LMFDB - Embedded newform 256.11.c.h.255.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,11,Mod(255,256)]

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, 0]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	256.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:	$162.651456684$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 573x^{2} + 26406 $ x^4 + 573*x^2 + 26406
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3 $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.3
Root			$7.10928i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.h.255.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+113.748i q^{3} +2478.67 q^{5} -24856.1i q^{7} +46110.3 q^{9} +O(q^{10})$	$q+113.748i q^{3} +2478.67 q^{5} -24856.1i q^{7} +46110.3 q^{9} -198508. i q^{11} +440436. q^{13} +281944. i q^{15} +2.05305e6 q^{17} +4.14273e6i q^{19} +2.82735e6 q^{21} +4.26728e6i q^{23} -3.62184e6 q^{25} +1.19617e7i q^{27} +2.50229e6 q^{29} +5.59162e7i q^{31} +2.25800e7 q^{33} -6.16101e7i q^{35} +9.82631e6 q^{37} +5.00989e7i q^{39} +7.52562e7 q^{41} -1.57668e8i q^{43} +1.14292e8 q^{45} +1.30737e8i q^{47} -3.35352e8 q^{49} +2.33532e8i q^{51} +7.15357e8 q^{53} -4.92035e8i q^{55} -4.71229e8 q^{57} +1.91372e8i q^{59} +1.27289e8 q^{61} -1.14612e9i q^{63} +1.09169e9 q^{65} -1.67019e9i q^{67} -4.85397e8 q^{69} +3.57334e8i q^{71} +1.33551e9 q^{73} -4.11979e8i q^{75} -4.93414e9 q^{77} -1.85555e9i q^{79} +1.36214e9 q^{81} +1.21966e9i q^{83} +5.08883e9 q^{85} +2.84632e8i q^{87} -6.21472e8 q^{89} -1.09475e10i q^{91} -6.36039e9 q^{93} +1.02684e10i q^{95} -1.01984e10 q^{97} -9.15325e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5600 q^{5} - 57180 q^{9}+O(q^{10}) $ 4 * q + 5600 * q^5 - 57180 * q^9	$ 4 q + 5600 q^{5} - 57180 q^{9} - 296352 q^{13} + 1895544 q^{17} - 12288 q^{21} - 26568420 q^{25} + 20066656 q^{29} - 978432 q^{33} + 154467936 q^{37} + 189395640 q^{41} + 180576480 q^{45} - 225809468 q^{49} + 831839968 q^{53} + 87436800 q^{57} - 936776736 q^{61} + 1805103360 q^{65} + 4959940608 q^{69} - 2975489720 q^{73} - 10845720576 q^{77} - 1910976444 q^{81} + 9467334720 q^{85} + 11059844040 q^{89} - 40085716992 q^{93} - 917012488 q^{97}+O(q^{100}) $ 4 * q + 5600 * q^5 - 57180 * q^9 - 296352 * q^13 + 1895544 * q^17 - 12288 * q^21 - 26568420 * q^25 + 20066656 * q^29 - 978432 * q^33 + 154467936 * q^37 + 189395640 * q^41 + 180576480 * q^45 - 225809468 * q^49 + 831839968 * q^53 + 87436800 * q^57 - 936776736 * q^61 + 1805103360 * q^65 + 4959940608 * q^69 - 2975489720 * q^73 - 10845720576 * q^77 - 1910976444 * q^81 + 9467334720 * q^85 + 11059844040 * q^89 - 40085716992 * q^93 - 917012488 * q^97

Display 100 coefficients 10 coefficients

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
$3$	113.748i	0.468101i	0.972224	+	0.234050i	$0.0751981\pi$
$3$	113.748i	0.468101i	−0.972224	+	0.234050i	$0.924802\pi$
$4$	0	0
$5$	2478.67	0.793173	0.396587	−	0.917997i	$-0.370195\pi$
$5$	2478.67	0.793173	0.396587	+	0.917997i	$0.370195\pi$
$6$	0	0
$7$	− 24856.1i	− 1.47892i	−0.673203	−	0.739458i	$-0.735082\pi$
$7$	− 24856.1i	− 1.47892i	0.673203	−	0.739458i	$-0.264918\pi$
$8$	0	0
$9$	46110.3	0.780882
$10$	0	0
$11$	− 198508.i	− 1.23258i	−0.787520	−	0.616289i	$-0.788635\pi$
$11$	− 198508.i	− 1.23258i	0.787520	−	0.616289i	$-0.211365\pi$
$12$	0	0
$13$	440436.	1.18622	0.593111	−	0.805121i	$-0.297900\pi$
$13$	440436.	1.18622	0.593111	+	0.805121i	$0.297900\pi$
$14$	0	0
$15$	281944.i	0.371285i
$16$	0	0
$17$	2.05305e6	1.44596	0.722979	−	0.690870i	$-0.242772\pi$
$17$	2.05305e6	1.44596	0.722979	+	0.690870i	$0.242772\pi$
$18$	0	0
$19$	4.14273e6i	1.67309i	0.547900	+	0.836544i	$0.315427\pi$
$19$	4.14273e6i	1.67309i	−0.547900	+	0.836544i	$0.684573\pi$
$20$	0	0
$21$	2.82735e6	0.692281
$22$	0	0
$23$	4.26728e6i	0.662998i	0.943456	+	0.331499i	$0.107554\pi$
$23$	4.26728e6i	0.662998i	−0.943456	+	0.331499i	$0.892446\pi$
$24$	0	0
$25$	−3.62184e6	−0.370876
$26$	0	0
$27$	1.19617e7i	0.833632i
$28$	0	0
$29$	2.50229e6	0.121997	0.0609984	−	0.998138i	$-0.480572\pi$
$29$	2.50229e6	0.121997	0.0609984	+	0.998138i	$0.480572\pi$
$30$	0	0
$31$	5.59162e7i	1.95312i	0.215240	+	0.976561i	$0.430947\pi$
$31$	5.59162e7i	1.95312i	−0.215240	+	0.976561i	$0.569053\pi$
$32$	0	0
$33$	2.25800e7	0.576970
$34$	0	0
$35$	− 6.16101e7i	− 1.17304i
$36$	0	0
$37$	9.82631e6	0.141704	0.0708520	−	0.997487i	$-0.477428\pi$
$37$	9.82631e6	0.141704	0.0708520	+	0.997487i	$0.477428\pi$
$38$	0	0
$39$	5.00989e7i	0.555271i
$40$	0	0
$41$	7.52562e7	0.649565	0.324783	−	0.945789i	$-0.394709\pi$
$41$	7.52562e7	0.649565	0.324783	+	0.945789i	$0.394709\pi$
$42$	0	0
$43$	− 1.57668e8i	− 1.07251i	−0.844057	−	0.536253i	$-0.819839\pi$
$43$	− 1.57668e8i	− 1.07251i	0.844057	−	0.536253i	$-0.180161\pi$
$44$	0	0
$45$	1.14292e8	0.619374
$46$	0	0
$47$	1.30737e8i	0.570047i	0.958521	+	0.285023i	$0.0920014\pi$
$47$	1.30737e8i	0.570047i	−0.958521	+	0.285023i	$0.907999\pi$
$48$	0	0
$49$	−3.35352e8	−1.18719
$50$	0	0
$51$	2.33532e8i	0.676854i
$52$	0	0
$53$	7.15357e8	1.71058	0.855290	−	0.518150i	$-0.173379\pi$
$53$	7.15357e8	1.71058	0.855290	+	0.518150i	$0.173379\pi$
$54$	0	0
$55$	− 4.92035e8i	− 0.977647i
$56$	0	0
$57$	−4.71229e8	−0.783173
$58$	0	0
$59$	1.91372e8i	0.267681i	0.991003	+	0.133841i	$0.0427310\pi$
$59$	1.91372e8i	0.267681i	−0.991003	+	0.133841i	$0.957269\pi$
$60$	0	0
$61$	1.27289e8	0.150710	0.0753552	−	0.997157i	$-0.475991\pi$
$61$	1.27289e8	0.150710	0.0753552	+	0.997157i	$0.475991\pi$
$62$	0	0
$63$	− 1.14612e9i	− 1.15486i
$64$	0	0
$65$	1.09169e9	0.940879
$66$	0	0
$67$	− 1.67019e9i	− 1.23706i	−0.785761	−	0.618530i	$-0.787729\pi$
$67$	− 1.67019e9i	− 1.23706i	0.785761	−	0.618530i	$-0.212271\pi$
$68$	0	0
$69$	−4.85397e8	−0.310350
$70$	0	0
$71$	3.57334e8i	0.198053i	0.995085	+	0.0990266i	$0.0315729\pi$
$71$	3.57334e8i	0.198053i	−0.995085	+	0.0990266i	$0.968427\pi$
$72$	0	0
$73$	1.33551e9	0.644218	0.322109	−	0.946703i	$-0.395608\pi$
$73$	1.33551e9	0.644218	0.322109	+	0.946703i	$0.395608\pi$
$74$	0	0
$75$	− 4.11979e8i	− 0.173608i
$76$	0	0
$77$	−4.93414e9	−1.82288
$78$	0	0
$79$	− 1.85555e9i	− 0.603027i	−0.953462	−	0.301513i	$-0.902508\pi$
$79$	− 1.85555e9i	− 0.603027i	0.953462	−	0.301513i	$-0.0974918\pi$
$80$	0	0
$81$	1.36214e9	0.390658
$82$	0	0
$83$	1.21966e9i	0.309633i	0.987943	+	0.154816i	$0.0494786\pi$
$83$	1.21966e9i	0.309633i	−0.987943	+	0.154816i	$0.950521\pi$
$84$	0	0
$85$	5.08883e9	1.14689
$86$	0	0
$87$	2.84632e8i	0.0571068i
$88$	0	0
$89$	−6.21472e8	−0.111294	−0.0556469	−	0.998451i	$-0.517722\pi$
$89$	−6.21472e8	−0.111294	−0.0556469	+	0.998451i	$0.517722\pi$
$90$	0	0
$91$	− 1.09475e10i	− 1.75432i
$92$	0	0
$93$	−6.36039e9	−0.914258
$94$	0	0
$95$	1.02684e10i	1.32705i
$96$	0	0
$97$	−1.01984e10	−1.18761	−0.593805	−	0.804609i	$-0.702375\pi$
$97$	−1.01984e10	−1.18761	−0.593805	+	0.804609i	$0.702375\pi$
$98$	0	0
$99$	− 9.15325e9i	− 0.962497i
$100$	0	0
$101$	9.61479e9	0.914814	0.457407	−	0.889257i	$-0.348778\pi$
$101$	9.61479e9	0.914814	0.457407	+	0.889257i	$0.348778\pi$
$102$	0	0
$103$	1.70958e10i	1.47470i	0.675511	+	0.737350i	$0.263923\pi$
$103$	1.70958e10i	1.47470i	−0.675511	+	0.737350i	$0.736077\pi$
$104$	0	0
$105$	7.00805e9	0.549099
$106$	0	0
$107$	1.96185e10i	1.39877i	0.714745	+	0.699385i	$0.246543\pi$
$107$	1.96185e10i	1.39877i	−0.714745	+	0.699385i	$0.753457\pi$
$108$	0	0
$109$	−2.72654e10	−1.77207	−0.886033	−	0.463623i	$-0.846549\pi$
$109$	−2.72654e10	−1.77207	−0.886033	+	0.463623i	$0.846549\pi$
$110$	0	0
$111$	1.11773e9i	0.0663317i
$112$	0	0
$113$	2.24716e10	1.21967	0.609833	−	0.792530i	$-0.291236\pi$
$113$	2.24716e10	1.21967	0.609833	+	0.792530i	$0.291236\pi$
$114$	0	0
$115$	1.05772e10i	0.525872i
$116$	0	0
$117$	2.03086e10	0.926298
$118$	0	0
$119$	− 5.10310e10i	− 2.13845i
$120$	0	0
$121$	−1.34679e10	−0.519247
$122$	0	0
$123$	8.56027e9i	0.304062i
$124$	0	0
$125$	−3.31831e10	−1.08734
$126$	0	0
$127$	− 1.50881e9i	− 0.0456685i	−0.999739	−	0.0228343i	$-0.992731\pi$
$127$	− 1.50881e9i	− 0.0456685i	0.999739	−	0.0228343i	$-0.00726901\pi$
$128$	0	0
$129$	1.79344e10	0.502041
$130$	0	0
$131$	1.84786e10i	0.478974i	0.970900	+	0.239487i	$0.0769793\pi$
$131$	1.84786e10i	0.478974i	−0.970900	+	0.239487i	$0.923021\pi$
$132$	0	0
$133$	1.02972e11	2.47436
$134$	0	0
$135$	2.96491e10i	0.661214i
$136$	0	0
$137$	4.39219e10	0.910077	0.455038	−	0.890472i	$-0.349626\pi$
$137$	4.39219e10	0.910077	0.455038	+	0.890472i	$0.349626\pi$
$138$	0	0
$139$	− 8.34667e10i	− 1.60857i	−0.594246	−	0.804284i	$-0.702549\pi$
$139$	− 8.34667e10i	− 1.60857i	0.594246	−	0.804284i	$-0.297451\pi$
$140$	0	0
$141$	−1.48712e10	−0.266839
$142$	0	0
$143$	− 8.74299e10i	− 1.46211i
$144$	0	0
$145$	6.20235e9	0.0967646
$146$	0	0
$147$	− 3.81458e10i	− 0.555725i
$148$	0	0
$149$	2.01015e10	0.273714	0.136857	−	0.990591i	$-0.456300\pi$
$149$	2.01015e10	0.273714	0.136857	+	0.990591i	$0.456300\pi$
$150$	0	0
$151$	− 4.37020e10i	− 0.556694i	−0.960481	−	0.278347i	$-0.910213\pi$
$151$	− 4.37020e10i	− 0.556694i	0.960481	−	0.278347i	$-0.0897865\pi$
$152$	0	0
$153$	9.46669e10	1.12912
$154$	0	0
$155$	1.38598e11i	1.54916i
$156$	0	0
$157$	2.39037e10	0.250592	0.125296	−	0.992119i	$-0.460012\pi$
$157$	2.39037e10	0.250592	0.125296	+	0.992119i	$0.460012\pi$
$158$	0	0
$159$	8.13707e10i	0.800724i
$160$	0	0
$161$	1.06068e11	0.980518
$162$	0	0
$163$	9.99893e10i	0.868991i	0.900674	+	0.434496i	$0.143073\pi$
$163$	9.99893e10i	0.868991i	−0.900674	+	0.434496i	$0.856927\pi$
$164$	0	0
$165$	5.59682e10	0.457637
$166$	0	0
$167$	1.81417e11i	1.39667i	0.715770	+	0.698336i	$0.246076\pi$
$167$	1.81417e11i	1.39667i	−0.715770	+	0.698336i	$0.753924\pi$
$168$	0	0
$169$	5.61250e10	0.407121
$170$	0	0
$171$	1.91022e11i	1.30648i
$172$	0	0
$173$	−2.70487e11	−1.74549	−0.872743	−	0.488179i	$-0.837661\pi$
$173$	−2.70487e11	−1.74549	−0.872743	+	0.488179i	$0.837661\pi$
$174$	0	0
$175$	9.00250e10i	0.548495i
$176$	0	0
$177$	−2.17683e10	−0.125302
$178$	0	0
$179$	− 1.71231e10i	− 0.0931790i	−0.998914	−	0.0465895i	$-0.985165\pi$
$179$	− 1.71231e10i	− 0.0931790i	0.998914	−	0.0465895i	$-0.0148353\pi$
$180$	0	0
$181$	−2.17519e11	−1.11971	−0.559853	−	0.828592i	$-0.689142\pi$
$181$	−2.17519e11	−1.11971	−0.559853	+	0.828592i	$0.689142\pi$
$182$	0	0
$183$	1.44790e10i	0.0705476i
$184$	0	0
$185$	2.43561e10	0.112396
$186$	0	0
$187$	− 4.07547e11i	− 1.78225i
$188$	0	0
$189$	2.97322e11	1.23287
$190$	0	0
$191$	− 5.66351e10i	− 0.222802i	−0.993776	−	0.111401i	$-0.964466\pi$
$191$	− 5.66351e10i	− 0.222802i	0.993776	−	0.111401i	$-0.0355338\pi$
$192$	0	0
$193$	3.84201e11	1.43473	0.717367	−	0.696695i	$-0.245347\pi$
$193$	3.84201e11	1.43473	0.717367	+	0.696695i	$0.245347\pi$
$194$	0	0
$195$	1.24178e11i	0.440426i
$196$	0	0
$197$	−3.92596e11	−1.32317	−0.661583	−	0.749872i	$-0.730115\pi$
$197$	−3.92596e11	−1.32317	−0.661583	+	0.749872i	$0.730115\pi$
$198$	0	0
$199$	− 1.37156e11i	− 0.439492i	−0.975557	−	0.219746i	$-0.929477\pi$
$199$	− 1.37156e11i	− 0.439492i	0.975557	−	0.219746i	$-0.0705228\pi$
$200$	0	0
$201$	1.89981e11	0.579068
$202$	0	0
$203$	− 6.21974e10i	− 0.180423i
$204$	0	0
$205$	1.86535e11	0.515218
$206$	0	0
$207$	1.96766e11i	0.517723i
$208$	0	0
$209$	8.22364e11	2.06221
$210$	0	0
$211$	− 4.12521e11i	− 0.986355i	−0.869929	−	0.493178i	$-0.835835\pi$
$211$	− 4.12521e11i	− 0.986355i	0.869929	−	0.493178i	$-0.164165\pi$
$212$	0	0
$213$	−4.06461e10	−0.0927089
$214$	0	0
$215$	− 3.90805e11i	− 0.850684i
$216$	0	0
$217$	1.38986e12	2.88850
$218$	0	0
$219$	1.51912e11i	0.301559i
$220$	0	0
$221$	9.04238e11	1.71523
$222$	0	0
$223$	− 3.18484e11i	− 0.577514i	−0.957402	−	0.288757i	$-0.906758\pi$
$223$	− 3.18484e11i	− 0.577514i	0.957402	−	0.288757i	$-0.0932420\pi$
$224$	0	0
$225$	−1.67004e11	−0.289611
$226$	0	0
$227$	− 5.17419e11i	− 0.858446i	−0.903198	−	0.429223i	$-0.858787\pi$
$227$	− 5.17419e11i	− 0.858446i	0.903198	−	0.429223i	$-0.141213\pi$
$228$	0	0
$229$	−5.46665e11	−0.868048	−0.434024	−	0.900901i	$-0.642907\pi$
$229$	−5.46665e11	−0.868048	−0.434024	+	0.900901i	$0.642907\pi$
$230$	0	0
$231$	− 5.61251e11i	− 0.853290i
$232$	0	0
$233$	1.60623e11	0.233898	0.116949	−	0.993138i	$-0.462689\pi$
$233$	1.60623e11	0.233898	0.116949	+	0.993138i	$0.462689\pi$
$234$	0	0
$235$	3.24054e11i	0.452146i
$236$	0	0
$237$	2.11066e11	0.282277
$238$	0	0
$239$	− 8.13513e11i	− 1.04322i	−0.853185	−	0.521609i	$-0.825332\pi$
$239$	− 8.13513e11i	− 1.04322i	0.853185	−	0.521609i	$-0.174668\pi$
$240$	0	0
$241$	3.78475e11	0.465534	0.232767	−	0.972532i	$-0.425222\pi$
$241$	3.78475e11	0.465534	0.232767	+	0.972532i	$0.425222\pi$
$242$	0	0
$243$	8.61268e11i	1.01650i
$244$	0	0
$245$	−8.31226e11	−0.941648
$246$	0	0
$247$	1.82461e12i	1.98465i
$248$	0	0
$249$	−1.38734e11	−0.144939
$250$	0	0
$251$	− 1.08609e12i	− 1.09018i	−0.838377	−	0.545090i	$-0.816495\pi$
$251$	− 1.08609e12i	− 1.09018i	0.838377	−	0.545090i	$-0.183505\pi$
$252$	0	0
$253$	8.47089e11	0.817197
$254$	0	0
$255$	5.78847e11i	0.536862i
$256$	0	0
$257$	1.66097e12	1.48148	0.740740	−	0.671792i	$-0.234475\pi$
$257$	1.66097e12	1.48148	0.740740	+	0.671792i	$0.234475\pi$
$258$	0	0
$259$	− 2.44244e11i	− 0.209568i
$260$	0	0
$261$	1.15381e11	0.0952651
$262$	0	0
$263$	− 1.22416e12i	− 0.972881i	−0.873714	−	0.486441i	$-0.838295\pi$
$263$	− 1.22416e12i	− 0.972881i	0.873714	−	0.486441i	$-0.161705\pi$
$264$	0	0
$265$	1.77313e12	1.35679
$266$	0	0
$267$	− 7.06914e10i	− 0.0520967i
$268$	0	0
$269$	1.65260e12	1.17329	0.586646	−	0.809843i	$-0.300448\pi$
$269$	1.65260e12	1.17329	0.586646	+	0.809843i	$0.300448\pi$
$270$	0	0
$271$	1.96077e12i	1.34147i	0.741698	+	0.670734i	$0.234021\pi$
$271$	1.96077e12i	1.34147i	−0.741698	+	0.670734i	$0.765979\pi$
$272$	0	0
$273$	1.24526e12	0.821199
$274$	0	0
$275$	7.18964e11i	0.457134i
$276$	0	0
$277$	−1.67161e11	−0.102503	−0.0512514	−	0.998686i	$-0.516321\pi$
$277$	−1.67161e11	−0.102503	−0.0512514	+	0.998686i	$0.516321\pi$
$278$	0	0
$279$	2.57831e12i	1.52516i
$280$	0	0
$281$	7.62988e11	0.435498	0.217749	−	0.976005i	$-0.430129\pi$
$281$	7.62988e11	0.435498	0.217749	+	0.976005i	$0.430129\pi$
$282$	0	0
$283$	8.63651e11i	0.475780i	0.971292	+	0.237890i	$0.0764558\pi$
$283$	8.63651e11i	0.475780i	−0.971292	+	0.237890i	$0.923544\pi$
$284$	0	0
$285$	−1.16802e12	−0.621192
$286$	0	0
$287$	− 1.87058e12i	− 0.960652i
$288$	0	0
$289$	2.19903e12	1.09079
$290$	0	0
$291$	− 1.16005e12i	− 0.555921i
$292$	0	0
$293$	2.56454e12	1.18760	0.593801	−	0.804612i	$-0.297627\pi$
$293$	2.56454e12	1.18760	0.593801	+	0.804612i	$0.297627\pi$
$294$	0	0
$295$	4.74347e11i	0.212318i
$296$	0	0
$297$	2.37449e12	1.02752
$298$	0	0
$299$	1.87946e12i	0.786462i
$300$	0	0
$301$	−3.91901e12	−1.58615
$302$	0	0
$303$	1.09367e12i	0.428225i
$304$	0	0
$305$	3.15508e11	0.119539
$306$	0	0
$307$	− 3.36330e12i	− 1.23331i	−0.787232	−	0.616657i	$-0.788486\pi$
$307$	− 3.36330e12i	− 1.23331i	0.787232	−	0.616657i	$-0.211514\pi$
$308$	0	0
$309$	−1.94462e12	−0.690308
$310$	0	0
$311$	− 5.35626e12i	− 1.84102i	−0.390714	−	0.920512i	$-0.627772\pi$
$311$	− 5.35626e12i	− 1.84102i	0.390714	−	0.920512i	$-0.372228\pi$
$312$	0	0
$313$	2.45540e11	0.0817335	0.0408667	−	0.999165i	$-0.486988\pi$
$313$	2.45540e11	0.0817335	0.0408667	+	0.999165i	$0.486988\pi$
$314$	0	0
$315$	− 2.84086e12i	− 0.916002i
$316$	0	0
$317$	−3.99276e12	−1.24732	−0.623659	−	0.781697i	$-0.714355\pi$
$317$	−3.99276e12	−1.24732	−0.623659	+	0.781697i	$0.714355\pi$
$318$	0	0
$319$	− 4.96725e11i	− 0.150370i
$320$	0	0
$321$	−2.23157e12	−0.654765
$322$	0	0
$323$	8.50524e12i	2.41921i
$324$	0	0
$325$	−1.59519e12	−0.439942
$326$	0	0
$327$	− 3.10140e12i	− 0.829505i
$328$	0	0
$329$	3.24963e12	0.843051
$330$	0	0
$331$	4.34999e12i	1.09483i	0.836860	+	0.547417i	$0.184389\pi$
$331$	4.34999e12i	1.09483i	−0.836860	+	0.547417i	$0.815611\pi$
$332$	0	0
$333$	4.53094e11	0.110654
$334$	0	0
$335$	− 4.13983e12i	− 0.981202i
$336$	0	0
$337$	2.27886e12	0.524285	0.262142	−	0.965029i	$-0.415571\pi$
$337$	2.27886e12	0.524285	0.262142	+	0.965029i	$0.415571\pi$
$338$	0	0
$339$	2.55611e12i	0.570927i
$340$	0	0
$341$	1.10998e13	2.40737
$342$	0	0
$343$	1.31432e12i	0.276840i
$344$	0	0
$345$	−1.20314e12	−0.246161
$346$	0	0
$347$	− 3.62893e11i	− 0.0721326i	−0.999349	−	0.0360663i	$-0.988517\pi$
$347$	− 3.62893e11i	− 0.0721326i	0.999349	−	0.0360663i	$-0.0114828\pi$
$348$	0	0
$349$	−7.99809e12	−1.54475	−0.772377	−	0.635165i	$-0.780932\pi$
$349$	−7.99809e12	−1.54475	−0.772377	+	0.635165i	$0.780932\pi$
$350$	0	0
$351$	5.26836e12i	0.988872i
$352$	0	0
$353$	−5.41287e11	−0.0987538	−0.0493769	−	0.998780i	$-0.515724\pi$
$353$	−5.41287e11	−0.0987538	−0.0493769	+	0.998780i	$0.515724\pi$
$354$	0	0
$355$	8.85710e11i	0.157091i
$356$	0	0
$357$	5.80469e12	1.00101
$358$	0	0
$359$	8.65405e12i	1.45127i	0.688081	+	0.725633i	$0.258453\pi$
$359$	8.65405e12i	1.45127i	−0.688081	+	0.725633i	$0.741547\pi$
$360$	0	0
$361$	−1.10311e13	−1.79922
$362$	0	0
$363$	− 1.53196e12i	− 0.243060i
$364$	0	0
$365$	3.31028e12	0.510977
$366$	0	0
$367$	− 3.02092e12i	− 0.453742i	−0.973925	−	0.226871i	$-0.927150\pi$
$367$	− 3.02092e12i	− 0.453742i	0.973925	−	0.226871i	$-0.0728495\pi$
$368$	0	0
$369$	3.47008e12	0.507234
$370$	0	0
$371$	− 1.77810e13i	− 2.52980i
$372$	0	0
$373$	5.98678e12	0.829181	0.414590	−	0.910008i	$-0.363925\pi$
$373$	5.98678e12	0.829181	0.414590	+	0.910008i	$0.363925\pi$
$374$	0	0
$375$	− 3.77452e12i	− 0.508986i
$376$	0	0
$377$	1.10210e12	0.144715
$378$	0	0
$379$	− 2.64321e12i	− 0.338015i	−0.985615	−	0.169007i	$-0.945944\pi$
$379$	− 2.64321e12i	− 0.338015i	0.985615	−	0.169007i	$-0.0540562\pi$
$380$	0	0
$381$	1.71625e11	0.0213775
$382$	0	0
$383$	− 1.59866e12i	− 0.193983i	−0.995285	−	0.0969913i	$-0.969078\pi$
$383$	− 1.59866e12i	− 0.193983i	0.995285	−	0.0969913i	$-0.0309219\pi$
$384$	0	0
$385$	−1.22301e13	−1.44586
$386$	0	0
$387$	− 7.27010e12i	− 0.837501i
$388$	0	0
$389$	−5.61283e12	−0.630136	−0.315068	−	0.949069i	$-0.602027\pi$
$389$	−5.61283e12	−0.630136	−0.315068	+	0.949069i	$0.602027\pi$
$390$	0	0
$391$	8.76096e12i	0.958667i
$392$	0	0
$393$	−2.10191e12	−0.224208
$394$	0	0
$395$	− 4.59928e12i	− 0.478305i
$396$	0	0
$397$	−2.88680e12	−0.292728	−0.146364	−	0.989231i	$-0.546757\pi$
$397$	−2.88680e12	−0.292728	−0.146364	+	0.989231i	$0.546757\pi$
$398$	0	0
$399$	1.17129e13i	1.15825i
$400$	0	0
$401$	−1.08786e13	−1.04918	−0.524591	−	0.851354i	$-0.675782\pi$
$401$	−1.08786e13	−1.04918	−0.524591	+	0.851354i	$0.675782\pi$
$402$	0	0
$403$	2.46275e13i	2.31683i
$404$	0	0
$405$	3.37629e12	0.309859
$406$	0	0
$407$	− 1.95060e12i	− 0.174661i
$408$	0	0
$409$	2.52570e12	0.220681	0.110340	−	0.993894i	$-0.464806\pi$
$409$	2.52570e12	0.220681	0.110340	+	0.993894i	$0.464806\pi$
$410$	0	0
$411$	4.99605e12i	0.426008i
$412$	0	0
$413$	4.75677e12	0.395878
$414$	0	0
$415$	3.02312e12i	0.245592i
$416$	0	0
$417$	9.49421e12	0.752971
$418$	0	0
$419$	− 2.22343e12i	− 0.172168i	−0.996288	−	0.0860842i	$-0.972565\pi$
$419$	− 2.22343e12i	− 0.172168i	0.996288	−	0.0860842i	$-0.0274354\pi$
$420$	0	0
$421$	−1.11161e13	−0.840507	−0.420253	−	0.907407i	$-0.638059\pi$
$421$	−1.11161e13	−0.840507	−0.420253	+	0.907407i	$0.638059\pi$
$422$	0	0
$423$	6.02834e12i	0.445139i
$424$	0	0
$425$	−7.43583e12	−0.536272
$426$	0	0
$427$	− 3.16392e12i	− 0.222888i
$428$	0	0
$429$	9.94502e12	0.684414
$430$	0	0
$431$	− 2.51262e13i	− 1.68943i	−0.535216	−	0.844715i	$-0.679770\pi$
$431$	− 2.51262e13i	− 1.68943i	0.535216	−	0.844715i	$-0.320230\pi$
$432$	0	0
$433$	−1.42658e13	−0.937253	−0.468627	−	0.883396i	$-0.655251\pi$
$433$	−1.42658e13	−0.937253	−0.468627	+	0.883396i	$0.655251\pi$
$434$	0	0
$435$	7.05508e11i	0.0452956i
$436$	0	0
$437$	−1.76782e13	−1.10925
$438$	0	0
$439$	− 1.97925e13i	− 1.21389i	−0.794745	−	0.606944i	$-0.792395\pi$
$439$	− 1.97925e13i	− 1.21389i	0.794745	−	0.606944i	$-0.207605\pi$
$440$	0	0
$441$	−1.54632e13	−0.927056
$442$	0	0
$443$	− 1.31931e13i	− 0.773262i	−0.922234	−	0.386631i	$-0.873639\pi$
$443$	− 1.31931e13i	− 0.773262i	0.922234	−	0.386631i	$-0.126361\pi$
$444$	0	0
$445$	−1.54042e12	−0.0882753
$446$	0	0
$447$	2.28652e12i	0.128126i
$448$	0	0
$449$	−3.10633e13	−1.70222	−0.851110	−	0.524988i	$-0.824070\pi$
$449$	−3.10633e13	−1.70222	−0.851110	+	0.524988i	$0.824070\pi$
$450$	0	0
$451$	− 1.49389e13i	− 0.800639i
$452$	0	0
$453$	4.97104e12	0.260589
$454$	0	0
$455$	− 2.71353e13i	− 1.39148i
$456$	0	0
$457$	−3.06537e13	−1.53781	−0.768904	−	0.639364i	$-0.779198\pi$
$457$	−3.06537e13	−1.53781	−0.768904	+	0.639364i	$0.779198\pi$
$458$	0	0
$459$	2.45580e13i	1.20540i
$460$	0	0
$461$	1.45855e13	0.700514	0.350257	−	0.936654i	$-0.386094\pi$
$461$	1.45855e13	0.700514	0.350257	+	0.936654i	$0.386094\pi$
$462$	0	0
$463$	1.40677e13i	0.661176i	0.943775	+	0.330588i	$0.107247\pi$
$463$	1.40677e13i	0.661176i	−0.943775	+	0.330588i	$0.892753\pi$
$464$	0	0
$465$	−1.57653e13	−0.725165
$466$	0	0
$467$	− 2.36434e13i	− 1.06445i	−0.846602	−	0.532226i	$-0.821356\pi$
$467$	− 2.36434e13i	− 1.06445i	0.846602	−	0.532226i	$-0.178644\pi$
$468$	0	0
$469$	−4.15144e13	−1.82951
$470$	0	0
$471$	2.71901e12i	0.117302i
$472$	0	0
$473$	−3.12982e13	−1.32195
$474$	0	0
$475$	− 1.50043e13i	− 0.620509i
$476$	0	0
$477$	3.29853e13	1.33576
$478$	0	0
$479$	9.58041e12i	0.379933i	0.981791	+	0.189966i	$0.0608379\pi$
$479$	9.58041e12i	0.379933i	−0.981791	+	0.189966i	$0.939162\pi$
$480$	0	0
$481$	4.32786e12	0.168092
$482$	0	0
$483$	1.20651e13i	0.458981i
$484$	0	0
$485$	−2.52785e13	−0.941980
$486$	0	0
$487$	2.00796e13i	0.733009i	0.930416	+	0.366505i	$0.119446\pi$
$487$	2.00796e13i	0.733009i	−0.930416	+	0.366505i	$0.880554\pi$
$488$	0	0
$489$	−1.13736e13	−0.406775
$490$	0	0
$491$	1.29905e13i	0.455216i	0.973753	+	0.227608i	$0.0730904\pi$
$491$	1.29905e13i	0.455216i	−0.973753	+	0.227608i	$0.926910\pi$
$492$	0	0
$493$	5.13734e12	0.176402
$494$	0	0
$495$	− 2.26879e13i	− 0.763427i
$496$	0	0
$497$	8.88193e12	0.292904
$498$	0	0
$499$	2.83436e13i	0.916120i	0.888921	+	0.458060i	$0.151455\pi$
$499$	2.83436e13i	0.916120i	−0.888921	+	0.458060i	$0.848545\pi$
$500$	0	0
$501$	−2.06359e13	−0.653783
$502$	0	0
$503$	4.73557e13i	1.47073i	0.677673	+	0.735364i	$0.262989\pi$
$503$	4.73557e13i	1.47073i	−0.677673	+	0.735364i	$0.737011\pi$
$504$	0	0
$505$	2.38319e13	0.725606
$506$	0	0
$507$	6.38414e12i	0.190573i
$508$	0	0
$509$	−4.22980e12	−0.123803	−0.0619015	−	0.998082i	$-0.519716\pi$
$509$	−4.22980e12	−0.123803	−0.0619015	+	0.998082i	$0.519716\pi$
$510$	0	0
$511$	− 3.31956e13i	− 0.952744i
$512$	0	0
$513$	−4.95541e13	−1.39474
$514$	0	0
$515$	4.23748e13i	1.16969i
$516$	0	0
$517$	2.59524e13	0.702627
$518$	0	0
$519$	− 3.07675e13i	− 0.817063i
$520$	0	0
$521$	2.01599e13	0.525171	0.262585	−	0.964909i	$-0.415425\pi$
$521$	2.01599e13	0.525171	0.262585	+	0.964909i	$0.415425\pi$
$522$	0	0
$523$	− 4.21335e13i	− 1.07676i	−0.842702	−	0.538380i	$-0.819037\pi$
$523$	− 4.21335e13i	− 1.07676i	0.842702	−	0.538380i	$-0.180963\pi$
$524$	0	0
$525$	−1.02402e13	−0.256751
$526$	0	0
$527$	1.14799e14i	2.82413i
$528$	0	0
$529$	2.32168e13	0.560433
$530$	0	0
$531$	8.82421e12i	0.209028i
$532$	0	0
$533$	3.31455e13	0.770528
$534$	0	0
$535$	4.86277e13i	1.10947i
$536$	0	0
$537$	1.94773e12	0.0436172
$538$	0	0
$539$	6.65700e13i	1.46331i
$540$	0	0
$541$	4.54604e13	0.980949	0.490475	−	0.871455i	$-0.336823\pi$
$541$	4.54604e13	0.980949	0.490475	+	0.871455i	$0.336823\pi$
$542$	0	0
$543$	− 2.47424e13i	− 0.524135i
$544$	0	0
$545$	−6.75819e13	−1.40555
$546$	0	0
$547$	− 6.66999e13i	− 1.36204i	−0.732267	−	0.681018i	$-0.761538\pi$
$547$	− 6.66999e13i	− 1.36204i	0.732267	−	0.681018i	$-0.238462\pi$
$548$	0	0
$549$	5.86935e12	0.117687
$550$	0	0
$551$	1.03663e13i	0.204111i
$552$	0	0
$553$	−4.61217e13	−0.891826
$554$	0	0
$555$	2.77047e12i	0.0526125i
$556$	0	0
$557$	2.78545e13	0.519540	0.259770	−	0.965671i	$-0.416353\pi$
$557$	2.78545e13	0.519540	0.259770	+	0.965671i	$0.416353\pi$
$558$	0	0
$559$	− 6.94424e13i	− 1.27223i
$560$	0	0
$561$	4.63579e13	0.834275
$562$	0	0
$563$	1.01955e13i	0.180247i	0.995931	+	0.0901236i	$0.0287262\pi$
$563$	1.01955e13i	0.180247i	−0.995931	+	0.0901236i	$0.971274\pi$
$564$	0	0
$565$	5.56995e13	0.967407
$566$	0	0
$567$	− 3.38575e13i	− 0.577750i
$568$	0	0
$569$	−6.70305e13	−1.12386	−0.561929	−	0.827185i	$-0.689941\pi$
$569$	−6.70305e13	−1.12386	−0.561929	+	0.827185i	$0.689941\pi$
$570$	0	0
$571$	7.59908e13i	1.25193i	0.779851	+	0.625965i	$0.215295\pi$
$571$	7.59908e13i	1.25193i	−0.779851	+	0.625965i	$0.784705\pi$
$572$	0	0
$573$	6.44215e12	0.104294
$574$	0	0
$575$	− 1.54554e13i	− 0.245890i
$576$	0	0
$577$	−6.61883e12	−0.103491	−0.0517454	−	0.998660i	$-0.516478\pi$
$577$	−6.61883e12	−0.103491	−0.0517454	+	0.998660i	$0.516478\pi$
$578$	0	0
$579$	4.37022e13i	0.671600i
$580$	0	0
$581$	3.03159e13	0.457921
$582$	0	0
$583$	− 1.42004e14i	− 2.10842i
$584$	0	0
$585$	5.03383e13	0.734715
$586$	0	0
$587$	− 1.06868e14i	− 1.53341i	−0.642000	−	0.766705i	$-0.721895\pi$
$587$	− 1.06868e14i	− 1.53341i	0.642000	−	0.766705i	$-0.278105\pi$
$588$	0	0
$589$	−2.31646e14	−3.26774
$590$	0	0
$591$	− 4.46571e13i	− 0.619375i
$592$	0	0
$593$	1.11343e14	1.51841	0.759203	−	0.650854i	$-0.225589\pi$
$593$	1.11343e14	1.51841	0.759203	+	0.650854i	$0.225589\pi$
$594$	0	0
$595$	− 1.26489e14i	− 1.69616i
$596$	0	0
$597$	1.56013e13	0.205726
$598$	0	0
$599$	6.20791e13i	0.805029i	0.915414	+	0.402514i	$0.131864\pi$
$599$	6.20791e13i	0.805029i	−0.915414	+	0.402514i	$0.868136\pi$
$600$	0	0
$601$	9.54069e13	1.21677	0.608384	−	0.793643i	$-0.291818\pi$
$601$	9.54069e13	1.21677	0.608384	+	0.793643i	$0.291818\pi$
$602$	0	0
$603$	− 7.70127e13i	− 0.965997i
$604$	0	0
$605$	−3.33825e13	−0.411853
$606$	0	0
$607$	7.96810e13i	0.966966i	0.875354	+	0.483483i	$0.160629\pi$
$607$	7.96810e13i	0.966966i	−0.875354	+	0.483483i	$0.839371\pi$
$608$	0	0
$609$	7.07485e12	0.0844561
$610$	0	0
$611$	5.75814e13i	0.676202i
$612$	0	0
$613$	1.71208e13	0.197798	0.0988988	−	0.995097i	$-0.468468\pi$
$613$	1.71208e13	0.197798	0.0988988	+	0.995097i	$0.468468\pi$
$614$	0	0
$615$	2.12181e13i	0.241174i
$616$	0	0
$617$	1.31881e13	0.147488	0.0737439	−	0.997277i	$-0.476505\pi$
$617$	1.31881e13	0.147488	0.0737439	+	0.997277i	$0.476505\pi$
$618$	0	0
$619$	− 5.36881e13i	− 0.590779i	−0.955377	−	0.295389i	$-0.904551\pi$
$619$	− 5.36881e13i	− 0.590779i	0.955377	−	0.295389i	$-0.0954494\pi$
$620$	0	0
$621$	−5.10440e13	−0.552696
$622$	0	0
$623$	1.54474e13i	0.164594i
$624$	0	0
$625$	−4.68802e13	−0.491574
$626$	0	0
$627$	9.35427e13i	0.965322i
$628$	0	0
$629$	2.01739e13	0.204898
$630$	0	0
$631$	− 1.52794e14i	− 1.52742i	−0.645557	−	0.763712i	$-0.723375\pi$
$631$	− 1.52794e14i	− 1.52742i	0.645557	−	0.763712i	$-0.276625\pi$
$632$	0	0
$633$	4.69236e13	0.461714
$634$	0	0
$635$	− 3.73985e12i	− 0.0362231i
$636$	0	0
$637$	−1.47701e14	−1.40827
$638$	0	0
$639$	1.64768e13i	0.154656i
$640$	0	0
$641$	1.49415e14	1.38071	0.690357	−	0.723469i	$-0.257453\pi$
$641$	1.49415e14	1.38071	0.690357	+	0.723469i	$0.257453\pi$
$642$	0	0
$643$	− 1.27252e14i	− 1.15774i	−0.815420	−	0.578870i	$-0.803494\pi$
$643$	− 1.27252e14i	− 1.15774i	0.815420	−	0.578870i	$-0.196506\pi$
$644$	0	0
$645$	4.44535e13	0.398206
$646$	0	0
$647$	1.52008e14i	1.34074i	0.742028	+	0.670369i	$0.233864\pi$
$647$	1.52008e14i	1.34074i	−0.742028	+	0.670369i	$0.766136\pi$
$648$	0	0
$649$	3.79888e13	0.329938
$650$	0	0
$651$	1.58095e14i	1.35211i
$652$	0	0
$653$	7.24002e13	0.609781	0.304891	−	0.952387i	$-0.401380\pi$
$653$	7.24002e13	0.609781	0.304891	+	0.952387i	$0.401380\pi$
$654$	0	0
$655$	4.58022e13i	0.379909i
$656$	0	0
$657$	6.15808e13	0.503058
$658$	0	0
$659$	− 1.33873e12i	− 0.0107712i	−0.999985	−	0.00538562i	$-0.998286\pi$
$659$	− 1.33873e12i	− 0.0107712i	0.999985	−	0.00538562i	$-0.00171431\pi$
$660$	0	0
$661$	1.30078e14	1.03085	0.515427	−	0.856934i	$-0.327634\pi$
$661$	1.30078e14	1.03085	0.515427	+	0.856934i	$0.327634\pi$
$662$	0	0
$663$	1.02856e14i	0.802898i
$664$	0	0
$665$	2.55234e14	1.96259
$666$	0	0
$667$	1.06780e13i	0.0808836i
$668$	0	0
$669$	3.62270e13	0.270335
$670$	0	0
$671$	− 2.52679e13i	− 0.185762i
$672$	0	0
$673$	−7.41289e13	−0.536923	−0.268462	−	0.963290i	$-0.586515\pi$
$673$	−7.41289e13	−0.536923	−0.268462	+	0.963290i	$0.586515\pi$
$674$	0	0
$675$	− 4.33234e13i	− 0.309174i
$676$	0	0
$677$	1.68609e13	0.118560	0.0592798	−	0.998241i	$-0.481120\pi$
$677$	1.68609e13	0.118560	0.0592798	+	0.998241i	$0.481120\pi$
$678$	0	0
$679$	2.53493e14i	1.75638i
$680$	0	0
$681$	5.88556e13	0.401839
$682$	0	0
$683$	4.07856e13i	0.274412i	0.990543	+	0.137206i	$0.0438122\pi$
$683$	4.07856e13i	0.274412i	−0.990543	+	0.137206i	$0.956188\pi$
$684$	0	0
$685$	1.08868e14	0.721848
$686$	0	0
$687$	− 6.21823e13i	− 0.406334i
$688$	0	0
$689$	3.15069e14	2.02913
$690$	0	0
$691$	1.60520e14i	1.01892i	0.860494	+	0.509460i	$0.170155\pi$
$691$	1.60520e14i	1.01892i	−0.860494	+	0.509460i	$0.829845\pi$
$692$	0	0
$693$	−2.27515e14	−1.42345
$694$	0	0
$695$	− 2.06886e14i	− 1.27587i
$696$	0	0
$697$	1.54505e14	0.939244
$698$	0	0
$699$	1.82706e13i	0.109488i
$700$	0	0
$701$	−1.46039e14	−0.862740	−0.431370	−	0.902175i	$-0.641970\pi$
$701$	−1.46039e14	−0.862740	−0.431370	+	0.902175i	$0.641970\pi$
$702$	0	0
$703$	4.07078e13i	0.237083i
$704$	0	0
$705$	−3.68607e13	−0.211650
$706$	0	0
$707$	− 2.38987e14i	− 1.35293i
$708$	0	0
$709$	1.30348e14	0.727568	0.363784	−	0.931483i	$-0.381485\pi$
$709$	1.30348e14	0.727568	0.363784	+	0.931483i	$0.381485\pi$
$710$	0	0
$711$	− 8.55598e13i	− 0.470893i
$712$	0	0
$713$	−2.38610e14	−1.29492
$714$	0	0
$715$	− 2.16710e14i	− 1.15971i
$716$	0	0
$717$	9.25358e13	0.488331
$718$	0	0
$719$	6.68110e13i	0.347699i	0.984772	+	0.173849i	$0.0556206\pi$
$719$	6.68110e13i	0.347699i	−0.984772	+	0.173849i	$0.944379\pi$
$720$	0	0
$721$	4.24936e14	2.18096
$722$	0	0
$723$	4.30509e13i	0.217917i
$724$	0	0
$725$	−9.06291e12	−0.0452457
$726$	0	0
$727$	− 7.15420e13i	− 0.352281i	−0.984365	−	0.176140i	$-0.943639\pi$
$727$	− 7.15420e13i	− 0.352281i	0.984365	−	0.176140i	$-0.0563613\pi$
$728$	0	0
$729$	−1.75349e13	−0.0851659
$730$	0	0
$731$	− 3.23700e14i	− 1.55080i
$732$	0	0
$733$	1.08237e14	0.511510	0.255755	−	0.966742i	$-0.417676\pi$
$733$	1.08237e14	0.511510	0.255755	+	0.966742i	$0.417676\pi$
$734$	0	0
$735$	− 9.45507e13i	− 0.440786i
$736$	0	0
$737$	−3.31545e14	−1.52477
$738$	0	0
$739$	2.75969e14i	1.25210i	0.779784	+	0.626049i	$0.215329\pi$
$739$	2.75969e14i	1.25210i	−0.779784	+	0.626049i	$0.784671\pi$
$740$	0	0
$741$	−2.07546e14	−0.929017
$742$	0	0
$743$	− 5.40557e12i	− 0.0238725i	−0.999929	−	0.0119362i	$-0.996200\pi$
$743$	− 5.40557e12i	− 0.0238725i	0.999929	−	0.0119362i	$-0.00379951\pi$
$744$	0	0
$745$	4.98249e13	0.217103
$746$	0	0
$747$	5.62387e13i	0.241787i
$748$	0	0
$749$	4.87640e14	2.06866
$750$	0	0
$751$	− 3.14038e14i	− 1.31456i	−0.753644	−	0.657282i	$-0.771706\pi$
$751$	− 3.14038e14i	− 1.31456i	0.753644	−	0.657282i	$-0.228294\pi$
$752$	0	0
$753$	1.23541e14	0.510314
$754$	0	0
$755$	− 1.08323e14i	− 0.441555i
$756$	0	0
$757$	−8.11946e13	−0.326624	−0.163312	−	0.986574i	$-0.552218\pi$
$757$	−8.11946e13	−0.326624	−0.163312	+	0.986574i	$0.552218\pi$
$758$	0	0
$759$	9.63551e13i	0.382530i
$760$	0	0
$761$	3.78868e14	1.48445	0.742224	−	0.670152i	$-0.233771\pi$
$761$	3.78868e14	1.48445	0.742224	+	0.670152i	$0.233771\pi$
$762$	0	0
$763$	6.77713e14i	2.62073i
$764$	0	0
$765$	2.34647e14	0.895589
$766$	0	0
$767$	8.42870e13i	0.317529i
$768$	0	0
$769$	−1.05125e14	−0.390907	−0.195453	−	0.980713i	$-0.562618\pi$
$769$	−1.05125e14	−0.390907	−0.195453	+	0.980713i	$0.562618\pi$
$770$	0	0
$771$	1.88933e14i	0.693482i
$772$	0	0
$773$	−5.26243e14	−1.90673	−0.953364	−	0.301824i	$-0.902405\pi$
$773$	−5.26243e14	−1.90673	−0.953364	+	0.301824i	$0.902405\pi$
$774$	0	0
$775$	− 2.02520e14i	− 0.724367i
$776$	0	0
$777$	2.77824e13	0.0980990
$778$	0	0
$779$	3.11766e14i	1.08678i
$780$	0	0
$781$	7.09335e13	0.244116
$782$	0	0
$783$	2.99317e13i	0.101700i
$784$	0	0
$785$	5.92494e13	0.198763
$786$	0	0
$787$	2.14499e14i	0.710480i	0.934775	+	0.355240i	$0.115601\pi$
$787$	2.14499e14i	0.710480i	−0.934775	+	0.355240i	$0.884399\pi$
$788$	0	0
$789$	1.39246e14	0.455406
$790$	0	0
$791$	− 5.58556e14i	− 1.80378i
$792$	0	0
$793$	5.60628e13	0.178776
$794$	0	0
$795$	2.01691e14i	0.635112i
$796$	0	0
$797$	6.77376e13	0.210639	0.105319	−	0.994438i	$-0.466414\pi$
$797$	6.77376e13	0.210639	0.105319	+	0.994438i	$0.466414\pi$
$798$	0	0
$799$	2.68411e14i	0.824264i
$800$	0	0
$801$	−2.86562e13	−0.0869074
$802$	0	0
$803$	− 2.65109e14i	− 0.794049i
$804$	0	0
$805$	2.62908e14	0.777721
$806$	0	0
$807$	1.87981e14i	0.549219i
$808$	0	0
$809$	1.96726e14	0.567701	0.283851	−	0.958868i	$-0.408388\pi$
$809$	1.96726e14	0.567701	0.283851	+	0.958868i	$0.408388\pi$
$810$	0	0
$811$	− 5.47676e14i	− 1.56106i	−0.625119	−	0.780530i	$-0.714949\pi$
$811$	− 5.47676e14i	− 1.56106i	0.625119	−	0.780530i	$-0.285051\pi$
$812$	0	0
$813$	−2.23035e14	−0.627942
$814$	0	0
$815$	2.47840e14i	0.689261i
$816$	0	0
$817$	6.53174e14	1.79440
$818$	0	0
$819$	− 5.04794e14i	− 1.36992i
$820$	0	0
$821$	−7.02777e14	−1.88409	−0.942045	−	0.335486i	$-0.891100\pi$
$821$	−7.02777e14	−1.88409	−0.942045	+	0.335486i	$0.891100\pi$
$822$	0	0
$823$	7.08819e14i	1.87731i	0.344855	+	0.938656i	$0.387928\pi$
$823$	7.08819e14i	1.87731i	−0.344855	+	0.938656i	$0.612072\pi$
$824$	0	0
$825$	−8.17810e13	−0.213985
$826$	0	0
$827$	− 5.17551e14i	− 1.33790i	−0.743305	−	0.668952i	$-0.766743\pi$
$827$	− 5.17551e14i	− 1.33790i	0.743305	−	0.668952i	$-0.233257\pi$
$828$	0	0
$829$	−1.93138e14	−0.493283	−0.246641	−	0.969107i	$-0.579327\pi$
$829$	−1.93138e14	−0.493283	−0.246641	+	0.969107i	$0.579327\pi$
$830$	0	0
$831$	− 1.90143e13i	− 0.0479816i
$832$	0	0
$833$	−6.88496e14	−1.71663
$834$	0	0
$835$	4.49671e14i	1.10780i
$836$	0	0
$837$	−6.68854e14	−1.62819
$838$	0	0
$839$	2.54851e14i	0.613023i	0.951867	+	0.306512i	$0.0991618\pi$
$839$	2.54851e14i	0.613023i	−0.951867	+	0.306512i	$0.900838\pi$
$840$	0	0
$841$	−4.14446e14	−0.985117
$842$	0	0
$843$	8.67888e13i	0.203857i
$844$	0	0
$845$	1.39115e14	0.322917
$846$	0	0
$847$	3.34761e14i	0.767923i
$848$	0	0
$849$	−9.82390e13	−0.222713
$850$	0	0
$851$	4.19317e13i	0.0939495i
$852$	0	0
$853$	8.31517e14	1.84131	0.920653	−	0.390383i	$-0.127657\pi$
$853$	8.31517e14	1.84131	0.920653	+	0.390383i	$0.127657\pi$
$854$	0	0
$855$	4.73481e14i	1.03627i
$856$	0	0
$857$	−6.93268e14	−1.49968	−0.749838	−	0.661622i	$-0.769868\pi$
$857$	−6.93268e14	−1.49968	−0.749838	+	0.661622i	$0.769868\pi$
$858$	0	0
$859$	− 5.92737e14i	− 1.26735i	−0.773600	−	0.633675i	$-0.781546\pi$
$859$	− 5.92737e14i	− 1.26735i	0.773600	−	0.633675i	$-0.218454\pi$
$860$	0	0
$861$	2.12775e14	0.449682
$862$	0	0
$863$	− 7.73966e14i	− 1.61684i	−0.588605	−	0.808421i	$-0.700322\pi$
$863$	− 7.73966e14i	− 1.61684i	0.588605	−	0.808421i	$-0.299678\pi$
$864$	0	0
$865$	−6.70448e14	−1.38447
$866$	0	0
$867$	2.50136e14i	0.510601i
$868$	0	0
$869$	−3.68341e14	−0.743277
$870$	0	0
$871$	− 7.35609e14i	− 1.46743i
$872$	0	0
$873$	−4.70252e14	−0.927383
$874$	0	0
$875$	8.24802e14i	1.60809i
$876$	0	0
$877$	−5.15078e14	−0.992830	−0.496415	−	0.868085i	$-0.665351\pi$
$877$	−5.15078e14	−0.992830	−0.496415	+	0.868085i	$0.665351\pi$
$878$	0	0
$879$	2.91712e14i	0.555918i
$880$	0	0
$881$	−4.47316e13	−0.0842821	−0.0421410	−	0.999112i	$-0.513418\pi$
$881$	−4.47316e13	−0.0842821	−0.0421410	+	0.999112i	$0.513418\pi$
$882$	0	0
$883$	− 7.32234e14i	− 1.36410i	−0.731306	−	0.682050i	$-0.761089\pi$
$883$	− 7.32234e14i	− 1.36410i	0.731306	−	0.682050i	$-0.238911\pi$
$884$	0	0
$885$	−5.39563e13	−0.0993860
$886$	0	0
$887$	2.14439e14i	0.390559i	0.980748	+	0.195279i	$0.0625614\pi$
$887$	2.14439e14i	0.390559i	−0.980748	+	0.195279i	$0.937439\pi$
$888$	0	0
$889$	−3.75033e13	−0.0675399
$890$	0	0
$891$	− 2.70396e14i	− 0.481516i
$892$	0	0
$893$	−5.41610e14	−0.953738
$894$	0	0
$895$	− 4.24425e13i	− 0.0739071i
$896$	0	0
$897$	−2.13786e14	−0.368144
$898$	0	0
$899$	1.39919e14i	0.238275i
$900$	0	0
$901$	1.46867e15	2.47343
$902$	0	0
$903$	− 4.45781e14i	− 0.742477i
$904$	0	0
$905$	−5.39156e14	−0.888120
$906$	0	0
$907$	2.13922e14i	0.348513i	0.984700	+	0.174256i	$0.0557522\pi$
$907$	2.13922e14i	0.348513i	−0.984700	+	0.174256i	$0.944248\pi$
$908$	0	0
$909$	4.43341e14	0.714362
$910$	0	0
$911$	− 1.10061e15i	− 1.75406i	−0.480440	−	0.877028i	$-0.659523\pi$
$911$	− 1.10061e15i	− 1.75406i	0.480440	−	0.877028i	$-0.340477\pi$
$912$	0	0
$913$	2.42111e14	0.381647
$914$	0	0
$915$	3.58885e13i	0.0559565i
$916$	0	0
$917$	4.59306e14	0.708362
$918$	0	0
$919$	3.61516e14i	0.551506i	0.961229	+	0.275753i	$0.0889271\pi$
$919$	3.61516e14i	0.551506i	−0.961229	+	0.275753i	$0.911073\pi$
$920$	0	0
$921$	3.82570e14	0.577316
$922$	0	0
$923$	1.57382e14i	0.234935i
$924$	0	0
$925$	−3.55893e13	−0.0525547
$926$	0	0
$927$	7.88293e14i	1.15157i
$928$	0	0
$929$	−1.29839e15	−1.87641	−0.938204	−	0.346083i	$-0.887512\pi$
$929$	−1.29839e15	−1.87641	−0.938204	+	0.346083i	$0.887512\pi$
$930$	0	0
$931$	− 1.38927e15i	− 1.98628i
$932$	0	0
$933$	6.09266e14	0.861785
$934$	0	0
$935$	− 1.01017e15i	− 1.41364i
$936$	0	0
$937$	−1.13914e14	−0.157718	−0.0788588	−	0.996886i	$-0.525128\pi$
$937$	−1.13914e14	−0.157718	−0.0788588	+	0.996886i	$0.525128\pi$
$938$	0	0
$939$	2.79298e13i	0.0382595i
$940$	0	0
$941$	−5.62431e14	−0.762292	−0.381146	−	0.924515i	$-0.624470\pi$
$941$	−5.62431e14	−0.762292	−0.381146	+	0.924515i	$0.624470\pi$
$942$	0	0
$943$	3.21139e14i	0.430661i
$944$	0	0
$945$	7.36961e14	0.977880
$946$	0	0
$947$	1.04435e15i	1.37119i	0.727983	+	0.685595i	$0.240458\pi$
$947$	1.04435e15i	1.37119i	−0.727983	+	0.685595i	$0.759542\pi$
$948$	0	0
$949$	5.88206e14	0.764185
$950$	0	0
$951$	− 4.54171e14i	− 0.583870i
$952$	0	0
$953$	6.17432e14	0.785461	0.392731	−	0.919654i	$-0.371530\pi$
$953$	6.17432e14	0.785461	0.392731	+	0.919654i	$0.371530\pi$
$954$	0	0
$955$	− 1.40379e14i	− 0.176720i
$956$	0	0
$957$	5.65017e13	0.0703885
$958$	0	0
$959$	− 1.09173e15i	− 1.34593i
$960$	0	0
$961$	−2.30700e15	−2.81469
$962$	0	0
$963$	9.04614e14i	1.09227i
$964$	0	0
$965$	9.52305e14	1.13799
$966$	0	0
$967$	9.53378e14i	1.12754i	0.825931	+	0.563771i	$0.190650\pi$
$967$	9.53378e14i	1.12754i	−0.825931	+	0.563771i	$0.809350\pi$
$968$	0	0
$969$	−9.67458e14	−1.13244
$970$	0	0
$971$	− 1.09936e15i	− 1.27363i	−0.771015	−	0.636817i	$-0.780251\pi$
$971$	− 1.09936e15i	− 1.27363i	0.771015	−	0.636817i	$-0.219749\pi$
$972$	0	0
$973$	−2.07466e15	−2.37893
$974$	0	0
$975$	− 1.81450e14i	− 0.205937i
$976$	0	0
$977$	8.24134e14	0.925817	0.462908	−	0.886406i	$-0.346806\pi$
$977$	8.24134e14	0.925817	0.462908	+	0.886406i	$0.346806\pi$
$978$	0	0
$979$	1.23367e14i	0.137178i
$980$	0	0
$981$	−1.25722e15	−1.38377
$982$	0	0
$983$	− 3.34964e14i	− 0.364948i	−0.983211	−	0.182474i	$-0.941590\pi$
$983$	− 3.34964e14i	− 0.364948i	0.983211	−	0.182474i	$-0.0584105\pi$
$984$	0	0
$985$	−9.73113e14	−1.04950
$986$	0	0
$987$	3.69640e14i	0.394633i
$988$	0	0
$989$	6.72812e14	0.711070
$990$	0	0
$991$	2.26144e14i	0.236601i	0.992978	+	0.118300i	$0.0377446\pi$
$991$	2.26144e14i	0.236601i	−0.992978	+	0.118300i	$0.962255\pi$
$992$	0	0
$993$	−4.94805e14	−0.512493
$994$	0	0
$995$	− 3.39965e14i	− 0.348593i
$996$	0	0
$997$	4.45062e14	0.451798	0.225899	−	0.974151i	$-0.427468\pi$
$997$	4.45062e14	0.451798	0.225899	+	0.974151i	$0.427468\pi$
$998$	0	0
$999$	1.17539e14i	0.118129i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.h.255.3		4
4.3	odd	2	inner	256.11.c.h.255.2		4
8.3	odd	2		256.11.c.e.255.3		4
8.5	even	2		256.11.c.e.255.2		4
16.3	odd	4		128.11.d.e.63.5	yes	8
16.5	even	4		128.11.d.e.63.6	yes	8
16.11	odd	4		128.11.d.e.63.4	yes	8
16.13	even	4		128.11.d.e.63.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.11.d.e.63.3	✓	8	16.13	even	4
128.11.d.e.63.4	yes	8	16.11	odd	4
128.11.d.e.63.5	yes	8	16.3	odd	4
128.11.d.e.63.6	yes	8	16.5	even	4
256.11.c.e.255.2		4	8.5	even	2
256.11.c.e.255.3		4	8.3	odd	2
256.11.c.h.255.2		4	4.3	odd	2	inner
256.11.c.h.255.3		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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+113.748i q^{3} +2478.67 q^{5} -24856.1i q^{7} +46110.3 q^{9} +O(q^{10})\)	\(q+113.748i q^{3} +2478.67 q^{5} -24856.1i q^{7} +46110.3 q^{9} -198508. i q^{11} +440436. q^{13} +281944. i q^{15} +2.05305e6 q^{17} +4.14273e6i q^{19} +2.82735e6 q^{21} +4.26728e6i q^{23} -3.62184e6 q^{25} +1.19617e7i q^{27} +2.50229e6 q^{29} +5.59162e7i q^{31} +2.25800e7 q^{33} -6.16101e7i q^{35} +9.82631e6 q^{37} +5.00989e7i q^{39} +7.52562e7 q^{41} -1.57668e8i q^{43} +1.14292e8 q^{45} +1.30737e8i q^{47} -3.35352e8 q^{49} +2.33532e8i q^{51} +7.15357e8 q^{53} -4.92035e8i q^{55} -4.71229e8 q^{57} +1.91372e8i q^{59} +1.27289e8 q^{61} -1.14612e9i q^{63} +1.09169e9 q^{65} -1.67019e9i q^{67} -4.85397e8 q^{69} +3.57334e8i q^{71} +1.33551e9 q^{73} -4.11979e8i q^{75} -4.93414e9 q^{77} -1.85555e9i q^{79} +1.36214e9 q^{81} +1.21966e9i q^{83} +5.08883e9 q^{85} +2.84632e8i q^{87} -6.21472e8 q^{89} -1.09475e10i q^{91} -6.36039e9 q^{93} +1.02684e10i q^{95} -1.01984e10 q^{97} -9.15325e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5600 q^{5} - 57180 q^{9}+O(q^{10}) \) 4 * q + 5600 * q^5 - 57180 * q^9	\( 4 q + 5600 q^{5} - 57180 q^{9} - 296352 q^{13} + 1895544 q^{17} - 12288 q^{21} - 26568420 q^{25} + 20066656 q^{29} - 978432 q^{33} + 154467936 q^{37} + 189395640 q^{41} + 180576480 q^{45} - 225809468 q^{49} + 831839968 q^{53} + 87436800 q^{57} - 936776736 q^{61} + 1805103360 q^{65} + 4959940608 q^{69} - 2975489720 q^{73} - 10845720576 q^{77} - 1910976444 q^{81} + 9467334720 q^{85} + 11059844040 q^{89} - 40085716992 q^{93} - 917012488 q^{97}+O(q^{100}) \) 4 * q + 5600 * q^5 - 57180 * q^9 - 296352 * q^13 + 1895544 * q^17 - 12288 * q^21 - 26568420 * q^25 + 20066656 * q^29 - 978432 * q^33 + 154467936 * q^37 + 189395640 * q^41 + 180576480 * q^45 - 225809468 * q^49 + 831839968 * q^53 + 87436800 * q^57 - 936776736 * q^61 + 1805103360 * q^65 + 4959940608 * q^69 - 2975489720 * q^73 - 10845720576 * q^77 - 1910976444 * q^81 + 9467334720 * q^85 + 11059844040 * q^89 - 40085716992 * q^93 - 917012488 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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