LMFDB - Embedded newform 64.22.b.c.33.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,22,Mod(33,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("64.33");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	64.b (of order $2$, degree $1$, 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:	$178.865500344$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.9
Character	$\chi$	$=$	64.33
Dual form			64.22.b.c.33.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-58474.4i q^{3} -3.08952e7i q^{5} +2.12266e7 q^{7} +7.04110e9 q^{9} +O(q^{10})$	$q-58474.4i q^{3} -3.08952e7i q^{5} +2.12266e7 q^{7} +7.04110e9 q^{9} -3.98975e10i q^{11} +5.30664e11i q^{13} -1.80658e12 q^{15} +8.07841e12 q^{17} +3.47688e12i q^{19} -1.24121e12i q^{21} -1.48025e14 q^{23} -4.77675e14 q^{25} -1.02339e15i q^{27} +7.04791e14i q^{29} +2.40109e15 q^{31} -2.33298e15 q^{33} -6.55798e14i q^{35} -3.01695e16i q^{37} +3.10302e16 q^{39} -1.09745e17 q^{41} -2.78962e17i q^{43} -2.17536e17i q^{45} +2.62982e17 q^{47} -5.58095e17 q^{49} -4.72380e17i q^{51} +2.30487e17i q^{53} -1.23264e18 q^{55} +2.03308e17 q^{57} -4.53608e17i q^{59} -5.41401e18i q^{61} +1.49458e17 q^{63} +1.63949e19 q^{65} +1.03505e19i q^{67} +8.65568e18i q^{69} -4.98513e19 q^{71} +2.15803e19 q^{73} +2.79317e19i q^{75} -8.46886e17i q^{77} -5.01272e18 q^{79} +1.38105e19 q^{81} -2.43571e20i q^{83} -2.49584e20i q^{85} +4.12122e19 q^{87} -2.49270e20 q^{89} +1.12642e19i q^{91} -1.40402e20i q^{93} +1.07419e20 q^{95} -1.02418e20 q^{97} -2.80922e20i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 77960422492 q^{9}+O(q^{10}) $ 28 * q - 77960422492 * q^9	$ 28 q - 77960422492 q^{9} + 16832040195288 q^{17} + 202504130118092 q^{25} - 55\!\cdots\!92 q^{33}+ \cdots - 19\!\cdots\!16 q^{97}+O(q^{100}) $ 28 * q - 77960422492 * q^9 + 16832040195288 * q^17 + 202504130118092 * q^25 - 55633095046450992 * q^33 + 265230847130533848 * q^41 + 5944576136704810940 * q^49 - 4544453360963256304 * q^57 - 63340883112821103360 * q^65 + 2760069698648092472 * q^73 + 934709246686586470348 * q^81 - 375882191672218968072 * q^89 - 1953791530192678912616 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$.

$n$	$5$	$63$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 58474.4i	− 0.571732i	−0.958270	−	0.285866i	$-0.907719\pi$
$3$	− 58474.4i	− 0.571732i	0.958270	−	0.285866i	$-0.0922812\pi$
$4$	0	0
$5$	− 3.08952e7i	− 1.41483i	−0.706796	−	0.707417i	$-0.749860\pi$
$5$	− 3.08952e7i	− 1.41483i	0.706796	−	0.707417i	$-0.250140\pi$
$6$	0	0
$7$	2.12266e7	0.0284021	0.0142010	−	0.999899i	$-0.495480\pi$
$7$	2.12266e7	0.0284021	0.0142010	+	0.999899i	$0.495480\pi$
$8$	0	0
$9$	7.04110e9	0.673123
$10$	0	0
$11$	− 3.98975e10i	− 0.463791i	−0.972741	−	0.231895i	$-0.925507\pi$
$11$	− 3.98975e10i	− 0.463791i	0.972741	−	0.231895i	$-0.0744927\pi$
$12$	0	0
$13$	5.30664e11i	1.06761i	0.845606	+	0.533807i	$0.179239\pi$
$13$	5.30664e11i	1.06761i	−0.845606	+	0.533807i	$0.820761\pi$
$14$	0	0
$15$	−1.80658e12	−0.808906
$16$	0	0
$17$	8.07841e12	0.971879	0.485940	−	0.873992i	$-0.338477\pi$
$17$	8.07841e12	0.971879	0.485940	+	0.873992i	$0.338477\pi$
$18$	0	0
$19$	3.47688e12i	0.130100i	0.997882	+	0.0650499i	$0.0207207\pi$
$19$	3.47688e12i	0.130100i	−0.997882	+	0.0650499i	$0.979279\pi$
$20$	0	0
$21$	− 1.24121e12i	− 0.0162384i
$22$	0	0
$23$	−1.48025e14	−0.745063	−0.372532	−	0.928019i	$-0.621510\pi$
$23$	−1.48025e14	−0.745063	−0.372532	+	0.928019i	$0.621510\pi$
$24$	0	0
$25$	−4.77675e14	−1.00176
$26$	0	0
$27$	− 1.02339e15i	− 0.956578i
$28$	0	0
$29$	7.04791e14i	0.311086i	0.987829	+	0.155543i	$0.0497128\pi$
$29$	7.04791e14i	0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$30$	0	0
$31$	2.40109e15	0.526150	0.263075	−	0.964775i	$-0.415263\pi$
$31$	2.40109e15	0.526150	0.263075	+	0.964775i	$0.415263\pi$
$32$	0	0
$33$	−2.33298e15	−0.265164
$34$	0	0
$35$	− 6.55798e14i	− 0.0401842i
$36$	0	0
$37$	− 3.01695e16i	− 1.03146i	−0.856753	−	0.515728i	$-0.827522\pi$
$37$	− 3.01695e16i	− 1.03146i	0.856753	−	0.515728i	$-0.172478\pi$
$38$	0	0
$39$	3.10302e16	0.610389
$40$	0	0
$41$	−1.09745e17	−1.27689	−0.638447	−	0.769666i	$-0.720423\pi$
$41$	−1.09745e17	−1.27689	−0.638447	+	0.769666i	$0.720423\pi$
$42$	0	0
$43$	− 2.78962e17i	− 1.96846i	−0.176907	−	0.984228i	$-0.556609\pi$
$43$	− 2.78962e17i	− 1.96846i	0.176907	−	0.984228i	$-0.443391\pi$
$44$	0	0
$45$	− 2.17536e17i	− 0.952357i
$46$	0	0
$47$	2.62982e17	0.729286	0.364643	−	0.931147i	$-0.381191\pi$
$47$	2.62982e17	0.729286	0.364643	+	0.931147i	$0.381191\pi$
$48$	0	0
$49$	−5.58095e17	−0.999193
$50$	0	0
$51$	− 4.72380e17i	− 0.555654i
$52$	0	0
$53$	2.30487e17i	0.181030i	0.995895	+	0.0905150i	$0.0288513\pi$
$53$	2.30487e17i	0.181030i	−0.995895	+	0.0905150i	$0.971149\pi$
$54$	0	0
$55$	−1.23264e18	−0.656187
$56$	0	0
$57$	2.03308e17	0.0743823
$58$	0	0
$59$	− 4.53608e17i	− 0.115541i	−0.998330	−	0.0577703i	$-0.981601\pi$
$59$	− 4.53608e17i	− 0.115541i	0.998330	−	0.0577703i	$-0.0183991\pi$
$60$	0	0
$61$	− 5.41401e18i	− 0.971752i	−0.874028	−	0.485876i	$-0.838501\pi$
$61$	− 5.41401e18i	− 0.971752i	0.874028	−	0.485876i	$-0.161499\pi$
$62$	0	0
$63$	1.49458e17	0.0191181
$64$	0	0
$65$	1.63949e19	1.51050
$66$	0	0
$67$	1.03505e19i	0.693707i	0.937919	+	0.346854i	$0.112750\pi$
$67$	1.03505e19i	0.693707i	−0.937919	+	0.346854i	$0.887250\pi$
$68$	0	0
$69$	8.65568e18i	0.425976i
$70$	0	0
$71$	−4.98513e19	−1.81746	−0.908729	−	0.417388i	$-0.862946\pi$
$71$	−4.98513e19	−1.81746	−0.908729	+	0.417388i	$0.862946\pi$
$72$	0	0
$73$	2.15803e19	0.587715	0.293857	−	0.955849i	$-0.405061\pi$
$73$	2.15803e19	0.587715	0.293857	+	0.955849i	$0.405061\pi$
$74$	0	0
$75$	2.79317e19i	0.572736i
$76$	0	0
$77$	− 8.46886e17i	− 0.0131726i
$78$	0	0
$79$	−5.01272e18	−0.0595648	−0.0297824	−	0.999556i	$-0.509481\pi$
$79$	−5.01272e18	−0.0595648	−0.0297824	+	0.999556i	$0.509481\pi$
$80$	0	0
$81$	1.38105e19	0.126217
$82$	0	0
$83$	− 2.43571e20i	− 1.72308i	−0.507690	−	0.861540i	$-0.669500\pi$
$83$	− 2.43571e20i	− 1.72308i	0.507690	−	0.861540i	$-0.330500\pi$
$84$	0	0
$85$	− 2.49584e20i	− 1.37505i
$86$	0	0
$87$	4.12122e19	0.177858
$88$	0	0
$89$	−2.49270e20	−0.847373	−0.423687	−	0.905809i	$-0.639264\pi$
$89$	−2.49270e20	−0.847373	−0.423687	+	0.905809i	$0.639264\pi$
$90$	0	0
$91$	1.12642e19i	0.0303224i
$92$	0	0
$93$	− 1.40402e20i	− 0.300817i
$94$	0	0
$95$	1.07419e20	0.184070
$96$	0	0
$97$	−1.02418e20	−0.141017	−0.0705086	−	0.997511i	$-0.522462\pi$
$97$	−1.02418e20	−0.141017	−0.0705086	+	0.997511i	$0.522462\pi$
$98$	0	0
$99$	− 2.80922e20i	− 0.312188i
$100$	0	0
$101$	6.07686e20i	0.547400i	0.961815	+	0.273700i	$0.0882475\pi$
$101$	6.07686e20i	0.547400i	−0.961815	+	0.273700i	$0.911752\pi$
$102$	0	0
$103$	4.95376e20	0.363198	0.181599	−	0.983373i	$-0.441873\pi$
$103$	4.95376e20	0.363198	0.181599	+	0.983373i	$0.441873\pi$
$104$	0	0
$105$	−3.83474e19	−0.0229746
$106$	0	0
$107$	8.52394e20i	0.418901i	0.977819	+	0.209450i	$0.0671674\pi$
$107$	8.52394e20i	0.418901i	−0.977819	+	0.209450i	$0.932833\pi$
$108$	0	0
$109$	1.73806e21i	0.703211i	0.936148	+	0.351606i	$0.114364\pi$
$109$	1.73806e21i	0.703211i	−0.936148	+	0.351606i	$0.885636\pi$
$110$	0	0
$111$	−1.76414e21	−0.589716
$112$	0	0
$113$	1.13271e21	0.313904	0.156952	−	0.987606i	$-0.449833\pi$
$113$	1.13271e21	0.313904	0.156952	+	0.987606i	$0.449833\pi$
$114$	0	0
$115$	4.57326e21i	1.05414i
$116$	0	0
$117$	3.73646e21i	0.718635i
$118$	0	0
$119$	1.71477e20	0.0276034
$120$	0	0
$121$	5.80844e21	0.784898
$122$	0	0
$123$	6.41728e21i	0.730041i
$124$	0	0
$125$	2.58809e19i	0.00248556i
$126$	0	0
$127$	1.57363e22	1.27928	0.639638	−	0.768677i	$-0.279084\pi$
$127$	1.57363e22	1.27928	0.639638	+	0.768677i	$0.279084\pi$
$128$	0	0
$129$	−1.63121e22	−1.12543
$130$	0	0
$131$	− 2.61980e22i	− 1.53787i	−0.639329	−	0.768933i	$-0.720788\pi$
$131$	− 2.61980e22i	− 1.53787i	0.639329	−	0.768933i	$-0.279212\pi$
$132$	0	0
$133$	7.38022e19i	0.00369511i
$134$	0	0
$135$	−3.16177e22	−1.35340
$136$	0	0
$137$	−3.07469e22	−1.12781	−0.563905	−	0.825840i	$-0.690702\pi$
$137$	−3.07469e22	−1.12781	−0.563905	+	0.825840i	$0.690702\pi$
$138$	0	0
$139$	− 3.05291e22i	− 0.961741i	−0.876792	−	0.480870i	$-0.840321\pi$
$139$	− 3.05291e22i	− 0.961741i	0.876792	−	0.480870i	$-0.159679\pi$
$140$	0	0
$141$	− 1.53777e22i	− 0.416956i
$142$	0	0
$143$	2.11721e22	0.495149
$144$	0	0
$145$	2.17746e22	0.440136
$146$	0	0
$147$	3.26343e22i	0.571271i
$148$	0	0
$149$	− 1.90151e22i	− 0.288831i	−0.989517	−	0.144415i	$-0.953870\pi$
$149$	− 1.90151e22i	− 0.288831i	0.989517	−	0.144415i	$-0.0461302\pi$
$150$	0	0
$151$	−1.03229e23	−1.36314	−0.681572	−	0.731751i	$-0.738703\pi$
$151$	−1.03229e23	−1.36314	−0.681572	+	0.731751i	$0.738703\pi$
$152$	0	0
$153$	5.68809e22	0.654194
$154$	0	0
$155$	− 7.41820e22i	− 0.744416i
$156$	0	0
$157$	1.03023e23i	0.903622i	0.892114	+	0.451811i	$0.149222\pi$
$157$	1.03023e23i	0.903622i	−0.892114	+	0.451811i	$0.850778\pi$
$158$	0	0
$159$	1.34776e22	0.103501
$160$	0	0
$161$	−3.14206e21	−0.0211613
$162$	0	0
$163$	1.22180e23i	0.722821i	0.932407	+	0.361410i	$0.117705\pi$
$163$	1.22180e23i	0.722821i	−0.932407	+	0.361410i	$0.882295\pi$
$164$	0	0
$165$	7.20778e22i	0.375163i
$166$	0	0
$167$	−3.83428e23	−1.75857	−0.879287	−	0.476292i	$-0.841981\pi$
$167$	−3.83428e23	−1.75857	−0.879287	+	0.476292i	$0.841981\pi$
$168$	0	0
$169$	−3.45394e22	−0.139799
$170$	0	0
$171$	2.44811e22i	0.0875732i
$172$	0	0
$173$	1.98387e23i	0.628099i	0.949407	+	0.314050i	$0.101686\pi$
$173$	1.98387e23i	0.628099i	−0.949407	+	0.314050i	$0.898314\pi$
$174$	0	0
$175$	−1.01394e22	−0.0284520
$176$	0	0
$177$	−2.65245e22	−0.0660583
$178$	0	0
$179$	− 5.25357e23i	− 1.16278i	−0.813625	−	0.581390i	$-0.802509\pi$
$179$	− 5.25357e23i	− 1.16278i	0.813625	−	0.581390i	$-0.197491\pi$
$180$	0	0
$181$	7.47778e23i	1.47281i	0.676539	+	0.736407i	$0.263479\pi$
$181$	7.47778e23i	1.47281i	−0.676539	+	0.736407i	$0.736521\pi$
$182$	0	0
$183$	−3.16581e23	−0.555582
$184$	0	0
$185$	−9.32093e23	−1.45934
$186$	0	0
$187$	− 3.22308e23i	− 0.450749i
$188$	0	0
$189$	− 2.17230e22i	− 0.0271688i
$190$	0	0
$191$	1.09659e24	1.22799	0.613994	−	0.789311i	$-0.289562\pi$
$191$	1.09659e24	1.22799	0.613994	+	0.789311i	$0.289562\pi$
$192$	0	0
$193$	−1.85770e24	−1.86476	−0.932381	−	0.361476i	$-0.882273\pi$
$193$	−1.85770e24	−1.86476	−0.932381	+	0.361476i	$0.882273\pi$
$194$	0	0
$195$	− 9.58684e23i	− 0.863599i
$196$	0	0
$197$	5.59976e23i	0.453183i	0.973990	+	0.226592i	$0.0727582\pi$
$197$	5.59976e23i	0.453183i	−0.973990	+	0.226592i	$0.927242\pi$
$198$	0	0
$199$	−2.30633e24	−1.67866	−0.839332	−	0.543620i	$-0.817053\pi$
$199$	−2.30633e24	−1.67866	−0.839332	+	0.543620i	$0.817053\pi$
$200$	0	0
$201$	6.05239e23	0.396615
$202$	0	0
$203$	1.49603e22i	0.00883550i
$204$	0	0
$205$	3.39060e24i	1.80659i
$206$	0	0
$207$	−1.04226e24	−0.501519
$208$	0	0
$209$	1.38719e23	0.0603391
$210$	0	0
$211$	− 8.49269e23i	− 0.334256i	−0.985935	−	0.167128i	$-0.946551\pi$
$211$	− 8.49269e23i	− 0.334256i	0.985935	−	0.167128i	$-0.0534494\pi$
$212$	0	0
$213$	2.91503e24i	1.03910i
$214$	0	0
$215$	−8.61857e24	−2.78504
$216$	0	0
$217$	5.09668e22	0.0149438
$218$	0	0
$219$	− 1.26189e24i	− 0.336015i
$220$	0	0
$221$	4.28692e24i	1.03759i
$222$	0	0
$223$	−3.20411e24	−0.705514	−0.352757	−	0.935715i	$-0.614756\pi$
$223$	−3.20411e24	−0.705514	−0.352757	+	0.935715i	$0.614756\pi$
$224$	0	0
$225$	−3.36336e24	−0.674305
$226$	0	0
$227$	− 7.05178e23i	− 0.128833i	−0.997923	−	0.0644166i	$-0.979481\pi$
$227$	− 7.05178e23i	− 0.128833i	0.997923	−	0.0644166i	$-0.0205186\pi$
$228$	0	0
$229$	8.70284e23i	0.145007i	0.997368	+	0.0725034i	$0.0230988\pi$
$229$	8.70284e23i	0.145007i	−0.997368	+	0.0725034i	$0.976901\pi$
$230$	0	0
$231$	−4.95211e22	−0.00753121
$232$	0	0
$233$	−1.06277e24	−0.147639	−0.0738196	−	0.997272i	$-0.523519\pi$
$233$	−1.06277e24	−0.147639	−0.0738196	+	0.997272i	$0.523519\pi$
$234$	0	0
$235$	− 8.12487e24i	− 1.03182i
$236$	0	0
$237$	2.93116e23i	0.0340551i
$238$	0	0
$239$	−1.74415e25	−1.85527	−0.927634	−	0.373490i	$-0.878161\pi$
$239$	−1.74415e25	−1.85527	−0.927634	+	0.373490i	$0.878161\pi$
$240$	0	0
$241$	−6.36726e24	−0.620546	−0.310273	−	0.950648i	$-0.600420\pi$
$241$	−6.36726e24	−0.620546	−0.310273	+	0.950648i	$0.600420\pi$
$242$	0	0
$243$	− 1.15125e25i	− 1.02874i
$244$	0	0
$245$	1.72425e25i	1.41369i
$246$	0	0
$247$	−1.84505e24	−0.138896
$248$	0	0
$249$	−1.42427e25	−0.985140
$250$	0	0
$251$	− 1.81850e25i	− 1.15649i	−0.815865	−	0.578243i	$-0.803739\pi$
$251$	− 1.81850e25i	− 1.15649i	0.815865	−	0.578243i	$-0.196261\pi$
$252$	0	0
$253$	5.90583e24i	0.345553i
$254$	0	0
$255$	−1.45943e25	−0.786159
$256$	0	0
$257$	−2.68614e25	−1.33300	−0.666501	−	0.745504i	$-0.732209\pi$
$257$	−2.68614e25	−1.33300	−0.666501	+	0.745504i	$0.732209\pi$
$258$	0	0
$259$	− 6.40395e23i	− 0.0292955i
$260$	0	0
$261$	4.96250e24i	0.209399i
$262$	0	0
$263$	2.26927e25	0.883793	0.441896	−	0.897066i	$-0.354306\pi$
$263$	2.26927e25	0.883793	0.441896	+	0.897066i	$0.354306\pi$
$264$	0	0
$265$	7.12095e24	0.256128
$266$	0	0
$267$	1.45759e25i	0.484470i
$268$	0	0
$269$	4.22722e25i	1.29914i	0.760302	+	0.649570i	$0.225051\pi$
$269$	4.22722e25i	1.29914i	−0.760302	+	0.649570i	$0.774949\pi$
$270$	0	0
$271$	−4.93776e25	−1.40395	−0.701976	−	0.712200i	$-0.747699\pi$
$271$	−4.93776e25	−1.40395	−0.701976	+	0.712200i	$0.747699\pi$
$272$	0	0
$273$	6.58665e23	0.0173363
$274$	0	0
$275$	1.90580e25i	0.464606i
$276$	0	0
$277$	4.80117e25i	1.08470i	0.840153	+	0.542349i	$0.182465\pi$
$277$	4.80117e25i	1.08470i	−0.840153	+	0.542349i	$0.817535\pi$
$278$	0	0
$279$	1.69063e25	0.354164
$280$	0	0
$281$	6.72951e25	1.30788	0.653938	−	0.756548i	$-0.273116\pi$
$281$	6.72951e25	1.30788	0.653938	+	0.756548i	$0.273116\pi$
$282$	0	0
$283$	4.16465e25i	0.751314i	0.926759	+	0.375657i	$0.122583\pi$
$283$	4.16465e25i	0.751314i	−0.926759	+	0.375657i	$0.877417\pi$
$284$	0	0
$285$	− 6.28125e24i	− 0.105239i
$286$	0	0
$287$	−2.32951e24	−0.0362664
$288$	0	0
$289$	−3.83121e24	−0.0554509
$290$	0	0
$291$	5.98881e24i	0.0806240i
$292$	0	0
$293$	− 9.52664e25i	− 1.19352i	−0.802420	−	0.596760i	$-0.796455\pi$
$293$	− 9.52664e25i	− 1.19352i	0.802420	−	0.596760i	$-0.203545\pi$
$294$	0	0
$295$	−1.40143e25	−0.163471
$296$	0	0
$297$	−4.08305e25	−0.443652
$298$	0	0
$299$	− 7.85516e25i	− 0.795440i
$300$	0	0
$301$	− 5.92139e24i	− 0.0559082i
$302$	0	0
$303$	3.55341e25	0.312966
$304$	0	0
$305$	−1.67267e26	−1.37487
$306$	0	0
$307$	− 7.02891e25i	− 0.539430i	−0.962940	−	0.269715i	$-0.913071\pi$
$307$	− 7.02891e25i	− 0.539430i	0.962940	−	0.269715i	$-0.0869294\pi$
$308$	0	0
$309$	− 2.89668e25i	− 0.207652i
$310$	0	0
$311$	1.71948e26	1.15189	0.575947	−	0.817487i	$-0.304634\pi$
$311$	1.71948e26	1.15189	0.575947	+	0.817487i	$0.304634\pi$
$312$	0	0
$313$	2.75481e26	1.72535	0.862673	−	0.505763i	$-0.168789\pi$
$313$	2.75481e26	1.72535	0.862673	+	0.505763i	$0.168789\pi$
$314$	0	0
$315$	− 4.61754e24i	− 0.0270489i
$316$	0	0
$317$	− 3.17502e26i	− 1.74030i	−0.492785	−	0.870151i	$-0.664021\pi$
$317$	− 3.17502e26i	− 1.74030i	0.492785	−	0.870151i	$-0.335979\pi$
$318$	0	0
$319$	2.81194e25	0.144279
$320$	0	0
$321$	4.98432e25	0.239499
$322$	0	0
$323$	2.80877e25i	0.126441i
$324$	0	0
$325$	− 2.53485e26i	− 1.06949i
$326$	0	0
$327$	1.01632e26	0.402048
$328$	0	0
$329$	5.58220e24	0.0207132
$330$	0	0
$331$	5.31850e26i	1.85180i	0.377764	+	0.925902i	$0.376693\pi$
$331$	5.31850e26i	1.85180i	−0.377764	+	0.925902i	$0.623307\pi$
$332$	0	0
$333$	− 2.12427e26i	− 0.694296i
$334$	0	0
$335$	3.19781e26	0.981481
$336$	0	0
$337$	3.81955e26	1.10128	0.550640	−	0.834743i	$-0.314384\pi$
$337$	3.81955e26	1.10128	0.550640	+	0.834743i	$0.314384\pi$
$338$	0	0
$339$	− 6.62347e25i	− 0.179469i
$340$	0	0
$341$	− 9.57973e25i	− 0.244024i
$342$	0	0
$343$	−2.37024e25	−0.0567812
$344$	0	0
$345$	2.67419e26	0.602686
$346$	0	0
$347$	− 4.52577e25i	− 0.0959915i	−0.998848	−	0.0479958i	$-0.984717\pi$
$347$	− 4.52577e25i	− 0.0959915i	0.998848	−	0.0479958i	$-0.0152834\pi$
$348$	0	0
$349$	3.10422e26i	0.619849i	0.950761	+	0.309924i	$0.100304\pi$
$349$	3.10422e26i	0.619849i	−0.950761	+	0.309924i	$0.899696\pi$
$350$	0	0
$351$	5.43074e26	1.02126
$352$	0	0
$353$	1.20007e26	0.212605	0.106303	−	0.994334i	$-0.466099\pi$
$353$	1.20007e26	0.212605	0.106303	+	0.994334i	$0.466099\pi$
$354$	0	0
$355$	1.54017e27i	2.57140i
$356$	0	0
$357$	− 1.00270e25i	− 0.0157817i
$358$	0	0
$359$	1.66813e25	0.0247593	0.0123797	−	0.999923i	$-0.496059\pi$
$359$	1.66813e25	0.0247593	0.0123797	+	0.999923i	$0.496059\pi$
$360$	0	0
$361$	7.02121e26	0.983074
$362$	0	0
$363$	− 3.39645e26i	− 0.448751i
$364$	0	0
$365$	− 6.66726e26i	− 0.831519i
$366$	0	0
$367$	1.56922e27	1.84794	0.923972	−	0.382459i	$-0.124923\pi$
$367$	1.56922e27	1.84794	0.923972	+	0.382459i	$0.124923\pi$
$368$	0	0
$369$	−7.72726e26	−0.859506
$370$	0	0
$371$	4.89245e24i	0.00514163i
$372$	0	0
$373$	− 9.56660e26i	− 0.950200i	−0.879932	−	0.475100i	$-0.842412\pi$
$373$	− 9.56660e26i	− 0.950200i	0.879932	−	0.475100i	$-0.157588\pi$
$374$	0	0
$375$	1.51337e24	0.00142107
$376$	0	0
$377$	−3.74007e26	−0.332120
$378$	0	0
$379$	− 1.34328e26i	− 0.112838i	−0.998407	−	0.0564191i	$-0.982032\pi$
$379$	− 1.34328e26i	− 0.112838i	0.998407	−	0.0564191i	$-0.0179683\pi$
$380$	0	0
$381$	− 9.20171e26i	− 0.731403i
$382$	0	0
$383$	1.51720e27	1.14144	0.570722	−	0.821143i	$-0.306663\pi$
$383$	1.51720e27	1.14144	0.570722	+	0.821143i	$0.306663\pi$
$384$	0	0
$385$	−2.61647e25	−0.0186371
$386$	0	0
$387$	− 1.96420e27i	− 1.32501i
$388$	0	0
$389$	− 1.86087e26i	− 0.118917i	−0.998231	−	0.0594585i	$-0.981063\pi$
$389$	− 1.86087e26i	− 0.118917i	0.998231	−	0.0594585i	$-0.0189374\pi$
$390$	0	0
$391$	−1.19581e27	−0.724112
$392$	0	0
$393$	−1.53191e27	−0.879248
$394$	0	0
$395$	1.54869e26i	0.0842743i
$396$	0	0
$397$	1.51158e27i	0.780068i	0.920801	+	0.390034i	$0.127537\pi$
$397$	1.51158e27i	0.780068i	−0.920801	+	0.390034i	$0.872463\pi$
$398$	0	0
$399$	4.31554e24	0.00211261
$400$	0	0
$401$	1.59392e27	0.740375	0.370187	−	0.928957i	$-0.379293\pi$
$401$	1.59392e27	0.740375	0.370187	+	0.928957i	$0.379293\pi$
$402$	0	0
$403$	1.27417e27i	0.561725i
$404$	0	0
$405$	− 4.26678e26i	− 0.178576i
$406$	0	0
$407$	−1.20369e27	−0.478379
$408$	0	0
$409$	3.51520e27	1.32695	0.663475	−	0.748198i	$-0.269081\pi$
$409$	3.51520e27	1.32695	0.663475	+	0.748198i	$0.269081\pi$
$410$	0	0
$411$	1.79791e27i	0.644805i
$412$	0	0
$413$	− 9.62854e24i	− 0.00328159i
$414$	0	0
$415$	−7.52517e27	−2.43787
$416$	0	0
$417$	−1.78517e27	−0.549858
$418$	0	0
$419$	− 2.78607e27i	− 0.816102i	−0.912959	−	0.408051i	$-0.866209\pi$
$419$	− 2.78607e27i	− 0.816102i	0.912959	−	0.408051i	$-0.133791\pi$
$420$	0	0
$421$	5.79123e27i	1.61365i	0.590793	+	0.806823i	$0.298815\pi$
$421$	5.79123e27i	1.61365i	−0.590793	+	0.806823i	$0.701185\pi$
$422$	0	0
$423$	1.85168e27	0.490899
$424$	0	0
$425$	−3.85885e27	−0.973587
$426$	0	0
$427$	− 1.14921e26i	− 0.0275998i
$428$	0	0
$429$	− 1.23803e27i	− 0.283093i
$430$	0	0
$431$	−4.53475e27	−0.987511	−0.493755	−	0.869601i	$-0.664376\pi$
$431$	−4.53475e27	−0.987511	−0.493755	+	0.869601i	$0.664376\pi$
$432$	0	0
$433$	6.83097e27	1.41697	0.708483	−	0.705727i	$-0.249380\pi$
$433$	6.83097e27	1.41697	0.708483	+	0.705727i	$0.249380\pi$
$434$	0	0
$435$	− 1.27326e27i	− 0.251640i
$436$	0	0
$437$	− 5.14666e26i	− 0.0969327i
$438$	0	0
$439$	3.75691e26	0.0674455	0.0337228	−	0.999431i	$-0.489264\pi$
$439$	3.75691e26	0.0674455	0.0337228	+	0.999431i	$0.489264\pi$
$440$	0	0
$441$	−3.92960e27	−0.672580
$442$	0	0
$443$	− 6.25638e27i	− 1.02114i	−0.859837	−	0.510568i	$-0.829435\pi$
$443$	− 6.25638e27i	− 1.02114i	0.859837	−	0.510568i	$-0.170565\pi$
$444$	0	0
$445$	7.70124e27i	1.19889i
$446$	0	0
$447$	−1.11190e27	−0.165134
$448$	0	0
$449$	−9.46475e27	−1.34129	−0.670644	−	0.741779i	$-0.733982\pi$
$449$	−9.46475e27	−1.34129	−0.670644	+	0.741779i	$0.733982\pi$
$450$	0	0
$451$	4.37855e27i	0.592212i
$452$	0	0
$453$	6.03622e27i	0.779353i
$454$	0	0
$455$	3.48008e26	0.0429012
$456$	0	0
$457$	−1.27693e28	−1.50330	−0.751649	−	0.659563i	$-0.770741\pi$
$457$	−1.27693e28	−1.50330	−0.751649	+	0.659563i	$0.770741\pi$
$458$	0	0
$459$	− 8.26734e27i	− 0.929678i
$460$	0	0
$461$	1.16045e28i	1.24671i	0.781937	+	0.623357i	$0.214232\pi$
$461$	1.16045e28i	1.24671i	−0.781937	+	0.623357i	$0.785768\pi$
$462$	0	0
$463$	−1.61296e28	−1.65585	−0.827927	−	0.560835i	$-0.810480\pi$
$463$	−1.61296e28	−1.65585	−0.827927	+	0.560835i	$0.810480\pi$
$464$	0	0
$465$	−4.33775e27	−0.425606
$466$	0	0
$467$	− 1.25883e28i	− 1.18070i	−0.807146	−	0.590352i	$-0.798989\pi$
$467$	− 1.25883e28i	− 1.18070i	0.807146	−	0.590352i	$-0.201011\pi$
$468$	0	0
$469$	2.19706e26i	0.0197027i
$470$	0	0
$471$	6.02419e27	0.516629
$472$	0	0
$473$	−1.11299e28	−0.912951
$474$	0	0
$475$	− 1.66082e27i	− 0.130328i
$476$	0	0
$477$	1.62288e27i	0.121855i
$478$	0	0
$479$	6.99593e27	0.502716	0.251358	−	0.967894i	$-0.419123\pi$
$479$	6.99593e27	0.502716	0.251358	+	0.967894i	$0.419123\pi$
$480$	0	0
$481$	1.60099e28	1.10120
$482$	0	0
$483$	1.83730e26i	0.0120986i
$484$	0	0
$485$	3.16421e27i	0.199516i
$486$	0	0
$487$	−1.16609e28	−0.704173	−0.352086	−	0.935968i	$-0.614528\pi$
$487$	−1.16609e28	−0.704173	−0.352086	+	0.935968i	$0.614528\pi$
$488$	0	0
$489$	7.14440e27	0.413260
$490$	0	0
$491$	8.61535e27i	0.477438i	0.971089	+	0.238719i	$0.0767275\pi$
$491$	8.61535e27i	0.477438i	−0.971089	+	0.238719i	$0.923272\pi$
$492$	0	0
$493$	5.69359e27i	0.302338i
$494$	0	0
$495$	−8.67914e27	−0.441694
$496$	0	0
$497$	−1.05817e27	−0.0516196
$498$	0	0
$499$	3.16732e28i	1.48128i	0.671904	+	0.740638i	$0.265477\pi$
$499$	3.16732e28i	1.48128i	−0.671904	+	0.740638i	$0.734523\pi$
$500$	0	0
$501$	2.24207e28i	1.00543i
$502$	0	0
$503$	−2.50128e28	−1.07572	−0.537858	−	0.843035i	$-0.680766\pi$
$503$	−2.50128e28	−1.07572	−0.537858	+	0.843035i	$0.680766\pi$
$504$	0	0
$505$	1.87746e28	0.774480
$506$	0	0
$507$	2.01967e27i	0.0799276i
$508$	0	0
$509$	− 2.52130e28i	− 0.957388i	−0.877982	−	0.478694i	$-0.841110\pi$
$509$	− 2.52130e28i	− 0.957388i	0.877982	−	0.478694i	$-0.158890\pi$
$510$	0	0
$511$	4.58075e26	0.0166923
$512$	0	0
$513$	3.55819e27	0.124451
$514$	0	0
$515$	− 1.53047e28i	− 0.513865i
$516$	0	0
$517$	− 1.04923e28i	− 0.338236i
$518$	0	0
$519$	1.16005e28	0.359104
$520$	0	0
$521$	9.46305e27	0.281342	0.140671	−	0.990056i	$-0.455074\pi$
$521$	9.46305e27	0.281342	0.140671	+	0.990056i	$0.455074\pi$
$522$	0	0
$523$	− 3.79515e28i	− 1.08383i	−0.840433	−	0.541915i	$-0.817699\pi$
$523$	− 3.79515e28i	− 1.08383i	0.840433	−	0.541915i	$-0.182301\pi$
$524$	0	0
$525$	5.92895e26i	0.0162669i
$526$	0	0
$527$	1.93970e28	0.511355
$528$	0	0
$529$	−1.75601e28	−0.444881
$530$	0	0
$531$	− 3.19390e27i	− 0.0777730i
$532$	0	0
$533$	− 5.82378e28i	− 1.36323i
$534$	0	0
$535$	2.63349e28	0.592675
$536$	0	0
$537$	−3.07199e28	−0.664799
$538$	0	0
$539$	2.22666e28i	0.463417i
$540$	0	0
$541$	− 7.62700e28i	− 1.52680i	−0.645925	−	0.763401i	$-0.723528\pi$
$541$	− 7.62700e28i	− 1.52680i	0.645925	−	0.763401i	$-0.276472\pi$
$542$	0	0
$543$	4.37259e28	0.842055
$544$	0	0
$545$	5.36976e28	0.994928
$546$	0	0
$547$	− 1.42433e28i	− 0.253948i	−0.991906	−	0.126974i	$-0.959474\pi$
$547$	− 1.42433e28i	− 0.253948i	0.991906	−	0.126974i	$-0.0405264\pi$
$548$	0	0
$549$	− 3.81206e28i	− 0.654108i
$550$	0	0
$551$	−2.45047e27	−0.0404723
$552$	0	0
$553$	−1.06403e26	−0.00169176
$554$	0	0
$555$	5.45035e28i	0.834350i
$556$	0	0
$557$	6.33843e28i	0.934334i	0.884169	+	0.467167i	$0.154725\pi$
$557$	6.33843e28i	0.934334i	−0.884169	+	0.467167i	$0.845275\pi$
$558$	0	0
$559$	1.48035e29	2.10155
$560$	0	0
$561$	−1.88468e28	−0.257707
$562$	0	0
$563$	7.76864e28i	1.02331i	0.859191	+	0.511655i	$0.170967\pi$
$563$	7.76864e28i	1.02331i	−0.859191	+	0.511655i	$0.829033\pi$
$564$	0	0
$565$	− 3.49954e28i	− 0.444122i
$566$	0	0
$567$	2.93149e26	0.00358481
$568$	0	0
$569$	1.04741e29	1.23434	0.617172	−	0.786828i	$-0.288278\pi$
$569$	1.04741e29	1.23434	0.617172	+	0.786828i	$0.288278\pi$
$570$	0	0
$571$	− 2.49611e28i	− 0.283520i	−0.989901	−	0.141760i	$-0.954724\pi$
$571$	− 2.49611e28i	− 0.283520i	0.989901	−	0.141760i	$-0.0452762\pi$
$572$	0	0
$573$	− 6.41224e28i	− 0.702080i
$574$	0	0
$575$	7.07079e28	0.746372
$576$	0	0
$577$	−3.29189e28	−0.335042	−0.167521	−	0.985868i	$-0.553576\pi$
$577$	−3.29189e28	−0.335042	−0.167521	+	0.985868i	$0.553576\pi$
$578$	0	0
$579$	1.08628e29i	1.06614i
$580$	0	0
$581$	− 5.17017e27i	− 0.0489390i
$582$	0	0
$583$	9.19586e27	0.0839600
$584$	0	0
$585$	1.15438e29	1.01675
$586$	0	0
$587$	7.49973e28i	0.637302i	0.947872	+	0.318651i	$0.103230\pi$
$587$	7.49973e28i	0.637302i	−0.947872	+	0.318651i	$0.896770\pi$
$588$	0	0
$589$	8.34829e27i	0.0684521i
$590$	0	0
$591$	3.27442e28	0.259099
$592$	0	0
$593$	8.69111e28	0.663744	0.331872	−	0.943324i	$-0.392320\pi$
$593$	8.69111e28	0.663744	0.331872	+	0.943324i	$0.392320\pi$
$594$	0	0
$595$	− 5.29781e27i	− 0.0390542i
$596$	0	0
$597$	1.34861e29i	0.959746i
$598$	0	0
$599$	−5.31706e28	−0.365334	−0.182667	−	0.983175i	$-0.558473\pi$
$599$	−5.31706e28	−0.365334	−0.182667	+	0.983175i	$0.558473\pi$
$600$	0	0
$601$	−8.06912e28	−0.535358	−0.267679	−	0.963508i	$-0.586257\pi$
$601$	−8.06912e28	−0.535358	−0.267679	+	0.963508i	$0.586257\pi$
$602$	0	0
$603$	7.28790e28i	0.466950i
$604$	0	0
$605$	− 1.79453e29i	− 1.11050i
$606$	0	0
$607$	−7.97713e28	−0.476832	−0.238416	−	0.971163i	$-0.576628\pi$
$607$	−7.97713e28	−0.476832	−0.238416	+	0.971163i	$0.576628\pi$
$608$	0	0
$609$	8.74793e26	0.00505154
$610$	0	0
$611$	1.39555e29i	0.778596i
$612$	0	0
$613$	− 1.72364e29i	− 0.929207i	−0.885519	−	0.464603i	$-0.846197\pi$
$613$	− 1.72364e29i	− 0.929207i	0.885519	−	0.464603i	$-0.153803\pi$
$614$	0	0
$615$	1.98263e29	1.03289
$616$	0	0
$617$	1.67430e29	0.843022	0.421511	−	0.906823i	$-0.361500\pi$
$617$	1.67430e29	0.843022	0.421511	+	0.906823i	$0.361500\pi$
$618$	0	0
$619$	− 3.01124e29i	− 1.46553i	−0.680483	−	0.732764i	$-0.738230\pi$
$619$	− 3.01124e29i	− 1.46553i	0.680483	−	0.732764i	$-0.261770\pi$
$620$	0	0
$621$	1.51487e29i	0.712711i
$622$	0	0
$623$	−5.29114e27	−0.0240672
$624$	0	0
$625$	−2.26974e29	−0.998240
$626$	0	0
$627$	− 8.11149e27i	− 0.0344978i
$628$	0	0
$629$	− 2.43722e29i	− 1.00245i
$630$	0	0
$631$	3.18935e29	1.26880	0.634400	−	0.773005i	$-0.281247\pi$
$631$	3.18935e29	1.26880	0.634400	+	0.773005i	$0.281247\pi$
$632$	0	0
$633$	−4.96605e28	−0.191105
$634$	0	0
$635$	− 4.86176e29i	− 1.80996i
$636$	0	0
$637$	− 2.96161e29i	− 1.06675i
$638$	0	0
$639$	−3.51008e29	−1.22337
$640$	0	0
$641$	−3.71908e29	−1.25437	−0.627186	−	0.778870i	$-0.715793\pi$
$641$	−3.71908e29	−1.25437	−0.627186	+	0.778870i	$0.715793\pi$
$642$	0	0
$643$	− 1.13039e29i	− 0.368990i	−0.982833	−	0.184495i	$-0.940935\pi$
$643$	− 1.13039e29i	− 0.368990i	0.982833	−	0.184495i	$-0.0590649\pi$
$644$	0	0
$645$	5.03965e29i	1.59230i
$646$	0	0
$647$	−3.79975e29	−1.16214	−0.581072	−	0.813852i	$-0.697367\pi$
$647$	−3.79975e29	−1.16214	−0.581072	+	0.813852i	$0.697367\pi$
$648$	0	0
$649$	−1.80978e28	−0.0535867
$650$	0	0
$651$	− 2.98025e27i	− 0.00854383i
$652$	0	0
$653$	5.07627e29i	1.40915i	0.709631	+	0.704573i	$0.248861\pi$
$653$	5.07627e29i	1.40915i	−0.709631	+	0.704573i	$0.751139\pi$
$654$	0	0
$655$	−8.09390e29	−2.17583
$656$	0	0
$657$	1.51949e29	0.395604
$658$	0	0
$659$	2.09449e29i	0.528180i	0.964498	+	0.264090i	$0.0850716\pi$
$659$	2.09449e29i	0.528180i	−0.964498	+	0.264090i	$0.914928\pi$
$660$	0	0
$661$	− 3.36129e29i	− 0.821090i	−0.911840	−	0.410545i	$-0.865338\pi$
$661$	− 3.36129e29i	− 0.821090i	0.911840	−	0.410545i	$-0.134662\pi$
$662$	0	0
$663$	2.50675e29	0.593224
$664$	0	0
$665$	2.28013e27	0.00522797
$666$	0	0
$667$	− 1.04327e29i	− 0.231779i
$668$	0	0
$669$	1.87358e29i	0.403365i
$670$	0	0
$671$	−2.16005e29	−0.450690
$672$	0	0
$673$	3.02000e29	0.610729	0.305364	−	0.952236i	$-0.401222\pi$
$673$	3.02000e29	0.610729	0.305364	+	0.952236i	$0.401222\pi$
$674$	0	0
$675$	4.88846e29i	0.958258i
$676$	0	0
$677$	9.72547e29i	1.84812i	0.382250	+	0.924059i	$0.375149\pi$
$677$	9.72547e29i	1.84812i	−0.382250	+	0.924059i	$0.624851\pi$
$678$	0	0
$679$	−2.17397e27	−0.00400518
$680$	0	0
$681$	−4.12349e28	−0.0736580
$682$	0	0
$683$	− 4.63752e28i	− 0.0803282i	−0.999193	−	0.0401641i	$-0.987212\pi$
$683$	− 4.63752e28i	− 0.0803282i	0.999193	−	0.0401641i	$-0.0127881\pi$
$684$	0	0
$685$	9.49932e29i	1.59566i
$686$	0	0
$687$	5.08893e28	0.0829050
$688$	0	0
$689$	−1.22311e29	−0.193270
$690$	0	0
$691$	− 5.94138e29i	− 0.910685i	−0.890316	−	0.455342i	$-0.849517\pi$
$691$	− 5.94138e29i	− 0.910685i	0.890316	−	0.455342i	$-0.150483\pi$
$692$	0	0
$693$	− 5.96301e27i	− 0.00886679i
$694$	0	0
$695$	−9.43201e29	−1.36070
$696$	0	0
$697$	−8.86566e29	−1.24099
$698$	0	0
$699$	6.21447e28i	0.0844100i
$700$	0	0
$701$	− 6.31875e29i	− 0.832898i	−0.909159	−	0.416449i	$-0.863275\pi$
$701$	− 6.31875e29i	− 0.832898i	0.909159	−	0.416449i	$-0.136725\pi$
$702$	0	0
$703$	1.04896e29	0.134192
$704$	0	0
$705$	−4.75097e29	−0.589924
$706$	0	0
$707$	1.28991e28i	0.0155473i
$708$	0	0
$709$	8.49724e29i	0.994243i	0.867681	+	0.497122i	$0.165610\pi$
$709$	8.49724e29i	0.994243i	−0.867681	+	0.497122i	$0.834390\pi$
$710$	0	0
$711$	−3.52951e28	−0.0400944
$712$	0	0
$713$	−3.55421e29	−0.392015
$714$	0	0
$715$	− 6.54117e29i	− 0.700554i
$716$	0	0
$717$	1.01988e30i	1.06072i
$718$	0	0
$719$	3.38717e29	0.342124	0.171062	−	0.985260i	$-0.445280\pi$
$719$	3.38717e29	0.342124	0.171062	+	0.985260i	$0.445280\pi$
$720$	0	0
$721$	1.05151e28	0.0103156
$722$	0	0
$723$	3.72322e29i	0.354786i
$724$	0	0
$725$	− 3.36661e29i	− 0.311633i
$726$	0	0
$727$	−1.20714e30	−1.08554	−0.542772	−	0.839880i	$-0.682625\pi$
$727$	−1.20714e30	−1.08554	−0.542772	+	0.839880i	$0.682625\pi$
$728$	0	0
$729$	−5.28726e29	−0.461947
$730$	0	0
$731$	− 2.25357e30i	− 1.91310i
$732$	0	0
$733$	− 9.72103e29i	− 0.801901i	−0.916100	−	0.400950i	$-0.868680\pi$
$733$	− 9.72103e29i	− 0.801901i	0.916100	−	0.400950i	$-0.131320\pi$
$734$	0	0
$735$	1.00824e30	0.808254
$736$	0	0
$737$	4.12959e29	0.321735
$738$	0	0
$739$	6.20974e29i	0.470226i	0.971968	+	0.235113i	$0.0755460\pi$
$739$	6.20974e29i	0.470226i	−0.971968	+	0.235113i	$0.924454\pi$
$740$	0	0
$741$	1.07888e29i	0.0794115i
$742$	0	0
$743$	−1.81337e28	−0.0129749	−0.00648745	−	0.999979i	$-0.502065\pi$
$743$	−1.81337e28	−0.0129749	−0.00648745	+	0.999979i	$0.502065\pi$
$744$	0	0
$745$	−5.87476e29	−0.408648
$746$	0	0
$747$	− 1.71501e30i	− 1.15984i
$748$	0	0
$749$	1.80934e28i	0.0118976i
$750$	0	0
$751$	−3.18101e28	−0.0203398	−0.0101699	−	0.999948i	$-0.503237\pi$
$751$	−3.18101e28	−0.0203398	−0.0101699	+	0.999948i	$0.503237\pi$
$752$	0	0
$753$	−1.06336e30	−0.661200
$754$	0	0
$755$	3.18926e30i	1.92862i
$756$	0	0
$757$	− 1.11901e30i	− 0.658156i	−0.944303	−	0.329078i	$-0.893262\pi$
$757$	− 1.11901e30i	− 0.658156i	0.944303	−	0.329078i	$-0.106738\pi$
$758$	0	0
$759$	3.45340e29	0.197564
$760$	0	0
$761$	1.87654e30	1.04429	0.522143	−	0.852858i	$-0.325133\pi$
$761$	1.87654e30	1.04429	0.522143	+	0.852858i	$0.325133\pi$
$762$	0	0
$763$	3.68930e28i	0.0199727i
$764$	0	0
$765$	− 1.75735e30i	− 0.925576i
$766$	0	0
$767$	2.40713e29	0.123353
$768$	0	0
$769$	3.46794e30	1.72920	0.864600	−	0.502461i	$-0.167572\pi$
$769$	3.46794e30	1.72920	0.864600	+	0.502461i	$0.167572\pi$
$770$	0	0
$771$	1.57070e30i	0.762120i
$772$	0	0
$773$	− 9.90087e29i	− 0.467508i	−0.972296	−	0.233754i	$-0.924899\pi$
$773$	− 9.90087e29i	− 0.467508i	0.972296	−	0.233754i	$-0.0751010\pi$
$774$	0	0
$775$	−1.14694e30	−0.527075
$776$	0	0
$777$	−3.74467e28	−0.0167492
$778$	0	0
$779$	− 3.81571e29i	− 0.166124i
$780$	0	0
$781$	1.98894e30i	0.842920i
$782$	0	0
$783$	7.21273e29	0.297578
$784$	0	0
$785$	3.18291e30	1.27848
$786$	0	0
$787$	3.83990e30i	1.50171i	0.660469	+	0.750854i	$0.270358\pi$
$787$	3.83990e30i	1.50171i	−0.660469	+	0.750854i	$0.729642\pi$
$788$	0	0
$789$	− 1.32694e30i	− 0.505293i
$790$	0	0
$791$	2.40436e28	0.00891551
$792$	0	0
$793$	2.87302e30	1.03746
$794$	0	0
$795$	− 4.16393e29i	− 0.146436i
$796$	0	0
$797$	− 3.81375e30i	− 1.30629i	−0.757233	−	0.653145i	$-0.773449\pi$
$797$	− 3.81375e30i	− 1.30629i	0.757233	−	0.653145i	$-0.226551\pi$
$798$	0	0
$799$	2.12447e30	0.708778
$800$	0	0
$801$	−1.75513e30	−0.570386
$802$	0	0
$803$	− 8.60998e29i	− 0.272577i
$804$	0	0
$805$	9.70746e28i	0.0299398i
$806$	0	0
$807$	2.47184e30	0.742760
$808$	0	0
$809$	−5.60615e30	−1.64137	−0.820683	−	0.571384i	$-0.806407\pi$
$809$	−5.60615e30	−1.64137	−0.820683	+	0.571384i	$0.806407\pi$
$810$	0	0
$811$	3.89260e29i	0.111051i	0.998457	+	0.0555253i	$0.0176834\pi$
$811$	3.89260e29i	0.111051i	−0.998457	+	0.0555253i	$0.982317\pi$
$812$	0	0
$813$	2.88732e30i	0.802685i
$814$	0	0
$815$	3.77477e30	1.02267
$816$	0	0
$817$	9.69916e29	0.256096
$818$	0	0
$819$	7.93121e28i	0.0204107i
$820$	0	0
$821$	− 3.60159e30i	− 0.903424i	−0.892164	−	0.451712i	$-0.850814\pi$
$821$	− 3.60159e30i	− 0.903424i	0.892164	−	0.451712i	$-0.149186\pi$
$822$	0	0
$823$	4.59953e30	1.12464	0.562322	−	0.826919i	$-0.309908\pi$
$823$	4.59953e30	1.12464	0.562322	+	0.826919i	$0.309908\pi$
$824$	0	0
$825$	1.11441e30	0.265630
$826$	0	0
$827$	− 5.37765e30i	− 1.24964i	−0.780769	−	0.624820i	$-0.785172\pi$
$827$	− 5.37765e30i	− 1.24964i	0.780769	−	0.624820i	$-0.214828\pi$
$828$	0	0
$829$	− 8.22155e30i	− 1.86265i	−0.364189	−	0.931325i	$-0.618654\pi$
$829$	− 8.22155e30i	− 1.86265i	0.364189	−	0.931325i	$-0.381346\pi$
$830$	0	0
$831$	2.80745e30	0.620157
$832$	0	0
$833$	−4.50852e30	−0.971095
$834$	0	0
$835$	1.18461e31i	2.48809i
$836$	0	0
$837$	− 2.45724e30i	− 0.503304i
$838$	0	0
$839$	6.06385e30	1.21129	0.605644	−	0.795735i	$-0.292915\pi$
$839$	6.06385e30	1.21129	0.605644	+	0.795735i	$0.292915\pi$
$840$	0	0
$841$	4.63611e30	0.903225
$842$	0	0
$843$	− 3.93504e30i	− 0.747755i
$844$	0	0
$845$	1.06710e30i	0.197793i
$846$	0	0
$847$	1.23293e29	0.0222927
$848$	0	0
$849$	2.43526e30	0.429550
$850$	0	0
$851$	4.46585e30i	0.768499i
$852$	0	0
$853$	5.09605e30i	0.855596i	0.903874	+	0.427798i	$0.140710\pi$
$853$	5.09605e30i	0.855596i	−0.903874	+	0.427798i	$0.859290\pi$
$854$	0	0
$855$	7.56347e29	0.123902
$856$	0	0
$857$	−1.42365e30	−0.227565	−0.113782	−	0.993506i	$-0.536297\pi$
$857$	−1.42365e30	−0.227565	−0.113782	+	0.993506i	$0.536297\pi$
$858$	0	0
$859$	− 5.88644e30i	− 0.918173i	−0.888391	−	0.459087i	$-0.848177\pi$
$859$	− 5.88644e30i	− 0.918173i	0.888391	−	0.459087i	$-0.151823\pi$
$860$	0	0
$861$	1.36217e29i	0.0207347i
$862$	0	0
$863$	−7.22008e30	−1.07258	−0.536289	−	0.844035i	$-0.680174\pi$
$863$	−7.22008e30	−1.07258	−0.536289	+	0.844035i	$0.680174\pi$
$864$	0	0
$865$	6.12919e30	0.888657
$866$	0	0
$867$	2.24027e29i	0.0317030i
$868$	0	0
$869$	1.99995e29i	0.0276256i
$870$	0	0
$871$	−5.49264e30	−0.740611
$872$	0	0
$873$	−7.21133e29	−0.0949218
$874$	0	0
$875$	5.49362e26i	0 7.05950e-5i
$876$	0	0
$877$	2.36275e30i	0.296430i	0.988955	+	0.148215i	$0.0473528\pi$
$877$	2.36275e30i	0.296430i	−0.988955	+	0.148215i	$0.952647\pi$
$878$	0	0
$879$	−5.57064e30	−0.682373
$880$	0	0
$881$	−3.17768e30	−0.380070	−0.190035	−	0.981777i	$-0.560860\pi$
$881$	−3.17768e30	−0.380070	−0.190035	+	0.981777i	$0.560860\pi$
$882$	0	0
$883$	1.00782e31i	1.17706i	0.808477	+	0.588528i	$0.200292\pi$
$883$	1.00782e31i	1.17706i	−0.808477	+	0.588528i	$0.799708\pi$
$884$	0	0
$885$	8.19478e29i	0.0934615i
$886$	0	0
$887$	1.14240e31	1.27239	0.636193	−	0.771530i	$-0.280508\pi$
$887$	1.14240e31	1.27239	0.636193	+	0.771530i	$0.280508\pi$
$888$	0	0
$889$	3.34028e29	0.0363341
$890$	0	0
$891$	− 5.51004e29i	− 0.0585381i
$892$	0	0
$893$	9.14356e29i	0.0948801i
$894$	0	0
$895$	−1.62310e31	−1.64514
$896$	0	0
$897$	−4.59325e30	−0.454778
$898$	0	0
$899$	1.69226e30i	0.163678i
$900$	0	0
$901$	1.86197e30i	0.175939i
$902$	0	0
$903$	−3.46250e29	−0.0319645
$904$	0	0
$905$	2.31027e31	2.08379
$906$	0	0
$907$	− 1.78350e31i	− 1.57180i	−0.618354	−	0.785900i	$-0.712200\pi$
$907$	− 1.78350e31i	− 1.57180i	0.618354	−	0.785900i	$-0.287800\pi$
$908$	0	0
$909$	4.27878e30i	0.368467i
$910$	0	0
$911$	−9.89486e30	−0.832658	−0.416329	−	0.909214i	$-0.636684\pi$
$911$	−9.89486e30	−0.832658	−0.416329	+	0.909214i	$0.636684\pi$
$912$	0	0
$913$	−9.71786e30	−0.799149
$914$	0	0
$915$	9.78082e30i	0.786056i
$916$	0	0
$917$	− 5.56092e29i	− 0.0436786i
$918$	0	0
$919$	−1.12977e31	−0.867315	−0.433657	−	0.901078i	$-0.642777\pi$
$919$	−1.12977e31	−0.867315	−0.433657	+	0.901078i	$0.642777\pi$
$920$	0	0
$921$	−4.11011e30	−0.308409
$922$	0	0
$923$	− 2.64543e31i	− 1.94034i
$924$	0	0
$925$	1.44112e31i	1.03327i
$926$	0	0
$927$	3.48799e30	0.244477
$928$	0	0
$929$	−1.78445e31	−1.22275	−0.611377	−	0.791340i	$-0.709384\pi$
$929$	−1.78445e31	−1.22275	−0.611377	+	0.791340i	$0.709384\pi$
$930$	0	0
$931$	− 1.94043e30i	− 0.129995i
$932$	0	0
$933$	− 1.00545e31i	− 0.658574i
$934$	0	0
$935$	−9.95777e30	−0.637735
$936$	0	0
$937$	−2.24721e31	−1.40727	−0.703635	−	0.710562i	$-0.748441\pi$
$937$	−2.24721e31	−1.40727	−0.703635	+	0.710562i	$0.748441\pi$
$938$	0	0
$939$	− 1.61086e31i	− 0.986435i
$940$	0	0
$941$	− 2.28052e29i	− 0.0136566i	−0.999977	−	0.00682831i	$-0.997826\pi$
$941$	− 2.28052e29i	− 0.0136566i	0.999977	−	0.00682831i	$-0.00217353\pi$
$942$	0	0
$943$	1.62450e31	0.951367
$944$	0	0
$945$	−6.71135e29	−0.0384393
$946$	0	0
$947$	1.85950e31i	1.04165i	0.853664	+	0.520824i	$0.174375\pi$
$947$	1.85950e31i	1.04165i	−0.853664	+	0.520824i	$0.825625\pi$
$948$	0	0
$949$	1.14519e31i	0.627453i
$950$	0	0
$951$	−1.85657e31	−0.994986
$952$	0	0
$953$	−1.45621e31	−0.763396	−0.381698	−	0.924287i	$-0.624660\pi$
$953$	−1.45621e31	−0.763396	−0.381698	+	0.924287i	$0.624660\pi$
$954$	0	0
$955$	− 3.38793e31i	− 1.73740i
$956$	0	0
$957$	− 1.64426e30i	− 0.0824889i
$958$	0	0
$959$	−6.52652e29	−0.0320322
$960$	0	0
$961$	−1.50603e31	−0.723166
$962$	0	0
$963$	6.00179e30i	0.281971i
$964$	0	0
$965$	5.73939e31i	2.63833i
$966$	0	0
$967$	−2.26268e30	−0.101776	−0.0508880	−	0.998704i	$-0.516205\pi$
$967$	−2.26268e30	−0.101776	−0.0508880	+	0.998704i	$0.516205\pi$
$968$	0	0
$969$	1.64241e30	0.0722906
$970$	0	0
$971$	− 2.08215e29i	− 0.00896829i	−0.999990	−	0.00448415i	$-0.998573\pi$
$971$	− 2.08215e29i	− 0.00896829i	0.999990	−	0.00448415i	$-0.00142735\pi$
$972$	0	0
$973$	− 6.48027e29i	− 0.0273154i
$974$	0	0
$975$	−1.48224e31	−0.611461
$976$	0	0
$977$	1.31496e31	0.530909	0.265455	−	0.964123i	$-0.414478\pi$
$977$	1.31496e31	0.530909	0.265455	+	0.964123i	$0.414478\pi$
$978$	0	0
$979$	9.94524e30i	0.393004i
$980$	0	0
$981$	1.22378e31i	0.473348i
$982$	0	0
$983$	−3.42515e31	−1.29678	−0.648392	−	0.761307i	$-0.724558\pi$
$983$	−3.42515e31	−1.29678	−0.648392	+	0.761307i	$0.724558\pi$
$984$	0	0
$985$	1.73005e31	0.641179
$986$	0	0
$987$	− 3.26415e29i	− 0.0118424i
$988$	0	0
$989$	4.12933e31i	1.46662i
$990$	0	0
$991$	5.15488e31	1.79244	0.896222	−	0.443606i	$-0.146301\pi$
$991$	5.15488e31	1.79244	0.896222	+	0.443606i	$0.146301\pi$
$992$	0	0
$993$	3.10996e31	1.05874
$994$	0	0
$995$	7.12544e31i	2.37503i
$996$	0	0
$997$	− 2.57511e31i	− 0.840419i	−0.907427	−	0.420210i	$-0.861957\pi$
$997$	− 2.57511e31i	− 0.840419i	0.907427	−	0.420210i	$-0.138043\pi$
$998$	0	0
$999$	−3.08751e31	−0.986667

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.22.b.c.33.9	✓	28
4.3	odd	2	inner	64.22.b.c.33.19	yes	28
8.3	odd	2	inner	64.22.b.c.33.10	yes	28
8.5	even	2	inner	64.22.b.c.33.20	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.22.b.c.33.9	✓	28	1.1	even	1	trivial
64.22.b.c.33.10	yes	28	8.3	odd	2	inner
64.22.b.c.33.19	yes	28	4.3	odd	2	inner
64.22.b.c.33.20	yes	28	8.5	even	2	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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-58474.4i q^{3} -3.08952e7i q^{5} +2.12266e7 q^{7} +7.04110e9 q^{9} +O(q^{10})\)	\(q-58474.4i q^{3} -3.08952e7i q^{5} +2.12266e7 q^{7} +7.04110e9 q^{9} -3.98975e10i q^{11} +5.30664e11i q^{13} -1.80658e12 q^{15} +8.07841e12 q^{17} +3.47688e12i q^{19} -1.24121e12i q^{21} -1.48025e14 q^{23} -4.77675e14 q^{25} -1.02339e15i q^{27} +7.04791e14i q^{29} +2.40109e15 q^{31} -2.33298e15 q^{33} -6.55798e14i q^{35} -3.01695e16i q^{37} +3.10302e16 q^{39} -1.09745e17 q^{41} -2.78962e17i q^{43} -2.17536e17i q^{45} +2.62982e17 q^{47} -5.58095e17 q^{49} -4.72380e17i q^{51} +2.30487e17i q^{53} -1.23264e18 q^{55} +2.03308e17 q^{57} -4.53608e17i q^{59} -5.41401e18i q^{61} +1.49458e17 q^{63} +1.63949e19 q^{65} +1.03505e19i q^{67} +8.65568e18i q^{69} -4.98513e19 q^{71} +2.15803e19 q^{73} +2.79317e19i q^{75} -8.46886e17i q^{77} -5.01272e18 q^{79} +1.38105e19 q^{81} -2.43571e20i q^{83} -2.49584e20i q^{85} +4.12122e19 q^{87} -2.49270e20 q^{89} +1.12642e19i q^{91} -1.40402e20i q^{93} +1.07419e20 q^{95} -1.02418e20 q^{97} -2.80922e20i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 77960422492 q^{9}+O(q^{10}) \) 28 * q - 77960422492 * q^9	\( 28 q - 77960422492 q^{9} + 16832040195288 q^{17} + 202504130118092 q^{25} - 55\!\cdots\!92 q^{33}+ \cdots - 19\!\cdots\!16 q^{97}+O(q^{100}) \) 28 * q - 77960422492 * q^9 + 16832040195288 * q^17 + 202504130118092 * q^25 - 55633095046450992 * q^33 + 265230847130533848 * q^41 + 5944576136704810940 * q^49 - 4544453360963256304 * q^57 - 63340883112821103360 * q^65 + 2760069698648092472 * q^73 + 934709246686586470348 * q^81 - 375882191672218968072 * q^89 - 1953791530192678912616 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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