LMFDB - Embedded newform 64.9.c.b.63.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,9,Mod(63,64)]

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([1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("64.63");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	64.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:	$26.0722310439$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-39}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 10 $ x^2 - x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			63.1
Root			$0.500000 + 3.12250i$ of defining polynomial
Character	$\chi$	$=$	64.63
Dual form			64.9.c.b.63.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-99.9200i q^{3} -610.000 q^{5} +1398.88i q^{7} -3423.00 q^{9} +18485.2i q^{11} +5470.00 q^{13} +60951.2i q^{15} +73090.0 q^{17} +19484.4i q^{19} +139776. q^{21} -237210. i q^{23} -18525.0 q^{25} -313549. i q^{27} +128222. q^{29} -67945.6i q^{31} +1.84704e6 q^{33} -853317. i q^{35} +3.47203e6 q^{37} -546562. i q^{39} +2.14688e6 q^{41} +5.92815e6i q^{43} +2.08803e6 q^{45} +7.62629e6i q^{47} +3.80794e6 q^{49} -7.30315e6i q^{51} -824290. q^{53} -1.12760e7i q^{55} +1.94688e6 q^{57} +3.72552e6i q^{59} +1.47461e7 q^{61} -4.78836e6i q^{63} -3.33670e6 q^{65} -1.52567e7i q^{67} -2.37020e7 q^{69} -1.19604e6i q^{71} -5.72563e6 q^{73} +1.85102e6i q^{75} -2.58586e7 q^{77} +3.59132e7i q^{79} -5.37881e7 q^{81} +5.19603e7i q^{83} -4.45849e7 q^{85} -1.28119e7i q^{87} -8.33242e7 q^{89} +7.65187e6i q^{91} -6.78912e6 q^{93} -1.18855e7i q^{95} +1.20619e8 q^{97} -6.32748e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 1220 q^{5} - 6846 q^{9} + 10940 q^{13} + 146180 q^{17} + 279552 q^{21} - 37050 q^{25} + 256444 q^{29} + 3694080 q^{33} + 6944060 q^{37} + 4293764 q^{41} + 4176060 q^{45} + 7615874 q^{49} - 1648580 q^{53}+ \cdots + 241238020 q^{97}+O(q^{100}) $ 2 * q - 1220 * q^5 - 6846 * q^9 + 10940 * q^13 + 146180 * q^17 + 279552 * q^21 - 37050 * q^25 + 256444 * q^29 + 3694080 * q^33 + 6944060 * q^37 + 4293764 * q^41 + 4176060 * q^45 + 7615874 * q^49 - 1648580 * q^53 + 3893760 * q^57 + 29492156 * q^61 - 6673400 * q^65 - 47404032 * q^69 - 11451260 * q^73 - 51717120 * q^77 - 107576190 * q^81 - 89169800 * q^85 - 166648444 * q^89 - 13578240 * q^93 + 241238020 * q^97

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
$3$	− 99.9200i	− 1.23358i	−0.787128	−	0.616790i	$-0.788433\pi$
$3$	− 99.9200i	− 1.23358i	0.787128	−	0.616790i	$-0.211567\pi$
$4$	0	0
$5$	−610.000	−0.976000	−0.488000	−	0.872844i	$-0.662273\pi$
$5$	−610.000	−0.976000	−0.488000	+	0.872844i	$0.662273\pi$
$6$	0	0
$7$	1398.88i	0.582624i	0.956628	+	0.291312i	$0.0940917\pi$
$7$	1398.88i	0.582624i	−0.956628	+	0.291312i	$0.905908\pi$
$8$	0	0
$9$	−3423.00	−0.521719
$10$	0	0
$11$	18485.2i	1.26256i	0.775554	+	0.631282i	$0.217471\pi$
$11$	18485.2i	1.26256i	−0.775554	+	0.631282i	$0.782529\pi$
$12$	0	0
$13$	5470.00	0.191520	0.0957600	−	0.995404i	$-0.469472\pi$
$13$	5470.00	0.191520	0.0957600	+	0.995404i	$0.469472\pi$
$14$	0	0
$15$	60951.2i	1.20397i
$16$	0	0
$17$	73090.0	0.875109	0.437555	−	0.899192i	$-0.355845\pi$
$17$	73090.0	0.875109	0.437555	+	0.899192i	$0.355845\pi$
$18$	0	0
$19$	19484.4i	0.149511i	0.997202	+	0.0747554i	$0.0238176\pi$
$19$	19484.4i	0.149511i	−0.997202	+	0.0747554i	$0.976182\pi$
$20$	0	0
$21$	139776.	0.718713
$22$	0	0
$23$	− 237210.i	− 0.847660i	−0.905742	−	0.423830i	$-0.860685\pi$
$23$	− 237210.i	− 0.847660i	0.905742	−	0.423830i	$-0.139315\pi$
$24$	0	0
$25$	−18525.0	−0.0474240
$26$	0	0
$27$	− 313549.i	− 0.589997i
$28$	0	0
$29$	128222.	0.181289	0.0906443	−	0.995883i	$-0.471107\pi$
$29$	128222.	0.181289	0.0906443	+	0.995883i	$0.471107\pi$
$30$	0	0
$31$	− 67945.6i	− 0.0735723i	−0.999323	−	0.0367862i	$-0.988288\pi$
$31$	− 67945.6i	− 0.0735723i	0.999323	−	0.0367862i	$-0.0117120\pi$
$32$	0	0
$33$	1.84704e6	1.55747
$34$	0	0
$35$	− 853317.i	− 0.568641i
$36$	0	0
$37$	3.47203e6	1.85258	0.926289	−	0.376813i	$-0.122980\pi$
$37$	3.47203e6	1.85258	0.926289	+	0.376813i	$0.122980\pi$
$38$	0	0
$39$	− 546562.i	− 0.236255i
$40$	0	0
$41$	2.14688e6	0.759754	0.379877	−	0.925037i	$-0.375966\pi$
$41$	2.14688e6	0.759754	0.379877	+	0.925037i	$0.375966\pi$
$42$	0	0
$43$	5.92815e6i	1.73399i	0.498321	+	0.866993i	$0.333950\pi$
$43$	5.92815e6i	1.73399i	−0.498321	+	0.866993i	$0.666050\pi$
$44$	0	0
$45$	2.08803e6	0.509198
$46$	0	0
$47$	7.62629e6i	1.56287i	0.623989	+	0.781433i	$0.285511\pi$
$47$	7.62629e6i	1.56287i	−0.623989	+	0.781433i	$0.714489\pi$
$48$	0	0
$49$	3.80794e6	0.660550
$50$	0	0
$51$	− 7.30315e6i	− 1.07952i
$52$	0	0
$53$	−824290.	−0.104466	−0.0522332	−	0.998635i	$-0.516634\pi$
$53$	−824290.	−0.104466	−0.0522332	+	0.998635i	$0.516634\pi$
$54$	0	0
$55$	− 1.12760e7i	− 1.23226i
$56$	0	0
$57$	1.94688e6	0.184433
$58$	0	0
$59$	3.72552e6i	0.307453i	0.988113	+	0.153726i	$0.0491274\pi$
$59$	3.72552e6i	0.307453i	−0.988113	+	0.153726i	$0.950873\pi$
$60$	0	0
$61$	1.47461e7	1.06502	0.532509	−	0.846424i	$-0.321249\pi$
$61$	1.47461e7	1.06502	0.532509	+	0.846424i	$0.321249\pi$
$62$	0	0
$63$	− 4.78836e6i	− 0.303966i
$64$	0	0
$65$	−3.33670e6	−0.186923
$66$	0	0
$67$	− 1.52567e7i	− 0.757113i	−0.925578	−	0.378557i	$-0.876421\pi$
$67$	− 1.52567e7i	− 0.757113i	0.925578	−	0.378557i	$-0.123579\pi$
$68$	0	0
$69$	−2.37020e7	−1.04566
$70$	0	0
$71$	− 1.19604e6i	− 0.0470666i	−0.999723	−	0.0235333i	$-0.992508\pi$
$71$	− 1.19604e6i	− 0.0470666i	0.999723	−	0.0235333i	$-0.00749158\pi$
$72$	0	0
$73$	−5.72563e6	−0.201619	−0.100810	−	0.994906i	$-0.532143\pi$
$73$	−5.72563e6	−0.201619	−0.100810	+	0.994906i	$0.532143\pi$
$74$	0	0
$75$	1.85102e6i	0.0585013i
$76$	0	0
$77$	−2.58586e7	−0.735600
$78$	0	0
$79$	3.59132e7i	0.922032i	0.887392	+	0.461016i	$0.152515\pi$
$79$	3.59132e7i	0.922032i	−0.887392	+	0.461016i	$0.847485\pi$
$80$	0	0
$81$	−5.37881e7	−1.24953
$82$	0	0
$83$	5.19603e7i	1.09486i	0.836851	+	0.547431i	$0.184394\pi$
$83$	5.19603e7i	1.09486i	−0.836851	+	0.547431i	$0.815606\pi$
$84$	0	0
$85$	−4.45849e7	−0.854107
$86$	0	0
$87$	− 1.28119e7i	− 0.223634i
$88$	0	0
$89$	−8.33242e7	−1.32804	−0.664020	−	0.747715i	$-0.731151\pi$
$89$	−8.33242e7	−1.32804	−0.664020	+	0.747715i	$0.731151\pi$
$90$	0	0
$91$	7.65187e6i	0.111584i
$92$	0	0
$93$	−6.78912e6	−0.0907573
$94$	0	0
$95$	− 1.18855e7i	− 0.145923i
$96$	0	0
$97$	1.20619e8	1.36248	0.681238	−	0.732062i	$-0.261442\pi$
$97$	1.20619e8	1.36248	0.681238	+	0.732062i	$0.261442\pi$
$98$	0	0
$99$	− 6.32748e7i	− 0.658704i
$100$	0	0
$101$	−2.77246e7	−0.266428	−0.133214	−	0.991087i	$-0.542530\pi$
$101$	−2.77246e7	−0.266428	−0.133214	+	0.991087i	$0.542530\pi$
$102$	0	0
$103$	1.04501e8i	0.928477i	0.885710	+	0.464238i	$0.153672\pi$
$103$	1.04501e8i	0.928477i	−0.885710	+	0.464238i	$0.846328\pi$
$104$	0	0
$105$	−8.52634e7	−0.701464
$106$	0	0
$107$	− 1.00328e8i	− 0.765394i	−0.923874	−	0.382697i	$-0.874995\pi$
$107$	− 1.00328e8i	− 0.765394i	0.923874	−	0.382697i	$-0.125005\pi$
$108$	0	0
$109$	5.90716e7	0.418478	0.209239	−	0.977865i	$-0.432901\pi$
$109$	5.90716e7	0.418478	0.209239	+	0.977865i	$0.432901\pi$
$110$	0	0
$111$	− 3.46925e8i	− 2.28530i
$112$	0	0
$113$	5.50849e7	0.337846	0.168923	−	0.985629i	$-0.445971\pi$
$113$	5.50849e7	0.337846	0.168923	+	0.985629i	$0.445971\pi$
$114$	0	0
$115$	1.44698e8i	0.827316i
$116$	0	0
$117$	−1.87238e7	−0.0999196
$118$	0	0
$119$	1.02244e8i	0.509859i
$120$	0	0
$121$	−1.27344e8	−0.594067
$122$	0	0
$123$	− 2.14516e8i	− 0.937217i
$124$	0	0
$125$	2.49581e8	1.02229
$126$	0	0
$127$	2.57160e8i	0.988529i	0.869312	+	0.494264i	$0.164562\pi$
$127$	2.57160e8i	0.988529i	−0.869312	+	0.494264i	$0.835438\pi$
$128$	0	0
$129$	5.92341e8	2.13901
$130$	0	0
$131$	3.12175e8i	1.06002i	0.847992	+	0.530009i	$0.177812\pi$
$131$	3.12175e8i	1.06002i	−0.847992	+	0.530009i	$0.822188\pi$
$132$	0	0
$133$	−2.72563e7	−0.0871085
$134$	0	0
$135$	1.91265e8i	0.575838i
$136$	0	0
$137$	2.21980e8	0.630132	0.315066	−	0.949070i	$-0.397973\pi$
$137$	2.21980e8	0.630132	0.315066	+	0.949070i	$0.397973\pi$
$138$	0	0
$139$	2.95030e8i	0.790328i	0.918611	+	0.395164i	$0.129312\pi$
$139$	2.95030e8i	0.790328i	−0.918611	+	0.395164i	$0.870688\pi$
$140$	0	0
$141$	7.62019e8	1.92792
$142$	0	0
$143$	1.01114e8i	0.241806i
$144$	0	0
$145$	−7.82154e7	−0.176938
$146$	0	0
$147$	− 3.80489e8i	− 0.814841i
$148$	0	0
$149$	−4.03603e8	−0.818859	−0.409429	−	0.912342i	$-0.634272\pi$
$149$	−4.03603e8	−0.818859	−0.409429	+	0.912342i	$0.634272\pi$
$150$	0	0
$151$	− 8.36985e8i	− 1.60994i	−0.593316	−	0.804970i	$-0.702181\pi$
$151$	− 8.36985e8i	− 1.60994i	0.593316	−	0.804970i	$-0.297819\pi$
$152$	0	0
$153$	−2.50187e8	−0.456561
$154$	0	0
$155$	4.14468e7i	0.0718066i
$156$	0	0
$157$	2.71319e8	0.446561	0.223281	−	0.974754i	$-0.428323\pi$
$157$	2.71319e8	0.446561	0.223281	+	0.974754i	$0.428323\pi$
$158$	0	0
$159$	8.23630e7i	0.128868i
$160$	0	0
$161$	3.31828e8	0.493867
$162$	0	0
$163$	− 5.78509e8i	− 0.819520i	−0.912193	−	0.409760i	$-0.865612\pi$
$163$	− 5.78509e8i	− 0.819520i	0.912193	−	0.409760i	$-0.134388\pi$
$164$	0	0
$165$	−1.12669e9	−1.52009
$166$	0	0
$167$	4.68118e8i	0.601852i	0.953647	+	0.300926i	$0.0972958\pi$
$167$	4.68118e8i	0.601852i	−0.953647	+	0.300926i	$0.902704\pi$
$168$	0	0
$169$	−7.85810e8	−0.963320
$170$	0	0
$171$	− 6.66951e7i	− 0.0780026i
$172$	0	0
$173$	2.06197e8	0.230196	0.115098	−	0.993354i	$-0.463282\pi$
$173$	2.06197e8	0.230196	0.115098	+	0.993354i	$0.463282\pi$
$174$	0	0
$175$	− 2.59142e7i	− 0.0276303i
$176$	0	0
$177$	3.72253e8	0.379268
$178$	0	0
$179$	− 1.41911e8i	− 0.138230i	−0.997609	−	0.0691152i	$-0.977982\pi$
$179$	− 1.41911e8i	− 0.138230i	0.997609	−	0.0691152i	$-0.0220176\pi$
$180$	0	0
$181$	−4.82566e8	−0.449616	−0.224808	−	0.974403i	$-0.572176\pi$
$181$	−4.82566e8	−0.449616	−0.224808	+	0.974403i	$0.572176\pi$
$182$	0	0
$183$	− 1.47343e9i	− 1.31379i
$184$	0	0
$185$	−2.11794e9	−1.80812
$186$	0	0
$187$	1.35108e9i	1.10488i
$188$	0	0
$189$	4.38617e8	0.343747
$190$	0	0
$191$	9.92461e8i	0.745727i	0.927886	+	0.372864i	$0.121624\pi$
$191$	9.92461e8i	0.745727i	−0.927886	+	0.372864i	$0.878376\pi$
$192$	0	0
$193$	1.17593e9	0.847526	0.423763	−	0.905773i	$-0.360709\pi$
$193$	1.17593e9	0.847526	0.423763	+	0.905773i	$0.360709\pi$
$194$	0	0
$195$	3.33403e8i	0.230585i
$196$	0	0
$197$	−1.70538e9	−1.13229	−0.566144	−	0.824306i	$-0.691565\pi$
$197$	−1.70538e9	−1.13229	−0.566144	+	0.824306i	$0.691565\pi$
$198$	0	0
$199$	2.49036e9i	1.58800i	0.607919	+	0.793999i	$0.292004\pi$
$199$	2.49036e9i	1.58800i	−0.607919	+	0.793999i	$0.707996\pi$
$200$	0	0
$201$	−1.52445e9	−0.933960
$202$	0	0
$203$	1.79367e8i	0.105623i
$204$	0	0
$205$	−1.30960e9	−0.741519
$206$	0	0
$207$	8.11970e8i	0.442241i
$208$	0	0
$209$	−3.60173e8	−0.188767
$210$	0	0
$211$	− 1.46774e9i	− 0.740491i	−0.928934	−	0.370245i	$-0.879274\pi$
$211$	− 1.46774e9i	− 0.740491i	0.928934	−	0.370245i	$-0.120726\pi$
$212$	0	0
$213$	−1.19508e8	−0.0580604
$214$	0	0
$215$	− 3.61617e9i	− 1.69237i
$216$	0	0
$217$	9.50477e7	0.0428650
$218$	0	0
$219$	5.72105e8i	0.248713i
$220$	0	0
$221$	3.99802e8	0.167601
$222$	0	0
$223$	− 1.47920e9i	− 0.598147i	−0.954230	−	0.299073i	$-0.903322\pi$
$223$	− 1.47920e9i	− 0.598147i	0.954230	−	0.299073i	$-0.0966776\pi$
$224$	0	0
$225$	6.34111e7	0.0247420
$226$	0	0
$227$	7.50054e8i	0.282481i	0.989975	+	0.141241i	$0.0451091\pi$
$227$	7.50054e8i	0.282481i	−0.989975	+	0.141241i	$0.954891\pi$
$228$	0	0
$229$	2.84784e9	1.03556	0.517778	−	0.855515i	$-0.326759\pi$
$229$	2.84784e9	1.03556	0.517778	+	0.855515i	$0.326759\pi$
$230$	0	0
$231$	2.58379e9i	0.907421i
$232$	0	0
$233$	2.20621e8	0.0748553	0.0374276	−	0.999299i	$-0.488084\pi$
$233$	2.20621e8	0.0748553	0.0374276	+	0.999299i	$0.488084\pi$
$234$	0	0
$235$	− 4.65204e9i	− 1.52536i
$236$	0	0
$237$	3.58845e9	1.13740
$238$	0	0
$239$	− 4.04493e9i	− 1.23971i	−0.784717	−	0.619855i	$-0.787192\pi$
$239$	− 4.04493e9i	− 1.23971i	0.784717	−	0.619855i	$-0.212808\pi$
$240$	0	0
$241$	6.17983e9	1.83193	0.915964	−	0.401260i	$-0.131427\pi$
$241$	6.17983e9	1.83193	0.915964	+	0.401260i	$0.131427\pi$
$242$	0	0
$243$	3.31731e9i	0.951395i
$244$	0	0
$245$	−2.32284e9	−0.644696
$246$	0	0
$247$	1.06580e8i	0.0286343i
$248$	0	0
$249$	5.19187e9	1.35060
$250$	0	0
$251$	− 5.21367e9i	− 1.31356i	−0.754084	−	0.656778i	$-0.771919\pi$
$251$	− 5.21367e9i	− 1.31356i	0.754084	−	0.656778i	$-0.228081\pi$
$252$	0	0
$253$	4.38487e9	1.07022
$254$	0	0
$255$	4.45492e9i	1.05361i
$256$	0	0
$257$	−6.13693e9	−1.40676	−0.703378	−	0.710816i	$-0.748326\pi$
$257$	−6.13693e9	−1.40676	−0.703378	+	0.710816i	$0.748326\pi$
$258$	0	0
$259$	4.85695e9i	1.07936i
$260$	0	0
$261$	−4.38904e8	−0.0945818
$262$	0	0
$263$	6.96916e9i	1.45666i	0.685228	+	0.728329i	$0.259703\pi$
$263$	6.96916e9i	1.45666i	−0.685228	+	0.728329i	$0.740297\pi$
$264$	0	0
$265$	5.02817e8	0.101959
$266$	0	0
$267$	8.32575e9i	1.63824i
$268$	0	0
$269$	−2.70720e9	−0.517025	−0.258513	−	0.966008i	$-0.583232\pi$
$269$	−2.70720e9	−0.517025	−0.258513	+	0.966008i	$0.583232\pi$
$270$	0	0
$271$	− 7.99032e9i	− 1.48145i	−0.671808	−	0.740725i	$-0.734482\pi$
$271$	− 7.99032e9i	− 1.48145i	0.671808	−	0.740725i	$-0.265518\pi$
$272$	0	0
$273$	7.64575e8	0.137648
$274$	0	0
$275$	− 3.42438e8i	− 0.0598758i
$276$	0	0
$277$	8.22965e9	1.39786	0.698928	−	0.715192i	$-0.253661\pi$
$277$	8.22965e9	1.39786	0.698928	+	0.715192i	$0.253661\pi$
$278$	0	0
$279$	2.32578e8i	0.0383841i
$280$	0	0
$281$	3.08105e9	0.494167	0.247083	−	0.968994i	$-0.420528\pi$
$281$	3.08105e9	0.494167	0.247083	+	0.968994i	$0.420528\pi$
$282$	0	0
$283$	− 1.17112e9i	− 0.182582i	−0.995824	−	0.0912908i	$-0.970901\pi$
$283$	− 1.17112e9i	− 0.182582i	0.995824	−	0.0912908i	$-0.0290993\pi$
$284$	0	0
$285$	−1.18760e9	−0.180007
$286$	0	0
$287$	3.00323e9i	0.442650i
$288$	0	0
$289$	−1.63361e9	−0.234184
$290$	0	0
$291$	− 1.20522e10i	− 1.68072i
$292$	0	0
$293$	−4.80980e9	−0.652614	−0.326307	−	0.945264i	$-0.605804\pi$
$293$	−4.80980e9	−0.652614	−0.326307	+	0.945264i	$0.605804\pi$
$294$	0	0
$295$	− 2.27256e9i	− 0.300074i
$296$	0	0
$297$	5.79601e9	0.744909
$298$	0	0
$299$	− 1.29754e9i	− 0.162344i
$300$	0	0
$301$	−8.29277e9	−1.01026
$302$	0	0
$303$	2.77025e9i	0.328661i
$304$	0	0
$305$	−8.99511e9	−1.03946
$306$	0	0
$307$	3.49176e9i	0.393089i	0.980495	+	0.196545i	$0.0629721\pi$
$307$	3.49176e9i	0.393089i	−0.980495	+	0.196545i	$0.937028\pi$
$308$	0	0
$309$	1.04417e10	1.14535
$310$	0	0
$311$	− 1.29807e10i	− 1.38757i	−0.720182	−	0.693785i	$-0.755942\pi$
$311$	− 1.29807e10i	− 1.38757i	0.720182	−	0.693785i	$-0.244058\pi$
$312$	0	0
$313$	−6.31165e9	−0.657606	−0.328803	−	0.944399i	$-0.606645\pi$
$313$	−6.31165e9	−0.657606	−0.328803	+	0.944399i	$0.606645\pi$
$314$	0	0
$315$	2.92090e9i	0.296671i
$316$	0	0
$317$	−1.65902e10	−1.64291	−0.821455	−	0.570273i	$-0.806837\pi$
$317$	−1.65902e10	−1.64291	−0.821455	+	0.570273i	$0.806837\pi$
$318$	0	0
$319$	2.37021e9i	0.228888i
$320$	0	0
$321$	−1.00247e10	−0.944175
$322$	0	0
$323$	1.42411e9i	0.130838i
$324$	0	0
$325$	−1.01332e8	−0.00908264
$326$	0	0
$327$	− 5.90243e9i	− 0.516226i
$328$	0	0
$329$	−1.06683e10	−0.910563
$330$	0	0
$331$	− 5.48640e9i	− 0.457062i	−0.973537	−	0.228531i	$-0.926608\pi$
$331$	− 5.48640e9i	− 0.457062i	0.973537	−	0.228531i	$-0.0733922\pi$
$332$	0	0
$333$	−1.18848e10	−0.966526
$334$	0	0
$335$	9.30657e9i	0.738942i
$336$	0	0
$337$	−3.56226e8	−0.0276189	−0.0138095	−	0.999905i	$-0.504396\pi$
$337$	−3.56226e8	−0.0276189	−0.0138095	+	0.999905i	$0.504396\pi$
$338$	0	0
$339$	− 5.50408e9i	− 0.416760i
$340$	0	0
$341$	1.25599e9	0.0928897
$342$	0	0
$343$	1.33911e10i	0.967476i
$344$	0	0
$345$	1.44582e10	1.02056
$346$	0	0
$347$	1.59731e10i	1.10172i	0.834599	+	0.550859i	$0.185700\pi$
$347$	1.59731e10i	1.10172i	−0.834599	+	0.550859i	$0.814300\pi$
$348$	0	0
$349$	−1.03634e10	−0.698553	−0.349277	−	0.937020i	$-0.613573\pi$
$349$	−1.03634e10	−0.698553	−0.349277	+	0.937020i	$0.613573\pi$
$350$	0	0
$351$	− 1.71511e9i	− 0.112996i
$352$	0	0
$353$	−1.30979e10	−0.843536	−0.421768	−	0.906704i	$-0.638590\pi$
$353$	−1.30979e10	−0.843536	−0.421768	+	0.906704i	$0.638590\pi$
$354$	0	0
$355$	7.29586e8i	0.0459370i
$356$	0	0
$357$	1.02162e10	0.628952
$358$	0	0
$359$	3.31454e9i	0.199547i	0.995010	+	0.0997737i	$0.0318119\pi$
$359$	3.31454e9i	0.199547i	−0.995010	+	0.0997737i	$0.968188\pi$
$360$	0	0
$361$	1.66039e10	0.977647
$362$	0	0
$363$	1.27242e10i	0.732829i
$364$	0	0
$365$	3.49263e9	0.196780
$366$	0	0
$367$	1.96628e10i	1.08388i	0.840418	+	0.541939i	$0.182309\pi$
$367$	1.96628e10i	1.08388i	−0.840418	+	0.541939i	$0.817691\pi$
$368$	0	0
$369$	−7.34878e9	−0.396378
$370$	0	0
$371$	− 1.15308e9i	− 0.0608646i
$372$	0	0
$373$	2.10063e10	1.08521	0.542606	−	0.839987i	$-0.317438\pi$
$373$	2.10063e10	1.08521	0.542606	+	0.839987i	$0.317438\pi$
$374$	0	0
$375$	− 2.49382e10i	− 1.26107i
$376$	0	0
$377$	7.01374e8	0.0347204
$378$	0	0
$379$	− 3.04816e9i	− 0.147734i	−0.997268	−	0.0738670i	$-0.976466\pi$
$379$	− 3.04816e9i	− 0.147734i	0.997268	−	0.0738670i	$-0.0235340\pi$
$380$	0	0
$381$	2.56955e10	1.21943
$382$	0	0
$383$	− 2.23357e10i	− 1.03802i	−0.854770	−	0.519008i	$-0.826302\pi$
$383$	− 2.23357e10i	− 1.03802i	0.854770	−	0.519008i	$-0.173698\pi$
$384$	0	0
$385$	1.57737e10	0.717945
$386$	0	0
$387$	− 2.02921e10i	− 0.904654i
$388$	0	0
$389$	−3.13680e10	−1.36990	−0.684948	−	0.728592i	$-0.740175\pi$
$389$	−3.13680e10	−1.36990	−0.684948	+	0.728592i	$0.740175\pi$
$390$	0	0
$391$	− 1.73377e10i	− 0.741795i
$392$	0	0
$393$	3.11926e10	1.30762
$394$	0	0
$395$	− 2.19071e10i	− 0.899904i
$396$	0	0
$397$	−7.65788e9	−0.308281	−0.154140	−	0.988049i	$-0.549261\pi$
$397$	−7.65788e9	−0.308281	−0.154140	+	0.988049i	$0.549261\pi$
$398$	0	0
$399$	2.72345e9i	0.107455i
$400$	0	0
$401$	−3.26120e10	−1.26125	−0.630623	−	0.776089i	$-0.717201\pi$
$401$	−3.26120e10	−1.26125	−0.630623	+	0.776089i	$0.717201\pi$
$402$	0	0
$403$	− 3.71662e8i	− 0.0140906i
$404$	0	0
$405$	3.28107e10	1.21954
$406$	0	0
$407$	6.41811e10i	2.33900i
$408$	0	0
$409$	2.26168e10	0.808236	0.404118	−	0.914707i	$-0.367578\pi$
$409$	2.26168e10	0.808236	0.404118	+	0.914707i	$0.367578\pi$
$410$	0	0
$411$	− 2.21802e10i	− 0.777318i
$412$	0	0
$413$	−5.21155e9	−0.179129
$414$	0	0
$415$	− 3.16958e10i	− 1.06858i
$416$	0	0
$417$	2.94794e10	0.974932
$418$	0	0
$419$	4.94503e10i	1.60440i	0.597054	+	0.802201i	$0.296338\pi$
$419$	4.94503e10i	1.60440i	−0.597054	+	0.802201i	$0.703662\pi$
$420$	0	0
$421$	3.34077e10	1.06345	0.531726	−	0.846916i	$-0.321543\pi$
$421$	3.34077e10	1.06345	0.531726	+	0.846916i	$0.321543\pi$
$422$	0	0
$423$	− 2.61048e10i	− 0.815378i
$424$	0	0
$425$	−1.35399e9	−0.0415012
$426$	0	0
$427$	2.06280e10i	0.620505i
$428$	0	0
$429$	1.01033e10	0.298287
$430$	0	0
$431$	− 3.06956e10i	− 0.889544i	−0.895644	−	0.444772i	$-0.853285\pi$
$431$	− 3.06956e10i	− 0.889544i	0.895644	−	0.444772i	$-0.146715\pi$
$432$	0	0
$433$	2.88433e9	0.0820529	0.0410265	−	0.999158i	$-0.486937\pi$
$433$	2.88433e9	0.0820529	0.0410265	+	0.999158i	$0.486937\pi$
$434$	0	0
$435$	7.81528e9i	0.218267i
$436$	0	0
$437$	4.62189e9	0.126734
$438$	0	0
$439$	6.92422e10i	1.86429i	0.362088	+	0.932144i	$0.382064\pi$
$439$	6.92422e10i	1.86429i	−0.362088	+	0.932144i	$0.617936\pi$
$440$	0	0
$441$	−1.30346e10	−0.344621
$442$	0	0
$443$	− 2.06609e10i	− 0.536455i	−0.963356	−	0.268228i	$-0.913562\pi$
$443$	− 2.06609e10i	− 0.536455i	0.963356	−	0.268228i	$-0.0864379\pi$
$444$	0	0
$445$	5.08278e10	1.29617
$446$	0	0
$447$	4.03280e10i	1.01013i
$448$	0	0
$449$	2.11092e10	0.519382	0.259691	−	0.965692i	$-0.416379\pi$
$449$	2.11092e10	0.519382	0.259691	+	0.965692i	$0.416379\pi$
$450$	0	0
$451$	3.96855e10i	0.959237i
$452$	0	0
$453$	−8.36315e10	−1.98599
$454$	0	0
$455$	− 4.66764e9i	− 0.108906i
$456$	0	0
$457$	−2.06831e10	−0.474188	−0.237094	−	0.971487i	$-0.576195\pi$
$457$	−2.06831e10	−0.474188	−0.237094	+	0.971487i	$0.576195\pi$
$458$	0	0
$459$	− 2.29173e10i	− 0.516312i
$460$	0	0
$461$	−7.65072e10	−1.69394	−0.846971	−	0.531640i	$-0.821576\pi$
$461$	−7.65072e10	−1.69394	−0.846971	+	0.531640i	$0.821576\pi$
$462$	0	0
$463$	3.41303e9i	0.0742704i	0.999310	+	0.0371352i	$0.0118232\pi$
$463$	3.41303e9i	0.0742704i	−0.999310	+	0.0371352i	$0.988177\pi$
$464$	0	0
$465$	4.14136e9	0.0885791
$466$	0	0
$467$	− 1.92903e10i	− 0.405576i	−0.979223	−	0.202788i	$-0.935000\pi$
$467$	− 1.92903e10i	− 0.405576i	0.979223	−	0.202788i	$-0.0650002\pi$
$468$	0	0
$469$	2.13423e10	0.441112
$470$	0	0
$471$	− 2.71102e10i	− 0.550869i
$472$	0	0
$473$	−1.09583e11	−2.18927
$474$	0	0
$475$	− 3.60948e8i	− 0.00709040i
$476$	0	0
$477$	2.82154e9	0.0545021
$478$	0	0
$479$	− 2.43887e10i	− 0.463282i	−0.972801	−	0.231641i	$-0.925590\pi$
$479$	− 2.43887e10i	− 0.463282i	0.972801	−	0.231641i	$-0.0744095\pi$
$480$	0	0
$481$	1.89920e10	0.354806
$482$	0	0
$483$	− 3.31563e10i	− 0.609224i
$484$	0	0
$485$	−7.35776e10	−1.32978
$486$	0	0
$487$	− 9.30801e10i	− 1.65478i	−0.561626	−	0.827391i	$-0.689824\pi$
$487$	− 9.30801e10i	− 1.65478i	0.561626	−	0.827391i	$-0.310176\pi$
$488$	0	0
$489$	−5.78046e10	−1.01094
$490$	0	0
$491$	2.12850e9i	0.0366225i	0.999832	+	0.0183113i	$0.00582898\pi$
$491$	2.12850e9i	0.0366225i	−0.999832	+	0.0183113i	$0.994171\pi$
$492$	0	0
$493$	9.37175e9	0.158647
$494$	0	0
$495$	3.85976e10i	0.642895i
$496$	0	0
$497$	1.67312e9	0.0274221
$498$	0	0
$499$	− 1.04101e10i	− 0.167901i	−0.996470	−	0.0839503i	$-0.973246\pi$
$499$	− 1.04101e10i	− 0.167901i	0.996470	−	0.0839503i	$-0.0267537\pi$
$500$	0	0
$501$	4.67744e10	0.742433
$502$	0	0
$503$	3.93019e10i	0.613962i	0.951716	+	0.306981i	$0.0993188\pi$
$503$	3.93019e10i	0.613962i	−0.951716	+	0.306981i	$0.900681\pi$
$504$	0	0
$505$	1.69120e10	0.260034
$506$	0	0
$507$	7.85181e10i	1.18833i
$508$	0	0
$509$	3.25113e10	0.484354	0.242177	−	0.970232i	$-0.422139\pi$
$509$	3.25113e10	0.484354	0.242177	+	0.970232i	$0.422139\pi$
$510$	0	0
$511$	− 8.00947e9i	− 0.117468i
$512$	0	0
$513$	6.10931e9	0.0882110
$514$	0	0
$515$	− 6.37455e10i	− 0.906194i
$516$	0	0
$517$	−1.40973e11	−1.97322
$518$	0	0
$519$	− 2.06032e10i	− 0.283965i
$520$	0	0
$521$	1.84550e9	0.0250475	0.0125237	−	0.999922i	$-0.496013\pi$
$521$	1.84550e9	0.0250475	0.0125237	+	0.999922i	$0.496013\pi$
$522$	0	0
$523$	− 6.23770e10i	− 0.833715i	−0.908972	−	0.416858i	$-0.863131\pi$
$523$	− 6.23770e10i	− 0.833715i	0.908972	−	0.416858i	$-0.136869\pi$
$524$	0	0
$525$	−2.58935e9	−0.0340842
$526$	0	0
$527$	− 4.96614e9i	− 0.0643838i
$528$	0	0
$529$	2.20424e10	0.281473
$530$	0	0
$531$	− 1.27524e10i	− 0.160404i
$532$	0	0
$533$	1.17434e10	0.145508
$534$	0	0
$535$	6.11998e10i	0.747025i
$536$	0	0
$537$	−1.41797e10	−0.170518
$538$	0	0
$539$	7.03905e10i	0.833986i
$540$	0	0
$541$	7.45917e10	0.870766	0.435383	−	0.900245i	$-0.356613\pi$
$541$	7.45917e10	0.870766	0.435383	+	0.900245i	$0.356613\pi$
$542$	0	0
$543$	4.82179e10i	0.554638i
$544$	0	0
$545$	−3.60337e10	−0.408435
$546$	0	0
$547$	− 1.41531e9i	− 0.0158089i	−0.999969	−	0.00790445i	$-0.997484\pi$
$547$	− 1.41531e9i	− 0.0158089i	0.999969	−	0.00790445i	$-0.00251609\pi$
$548$	0	0
$549$	−5.04758e10	−0.555641
$550$	0	0
$551$	2.49833e9i	0.0271046i
$552$	0	0
$553$	−5.02383e10	−0.537198
$554$	0	0
$555$	2.11624e11i	2.23046i
$556$	0	0
$557$	1.37543e11	1.42895	0.714475	−	0.699661i	$-0.246666\pi$
$557$	1.37543e11	1.42895	0.714475	+	0.699661i	$0.246666\pi$
$558$	0	0
$559$	3.24270e10i	0.332093i
$560$	0	0
$561$	1.35000e11	1.36296
$562$	0	0
$563$	1.06415e11i	1.05918i	0.848255	+	0.529589i	$0.177654\pi$
$563$	1.06415e11i	1.05918i	−0.848255	+	0.529589i	$0.822346\pi$
$564$	0	0
$565$	−3.36018e10	−0.329738
$566$	0	0
$567$	− 7.52431e10i	− 0.728005i
$568$	0	0
$569$	4.02429e10	0.383919	0.191960	−	0.981403i	$-0.438516\pi$
$569$	4.02429e10	0.383919	0.191960	+	0.981403i	$0.438516\pi$
$570$	0	0
$571$	− 1.50341e11i	− 1.41427i	−0.707077	−	0.707137i	$-0.749986\pi$
$571$	− 1.50341e11i	− 1.41427i	0.707077	−	0.707137i	$-0.250014\pi$
$572$	0	0
$573$	9.91667e10	0.919914
$574$	0	0
$575$	4.39432e9i	0.0401994i
$576$	0	0
$577$	4.96477e9	0.0447915	0.0223958	−	0.999749i	$-0.492871\pi$
$577$	4.96477e9	0.0447915	0.0223958	+	0.999749i	$0.492871\pi$
$578$	0	0
$579$	− 1.17499e11i	− 1.04549i
$580$	0	0
$581$	−7.26862e10	−0.637892
$582$	0	0
$583$	− 1.52372e10i	− 0.131895i
$584$	0	0
$585$	1.14215e10	0.0975216
$586$	0	0
$587$	− 1.53440e11i	− 1.29237i	−0.763181	−	0.646185i	$-0.776363\pi$
$587$	− 1.53440e11i	− 1.29237i	0.763181	−	0.646185i	$-0.223637\pi$
$588$	0	0
$589$	1.32388e9	0.0109999
$590$	0	0
$591$	1.70402e11i	1.39677i
$592$	0	0
$593$	2.06036e11	1.66619	0.833094	−	0.553131i	$-0.186567\pi$
$593$	2.06036e11	1.66619	0.833094	+	0.553131i	$0.186567\pi$
$594$	0	0
$595$	− 6.23689e10i	− 0.497623i
$596$	0	0
$597$	2.48837e11	1.95892
$598$	0	0
$599$	− 2.30634e11i	− 1.79150i	−0.444558	−	0.895750i	$-0.646639\pi$
$599$	− 2.30634e11i	− 1.79150i	0.444558	−	0.895750i	$-0.353361\pi$
$600$	0	0
$601$	1.01422e11	0.777382	0.388691	−	0.921368i	$-0.372927\pi$
$601$	1.01422e11	0.777382	0.388691	+	0.921368i	$0.372927\pi$
$602$	0	0
$603$	5.22236e10i	0.395001i
$604$	0	0
$605$	7.76795e10	0.579809
$606$	0	0
$607$	− 1.97883e11i	− 1.45765i	−0.684700	−	0.728825i	$-0.740067\pi$
$607$	− 1.97883e11i	− 1.45765i	0.684700	−	0.728825i	$-0.259933\pi$
$608$	0	0
$609$	1.79224e10	0.130294
$610$	0	0
$611$	4.17158e10i	0.299320i
$612$	0	0
$613$	−1.27158e11	−0.900538	−0.450269	−	0.892893i	$-0.648672\pi$
$613$	−1.27158e11	−0.900538	−0.450269	+	0.892893i	$0.648672\pi$
$614$	0	0
$615$	1.30855e11i	0.914723i
$616$	0	0
$617$	−5.06702e10	−0.349632	−0.174816	−	0.984601i	$-0.555933\pi$
$617$	−5.06702e10	−0.349632	−0.174816	+	0.984601i	$0.555933\pi$
$618$	0	0
$619$	7.06748e10i	0.481395i	0.970600	+	0.240698i	$0.0773762\pi$
$619$	7.06748e10i	0.481395i	−0.970600	+	0.240698i	$0.922624\pi$
$620$	0	0
$621$	−7.43769e10	−0.500117
$622$	0	0
$623$	− 1.16561e11i	− 0.773748i
$624$	0	0
$625$	−1.45008e11	−0.950327
$626$	0	0
$627$	3.59885e10i	0.232859i
$628$	0	0
$629$	2.53771e11	1.62121
$630$	0	0
$631$	1.65273e11i	1.04252i	0.853399	+	0.521259i	$0.174537\pi$
$631$	1.65273e11i	1.04252i	−0.853399	+	0.521259i	$0.825463\pi$
$632$	0	0
$633$	−1.46657e11	−0.913454
$634$	0	0
$635$	− 1.56868e11i	− 0.964804i
$636$	0	0
$637$	2.08294e10	0.126508
$638$	0	0
$639$	4.09405e9i	0.0245556i
$640$	0	0
$641$	1.12013e11	0.663490	0.331745	−	0.943369i	$-0.392363\pi$
$641$	1.12013e11	0.663490	0.331745	+	0.943369i	$0.392363\pi$
$642$	0	0
$643$	2.65913e11i	1.55559i	0.628518	+	0.777795i	$0.283662\pi$
$643$	2.65913e11i	1.55559i	−0.628518	+	0.777795i	$0.716338\pi$
$644$	0	0
$645$	−3.61328e11	−2.08767
$646$	0	0
$647$	2.71996e11i	1.55219i	0.630614	+	0.776097i	$0.282803\pi$
$647$	2.71996e11i	1.55219i	−0.630614	+	0.776097i	$0.717197\pi$
$648$	0	0
$649$	−6.88669e10	−0.388179
$650$	0	0
$651$	− 9.49716e9i	− 0.0528774i
$652$	0	0
$653$	−3.03789e11	−1.67078	−0.835391	−	0.549656i	$-0.814759\pi$
$653$	−3.03789e11	−1.67078	−0.835391	+	0.549656i	$0.814759\pi$
$654$	0	0
$655$	− 1.90427e11i	− 1.03458i
$656$	0	0
$657$	1.95988e10	0.105189
$658$	0	0
$659$	− 4.18575e10i	− 0.221938i	−0.993824	−	0.110969i	$-0.964605\pi$
$659$	− 4.18575e10i	− 0.221938i	0.993824	−	0.110969i	$-0.0353954\pi$
$660$	0	0
$661$	2.46529e11	1.29141	0.645703	−	0.763589i	$-0.276565\pi$
$661$	2.46529e11	1.29141	0.645703	+	0.763589i	$0.276565\pi$
$662$	0	0
$663$	− 3.99482e10i	− 0.206749i
$664$	0	0
$665$	1.66264e10	0.0850179
$666$	0	0
$667$	− 3.04155e10i	− 0.153671i
$668$	0	0
$669$	−1.47802e11	−0.737862
$670$	0	0
$671$	2.72584e11i	1.34465i
$672$	0	0
$673$	−3.15336e11	−1.53714	−0.768569	−	0.639767i	$-0.779031\pi$
$673$	−3.15336e11	−1.53714	−0.768569	+	0.639767i	$0.779031\pi$
$674$	0	0
$675$	5.80849e9i	0.0279800i
$676$	0	0
$677$	2.47236e10	0.117695	0.0588475	−	0.998267i	$-0.481257\pi$
$677$	2.47236e10	0.117695	0.0588475	+	0.998267i	$0.481257\pi$
$678$	0	0
$679$	1.68731e11i	0.793811i
$680$	0	0
$681$	7.49454e10	0.348463
$682$	0	0
$683$	− 7.20843e10i	− 0.331251i	−0.986189	−	0.165626i	$-0.947036\pi$
$683$	− 7.20843e10i	− 0.331251i	0.986189	−	0.165626i	$-0.0529644\pi$
$684$	0	0
$685$	−1.35408e11	−0.615009
$686$	0	0
$687$	− 2.84556e11i	− 1.27744i
$688$	0	0
$689$	−4.50887e9	−0.0200074
$690$	0	0
$691$	2.95424e11i	1.29578i	0.761732	+	0.647892i	$0.224349\pi$
$691$	2.95424e11i	1.29578i	−0.761732	+	0.647892i	$0.775651\pi$
$692$	0	0
$693$	8.85139e10	0.383776
$694$	0	0
$695$	− 1.79968e11i	− 0.771360i
$696$	0	0
$697$	1.56916e11	0.664867
$698$	0	0
$699$	− 2.20444e10i	− 0.0923400i
$700$	0	0
$701$	2.87925e11	1.19236	0.596180	−	0.802851i	$-0.296685\pi$
$701$	2.87925e11	1.19236	0.596180	+	0.802851i	$0.296685\pi$
$702$	0	0
$703$	6.76504e10i	0.276980i
$704$	0	0
$705$	−4.64831e11	−1.88165
$706$	0	0
$707$	− 3.87834e10i	− 0.155227i
$708$	0	0
$709$	−2.51685e11	−0.996030	−0.498015	−	0.867168i	$-0.665938\pi$
$709$	−2.51685e11	−0.996030	−0.498015	+	0.867168i	$0.665938\pi$
$710$	0	0
$711$	− 1.22931e11i	− 0.481042i
$712$	0	0
$713$	−1.61174e10	−0.0623643
$714$	0	0
$715$	− 6.16795e10i	− 0.236003i
$716$	0	0
$717$	−4.04170e11	−1.52928
$718$	0	0
$719$	− 1.38856e11i	− 0.519574i	−0.965666	−	0.259787i	$-0.916348\pi$
$719$	− 1.38856e11i	− 0.519574i	0.965666	−	0.259787i	$-0.0836524\pi$
$720$	0	0
$721$	−1.46184e11	−0.540953
$722$	0	0
$723$	− 6.17489e11i	− 2.25983i
$724$	0	0
$725$	−2.37531e9	−0.00859743
$726$	0	0
$727$	− 1.79083e11i	− 0.641088i	−0.947234	−	0.320544i	$-0.896134\pi$
$727$	− 1.79083e11i	− 0.641088i	0.947234	−	0.320544i	$-0.103866\pi$
$728$	0	0
$729$	−2.14381e10	−0.0759061
$730$	0	0
$731$	4.33289e11i	1.51743i
$732$	0	0
$733$	−2.17618e11	−0.753839	−0.376920	−	0.926246i	$-0.623017\pi$
$733$	−2.17618e11	−0.753839	−0.376920	+	0.926246i	$0.623017\pi$
$734$	0	0
$735$	2.32098e11i	0.795285i
$736$	0	0
$737$	2.82023e11	0.955904
$738$	0	0
$739$	− 4.84950e11i	− 1.62599i	−0.582268	−	0.812997i	$-0.697834\pi$
$739$	− 4.84950e11i	− 1.62599i	0.582268	−	0.812997i	$-0.302166\pi$
$740$	0	0
$741$	1.06494e10	0.0353227
$742$	0	0
$743$	2.03509e11i	0.667771i	0.942614	+	0.333886i	$0.108360\pi$
$743$	2.03509e11i	0.667771i	−0.942614	+	0.333886i	$0.891640\pi$
$744$	0	0
$745$	2.46198e11	0.799206
$746$	0	0
$747$	− 1.77860e11i	− 0.571210i
$748$	0	0
$749$	1.40346e11	0.445937
$750$	0	0
$751$	2.34693e11i	0.737804i	0.929468	+	0.368902i	$0.120266\pi$
$751$	2.34693e11i	0.737804i	−0.929468	+	0.368902i	$0.879734\pi$
$752$	0	0
$753$	−5.20950e11	−1.62038
$754$	0	0
$755$	5.10561e11i	1.57130i
$756$	0	0
$757$	3.84882e11	1.17204	0.586022	−	0.810295i	$-0.300693\pi$
$757$	3.84882e11	1.17204	0.586022	+	0.810295i	$0.300693\pi$
$758$	0	0
$759$	− 4.38136e11i	− 1.32021i
$760$	0	0
$761$	2.39209e11	0.713244	0.356622	−	0.934249i	$-0.383928\pi$
$761$	2.39209e11	0.713244	0.356622	+	0.934249i	$0.383928\pi$
$762$	0	0
$763$	8.26340e10i	0.243815i
$764$	0	0
$765$	1.52614e11	0.445604
$766$	0	0
$767$	2.03786e10i	0.0588833i
$768$	0	0
$769$	2.08457e11	0.596089	0.298045	−	0.954552i	$-0.403666\pi$
$769$	2.08457e11	0.596089	0.298045	+	0.954552i	$0.403666\pi$
$770$	0	0
$771$	6.13202e11i	1.73535i
$772$	0	0
$773$	5.54469e10	0.155296	0.0776478	−	0.996981i	$-0.475259\pi$
$773$	5.54469e10	0.155296	0.0776478	+	0.996981i	$0.475259\pi$
$774$	0	0
$775$	1.25869e9i	0.00348909i
$776$	0	0
$777$	4.85306e11	1.33147
$778$	0	0
$779$	4.18307e10i	0.113591i
$780$	0	0
$781$	2.21091e10	0.0594246
$782$	0	0
$783$	− 4.02039e10i	− 0.106960i
$784$	0	0
$785$	−1.65504e11	−0.435844
$786$	0	0
$787$	− 4.05908e11i	− 1.05811i	−0.848589	−	0.529053i	$-0.822547\pi$
$787$	− 4.05908e11i	− 1.05811i	0.848589	−	0.529053i	$-0.177453\pi$
$788$	0	0
$789$	6.96358e11	1.79690
$790$	0	0
$791$	7.70572e10i	0.196837i
$792$	0	0
$793$	8.06610e10	0.203972
$794$	0	0
$795$	− 5.02414e10i	− 0.125775i
$796$	0	0
$797$	3.09015e11	0.765855	0.382927	−	0.923778i	$-0.374916\pi$
$797$	3.09015e11	0.765855	0.382927	+	0.923778i	$0.374916\pi$
$798$	0	0
$799$	5.57406e11i	1.36768i
$800$	0	0
$801$	2.85219e11	0.692864
$802$	0	0
$803$	− 1.05839e11i	− 0.254557i
$804$	0	0
$805$	−2.02415e11	−0.482014
$806$	0	0
$807$	2.70504e11i	0.637792i
$808$	0	0
$809$	4.77958e11	1.11582	0.557912	−	0.829900i	$-0.311603\pi$
$809$	4.77958e11	1.11582	0.557912	+	0.829900i	$0.311603\pi$
$810$	0	0
$811$	6.37503e11i	1.47366i	0.676075	+	0.736832i	$0.263679\pi$
$811$	6.37503e11i	1.47366i	−0.676075	+	0.736832i	$0.736321\pi$
$812$	0	0
$813$	−7.98393e11	−1.82749
$814$	0	0
$815$	3.52890e11i	0.799851i
$816$	0	0
$817$	−1.15506e11	−0.259250
$818$	0	0
$819$	− 2.61924e10i	− 0.0582155i
$820$	0	0
$821$	8.43824e11	1.85729	0.928644	−	0.370973i	$-0.120976\pi$
$821$	8.43824e11	1.85729	0.928644	+	0.370973i	$0.120976\pi$
$822$	0	0
$823$	− 2.60916e11i	− 0.568723i	−0.958717	−	0.284362i	$-0.908218\pi$
$823$	− 2.60916e11i	− 0.568723i	0.958717	−	0.284362i	$-0.0917817\pi$
$824$	0	0
$825$	−3.42164e10	−0.0738616
$826$	0	0
$827$	− 2.78675e11i	− 0.595765i	−0.954602	−	0.297883i	$-0.903720\pi$
$827$	− 2.78675e11i	− 0.595765i	0.954602	−	0.297883i	$-0.0962804\pi$
$828$	0	0
$829$	−4.75156e11	−1.00605	−0.503023	−	0.864273i	$-0.667779\pi$
$829$	−4.75156e11	−1.00605	−0.503023	+	0.864273i	$0.667779\pi$
$830$	0	0
$831$	− 8.22306e11i	− 1.72437i
$832$	0	0
$833$	2.78322e11	0.578053
$834$	0	0
$835$	− 2.85552e11i	− 0.587408i
$836$	0	0
$837$	−2.13043e10	−0.0434075
$838$	0	0
$839$	− 3.35440e11i	− 0.676966i	−0.940973	−	0.338483i	$-0.890086\pi$
$839$	− 3.35440e11i	− 0.676966i	0.940973	−	0.338483i	$-0.109914\pi$
$840$	0	0
$841$	−4.83806e11	−0.967134
$842$	0	0
$843$	− 3.07858e11i	− 0.609594i
$844$	0	0
$845$	4.79344e11	0.940200
$846$	0	0
$847$	− 1.78138e11i	− 0.346117i
$848$	0	0
$849$	−1.17019e11	−0.225229
$850$	0	0
$851$	− 8.23600e11i	− 1.57036i
$852$	0	0
$853$	9.75408e10	0.184243	0.0921213	−	0.995748i	$-0.470635\pi$
$853$	9.75408e10	0.184243	0.0921213	+	0.995748i	$0.470635\pi$
$854$	0	0
$855$	4.06840e10i	0.0761306i
$856$	0	0
$857$	−7.94769e10	−0.147339	−0.0736695	−	0.997283i	$-0.523471\pi$
$857$	−7.94769e10	−0.147339	−0.0736695	+	0.997283i	$0.523471\pi$
$858$	0	0
$859$	4.15618e11i	0.763347i	0.924297	+	0.381673i	$0.124652\pi$
$859$	4.15618e11i	0.763347i	−0.924297	+	0.381673i	$0.875348\pi$
$860$	0	0
$861$	3.00083e11	0.546045
$862$	0	0
$863$	− 4.80012e11i	− 0.865383i	−0.901542	−	0.432692i	$-0.857564\pi$
$863$	− 4.80012e11i	− 0.865383i	0.901542	−	0.432692i	$-0.142436\pi$
$864$	0	0
$865$	−1.25780e11	−0.224671
$866$	0	0
$867$	1.63230e11i	0.288884i
$868$	0	0
$869$	−6.63863e11	−1.16412
$870$	0	0
$871$	− 8.34540e10i	− 0.145002i
$872$	0	0
$873$	−4.12879e11	−0.710830
$874$	0	0
$875$	3.49134e11i	0.595608i
$876$	0	0
$877$	−2.74155e11	−0.463444	−0.231722	−	0.972782i	$-0.574436\pi$
$877$	−2.74155e11	−0.463444	−0.231722	+	0.972782i	$0.574436\pi$
$878$	0	0
$879$	4.80595e11i	0.805051i
$880$	0	0
$881$	8.01838e11	1.33101	0.665507	−	0.746391i	$-0.268215\pi$
$881$	8.01838e11	1.33101	0.665507	+	0.746391i	$0.268215\pi$
$882$	0	0
$883$	9.95008e11i	1.63676i	0.574681	+	0.818378i	$0.305126\pi$
$883$	9.95008e11i	1.63676i	−0.574681	+	0.818378i	$0.694874\pi$
$884$	0	0
$885$	−2.27075e11	−0.370165
$886$	0	0
$887$	5.46038e11i	0.882122i	0.897477	+	0.441061i	$0.145398\pi$
$887$	5.46038e11i	0.882122i	−0.897477	+	0.441061i	$0.854602\pi$
$888$	0	0
$889$	−3.59736e11	−0.575940
$890$	0	0
$891$	− 9.94283e11i	− 1.57761i
$892$	0	0
$893$	−1.48594e11	−0.233665
$894$	0	0
$895$	8.65656e10i	0.134913i
$896$	0	0
$897$	−1.29650e11	−0.200264
$898$	0	0
$899$	− 8.71212e9i	− 0.0133378i
$900$	0	0
$901$	−6.02474e10	−0.0914195
$902$	0	0
$903$	8.28613e11i	1.24624i
$904$	0	0
$905$	2.94365e11	0.438826
$906$	0	0
$907$	− 6.55018e11i	− 0.967886i	−0.875100	−	0.483943i	$-0.839204\pi$
$907$	− 6.55018e11i	− 0.967886i	0.875100	−	0.483943i	$-0.160796\pi$
$908$	0	0
$909$	9.49014e10	0.139001
$910$	0	0
$911$	− 1.19425e11i	− 0.173389i	−0.996235	−	0.0866943i	$-0.972370\pi$
$911$	− 1.19425e11i	− 0.173389i	0.996235	−	0.0866943i	$-0.0276304\pi$
$912$	0	0
$913$	−9.60496e11	−1.38233
$914$	0	0
$915$	8.98791e11i	1.28225i
$916$	0	0
$917$	−4.36696e11	−0.617592
$918$	0	0
$919$	5.33989e10i	0.0748635i	0.999299	+	0.0374318i	$0.0119177\pi$
$919$	5.33989e10i	0.0748635i	−0.999299	+	0.0374318i	$0.988082\pi$
$920$	0	0
$921$	3.48897e11	0.484907
$922$	0	0
$923$	− 6.54235e9i	− 0.00901420i
$924$	0	0
$925$	−6.43194e10	−0.0878567
$926$	0	0
$927$	− 3.57707e11i	− 0.484404i
$928$	0	0
$929$	−6.30991e11	−0.847150	−0.423575	−	0.905861i	$-0.639225\pi$
$929$	−6.30991e11	−0.847150	−0.423575	+	0.905861i	$0.639225\pi$
$930$	0	0
$931$	7.41953e10i	0.0987593i
$932$	0	0
$933$	−1.29703e12	−1.71168
$934$	0	0
$935$	− 8.24161e11i	− 1.07836i
$936$	0	0
$937$	−8.41436e11	−1.09160	−0.545799	−	0.837916i	$-0.683774\pi$
$937$	−8.41436e11	−1.09160	−0.545799	+	0.837916i	$0.683774\pi$
$938$	0	0
$939$	6.30660e11i	0.811209i
$940$	0	0
$941$	4.52935e11	0.577666	0.288833	−	0.957379i	$-0.406733\pi$
$941$	4.52935e11	0.577666	0.288833	+	0.957379i	$0.406733\pi$
$942$	0	0
$943$	− 5.09262e11i	− 0.644013i
$944$	0	0
$945$	−2.67556e11	−0.335497
$946$	0	0
$947$	1.00309e12i	1.24721i	0.781739	+	0.623605i	$0.214333\pi$
$947$	1.00309e12i	1.24721i	−0.781739	+	0.623605i	$0.785667\pi$
$948$	0	0
$949$	−3.13192e10	−0.0386141
$950$	0	0
$951$	1.65769e12i	2.02666i
$952$	0	0
$953$	−1.39040e12	−1.68565	−0.842826	−	0.538186i	$-0.819110\pi$
$953$	−1.39040e12	−1.68565	−0.842826	+	0.538186i	$0.819110\pi$
$954$	0	0
$955$	− 6.05401e11i	− 0.727830i
$956$	0	0
$957$	2.36831e11	0.282352
$958$	0	0
$959$	3.10523e11i	0.367130i
$960$	0	0
$961$	8.48274e11	0.994587
$962$	0	0
$963$	3.43421e11i	0.399321i
$964$	0	0
$965$	−7.17318e11	−0.827185
$966$	0	0
$967$	− 9.01843e11i	− 1.03140i	−0.856771	−	0.515698i	$-0.827533\pi$
$967$	− 9.01843e11i	− 1.03140i	0.856771	−	0.515698i	$-0.172467\pi$
$968$	0	0
$969$	1.42297e11	0.161399
$970$	0	0
$971$	− 1.28411e12i	− 1.44452i	−0.691621	−	0.722261i	$-0.743103\pi$
$971$	− 1.28411e12i	− 1.44452i	0.691621	−	0.722261i	$-0.256897\pi$
$972$	0	0
$973$	−4.12712e11	−0.460464
$974$	0	0
$975$	1.01251e10i	0.0112042i
$976$	0	0
$977$	1.20120e12	1.31837	0.659183	−	0.751983i	$-0.270902\pi$
$977$	1.20120e12	1.31837	0.659183	+	0.751983i	$0.270902\pi$
$978$	0	0
$979$	− 1.54026e12i	− 1.67674i
$980$	0	0
$981$	−2.02202e11	−0.218328
$982$	0	0
$983$	3.77388e11i	0.404179i	0.979367	+	0.202090i	$0.0647733\pi$
$983$	3.77388e11i	0.404179i	−0.979367	+	0.202090i	$0.935227\pi$
$984$	0	0
$985$	1.04028e12	1.10511
$986$	0	0
$987$	1.06597e12i	1.12325i
$988$	0	0
$989$	1.40622e12	1.46983
$990$	0	0
$991$	− 3.35563e11i	− 0.347920i	−0.984753	−	0.173960i	$-0.944344\pi$
$991$	− 3.35563e11i	− 0.347920i	0.984753	−	0.173960i	$-0.0556563\pi$
$992$	0	0
$993$	−5.48200e11	−0.563822
$994$	0	0
$995$	− 1.51912e12i	− 1.54989i
$996$	0	0
$997$	1.03163e12	1.04410	0.522050	−	0.852915i	$-0.325167\pi$
$997$	1.03163e12	1.04410	0.522050	+	0.852915i	$0.325167\pi$
$998$	0	0
$999$	− 1.08865e12i	− 1.09302i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.9.c.b.63.1		2
4.3	odd	2	inner	64.9.c.b.63.2		2
8.3	odd	2		4.9.b.b.3.1	✓	2
8.5	even	2		4.9.b.b.3.2	yes	2
16.3	odd	4		256.9.d.e.127.2		4
16.5	even	4		256.9.d.e.127.1		4
16.11	odd	4		256.9.d.e.127.3		4
16.13	even	4		256.9.d.e.127.4		4
24.5	odd	2		36.9.d.b.19.1		2
24.11	even	2		36.9.d.b.19.2		2
40.3	even	4		100.9.d.b.99.2		4
40.13	odd	4		100.9.d.b.99.4		4
40.19	odd	2		100.9.b.c.51.2		2
40.27	even	4		100.9.d.b.99.3		4
40.29	even	2		100.9.b.c.51.1		2
40.37	odd	4		100.9.d.b.99.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.9.b.b.3.1	✓	2	8.3	odd	2
4.9.b.b.3.2	yes	2	8.5	even	2
36.9.d.b.19.1		2	24.5	odd	2
36.9.d.b.19.2		2	24.11	even	2
64.9.c.b.63.1		2	1.1	even	1	trivial
64.9.c.b.63.2		2	4.3	odd	2	inner
100.9.b.c.51.1		2	40.29	even	2
100.9.b.c.51.2		2	40.19	odd	2
100.9.d.b.99.1		4	40.37	odd	4
100.9.d.b.99.2		4	40.3	even	4
100.9.d.b.99.3		4	40.27	even	4
100.9.d.b.99.4		4	40.13	odd	4
256.9.d.e.127.1		4	16.5	even	4
256.9.d.e.127.2		4	16.3	odd	4
256.9.d.e.127.3		4	16.11	odd	4
256.9.d.e.127.4		4	16.13	even	4

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	64.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-99.9200i q^{3} -610.000 q^{5} +1398.88i q^{7} -3423.00 q^{9} +18485.2i q^{11} +5470.00 q^{13} +60951.2i q^{15} +73090.0 q^{17} +19484.4i q^{19} +139776. q^{21} -237210. i q^{23} -18525.0 q^{25} -313549. i q^{27} +128222. q^{29} -67945.6i q^{31} +1.84704e6 q^{33} -853317. i q^{35} +3.47203e6 q^{37} -546562. i q^{39} +2.14688e6 q^{41} +5.92815e6i q^{43} +2.08803e6 q^{45} +7.62629e6i q^{47} +3.80794e6 q^{49} -7.30315e6i q^{51} -824290. q^{53} -1.12760e7i q^{55} +1.94688e6 q^{57} +3.72552e6i q^{59} +1.47461e7 q^{61} -4.78836e6i q^{63} -3.33670e6 q^{65} -1.52567e7i q^{67} -2.37020e7 q^{69} -1.19604e6i q^{71} -5.72563e6 q^{73} +1.85102e6i q^{75} -2.58586e7 q^{77} +3.59132e7i q^{79} -5.37881e7 q^{81} +5.19603e7i q^{83} -4.45849e7 q^{85} -1.28119e7i q^{87} -8.33242e7 q^{89} +7.65187e6i q^{91} -6.78912e6 q^{93} -1.18855e7i q^{95} +1.20619e8 q^{97} -6.32748e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1220 q^{5} - 6846 q^{9} + 10940 q^{13} + 146180 q^{17} + 279552 q^{21} - 37050 q^{25} + 256444 q^{29} + 3694080 q^{33} + 6944060 q^{37} + 4293764 q^{41} + 4176060 q^{45} + 7615874 q^{49} - 1648580 q^{53}+ \cdots + 241238020 q^{97}+O(q^{100}) \) 2 * q - 1220 * q^5 - 6846 * q^9 + 10940 * q^13 + 146180 * q^17 + 279552 * q^21 - 37050 * q^25 + 256444 * q^29 + 3694080 * q^33 + 6944060 * q^37 + 4293764 * q^41 + 4176060 * q^45 + 7615874 * q^49 - 1648580 * q^53 + 3893760 * q^57 + 29492156 * q^61 - 6673400 * q^65 - 47404032 * q^69 - 11451260 * q^73 - 51717120 * q^77 - 107576190 * q^81 - 89169800 * q^85 - 166648444 * q^89 - 13578240 * q^93 + 241238020 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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