352.000i	0.155055	+	0.155055i
$23$	1813.02	+	1813.02i	0.714634	+	0.714634i	0.967501	−	0.252867i	$-0.0813736\pi$
$23$	1813.02	+	1813.02i	0.714634	+	0.714634i	−0.252867	+	0.967501i	$0.581374\pi$
$24$		0			0
$25$		0			0
$26$		−	1035.20i		−	0.300326i
$27$		0			0
$28$	832.000	−	832.000i	0.200553	−	0.200553i
$29$	−6998.94			−1.54539			−0.772694	−	0.634779i	$-0.781091\pi$
$29$	−6998.94			−1.54539			−0.772694	+	0.634779i	$0.781091\pi$
$30$		0			0
$31$	6784.00			1.26789			0.633945	−	0.773378i	$-0.281435\pi$
$31$	6784.00			1.26789			0.633945	+	0.773378i	$0.281435\pi$
$32$	724.077	−	724.077i	0.125000	−	0.125000i
$33$		0			0
$34$			1984.00i			0.294337i
$35$		0			0
$36$		0			0
$37$	−3723.00	−	3723.00i	−0.447084	−	0.447084i	0.447300	−	0.894384i	$-0.352386\pi$
$37$	−3723.00	−	3723.00i	−0.447084	−	0.447084i	−0.894384	+	0.447300i	$0.852386\pi$
$38$	5962.32	+	5962.32i	0.669818	+	0.669818i
$39$		0			0
$40$		0			0
$41$			12208.9i			1.13427i	0.823624	+	0.567136i	$0.191949\pi$
$41$			12208.9i			1.13427i	−0.823624	+	0.567136i	$0.808051\pi$
$42$		0			0
$43$	−3180.00	+	3180.00i	−0.262274	+	0.262274i	−0.825978	−	0.563703i	$-0.809376\pi$
$43$	−3180.00	+	3180.00i	−0.262274	+	0.262274i	0.563703	+	0.825978i	$0.309376\pi$
$44$	−1991.21			−0.155055
$45$		0			0
$46$	−10256.0			−0.714634
$47$	−7764.03	+	7764.03i	−0.512676	+	0.512676i	−0.915345	−	0.402670i	$-0.868082\pi$
$47$	−7764.03	+	7764.03i	−0.512676	+	0.512676i	0.402670	+	0.915345i	$0.368082\pi$
$48$		0			0
$49$		−	11399.0i		−	0.678229i
$50$		0			0
$51$		0			0
$52$	2928.00	+	2928.00i	0.150163	+	0.150163i
$53$	−22856.5	−	22856.5i	−1.11769	−	1.11769i	−0.992080	−	0.125608i	$-0.959912\pi$
$53$	−22856.5	−	22856.5i	−1.11769	−	1.11769i	−0.125608	−	0.992080i	$-0.540088\pi$
$54$		0			0
$55$		0			0
$56$			4706.50i			0.200553i
$57$		0			0
$58$	19796.0	−	19796.0i	0.772694	−	0.772694i
$59$	35847.5			1.34069			0.670345	−	0.742049i	$-0.266146\pi$
$59$	35847.5			1.34069			0.670345	+	0.742049i	$0.266146\pi$
$60$		0			0
$61$	13740.0			0.472783			0.236392	−	0.971658i	$-0.424035\pi$
$61$	13740.0			0.472783			0.236392	+	0.971658i	$0.424035\pi$
$62$	−19188.0	+	19188.0i	−0.633945	+	0.633945i
$63$		0			0
$64$			4096.00i			0.125000i
$65$		0			0
$66$		0			0
$67$	−34064.0	−	34064.0i	−0.927062	−	0.927062i	0.0704534	−	0.997515i	$-0.477555\pi$
$67$	−34064.0	−	34064.0i	−0.927062	−	0.927062i	−0.997515	+	0.0704534i	$0.977555\pi$
$68$	−5611.60	−	5611.60i	−0.147168	−	0.147168i
$69$		0			0
$70$		0			0
$71$			48032.3i			1.13081i	0.824815	+	0.565403i	$0.191279\pi$
$71$			48032.3i			1.13081i	−0.824815	+	0.565403i	$0.808721\pi$
$72$		0			0
$73$	29891.0	−	29891.0i	0.656498	−	0.656498i	−0.298052	−	0.954550i	$-0.596337\pi$
$73$	29891.0	−	29891.0i	0.656498	−	0.656498i	0.954550	+	0.298052i	$0.0963368\pi$
$74$	21060.5			0.447084
$75$		0			0
$76$	−33728.0			−0.669818
$77$	6471.44	−	6471.44i	0.124387	−	0.124387i
$78$		0			0
$79$		−	59328.0i		−	1.06953i	−0.845002	−	0.534764i	$-0.820401\pi$
$79$		−	59328.0i		−	1.06953i	0.845002	−	0.534764i	$-0.179599\pi$
$80$		0			0
$81$		0			0
$82$	−34532.0	−	34532.0i	−0.567136	−	0.567136i
$83$	−38132.9	−	38132.9i	−0.607581	−	0.607581i	0.334732	−	0.942313i	$-0.391354\pi$
$83$	−38132.9	−	38132.9i	−0.607581	−	0.607581i	−0.942313	+	0.334732i	$0.891354\pi$
$84$		0			0
$85$		0			0
$86$		−	17988.8i		−	0.262274i
$87$		0			0
$88$	5632.00	−	5632.00i	0.0775275	−	0.0775275i
$89$	17045.5			0.228105			0.114053	−	0.993475i	$-0.463617\pi$
$89$	17045.5			0.228105			0.114053	+	0.993475i	$0.463617\pi$
$90$		0			0
$91$	−19032.0			−0.240924
$92$	29008.3	−	29008.3i	0.357317	−	0.357317i
$93$		0			0
$94$		−	43920.0i		−	0.512676i
$95$		0			0
$96$		0			0
$97$	−96093.0	−	96093.0i	−1.03696	−	1.03696i	−0.999290	−	0.0376708i	$-0.988006\pi$
$97$	−96093.0	−	96093.0i	−1.03696	−	1.03696i	−0.0376708	−	0.999290i	$-0.511994\pi$
$98$	32241.2	+	32241.2i	0.339115	+	0.339115i
$99$		0			0
$100$		0			0
$101$			36615.4i			0.357158i	0.983926	+	0.178579i	$0.0571500\pi$
$101$			36615.4i			0.357158i	−0.983926	+	0.178579i	$0.942850\pi$
$102$		0			0
$103$	20528.0	−	20528.0i	0.190657	−	0.190657i	−0.605323	−	0.795980i	$-0.706956\pi$
$103$	20528.0	−	20528.0i	0.190657	−	0.190657i	0.795980	+	0.605323i	$0.206956\pi$
$104$	−16563.3			−0.150163
$105$		0			0
$106$	129296.			1.11769
$107$	52637.0	−	52637.0i	0.444459	−	0.444459i	−0.449048	−	0.893507i	$-0.648237\pi$
$107$	52637.0	−	52637.0i	0.444459	−	0.444459i	0.893507	+	0.449048i	$0.148237\pi$
$108$		0			0
$109$		−	166492.i		−	1.34223i	−0.741353	−	0.671115i	$-0.765816\pi$
$109$		−	166492.i		−	1.34223i	0.741353	−	0.671115i	$-0.234184\pi$
$110$		0			0
$111$		0			0
$112$	−13312.0	−	13312.0i	−0.100276	−	0.100276i
$113$	179182.	+	179182.i	1.32008	+	1.32008i	0.913711	+	0.406365i	$0.133204\pi$
$113$	179182.	+	179182.i	1.32008	+	1.32008i	0.406365	+	0.913711i	$0.366796\pi$
$114$		0			0
$115$		0			0
$116$			111983.i			0.772694i
$117$		0			0
$118$	−101392.	+	101392.i	−0.670345	+	0.670345i
$119$	36475.4			0.236120
$120$		0			0
$121$	145563.			0.903832
$122$	−38862.6	+	38862.6i	−0.236392	+	0.236392i
$123$		0			0
$124$		−	108544.i		−	0.633945i
$125$		0			0
$126$		0			0
$127$	211384.	+	211384.i	1.16295	+	1.16295i	0.983826	+	0.179129i	$0.0573280\pi$
$127$	211384.	+	211384.i	1.16295	+	1.16295i	0.179129	+	0.983826i	$0.442672\pi$
$128$	−11585.2	−	11585.2i	−0.0625000	−	0.0625000i
$129$		0			0
$130$		0			0
$131$		−	169592.i		−	0.863432i	−0.902009	−	0.431716i	$-0.857908\pi$
$131$		−	169592.i		−	0.863432i	0.902009	−	0.431716i	$-0.142092\pi$
$132$		0			0
$133$	109616.	−	109616.i	0.537335	−	0.537335i
$134$	192695.			0.927062
$135$		0			0
$136$	31744.0			0.147168
$137$	193044.	−	193044.i	0.878731	−	0.878731i	−0.114673	−	0.993403i	$-0.536582\pi$
$137$	193044.	−	193044.i	0.878731	−	0.878731i	0.993403	+	0.114673i	$0.0365819\pi$
$138$		0			0
$139$		−	309604.i		−	1.35916i	−0.733603	−	0.679578i	$-0.762163\pi$
$139$		−	309604.i		−	1.35916i	0.733603	−	0.679578i	$-0.237837\pi$
$140$		0			0
$141$		0			0
$142$	−135856.	−	135856.i	−0.565403	−	0.565403i
$143$	22774.5	+	22774.5i	0.0931341	+	0.0931341i
$144$		0			0
$145$		0			0
$146$			169089.i			0.656498i
$147$		0			0
$148$	−59568.0	+	59568.0i	−0.223542	+	0.223542i
$149$	−59078.8			−0.218005			−0.109002	−	0.994041i	$-0.534766\pi$
$149$	−59078.8			−0.218005			−0.109002	+	0.994041i	$0.534766\pi$
$150$		0			0
$151$	332872.			1.18805			0.594025	−	0.804446i	$-0.297538\pi$
$151$	332872.			1.18805			0.594025	+	0.804446i	$0.297538\pi$
$152$	95397.2	−	95397.2i	0.334909	−	0.334909i
$153$		0			0
$154$			36608.0i			0.124387i
$155$		0			0
$156$		0			0
$157$	−319869.	−	319869.i	−1.03567	−	1.03567i	−0.999340	−	0.0363342i	$-0.988432\pi$
$157$	−319869.	−	319869.i	−1.03567	−	1.03567i	−0.0363342	−	0.999340i	$-0.511568\pi$
$158$	167805.	+	167805.i	0.534764	+	0.534764i
$159$		0			0
$160$		0			0
$161$			188554.i			0.573286i
$162$		0			0
$163$	408844.	−	408844.i	1.20528	−	1.20528i	0.232744	−	0.972538i	$-0.425230\pi$
$163$	408844.	−	408844.i	1.20528	−	1.20528i	0.972538	−	0.232744i	$-0.0747703\pi$
$164$	195342.			0.567136
$165$		0			0
$166$	215712.			0.607581
$167$	97951.3	−	97951.3i	0.271781	−	0.271781i	−0.558036	−	0.829817i	$-0.688445\pi$
$167$	97951.3	−	97951.3i	0.271781	−	0.271781i	0.829817	+	0.558036i	$0.188445\pi$
$168$		0			0
$169$			304315.i			0.819609i
$170$		0			0
$171$		0			0
$172$	50880.0	+	50880.0i	0.131137	+	0.131137i
$173$	−171384.	−	171384.i	−0.435367	−	0.435367i	0.455082	−	0.890449i	$-0.349610\pi$
$173$	−171384.	−	171384.i	−0.435367	−	0.435367i	−0.890449	+	0.455082i	$0.849610\pi$
$174$		0			0
$175$		0			0
$176$			31859.4i			0.0775275i
$177$		0			0
$178$	−48212.0	+	48212.0i	−0.114053	+	0.114053i
$179$	716033.			1.67032			0.835162	−	0.550004i	$-0.185374\pi$
$179$	716033.			1.67032			0.835162	+	0.550004i	$0.185374\pi$
$180$		0			0
$181$	−18052.0			−0.0409571			−0.0204785	−	0.999790i	$-0.506519\pi$
$181$	−18052.0			−0.0409571			−0.0204785	+	0.999790i	$0.506519\pi$
$182$	53830.6	−	53830.6i	0.120462	−	0.120462i
$183$		0			0
$184$			164096.i			0.357317i
$185$		0			0
$186$		0			0
$187$	−43648.0	−	43648.0i	−0.0912768	−	0.0912768i
$188$	124225.	+	124225.i	0.256338	+	0.256338i
$189$		0			0
$190$		0			0
$191$		−	304260.i		−	0.603477i	−0.953391	−	0.301739i	$-0.902433\pi$
$191$		−	304260.i		−	0.603477i	0.953391	−	0.301739i	$-0.0975670\pi$
$192$		0			0
$193$	132339.	−	132339.i	0.255738	−	0.255738i	−0.567580	−	0.823318i	$-0.692120\pi$
$193$	132339.	−	132339.i	0.255738	−	0.255738i	0.823318	+	0.567580i	$0.192120\pi$
$194$	543584.			1.03696
$195$		0			0
$196$	−182384.			−0.339115
$197$	507237.	−	507237.i	0.931206	−	0.931206i	−0.0665755	−	0.997781i	$-0.521207\pi$
$197$	507237.	−	507237.i	0.931206	−	0.931206i	0.997781	+	0.0665755i	$0.0212073\pi$
$198$		0			0
$199$		−	990840.i		−	1.77366i	−0.462094	−	0.886831i	$-0.652902\pi$
$199$		−	990840.i		−	1.77366i	0.462094	−	0.886831i	$-0.347098\pi$
$200$		0			0
$201$		0			0
$202$	−103564.	−	103564.i	−0.178579	−	0.178579i
$203$	−363945.	−	363945.i	−0.619863	−	0.619863i
$204$		0			0
$205$		0			0
$206$			116124.i			0.190657i
$207$		0			0
$208$	46848.0	−	46848.0i	0.0750815	−	0.0750815i
$209$	−262342.			−0.415435
$210$		0			0
$211$	217076.			0.335665			0.167832	−	0.985816i	$-0.446323\pi$
$211$	217076.			0.335665			0.167832	+	0.985816i	$0.446323\pi$
$212$	−365704.	+	365704.i	−0.558844	+	0.558844i
$213$		0			0
$214$			297760.i			0.444459i
$215$		0			0
$216$		0			0
$217$	352768.	+	352768.i	0.508557	+	0.508557i
$218$	470910.	+	470910.i	0.671115	+	0.671115i
$219$		0			0
$220$		0			0
$221$			128365.i			0.176794i
$222$		0			0
$223$	−800004.	+	800004.i	−1.07728	+	1.07728i	−0.0805314	+	0.996752i	$0.525662\pi$
$223$	−800004.	+	800004.i	−1.07728	+	1.07728i	−0.996752	+	0.0805314i	$0.974338\pi$
$224$	75304.0			0.100276
$225$		0			0
$226$	−1.01361e6			−1.32008
$227$	−231286.	+	231286.i	−0.297910	+	0.297910i	−0.840195	−	0.542285i	$-0.817559\pi$
$227$	−231286.	+	231286.i	−0.297910	+	0.297910i	0.542285	+	0.840195i	$0.317559\pi$
$228$		0			0
$229$		−	185046.i		−	0.233180i	−0.993180	−	0.116590i	$-0.962804\pi$
$229$		−	185046.i		−	0.233180i	0.993180	−	0.116590i	$-0.0371963\pi$
$230$		0			0
$231$		0			0
$232$	−316736.	−	316736.i	−0.386347	−	0.386347i
$233$	59685.5	+	59685.5i	0.0720243	+	0.0720243i	0.742201	−	0.670177i	$-0.233782\pi$
$233$	59685.5	+	59685.5i	0.0720243	+	0.0720243i	−0.670177	+	0.742201i	$0.733782\pi$
$234$		0			0
$235$		0			0
$236$		−	573560.i		−	0.670345i
$237$		0			0
$238$	−103168.	+	103168.i	−0.118060	+	0.118060i
$239$	−612960.			−0.694124			−0.347062	−	0.937842i	$-0.612821\pi$
$239$	−612960.			−0.694124			−0.347062	+	0.937842i	$0.612821\pi$
$240$		0			0
$241$	1.28174e6			1.42154			0.710768	−	0.703426i	$-0.248347\pi$
$241$	1.28174e6			1.42154			0.710768	+	0.703426i	$0.248347\pi$
$242$	−411714.	+	411714.i	−0.451916	+	0.451916i
$243$		0			0
$244$		−	219840.i		−	0.236392i
$245$		0			0
$246$		0			0
$247$	385764.	+	385764.i	0.402327	+	0.402327i
$248$	307009.	+	307009.i	0.316973	+	0.316973i
$249$		0			0
$250$		0			0
$251$		−	101603.i		−	0.101794i	−0.998704	−	0.0508969i	$-0.983792\pi$
$251$		−	101603.i		−	0.101794i	0.998704	−	0.0508969i	$-0.0162080\pi$
$252$		0			0
$253$	225632.	−	225632.i	0.221615	−	0.221615i
$254$	−1.19577e6			−1.16295
$255$		0			0
$256$	65536.0			0.0625000
$257$	812060.	−	812060.i	0.766929	−	0.766929i	−0.210636	−	0.977565i	$-0.567553\pi$
$257$	812060.	−	812060.i	0.766929	−	0.766929i	0.977565	+	0.210636i	$0.0675533\pi$
$258$		0			0
$259$		−	387192.i		−	0.358655i
$260$		0			0
$261$		0			0
$262$	479680.	+	479680.i	0.431716	+	0.431716i
$263$	−885657.	−	885657.i	−0.789544	−	0.789544i	0.191876	−	0.981419i	$-0.438543\pi$
$263$	−885657.	−	885657.i	−0.789544	−	0.789544i	−0.981419	+	0.191876i	$0.938543\pi$
$264$		0			0
$265$		0			0
$266$			620082.i			0.537335i
$267$		0			0
$268$	−545024.	+	545024.i	−0.463531	+	0.463531i
$269$	−2.09847e6			−1.76816			−0.884080	−	0.467335i	$-0.845214\pi$
$269$	−2.09847e6			−1.76816			−0.884080	+	0.467335i	$0.845214\pi$
$270$		0			0
$271$	−1.52784e6			−1.26373			−0.631866	−	0.775078i	$-0.717711\pi$
$271$	−1.52784e6			−1.26373			−0.631866	+	0.775078i	$0.717711\pi$
$272$	−89785.6	+	89785.6i	−0.0735842	+	0.0735842i
$273$		0			0
$274$			1.09202e6i			0.878731i
$275$		0			0
$276$		0			0
$277$	−808071.	−	808071.i	−0.632776	−	0.632776i	0.315987	−	0.948763i	$-0.397664\pi$
$277$	−808071.	−	808071.i	−0.632776	−	0.632776i	−0.948763	+	0.315987i	$0.897664\pi$
$278$	875692.	+	875692.i	0.679578	+	0.679578i
$279$		0			0
$280$		0			0
$281$		−	487322.i		−	0.368172i	−0.982910	−	0.184086i	$-0.941068\pi$
$281$		−	487322.i		−	0.368172i	0.982910	−	0.184086i	$-0.0589325\pi$
$282$		0			0
$283$	401968.	−	401968.i	0.298350	−	0.298350i	−0.542018	−	0.840367i	$-0.682339\pi$
$283$	401968.	−	401968.i	0.298350	−	0.298350i	0.840367	+	0.542018i	$0.182339\pi$
$284$	768518.			0.565403
$285$		0			0
$286$	−128832.			−0.0931341
$287$	−634863.	+	634863.i	−0.454962	+	0.454962i
$288$		0			0
$289$			1.17384e6i			0.826732i
$290$		0			0
$291$		0			0
$292$	−478256.	−	478256.i	−0.328249	−	0.328249i
$293$	−1.12004e6	−	1.12004e6i	−0.762195	−	0.762195i	0.214524	−	0.976719i	$-0.431180\pi$
$293$	−1.12004e6	−	1.12004e6i	−0.762195	−	0.762195i	−0.976719	+	0.214524i	$0.931180\pi$
$294$		0			0
$295$		0			0
$296$		−	336967.i		−	0.223542i
$297$		0			0
$298$	167100.	−	167100.i	0.109002	−	0.109002i
$299$	−663566.			−0.429246
$300$		0			0
$301$	−330720.			−0.210399
$302$	−941504.	+	941504.i	−0.594025	+	0.594025i
$303$		0			0
$304$			539648.i			0.334909i
$305$		0			0
$306$		0			0
$307$	1.80918e6	+	1.80918e6i	1.09556	+	1.09556i	0.994923	+	0.100636i	$0.0320877\pi$
$307$	1.80918e6	+	1.80918e6i	1.09556	+	1.09556i	0.100636	+	0.994923i	$0.467912\pi$
$308$	−103543.	−	103543.i	−0.0621934	−	0.0621934i
$309$		0			0
$310$		0			0
$311$			170831.i			0.100154i	0.998745	+	0.0500768i	$0.0159466\pi$
$311$			170831.i			0.100154i	−0.998745	+	0.0500768i	$0.984053\pi$
$312$		0			0
$313$	713409.	−	713409.i	0.411602	−	0.411602i	−0.470694	−	0.882296i	$-0.655996\pi$
$313$	713409.	−	713409.i	0.411602	−	0.411602i	0.882296	+	0.470694i	$0.155996\pi$
$314$	1.80945e6			1.03567
$315$		0			0
$316$	−949248.			−0.534764
$317$	−1.89874e6	+	1.89874e6i	−1.06125	+	1.06125i	−0.0632533	+	0.997998i	$0.520148\pi$
$317$	−1.89874e6	+	1.89874e6i	−1.06125	+	1.06125i	−0.997998	+	0.0632533i	$0.979852\pi$
$318$		0			0
$319$			871024.i			0.479240i
$320$		0			0
$321$		0			0
$322$	−533312.	−	533312.i	−0.286643	−	0.286643i
$323$	−739328.	−	739328.i	−0.394304	−	0.394304i
$324$		0			0
$325$		0			0
$326$			2.31277e6i			1.20528i
$327$		0			0
$328$	−552512.	+	552512.i	−0.283568	+	0.283568i
$329$	−807459.			−0.411274
$330$		0			0
$331$	−2.07412e6			−1.04055			−0.520277	−	0.853997i	$-0.674171\pi$
$331$	−2.07412e6			−1.04055			−0.520277	+	0.853997i	$0.674171\pi$
$332$	−610126.	+	610126.i	−0.303790	+	0.303790i
$333$		0			0
$334$			554096.i			0.271781i
$335$		0			0
$336$		0			0
$337$	954609.	+	954609.i	0.457879	+	0.457879i	0.897959	−	0.440080i	$-0.145050\pi$
$337$	954609.	+	954609.i	0.457879	+	0.457879i	−0.440080	+	0.897959i	$0.645050\pi$
$338$	−860733.	−	860733.i	−0.409804	−	0.409804i
$339$		0			0
$340$		0			0
$341$		−	844274.i		−	0.393186i
$342$		0			0
$343$	1.46671e6	−	1.46671e6i	0.673146	−	0.673146i
$344$	−287821.			−0.131137
$345$		0			0
$346$	969496.			0.435367
$347$	−803681.	+	803681.i	−0.358311	+	0.358311i	−0.863190	−	0.504879i	$-0.831537\pi$
$347$	−803681.	+	803681.i	−0.358311	+	0.358311i	0.504879	+	0.863190i	$0.331537\pi$
$348$		0			0
$349$			3.59168e6i			1.57846i	0.614098	+	0.789230i	$0.289520\pi$
$349$			3.59168e6i			1.57846i	−0.614098	+	0.789230i	$0.710480\pi$
$350$		0			0
$351$		0			0
$352$	−90112.0	−	90112.0i	−0.0387638	−	0.0387638i
$353$	−1.06648e6	−	1.06648e6i	−0.455529	−	0.455529i	0.441655	−	0.897185i	$-0.354391\pi$
$353$	−1.06648e6	−	1.06648e6i	−0.455529	−	0.455529i	−0.897185	+	0.441655i	$0.854391\pi$
$354$		0			0
$355$		0			0
$356$		−	272728.i		−	0.114053i
$357$		0			0
$358$	−2.02525e6	+	2.02525e6i	−0.835162	+	0.835162i
$359$	−1.93336e6			−0.791729			−0.395865	−	0.918309i	$-0.629555\pi$
$359$	−1.93336e6			−0.791729			−0.395865	+	0.918309i	$0.629555\pi$
$360$		0			0
$361$	−1.96756e6			−0.794623
$362$	51058.8	−	51058.8i	0.0204785	−	0.0204785i
$363$		0			0
$364$			304512.i			0.120462i
$365$		0			0
$366$		0			0
$367$	−320192.	−	320192.i	−0.124092	−	0.124092i	0.642333	−	0.766426i	$-0.277967\pi$
$367$	−320192.	−	320192.i	−0.124092	−	0.124092i	−0.766426	+	0.642333i	$0.777967\pi$
$368$	−464134.	−	464134.i	−0.178658	−	0.178658i
$369$		0			0
$370$		0			0
$371$		−	2.37708e6i		−	0.896621i
$372$		0			0
$373$	194975.	−	194975.i	0.0725616	−	0.0725616i	−0.669895	−	0.742456i	$-0.733661\pi$
$373$	194975.	−	194975.i	0.0725616	−	0.0725616i	0.742456	+	0.669895i	$0.233661\pi$
$374$	246910.			0.0912768
$375$		0			0
$376$	−702720.			−0.256338
$377$	1.28081e6	−	1.28081e6i	0.464120	−	0.464120i
$378$		0			0
$379$			1.13974e6i			0.407575i	0.979015	+	0.203788i	$0.0653252\pi$
$379$			1.13974e6i			0.407575i	−0.979015	+	0.203788i	$0.934675\pi$
$380$		0			0
$381$		0			0
$382$	860576.	+	860576.i	0.301739	+	0.301739i
$383$	−893780.	−	893780.i	−0.311339	−	0.311339i	0.534089	−	0.845428i	$-0.320655\pi$
$383$	−893780.	−	893780.i	−0.311339	−	0.311339i	−0.845428	+	0.534089i	$0.820655\pi$
$384$		0			0
$385$		0			0
$386$			748622.i			0.255738i
$387$		0			0
$388$	−1.53749e6	+	1.53749e6i	−0.518480	+	0.518480i
$389$	−537776.			−0.180189			−0.0900943	−	0.995933i	$-0.528717\pi$
$389$	−537776.			−0.180189			−0.0900943	+	0.995933i	$0.528717\pi$
$390$		0			0
$391$	1.27174e6			0.420686
$392$	515860.	−	515860.i	0.169557	−	0.169557i
$393$		0			0
$394$			2.86937e6i			0.931206i
$395$		0			0
$396$		0			0
$397$	2.68409e6	+	2.68409e6i	0.854713	+	0.854713i	0.990709	−	0.135996i	$-0.0434235\pi$
$397$	2.68409e6	+	2.68409e6i	0.854713	+	0.854713i	−0.135996	+	0.990709i	$0.543424\pi$
$398$	2.80252e6	+	2.80252e6i	0.886831	+	0.886831i
$399$		0			0
$400$		0			0
$401$		−	4.27600e6i		−	1.32794i	−0.747761	−	0.663968i	$-0.768871\pi$
$401$		−	4.27600e6i		−	1.32794i	0.747761	−	0.663968i	$-0.231129\pi$
$402$		0			0
$403$	−1.24147e6	+	1.24147e6i	−0.380780	+	0.380780i
$404$	585846.			0.178579
$405$		0			0
$406$	2.05878e6			0.619863
$407$	−463330.	+	463330.i	−0.138645	+	0.138645i
$408$		0			0
$409$		−	2.75446e6i		−	0.814193i	−0.913385	−	0.407097i	$-0.866541\pi$
$409$		−	2.75446e6i		−	0.814193i	0.913385	−	0.407097i	$-0.133459\pi$
$410$		0			0
$411$		0			0
$412$	−328448.	−	328448.i	−0.0953287	−	0.0953287i
$413$	1.86407e6	+	1.86407e6i	0.537758	+	0.537758i
$414$		0			0
$415$		0			0
$416$			265012.i			0.0750815i
$417$		0			0
$418$	742016.	−	742016.i	0.207717	−	0.207717i
$419$	−5.76471e6			−1.60414			−0.802071	−	0.597229i	$-0.796268\pi$
$419$	−5.76471e6			−1.60414			−0.802071	+	0.597229i	$0.796268\pi$
$420$		0			0
$421$	−256330.			−0.0704846			−0.0352423	−	0.999379i	$-0.511220\pi$
$421$	−256330.			−0.0704846			−0.0352423	+	0.999379i	$0.511220\pi$
$422$	−613984.	+	613984.i	−0.167832	+	0.167832i
$423$		0			0
$424$		−	2.06874e6i		−	0.558844i
$425$		0			0
$426$		0			0
$427$	714480.	+	714480.i	0.189636	+	0.189636i
$428$	−842192.	−	842192.i	−0.222230	−	0.222230i
$429$		0			0
$430$		0			0
$431$			2.57507e6i			0.667723i	0.942622	+	0.333862i	$0.108352\pi$
$431$			2.57507e6i			0.667723i	−0.942622	+	0.333862i	$0.891648\pi$
$432$		0			0
$433$	−2.45903e6	+	2.45903e6i	−0.630295	+	0.630295i	−0.948142	−	0.317847i	$-0.897040\pi$
$433$	−2.45903e6	+	2.45903e6i	−0.630295	+	0.630295i	0.317847	+	0.948142i	$0.397040\pi$
$434$	−1.99556e6			−0.508557
$435$		0			0
$436$	−2.66387e6			−0.671115
$437$	3.82185e6	−	3.82185e6i	0.957348	−	0.957348i
$438$		0			0
$439$		−	282472.i		−	0.0699542i	−0.999388	−	0.0349771i	$-0.988864\pi$
$439$		−	282472.i		−	0.0699542i	0.999388	−	0.0349771i	$-0.0111358\pi$
$440$		0			0
$441$		0			0
$442$	−363072.	−	363072.i	−0.0883969	−	0.0883969i
$443$	2.56554e6	+	2.56554e6i	0.621112	+	0.621112i	0.945816	−	0.324704i	$-0.105265\pi$
$443$	2.56554e6	+	2.56554e6i	0.621112	+	0.621112i	−0.324704	+	0.945816i	$0.605265\pi$
$444$		0			0
$445$		0			0
$446$		−	4.52551e6i		−	1.07728i
$447$		0			0
$448$	−212992.	+	212992.i	−0.0501381	+	0.0501381i
$449$	3.48771e6			0.816440			0.408220	−	0.912884i	$-0.366150\pi$
$449$	3.48771e6			0.816440			0.408220	+	0.912884i	$0.366150\pi$
$450$		0			0
$451$	1.51941e6			0.351749
$452$	2.86692e6	−	2.86692e6i	0.660038	−	0.660038i
$453$		0			0
$454$		−	1.30835e6i		−	0.297910i
$455$		0			0
$456$		0			0
$457$	2.53370e6	+	2.53370e6i	0.567497	+	0.567497i	0.931427	−	0.363929i	$-0.118565\pi$
$457$	2.53370e6	+	2.53370e6i	0.567497	+	0.567497i	−0.363929	+	0.931427i	$0.618565\pi$
$458$	523389.	+	523389.i	0.116590	+	0.116590i
$459$		0			0
$460$		0			0
$461$		−	3.97168e6i		−	0.870406i	−0.900332	−	0.435203i	$-0.856677\pi$
$461$		−	3.97168e6i		−	0.870406i	0.900332	−	0.435203i	$-0.143323\pi$
$462$		0			0
$463$	−2.00226e6	+	2.00226e6i	−0.434077	+	0.434077i	−0.890013	−	0.455935i	$-0.849305\pi$
$463$	−2.00226e6	+	2.00226e6i	−0.434077	+	0.434077i	0.455935	+	0.890013i	$0.349305\pi$
$464$	1.79173e6			0.386347
$465$		0			0
$466$	−337632.			−0.0720243
$467$	2.66700e6	−	2.66700e6i	0.565889	−	0.565889i	−0.365085	−	0.930974i	$-0.618960\pi$
$467$	2.66700e6	−	2.66700e6i	0.565889	−	0.565889i	0.930974	+	0.365085i	$0.118960\pi$
$468$		0			0
$469$		−	3.54266e6i		−	0.743698i
$470$		0			0
$471$		0			0
$472$	1.62227e6	+	1.62227e6i	0.335173	+	0.335173i
$473$	395754.	+	395754.i	0.0813340	+	0.0813340i
$474$		0			0
$475$		0			0
$476$		−	583606.i		−	0.118060i
$477$		0			0
$478$	1.73371e6	−	1.73371e6i	0.347062	−	0.347062i
$479$	6.79230e6			1.35263			0.676314	−	0.736614i	$-0.263576\pi$
$479$	6.79230e6			1.35263			0.676314	+	0.736614i	$0.263576\pi$
$480$		0			0
$481$	1.36262e6			0.268542
$482$	−3.62531e6	+	3.62531e6i	−0.710768	+	0.710768i
$483$		0			0
$484$		−	2.32901e6i		−	0.451916i
$485$		0			0
$486$		0			0
$487$	−6.81905e6	−	6.81905e6i	−1.30287	−	1.30287i	−0.926448	−	0.376423i	$-0.877154\pi$
$487$	−6.81905e6	−	6.81905e6i	−1.30287	−	1.30287i	−0.376423	−	0.926448i	$-0.622846\pi$
$488$	621801.	+	621801.i	0.118196	+	0.118196i
$489$		0			0
$490$		0			0
$491$			2.09964e6i			0.393043i	0.980499	+	0.196522i	$0.0629646\pi$
$491$			2.09964e6i			0.393043i	−0.980499	+	0.196522i	$0.937035\pi$
$492$		0			0
$493$	−2.45470e6	+	2.45470e6i	−0.454864	+	0.454864i
$494$	−2.18221e6			−0.402327
$495$		0			0
$496$	−1.73670e6			−0.316973
$497$	−2.49768e6	+	2.49768e6i	−0.453572	+	0.453572i
$498$		0			0
$499$			3.39057e6i			0.609567i	0.952422	+	0.304784i	$0.0985841\pi$
$499$			3.39057e6i			0.609567i	−0.952422	+	0.304784i	$0.901416\pi$
$500$		0			0
$501$		0			0
$502$	287376.	+	287376.i	0.0508969	+	0.0508969i
$503$	−1.45919e6	−	1.45919e6i	−0.257153	−	0.257153i	0.566742	−	0.823895i	$-0.308204\pi$
$503$	−1.45919e6	−	1.45919e6i	−0.257153	−	0.257153i	−0.823895	+	0.566742i	$0.808204\pi$
$504$		0			0
$505$		0			0
$506$			1.27637e6i			0.221615i
$507$		0			0
$508$	3.38214e6	−	3.38214e6i	0.581477	−	0.581477i
$509$	7.03058e6			1.20281			0.601405	−	0.798944i	$-0.294608\pi$
$509$	7.03058e6			1.20281			0.601405	+	0.798944i	$0.294608\pi$
$510$		0			0
$511$	3.10866e6			0.526649
$512$	−185364.	+	185364.i	−0.0312500	+	0.0312500i
$513$		0			0
$514$			4.59370e6i			0.766929i
$515$		0			0
$516$		0			0
$517$	966240.	+	966240.i	0.158986	+	0.158986i
$518$	1.09514e6	+	1.09514e6i	0.179328	+	0.179328i
$519$		0			0
$520$		0			0
$521$			8.98178e6i			1.44967i	0.688924	+	0.724834i	$0.258083\pi$
$521$			8.98178e6i			1.44967i	−0.688924	+	0.724834i	$0.741917\pi$
$522$		0			0
$523$	6.11278e6	−	6.11278e6i	0.977202	−	0.977202i	−0.0225440	−	0.999746i	$-0.507177\pi$
$523$	6.11278e6	−	6.11278e6i	0.977202	−	0.977202i	0.999746	+	0.0225440i	$0.00717660\pi$
$524$	−2.71348e6			−0.431716
$525$		0			0
$526$	5.01003e6			0.789544
$527$	2.37932e6	−	2.37932e6i	0.373187	−	0.373187i
$528$		0			0
$529$			137753.i			0.0214024i
$530$		0			0
$531$		0			0
$532$	−1.75386e6	−	1.75386e6i	−0.268667	−	0.268667i
$533$	−2.23423e6	−	2.23423e6i	−0.340651	−	0.340651i
$534$		0			0
$535$		0			0
$536$		−	3.08312e6i		−	0.463531i
$537$		0			0
$538$	5.93536e6	−	5.93536e6i	0.884080	−	0.884080i
$539$	−1.41861e6			−0.210326
$540$		0			0
$541$	1.23123e7			1.80861			0.904307	−	0.426883i	$-0.140389\pi$
$541$	1.23123e7			1.80861			0.904307	+	0.426883i	$0.140389\pi$
$542$	4.32138e6	−	4.32138e6i	0.631866	−	0.631866i
$543$		0			0
$544$		−	507904.i		−	0.0735842i
$545$		0			0
$546$		0			0
$547$	−6.04163e6	−	6.04163e6i	−0.863348	−	0.863348i	0.128378	−	0.991725i	$-0.459023\pi$
$547$	−6.04163e6	−	6.04163e6i	−0.863348	−	0.863348i	−0.991725	+	0.128378i	$0.959023\pi$
$548$	−3.08871e6	−	3.08871e6i	−0.439365	−	0.439365i
$549$		0			0
$550$		0			0
$551$			1.47538e7i			2.07026i
$552$		0			0
$553$	3.08506e6	−	3.08506e6i	0.428993	−	0.428993i
$554$	4.57114e6			0.632776
$555$		0			0
$556$	−4.95366e6			−0.679578
$557$	−1.67984e6	+	1.67984e6i	−0.229419	+	0.229419i	−0.812450	−	0.583031i	$-0.801867\pi$
$557$	−1.67984e6	+	1.67984e6i	−0.229419	+	0.229419i	0.583031	+	0.812450i	$0.301867\pi$
$558$		0			0
$559$		−	1.16388e6i		−	0.157536i
$560$		0			0
$561$		0			0
$562$	1.37836e6	+	1.37836e6i	0.184086	+	0.184086i
$563$	1.54549e6	+	1.54549e6i	0.205492	+	0.205492i	0.802348	−	0.596856i	$-0.203584\pi$
$563$	1.54549e6	+	1.54549e6i	0.205492	+	0.205492i	−0.596856	+	0.802348i	$0.703584\pi$
$564$		0			0
$565$		0			0
$566$			2.27387e6i			0.298350i
$567$		0			0
$568$	−2.17370e6	+	2.17370e6i	−0.282701	+	0.282701i
$569$	1.07876e7			1.39683			0.698414	−	0.715694i	$-0.253889\pi$
$569$	1.07876e7			1.39683			0.698414	+	0.715694i	$0.253889\pi$
$570$		0			0
$571$	−5.00501e6			−0.642414			−0.321207	−	0.947009i	$-0.604088\pi$
$571$	−5.00501e6			−0.642414			−0.321207	+	0.947009i	$0.604088\pi$
$572$	364392.	−	364392.i	0.0465670	−	0.0465670i
$573$		0			0
$574$		−	3.59133e6i		−	0.454962i
$575$		0			0
$576$		0			0
$577$	−3.63956e6	−	3.63956e6i	−0.455103	−	0.455103i	0.441941	−	0.897044i	$-0.354290\pi$
$577$	−3.63956e6	−	3.63956e6i	−0.455103	−	0.455103i	−0.897044	+	0.441941i	$0.854290\pi$
$578$	−3.32012e6	−	3.32012e6i	−0.413366	−	0.413366i
$579$		0			0
$580$		0			0
$581$		−	3.96582e6i		−	0.487408i
$582$		0			0
$583$	−2.84451e6	+	2.84451e6i	−0.346606	+	0.346606i
$584$	2.70542e6			0.328249
$585$		0			0
$586$	6.33593e6			0.762195
$587$	−7.09648e6	+	7.09648e6i	−0.850056	+	0.850056i	−0.990140	−	0.140084i	$-0.955263\pi$
$587$	−7.09648e6	+	7.09648e6i	−0.850056	+	0.850056i	0.140084	+	0.990140i	$0.455263\pi$
$588$		0			0
$589$		−	1.43007e7i		−	1.69851i
$590$		0			0
$591$		0			0
$592$	953088.	+	953088.i	0.111771	+	0.111771i
$593$	5.45083e6	+	5.45083e6i	0.636540	+	0.636540i	0.949700	−	0.313160i	$-0.101388\pi$
$593$	5.45083e6	+	5.45083e6i	0.636540	+	0.636540i	−0.313160	+	0.949700i	$0.601388\pi$
$594$		0			0
$595$		0			0
$596$			945260.i			0.109002i
$597$		0			0
$598$	1.87685e6	−	1.87685e6i	0.214623	−	0.214623i
$599$	−305261.			−0.0347619			−0.0173810	−	0.999849i	$-0.505533\pi$
$599$	−305261.			−0.0347619			−0.0173810	+	0.999849i	$0.505533\pi$
$600$		0			0
$601$	−9.67187e6			−1.09226			−0.546128	−	0.837702i	$-0.683899\pi$
$601$	−9.67187e6			−1.09226			−0.546128	+	0.837702i	$0.683899\pi$
$602$	935417.	−	935417.i	0.105200	−	0.105200i
$603$		0			0
$604$		−	5.32595e6i		−	0.594025i
$605$		0			0
$606$		0			0
$607$	−3.87772e6	−	3.87772e6i	−0.427173	−	0.427173i	0.460491	−	0.887664i	$-0.347673\pi$
$607$	−3.87772e6	−	3.87772e6i	−0.427173	−	0.427173i	−0.887664	+	0.460491i	$0.847673\pi$
$608$	−1.52636e6	−	1.52636e6i	−0.167454	−	0.167454i
$609$		0			0
$610$		0			0
$611$		−	2.84164e6i		−	0.307940i
$612$		0			0
$613$	−2.14632e6	+	2.14632e6i	−0.230698	+	0.230698i	−0.812984	−	0.582286i	$-0.802158\pi$
$613$	−2.14632e6	+	2.14632e6i	−0.230698	+	0.230698i	0.582286	+	0.812984i	$0.302158\pi$
$614$	−1.02343e7			−1.09556
$615$		0			0
$616$	585728.			0.0621934
$617$	2.19835e6	−	2.19835e6i	0.232479	−	0.232479i	−0.581248	−	0.813727i	$-0.697435\pi$
$617$	2.19835e6	−	2.19835e6i	0.232479	−	0.232479i	0.813727	+	0.581248i	$0.197435\pi$
$618$		0			0
$619$		−	2.24533e6i		−	0.235534i	−0.993041	−	0.117767i	$-0.962426\pi$
$619$		−	2.24533e6i		−	0.235534i	0.993041	−	0.117767i	$-0.0375736\pi$
$620$		0			0
$621$		0			0
$622$	−483184.	−	483184.i	−0.0500768	−	0.0500768i
$623$	886367.	+	886367.i	0.0914942	+	0.0914942i
$624$		0			0
$625$		0			0
$626$			4.03565e6i			0.411602i
$627$		0			0
$628$	−5.11790e6	+	5.11790e6i	−0.517837	+	0.517837i
$629$	−2.61150e6			−0.263186
$630$		0			0
$631$	1.32882e7			1.32860			0.664300	−	0.747466i	$-0.268730\pi$
$631$	1.32882e7			1.32860			0.664300	+	0.747466i	$0.268730\pi$
$632$	2.68488e6	−	2.68488e6i	0.267382	−	0.267382i
$633$		0			0
$634$		−	1.07409e7i		−	1.06125i
$635$		0			0
$636$		0			0
$637$	2.08602e6	+	2.08602e6i	0.203690	+	0.203690i
$638$	−2.46363e6	−	2.46363e6i	−0.239620	−	0.239620i
$639$		0			0
$640$		0			0
$641$			241509.i			0.0232161i	0.999933	+	0.0116080i	$0.00369504\pi$
$641$			241509.i			0.0232161i	−0.999933	+	0.0116080i	$0.996305\pi$
$642$		0			0
$643$	1.31899e7	−	1.31899e7i	1.25810	−	1.25810i	0.306101	−	0.951999i	$-0.400976\pi$
$643$	1.31899e7	−	1.31899e7i	1.25810	−	1.25810i	0.951999	−	0.306101i	$-0.0990245\pi$
$644$	3.01687e6			0.286643
$645$		0			0
$646$	4.18227e6			0.394304
$647$	−2.08840e6	+	2.08840e6i	−0.196134	+	0.196134i	−0.798340	−	0.602206i	$-0.794288\pi$
$647$	−2.08840e6	+	2.08840e6i	−0.196134	+	0.196134i	0.602206	+	0.798340i	$0.294288\pi$
$648$		0			0
$649$		−	4.46125e6i		−	0.415762i
$650$		0			0
$651$		0			0
$652$	−6.54150e6	−	6.54150e6i	−0.602641	−	0.602641i
$653$	9.34448e6	+	9.34448e6i	0.857575	+	0.857575i	0.991052	−	0.133477i	$-0.0426143\pi$
$653$	9.34448e6	+	9.34448e6i	0.857575	+	0.857575i	−0.133477	+	0.991052i	$0.542614\pi$
$654$		0			0
$655$		0			0
$656$		−	3.12548e6i		−	0.283568i
$657$		0			0
$658$	2.28384e6	−	2.28384e6i	0.205637	−	0.205637i
$659$	−1.69591e6			−0.152121			−0.0760604	−	0.997103i	$-0.524234\pi$
$659$	−1.69591e6			−0.152121			−0.0760604	+	0.997103i	$0.524234\pi$
$660$		0			0
$661$	−7.77243e6			−0.691915			−0.345958	−	0.938250i	$-0.612446\pi$
$661$	−7.77243e6			−0.691915			−0.345958	+	0.938250i	$0.612446\pi$
$662$	5.86651e6	−	5.86651e6i	0.520277	−	0.520277i
$663$		0			0
$664$		−	3.45139e6i		−	0.303790i
$665$		0			0
$666$		0			0
$667$	−1.26892e7	−	1.26892e7i	−1.10439	−	1.10439i
$668$	−1.56722e6	−	1.56722e6i	−0.135890	−	0.135890i
$669$		0			0
$670$		0			0
$671$		−	1.70995e6i		−	0.146615i
$672$		0			0
$673$	−1.04638e7	+	1.04638e7i	−0.890540	+	0.890540i	−0.994574	−	0.104033i	$-0.966825\pi$
$673$	−1.04638e7	+	1.04638e7i	−0.890540	+	0.890540i	0.104033	+	0.994574i	$0.466825\pi$
$674$	−5.40008e6			−0.457879
$675$		0			0
$676$	4.86904e6			0.409804
$677$	2.25257e6	−	2.25257e6i	0.188889	−	0.188889i	−0.606327	−	0.795216i	$-0.707358\pi$
$677$	2.25257e6	−	2.25257e6i	0.188889	−	0.188889i	0.795216	+	0.606327i	$0.207358\pi$
$678$		0			0
$679$		−	9.99367e6i		−	0.831861i
$680$		0			0
$681$		0			0
$682$	2.38797e6	+	2.38797e6i	0.196593	+	0.196593i
$683$	9.66190e6	+	9.66190e6i	0.792521	+	0.792521i	0.981903	−	0.189382i	$-0.0606485\pi$
$683$	9.66190e6	+	9.66190e6i	0.792521	+	0.792521i	−0.189382	+	0.981903i	$0.560649\pi$
$684$		0			0
$685$		0			0
$686$			8.29698e6i			0.673146i
$687$		0			0
$688$	814080.	−	814080.i	0.0655686	−	0.0655686i
$689$	8.36549e6			0.671341
$690$		0			0
$691$	−2.36923e7			−1.88761			−0.943803	−	0.330508i	$-0.892780\pi$
$691$	−2.36923e7			−1.88761			−0.943803	+	0.330508i	$0.892780\pi$
$692$	−2.74215e6	+	2.74215e6i	−0.217684	+	0.217684i
$693$		0			0
$694$		−	4.54630e6i		−	0.358311i
$695$		0			0
$696$		0			0
$697$	4.28197e6	+	4.28197e6i	0.333858	+	0.333858i
$698$	−1.01588e7	−	1.01588e7i	−0.789230	−	0.789230i
$699$		0			0
$700$		0			0
$701$			5.35032e6i			0.411230i	0.978633	+	0.205615i	$0.0659194\pi$
$701$			5.35032e6i			0.411230i	−0.978633	+	0.205615i	$0.934081\pi$
$702$		0			0
$703$	−7.84808e6	+	7.84808e6i	−0.598929	+	0.598929i
$704$	509750.			0.0387638
$705$		0			0
$706$	6.03293e6			0.455529
$707$	−1.90400e6	+	1.90400e6i	−0.143258	+	0.143258i
$708$		0			0
$709$			1.20669e7i			0.901532i	0.892642	+	0.450766i	$0.148849\pi$
$709$			1.20669e7i			0.901532i	−0.892642	+	0.450766i	$0.851151\pi$
$710$		0			0
$711$		0			0
$712$	771392.	+	771392.i	0.0570263	+	0.0570263i
$713$	1.22995e7	+	1.22995e7i	0.906077	+	0.906077i
$714$		0			0
$715$		0			0
$716$		−	1.14565e7i		−	0.835162i
$717$		0			0
$718$	5.46837e6	−	5.46837e6i	0.395865	−	0.395865i
$719$	−6.03607e6			−0.435444			−0.217722	−	0.976011i	$-0.569863\pi$
$719$	−6.03607e6			−0.435444			−0.217722	+	0.976011i	$0.569863\pi$
$720$		0			0
$721$	2.13491e6			0.152947
$722$	5.56511e6	−	5.56511e6i	0.397311	−	0.397311i
$723$		0			0
$724$			288832.i			0.0204785i
$725$		0			0
$726$		0			0
$727$	−7.79026e6	−	7.79026e6i	−0.546658	−	0.546658i	0.378815	−	0.925473i	$-0.376332\pi$
$727$	−7.79026e6	−	7.79026e6i	−0.546658	−	0.546658i	−0.925473	+	0.378815i	$0.876332\pi$
$728$	−861290.	−	861290.i	−0.0602311	−	0.0602311i
$729$		0			0
$730$		0			0
$731$			2.23061e6i			0.154394i
$732$		0			0
$733$	1.98675e6	−	1.98675e6i	0.136579	−	0.136579i	−0.635512	−	0.772091i	$-0.719211\pi$
$733$	1.98675e6	−	1.98675e6i	0.136579	−	0.136579i	0.772091	+	0.635512i	$0.219211\pi$
$734$	1.81128e6			0.124092
$735$		0			0
$736$	2.62554e6			0.178658
$737$	−4.23929e6	+	4.23929e6i	−0.287491	+	0.287491i
$738$		0			0
$739$			1.35338e7i			0.911608i	0.890080	+	0.455804i	$0.150648\pi$
$739$			1.35338e7i			0.911608i	−0.890080	+	0.455804i	$0.849352\pi$
$740$		0			0
$741$		0			0
$742$	6.72339e6	+	6.72339e6i	0.448310	+	0.448310i
$743$	−1.97667e7	−	1.97667e7i	−1.31360	−	1.31360i	−0.918743	−	0.394856i	$-0.870794\pi$
$743$	−1.97667e7	−	1.97667e7i	−1.31360	−	1.31360i	−0.394856	−	0.918743i	$-0.629206\pi$
$744$		0			0
$745$		0			0
$746$			1.10295e6i			0.0725616i
$747$		0			0
$748$	−698368.	+	698368.i	−0.0456384	+	0.0456384i
$749$	5.47425e6			0.356550
$750$		0			0
$751$	1.12214e7			0.726016			0.363008	−	0.931786i	$-0.381750\pi$
$751$	1.12214e7			0.726016			0.363008	+	0.931786i	$0.381750\pi$
$752$	1.98759e6	−	1.98759e6i	0.128169	−	0.128169i
$753$		0			0
$754$			7.24534e6i			0.464120i
$755$		0			0
$756$		0			0
$757$	3.20785e6	+	3.20785e6i	0.203458	+	0.203458i	0.801480	−	0.598022i	$-0.204046\pi$
$757$	3.20785e6	+	3.20785e6i	0.203458	+	0.203458i	−0.598022	+	0.801480i	$0.704046\pi$
$758$	−3.22367e6	−	3.22367e6i	−0.203788	−	0.203788i
$759$		0			0
$760$		0			0
$761$		−	522228.i		−	0.0326888i	−0.999866	−	0.0163444i	$-0.994797\pi$
$761$		−	522228.i		−	0.0326888i	0.999866	−	0.0163444i	$-0.00520281\pi$
$762$		0			0
$763$	8.65758e6	−	8.65758e6i	0.538376	−	0.538376i
$764$	−4.86815e6			−0.301739
$765$		0			0
$766$	5.05598e6			0.311339
$767$	−6.56009e6	+	6.56009e6i	−0.402644	+	0.402644i
$768$		0			0
$769$			2.13153e7i			1.29980i	0.760020	+	0.649899i	$0.225189\pi$
$769$			2.13153e7i			1.29980i	−0.760020	+	0.649899i	$0.774811\pi$
$770$		0			0
$771$		0			0
$772$	−2.11742e6	−	2.11742e6i	−0.127869	−	0.127869i
$773$	−1.24996e6	−	1.24996e6i	−0.0752397	−	0.0752397i	0.668486	−	0.743725i	$-0.266943\pi$
$773$	−1.24996e6	−	1.24996e6i	−0.0752397	−	0.0752397i	−0.743725	+	0.668486i	$0.766943\pi$
$774$		0			0
$775$		0			0
$776$		−	8.69735e6i		−	0.518480i
$777$		0			0
$778$	1.52106e6	−	1.52106e6i	0.0900943	−	0.0900943i
$779$	2.57364e7			1.51951
$780$		0			0
$781$	5.97766e6			0.350674
$782$	−3.59704e6	+	3.59704e6i	−0.210343	+	0.210343i
$783$		0			0
$784$			2.91814e6i			0.169557i
$785$		0			0
$786$		0			0
$787$	2.01615e7	+	2.01615e7i	1.16034	+	1.16034i	0.984400	+	0.175943i	$0.0562974\pi$
$787$	2.01615e7	+	2.01615e7i	1.16034	+	1.16034i	0.175943	+	0.984400i	$0.443703\pi$
$788$	−8.11580e6	−	8.11580e6i	−0.465603	−	0.465603i
$789$		0			0
$790$		0			0
$791$			1.86350e7i			1.05898i
$792$		0			0
$793$	−2.51442e6	+	2.51442e6i	−0.141989	+	0.141989i
$794$	−1.51835e7			−0.854713
$795$		0			0
$796$	−1.58534e7			−0.886831
$797$	3.94436e6	−	3.94436e6i	0.219954	−	0.219954i	−0.588525	−	0.808479i	$-0.700291\pi$
$797$	3.94436e6	−	3.94436e6i	0.219954	−	0.219954i	0.808479	+	0.588525i	$0.200291\pi$
$798$		0			0
$799$			5.44608e6i			0.301798i
$800$		0			0
$801$		0			0
$802$	1.20944e7	+	1.20944e7i	0.663968	+	0.663968i
$803$	−3.71996e6	−	3.71996e6i	−0.203587	−	0.203587i
$804$		0			0
$805$		0			0
$806$		−	7.02283e6i		−	0.380780i
$807$		0			0
$808$	−1.65702e6	+	1.65702e6i	−0.0892895	+	0.0892895i
$809$	−1.08001e6			−0.0580171			−0.0290085	−	0.999579i	$-0.509235\pi$
$809$	−1.08001e6			−0.0580171			−0.0290085	+	0.999579i	$0.509235\pi$
$810$		0			0
$811$	−2.61743e7			−1.39740			−0.698702	−	0.715412i	$-0.746239\pi$
$811$	−2.61743e7			−1.39740			−0.698702	+	0.715412i	$0.746239\pi$
$812$	−5.82312e6	+	5.82312e6i	−0.309931	+	0.309931i
$813$		0			0
$814$		−	2.62099e6i		−	0.138645i
$815$		0			0
$816$		0			0
$817$	6.70344e6	+	6.70344e6i	0.351352	+	0.351352i
$818$	7.79078e6	+	7.79078e6i	0.407097	+	0.407097i
$819$		0			0
$820$		0			0
$821$		−	1.81466e7i		−	0.939589i	−0.882776	−	0.469795i	$-0.844328\pi$
$821$		−	1.81466e7i		−	0.939589i	0.882776	−	0.469795i	$-0.155672\pi$
$822$		0			0
$823$	−1.63046e7	+	1.63046e7i	−0.839095	+	0.839095i	−0.988740	−	0.149645i	$-0.952187\pi$
$823$	−1.63046e7	+	1.63046e7i	−0.839095	+	0.839095i	0.149645	+	0.988740i	$0.452187\pi$
$824$	1.85798e6			0.0953287
$825$		0			0
$826$	−1.05448e7			−0.537758
$827$	−1.89659e7	+	1.89659e7i	−0.964295	+	0.964295i	−0.999384	−	0.0350892i	$-0.988828\pi$
$827$	−1.89659e7	+	1.89659e7i	−0.964295	+	0.964295i	0.0350892	+	0.999384i	$0.488828\pi$
$828$		0			0
$829$			1.48454e7i			0.750251i	0.926974	+	0.375126i	$0.122400\pi$
$829$			1.48454e7i			0.750251i	−0.926974	+	0.375126i	$0.877600\pi$
$830$		0			0
$831$		0			0
$832$	−749568.	−	749568.i	−0.0375407	−	0.0375407i
$833$	−3.99791e6	−	3.99791e6i	−0.199628	−	0.199628i
$834$		0			0
$835$		0			0
$836$			4.19748e6i			0.207717i
$837$		0			0
$838$	1.63051e7	−	1.63051e7i	0.802071	−	0.802071i
$839$	−958537.			−0.0470115			−0.0235057	−	0.999724i	$-0.507483\pi$
$839$	−958537.			−0.0470115			−0.0235057	+	0.999724i	$0.507483\pi$
$840$		0			0
$841$	2.84741e7			1.38822
$842$	725011.	−	725011.i	0.0352423	−	0.0352423i
$843$		0			0
$844$		−	3.47322e6i		−	0.167832i
$845$		0			0
$846$		0			0
$847$	7.56928e6	+	7.56928e6i	0.362532	+	0.362532i
$848$	5.85127e6	+	5.85127e6i	0.279422	+	0.279422i
$849$		0			0
$850$		0			0
$851$		−	1.34998e7i		−	0.639002i
$852$		0			0
$853$	−1.97155e7	+	1.97155e7i	−0.927757	+	0.927757i	−0.997561	−	0.0698035i	$-0.977763\pi$
$853$	−1.97155e7	+	1.97155e7i	−0.927757	+	0.927757i	0.0698035	+	0.997561i	$0.477763\pi$
$854$	−4.04171e6			−0.189636
$855$		0			0
$856$	4.76416e6			0.222230
$857$	2.65954e7	−	2.65954e7i	1.23696	−	1.23696i	0.275717	−	0.961239i	$-0.411085\pi$
$857$	2.65954e7	−	2.65954e7i	1.23696	−	1.23696i	0.961239	−	0.275717i	$-0.0889153\pi$
$858$		0			0
$859$			7.29136e6i			0.337152i	0.985689	+	0.168576i	$0.0539169\pi$
$859$			7.29136e6i			0.337152i	−0.985689	+	0.168576i	$0.946083\pi$
$860$		0			0
$861$		0			0
$862$	−7.28341e6	−	7.28341e6i	−0.333862	−	0.333862i
$863$	1.33811e7	+	1.33811e7i	0.611597	+	0.611597i	0.943362	−	0.331765i	$-0.107644\pi$
$863$	1.33811e7	+	1.33811e7i	0.611597	+	0.611597i	−0.331765	+	0.943362i	$0.607644\pi$
$864$		0			0
$865$		0			0
$866$		−	1.39104e7i		−	0.630295i
$867$		0			0
$868$	5.64429e6	−	5.64429e6i	0.254279	−	0.254279i
$869$	−7.38342e6			−0.331671
$870$		0			0
$871$	1.24674e7			0.556841
$872$	7.53457e6	−	7.53457e6i	0.335558	−	0.335558i
$873$		0			0
$874$			2.16196e7i			0.957348i
$875$		0			0
$876$		0			0
$877$	−1.21873e7	−	1.21873e7i	−0.535066	−	0.535066i	0.387009	−	0.922076i	$-0.373508\pi$
$877$	−1.21873e7	−	1.21873e7i	−0.535066	−	0.535066i	−0.922076	+	0.387009i	$0.873508\pi$
$878$	798951.	+	798951.i	0.0349771	+	0.0349771i
$879$		0			0
$880$		0			0
$881$		−	1.93700e7i		−	0.840794i	−0.907340	−	0.420397i	$-0.861891\pi$
$881$		−	1.93700e7i		−	0.840794i	0.907340	−	0.420397i	$-0.138109\pi$
$882$		0			0
$883$	−3.04008e6	+	3.04008e6i	−0.131215	+	0.131215i	−0.769664	−	0.638449i	$-0.779576\pi$
$883$	−3.04008e6	+	3.04008e6i	−0.131215	+	0.131215i	0.638449	+	0.769664i	$0.279576\pi$
$884$	2.05385e6			0.0883969
$885$		0			0
$886$	−1.45129e7			−0.621112
$887$	1.74129e6	−	1.74129e6i	0.0743125	−	0.0743125i	−0.668974	−	0.743286i	$-0.733266\pi$
$887$	1.74129e6	−	1.74129e6i	0.0743125	−	0.0743125i	0.743286	+	0.668974i	$0.233266\pi$
$888$		0			0
$889$			2.19839e7i			0.932934i
$890$		0			0
$891$		0			0
$892$	1.28001e7	+	1.28001e7i	0.538642	+	0.538642i
$893$	1.63666e7	+	1.63666e7i	0.686798	+	0.686798i
$894$		0			0
$895$		0			0
$896$		−	1.20486e6i		−	0.0501381i
$897$		0			0
$898$	−9.86472e6	+	9.86472e6i	−0.408220	+	0.408220i
$899$	−4.74808e7			−1.95938
$900$		0			0
$901$	−1.60327e7			−0.657953
$902$	−4.29753e6	+	4.29753e6i	−0.175875	+	0.175875i
$903$		0			0
$904$			1.62177e7i			0.660038i
$905$		0			0
$906$		0			0
$907$	9.57742e6	+	9.57742e6i	0.386572	+	0.386572i	0.873463	−	0.486891i	$-0.161869\pi$
$907$	9.57742e6	+	9.57742e6i	0.386572	+	0.386572i	−0.486891	+	0.873463i	$0.661869\pi$
$908$	3.70058e6	+	3.70058e6i	0.148955	+	0.148955i
$909$		0			0
$910$		0			0
$911$			3.24413e7i			1.29510i	0.762024	+	0.647549i	$0.224206\pi$
$911$			3.24413e7i			1.29510i	−0.762024	+	0.647549i	$0.775794\pi$
$912$		0			0
$913$	−4.74566e6	+	4.74566e6i	−0.188417	+	0.188417i
$914$	−1.43327e7			−0.567497
$915$		0			0
$916$	−2.96074e6			−0.116590
$917$	8.81881e6	−	8.81881e6i	0.346327	−	0.346327i
$918$		0			0
$919$		−	3.54533e7i		−	1.38474i	−0.721543	−	0.692370i	$-0.756567\pi$
$919$		−	3.54533e7i		−	1.38474i	0.721543	−	0.692370i	$-0.243433\pi$
$920$		0			0
$921$		0			0
$922$	1.12336e7	+	1.12336e7i	0.435203	+	0.435203i
$923$	−8.78992e6	−	8.78992e6i	−0.339610	−	0.339610i
$924$		0			0
$925$		0			0
$926$		−	1.13265e7i		−	0.434077i
$927$		0			0
$928$	−5.06778e6	+	5.06778e6i	−0.193173	+	0.193173i
$929$	−3.41147e7			−1.29689			−0.648443	−	0.761263i	$-0.724580\pi$
$929$	−3.41147e7			−1.29689			−0.648443	+	0.761263i	$0.724580\pi$
$930$		0			0
$931$	−2.40291e7			−0.908580
$932$	954968.	−	954968.i	0.0360121	−	0.0360121i
$933$		0			0
$934$			1.50868e7i			0.565889i
$935$		0			0
$936$		0			0
$937$	1.40765e7	+	1.40765e7i	0.523778	+	0.523778i	0.918710	−	0.394933i	$-0.129232\pi$
$937$	1.40765e7	+	1.40765e7i	0.523778	+	0.523778i	−0.394933	+	0.918710i	$0.629232\pi$
$938$	1.00201e7	+	1.00201e7i	0.371849	+	0.371849i
$939$		0			0
$940$		0			0
$941$		−	4.32932e7i		−	1.59384i	−0.604083	−	0.796922i	$-0.706460\pi$
$941$		−	4.32932e7i		−	1.59384i	0.604083	−	0.796922i	$-0.293540\pi$
$942$		0			0
$943$	−2.21350e7	+	2.21350e7i	−0.810589	+	0.810589i
$944$	−9.17696e6			−0.335173
$945$		0			0
$946$	−2.23872e6			−0.0813340
$947$	−3.43867e7	+	3.43867e7i	−1.24599	+	1.24599i	−0.288521	+	0.957474i	$0.593164\pi$
$947$	−3.43867e7	+	3.43867e7i	−1.24599	+	1.24599i	−0.957474	+	0.288521i	$0.906836\pi$
$948$		0			0
$949$			1.09401e7i			0.394327i
$950$		0			0
$951$		0			0
$952$	1.65069e6	+	1.65069e6i	0.0590300	+	0.0590300i
$953$	−2.36832e6	−	2.36832e6i	−0.0844712	−	0.0844712i	0.663609	−	0.748080i	$-0.269024\pi$
$953$	−2.36832e6	−	2.36832e6i	−0.0844712	−	0.0844712i	−0.748080	+	0.663609i	$0.769024\pi$
$954$		0			0
$955$		0			0
$956$			9.80736e6i			0.347062i
$957$		0			0
$958$	−1.92115e7	+	1.92115e7i	−0.676314	+	0.676314i
$959$	2.00766e7			0.704927
$960$		0			0
$961$	1.73935e7			0.607545
$962$	−3.85407e6	+	3.85407e6i	−0.134271	+	0.134271i
$963$		0			0
$964$		−	2.05079e7i		−	0.710768i
$965$		0			0
$966$		0			0
$967$	−6.30394e6	−	6.30394e6i	−0.216793	−	0.216793i	0.590352	−	0.807146i	$-0.298989\pi$
$967$	−6.30394e6	−	6.30394e6i	−0.216793	−	0.216793i	−0.807146	+	0.590352i	$0.798989\pi$
$968$	6.58743e6	+	6.58743e6i	0.225958	+	0.225958i
$969$		0			0
$970$		0			0
$971$		−	3.12139e6i		−	0.106243i	−0.998588	−	0.0531215i	$-0.983083\pi$
$971$		−	3.12139e6i		−	0.106243i	0.998588	−	0.0531215i	$-0.0169170\pi$
$972$		0			0
$973$	1.60994e7	−	1.60994e7i	0.545165	−	0.545165i
$974$	3.85744e7			1.30287
$975$		0			0
$976$	−3.51744e6			−0.118196
$977$	3.00825e7	−	3.00825e7i	1.00827	−	1.00827i	0.00830587	−	0.999966i	$-0.497356\pi$
$977$	3.00825e7	−	3.00825e7i	1.00827	−	1.00827i	0.999966	−	0.00830587i	$-0.00264387\pi$
$978$		0			0
$979$		−	2.12133e6i		−	0.0707377i
$980$		0			0
$981$		0			0
$982$	−5.93867e6	−	5.93867e6i	−0.196522	−	0.196522i
$983$	−3.86947e7	−	3.86947e7i	−1.27723	−	1.27723i	−0.942213	−	0.335013i	$-0.891259\pi$
$983$	−3.86947e7	−	3.86947e7i	−1.27723	−	1.27723i	−0.335013	−	0.942213i	$-0.608741\pi$
$984$		0			0
$985$		0			0
$986$		−	1.38859e7i		−	0.454864i
$987$		0			0
$988$	6.17222e6	−	6.17222e6i	0.201164	−	0.201164i
$989$	−1.15308e7			−0.374860
$990$		0			0
$991$	−4.81781e7			−1.55835			−0.779176	−	0.626805i	$-0.784362\pi$
$991$	−4.81781e7			−1.55835			−0.779176	+	0.626805i	$0.784362\pi$
$992$	4.91214e6	−	4.91214e6i	0.158486	−	0.158486i
$993$		0			0
$994$		−	1.41290e7i		−	0.453572i
$995$		0			0
$996$		0			0
$997$	9.12659e6	+	9.12659e6i	0.290784	+	0.290784i	0.837390	−	0.546606i	$-0.184080\pi$
$997$	9.12659e6	+	9.12659e6i	0.290784	+	0.290784i	−0.546606	+	0.837390i	$0.684080\pi$
$998$	−9.58999e6	−	9.58999e6i	−0.304784	−	0.304784i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.f.d.107.1		4
3.2	odd	2	inner	450.6.f.d.107.2		4
5.2	odd	4		90.6.f.a.53.1	yes	4
5.3	odd	4	inner	450.6.f.d.143.2		4
5.4	even	2		90.6.f.a.17.2	yes	4
15.2	even	4		90.6.f.a.53.2	yes	4
15.8	even	4	inner	450.6.f.d.143.1		4
15.14	odd	2		90.6.f.a.17.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.6.f.a.17.1	✓	4	15.14	odd	2
90.6.f.a.17.2	yes	4	5.4	even	2
90.6.f.a.53.1	yes	4	5.2	odd	4
90.6.f.a.53.2	yes	4	15.2	even	4
450.6.f.d.107.1		4	1.1	even	1	trivial
450.6.f.d.107.2		4	3.2	odd	2	inner
450.6.f.d.143.1		4	15.8	even	4	inner
450.6.f.d.143.2		4	5.3	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(52.0000 + 52.0000i) q^{7} +(45.2548 + 45.2548i) q^{8} +O(q^{10})\)	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(52.0000 + 52.0000i) q^{7} +(45.2548 + 45.2548i) q^{8} -124.451i q^{11} +(-183.000 + 183.000i) q^{13} -294.156 q^{14} -256.000 q^{16} +(350.725 - 350.725i) q^{17} -2108.00i q^{19} +(352.000 + 352.000i) q^{22} +(1813.02 + 1813.02i) q^{23} -1035.20i q^{26} +(832.000 - 832.000i) q^{28} -6998.94 q^{29} +6784.00 q^{31} +(724.077 - 724.077i) q^{32} +1984.00i q^{34} +(-3723.00 - 3723.00i) q^{37} +(5962.32 + 5962.32i) q^{38} +12208.9i q^{41} +(-3180.00 + 3180.00i) q^{43} -1991.21 q^{44} -10256.0 q^{46} +(-7764.03 + 7764.03i) q^{47} -11399.0i q^{49} +(2928.00 + 2928.00i) q^{52} +(-22856.5 - 22856.5i) q^{53} +4706.50i q^{56} +(19796.0 - 19796.0i) q^{58} +35847.5 q^{59} +13740.0 q^{61} +(-19188.0 + 19188.0i) q^{62} +4096.00i q^{64} +(-34064.0 - 34064.0i) q^{67} +(-5611.60 - 5611.60i) q^{68} +48032.3i q^{71} +(29891.0 - 29891.0i) q^{73} +21060.5 q^{74} -33728.0 q^{76} +(6471.44 - 6471.44i) q^{77} -59328.0i q^{79} +(-34532.0 - 34532.0i) q^{82} +(-38132.9 - 38132.9i) q^{83} -17988.8i q^{86} +(5632.00 - 5632.00i) q^{88} +17045.5 q^{89} -19032.0 q^{91} +(29008.3 - 29008.3i) q^{92} -43920.0i q^{94} +(-96093.0 - 96093.0i) q^{97} +(32241.2 + 32241.2i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 208 q^{7}+O(q^{10}) \) 4 * q + 208 * q^7	\( 4 q + 208 q^{7} - 732 q^{13} - 1024 q^{16} + 1408 q^{22} + 3328 q^{28} + 27136 q^{31} - 14892 q^{37} - 12720 q^{43} - 41024 q^{46} + 11712 q^{52} + 79184 q^{58} + 54960 q^{61} - 136256 q^{67} + 119564 q^{73} - 134912 q^{76} - 138128 q^{82} + 22528 q^{88} - 76128 q^{91} - 384372 q^{97}+O(q^{100}) \) 4 * q + 208 * q^7 - 732 * q^13 - 1024 * q^16 + 1408 * q^22 + 3328 * q^28 + 27136 * q^31 - 14892 * q^37 - 12720 * q^43 - 41024 * q^46 + 11712 * q^52 + 79184 * q^58 + 54960 * q^61 - 136256 * q^67 + 119564 * q^73 - 134912 * q^76 - 138128 * q^82 + 22528 * q^88 - 76128 * q^91 - 384372 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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