LMFDB - Embedded newform 256.9.d.e.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,9,Mod(127,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("256.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	256.d (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:	$104.288924176$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{39})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 19x^{2} + 100 $ x^4 - 19*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$3.12250 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.127
Dual form			256.9.d.e.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-99.9200 q^{3} +610.000i q^{5} +1398.88i q^{7} +3423.00 q^{9} -18485.2 q^{11} +5470.00i q^{13} -60951.2i q^{15} +73090.0 q^{17} +19484.4 q^{19} -139776. i q^{21} -237210. i q^{23} +18525.0 q^{25} +313549. q^{27} +128222. i q^{29} +67945.6i q^{31} +1.84704e6 q^{33} -853317. q^{35} -3.47203e6i q^{37} -546562. i q^{39} -2.14688e6 q^{41} -5.92815e6 q^{43} +2.08803e6i q^{45} -7.62629e6i q^{47} +3.80794e6 q^{49} -7.30315e6 q^{51} +824290. i q^{53} -1.12760e7i q^{55} -1.94688e6 q^{57} -3.72552e6 q^{59} +1.47461e7i q^{61} +4.78836e6i q^{63} -3.33670e6 q^{65} -1.52567e7 q^{67} +2.37020e7i q^{69} -1.19604e6i q^{71} +5.72563e6 q^{73} -1.85102e6 q^{75} -2.58586e7i q^{77} -3.59132e7i q^{79} -5.37881e7 q^{81} +5.19603e7 q^{83} +4.45849e7i q^{85} -1.28119e7i q^{87} +8.33242e7 q^{89} -7.65187e6 q^{91} -6.78912e6i q^{93} +1.18855e7i q^{95} +1.20619e8 q^{97} -6.32748e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 13692 q^{9} + 292360 q^{17} + 74100 q^{25} + 7388160 q^{33} - 8587528 q^{41} + 15231748 q^{49} - 7787520 q^{57} - 13346800 q^{65} + 22902520 q^{73} - 215152380 q^{81} + 333296888 q^{89} + 482476040 q^{97}+O(q^{100}) $ 4 * q + 13692 * q^9 + 292360 * q^17 + 74100 * q^25 + 7388160 * q^33 - 8587528 * q^41 + 15231748 * q^49 - 7787520 * q^57 - 13346800 * q^65 + 22902520 * q^73 - 215152380 * q^81 + 333296888 * q^89 + 482476040 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$-1$

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
$3$	−99.9200	−1.23358	−0.616790	−	0.787128i	$-0.711567\pi$
$3$	−99.9200	−1.23358	−0.616790	+	0.787128i	$0.711567\pi$
$4$	0	0
$5$	610.000i	0.976000i	0.872844	+	0.488000i	$0.162273\pi$
$5$	610.000i	0.976000i	−0.872844	+	0.488000i	$0.837727\pi$
$6$	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
$11$	−18485.2	−1.26256	−0.631282	−	0.775554i	$-0.717471\pi$
$11$	−18485.2	−1.26256	−0.631282	+	0.775554i	$0.717471\pi$
$12$	0	0
$13$	5470.00i	0.191520i	0.995404	+	0.0957600i	$0.0305281\pi$
$13$	5470.00i	0.191520i	−0.995404	+	0.0957600i	$0.969472\pi$
$14$	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
$19$	19484.4	0.149511	0.0747554	−	0.997202i	$-0.476182\pi$
$19$	19484.4	0.149511	0.0747554	+	0.997202i	$0.476182\pi$
$20$	0	0
$21$	− 139776.i	− 0.718713i
$22$	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.	0.589997
$28$	0	0
$29$	128222.i	0.181289i	0.995883	+	0.0906443i	$0.0288926\pi$
$29$	128222.i	0.181289i	−0.995883	+	0.0906443i	$0.971107\pi$
$30$	0	0
$31$	67945.6i	0.0735723i	0.999323	+	0.0367862i	$0.0117120\pi$
$31$	67945.6i	0.0735723i	−0.999323	+	0.0367862i	$0.988288\pi$
$32$	0	0
$33$	1.84704e6	1.55747
$34$	0	0
$35$	−853317.	−0.568641
$36$	0	0
$37$	− 3.47203e6i	− 1.85258i	−0.376813	−	0.926289i	$-0.622980\pi$
$37$	− 3.47203e6i	− 1.85258i	0.376813	−	0.926289i	$-0.377020\pi$
$38$	0	0
$39$	− 546562.i	− 0.236255i
$40$	0	0
$41$	−2.14688e6	−0.759754	−0.379877	−	0.925037i	$-0.624034\pi$
$41$	−2.14688e6	−0.759754	−0.379877	+	0.925037i	$0.624034\pi$
$42$	0	0
$43$	−5.92815e6	−1.73399	−0.866993	−	0.498321i	$-0.833950\pi$
$43$	−5.92815e6	−1.73399	−0.866993	+	0.498321i	$0.833950\pi$
$44$	0	0
$45$	2.08803e6i	0.509198i
$46$	0	0
$47$	− 7.62629e6i	− 1.56287i	−0.623989	−	0.781433i	$-0.714489\pi$
$47$	− 7.62629e6i	− 1.56287i	0.623989	−	0.781433i	$-0.285511\pi$
$48$	0	0
$49$	3.80794e6	0.660550
$50$	0	0
$51$	−7.30315e6	−1.07952
$52$	0	0
$53$	824290.i	0.104466i	0.998635	+	0.0522332i	$0.0166339\pi$
$53$	824290.i	0.104466i	−0.998635	+	0.0522332i	$0.983366\pi$
$54$	0	0
$55$	− 1.12760e7i	− 1.23226i
$56$	0	0
$57$	−1.94688e6	−0.184433
$58$	0	0
$59$	−3.72552e6	−0.307453	−0.153726	−	0.988113i	$-0.549127\pi$
$59$	−3.72552e6	−0.307453	−0.153726	+	0.988113i	$0.549127\pi$
$60$	0	0
$61$	1.47461e7i	1.06502i	0.846424	+	0.532509i	$0.178751\pi$
$61$	1.47461e7i	1.06502i	−0.846424	+	0.532509i	$0.821249\pi$
$62$	0	0
$63$	4.78836e6i	0.303966i
$64$	0	0
$65$	−3.33670e6	−0.186923
$66$	0	0
$67$	−1.52567e7	−0.757113	−0.378557	−	0.925578i	$-0.623579\pi$
$67$	−1.52567e7	−0.757113	−0.378557	+	0.925578i	$0.623579\pi$
$68$	0	0
$69$	2.37020e7i	1.04566i
$70$	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	$-0.467857\pi$
$73$	5.72563e6	0.201619	0.100810	+	0.994906i	$0.467857\pi$
$74$	0	0
$75$	−1.85102e6	−0.0585013
$76$	0	0
$77$	− 2.58586e7i	− 0.735600i
$78$	0	0
$79$	− 3.59132e7i	− 0.922032i	−0.887392	−	0.461016i	$-0.847485\pi$
$79$	− 3.59132e7i	− 0.922032i	0.887392	−	0.461016i	$-0.152515\pi$
$80$	0	0
$81$	−5.37881e7	−1.24953
$82$	0	0
$83$	5.19603e7	1.09486	0.547431	−	0.836851i	$-0.315606\pi$
$83$	5.19603e7	1.09486	0.547431	+	0.836851i	$0.315606\pi$
$84$	0	0
$85$	4.45849e7i	0.854107i
$86$	0	0
$87$	− 1.28119e7i	− 0.223634i
$88$	0	0
$89$	8.33242e7	1.32804	0.664020	−	0.747715i	$-0.268849\pi$
$89$	8.33242e7	1.32804	0.664020	+	0.747715i	$0.268849\pi$
$90$	0	0
$91$	−7.65187e6	−0.111584
$92$	0	0
$93$	− 6.78912e6i	− 0.0907573i
$94$	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
$99$	−6.32748e7	−0.658704
$100$	0	0
$101$	2.77246e7i	0.266428i	0.991087	+	0.133214i	$0.0425298\pi$
$101$	2.77246e7i	0.266428i	−0.991087	+	0.133214i	$0.957470\pi$
$102$	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
$107$	1.00328e8	0.765394	0.382697	−	0.923874i	$-0.374995\pi$
$107$	1.00328e8	0.765394	0.382697	+	0.923874i	$0.374995\pi$
$108$	0	0
$109$	5.90716e7i	0.418478i	0.977865	+	0.209239i	$0.0670987\pi$
$109$	5.90716e7i	0.418478i	−0.977865	+	0.209239i	$0.932901\pi$
$110$	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
$115$	1.44698e8	0.827316
$116$	0	0
$117$	1.87238e7i	0.0999196i
$118$	0	0
$119$	1.02244e8i	0.509859i
$120$	0	0
$121$	1.27344e8	0.594067
$122$	0	0
$123$	2.14516e8	0.937217
$124$	0	0
$125$	2.49581e8i	1.02229i
$126$	0	0
$127$	− 2.57160e8i	− 0.988529i	−0.869312	−	0.494264i	$-0.835438\pi$
$127$	− 2.57160e8i	− 0.988529i	0.869312	−	0.494264i	$-0.164562\pi$
$128$	0	0
$129$	5.92341e8	2.13901
$130$	0	0
$131$	3.12175e8	1.06002	0.530009	−	0.847992i	$-0.322188\pi$
$131$	3.12175e8	1.06002	0.530009	+	0.847992i	$0.322188\pi$
$132$	0	0
$133$	2.72563e7i	0.0871085i
$134$	0	0
$135$	1.91265e8i	0.575838i
$136$	0	0
$137$	−2.21980e8	−0.630132	−0.315066	−	0.949070i	$-0.602027\pi$
$137$	−2.21980e8	−0.630132	−0.315066	+	0.949070i	$0.602027\pi$
$138$	0	0
$139$	−2.95030e8	−0.790328	−0.395164	−	0.918611i	$-0.629312\pi$
$139$	−2.95030e8	−0.790328	−0.395164	+	0.918611i	$0.629312\pi$
$140$	0	0
$141$	7.62019e8i	1.92792i
$142$	0	0
$143$	− 1.01114e8i	− 0.241806i
$144$	0	0
$145$	−7.82154e7	−0.176938
$146$	0	0
$147$	−3.80489e8	−0.814841
$148$	0	0
$149$	4.03603e8i	0.818859i	0.912342	+	0.409429i	$0.134272\pi$
$149$	4.03603e8i	0.818859i	−0.912342	+	0.409429i	$0.865728\pi$
$150$	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
$155$	−4.14468e7	−0.0718066
$156$	0	0
$157$	2.71319e8i	0.446561i	0.974754	+	0.223281i	$0.0716767\pi$
$157$	2.71319e8i	0.446561i	−0.974754	+	0.223281i	$0.928323\pi$
$158$	0	0
$159$	− 8.23630e7i	− 0.128868i
$160$	0	0
$161$	3.31828e8	0.493867
$162$	0	0
$163$	−5.78509e8	−0.819520	−0.409760	−	0.912193i	$-0.634388\pi$
$163$	−5.78509e8	−0.819520	−0.409760	+	0.912193i	$0.634388\pi$
$164$	0	0
$165$	1.12669e9i	1.52009i
$166$	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
$171$	6.66951e7	0.0780026
$172$	0	0
$173$	2.06197e8i	0.230196i	0.993354	+	0.115098i	$0.0367182\pi$
$173$	2.06197e8i	0.230196i	−0.993354	+	0.115098i	$0.963282\pi$
$174$	0	0
$175$	2.59142e7i	0.0276303i
$176$	0	0
$177$	3.72253e8	0.379268
$178$	0	0
$179$	−1.41911e8	−0.138230	−0.0691152	−	0.997609i	$-0.522018\pi$
$179$	−1.41911e8	−0.138230	−0.0691152	+	0.997609i	$0.522018\pi$
$180$	0	0
$181$	4.82566e8i	0.449616i	0.974403	+	0.224808i	$0.0721755\pi$
$181$	4.82566e8i	0.449616i	−0.974403	+	0.224808i	$0.927824\pi$
$182$	0	0
$183$	− 1.47343e9i	− 1.31379i
$184$	0	0
$185$	2.11794e9	1.80812
$186$	0	0
$187$	−1.35108e9	−1.10488
$188$	0	0
$189$	4.38617e8i	0.343747i
$190$	0	0
$191$	− 9.92461e8i	− 0.745727i	−0.927886	−	0.372864i	$-0.878376\pi$
$191$	− 9.92461e8i	− 0.745727i	0.927886	−	0.372864i	$-0.121624\pi$
$192$	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
$195$	3.33403e8	0.230585
$196$	0	0
$197$	1.70538e9i	1.13229i	0.824306	+	0.566144i	$0.191565\pi$
$197$	1.70538e9i	1.13229i	−0.824306	+	0.566144i	$0.808435\pi$
$198$	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
$203$	−1.79367e8	−0.105623
$204$	0	0
$205$	− 1.30960e9i	− 0.741519i
$206$	0	0
$207$	− 8.11970e8i	− 0.442241i
$208$	0	0
$209$	−3.60173e8	−0.188767
$210$	0	0
$211$	−1.46774e9	−0.740491	−0.370245	−	0.928934i	$-0.620726\pi$
$211$	−1.46774e9	−0.740491	−0.370245	+	0.928934i	$0.620726\pi$
$212$	0	0
$213$	1.19508e8i	0.0580604i
$214$	0	0
$215$	− 3.61617e9i	− 1.69237i
$216$	0	0
$217$	−9.50477e7	−0.0428650
$218$	0	0
$219$	−5.72105e8	−0.248713
$220$	0	0
$221$	3.99802e8i	0.167601i
$222$	0	0
$223$	1.47920e9i	0.598147i	0.954230	+	0.299073i	$0.0966776\pi$
$223$	1.47920e9i	0.598147i	−0.954230	+	0.299073i	$0.903322\pi$
$224$	0	0
$225$	6.34111e7	0.0247420
$226$	0	0
$227$	7.50054e8	0.282481	0.141241	−	0.989975i	$-0.454891\pi$
$227$	7.50054e8	0.282481	0.141241	+	0.989975i	$0.454891\pi$
$228$	0	0
$229$	− 2.84784e9i	− 1.03556i	−0.855515	−	0.517778i	$-0.826759\pi$
$229$	− 2.84784e9i	− 1.03556i	0.855515	−	0.517778i	$-0.173241\pi$
$230$	0	0
$231$	2.58379e9i	0.907421i
$232$	0	0
$233$	−2.20621e8	−0.0748553	−0.0374276	−	0.999299i	$-0.511916\pi$
$233$	−2.20621e8	−0.0748553	−0.0374276	+	0.999299i	$0.511916\pi$
$234$	0	0
$235$	4.65204e9	1.52536
$236$	0	0
$237$	3.58845e9i	1.13740i
$238$	0	0
$239$	4.04493e9i	1.23971i	0.784717	+	0.619855i	$0.212808\pi$
$239$	4.04493e9i	1.23971i	−0.784717	+	0.619855i	$0.787192\pi$
$240$	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
$243$	3.31731e9	0.951395
$244$	0	0
$245$	2.32284e9i	0.644696i
$246$	0	0
$247$	1.06580e8i	0.0286343i
$248$	0	0
$249$	−5.19187e9	−1.35060
$250$	0	0
$251$	5.21367e9	1.31356	0.656778	−	0.754084i	$-0.271919\pi$
$251$	5.21367e9	1.31356	0.656778	+	0.754084i	$0.271919\pi$
$252$	0	0
$253$	4.38487e9i	1.07022i
$254$	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
$259$	4.85695e9	1.07936
$260$	0	0
$261$	4.38904e8i	0.0945818i
$262$	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
$267$	−8.32575e9	−1.63824
$268$	0	0
$269$	− 2.70720e9i	− 0.517025i	−0.966008	−	0.258513i	$-0.916768\pi$
$269$	− 2.70720e9i	− 0.517025i	0.966008	−	0.258513i	$-0.0832324\pi$
$270$	0	0
$271$	7.99032e9i	1.48145i	0.671808	+	0.740725i	$0.265518\pi$
$271$	7.99032e9i	1.48145i	−0.671808	+	0.740725i	$0.734482\pi$
$272$	0	0
$273$	7.64575e8	0.137648
$274$	0	0
$275$	−3.42438e8	−0.0598758
$276$	0	0
$277$	− 8.22965e9i	− 1.39786i	−0.715192	−	0.698928i	$-0.753661\pi$
$277$	− 8.22965e9i	− 1.39786i	0.715192	−	0.698928i	$-0.246339\pi$
$278$	0	0
$279$	2.32578e8i	0.0383841i
$280$	0	0
$281$	−3.08105e9	−0.494167	−0.247083	−	0.968994i	$-0.579472\pi$
$281$	−3.08105e9	−0.494167	−0.247083	+	0.968994i	$0.579472\pi$
$282$	0	0
$283$	1.17112e9	0.182582	0.0912908	−	0.995824i	$-0.470901\pi$
$283$	1.17112e9	0.182582	0.0912908	+	0.995824i	$0.470901\pi$
$284$	0	0
$285$	− 1.18760e9i	− 0.180007i
$286$	0	0
$287$	− 3.00323e9i	− 0.442650i
$288$	0	0
$289$	−1.63361e9	−0.234184
$290$	0	0
$291$	−1.20522e10	−1.68072
$292$	0	0
$293$	4.80980e9i	0.652614i	0.945264	+	0.326307i	$0.105804\pi$
$293$	4.80980e9i	0.652614i	−0.945264	+	0.326307i	$0.894196\pi$
$294$	0	0
$295$	− 2.27256e9i	− 0.300074i
$296$	0	0
$297$	−5.79601e9	−0.744909
$298$	0	0
$299$	1.29754e9	0.162344
$300$	0	0
$301$	− 8.29277e9i	− 1.01026i
$302$	0	0
$303$	− 2.77025e9i	− 0.328661i
$304$	0	0
$305$	−8.99511e9	−1.03946
$306$	0	0
$307$	3.49176e9	0.393089	0.196545	−	0.980495i	$-0.437028\pi$
$307$	3.49176e9	0.393089	0.196545	+	0.980495i	$0.437028\pi$
$308$	0	0
$309$	− 1.04417e10i	− 1.14535i
$310$	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	$-0.393355\pi$
$313$	6.31165e9	0.657606	0.328803	+	0.944399i	$0.393355\pi$
$314$	0	0
$315$	−2.92090e9	−0.296671
$316$	0	0
$317$	− 1.65902e10i	− 1.64291i	−0.570273	−	0.821455i	$-0.693163\pi$
$317$	− 1.65902e10i	− 1.64291i	0.570273	−	0.821455i	$-0.306837\pi$
$318$	0	0
$319$	− 2.37021e9i	− 0.228888i
$320$	0	0
$321$	−1.00247e10	−0.944175
$322$	0	0
$323$	1.42411e9	0.130838
$324$	0	0
$325$	1.01332e8i	0.00908264i
$326$	0	0
$327$	− 5.90243e9i	− 0.516226i
$328$	0	0
$329$	1.06683e10	0.910563
$330$	0	0
$331$	5.48640e9	0.457062	0.228531	−	0.973537i	$-0.426608\pi$
$331$	5.48640e9	0.457062	0.228531	+	0.973537i	$0.426608\pi$
$332$	0	0
$333$	− 1.18848e10i	− 0.966526i
$334$	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
$339$	−5.50408e9	−0.416760
$340$	0	0
$341$	− 1.25599e9i	− 0.0928897i
$342$	0	0
$343$	1.33911e10i	0.967476i
$344$	0	0
$345$	−1.44582e10	−1.02056
$346$	0	0
$347$	−1.59731e10	−1.10172	−0.550859	−	0.834599i	$-0.685700\pi$
$347$	−1.59731e10	−1.10172	−0.550859	+	0.834599i	$0.685700\pi$
$348$	0	0
$349$	− 1.03634e10i	− 0.698553i	−0.937020	−	0.349277i	$-0.886427\pi$
$349$	− 1.03634e10i	− 0.698553i	0.937020	−	0.349277i	$-0.113573\pi$
$350$	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
$355$	7.29586e8	0.0459370
$356$	0	0
$357$	− 1.02162e10i	− 0.628952i
$358$	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
$363$	−1.27242e10	−0.732829
$364$	0	0
$365$	3.49263e9i	0.196780i
$366$	0	0
$367$	− 1.96628e10i	− 1.08388i	−0.840418	−	0.541939i	$-0.817691\pi$
$367$	− 1.96628e10i	− 1.08388i	0.840418	−	0.541939i	$-0.182309\pi$
$368$	0	0
$369$	−7.34878e9	−0.396378
$370$	0	0
$371$	−1.15308e9	−0.0608646
$372$	0	0
$373$	− 2.10063e10i	− 1.08521i	−0.839987	−	0.542606i	$-0.817438\pi$
$373$	− 2.10063e10i	− 1.08521i	0.839987	−	0.542606i	$-0.182562\pi$
$374$	0	0
$375$	− 2.49382e10i	− 1.26107i
$376$	0	0
$377$	−7.01374e8	−0.0347204
$378$	0	0
$379$	3.04816e9	0.147734	0.0738670	−	0.997268i	$-0.476466\pi$
$379$	3.04816e9	0.147734	0.0738670	+	0.997268i	$0.476466\pi$
$380$	0	0
$381$	2.56955e10i	1.21943i
$382$	0	0
$383$	2.23357e10i	1.03802i	0.854770	+	0.519008i	$0.173698\pi$
$383$	2.23357e10i	1.03802i	−0.854770	+	0.519008i	$0.826302\pi$
$384$	0	0
$385$	1.57737e10	0.717945
$386$	0	0
$387$	−2.02921e10	−0.904654
$388$	0	0
$389$	3.13680e10i	1.36990i	0.728592	+	0.684948i	$0.240175\pi$
$389$	3.13680e10i	1.36990i	−0.728592	+	0.684948i	$0.759825\pi$
$390$	0	0
$391$	− 1.73377e10i	− 0.741795i
$392$	0	0
$393$	−3.11926e10	−1.30762
$394$	0	0
$395$	2.19071e10	0.899904
$396$	0	0
$397$	− 7.65788e9i	− 0.308281i	−0.988049	−	0.154140i	$-0.950739\pi$
$397$	− 7.65788e9i	− 0.308281i	0.988049	−	0.154140i	$-0.0492608\pi$
$398$	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
$403$	−3.71662e8	−0.0140906
$404$	0	0
$405$	− 3.28107e10i	− 1.21954i
$406$	0	0
$407$	6.41811e10i	2.33900i
$408$	0	0
$409$	−2.26168e10	−0.808236	−0.404118	−	0.914707i	$-0.632422\pi$
$409$	−2.26168e10	−0.808236	−0.404118	+	0.914707i	$0.632422\pi$
$410$	0	0
$411$	2.21802e10	0.777318
$412$	0	0
$413$	− 5.21155e9i	− 0.179129i
$414$	0	0
$415$	3.16958e10i	1.06858i
$416$	0	0
$417$	2.94794e10	0.974932
$418$	0	0
$419$	4.94503e10	1.60440	0.802201	−	0.597054i	$-0.203662\pi$
$419$	4.94503e10	1.60440	0.802201	+	0.597054i	$0.203662\pi$
$420$	0	0
$421$	− 3.34077e10i	− 1.06345i	−0.846916	−	0.531726i	$-0.821543\pi$
$421$	− 3.34077e10i	− 1.06345i	0.846916	−	0.531726i	$-0.178457\pi$
$422$	0	0
$423$	− 2.61048e10i	− 0.815378i
$424$	0	0
$425$	1.35399e9	0.0415012
$426$	0	0
$427$	−2.06280e10	−0.620505
$428$	0	0
$429$	1.01033e10i	0.298287i
$430$	0	0
$431$	3.06956e10i	0.889544i	0.895644	+	0.444772i	$0.146715\pi$
$431$	3.06956e10i	0.889544i	−0.895644	+	0.444772i	$0.853285\pi$
$432$	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
$435$	7.81528e9	0.218267
$436$	0	0
$437$	− 4.62189e9i	− 0.126734i
$438$	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
$443$	2.06609e10	0.536455	0.268228	−	0.963356i	$-0.413562\pi$
$443$	2.06609e10	0.536455	0.268228	+	0.963356i	$0.413562\pi$
$444$	0	0
$445$	5.08278e10i	1.29617i
$446$	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
$451$	3.96855e10	0.959237
$452$	0	0
$453$	8.36315e10i	1.98599i
$454$	0	0
$455$	− 4.66764e9i	− 0.108906i
$456$	0	0
$457$	2.06831e10	0.474188	0.237094	−	0.971487i	$-0.423805\pi$
$457$	2.06831e10	0.474188	0.237094	+	0.971487i	$0.423805\pi$
$458$	0	0
$459$	2.29173e10	0.516312
$460$	0	0
$461$	− 7.65072e10i	− 1.69394i	−0.531640	−	0.846971i	$-0.678424\pi$
$461$	− 7.65072e10i	− 1.69394i	0.531640	−	0.846971i	$-0.321576\pi$
$462$	0	0
$463$	− 3.41303e9i	− 0.0742704i	−0.999310	−	0.0371352i	$-0.988177\pi$
$463$	− 3.41303e9i	− 0.0742704i	0.999310	−	0.0371352i	$-0.0118232\pi$
$464$	0	0
$465$	4.14136e9	0.0885791
$466$	0	0
$467$	−1.92903e10	−0.405576	−0.202788	−	0.979223i	$-0.565000\pi$
$467$	−1.92903e10	−0.405576	−0.202788	+	0.979223i	$0.565000\pi$
$468$	0	0
$469$	− 2.13423e10i	− 0.441112i
$470$	0	0
$471$	− 2.71102e10i	− 0.550869i
$472$	0	0
$473$	1.09583e11	2.18927
$474$	0	0
$475$	3.60948e8	0.00709040
$476$	0	0
$477$	2.82154e9i	0.0545021i
$478$	0	0
$479$	2.43887e10i	0.463282i	0.972801	+	0.231641i	$0.0744095\pi$
$479$	2.43887e10i	0.463282i	−0.972801	+	0.231641i	$0.925590\pi$
$480$	0	0
$481$	1.89920e10	0.354806
$482$	0	0
$483$	−3.31563e10	−0.609224
$484$	0	0
$485$	7.35776e10i	1.32978i
$486$	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
$491$	−2.12850e9	−0.0366225	−0.0183113	−	0.999832i	$-0.505829\pi$
$491$	−2.12850e9	−0.0366225	−0.0183113	+	0.999832i	$0.505829\pi$
$492$	0	0
$493$	9.37175e9i	0.158647i
$494$	0	0
$495$	− 3.85976e10i	− 0.642895i
$496$	0	0
$497$	1.67312e9	0.0274221
$498$	0	0
$499$	−1.04101e10	−0.167901	−0.0839503	−	0.996470i	$-0.526754\pi$
$499$	−1.04101e10	−0.167901	−0.0839503	+	0.996470i	$0.526754\pi$
$500$	0	0
$501$	− 4.67744e10i	− 0.742433i
$502$	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
$507$	−7.85181e10	−1.18833
$508$	0	0
$509$	3.25113e10i	0.484354i	0.970232	+	0.242177i	$0.0778615\pi$
$509$	3.25113e10i	0.484354i	−0.970232	+	0.242177i	$0.922139\pi$
$510$	0	0
$511$	8.00947e9i	0.117468i
$512$	0	0
$513$	6.10931e9	0.0882110
$514$	0	0
$515$	−6.37455e10	−0.906194
$516$	0	0
$517$	1.40973e11i	1.97322i
$518$	0	0
$519$	− 2.06032e10i	− 0.283965i
$520$	0	0
$521$	−1.84550e9	−0.0250475	−0.0125237	−	0.999922i	$-0.503987\pi$
$521$	−1.84550e9	−0.0250475	−0.0125237	+	0.999922i	$0.503987\pi$
$522$	0	0
$523$	6.23770e10	0.833715	0.416858	−	0.908972i	$-0.363131\pi$
$523$	6.23770e10	0.833715	0.416858	+	0.908972i	$0.363131\pi$
$524$	0	0
$525$	− 2.58935e9i	− 0.0340842i
$526$	0	0
$527$	4.96614e9i	0.0643838i
$528$	0	0
$529$	2.20424e10	0.281473
$530$	0	0
$531$	−1.27524e10	−0.160404
$532$	0	0
$533$	− 1.17434e10i	− 0.145508i
$534$	0	0
$535$	6.11998e10i	0.747025i
$536$	0	0
$537$	1.41797e10	0.170518
$538$	0	0
$539$	−7.03905e10	−0.833986
$540$	0	0
$541$	7.45917e10i	0.870766i	0.900245	+	0.435383i	$0.143387\pi$
$541$	7.45917e10i	0.870766i	−0.900245	+	0.435383i	$0.856613\pi$
$542$	0	0
$543$	− 4.82179e10i	− 0.554638i
$544$	0	0
$545$	−3.60337e10	−0.408435
$546$	0	0
$547$	−1.41531e9	−0.0158089	−0.00790445	−	0.999969i	$-0.502516\pi$
$547$	−1.41531e9	−0.0158089	−0.00790445	+	0.999969i	$0.502516\pi$
$548$	0	0
$549$	5.04758e10i	0.555641i
$550$	0	0
$551$	2.49833e9i	0.0271046i
$552$	0	0
$553$	5.02383e10	0.537198
$554$	0	0
$555$	−2.11624e11	−2.23046
$556$	0	0
$557$	1.37543e11i	1.42895i	0.699661	+	0.714475i	$0.253334\pi$
$557$	1.37543e11i	1.42895i	−0.699661	+	0.714475i	$0.746666\pi$
$558$	0	0
$559$	− 3.24270e10i	− 0.332093i
$560$	0	0
$561$	1.35000e11	1.36296
$562$	0	0
$563$	1.06415e11	1.05918	0.529589	−	0.848255i	$-0.322346\pi$
$563$	1.06415e11	1.05918	0.529589	+	0.848255i	$0.322346\pi$
$564$	0	0
$565$	3.36018e10i	0.329738i
$566$	0	0
$567$	− 7.52431e10i	− 0.728005i
$568$	0	0
$569$	−4.02429e10	−0.383919	−0.191960	−	0.981403i	$-0.561484\pi$
$569$	−4.02429e10	−0.383919	−0.191960	+	0.981403i	$0.561484\pi$
$570$	0	0
$571$	1.50341e11	1.41427	0.707137	−	0.707077i	$-0.249986\pi$
$571$	1.50341e11	1.41427	0.707137	+	0.707077i	$0.249986\pi$
$572$	0	0
$573$	9.91667e10i	0.919914i
$574$	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
$579$	−1.17499e11	−1.04549
$580$	0	0
$581$	7.26862e10i	0.637892i
$582$	0	0
$583$	− 1.52372e10i	− 0.131895i
$584$	0	0
$585$	−1.14215e10	−0.0975216
$586$	0	0
$587$	1.53440e11	1.29237	0.646185	−	0.763181i	$-0.276363\pi$
$587$	1.53440e11	1.29237	0.646185	+	0.763181i	$0.276363\pi$
$588$	0	0
$589$	1.32388e9i	0.0109999i
$590$	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
$595$	−6.23689e10	−0.497623
$596$	0	0
$597$	− 2.48837e11i	− 1.95892i
$598$	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	$-0.627073\pi$
$601$	−1.01422e11	−0.777382	−0.388691	+	0.921368i	$0.627073\pi$
$602$	0	0
$603$	−5.22236e10	−0.395001
$604$	0	0
$605$	7.76795e10i	0.579809i
$606$	0	0
$607$	1.97883e11i	1.45765i	0.684700	+	0.728825i	$0.259933\pi$
$607$	1.97883e11i	1.45765i	−0.684700	+	0.728825i	$0.740067\pi$
$608$	0	0
$609$	1.79224e10	0.130294
$610$	0	0
$611$	4.17158e10	0.299320
$612$	0	0
$613$	1.27158e11i	0.900538i	0.892893	+	0.450269i	$0.148672\pi$
$613$	1.27158e11i	0.900538i	−0.892893	+	0.450269i	$0.851328\pi$
$614$	0	0
$615$	1.30855e11i	0.914723i
$616$	0	0
$617$	5.06702e10	0.349632	0.174816	−	0.984601i	$-0.444067\pi$
$617$	5.06702e10	0.349632	0.174816	+	0.984601i	$0.444067\pi$
$618$	0	0
$619$	−7.06748e10	−0.481395	−0.240698	−	0.970600i	$-0.577376\pi$
$619$	−7.06748e10	−0.481395	−0.240698	+	0.970600i	$0.577376\pi$
$620$	0	0
$621$	− 7.43769e10i	− 0.500117i
$622$	0	0
$623$	1.16561e11i	0.773748i
$624$	0	0
$625$	−1.45008e11	−0.950327
$626$	0	0
$627$	3.59885e10	0.232859
$628$	0	0
$629$	− 2.53771e11i	− 1.62121i
$630$	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
$635$	1.56868e11	0.964804
$636$	0	0
$637$	2.08294e10i	0.126508i
$638$	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
$643$	2.65913e11	1.55559	0.777795	−	0.628518i	$-0.216338\pi$
$643$	2.65913e11	1.55559	0.777795	+	0.628518i	$0.216338\pi$
$644$	0	0
$645$	3.61328e11i	2.08767i
$646$	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
$651$	9.49716e9	0.0528774
$652$	0	0
$653$	− 3.03789e11i	− 1.67078i	−0.549656	−	0.835391i	$-0.685241\pi$
$653$	− 3.03789e11i	− 1.67078i	0.549656	−	0.835391i	$-0.314759\pi$
$654$	0	0
$655$	1.90427e11i	1.03458i
$656$	0	0
$657$	1.95988e10	0.105189
$658$	0	0
$659$	−4.18575e10	−0.221938	−0.110969	−	0.993824i	$-0.535395\pi$
$659$	−4.18575e10	−0.221938	−0.110969	+	0.993824i	$0.535395\pi$
$660$	0	0
$661$	− 2.46529e11i	− 1.29141i	−0.763589	−	0.645703i	$-0.776565\pi$
$661$	− 2.46529e11i	− 1.29141i	0.763589	−	0.645703i	$-0.223435\pi$
$662$	0	0
$663$	− 3.99482e10i	− 0.206749i
$664$	0	0
$665$	−1.66264e10	−0.0850179
$666$	0	0
$667$	3.04155e10	0.153671
$668$	0	0
$669$	− 1.47802e11i	− 0.737862i
$670$	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
$675$	5.80849e9	0.0279800
$676$	0	0
$677$	− 2.47236e10i	− 0.117695i	−0.998267	−	0.0588475i	$-0.981257\pi$
$677$	− 2.47236e10i	− 0.117695i	0.998267	−	0.0588475i	$-0.0187426\pi$
$678$	0	0
$679$	1.68731e11i	0.793811i
$680$	0	0
$681$	−7.49454e10	−0.348463
$682$	0	0
$683$	7.20843e10	0.331251	0.165626	−	0.986189i	$-0.447036\pi$
$683$	7.20843e10	0.331251	0.165626	+	0.986189i	$0.447036\pi$
$684$	0	0
$685$	− 1.35408e11i	− 0.615009i
$686$	0	0
$687$	2.84556e11i	1.27744i
$688$	0	0
$689$	−4.50887e9	−0.0200074
$690$	0	0
$691$	2.95424e11	1.29578	0.647892	−	0.761732i	$-0.275651\pi$
$691$	2.95424e11	1.29578	0.647892	+	0.761732i	$0.275651\pi$
$692$	0	0
$693$	− 8.85139e10i	− 0.383776i
$694$	0	0
$695$	− 1.79968e11i	− 0.771360i
$696$	0	0
$697$	−1.56916e11	−0.664867
$698$	0	0
$699$	2.20444e10	0.0923400
$700$	0	0
$701$	2.87925e11i	1.19236i	0.802851	+	0.596180i	$0.203315\pi$
$701$	2.87925e11i	1.19236i	−0.802851	+	0.596180i	$0.796685\pi$
$702$	0	0
$703$	− 6.76504e10i	− 0.276980i
$704$	0	0
$705$	−4.64831e11	−1.88165
$706$	0	0
$707$	−3.87834e10	−0.155227
$708$	0	0
$709$	2.51685e11i	0.996030i	0.867168	+	0.498015i	$0.165938\pi$
$709$	2.51685e11i	0.996030i	−0.867168	+	0.498015i	$0.834062\pi$
$710$	0	0
$711$	− 1.22931e11i	− 0.481042i
$712$	0	0
$713$	1.61174e10	0.0623643
$714$	0	0
$715$	6.16795e10	0.236003
$716$	0	0
$717$	− 4.04170e11i	− 1.52928i
$718$	0	0
$719$	1.38856e11i	0.519574i	0.965666	+	0.259787i	$0.0836524\pi$
$719$	1.38856e11i	0.519574i	−0.965666	+	0.259787i	$0.916348\pi$
$720$	0	0
$721$	−1.46184e11	−0.540953
$722$	0	0
$723$	−6.17489e11	−2.25983
$724$	0	0
$725$	2.37531e9i	0.00859743i
$726$	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
$731$	−4.33289e11	−1.51743
$732$	0	0
$733$	− 2.17618e11i	− 0.753839i	−0.926246	−	0.376920i	$-0.876983\pi$
$733$	− 2.17618e11i	− 0.753839i	0.926246	−	0.376920i	$-0.123017\pi$
$734$	0	0
$735$	− 2.32098e11i	− 0.795285i
$736$	0	0
$737$	2.82023e11	0.955904
$738$	0	0
$739$	−4.84950e11	−1.62599	−0.812997	−	0.582268i	$-0.802166\pi$
$739$	−4.84950e11	−1.62599	−0.812997	+	0.582268i	$0.802166\pi$
$740$	0	0
$741$	− 1.06494e10i	− 0.0353227i
$742$	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
$747$	1.77860e11	0.571210
$748$	0	0
$749$	1.40346e11i	0.445937i
$750$	0	0
$751$	− 2.34693e11i	− 0.737804i	−0.929468	−	0.368902i	$-0.879734\pi$
$751$	− 2.34693e11i	− 0.737804i	0.929468	−	0.368902i	$-0.120266\pi$
$752$	0	0
$753$	−5.20950e11	−1.62038
$754$	0	0
$755$	5.10561e11	1.57130
$756$	0	0
$757$	− 3.84882e11i	− 1.17204i	−0.810295	−	0.586022i	$-0.800693\pi$
$757$	− 3.84882e11i	− 1.17204i	0.810295	−	0.586022i	$-0.199307\pi$
$758$	0	0
$759$	− 4.38136e11i	− 1.32021i
$760$	0	0
$761$	−2.39209e11	−0.713244	−0.356622	−	0.934249i	$-0.616072\pi$
$761$	−2.39209e11	−0.713244	−0.356622	+	0.934249i	$0.616072\pi$
$762$	0	0
$763$	−8.26340e10	−0.243815
$764$	0	0
$765$	1.52614e11i	0.445604i
$766$	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
$771$	6.13202e11	1.73535
$772$	0	0
$773$	− 5.54469e10i	− 0.155296i	−0.996981	−	0.0776478i	$-0.975259\pi$
$773$	− 5.54469e10i	− 0.155296i	0.996981	−	0.0776478i	$-0.0247410\pi$
$774$	0	0
$775$	1.25869e9i	0.00348909i
$776$	0	0
$777$	−4.85306e11	−1.33147
$778$	0	0
$779$	−4.18307e10	−0.113591
$780$	0	0
$781$	2.21091e10i	0.0594246i
$782$	0	0
$783$	4.02039e10i	0.106960i
$784$	0	0
$785$	−1.65504e11	−0.435844
$786$	0	0
$787$	−4.05908e11	−1.05811	−0.529053	−	0.848589i	$-0.677453\pi$
$787$	−4.05908e11	−1.05811	−0.529053	+	0.848589i	$0.677453\pi$
$788$	0	0
$789$	− 6.96358e11i	− 1.79690i
$790$	0	0
$791$	7.70572e10i	0.196837i
$792$	0	0
$793$	−8.06610e10	−0.203972
$794$	0	0
$795$	5.02414e10	0.125775
$796$	0	0
$797$	3.09015e11i	0.765855i	0.923778	+	0.382927i	$0.125084\pi$
$797$	3.09015e11i	0.765855i	−0.923778	+	0.382927i	$0.874916\pi$
$798$	0	0
$799$	− 5.57406e11i	− 1.36768i
$800$	0	0
$801$	2.85219e11	0.692864
$802$	0	0
$803$	−1.05839e11	−0.254557
$804$	0	0
$805$	2.02415e11i	0.482014i
$806$	0	0
$807$	2.70504e11i	0.637792i
$808$	0	0
$809$	−4.77958e11	−1.11582	−0.557912	−	0.829900i	$-0.688397\pi$
$809$	−4.77958e11	−1.11582	−0.557912	+	0.829900i	$0.688397\pi$
$810$	0	0
$811$	−6.37503e11	−1.47366	−0.736832	−	0.676075i	$-0.763679\pi$
$811$	−6.37503e11	−1.47366	−0.736832	+	0.676075i	$0.763679\pi$
$812$	0	0
$813$	− 7.98393e11i	− 1.82749i
$814$	0	0
$815$	− 3.52890e11i	− 0.799851i
$816$	0	0
$817$	−1.15506e11	−0.259250
$818$	0	0
$819$	−2.61924e10	−0.0582155
$820$	0	0
$821$	− 8.43824e11i	− 1.85729i	−0.370973	−	0.928644i	$-0.620976\pi$
$821$	− 8.43824e11i	− 1.85729i	0.370973	−	0.928644i	$-0.379024\pi$
$822$	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
$827$	2.78675e11	0.595765	0.297883	−	0.954602i	$-0.403720\pi$
$827$	2.78675e11	0.595765	0.297883	+	0.954602i	$0.403720\pi$
$828$	0	0
$829$	− 4.75156e11i	− 1.00605i	−0.864273	−	0.503023i	$-0.832221\pi$
$829$	− 4.75156e11i	− 1.00605i	0.864273	−	0.503023i	$-0.167779\pi$
$830$	0	0
$831$	8.22306e11i	1.72437i
$832$	0	0
$833$	2.78322e11	0.578053
$834$	0	0
$835$	−2.85552e11	−0.587408
$836$	0	0
$837$	2.13043e10i	0.0434075i
$838$	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
$843$	3.07858e11	0.609594
$844$	0	0
$845$	4.79344e11i	0.940200i
$846$	0	0
$847$	1.78138e11i	0.346117i
$848$	0	0
$849$	−1.17019e11	−0.225229
$850$	0	0
$851$	−8.23600e11	−1.57036
$852$	0	0
$853$	− 9.75408e10i	− 0.184243i	−0.995748	−	0.0921213i	$-0.970635\pi$
$853$	− 9.75408e10i	− 0.184243i	0.995748	−	0.0921213i	$-0.0293648\pi$
$854$	0	0
$855$	4.06840e10i	0.0761306i
$856$	0	0
$857$	7.94769e10	0.147339	0.0736695	−	0.997283i	$-0.476529\pi$
$857$	7.94769e10	0.147339	0.0736695	+	0.997283i	$0.476529\pi$
$858$	0	0
$859$	−4.15618e11	−0.763347	−0.381673	−	0.924297i	$-0.624652\pi$
$859$	−4.15618e11	−0.763347	−0.381673	+	0.924297i	$0.624652\pi$
$860$	0	0
$861$	3.00083e11i	0.546045i
$862$	0	0
$863$	4.80012e11i	0.865383i	0.901542	+	0.432692i	$0.142436\pi$
$863$	4.80012e11i	0.865383i	−0.901542	+	0.432692i	$0.857564\pi$
$864$	0	0
$865$	−1.25780e11	−0.224671
$866$	0	0
$867$	1.63230e11	0.288884
$868$	0	0
$869$	6.63863e11i	1.16412i
$870$	0	0
$871$	− 8.34540e10i	− 0.145002i
$872$	0	0
$873$	4.12879e11	0.710830
$874$	0	0
$875$	−3.49134e11	−0.595608
$876$	0	0
$877$	− 2.74155e11i	− 0.463444i	−0.972782	−	0.231722i	$-0.925564\pi$
$877$	− 2.74155e11i	− 0.463444i	0.972782	−	0.231722i	$-0.0744360\pi$
$878$	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
$883$	9.95008e11	1.63676	0.818378	−	0.574681i	$-0.194874\pi$
$883$	9.95008e11	1.63676	0.818378	+	0.574681i	$0.194874\pi$
$884$	0	0
$885$	2.27075e11i	0.370165i
$886$	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
$891$	9.94283e11	1.57761
$892$	0	0
$893$	− 1.48594e11i	− 0.233665i
$894$	0	0
$895$	− 8.65656e10i	− 0.134913i
$896$	0	0
$897$	−1.29650e11	−0.200264
$898$	0	0
$899$	−8.71212e9	−0.0133378
$900$	0	0
$901$	6.02474e10i	0.0914195i
$902$	0	0
$903$	8.28613e11i	1.24624i
$904$	0	0
$905$	−2.94365e11	−0.438826
$906$	0	0
$907$	6.55018e11	0.967886	0.483943	−	0.875100i	$-0.339204\pi$
$907$	6.55018e11	0.967886	0.483943	+	0.875100i	$0.339204\pi$
$908$	0	0
$909$	9.49014e10i	0.139001i
$910$	0	0
$911$	1.19425e11i	0.173389i	0.996235	+	0.0866943i	$0.0276304\pi$
$911$	1.19425e11i	0.173389i	−0.996235	+	0.0866943i	$0.972370\pi$
$912$	0	0
$913$	−9.60496e11	−1.38233
$914$	0	0
$915$	8.98791e11	1.28225
$916$	0	0
$917$	4.36696e11i	0.617592i
$918$	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
$923$	6.54235e9	0.00901420
$924$	0	0
$925$	− 6.43194e10i	− 0.0878567i
$926$	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
$931$	7.41953e10	0.0987593
$932$	0	0
$933$	1.29703e12i	1.71168i
$934$	0	0
$935$	− 8.24161e11i	− 1.07836i
$936$	0	0
$937$	8.41436e11	1.09160	0.545799	−	0.837916i	$-0.316226\pi$
$937$	8.41436e11	1.09160	0.545799	+	0.837916i	$0.316226\pi$
$938$	0	0
$939$	−6.30660e11	−0.811209
$940$	0	0
$941$	4.52935e11i	0.577666i	0.957379	+	0.288833i	$0.0932673\pi$
$941$	4.52935e11i	0.577666i	−0.957379	+	0.288833i	$0.906733\pi$
$942$	0	0
$943$	5.09262e11i	0.644013i
$944$	0	0
$945$	−2.67556e11	−0.335497
$946$	0	0
$947$	1.00309e12	1.24721	0.623605	−	0.781739i	$-0.285667\pi$
$947$	1.00309e12	1.24721	0.623605	+	0.781739i	$0.285667\pi$
$948$	0	0
$949$	3.13192e10i	0.0386141i
$950$	0	0
$951$	1.65769e12i	2.02666i
$952$	0	0
$953$	1.39040e12	1.68565	0.842826	−	0.538186i	$-0.180890\pi$
$953$	1.39040e12	1.68565	0.842826	+	0.538186i	$0.180890\pi$
$954$	0	0
$955$	6.05401e11	0.727830
$956$	0	0
$957$	2.36831e11i	0.282352i
$958$	0	0
$959$	− 3.10523e11i	− 0.367130i
$960$	0	0
$961$	8.48274e11	0.994587
$962$	0	0
$963$	3.43421e11	0.399321
$964$	0	0
$965$	7.17318e11i	0.827185i
$966$	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
$971$	1.28411e12	1.44452	0.722261	−	0.691621i	$-0.243103\pi$
$971$	1.28411e12	1.44452	0.722261	+	0.691621i	$0.243103\pi$
$972$	0	0
$973$	− 4.12712e11i	− 0.460464i
$974$	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
$979$	−1.54026e12	−1.67674
$980$	0	0
$981$	2.02202e11i	0.218328i
$982$	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
$987$	−1.06597e12	−1.12325
$988$	0	0
$989$	1.40622e12i	1.46983i
$990$	0	0
$991$	3.35563e11i	0.347920i	0.984753	+	0.173960i	$0.0556563\pi$
$991$	3.35563e11i	0.347920i	−0.984753	+	0.173960i	$0.944344\pi$
$992$	0	0
$993$	−5.48200e11	−0.563822
$994$	0	0
$995$	−1.51912e12	−1.54989
$996$	0	0
$997$	− 1.03163e12i	− 1.04410i	−0.852915	−	0.522050i	$-0.825167\pi$
$997$	− 1.03163e12i	− 1.04410i	0.852915	−	0.522050i	$-0.174833\pi$
$998$	0	0
$999$	− 1.08865e12i	− 1.09302i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.9.d.e.127.2		4
4.3	odd	2	inner	256.9.d.e.127.4		4
8.3	odd	2	inner	256.9.d.e.127.1		4
8.5	even	2	inner	256.9.d.e.127.3		4
16.3	odd	4		4.9.b.b.3.2	yes	2
16.5	even	4		64.9.c.b.63.2		2
16.11	odd	4		64.9.c.b.63.1		2
16.13	even	4		4.9.b.b.3.1	✓	2
48.29	odd	4		36.9.d.b.19.2		2
48.35	even	4		36.9.d.b.19.1		2
80.3	even	4		100.9.d.b.99.4		4
80.13	odd	4		100.9.d.b.99.2		4
80.19	odd	4		100.9.b.c.51.1		2
80.29	even	4		100.9.b.c.51.2		2
80.67	even	4		100.9.d.b.99.1		4
80.77	odd	4		100.9.d.b.99.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.9.b.b.3.1	✓	2	16.13	even	4
4.9.b.b.3.2	yes	2	16.3	odd	4
36.9.d.b.19.1		2	48.35	even	4
36.9.d.b.19.2		2	48.29	odd	4
64.9.c.b.63.1		2	16.11	odd	4
64.9.c.b.63.2		2	16.5	even	4
100.9.b.c.51.1		2	80.19	odd	4
100.9.b.c.51.2		2	80.29	even	4
100.9.d.b.99.1		4	80.67	even	4
100.9.d.b.99.2		4	80.13	odd	4
100.9.d.b.99.3		4	80.77	odd	4
100.9.d.b.99.4		4	80.3	even	4
256.9.d.e.127.1		4	8.3	odd	2	inner
256.9.d.e.127.2		4	1.1	even	1	trivial
256.9.d.e.127.3		4	8.5	even	2	inner
256.9.d.e.127.4		4	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	256.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-99.9200 q^{3} +610.000i q^{5} +1398.88i q^{7} +3423.00 q^{9} -18485.2 q^{11} +5470.00i q^{13} -60951.2i q^{15} +73090.0 q^{17} +19484.4 q^{19} -139776. i q^{21} -237210. i q^{23} +18525.0 q^{25} +313549. q^{27} +128222. i q^{29} +67945.6i q^{31} +1.84704e6 q^{33} -853317. q^{35} -3.47203e6i q^{37} -546562. i q^{39} -2.14688e6 q^{41} -5.92815e6 q^{43} +2.08803e6i q^{45} -7.62629e6i q^{47} +3.80794e6 q^{49} -7.30315e6 q^{51} +824290. i q^{53} -1.12760e7i q^{55} -1.94688e6 q^{57} -3.72552e6 q^{59} +1.47461e7i q^{61} +4.78836e6i q^{63} -3.33670e6 q^{65} -1.52567e7 q^{67} +2.37020e7i q^{69} -1.19604e6i q^{71} +5.72563e6 q^{73} -1.85102e6 q^{75} -2.58586e7i q^{77} -3.59132e7i q^{79} -5.37881e7 q^{81} +5.19603e7 q^{83} +4.45849e7i q^{85} -1.28119e7i q^{87} +8.33242e7 q^{89} -7.65187e6 q^{91} -6.78912e6i q^{93} +1.18855e7i q^{95} +1.20619e8 q^{97} -6.32748e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 13692 q^{9} + 292360 q^{17} + 74100 q^{25} + 7388160 q^{33} - 8587528 q^{41} + 15231748 q^{49} - 7787520 q^{57} - 13346800 q^{65} + 22902520 q^{73} - 215152380 q^{81} + 333296888 q^{89} + 482476040 q^{97}+O(q^{100}) \) 4 * q + 13692 * q^9 + 292360 * q^17 + 74100 * q^25 + 7388160 * q^33 - 8587528 * q^41 + 15231748 * q^49 - 7787520 * q^57 - 13346800 * q^65 + 22902520 * q^73 - 215152380 * q^81 + 333296888 * q^89 + 482476040 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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