LMFDB - Embedded newform 100.18.c.a.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,18,Mod(49,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("100.49");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	100.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:	$183.222087345$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{9361})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4681x^{2} + 5475600 $ x^4 + 4681*x^2 + 5475600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$48.8761i$ of defining polynomial
Character	$\chi$	$=$	100.49
Dual form			100.18.c.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15636.4i q^{3} -2.37095e7i q^{7} -1.15358e8 q^{9} +O(q^{10})$	$q+15636.4i q^{3} -2.37095e7i q^{7} -1.15358e8 q^{9} +1.33786e9 q^{11} -3.68345e9i q^{13} +1.70006e10i q^{17} +2.59141e9 q^{19} +3.70732e11 q^{21} -1.96178e11i q^{23} +2.15505e11i q^{27} +1.36209e12 q^{29} -2.25575e12 q^{31} +2.09194e13i q^{33} -1.28643e12i q^{37} +5.75961e13 q^{39} +3.74268e13 q^{41} -2.61921e13i q^{43} -1.94586e14i q^{47} -3.29509e14 q^{49} -2.65829e14 q^{51} -4.69769e14i q^{53} +4.05205e13i q^{57} -1.58789e15 q^{59} -2.16562e15 q^{61} +2.73508e15i q^{63} -1.54302e15i q^{67} +3.06753e15 q^{69} -4.51083e15 q^{71} -6.24410e15i q^{73} -3.17201e16i q^{77} +7.95599e14 q^{79} -1.82671e16 q^{81} +2.62699e16i q^{83} +2.12982e16i q^{87} -3.01291e16 q^{89} -8.73327e16 q^{91} -3.52719e16i q^{93} -3.38333e16i q^{97} -1.54333e17 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 898349364 q^{9}+O(q^{10}) $ 4 * q - 898349364 * q^9	$ 4 q - 898349364 q^{9} + 2519296560 q^{11} - 202267267664 q^{19} + 670860415104 q^{21} - 4674311165304 q^{29} + 557672226944 q^{31} + 216895994347296 q^{39} + 8333184631464 q^{41} - 199140653483748 q^{49} - 79913333836512 q^{51} - 56\!\cdots\!48 q^{59}+ \cdots - 25\!\cdots\!60 q^{99}+O(q^{100}) $ 4 * q - 898349364 * q^9 + 2519296560 * q^11 - 202267267664 * q^19 + 670860415104 * q^21 - 4674311165304 * q^29 + 557672226944 * q^31 + 216895994347296 * q^39 + 8333184631464 * q^41 - 199140653483748 * q^49 - 79913333836512 * q^51 - 5671808395627248 * q^59 - 135088069988872 * q^61 + 20376604200210048 * q^69 + 7764490986430752 * q^71 + 29968543068131008 * q^79 + 66760460575058916 * q^81 - 103818015917692776 * q^89 - 166910338800924224 * q^91 - 256446542618267760 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$.

$n$	$51$	$77$
$\chi(n)$	$1$	$-1$

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
$3$	15636.4i	1.37596i	0.725728	+	0.687982i	$0.241503\pi$
$3$	15636.4i	1.37596i	−0.725728	+	0.687982i	$0.758497\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.37095e7i	− 1.55449i	−0.629196	−	0.777246i	$-0.716616\pi$
$7$	− 2.37095e7i	− 1.55449i	0.629196	−	0.777246i	$-0.283384\pi$
$8$	0	0
$9$	−1.15358e8	−0.893277
$10$	0	0
$11$	1.33786e9	1.88180	0.940902	−	0.338679i	$-0.109980\pi$
$11$	1.33786e9	1.88180	0.940902	+	0.338679i	$0.109980\pi$
$12$	0	0
$13$	− 3.68345e9i	− 1.25238i	−0.779670	−	0.626191i	$-0.784613\pi$
$13$	− 3.68345e9i	− 1.25238i	0.779670	−	0.626191i	$-0.215387\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.70006e10i	0.591082i	0.955330	+	0.295541i	$0.0955000\pi$
$17$	1.70006e10i	0.591082i	−0.955330	+	0.295541i	$0.904500\pi$
$18$	0	0
$19$	2.59141e9	0.0350051	0.0175025	−	0.999847i	$-0.494428\pi$
$19$	2.59141e9	0.0350051	0.0175025	+	0.999847i	$0.494428\pi$
$20$	0	0
$21$	3.70732e11	2.13893
$22$	0	0
$23$	− 1.96178e11i	− 0.522353i	−0.965291	−	0.261177i	$-0.915889\pi$
$23$	− 1.96178e11i	− 0.522353i	0.965291	−	0.261177i	$-0.0841105\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.15505e11i	0.146847i
$28$	0	0
$29$	1.36209e12	0.505618	0.252809	−	0.967516i	$-0.418646\pi$
$29$	1.36209e12	0.505618	0.252809	+	0.967516i	$0.418646\pi$
$30$	0	0
$31$	−2.25575e12	−0.475025	−0.237512	−	0.971384i	$-0.576332\pi$
$31$	−2.25575e12	−0.475025	−0.237512	+	0.971384i	$0.576332\pi$
$32$	0	0
$33$	2.09194e13i	2.58929i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.28643e12i	− 0.0602105i	−0.999547	−	0.0301052i	$-0.990416\pi$
$37$	− 1.28643e12i	− 0.0602105i	0.999547	−	0.0301052i	$-0.00958424\pi$
$38$	0	0
$39$	5.75961e13	1.72323
$40$	0	0
$41$	3.74268e13	0.732014	0.366007	−	0.930612i	$-0.380725\pi$
$41$	3.74268e13	0.732014	0.366007	+	0.930612i	$0.380725\pi$
$42$	0	0
$43$	− 2.61921e13i	− 0.341734i	−0.985294	−	0.170867i	$-0.945343\pi$
$43$	− 2.61921e13i	− 0.341734i	0.985294	−	0.170867i	$-0.0546568\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.94586e14i	− 1.19201i	−0.802980	−	0.596006i	$-0.796754\pi$
$47$	− 1.94586e14i	− 1.19201i	0.802980	−	0.596006i	$-0.203246\pi$
$48$	0	0
$49$	−3.29509e14	−1.41645
$50$	0	0
$51$	−2.65829e14	−0.813308
$52$	0	0
$53$	− 4.69769e14i	− 1.03643i	−0.855251	−	0.518214i	$-0.826597\pi$
$53$	− 4.69769e14i	− 1.03643i	0.855251	−	0.518214i	$-0.173403\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.05205e13i	0.0481657i
$58$	0	0
$59$	−1.58789e15	−1.40792	−0.703962	−	0.710238i	$-0.748587\pi$
$59$	−1.58789e15	−1.40792	−0.703962	+	0.710238i	$0.748587\pi$
$60$	0	0
$61$	−2.16562e15	−1.44637	−0.723183	−	0.690656i	$-0.757322\pi$
$61$	−2.16562e15	−1.44637	−0.723183	+	0.690656i	$0.757322\pi$
$62$	0	0
$63$	2.73508e15i	1.38859i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.54302e15i	− 0.464232i	−0.972688	−	0.232116i	$-0.925435\pi$
$67$	− 1.54302e15i	− 0.464232i	0.972688	−	0.232116i	$-0.0745649\pi$
$68$	0	0
$69$	3.06753e15	0.718739
$70$	0	0
$71$	−4.51083e15	−0.829011	−0.414506	−	0.910047i	$-0.636046\pi$
$71$	−4.51083e15	−0.829011	−0.414506	+	0.910047i	$0.636046\pi$
$72$	0	0
$73$	− 6.24410e15i	− 0.906203i	−0.891459	−	0.453101i	$-0.850318\pi$
$73$	− 6.24410e15i	− 0.906203i	0.891459	−	0.453101i	$-0.149682\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.17201e16i	− 2.92525i
$78$	0	0
$79$	7.95599e14	0.0590016	0.0295008	−	0.999565i	$-0.490608\pi$
$79$	7.95599e14	0.0590016	0.0295008	+	0.999565i	$0.490608\pi$
$80$	0	0
$81$	−1.82671e16	−1.09533
$82$	0	0
$83$	2.62699e16i	1.28025i	0.768271	+	0.640125i	$0.221117\pi$
$83$	2.62699e16i	1.28025i	−0.768271	+	0.640125i	$0.778883\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.12982e16i	0.695712i
$88$	0	0
$89$	−3.01291e16	−0.811280	−0.405640	−	0.914033i	$-0.632951\pi$
$89$	−3.01291e16	−0.811280	−0.405640	+	0.914033i	$0.632951\pi$
$90$	0	0
$91$	−8.73327e16	−1.94682
$92$	0	0
$93$	− 3.52719e16i	− 0.653617i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.38333e16i	− 0.438314i	−0.975690	−	0.219157i	$-0.929669\pi$
$97$	− 3.38333e16i	− 0.438314i	0.975690	−	0.219157i	$-0.0703307\pi$
$98$	0	0
$99$	−1.54333e17	−1.68097
$100$	0	0
$101$	5.50280e16	0.505653	0.252826	−	0.967512i	$-0.418640\pi$
$101$	5.50280e16	0.505653	0.252826	+	0.967512i	$0.418640\pi$
$102$	0	0
$103$	− 2.99690e16i	− 0.233107i	−0.993184	−	0.116554i	$-0.962815\pi$
$103$	− 2.99690e16i	− 0.233107i	0.993184	−	0.116554i	$-0.0371847\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.83477e17i	− 1.03233i	−0.856489	−	0.516166i	$-0.827359\pi$
$107$	− 1.83477e17i	− 1.03233i	0.856489	−	0.516166i	$-0.172641\pi$
$108$	0	0
$109$	2.65387e17	1.27572	0.637859	−	0.770153i	$-0.279820\pi$
$109$	2.65387e17	1.27572	0.637859	+	0.770153i	$0.279820\pi$
$110$	0	0
$111$	2.01152e16	0.0828475
$112$	0	0
$113$	− 1.05318e17i	− 0.372681i	−0.982485	−	0.186341i	$-0.940337\pi$
$113$	− 1.05318e17i	− 0.372681i	0.982485	−	0.186341i	$-0.0596628\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.24915e17i	1.11872i
$118$	0	0
$119$	4.03075e17	0.918833
$120$	0	0
$121$	1.28444e18	2.54119
$122$	0	0
$123$	5.85221e17i	1.00723i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.92218e17i	1.03875i	0.854545	+	0.519377i	$0.173836\pi$
$127$	7.92218e17i	1.03875i	−0.854545	+	0.519377i	$0.826164\pi$
$128$	0	0
$129$	4.09551e17	0.470213
$130$	0	0
$131$	−7.17088e17	−0.722381	−0.361191	−	0.932492i	$-0.617630\pi$
$131$	−7.17088e17	−0.722381	−0.361191	+	0.932492i	$0.617630\pi$
$132$	0	0
$133$	− 6.14411e16i	− 0.0544152i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.94843e18i	1.34141i	0.741726	+	0.670703i	$0.234008\pi$
$137$	1.94843e18i	1.34141i	−0.741726	+	0.670703i	$0.765992\pi$
$138$	0	0
$139$	−1.23071e18	−0.749085	−0.374543	−	0.927210i	$-0.622200\pi$
$139$	−1.23071e18	−0.749085	−0.374543	+	0.927210i	$0.622200\pi$
$140$	0	0
$141$	3.04264e18	1.64016
$142$	0	0
$143$	− 4.92796e18i	− 2.35674i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 5.15235e18i	− 1.94898i
$148$	0	0
$149$	−2.33643e18	−0.787898	−0.393949	−	0.919132i	$-0.628891\pi$
$149$	−2.33643e18	−0.787898	−0.393949	+	0.919132i	$0.628891\pi$
$150$	0	0
$151$	5.57631e17	0.167897	0.0839485	−	0.996470i	$-0.473247\pi$
$151$	5.57631e17	0.167897	0.0839485	+	0.996470i	$0.473247\pi$
$152$	0	0
$153$	− 1.96115e18i	− 0.528000i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.14946e18i	1.11331i	0.830745	+	0.556653i	$0.187915\pi$
$157$	5.14946e18i	1.11331i	−0.830745	+	0.556653i	$0.812085\pi$
$158$	0	0
$159$	7.34551e18	1.42609
$160$	0	0
$161$	−4.65129e18	−0.811994
$162$	0	0
$163$	1.04483e19i	1.64230i	0.570715	+	0.821148i	$0.306666\pi$
$163$	1.04483e19i	1.64230i	−0.570715	+	0.821148i	$0.693334\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.10713e19i	− 1.41614i	−0.706141	−	0.708071i	$-0.749565\pi$
$167$	− 1.10713e19i	− 1.41614i	0.706141	−	0.708071i	$-0.250435\pi$
$168$	0	0
$169$	−4.91741e18	−0.568459
$170$	0	0
$171$	−2.98940e17	−0.0312692
$172$	0	0
$173$	− 1.38307e18i	− 0.131055i	−0.997851	−	0.0655274i	$-0.979127\pi$
$173$	− 1.38307e18i	− 0.131055i	0.997851	−	0.0655274i	$-0.0208730\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2.48290e19i	− 1.93725i
$178$	0	0
$179$	−1.50209e19	−1.06524	−0.532618	−	0.846356i	$-0.678792\pi$
$179$	−1.50209e19	−1.06524	−0.532618	+	0.846356i	$0.678792\pi$
$180$	0	0
$181$	−3.60783e17	−0.0232798	−0.0116399	−	0.999932i	$-0.503705\pi$
$181$	−3.60783e17	−0.0232798	−0.0116399	+	0.999932i	$0.503705\pi$
$182$	0	0
$183$	− 3.38625e19i	− 1.99015i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.27445e19i	1.11230i
$188$	0	0
$189$	5.10952e18	0.228273
$190$	0	0
$191$	8.97904e18	0.366814	0.183407	−	0.983037i	$-0.441287\pi$
$191$	8.97904e18	0.366814	0.183407	+	0.983037i	$0.441287\pi$
$192$	0	0
$193$	− 2.87785e19i	− 1.07605i	−0.842930	−	0.538023i	$-0.819171\pi$
$193$	− 2.87785e19i	− 1.07605i	0.842930	−	0.538023i	$-0.180829\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.29790e19i	− 1.66395i	−0.554811	−	0.831977i	$-0.687209\pi$
$197$	− 5.29790e19i	− 1.66395i	0.554811	−	0.831977i	$-0.312791\pi$
$198$	0	0
$199$	−2.52925e19	−0.729021	−0.364510	−	0.931199i	$-0.618764\pi$
$199$	−2.52925e19	−0.729021	−0.364510	+	0.931199i	$0.618764\pi$
$200$	0	0
$201$	2.41273e19	0.638767
$202$	0	0
$203$	− 3.22944e19i	− 0.785979i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.26307e19i	0.466606i
$208$	0	0
$209$	3.46696e18	0.0658727
$210$	0	0
$211$	6.87338e19	1.20440	0.602198	−	0.798346i	$-0.294292\pi$
$211$	6.87338e19	1.20440	0.602198	+	0.798346i	$0.294292\pi$
$212$	0	0
$213$	− 7.05333e19i	− 1.14069i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.34826e19i	0.738423i
$218$	0	0
$219$	9.76354e19	1.24690
$220$	0	0
$221$	6.26208e19	0.740260
$222$	0	0
$223$	− 4.40564e18i	− 0.0482412i	−0.999709	−	0.0241206i	$-0.992321\pi$
$223$	− 4.40564e18i	− 0.0482412i	0.999709	−	0.0241206i	$-0.00767857\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.27911e19i	− 0.308700i	−0.988016	−	0.154350i	$-0.950672\pi$
$227$	− 3.27911e19i	− 0.308700i	0.988016	−	0.154350i	$-0.0493283\pi$
$228$	0	0
$229$	−9.54497e19	−0.834013	−0.417006	−	0.908904i	$-0.636921\pi$
$229$	−9.54497e19	−0.834013	−0.417006	+	0.908904i	$0.636921\pi$
$230$	0	0
$231$	4.95989e20	4.02504
$232$	0	0
$233$	− 2.07634e20i	− 1.56593i	−0.622066	−	0.782965i	$-0.713706\pi$
$233$	− 2.07634e20i	− 1.56593i	0.622066	−	0.782965i	$-0.286294\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.24403e19i	0.0811841i
$238$	0	0
$239$	2.76425e20	1.67956	0.839780	−	0.542927i	$-0.182684\pi$
$239$	2.76425e20	1.67956	0.839780	+	0.542927i	$0.182684\pi$
$240$	0	0
$241$	2.35403e20	1.33250	0.666249	−	0.745730i	$-0.267899\pi$
$241$	2.35403e20	1.33250	0.666249	+	0.745730i	$0.267899\pi$
$242$	0	0
$243$	− 2.57802e20i	− 1.36029i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 9.54535e18i	− 0.0438397i
$248$	0	0
$249$	−4.10768e20	−1.76158
$250$	0	0
$251$	−3.91593e20	−1.56895	−0.784473	−	0.620163i	$-0.787067\pi$
$251$	−3.91593e20	−1.56895	−0.784473	+	0.620163i	$0.787067\pi$
$252$	0	0
$253$	− 2.62460e20i	− 0.982967i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.14828e20i	− 1.35968i	−0.733361	−	0.679839i	$-0.762049\pi$
$257$	− 4.14828e20i	− 1.35968i	0.733361	−	0.679839i	$-0.237951\pi$
$258$	0	0
$259$	−3.05006e19	−0.0935968
$260$	0	0
$261$	−1.57128e20	−0.451657
$262$	0	0
$263$	9.82407e19i	0.264647i	0.991207	+	0.132324i	$0.0422438\pi$
$263$	9.82407e19i	0.264647i	−0.991207	+	0.132324i	$0.957756\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.71112e20i	− 1.11629i
$268$	0	0
$269$	−5.22922e20	−1.16290	−0.581450	−	0.813582i	$-0.697514\pi$
$269$	−5.22922e20	−1.16290	−0.581450	+	0.813582i	$0.697514\pi$
$270$	0	0
$271$	4.47142e20	0.933698	0.466849	−	0.884337i	$-0.345389\pi$
$271$	4.47142e20	0.933698	0.466849	+	0.884337i	$0.345389\pi$
$272$	0	0
$273$	− 1.36557e21i	− 2.67875i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.36272e20i	0.236226i	0.993000	+	0.118113i	$0.0376845\pi$
$277$	1.36272e20i	0.236226i	−0.993000	+	0.118113i	$0.962316\pi$
$278$	0	0
$279$	2.60219e20	0.424329
$280$	0	0
$281$	1.26390e20	0.193958	0.0969790	−	0.995286i	$-0.469082\pi$
$281$	1.26390e20	0.193958	0.0969790	+	0.995286i	$0.469082\pi$
$282$	0	0
$283$	− 9.81664e19i	− 0.141833i	−0.997482	−	0.0709167i	$-0.977408\pi$
$283$	− 9.81664e19i	− 0.141833i	0.997482	−	0.0709167i	$-0.0225924\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.87370e20i	− 1.13791i
$288$	0	0
$289$	5.38220e20	0.650622
$290$	0	0
$291$	5.29033e20	0.603104
$292$	0	0
$293$	1.01600e21i	1.09275i	0.837541	+	0.546374i	$0.183992\pi$
$293$	1.01600e21i	1.09275i	−0.837541	+	0.546374i	$0.816008\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.88317e20i	0.276338i
$298$	0	0
$299$	−7.22613e20	−0.654185
$300$	0	0
$301$	−6.21000e20	−0.531223
$302$	0	0
$303$	8.60442e20i	0.695760i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.04896e21i	− 0.758720i	−0.925249	−	0.379360i	$-0.876144\pi$
$307$	− 1.04896e21i	− 0.758720i	0.925249	−	0.379360i	$-0.123856\pi$
$308$	0	0
$309$	4.68609e20	0.320747
$310$	0	0
$311$	−1.73873e21	−1.12660	−0.563298	−	0.826254i	$-0.690468\pi$
$311$	−1.73873e21	−1.12660	−0.563298	+	0.826254i	$0.690468\pi$
$312$	0	0
$313$	− 2.56401e21i	− 1.57323i	−0.617443	−	0.786616i	$-0.711831\pi$
$313$	− 2.56401e21i	− 1.57323i	0.617443	−	0.786616i	$-0.288169\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.06492e20i	0.168817i	0.996431	+	0.0844084i	$0.0269000\pi$
$317$	3.06492e20i	0.168817i	−0.996431	+	0.0844084i	$0.973100\pi$
$318$	0	0
$319$	1.82229e21	0.951474
$320$	0	0
$321$	2.86892e21	1.42045
$322$	0	0
$323$	4.40556e19i	0.0206909i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.14971e21i	1.75534i
$328$	0	0
$329$	−4.61354e21	−1.85297
$330$	0	0
$331$	4.66802e21	1.78072	0.890359	−	0.455259i	$-0.150453\pi$
$331$	4.66802e21	1.78072	0.890359	+	0.455259i	$0.150453\pi$
$332$	0	0
$333$	1.48400e20i	0.0537846i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.64491e21i	0.538626i	0.963053	+	0.269313i	$0.0867967\pi$
$337$	1.64491e21i	0.538626i	−0.963053	+	0.269313i	$0.913203\pi$
$338$	0	0
$339$	1.64681e21	0.512796
$340$	0	0
$341$	−3.01789e21	−0.893904
$342$	0	0
$343$	2.29694e21i	0.647365i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.19804e20i	0.234906i	0.993078	+	0.117453i	$0.0374730\pi$
$347$	9.19804e20i	0.234906i	−0.993078	+	0.117453i	$0.962527\pi$
$348$	0	0
$349$	1.23910e21	0.301363	0.150681	−	0.988582i	$-0.451853\pi$
$349$	1.23910e21	0.301363	0.150681	+	0.988582i	$0.451853\pi$
$350$	0	0
$351$	7.93804e20	0.183909
$352$	0	0
$353$	− 2.26089e21i	− 0.499109i	−0.968361	−	0.249554i	$-0.919716\pi$
$353$	− 2.26089e21i	− 0.499109i	0.968361	−	0.249554i	$-0.0802841\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.30266e21i	1.26428i
$358$	0	0
$359$	9.96795e21	1.90679	0.953395	−	0.301725i	$-0.0975624\pi$
$359$	9.96795e21	1.90679	0.953395	+	0.301725i	$0.0975624\pi$
$360$	0	0
$361$	−5.47367e21	−0.998775
$362$	0	0
$363$	2.00840e22i	3.49658i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5.17139e21i	− 0.820248i	−0.912030	−	0.410124i	$-0.865485\pi$
$367$	− 5.17139e21i	− 0.820248i	0.912030	−	0.410124i	$-0.134515\pi$
$368$	0	0
$369$	−4.31748e21	−0.653892
$370$	0	0
$371$	−1.11380e22	−1.61112
$372$	0	0
$373$	1.06342e22i	1.46954i	0.678317	+	0.734769i	$0.262710\pi$
$373$	1.06342e22i	1.46954i	−0.678317	+	0.734769i	$0.737290\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.01719e21i	− 0.633226i
$378$	0	0
$379$	6.81105e21	0.821827	0.410914	−	0.911674i	$-0.365210\pi$
$379$	6.81105e21	0.821827	0.410914	+	0.911674i	$0.365210\pi$
$380$	0	0
$381$	−1.23875e22	−1.42929
$382$	0	0
$383$	− 6.65462e21i	− 0.734402i	−0.930142	−	0.367201i	$-0.880316\pi$
$383$	− 6.65462e21i	− 0.734402i	0.930142	−	0.367201i	$-0.119684\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.02146e21i	0.305263i
$388$	0	0
$389$	−1.61442e22	−1.56115	−0.780574	−	0.625063i	$-0.785073\pi$
$389$	−1.61442e22	−1.56115	−0.780574	+	0.625063i	$0.785073\pi$
$390$	0	0
$391$	3.33515e21	0.308754
$392$	0	0
$393$	− 1.12127e22i	− 0.993971i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.02258e21i	− 0.0831725i	−0.999135	−	0.0415862i	$-0.986759\pi$
$397$	− 1.02258e21i	− 0.0831725i	0.999135	−	0.0415862i	$-0.0132411\pi$
$398$	0	0
$399$	9.60719e20	0.0748733
$400$	0	0
$401$	−1.32825e22	−0.992095	−0.496047	−	0.868295i	$-0.665216\pi$
$401$	−1.32825e22	−0.992095	−0.496047	+	0.868295i	$0.665216\pi$
$402$	0	0
$403$	8.30895e21i	0.594912i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.72107e21i	− 0.113304i
$408$	0	0
$409$	−2.58540e22	−1.63260	−0.816300	−	0.577628i	$-0.803978\pi$
$409$	−2.58540e22	−1.63260	−0.816300	+	0.577628i	$0.803978\pi$
$410$	0	0
$411$	−3.04665e22	−1.84573
$412$	0	0
$413$	3.76481e22i	2.18861i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.92440e22i	− 1.03071i
$418$	0	0
$419$	−1.13941e22	−0.585951	−0.292975	−	0.956120i	$-0.594645\pi$
$419$	−1.13941e22	−0.585951	−0.292975	+	0.956120i	$0.594645\pi$
$420$	0	0
$421$	−2.96045e22	−1.46204	−0.731021	−	0.682354i	$-0.760956\pi$
$421$	−2.96045e22	−1.46204	−0.731021	+	0.682354i	$0.760956\pi$
$422$	0	0
$423$	2.24471e22i	1.06480i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.13457e22i	2.24837i
$428$	0	0
$429$	7.70557e22	3.24278
$430$	0	0
$431$	1.85742e22	0.751369	0.375684	−	0.926748i	$-0.377408\pi$
$431$	1.85742e22	0.751369	0.375684	+	0.926748i	$0.377408\pi$
$432$	0	0
$433$	1.71171e21i	0.0665707i	0.999446	+	0.0332854i	$0.0105970\pi$
$433$	1.71171e21i	0.0665707i	−0.999446	+	0.0332854i	$0.989403\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.08379e20i	− 0.0182850i
$438$	0	0
$439$	−3.04484e22	−1.05346	−0.526728	−	0.850034i	$-0.676581\pi$
$439$	−3.04484e22	−1.05346	−0.526728	+	0.850034i	$0.676581\pi$
$440$	0	0
$441$	3.80115e22	1.26528
$442$	0	0
$443$	− 2.23180e22i	− 0.714864i	−0.933939	−	0.357432i	$-0.883652\pi$
$443$	− 2.23180e22i	− 0.714864i	0.933939	−	0.357432i	$-0.116348\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.65335e22i	− 1.08412i
$448$	0	0
$449$	5.77731e22	1.65056	0.825281	−	0.564723i	$-0.191017\pi$
$449$	5.77731e22	1.65056	0.825281	+	0.564723i	$0.191017\pi$
$450$	0	0
$451$	5.00720e22	1.37751
$452$	0	0
$453$	8.71936e21i	0.231020i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.47877e22i	1.10121i	0.834766	+	0.550606i	$0.185603\pi$
$457$	4.47877e22i	1.10121i	−0.834766	+	0.550606i	$0.814397\pi$
$458$	0	0
$459$	−3.66372e21	−0.0867989
$460$	0	0
$461$	7.44258e22	1.69928	0.849641	−	0.527361i	$-0.176818\pi$
$461$	7.44258e22	1.69928	0.849641	+	0.527361i	$0.176818\pi$
$462$	0	0
$463$	− 3.47065e22i	− 0.763787i	−0.924206	−	0.381893i	$-0.875272\pi$
$463$	− 3.47065e22i	− 0.763787i	0.924206	−	0.381893i	$-0.124728\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.07656e22i	0.629321i	0.949204	+	0.314661i	$0.101891\pi$
$467$	3.07656e22i	0.629321i	−0.949204	+	0.314661i	$0.898109\pi$
$468$	0	0
$469$	−3.65842e22	−0.721646
$470$	0	0
$471$	−8.05192e22	−1.53187
$472$	0	0
$473$	− 3.50414e22i	− 0.643076i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.41916e22i	0.925817i
$478$	0	0
$479$	2.50857e22	0.413595	0.206797	−	0.978384i	$-0.433696\pi$
$479$	2.50857e22	0.413595	0.206797	+	0.978384i	$0.433696\pi$
$480$	0	0
$481$	−4.73851e21	−0.0754065
$482$	0	0
$483$	− 7.27295e22i	− 1.11728i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.25508e23i	− 1.79752i	−0.438439	−	0.898761i	$-0.644468\pi$
$487$	− 1.25508e23i	− 1.79752i	0.438439	−	0.898761i	$-0.355532\pi$
$488$	0	0
$489$	−1.63374e23	−2.25974
$490$	0	0
$491$	5.29654e22	0.707619	0.353809	−	0.935318i	$-0.384886\pi$
$491$	5.29654e22	0.707619	0.353809	+	0.935318i	$0.384886\pi$
$492$	0	0
$493$	2.31563e22i	0.298862i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.06949e23i	1.28869i
$498$	0	0
$499$	−2.81497e22	−0.327807	−0.163904	−	0.986476i	$-0.552409\pi$
$499$	−2.81497e22	−0.327807	−0.163904	+	0.986476i	$0.552409\pi$
$500$	0	0
$501$	1.73115e23	1.94856
$502$	0	0
$503$	− 5.42160e22i	− 0.589929i	−0.955508	−	0.294964i	$-0.904692\pi$
$503$	− 5.42160e22i	− 0.589929i	0.955508	−	0.294964i	$-0.0953078\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7.68907e22i	− 0.782179i
$508$	0	0
$509$	1.78224e23	1.75333	0.876666	−	0.481099i	$-0.159762\pi$
$509$	1.78224e23	1.75333	0.876666	+	0.481099i	$0.159762\pi$
$510$	0	0
$511$	−1.48044e23	−1.40869
$512$	0	0
$513$	5.58463e20i	0.00514040i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.60330e23i	− 2.24313i
$518$	0	0
$519$	2.16263e22	0.180327
$520$	0	0
$521$	−1.94263e22	−0.156773	−0.0783863	−	0.996923i	$-0.524977\pi$
$521$	−1.94263e22	−0.156773	−0.0783863	+	0.996923i	$0.524977\pi$
$522$	0	0
$523$	− 1.65768e23i	− 1.29490i	−0.762107	−	0.647451i	$-0.775835\pi$
$523$	− 1.65768e23i	− 1.29490i	0.762107	−	0.647451i	$-0.224165\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.83491e22i	− 0.280779i
$528$	0	0
$529$	1.02564e23	0.727147
$530$	0	0
$531$	1.83176e23	1.25767
$532$	0	0
$533$	− 1.37860e23i	− 0.916761i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.34874e23i	− 1.46573i
$538$	0	0
$539$	−4.40839e23	−2.66548
$540$	0	0
$541$	2.94676e22	0.172650	0.0863251	−	0.996267i	$-0.472488\pi$
$541$	2.94676e22	0.172650	0.0863251	+	0.996267i	$0.472488\pi$
$542$	0	0
$543$	− 5.64137e21i	− 0.0320321i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.00474e22i	− 0.427027i	−0.976940	−	0.213513i	$-0.931509\pi$
$547$	− 8.00474e22i	− 0.427027i	0.976940	−	0.213513i	$-0.0684907\pi$
$548$	0	0
$549$	2.49821e23	1.29201
$550$	0	0
$551$	3.52974e21	0.0176992
$552$	0	0
$553$	− 1.88632e22i	− 0.0917176i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.60669e23i	0.734790i	0.930065	+	0.367395i	$0.119750\pi$
$557$	1.60669e23i	0.734790i	−0.930065	+	0.367395i	$0.880250\pi$
$558$	0	0
$559$	−9.64772e22	−0.427981
$560$	0	0
$561$	−3.55643e23	−1.53049
$562$	0	0
$563$	− 9.05044e22i	− 0.377875i	−0.981989	−	0.188937i	$-0.939496\pi$
$563$	− 9.05044e22i	− 0.377875i	0.981989	−	0.188937i	$-0.0605044\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.33103e23i	1.70269i
$568$	0	0
$569$	−4.23350e23	−1.61527	−0.807635	−	0.589683i	$-0.799253\pi$
$569$	−4.23350e23	−1.61527	−0.807635	+	0.589683i	$0.799253\pi$
$570$	0	0
$571$	1.34739e21	0.00498985	0.00249493	−	0.999997i	$-0.499206\pi$
$571$	1.34739e21	0.00498985	0.00249493	+	0.999997i	$0.499206\pi$
$572$	0	0
$573$	1.40400e23i	0.504723i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.74144e22i	0.160663i	0.996768	+	0.0803315i	$0.0255979\pi$
$577$	4.74144e22i	0.160663i	−0.996768	+	0.0803315i	$0.974402\pi$
$578$	0	0
$579$	4.49993e23	1.48060
$580$	0	0
$581$	6.22846e23	1.99014
$582$	0	0
$583$	− 6.28487e23i	− 1.95036i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.06143e23i	− 0.310790i	−0.987852	−	0.155395i	$-0.950335\pi$
$587$	− 1.06143e23i	− 0.310790i	0.987852	−	0.155395i	$-0.0496651\pi$
$588$	0	0
$589$	−5.84558e21	−0.0166283
$590$	0	0
$591$	8.28403e23	2.28954
$592$	0	0
$593$	− 3.00230e23i	− 0.806285i	−0.915137	−	0.403142i	$-0.867918\pi$
$593$	− 3.00230e23i	− 0.806285i	0.915137	−	0.403142i	$-0.132082\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3.95484e23i	− 1.00311i
$598$	0	0
$599$	−1.55455e23	−0.383246	−0.191623	−	0.981469i	$-0.561375\pi$
$599$	−1.55455e23	−0.383246	−0.191623	+	0.981469i	$0.561375\pi$
$600$	0	0
$601$	−7.11005e22	−0.170388	−0.0851941	−	0.996364i	$-0.527151\pi$
$601$	−7.11005e22	−0.170388	−0.0851941	+	0.996364i	$0.527151\pi$
$602$	0	0
$603$	1.77999e23i	0.414688i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.07763e23i	0.457576i	0.973476	+	0.228788i	$0.0734763\pi$
$607$	2.07763e23i	0.457576i	−0.973476	+	0.228788i	$0.926524\pi$
$608$	0	0
$609$	5.04970e23	1.08148
$610$	0	0
$611$	−7.16749e23	−1.49285
$612$	0	0
$613$	5.47051e23i	1.10819i	0.832454	+	0.554095i	$0.186936\pi$
$613$	5.47051e23i	1.10819i	−0.832454	+	0.554095i	$0.813064\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.79656e23i	1.11108i	0.831489	+	0.555542i	$0.187489\pi$
$617$	5.79656e23i	1.11108i	−0.831489	+	0.555542i	$0.812511\pi$
$618$	0	0
$619$	5.83066e23	1.08729	0.543647	−	0.839314i	$-0.317043\pi$
$619$	5.83066e23	1.08729	0.543647	+	0.839314i	$0.317043\pi$
$620$	0	0
$621$	4.22775e22	0.0767062
$622$	0	0
$623$	7.14345e23i	1.26113i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.42109e22i	0.0906385i
$628$	0	0
$629$	2.18701e22	0.0355894
$630$	0	0
$631$	9.51360e23	1.50694	0.753469	−	0.657483i	$-0.228379\pi$
$631$	9.51360e23	1.50694	0.753469	+	0.657483i	$0.228379\pi$
$632$	0	0
$633$	1.07475e24i	1.65721i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.21373e24i	1.77393i
$638$	0	0
$639$	5.20360e23	0.740537
$640$	0	0
$641$	−9.77138e23	−1.35414	−0.677069	−	0.735920i	$-0.736750\pi$
$641$	−9.77138e23	−1.35414	−0.677069	+	0.735920i	$0.736750\pi$
$642$	0	0
$643$	5.56439e23i	0.750973i	0.926828	+	0.375486i	$0.122524\pi$
$643$	5.56439e23i	0.750973i	−0.926828	+	0.375486i	$0.877476\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.15984e24i	− 1.48495i	−0.669871	−	0.742477i	$-0.733651\pi$
$647$	− 1.15984e24i	− 1.48495i	0.669871	−	0.742477i	$-0.266349\pi$
$648$	0	0
$649$	−2.12439e24	−2.64944
$650$	0	0
$651$	−8.36278e23	−1.01604
$652$	0	0
$653$	− 9.87432e23i	− 1.16881i	−0.811461	−	0.584407i	$-0.801327\pi$
$653$	− 9.87432e23i	− 1.16881i	0.811461	−	0.584407i	$-0.198673\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.20306e23i	0.809490i
$658$	0	0
$659$	4.97449e23	0.544782	0.272391	−	0.962187i	$-0.412186\pi$
$659$	4.97449e23	0.544782	0.272391	+	0.962187i	$0.412186\pi$
$660$	0	0
$661$	−1.16033e24	−1.23843	−0.619214	−	0.785222i	$-0.712549\pi$
$661$	−1.16033e24	−1.23843	−0.619214	+	0.785222i	$0.712549\pi$
$662$	0	0
$663$	9.79167e23i	1.01857i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.67212e23i	− 0.264111i
$668$	0	0
$669$	6.88886e22	0.0663781
$670$	0	0
$671$	−2.89730e24	−2.72178
$672$	0	0
$673$	− 1.13175e24i	− 1.03663i	−0.855190	−	0.518315i	$-0.826559\pi$
$673$	− 1.13175e24i	− 1.03663i	0.855190	−	0.518315i	$-0.173441\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.09418e24i	− 0.952983i	−0.879179	−	0.476492i	$-0.841908\pi$
$677$	− 1.09418e24i	− 0.952983i	0.879179	−	0.476492i	$-0.158092\pi$
$678$	0	0
$679$	−8.02171e23	−0.681356
$680$	0	0
$681$	5.12736e23	0.424760
$682$	0	0
$683$	1.05400e22i	0.00851660i	0.999991	+	0.00425830i	$0.00135546\pi$
$683$	1.05400e22i	0.00851660i	−0.999991	+	0.00425830i	$0.998645\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.49249e24i	− 1.14757i
$688$	0	0
$689$	−1.73037e24	−1.29800
$690$	0	0
$691$	−1.61636e24	−1.18297	−0.591485	−	0.806316i	$-0.701458\pi$
$691$	−1.61636e24	−1.18297	−0.591485	+	0.806316i	$0.701458\pi$
$692$	0	0
$693$	3.65916e24i	2.61306i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.36277e23i	0.432681i
$698$	0	0
$699$	3.24665e24	2.15466
$700$	0	0
$701$	1.86841e23	0.121024	0.0605118	−	0.998167i	$-0.480727\pi$
$701$	1.86841e23	0.121024	0.0605118	+	0.998167i	$0.480727\pi$
$702$	0	0
$703$	− 3.33368e21i	− 0.00210767i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.30469e24i	− 0.786033i
$708$	0	0
$709$	2.18773e23	0.128677	0.0643385	−	0.997928i	$-0.479506\pi$
$709$	2.18773e23	0.128677	0.0643385	+	0.997928i	$0.479506\pi$
$710$	0	0
$711$	−9.17786e22	−0.0527048
$712$	0	0
$713$	4.42529e23i	0.248131i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.32230e24i	2.31101i
$718$	0	0
$719$	1.15548e24	0.603345	0.301673	−	0.953412i	$-0.402455\pi$
$719$	1.15548e24	0.603345	0.301673	+	0.953412i	$0.402455\pi$
$720$	0	0
$721$	−7.10550e23	−0.362364
$722$	0	0
$723$	3.68086e24i	1.83347i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.57207e24i	1.69777i	0.528581	+	0.848883i	$0.322724\pi$
$727$	3.57207e24i	1.69777i	−0.528581	+	0.848883i	$0.677276\pi$
$728$	0	0
$729$	1.67208e24	0.776379
$730$	0	0
$731$	4.45280e23	0.201993
$732$	0	0
$733$	− 4.90391e23i	− 0.217350i	−0.994077	−	0.108675i	$-0.965339\pi$
$733$	− 4.90391e23i	− 0.217350i	0.994077	−	0.108675i	$-0.0346607\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.06435e24i	− 0.873594i
$738$	0	0
$739$	−2.17372e24	−0.898928	−0.449464	−	0.893298i	$-0.648385\pi$
$739$	−2.17372e24	−0.898928	−0.449464	+	0.893298i	$0.648385\pi$
$740$	0	0
$741$	1.49255e23	0.0603219
$742$	0	0
$743$	3.08963e24i	1.22040i	0.792248	+	0.610199i	$0.208911\pi$
$743$	3.08963e24i	1.22040i	−0.792248	+	0.610199i	$0.791089\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.03044e24i	− 1.14362i
$748$	0	0
$749$	−4.35014e24	−1.60475
$750$	0	0
$751$	4.58172e24	1.65230	0.826150	−	0.563451i	$-0.190526\pi$
$751$	4.58172e24	1.65230	0.826150	+	0.563451i	$0.190526\pi$
$752$	0	0
$753$	− 6.12312e24i	− 2.15881i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.43112e23i	− 0.115644i	−0.998327	−	0.0578218i	$-0.981584\pi$
$757$	− 3.43112e23i	− 0.115644i	0.998327	−	0.0578218i	$-0.0184155\pi$
$758$	0	0
$759$	4.10394e24	1.35253
$760$	0	0
$761$	3.01271e24	0.970930	0.485465	−	0.874256i	$-0.338650\pi$
$761$	3.01271e24	0.970930	0.485465	+	0.874256i	$0.338650\pi$
$762$	0	0
$763$	− 6.29219e24i	− 1.98309i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.84893e24i	1.76326i
$768$	0	0
$769$	1.62418e23	0.0478918	0.0239459	−	0.999713i	$-0.492377\pi$
$769$	1.62418e23	0.0478918	0.0239459	+	0.999713i	$0.492377\pi$
$770$	0	0
$771$	6.48643e24	1.87087
$772$	0	0
$773$	− 4.43975e24i	− 1.25266i	−0.779559	−	0.626329i	$-0.784557\pi$
$773$	− 4.43975e24i	− 1.25266i	0.779559	−	0.626329i	$-0.215443\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.76921e23i	− 0.128786i
$778$	0	0
$779$	9.69883e22	0.0256242
$780$	0	0
$781$	−6.03488e24	−1.56004
$782$	0	0
$783$	2.93537e23i	0.0742486i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.95546e24i	− 0.715880i	−0.933745	−	0.357940i	$-0.883479\pi$
$787$	− 2.95546e24i	− 0.715880i	0.933745	−	0.357940i	$-0.116521\pi$
$788$	0	0
$789$	−1.53613e24	−0.364145
$790$	0	0
$791$	−2.49705e24	−0.579331
$792$	0	0
$793$	7.97695e24i	1.81140i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.01045e24i	1.30771i	0.756620	+	0.653855i	$0.226849\pi$
$797$	6.01045e24i	1.30771i	−0.756620	+	0.653855i	$0.773151\pi$
$798$	0	0
$799$	3.30808e24	0.704577
$800$	0	0
$801$	3.47563e24	0.724698
$802$	0	0
$803$	− 8.35376e24i	− 1.70530i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 8.17663e24i	− 1.60011i
$808$	0	0
$809$	−7.16704e24	−1.37334	−0.686668	−	0.726971i	$-0.740928\pi$
$809$	−7.16704e24	−1.37334	−0.686668	+	0.726971i	$0.740928\pi$
$810$	0	0
$811$	5.73123e24	1.07540	0.537701	−	0.843136i	$-0.319293\pi$
$811$	5.73123e24	1.07540	0.537701	+	0.843136i	$0.319293\pi$
$812$	0	0
$813$	6.99170e24i	1.28473i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.78745e22i	− 0.0119624i
$818$	0	0
$819$	1.00745e25	1.73905
$820$	0	0
$821$	2.45653e24	0.415341	0.207670	−	0.978199i	$-0.433412\pi$
$821$	2.45653e24	0.415341	0.207670	+	0.978199i	$0.433412\pi$
$822$	0	0
$823$	− 1.00636e25i	− 1.66670i	−0.552749	−	0.833348i	$-0.686421\pi$
$823$	− 1.00636e25i	− 1.66670i	0.552749	−	0.833348i	$-0.313579\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.31323e24i	− 0.367639i	−0.982960	−	0.183820i	$-0.941154\pi$
$827$	− 2.31323e24i	− 0.367639i	0.982960	−	0.183820i	$-0.0588462\pi$
$828$	0	0
$829$	−5.49125e24	−0.854983	−0.427491	−	0.904019i	$-0.640603\pi$
$829$	−5.49125e24	−0.854983	−0.427491	+	0.904019i	$0.640603\pi$
$830$	0	0
$831$	−2.13080e24	−0.325038
$832$	0	0
$833$	− 5.60185e24i	− 0.837237i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.86126e23i	− 0.0697561i
$838$	0	0
$839$	−2.39004e24	−0.336069	−0.168035	−	0.985781i	$-0.553742\pi$
$839$	−2.39004e24	−0.336069	−0.168035	+	0.985781i	$0.553742\pi$
$840$	0	0
$841$	−5.40186e24	−0.744351
$842$	0	0
$843$	1.97628e24i	0.266879i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.04533e25i	− 3.95026i
$848$	0	0
$849$	1.53497e24	0.195158
$850$	0	0
$851$	−2.52370e23	−0.0314511
$852$	0	0
$853$	− 7.04660e24i	− 0.860821i	−0.902633	−	0.430410i	$-0.858369\pi$
$853$	− 7.04660e24i	− 0.860821i	0.902633	−	0.430410i	$-0.141631\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.71395e24i	0.788208i	0.919066	+	0.394104i	$0.128945\pi$
$857$	6.71395e24i	0.788208i	−0.919066	+	0.394104i	$0.871055\pi$
$858$	0	0
$859$	−8.58583e24	−0.988189	−0.494095	−	0.869408i	$-0.664500\pi$
$859$	−8.58583e24	−0.988189	−0.494095	+	0.869408i	$0.664500\pi$
$860$	0	0
$861$	1.38753e25	1.56572
$862$	0	0
$863$	− 5.52514e24i	− 0.611296i	−0.952145	−	0.305648i	$-0.901127\pi$
$863$	− 5.52514e24i	− 0.611296i	0.952145	−	0.305648i	$-0.0988731\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.41585e24i	0.895232i
$868$	0	0
$869$	1.06440e24	0.111030
$870$	0	0
$871$	−5.68364e24	−0.581396
$872$	0	0
$873$	3.90294e24i	0.391536i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.54824e24i	0.631871i	0.948781	+	0.315935i	$0.102318\pi$
$877$	6.54824e24i	0.631871i	−0.948781	+	0.315935i	$0.897682\pi$
$878$	0	0
$879$	−1.58867e25	−1.50358
$880$	0	0
$881$	5.78483e24	0.537026	0.268513	−	0.963276i	$-0.413468\pi$
$881$	5.78483e24	0.537026	0.268513	+	0.963276i	$0.413468\pi$
$882$	0	0
$883$	− 1.75999e25i	− 1.60267i	−0.598213	−	0.801337i	$-0.704122\pi$
$883$	− 1.75999e25i	− 1.60267i	0.598213	−	0.801337i	$-0.295878\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.06620e25i	− 0.934300i	−0.884178	−	0.467150i	$-0.845281\pi$
$887$	− 1.06620e25i	− 0.934300i	0.884178	−	0.467150i	$-0.154719\pi$
$888$	0	0
$889$	1.87831e25	1.61474
$890$	0	0
$891$	−2.44389e25	−2.06120
$892$	0	0
$893$	− 5.04254e23i	− 0.0417265i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.12991e25i	− 0.900136i
$898$	0	0
$899$	−3.07253e24	−0.240181
$900$	0	0
$901$	7.98634e24	0.612615
$902$	0	0
$903$	− 9.71023e24i	− 0.730943i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.63230e25i	1.18342i	0.806152	+	0.591709i	$0.201547\pi$
$907$	1.63230e25i	1.18342i	−0.806152	+	0.591709i	$0.798453\pi$
$908$	0	0
$909$	−6.34792e24	−0.451688
$910$	0	0
$911$	1.45971e25	1.01944	0.509719	−	0.860341i	$-0.329749\pi$
$911$	1.45971e25	1.01944	0.509719	+	0.860341i	$0.329749\pi$
$912$	0	0
$913$	3.51456e25i	2.40918i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.70018e25i	1.12294i
$918$	0	0
$919$	−1.83968e24	−0.119278	−0.0596390	−	0.998220i	$-0.518995\pi$
$919$	−1.83968e24	−0.119278	−0.0596390	+	0.998220i	$0.518995\pi$
$920$	0	0
$921$	1.64019e25	1.04397
$922$	0	0
$923$	1.66154e25i	1.03824i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.45716e24i	0.208230i
$928$	0	0
$929$	7.16411e24	0.423671	0.211835	−	0.977305i	$-0.432056\pi$
$929$	7.16411e24	0.423671	0.211835	+	0.977305i	$0.432056\pi$
$930$	0	0
$931$	−8.53894e23	−0.0495829
$932$	0	0
$933$	− 2.71875e25i	− 1.55016i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.68079e25i	0.924119i	0.886849	+	0.462059i	$0.152889\pi$
$937$	1.68079e25i	0.924119i	−0.886849	+	0.462059i	$0.847111\pi$
$938$	0	0
$939$	4.00920e25	2.16471
$940$	0	0
$941$	2.99139e25	1.58621	0.793105	−	0.609085i	$-0.208463\pi$
$941$	2.99139e25	1.58621	0.793105	+	0.609085i	$0.208463\pi$
$942$	0	0
$943$	− 7.34232e24i	− 0.382370i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.00925e25i	− 0.507020i	−0.967333	−	0.253510i	$-0.918415\pi$
$947$	− 1.00925e25i	− 0.507020i	0.967333	−	0.253510i	$-0.0815851\pi$
$948$	0	0
$949$	−2.29998e25	−1.13491
$950$	0	0
$951$	−4.79245e24	−0.232286
$952$	0	0
$953$	9.95941e24i	0.474181i	0.971488	+	0.237090i	$0.0761938\pi$
$953$	9.95941e24i	0.474181i	−0.971488	+	0.237090i	$0.923806\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.84941e25i	1.30919i
$958$	0	0
$959$	4.61963e25	2.08521
$960$	0	0
$961$	−1.74617e25	−0.774351
$962$	0	0
$963$	2.11655e25i	0.922158i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.62055e24i	0.194343i	0.995268	+	0.0971715i	$0.0309795\pi$
$967$	4.62055e24i	0.194343i	−0.995268	+	0.0971715i	$0.969020\pi$
$968$	0	0
$969$	−6.88872e23	−0.0284699
$970$	0	0
$971$	−1.50646e24	−0.0611780	−0.0305890	−	0.999532i	$-0.509738\pi$
$971$	−1.50646e24	−0.0611780	−0.0305890	+	0.999532i	$0.509738\pi$
$972$	0	0
$973$	2.91796e25i	1.16445i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.16292e25i	0.833560i	0.909007	+	0.416780i	$0.136841\pi$
$977$	2.16292e25i	0.833560i	−0.909007	+	0.416780i	$0.863159\pi$
$978$	0	0
$979$	−4.03087e25	−1.52667
$980$	0	0
$981$	−3.06145e25	−1.13957
$982$	0	0
$983$	1.09543e25i	0.400754i	0.979719	+	0.200377i	$0.0642167\pi$
$983$	1.09543e25i	0.400754i	−0.979719	+	0.200377i	$0.935783\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 7.21393e25i	− 2.54962i
$988$	0	0
$989$	−5.13831e24	−0.178506
$990$	0	0
$991$	−2.10377e25	−0.718410	−0.359205	−	0.933259i	$-0.616952\pi$
$991$	−2.10377e25	−0.718410	−0.359205	+	0.933259i	$0.616952\pi$
$992$	0	0
$993$	7.29913e25i	2.45020i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.15374e25i	0.698690i	0.936994	+	0.349345i	$0.113596\pi$
$997$	2.15374e25i	0.698690i	−0.936994	+	0.349345i	$0.886404\pi$
$998$	0	0
$999$	2.77233e23	0.00884175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.18.c.a.49.3		4
5.2	odd	4		4.18.a.a.1.2	✓	2
5.3	odd	4		100.18.a.b.1.1		2
5.4	even	2	inner	100.18.c.a.49.2		4
15.2	even	4		36.18.a.d.1.2		2
20.7	even	4		16.18.a.e.1.1		2
40.27	even	4		64.18.a.g.1.2		2
40.37	odd	4		64.18.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.18.a.a.1.2	✓	2	5.2	odd	4
16.18.a.e.1.1		2	20.7	even	4
36.18.a.d.1.2		2	15.2	even	4
64.18.a.g.1.2		2	40.27	even	4
64.18.a.l.1.1		2	40.37	odd	4
100.18.a.b.1.1		2	5.3	odd	4
100.18.c.a.49.2		4	5.4	even	2	inner
100.18.c.a.49.3		4	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:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	100.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15636.4i q^{3} -2.37095e7i q^{7} -1.15358e8 q^{9} +O(q^{10})\)	\(q+15636.4i q^{3} -2.37095e7i q^{7} -1.15358e8 q^{9} +1.33786e9 q^{11} -3.68345e9i q^{13} +1.70006e10i q^{17} +2.59141e9 q^{19} +3.70732e11 q^{21} -1.96178e11i q^{23} +2.15505e11i q^{27} +1.36209e12 q^{29} -2.25575e12 q^{31} +2.09194e13i q^{33} -1.28643e12i q^{37} +5.75961e13 q^{39} +3.74268e13 q^{41} -2.61921e13i q^{43} -1.94586e14i q^{47} -3.29509e14 q^{49} -2.65829e14 q^{51} -4.69769e14i q^{53} +4.05205e13i q^{57} -1.58789e15 q^{59} -2.16562e15 q^{61} +2.73508e15i q^{63} -1.54302e15i q^{67} +3.06753e15 q^{69} -4.51083e15 q^{71} -6.24410e15i q^{73} -3.17201e16i q^{77} +7.95599e14 q^{79} -1.82671e16 q^{81} +2.62699e16i q^{83} +2.12982e16i q^{87} -3.01291e16 q^{89} -8.73327e16 q^{91} -3.52719e16i q^{93} -3.38333e16i q^{97} -1.54333e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 898349364 q^{9}+O(q^{10}) \) 4 * q - 898349364 * q^9	\( 4 q - 898349364 q^{9} + 2519296560 q^{11} - 202267267664 q^{19} + 670860415104 q^{21} - 4674311165304 q^{29} + 557672226944 q^{31} + 216895994347296 q^{39} + 8333184631464 q^{41} - 199140653483748 q^{49} - 79913333836512 q^{51} - 56\!\cdots\!48 q^{59}+ \cdots - 25\!\cdots\!60 q^{99}+O(q^{100}) \) 4 * q - 898349364 * q^9 + 2519296560 * q^11 - 202267267664 * q^19 + 670860415104 * q^21 - 4674311165304 * q^29 + 557672226944 * q^31 + 216895994347296 * q^39 + 8333184631464 * q^41 - 199140653483748 * q^49 - 79913333836512 * q^51 - 5671808395627248 * q^59 - 135088069988872 * q^61 + 20376604200210048 * q^69 + 7764490986430752 * q^71 + 29968543068131008 * q^79 + 66760460575058916 * q^81 - 103818015917692776 * q^89 - 166910338800924224 * q^91 - 256446542618267760 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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