LMFDB - Embedded newform 400.10.c.g.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$206.014334466$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.10.c.g.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+60.0000i q^{3} -4344.00i q^{7} +16083.0 q^{9} +O(q^{10})$	$q+60.0000i q^{3} -4344.00i q^{7} +16083.0 q^{9} -93644.0 q^{11} -12242.0i q^{13} +319598. i q^{17} -553516. q^{19} +260640. q^{21} +712936. i q^{23} +2.14596e6i q^{27} -2.07584e6 q^{29} +6.42045e6 q^{31} -5.61864e6i q^{33} +1.81978e7i q^{37} +734520. q^{39} +9.03383e6 q^{41} -1.95947e7i q^{43} -1.84842e7i q^{47} +2.14833e7 q^{49} -1.91759e7 q^{51} +1.02558e7i q^{53} -3.32110e7i q^{57} +1.21667e8 q^{59} -4.59490e7 q^{61} -6.98646e7i q^{63} +5.05354e7i q^{67} -4.27762e7 q^{69} -2.67045e8 q^{71} -1.76213e8i q^{73} +4.06790e8i q^{77} -2.69686e8 q^{79} +1.87804e8 q^{81} +2.27033e8i q^{83} -1.24550e8i q^{87} -7.21416e7 q^{89} -5.31792e7 q^{91} +3.85227e8i q^{93} -2.28777e8i q^{97} -1.50608e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32166 q^{9}+O(q^{10}) $ 2 * q + 32166 * q^9	$ 2 q + 32166 q^{9} - 187288 q^{11} - 1107032 q^{19} + 521280 q^{21} - 4151676 q^{29} + 12840896 q^{31} + 1469040 q^{39} + 18067668 q^{41} + 42966542 q^{49} - 38351760 q^{51} + 243333112 q^{59} - 91897924 q^{61} - 85552320 q^{69} - 534089360 q^{71} - 539371360 q^{79} + 375608178 q^{81} - 144283188 q^{89} - 106358496 q^{91} - 3012152904 q^{99}+O(q^{100}) $ 2 * q + 32166 * q^9 - 187288 * q^11 - 1107032 * q^19 + 521280 * q^21 - 4151676 * q^29 + 12840896 * q^31 + 1469040 * q^39 + 18067668 * q^41 + 42966542 * q^49 - 38351760 * q^51 + 243333112 * q^59 - 91897924 * q^61 - 85552320 * q^69 - 534089360 * q^71 - 539371360 * q^79 + 375608178 * q^81 - 144283188 * q^89 - 106358496 * q^91 - 3012152904 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	60.0000i	0.427667i	0.976870	+	0.213833i	$0.0685950\pi$
$3$	60.0000i	0.427667i	−0.976870	+	0.213833i	$0.931405\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4344.00i	− 0.683831i	−0.939731	−	0.341915i	$-0.888924\pi$
$7$	− 4344.00i	− 0.683831i	0.939731	−	0.341915i	$-0.111076\pi$
$8$	0	0
$9$	16083.0	0.817101
$10$	0	0
$11$	−93644.0	−1.92847	−0.964235	−	0.265049i	$-0.914612\pi$
$11$	−93644.0	−1.92847	−0.964235	+	0.265049i	$0.914612\pi$
$12$	0	0
$13$	− 12242.0i	− 0.118880i	−0.998232	−	0.0594398i	$-0.981069\pi$
$13$	− 12242.0i	− 0.118880i	0.998232	−	0.0594398i	$-0.0189314\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	319598.i	0.928077i	0.885815	+	0.464038i	$0.153600\pi$
$17$	319598.i	0.928077i	−0.885815	+	0.464038i	$0.846400\pi$
$18$	0	0
$19$	−553516.	−0.974404	−0.487202	−	0.873289i	$-0.661982\pi$
$19$	−553516.	−0.974404	−0.487202	+	0.873289i	$0.661982\pi$
$20$	0	0
$21$	260640.	0.292452
$22$	0	0
$23$	712936.i	0.531221i	0.964080	+	0.265611i	$0.0855735\pi$
$23$	712936.i	0.531221i	−0.964080	+	0.265611i	$0.914426\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.14596e6i	0.777114i
$28$	0	0
$29$	−2.07584e6	−0.545007	−0.272504	−	0.962155i	$-0.587852\pi$
$29$	−2.07584e6	−0.545007	−0.272504	+	0.962155i	$0.587852\pi$
$30$	0	0
$31$	6.42045e6	1.24864	0.624321	−	0.781168i	$-0.285376\pi$
$31$	6.42045e6	1.24864	0.624321	+	0.781168i	$0.285376\pi$
$32$	0	0
$33$	− 5.61864e6i	− 0.824743i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.81978e7i	1.59628i	0.602470	+	0.798142i	$0.294183\pi$
$37$	1.81978e7i	1.59628i	−0.602470	+	0.798142i	$0.705817\pi$
$38$	0	0
$39$	734520.	0.0508409
$40$	0	0
$41$	9.03383e6	0.499281	0.249640	−	0.968339i	$-0.419688\pi$
$41$	9.03383e6	0.499281	0.249640	+	0.968339i	$0.419688\pi$
$42$	0	0
$43$	− 1.95947e7i	− 0.874040i	−0.899452	−	0.437020i	$-0.856034\pi$
$43$	− 1.95947e7i	− 0.874040i	0.899452	−	0.437020i	$-0.143966\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.84842e7i	− 0.552535i	−0.961081	−	0.276267i	$-0.910902\pi$
$47$	− 1.84842e7i	− 0.552535i	0.961081	−	0.276267i	$-0.0890976\pi$
$48$	0	0
$49$	2.14833e7	0.532375
$50$	0	0
$51$	−1.91759e7	−0.396908
$52$	0	0
$53$	1.02558e7i	0.178536i	0.996008	+	0.0892682i	$0.0284528\pi$
$53$	1.02558e7i	0.178536i	−0.996008	+	0.0892682i	$0.971547\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.32110e7i	− 0.416720i
$58$	0	0
$59$	1.21667e8	1.30719	0.653593	−	0.756847i	$-0.273261\pi$
$59$	1.21667e8	1.30719	0.653593	+	0.756847i	$0.273261\pi$
$60$	0	0
$61$	−4.59490e7	−0.424905	−0.212452	−	0.977171i	$-0.568145\pi$
$61$	−4.59490e7	−0.424905	−0.212452	+	0.977171i	$0.568145\pi$
$62$	0	0
$63$	− 6.98646e7i	− 0.558759i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.05354e7i	0.306379i	0.988197	+	0.153190i	$0.0489545\pi$
$67$	5.05354e7i	0.306379i	−0.988197	+	0.153190i	$0.951045\pi$
$68$	0	0
$69$	−4.27762e7	−0.227186
$70$	0	0
$71$	−2.67045e8	−1.24716	−0.623579	−	0.781760i	$-0.714322\pi$
$71$	−2.67045e8	−1.24716	−0.623579	+	0.781760i	$0.714322\pi$
$72$	0	0
$73$	− 1.76213e8i	− 0.726250i	−0.931740	−	0.363125i	$-0.881710\pi$
$73$	− 1.76213e8i	− 0.726250i	0.931740	−	0.363125i	$-0.118290\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.06790e8i	1.31875i
$78$	0	0
$79$	−2.69686e8	−0.778997	−0.389499	−	0.921027i	$-0.627352\pi$
$79$	−2.69686e8	−0.778997	−0.389499	+	0.921027i	$0.627352\pi$
$80$	0	0
$81$	1.87804e8	0.484755
$82$	0	0
$83$	2.27033e8i	0.525094i	0.964919	+	0.262547i	$0.0845624\pi$
$83$	2.27033e8i	0.525094i	−0.964919	+	0.262547i	$0.915438\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.24550e8i	− 0.233082i
$88$	0	0
$89$	−7.21416e7	−0.121880	−0.0609398	−	0.998141i	$-0.519410\pi$
$89$	−7.21416e7	−0.121880	−0.0609398	+	0.998141i	$0.519410\pi$
$90$	0	0
$91$	−5.31792e7	−0.0812935
$92$	0	0
$93$	3.85227e8i	0.534003i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.28777e8i	− 0.262385i	−0.991357	−	0.131192i	$-0.958119\pi$
$97$	− 2.28777e8i	− 0.262385i	0.991357	−	0.131192i	$-0.0418806\pi$
$98$	0	0
$99$	−1.50608e9	−1.57575
$100$	0	0
$101$	−8.03256e8	−0.768082	−0.384041	−	0.923316i	$-0.625468\pi$
$101$	−8.03256e8	−0.768082	−0.384041	+	0.923316i	$0.625468\pi$
$102$	0	0
$103$	− 7.81726e8i	− 0.684363i	−0.939634	−	0.342182i	$-0.888834\pi$
$103$	− 7.81726e8i	− 0.684363i	0.939634	−	0.342182i	$-0.111166\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.00756e9i	− 0.743093i	−0.928414	−	0.371546i	$-0.878828\pi$
$107$	− 1.00756e9i	− 0.743093i	0.928414	−	0.371546i	$-0.121172\pi$
$108$	0	0
$109$	4.80692e8	0.326173	0.163086	−	0.986612i	$-0.447855\pi$
$109$	4.80692e8	0.326173	0.163086	+	0.986612i	$0.447855\pi$
$110$	0	0
$111$	−1.09187e9	−0.682678
$112$	0	0
$113$	− 2.89781e9i	− 1.67193i	−0.548786	−	0.835963i	$-0.684910\pi$
$113$	− 2.89781e9i	− 1.67193i	0.548786	−	0.835963i	$-0.315090\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.96888e8i	− 0.0971366i
$118$	0	0
$119$	1.38833e9	0.634647
$120$	0	0
$121$	6.41125e9	2.71900
$122$	0	0
$123$	5.42030e8i	0.213526i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.24330e9i	− 1.44740i	−0.690117	−	0.723698i	$-0.742441\pi$
$127$	− 4.24330e9i	− 1.44740i	0.690117	−	0.723698i	$-0.257559\pi$
$128$	0	0
$129$	1.17568e9	0.373798
$130$	0	0
$131$	−2.89728e9	−0.859546	−0.429773	−	0.902937i	$-0.641406\pi$
$131$	−2.89728e9	−0.859546	−0.429773	+	0.902937i	$0.641406\pi$
$132$	0	0
$133$	2.40447e9i	0.666327i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.35617e9i	0.571432i	0.958314	+	0.285716i	$0.0922314\pi$
$137$	2.35617e9i	0.571432i	−0.958314	+	0.285716i	$0.907769\pi$
$138$	0	0
$139$	−2.71527e9	−0.616946	−0.308473	−	0.951233i	$-0.599818\pi$
$139$	−2.71527e9	−0.616946	−0.308473	+	0.951233i	$0.599818\pi$
$140$	0	0
$141$	1.10905e9	0.236301
$142$	0	0
$143$	1.14639e9i	0.229256i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.28900e9i	0.227679i
$148$	0	0
$149$	−1.67402e9	−0.278242	−0.139121	−	0.990275i	$-0.544428\pi$
$149$	−1.67402e9	−0.278242	−0.139121	+	0.990275i	$0.544428\pi$
$150$	0	0
$151$	−5.32709e9	−0.833860	−0.416930	−	0.908938i	$-0.636894\pi$
$151$	−5.32709e9	−0.833860	−0.416930	+	0.908938i	$0.636894\pi$
$152$	0	0
$153$	5.14009e9i	0.758333i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.15835e10i	1.52156i	0.649008	+	0.760782i	$0.275184\pi$
$157$	1.15835e10i	1.52156i	−0.649008	+	0.760782i	$0.724816\pi$
$158$	0	0
$159$	−6.15346e8	−0.0763541
$160$	0	0
$161$	3.09699e9	0.363265
$162$	0	0
$163$	− 9.48418e8i	− 0.105234i	−0.998615	−	0.0526169i	$-0.983244\pi$
$163$	− 9.48418e8i	− 0.105234i	0.998615	−	0.0526169i	$-0.0167562\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.44718e10i	− 1.43978i	−0.694087	−	0.719891i	$-0.744192\pi$
$167$	− 1.44718e10i	− 1.43978i	0.694087	−	0.719891i	$-0.255808\pi$
$168$	0	0
$169$	1.04546e10	0.985868
$170$	0	0
$171$	−8.90220e9	−0.796186
$172$	0	0
$173$	− 1.39886e10i	− 1.18732i	−0.804717	−	0.593658i	$-0.797683\pi$
$173$	− 1.39886e10i	− 1.18732i	0.804717	−	0.593658i	$-0.202317\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.29999e9i	0.559040i
$178$	0	0
$179$	−4.54924e9	−0.331207	−0.165604	−	0.986192i	$-0.552957\pi$
$179$	−4.54924e9	−0.331207	−0.165604	+	0.986192i	$0.552957\pi$
$180$	0	0
$181$	1.56484e10	1.08372	0.541859	−	0.840469i	$-0.317721\pi$
$181$	1.56484e10	1.08372	0.541859	+	0.840469i	$0.317721\pi$
$182$	0	0
$183$	− 2.75694e9i	− 0.181718i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.99284e10i	− 1.78977i
$188$	0	0
$189$	9.32205e9	0.531414
$190$	0	0
$191$	−2.02052e10	−1.09853	−0.549267	−	0.835647i	$-0.685093\pi$
$191$	−2.02052e10	−1.09853	−0.549267	+	0.835647i	$0.685093\pi$
$192$	0	0
$193$	− 7.10827e9i	− 0.368770i	−0.982854	−	0.184385i	$-0.940971\pi$
$193$	− 7.10827e9i	− 0.368770i	0.982854	−	0.184385i	$-0.0590294\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.25924e10i	− 1.06872i	−0.845257	−	0.534359i	$-0.820553\pi$
$197$	− 2.25924e10i	− 1.06872i	0.845257	−	0.534359i	$-0.179447\pi$
$198$	0	0
$199$	3.55506e10	1.60697	0.803485	−	0.595325i	$-0.202977\pi$
$199$	3.55506e10	1.60697	0.803485	+	0.595325i	$0.202977\pi$
$200$	0	0
$201$	−3.03213e9	−0.131028
$202$	0	0
$203$	9.01744e9i	0.372693i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.14661e10i	0.434061i
$208$	0	0
$209$	5.18335e10	1.87911
$210$	0	0
$211$	5.58480e9	0.193971	0.0969854	−	0.995286i	$-0.469080\pi$
$211$	5.58480e9	0.193971	0.0969854	+	0.995286i	$0.469080\pi$
$212$	0	0
$213$	− 1.60227e10i	− 0.533368i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.78904e10i	− 0.853859i
$218$	0	0
$219$	1.05728e10	0.310593
$220$	0	0
$221$	3.91252e9	0.110329
$222$	0	0
$223$	− 4.74713e10i	− 1.28546i	−0.766092	−	0.642731i	$-0.777801\pi$
$223$	− 4.74713e10i	− 1.28546i	0.766092	−	0.642731i	$-0.222199\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.37702e10i	− 0.844146i	−0.906562	−	0.422073i	$-0.861303\pi$
$227$	− 3.37702e10i	− 0.844146i	0.906562	−	0.422073i	$-0.138697\pi$
$228$	0	0
$229$	−7.28989e9	−0.175171	−0.0875854	−	0.996157i	$-0.527915\pi$
$229$	−7.28989e9	−0.175171	−0.0875854	+	0.996157i	$0.527915\pi$
$230$	0	0
$231$	−2.44074e10	−0.563984
$232$	0	0
$233$	6.79739e10i	1.51092i	0.655197	+	0.755458i	$0.272586\pi$
$233$	6.79739e10i	1.51092i	−0.655197	+	0.755458i	$0.727414\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.61811e10i	− 0.333151i
$238$	0	0
$239$	−3.11283e10	−0.617114	−0.308557	−	0.951206i	$-0.599846\pi$
$239$	−3.11283e10	−0.617114	−0.308557	+	0.951206i	$0.599846\pi$
$240$	0	0
$241$	1.42372e10	0.271861	0.135931	−	0.990718i	$-0.456598\pi$
$241$	1.42372e10	0.271861	0.135931	+	0.990718i	$0.456598\pi$
$242$	0	0
$243$	5.35072e10i	0.984428i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.77614e9i	0.115837i
$248$	0	0
$249$	−1.36220e10	−0.224565
$250$	0	0
$251$	5.78389e10	0.919789	0.459894	−	0.887974i	$-0.347887\pi$
$251$	5.78389e10	0.919789	0.459894	+	0.887974i	$0.347887\pi$
$252$	0	0
$253$	− 6.67622e10i	− 1.02444i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.87176e10i	0.267641i	0.991006	+	0.133820i	$0.0427245\pi$
$257$	1.87176e10i	0.267641i	−0.991006	+	0.133820i	$0.957276\pi$
$258$	0	0
$259$	7.90510e10	1.09159
$260$	0	0
$261$	−3.33857e10	−0.445326
$262$	0	0
$263$	2.80437e10i	0.361439i	0.983535	+	0.180719i	$0.0578426\pi$
$263$	2.80437e10i	0.361439i	−0.983535	+	0.180719i	$0.942157\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.32850e9i	− 0.0521238i
$268$	0	0
$269$	4.46600e10	0.520036	0.260018	−	0.965604i	$-0.416271\pi$
$269$	4.46600e10	0.520036	0.260018	+	0.965604i	$0.416271\pi$
$270$	0	0
$271$	1.03375e11	1.16427	0.582137	−	0.813090i	$-0.302217\pi$
$271$	1.03375e11	1.16427	0.582137	+	0.813090i	$0.302217\pi$
$272$	0	0
$273$	− 3.19075e9i	− 0.0347665i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.81403e11i	− 1.85133i	−0.378341	−	0.925666i	$-0.623505\pi$
$277$	− 1.81403e11i	− 1.85133i	0.378341	−	0.925666i	$-0.376495\pi$
$278$	0	0
$279$	1.03260e11	1.02027
$280$	0	0
$281$	−1.25487e11	−1.20066	−0.600332	−	0.799751i	$-0.704965\pi$
$281$	−1.25487e11	−1.20066	−0.600332	+	0.799751i	$0.704965\pi$
$282$	0	0
$283$	− 1.33561e11i	− 1.23777i	−0.785481	−	0.618886i	$-0.787584\pi$
$283$	− 1.33561e11i	− 1.23777i	0.785481	−	0.618886i	$-0.212416\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.92430e10i	− 0.341423i
$288$	0	0
$289$	1.64450e10	0.138673
$290$	0	0
$291$	1.37266e10	0.112213
$292$	0	0
$293$	− 3.50635e9i	− 0.0277940i	−0.999903	−	0.0138970i	$-0.995576\pi$
$293$	− 3.50635e9i	− 0.0277940i	0.999903	−	0.0138970i	$-0.00442369\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.00956e11i	− 1.49864i
$298$	0	0
$299$	8.72776e9	0.0631513
$300$	0	0
$301$	−8.51195e10	−0.597695
$302$	0	0
$303$	− 4.81954e10i	− 0.328483i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.94357e11i	− 1.89126i	−0.325246	−	0.945629i	$-0.605447\pi$
$307$	− 2.94357e11i	− 1.89126i	0.325246	−	0.945629i	$-0.394553\pi$
$308$	0	0
$309$	4.69035e10	0.292680
$310$	0	0
$311$	−2.40305e10	−0.145660	−0.0728301	−	0.997344i	$-0.523203\pi$
$311$	−2.40305e10	−0.145660	−0.0728301	+	0.997344i	$0.523203\pi$
$312$	0	0
$313$	− 2.55229e11i	− 1.50308i	−0.659689	−	0.751539i	$-0.729312\pi$
$313$	− 2.55229e11i	− 1.50308i	0.659689	−	0.751539i	$-0.270688\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.30255e11i	− 1.28069i	−0.768089	−	0.640343i	$-0.778792\pi$
$317$	− 2.30255e11i	− 1.28069i	0.768089	−	0.640343i	$-0.221208\pi$
$318$	0	0
$319$	1.94390e11	1.05103
$320$	0	0
$321$	6.04535e10	0.317796
$322$	0	0
$323$	− 1.76903e11i	− 0.904322i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.88415e10i	0.139493i
$328$	0	0
$329$	−8.02953e10	−0.377840
$330$	0	0
$331$	1.21212e11	0.555035	0.277518	−	0.960721i	$-0.410488\pi$
$331$	1.21212e11	0.555035	0.277518	+	0.960721i	$0.410488\pi$
$332$	0	0
$333$	2.92674e11i	1.30432i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.52249e11i	− 1.06536i	−0.846318	−	0.532678i	$-0.821186\pi$
$337$	− 2.52249e11i	− 1.06536i	0.846318	−	0.532678i	$-0.178814\pi$
$338$	0	0
$339$	1.73869e11	0.715027
$340$	0	0
$341$	−6.01236e11	−2.40797
$342$	0	0
$343$	− 2.68619e11i	− 1.04789i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.99996e11i	1.11079i	0.831585	+	0.555397i	$0.187434\pi$
$347$	2.99996e11i	1.11079i	−0.831585	+	0.555397i	$0.812566\pi$
$348$	0	0
$349$	1.25625e11	0.453275	0.226638	−	0.973979i	$-0.427227\pi$
$349$	1.25625e11	0.453275	0.226638	+	0.973979i	$0.427227\pi$
$350$	0	0
$351$	2.62708e10	0.0923830
$352$	0	0
$353$	− 4.31672e11i	− 1.47968i	−0.672782	−	0.739841i	$-0.734901\pi$
$353$	− 4.31672e11i	− 1.47968i	0.672782	−	0.739841i	$-0.265099\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.33000e10i	0.271418i
$358$	0	0
$359$	1.83615e11	0.583421	0.291711	−	0.956507i	$-0.405776\pi$
$359$	1.83615e11	0.583421	0.291711	+	0.956507i	$0.405776\pi$
$360$	0	0
$361$	−1.63077e10	−0.0505372
$362$	0	0
$363$	3.84675e11i	1.16282i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.77185e11i	1.08532i	0.839953	+	0.542659i	$0.182582\pi$
$367$	3.77185e11i	1.08532i	−0.839953	+	0.542659i	$0.817418\pi$
$368$	0	0
$369$	1.45291e11	0.407963
$370$	0	0
$371$	4.45510e10	0.122089
$372$	0	0
$373$	2.69400e11i	0.720623i	0.932832	+	0.360312i	$0.117330\pi$
$373$	2.69400e11i	0.720623i	−0.932832	+	0.360312i	$0.882670\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.54124e10i	0.0647903i
$378$	0	0
$379$	−2.04102e11	−0.508124	−0.254062	−	0.967188i	$-0.581767\pi$
$379$	−2.04102e11	−0.508124	−0.254062	+	0.967188i	$0.581767\pi$
$380$	0	0
$381$	2.54598e11	0.619003
$382$	0	0
$383$	4.10631e10i	0.0975117i	0.998811	+	0.0487559i	$0.0155256\pi$
$383$	4.10631e10i	0.0975117i	−0.998811	+	0.0487559i	$0.984474\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3.15142e11i	− 0.714179i
$388$	0	0
$389$	2.86342e11	0.634032	0.317016	−	0.948420i	$-0.397319\pi$
$389$	2.86342e11	0.634032	0.317016	+	0.948420i	$0.397319\pi$
$390$	0	0
$391$	−2.27853e11	−0.493014
$392$	0	0
$393$	− 1.73837e11i	− 0.367599i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3.73016e11i	− 0.753651i	−0.926284	−	0.376826i	$-0.877016\pi$
$397$	− 3.73016e11i	− 0.753651i	0.926284	−	0.376826i	$-0.122984\pi$
$398$	0	0
$399$	−1.44268e11	−0.284966
$400$	0	0
$401$	4.70676e11	0.909018	0.454509	−	0.890742i	$-0.349815\pi$
$401$	4.70676e11	0.909018	0.454509	+	0.890742i	$0.349815\pi$
$402$	0	0
$403$	− 7.85991e10i	− 0.148438i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.70411e12i	− 3.07838i
$408$	0	0
$409$	8.60520e11	1.52057	0.760284	−	0.649590i	$-0.225060\pi$
$409$	8.60520e11	1.52057	0.760284	+	0.649590i	$0.225060\pi$
$410$	0	0
$411$	−1.41370e11	−0.244382
$412$	0	0
$413$	− 5.28520e11i	− 0.893894i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.62916e11i	− 0.263847i
$418$	0	0
$419$	8.46565e11	1.34183	0.670914	−	0.741535i	$-0.265902\pi$
$419$	8.46565e11	1.34183	0.670914	+	0.741535i	$0.265902\pi$
$420$	0	0
$421$	−2.27835e11	−0.353468	−0.176734	−	0.984259i	$-0.556553\pi$
$421$	−2.27835e11	−0.353468	−0.176734	+	0.984259i	$0.556553\pi$
$422$	0	0
$423$	− 2.97281e11i	− 0.451477i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.99602e11i	0.290563i
$428$	0	0
$429$	−6.87834e10	−0.0980451
$430$	0	0
$431$	6.47351e11	0.903633	0.451817	−	0.892111i	$-0.350776\pi$
$431$	6.47351e11	0.903633	0.451817	+	0.892111i	$0.350776\pi$
$432$	0	0
$433$	5.69898e11i	0.779114i	0.921002	+	0.389557i	$0.127372\pi$
$433$	5.69898e11i	0.779114i	−0.921002	+	0.389557i	$0.872628\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.94621e11i	− 0.517624i
$438$	0	0
$439$	5.98042e11	0.768496	0.384248	−	0.923230i	$-0.374461\pi$
$439$	5.98042e11	0.768496	0.384248	+	0.923230i	$0.374461\pi$
$440$	0	0
$441$	3.45515e11	0.435005
$442$	0	0
$443$	− 3.10867e11i	− 0.383494i	−0.981444	−	0.191747i	$-0.938585\pi$
$443$	− 3.10867e11i	− 0.383494i	0.981444	−	0.191747i	$-0.0614152\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.00441e11i	− 0.118995i
$448$	0	0
$449$	−7.47114e11	−0.867517	−0.433759	−	0.901029i	$-0.642813\pi$
$449$	−7.47114e11	−0.867517	−0.433759	+	0.901029i	$0.642813\pi$
$450$	0	0
$451$	−8.45964e11	−0.962848
$452$	0	0
$453$	− 3.19625e11i	− 0.356615i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.54275e12i	1.65452i	0.561819	+	0.827260i	$0.310102\pi$
$457$	1.54275e12i	1.65452i	−0.561819	+	0.827260i	$0.689898\pi$
$458$	0	0
$459$	−6.85845e11	−0.721221
$460$	0	0
$461$	1.62766e12	1.67846	0.839230	−	0.543777i	$-0.183006\pi$
$461$	1.62766e12	1.67846	0.839230	+	0.543777i	$0.183006\pi$
$462$	0	0
$463$	1.11591e12i	1.12854i	0.825592	+	0.564268i	$0.190842\pi$
$463$	1.11591e12i	1.12854i	−0.825592	+	0.564268i	$0.809158\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.30194e11i	− 0.515832i	−0.966167	−	0.257916i	$-0.916964\pi$
$467$	− 5.30194e11i	− 0.515832i	0.966167	−	0.257916i	$-0.0830358\pi$
$468$	0	0
$469$	2.19526e11	0.209512
$470$	0	0
$471$	−6.95008e11	−0.650722
$472$	0	0
$473$	1.83493e12i	1.68556i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.64943e11i	0.145882i
$478$	0	0
$479$	−2.10019e12	−1.82284	−0.911422	−	0.411473i	$-0.865014\pi$
$479$	−2.10019e12	−1.82284	−0.911422	+	0.411473i	$0.865014\pi$
$480$	0	0
$481$	2.22777e11	0.189766
$482$	0	0
$483$	1.85820e11i	0.155357i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.05307e12i	− 0.848351i	−0.905580	−	0.424176i	$-0.860564\pi$
$487$	− 1.05307e12i	− 0.848351i	0.905580	−	0.424176i	$-0.139436\pi$
$488$	0	0
$489$	5.69051e10	0.0450050
$490$	0	0
$491$	−2.10556e12	−1.63494	−0.817470	−	0.575971i	$-0.804624\pi$
$491$	−2.10556e12	−1.63494	−0.817470	+	0.575971i	$0.804624\pi$
$492$	0	0
$493$	− 6.63434e11i	− 0.505809i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.16004e12i	0.852845i
$498$	0	0
$499$	2.88807e11	0.208523	0.104262	−	0.994550i	$-0.466752\pi$
$499$	2.88807e11	0.208523	0.104262	+	0.994550i	$0.466752\pi$
$500$	0	0
$501$	8.68305e11	0.615747
$502$	0	0
$503$	− 5.17681e11i	− 0.360584i	−0.983613	−	0.180292i	$-0.942296\pi$
$503$	− 5.17681e11i	− 0.360584i	0.983613	−	0.180292i	$-0.0577042\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.27278e11i	0.421623i
$508$	0	0
$509$	−6.01747e11	−0.397360	−0.198680	−	0.980064i	$-0.563665\pi$
$509$	−6.01747e11	−0.397360	−0.198680	+	0.980064i	$0.563665\pi$
$510$	0	0
$511$	−7.65471e11	−0.496632
$512$	0	0
$513$	− 1.18782e12i	− 0.757223i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.73093e12i	1.06555i
$518$	0	0
$519$	8.39316e11	0.507776
$520$	0	0
$521$	−1.67285e11	−0.0994691	−0.0497345	−	0.998762i	$-0.515838\pi$
$521$	−1.67285e11	−0.0994691	−0.0497345	+	0.998762i	$0.515838\pi$
$522$	0	0
$523$	1.57966e12i	0.923222i	0.887083	+	0.461611i	$0.152728\pi$
$523$	1.57966e12i	0.923222i	−0.887083	+	0.461611i	$0.847272\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.05196e12i	1.15884i
$528$	0	0
$529$	1.29287e12	0.717804
$530$	0	0
$531$	1.95676e12	1.06810
$532$	0	0
$533$	− 1.10592e11i	− 0.0593543i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.72954e11i	− 0.141646i
$538$	0	0
$539$	−2.01178e12	−1.02667
$540$	0	0
$541$	−3.08736e12	−1.54953	−0.774765	−	0.632250i	$-0.782132\pi$
$541$	−3.08736e12	−1.54953	−0.774765	+	0.632250i	$0.782132\pi$
$542$	0	0
$543$	9.38904e11i	0.463470i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.62136e11i	0.125194i	0.998039	+	0.0625969i	$0.0199383\pi$
$547$	2.62136e11i	0.125194i	−0.998039	+	0.0625969i	$0.980062\pi$
$548$	0	0
$549$	−7.38997e11	−0.347190
$550$	0	0
$551$	1.14901e12	0.531057
$552$	0	0
$553$	1.17151e12i	0.532702i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.64238e11i	− 0.160338i	−0.996781	−	0.0801691i	$-0.974454\pi$
$557$	− 3.64238e11i	− 0.160338i	0.996781	−	0.0801691i	$-0.0255460\pi$
$558$	0	0
$559$	−2.39879e11	−0.103906
$560$	0	0
$561$	1.79571e12	0.765425
$562$	0	0
$563$	− 3.04052e12i	− 1.27544i	−0.770267	−	0.637721i	$-0.779877\pi$
$563$	− 3.04052e12i	− 1.27544i	0.770267	−	0.637721i	$-0.220123\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 8.15821e11i	− 0.331491i
$568$	0	0
$569$	7.35845e11	0.294294	0.147147	−	0.989115i	$-0.452991\pi$
$569$	7.35845e11	0.294294	0.147147	+	0.989115i	$0.452991\pi$
$570$	0	0
$571$	−1.44618e12	−0.569324	−0.284662	−	0.958628i	$-0.591881\pi$
$571$	−1.44618e12	−0.569324	−0.284662	+	0.958628i	$0.591881\pi$
$572$	0	0
$573$	− 1.21231e12i	− 0.469806i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.26945e12i	0.852371i	0.904636	+	0.426186i	$0.140143\pi$
$577$	2.26945e12i	0.852371i	−0.904636	+	0.426186i	$0.859857\pi$
$578$	0	0
$579$	4.26496e11	0.157711
$580$	0	0
$581$	9.86229e11	0.359075
$582$	0	0
$583$	− 9.60391e11i	− 0.344302i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.41977e12i	1.18885i	0.804153	+	0.594423i	$0.202619\pi$
$587$	3.41977e12i	1.18885i	−0.804153	+	0.594423i	$0.797381\pi$
$588$	0	0
$589$	−3.55382e12	−1.21668
$590$	0	0
$591$	1.35554e12	0.457056
$592$	0	0
$593$	− 1.32482e12i	− 0.439959i	−0.975505	−	0.219979i	$-0.929401\pi$
$593$	− 1.32482e12i	− 0.439959i	0.975505	−	0.219979i	$-0.0705990\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.13303e12i	0.687248i
$598$	0	0
$599$	−4.19936e12	−1.33279	−0.666395	−	0.745599i	$-0.732164\pi$
$599$	−4.19936e12	−1.33279	−0.666395	+	0.745599i	$0.732164\pi$
$600$	0	0
$601$	1.05682e12	0.330418	0.165209	−	0.986259i	$-0.447170\pi$
$601$	1.05682e12	0.330418	0.165209	+	0.986259i	$0.447170\pi$
$602$	0	0
$603$	8.12761e11i	0.250343i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.97096e12i	1.78523i	0.450816	+	0.892617i	$0.351133\pi$
$607$	5.97096e12i	1.78523i	−0.450816	+	0.892617i	$0.648867\pi$
$608$	0	0
$609$	−5.41046e11	−0.159388
$610$	0	0
$611$	−2.26283e11	−0.0656851
$612$	0	0
$613$	2.80650e12i	0.802774i	0.915909	+	0.401387i	$0.131472\pi$
$613$	2.80650e12i	0.802774i	−0.915909	+	0.401387i	$0.868528\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.48302e12i	− 0.411968i	−0.978555	−	0.205984i	$-0.933961\pi$
$617$	− 1.48302e12i	− 0.411968i	0.978555	−	0.205984i	$-0.0660394\pi$
$618$	0	0
$619$	1.53469e12	0.420158	0.210079	−	0.977684i	$-0.432628\pi$
$619$	1.53469e12	0.420158	0.210079	+	0.977684i	$0.432628\pi$
$620$	0	0
$621$	−1.52993e12	−0.412819
$622$	0	0
$623$	3.13383e11i	0.0833450i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.11001e12i	0.803632i
$628$	0	0
$629$	−5.81597e12	−1.48147
$630$	0	0
$631$	4.43498e12	1.11368	0.556839	−	0.830620i	$-0.312014\pi$
$631$	4.43498e12	1.11368	0.556839	+	0.830620i	$0.312014\pi$
$632$	0	0
$633$	3.35088e11i	0.0829549i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.62998e11i	− 0.0632886i
$638$	0	0
$639$	−4.29488e12	−1.01905
$640$	0	0
$641$	−4.56257e12	−1.06745	−0.533725	−	0.845658i	$-0.679208\pi$
$641$	−4.56257e12	−1.06745	−0.533725	+	0.845658i	$0.679208\pi$
$642$	0	0
$643$	3.32818e12i	0.767818i	0.923371	+	0.383909i	$0.125422\pi$
$643$	3.32818e12i	0.767818i	−0.923371	+	0.383909i	$0.874578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.31374e12i	0.519093i	0.965731	+	0.259547i	$0.0835731\pi$
$647$	2.31374e12i	0.519093i	−0.965731	+	0.259547i	$0.916427\pi$
$648$	0	0
$649$	−1.13933e13	−2.52087
$650$	0	0
$651$	1.67343e12	0.365167
$652$	0	0
$653$	7.03697e12i	1.51453i	0.653110	+	0.757263i	$0.273464\pi$
$653$	7.03697e12i	1.51453i	−0.653110	+	0.757263i	$0.726536\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.83404e12i	− 0.593419i
$658$	0	0
$659$	2.20320e12	0.455060	0.227530	−	0.973771i	$-0.426935\pi$
$659$	2.20320e12	0.455060	0.227530	+	0.973771i	$0.426935\pi$
$660$	0	0
$661$	−7.29570e12	−1.48648	−0.743242	−	0.669022i	$-0.766713\pi$
$661$	−7.29570e12	−1.48648	−0.743242	+	0.669022i	$0.766713\pi$
$662$	0	0
$663$	2.34751e11i	0.0471842i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.47994e12i	− 0.289519i
$668$	0	0
$669$	2.84828e12	0.549749
$670$	0	0
$671$	4.30284e12	0.819416
$672$	0	0
$673$	− 4.47079e12i	− 0.840073i	−0.907507	−	0.420036i	$-0.862017\pi$
$673$	− 4.47079e12i	− 0.840073i	0.907507	−	0.420036i	$-0.137983\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.42095e12i	0.625890i	0.949771	+	0.312945i	$0.101316\pi$
$677$	3.42095e12i	0.625890i	−0.949771	+	0.312945i	$0.898684\pi$
$678$	0	0
$679$	−9.93805e11	−0.179427
$680$	0	0
$681$	2.02621e12	0.361013
$682$	0	0
$683$	9.18730e12i	1.61546i	0.589556	+	0.807728i	$0.299303\pi$
$683$	9.18730e12i	1.61546i	−0.589556	+	0.807728i	$0.700697\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 4.37394e11i	− 0.0749147i
$688$	0	0
$689$	1.25551e11	0.0212243
$690$	0	0
$691$	1.88811e12	0.315047	0.157524	−	0.987515i	$-0.449649\pi$
$691$	1.88811e12	0.315047	0.157524	+	0.987515i	$0.449649\pi$
$692$	0	0
$693$	6.54240e12i	1.07755i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.88720e12i	0.463371i
$698$	0	0
$699$	−4.07843e12	−0.646169
$700$	0	0
$701$	1.61907e12	0.253242	0.126621	−	0.991951i	$-0.459587\pi$
$701$	1.61907e12	0.253242	0.126621	+	0.991951i	$0.459587\pi$
$702$	0	0
$703$	− 1.00727e13i	− 1.55542i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.48934e12i	0.525238i
$708$	0	0
$709$	1.06375e13	1.58099	0.790497	−	0.612466i	$-0.209822\pi$
$709$	1.06375e13	1.58099	0.790497	+	0.612466i	$0.209822\pi$
$710$	0	0
$711$	−4.33735e12	−0.636520
$712$	0	0
$713$	4.57737e12i	0.663305i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.86770e12i	− 0.263919i
$718$	0	0
$719$	1.32770e13	1.85276	0.926380	−	0.376589i	$-0.122903\pi$
$719$	1.32770e13	1.85276	0.926380	+	0.376589i	$0.122903\pi$
$720$	0	0
$721$	−3.39582e12	−0.467989
$722$	0	0
$723$	8.54230e11i	0.116266i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.60017e12i	0.345221i	0.984990	+	0.172611i	$0.0552202\pi$
$727$	2.60017e12i	0.345221i	−0.984990	+	0.172611i	$0.944780\pi$
$728$	0	0
$729$	4.86117e11	0.0637481
$730$	0	0
$731$	6.26244e12	0.811176
$732$	0	0
$733$	− 1.14818e13i	− 1.46906i	−0.678574	−	0.734532i	$-0.737402\pi$
$733$	− 1.14818e13i	− 1.46906i	0.678574	−	0.734532i	$-0.262598\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.73234e12i	− 0.590843i
$738$	0	0
$739$	−7.75984e12	−0.957090	−0.478545	−	0.878063i	$-0.658836\pi$
$739$	−7.75984e12	−0.957090	−0.478545	+	0.878063i	$0.658836\pi$
$740$	0	0
$741$	−4.06569e11	−0.0495395
$742$	0	0
$743$	2.58115e12i	0.310717i	0.987858	+	0.155358i	$0.0496532\pi$
$743$	2.58115e12i	0.310717i	−0.987858	+	0.155358i	$0.950347\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.65136e12i	0.429055i
$748$	0	0
$749$	−4.37683e12	−0.508150
$750$	0	0
$751$	8.39208e12	0.962697	0.481349	−	0.876529i	$-0.340147\pi$
$751$	8.39208e12	0.962697	0.481349	+	0.876529i	$0.340147\pi$
$752$	0	0
$753$	3.47033e12i	0.393363i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 8.15875e12i	− 0.903009i	−0.892269	−	0.451505i	$-0.850887\pi$
$757$	− 8.15875e12i	− 0.903009i	0.892269	−	0.451505i	$-0.149113\pi$
$758$	0	0
$759$	4.00573e12	0.438121
$760$	0	0
$761$	−6.27433e12	−0.678167	−0.339083	−	0.940756i	$-0.610117\pi$
$761$	−6.27433e12	−0.678167	−0.339083	+	0.940756i	$0.610117\pi$
$762$	0	0
$763$	− 2.08813e12i	− 0.223047i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.48944e12i	− 0.155398i
$768$	0	0
$769$	−6.12027e12	−0.631106	−0.315553	−	0.948908i	$-0.602190\pi$
$769$	−6.12027e12	−0.631106	−0.315553	+	0.948908i	$0.602190\pi$
$770$	0	0
$771$	−1.12306e12	−0.114461
$772$	0	0
$773$	− 6.62875e12i	− 0.667765i	−0.942615	−	0.333883i	$-0.891641\pi$
$773$	− 6.62875e12i	− 0.667765i	0.942615	−	0.333883i	$-0.108359\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.74306e12i	0.466836i
$778$	0	0
$779$	−5.00037e12	−0.486501
$780$	0	0
$781$	2.50071e13	2.40511
$782$	0	0
$783$	− 4.45467e12i	− 0.423533i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.19503e13i	1.11043i	0.831706	+	0.555216i	$0.187364\pi$
$787$	1.19503e13i	1.11043i	−0.831706	+	0.555216i	$0.812636\pi$
$788$	0	0
$789$	−1.68262e12	−0.154575
$790$	0	0
$791$	−1.25881e13	−1.14331
$792$	0	0
$793$	5.62507e11i	0.0505125i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.18887e13i	1.04369i	0.853041	+	0.521844i	$0.174756\pi$
$797$	1.18887e13i	1.04369i	−0.853041	+	0.521844i	$0.825244\pi$
$798$	0	0
$799$	5.90751e12	0.512795
$800$	0	0
$801$	−1.16025e12	−0.0995879
$802$	0	0
$803$	1.65013e13i	1.40055i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.67960e12i	0.222402i
$808$	0	0
$809$	1.68063e13	1.37944	0.689722	−	0.724074i	$-0.257733\pi$
$809$	1.68063e13	1.37944	0.689722	+	0.724074i	$0.257733\pi$
$810$	0	0
$811$	−1.98473e13	−1.61104	−0.805521	−	0.592567i	$-0.798114\pi$
$811$	−1.98473e13	−1.61104	−0.805521	+	0.592567i	$0.798114\pi$
$812$	0	0
$813$	6.20252e12i	0.497922i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.08460e13i	0.851668i
$818$	0	0
$819$	−8.55282e11	−0.0664250
$820$	0	0
$821$	4.83992e12	0.371787	0.185893	−	0.982570i	$-0.440482\pi$
$821$	4.83992e12	0.371787	0.185893	+	0.982570i	$0.440482\pi$
$822$	0	0
$823$	− 8.41664e12i	− 0.639499i	−0.947502	−	0.319749i	$-0.896401\pi$
$823$	− 8.41664e12i	− 0.639499i	0.947502	−	0.319749i	$-0.103599\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.16658e13i	− 1.61064i	−0.592839	−	0.805321i	$-0.701993\pi$
$827$	− 2.16658e13i	− 1.61064i	0.592839	−	0.805321i	$-0.298007\pi$
$828$	0	0
$829$	3.67734e12	0.270420	0.135210	−	0.990817i	$-0.456829\pi$
$829$	3.67734e12	0.270420	0.135210	+	0.990817i	$0.456829\pi$
$830$	0	0
$831$	1.08842e13	0.791754
$832$	0	0
$833$	6.86601e12i	0.494085i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.37780e13i	0.970337i
$838$	0	0
$839$	−1.46942e13	−1.02381	−0.511903	−	0.859043i	$-0.671059\pi$
$839$	−1.46942e13	−1.02381	−0.511903	+	0.859043i	$0.671059\pi$
$840$	0	0
$841$	−1.01980e13	−0.702967
$842$	0	0
$843$	− 7.52924e12i	− 0.513484i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.78505e13i	− 1.85933i
$848$	0	0
$849$	8.01366e12	0.529354
$850$	0	0
$851$	−1.29738e13	−0.847979
$852$	0	0
$853$	− 1.99845e13i	− 1.29248i	−0.763136	−	0.646238i	$-0.776341\pi$
$853$	− 1.99845e13i	− 1.29248i	0.763136	−	0.646238i	$-0.223659\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.11989e13i	− 1.34245i	−0.741252	−	0.671226i	$-0.765768\pi$
$857$	− 2.11989e13i	− 1.34245i	0.741252	−	0.671226i	$-0.234232\pi$
$858$	0	0
$859$	2.51809e13	1.57798	0.788992	−	0.614404i	$-0.210603\pi$
$859$	2.51809e13	1.57798	0.788992	+	0.614404i	$0.210603\pi$
$860$	0	0
$861$	2.35458e12	0.146016
$862$	0	0
$863$	− 2.15905e13i	− 1.32500i	−0.749064	−	0.662498i	$-0.769496\pi$
$863$	− 2.15905e13i	− 1.32500i	0.749064	−	0.662498i	$-0.230504\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.86700e11i	0.0593061i
$868$	0	0
$869$	2.52544e13	1.50227
$870$	0	0
$871$	6.18655e11	0.0364222
$872$	0	0
$873$	− 3.67941e12i	− 0.214395i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.63182e13i	− 1.50230i	−0.660129	−	0.751152i	$-0.729498\pi$
$877$	− 2.63182e13i	− 1.50230i	0.660129	−	0.751152i	$-0.270502\pi$
$878$	0	0
$879$	2.10381e11	0.0118866
$880$	0	0
$881$	4.23513e12	0.236851	0.118425	−	0.992963i	$-0.462215\pi$
$881$	4.23513e12	0.236851	0.118425	+	0.992963i	$0.462215\pi$
$882$	0	0
$883$	2.56557e13i	1.42024i	0.704081	+	0.710119i	$0.251359\pi$
$883$	2.56557e13i	1.42024i	−0.704081	+	0.710119i	$0.748641\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.14044e13i	− 1.70347i	−0.523973	−	0.851735i	$-0.675551\pi$
$887$	− 3.14044e13i	− 1.70347i	0.523973	−	0.851735i	$-0.324449\pi$
$888$	0	0
$889$	−1.84329e13	−0.989774
$890$	0	0
$891$	−1.75867e13	−0.934836
$892$	0	0
$893$	1.02313e13i	0.538392i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.23666e11i	0.0270077i
$898$	0	0
$899$	−1.33278e13	−0.680519
$900$	0	0
$901$	−3.27772e12	−0.165695
$902$	0	0
$903$	− 5.10717e12i	− 0.255615i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.87001e13i	− 1.89880i	−0.314070	−	0.949400i	$-0.601693\pi$
$907$	− 3.87001e13i	− 1.89880i	0.314070	−	0.949400i	$-0.398307\pi$
$908$	0	0
$909$	−1.29188e13	−0.627601
$910$	0	0
$911$	−1.73436e13	−0.834269	−0.417135	−	0.908845i	$-0.636966\pi$
$911$	−1.73436e13	−0.834269	−0.417135	+	0.908845i	$0.636966\pi$
$912$	0	0
$913$	− 2.12602e13i	− 1.01263i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.25858e13i	0.587784i
$918$	0	0
$919$	−1.75232e12	−0.0810387	−0.0405194	−	0.999179i	$-0.512901\pi$
$919$	−1.75232e12	−0.0810387	−0.0405194	+	0.999179i	$0.512901\pi$
$920$	0	0
$921$	1.76614e13	0.808829
$922$	0	0
$923$	3.26916e12i	0.148262i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.25725e13i	− 0.559194i
$928$	0	0
$929$	1.99977e13	0.880865	0.440432	−	0.897786i	$-0.354825\pi$
$929$	1.99977e13	0.880865	0.440432	+	0.897786i	$0.354825\pi$
$930$	0	0
$931$	−1.18913e13	−0.518749
$932$	0	0
$933$	− 1.44183e12i	− 0.0622940i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.62137e12i	− 0.195859i	−0.995193	−	0.0979293i	$-0.968778\pi$
$937$	− 4.62137e12i	− 0.195859i	0.995193	−	0.0979293i	$-0.0312219\pi$
$938$	0	0
$939$	1.53138e13	0.642816
$940$	0	0
$941$	1.64959e13	0.685838	0.342919	−	0.939365i	$-0.388584\pi$
$941$	1.64959e13	0.685838	0.342919	+	0.939365i	$0.388584\pi$
$942$	0	0
$943$	6.44055e12i	0.265228i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.99606e13i	0.806490i	0.915092	+	0.403245i	$0.132118\pi$
$947$	1.99606e13i	0.806490i	−0.915092	+	0.403245i	$0.867882\pi$
$948$	0	0
$949$	−2.15720e12	−0.0863363
$950$	0	0
$951$	1.38153e13	0.547707
$952$	0	0
$953$	− 8.83087e12i	− 0.346805i	−0.984851	−	0.173402i	$-0.944524\pi$
$953$	− 8.83087e12i	− 0.346805i	0.984851	−	0.173402i	$-0.0554761\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.16634e13i	0.449491i
$958$	0	0
$959$	1.02352e13	0.390763
$960$	0	0
$961$	1.47825e13	0.559105
$962$	0	0
$963$	− 1.62046e13i	− 0.607182i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.03562e13i	0.748647i	0.927298	+	0.374323i	$0.122125\pi$
$967$	2.03562e13i	0.748647i	−0.927298	+	0.374323i	$0.877875\pi$
$968$	0	0
$969$	1.06142e13	0.386748
$970$	0	0
$971$	−7.66888e12	−0.276851	−0.138425	−	0.990373i	$-0.544204\pi$
$971$	−7.66888e12	−0.276851	−0.138425	+	0.990373i	$0.544204\pi$
$972$	0	0
$973$	1.17951e13i	0.421886i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.23017e12i	− 0.218763i	−0.994000	−	0.109382i	$-0.965113\pi$
$977$	− 6.23017e12i	− 0.218763i	0.994000	−	0.109382i	$-0.0348871\pi$
$978$	0	0
$979$	6.75563e12	0.235041
$980$	0	0
$981$	7.73097e12	0.266516
$982$	0	0
$983$	− 5.22800e13i	− 1.78585i	−0.450207	−	0.892924i	$-0.648650\pi$
$983$	− 5.22800e13i	− 1.78585i	0.450207	−	0.892924i	$-0.351350\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.81772e12i	− 0.161590i
$988$	0	0
$989$	1.39698e13	0.464308
$990$	0	0
$991$	3.73672e13	1.23072	0.615359	−	0.788247i	$-0.289011\pi$
$991$	3.73672e13	1.23072	0.615359	+	0.788247i	$0.289011\pi$
$992$	0	0
$993$	7.27273e12i	0.237370i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 3.66230e13i	− 1.17389i	−0.809628	−	0.586943i	$-0.800331\pi$
$997$	− 3.66230e13i	− 1.17389i	0.809628	−	0.586943i	$-0.199669\pi$
$998$	0	0
$999$	−3.90517e13	−1.24049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.c.g.49.2		2
4.3	odd	2		200.10.c.b.49.1		2
5.2	odd	4		16.10.a.c.1.1		1
5.3	odd	4		400.10.a.d.1.1		1
5.4	even	2	inner	400.10.c.g.49.1		2
15.2	even	4		144.10.a.n.1.1		1
20.3	even	4		200.10.a.b.1.1		1
20.7	even	4		8.10.a.a.1.1	✓	1
20.19	odd	2		200.10.c.b.49.2		2
40.27	even	4		64.10.a.f.1.1		1
40.37	odd	4		64.10.a.d.1.1		1
60.47	odd	4		72.10.a.e.1.1		1
80.27	even	4		256.10.b.i.129.2		2
80.37	odd	4		256.10.b.c.129.1		2
80.67	even	4		256.10.b.i.129.1		2
80.77	odd	4		256.10.b.c.129.2		2
140.27	odd	4		392.10.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.a.a.1.1	✓	1	20.7	even	4
16.10.a.c.1.1		1	5.2	odd	4
64.10.a.d.1.1		1	40.37	odd	4
64.10.a.f.1.1		1	40.27	even	4
72.10.a.e.1.1		1	60.47	odd	4
144.10.a.n.1.1		1	15.2	even	4
200.10.a.b.1.1		1	20.3	even	4
200.10.c.b.49.1		2	4.3	odd	2
200.10.c.b.49.2		2	20.19	odd	2
256.10.b.c.129.1		2	80.37	odd	4
256.10.b.c.129.2		2	80.77	odd	4
256.10.b.i.129.1		2	80.67	even	4
256.10.b.i.129.2		2	80.27	even	4
392.10.a.b.1.1		1	140.27	odd	4
400.10.a.d.1.1		1	5.3	odd	4
400.10.c.g.49.1		2	5.4	even	2	inner
400.10.c.g.49.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+60.0000i q^{3} -4344.00i q^{7} +16083.0 q^{9} +O(q^{10})\)	\(q+60.0000i q^{3} -4344.00i q^{7} +16083.0 q^{9} -93644.0 q^{11} -12242.0i q^{13} +319598. i q^{17} -553516. q^{19} +260640. q^{21} +712936. i q^{23} +2.14596e6i q^{27} -2.07584e6 q^{29} +6.42045e6 q^{31} -5.61864e6i q^{33} +1.81978e7i q^{37} +734520. q^{39} +9.03383e6 q^{41} -1.95947e7i q^{43} -1.84842e7i q^{47} +2.14833e7 q^{49} -1.91759e7 q^{51} +1.02558e7i q^{53} -3.32110e7i q^{57} +1.21667e8 q^{59} -4.59490e7 q^{61} -6.98646e7i q^{63} +5.05354e7i q^{67} -4.27762e7 q^{69} -2.67045e8 q^{71} -1.76213e8i q^{73} +4.06790e8i q^{77} -2.69686e8 q^{79} +1.87804e8 q^{81} +2.27033e8i q^{83} -1.24550e8i q^{87} -7.21416e7 q^{89} -5.31792e7 q^{91} +3.85227e8i q^{93} -2.28777e8i q^{97} -1.50608e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32166 q^{9}+O(q^{10}) \) 2 * q + 32166 * q^9	\( 2 q + 32166 q^{9} - 187288 q^{11} - 1107032 q^{19} + 521280 q^{21} - 4151676 q^{29} + 12840896 q^{31} + 1469040 q^{39} + 18067668 q^{41} + 42966542 q^{49} - 38351760 q^{51} + 243333112 q^{59} - 91897924 q^{61} - 85552320 q^{69} - 534089360 q^{71} - 539371360 q^{79} + 375608178 q^{81} - 144283188 q^{89} - 106358496 q^{91} - 3012152904 q^{99}+O(q^{100}) \) 2 * q + 32166 * q^9 - 187288 * q^11 - 1107032 * q^19 + 521280 * q^21 - 4151676 * q^29 + 12840896 * q^31 + 1469040 * q^39 + 18067668 * q^41 + 42966542 * q^49 - 38351760 * q^51 + 243333112 * q^59 - 91897924 * q^61 - 85552320 * q^69 - 534089360 * q^71 - 539371360 * q^79 + 375608178 * q^81 - 144283188 * q^89 - 106358496 * q^91 - 3012152904 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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