LMFDB - Embedded newform 256.11.c.e.255.4

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			255.4
Root			$22.8573i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.e.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+365.717i q^{3} -321.334 q^{5} -7747.76i q^{7} -74700.3 q^{9} +O(q^{10})$	$q+365.717i q^{3} -321.334 q^{5} -7747.76i q^{7} -74700.3 q^{9} +63079.2i q^{11} +588612. q^{13} -117518. i q^{15} -1.10528e6 q^{17} -1.40805e6i q^{19} +2.83349e6 q^{21} +8.10835e6i q^{23} -9.66237e6 q^{25} -5.72395e6i q^{27} -7.53103e6 q^{29} -3.74127e7i q^{31} -2.30692e7 q^{33} +2.48962e6i q^{35} -6.74077e7 q^{37} +2.15266e8i q^{39} +1.94417e7 q^{41} -3.98511e7i q^{43} +2.40038e7 q^{45} -3.13565e8i q^{47} +2.22447e8 q^{49} -4.04221e8i q^{51} +2.99437e8 q^{53} -2.02695e7i q^{55} +5.14948e8 q^{57} -9.75283e8i q^{59} +5.95678e8 q^{61} +5.78760e8i q^{63} -1.89141e8 q^{65} -1.47130e9i q^{67} -2.96537e9 q^{69} -2.76443e9i q^{71} -2.82326e9 q^{73} -3.53370e9i q^{75} +4.88723e8 q^{77} +3.76461e9i q^{79} -2.31763e9 q^{81} -6.30934e9i q^{83} +3.55164e8 q^{85} -2.75423e9i q^{87} +6.15139e9 q^{89} -4.56042e9i q^{91} +1.36825e10 q^{93} +4.52454e8i q^{95} +9.73991e9 q^{97} -4.71204e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	365.717i	1.50501i	0.658586	+	0.752505i	$0.271155\pi$
$3$	365.717i	1.50501i	−0.658586	+	0.752505i	$0.728845\pi$
$4$	0	0
$5$	−321.334	−0.102827	−0.0514135	−	0.998677i	$-0.516373\pi$
$5$	−321.334	−0.102827	−0.0514135	+	0.998677i	$0.516373\pi$
$6$	0	0
$7$	− 7747.76i	− 0.460984i	−0.973074	−	0.230492i	$-0.925966\pi$
$7$	− 7747.76i	− 0.460984i	0.973074	−	0.230492i	$-0.0740336\pi$
$8$	0	0
$9$	−74700.3	−1.26506
$10$	0	0
$11$	63079.2i	0.391672i	0.980637	+	0.195836i	$0.0627421\pi$
$11$	63079.2i	0.391672i	−0.980637	+	0.195836i	$0.937258\pi$
$12$	0	0
$13$	588612.	1.58530	0.792651	−	0.609676i	$-0.208700\pi$
$13$	588612.	1.58530	0.792651	+	0.609676i	$0.208700\pi$
$14$	0	0
$15$	− 117518.i	− 0.154756i
$16$	0	0
$17$	−1.10528e6	−0.778445	−0.389223	−	0.921144i	$-0.627256\pi$
$17$	−1.10528e6	−0.778445	−0.389223	+	0.921144i	$0.627256\pi$
$18$	0	0
$19$	− 1.40805e6i	− 0.568656i	−0.958727	−	0.284328i	$-0.908230\pi$
$19$	− 1.40805e6i	− 0.568656i	0.958727	−	0.284328i	$-0.0917704\pi$
$20$	0	0
$21$	2.83349e6	0.693786
$22$	0	0
$23$	8.10835e6i	1.25978i	0.776686	+	0.629888i	$0.216899\pi$
$23$	8.10835e6i	1.25978i	−0.776686	+	0.629888i	$0.783101\pi$
$24$	0	0
$25$	−9.66237e6	−0.989427
$26$	0	0
$27$	− 5.72395e6i	− 0.398912i
$28$	0	0
$29$	−7.53103e6	−0.367168	−0.183584	−	0.983004i	$-0.558770\pi$
$29$	−7.53103e6	−0.367168	−0.183584	+	0.983004i	$0.558770\pi$
$30$	0	0
$31$	− 3.74127e7i	− 1.30680i	−0.757011	−	0.653402i	$-0.773341\pi$
$31$	− 3.74127e7i	− 1.30680i	0.757011	−	0.653402i	$-0.226659\pi$
$32$	0	0
$33$	−2.30692e7	−0.589471
$34$	0	0
$35$	2.48962e6i	0.0474016i
$36$	0	0
$37$	−6.74077e7	−0.972077	−0.486038	−	0.873937i	$-0.661558\pi$
$37$	−6.74077e7	−0.972077	−0.486038	+	0.873937i	$0.661558\pi$
$38$	0	0
$39$	2.15266e8i	2.38590i
$40$	0	0
$41$	1.94417e7	0.167809	0.0839043	−	0.996474i	$-0.473261\pi$
$41$	1.94417e7	0.167809	0.0839043	+	0.996474i	$0.473261\pi$
$42$	0	0
$43$	− 3.98511e7i	− 0.271080i	−0.990772	−	0.135540i	$-0.956723\pi$
$43$	− 3.98511e7i	− 0.271080i	0.990772	−	0.135540i	$-0.0432770\pi$
$44$	0	0
$45$	2.40038e7	0.130082
$46$	0	0
$47$	− 3.13565e8i	− 1.36722i	−0.729848	−	0.683610i	$-0.760409\pi$
$47$	− 3.13565e8i	− 1.36722i	0.729848	−	0.683610i	$-0.239591\pi$
$48$	0	0
$49$	2.22447e8	0.787494
$50$	0	0
$51$	− 4.04221e8i	− 1.17157i
$52$	0	0
$53$	2.99437e8	0.716021	0.358011	−	0.933718i	$-0.383455\pi$
$53$	2.99437e8	0.716021	0.358011	+	0.933718i	$0.383455\pi$
$54$	0	0
$55$	− 2.02695e7i	− 0.0402745i
$56$	0	0
$57$	5.14948e8	0.855832
$58$	0	0
$59$	− 9.75283e8i	− 1.36418i	−0.731270	−	0.682089i	$-0.761072\pi$
$59$	− 9.75283e8i	− 1.36418i	0.731270	−	0.682089i	$-0.238928\pi$
$60$	0	0
$61$	5.95678e8	0.705281	0.352641	−	0.935759i	$-0.385284\pi$
$61$	5.95678e8	0.705281	0.352641	+	0.935759i	$0.385284\pi$
$62$	0	0
$63$	5.78760e8i	0.583171i
$64$	0	0
$65$	−1.89141e8	−0.163012
$66$	0	0
$67$	− 1.47130e9i	− 1.08975i	−0.838517	−	0.544876i	$-0.816577\pi$
$67$	− 1.47130e9i	− 1.08975i	0.838517	−	0.544876i	$-0.183423\pi$
$68$	0	0
$69$	−2.96537e9	−1.89598
$70$	0	0
$71$	− 2.76443e9i	− 1.53220i	−0.642724	−	0.766098i	$-0.722196\pi$
$71$	− 2.76443e9i	− 1.53220i	0.642724	−	0.766098i	$-0.277804\pi$
$72$	0	0
$73$	−2.82326e9	−1.36187	−0.680935	−	0.732344i	$-0.738427\pi$
$73$	−2.82326e9	−1.36187	−0.680935	+	0.732344i	$0.738427\pi$
$74$	0	0
$75$	− 3.53370e9i	− 1.48910i
$76$	0	0
$77$	4.88723e8	0.180555
$78$	0	0
$79$	3.76461e9i	1.22345i	0.791072	+	0.611723i	$0.209523\pi$
$79$	3.76461e9i	1.22345i	−0.791072	+	0.611723i	$0.790477\pi$
$80$	0	0
$81$	−2.31763e9	−0.664689
$82$	0	0
$83$	− 6.30934e9i	− 1.60175i	−0.598834	−	0.800873i	$-0.704369\pi$
$83$	− 6.30934e9i	− 1.60175i	0.598834	−	0.800873i	$-0.295631\pi$
$84$	0	0
$85$	3.55164e8	0.0800451
$86$	0	0
$87$	− 2.75423e9i	− 0.552591i
$88$	0	0
$89$	6.15139e9	1.10160	0.550799	−	0.834638i	$-0.314323\pi$
$89$	6.15139e9	1.10160	0.550799	+	0.834638i	$0.314323\pi$
$90$	0	0
$91$	− 4.56042e9i	− 0.730799i
$92$	0	0
$93$	1.36825e10	1.96675
$94$	0	0
$95$	4.52454e8i	0.0584731i
$96$	0	0
$97$	9.73991e9	1.13422	0.567108	−	0.823643i	$-0.308062\pi$
$97$	9.73991e9	1.13422	0.567108	+	0.823643i	$0.308062\pi$
$98$	0	0
$99$	− 4.71204e9i	− 0.495487i
$100$	0	0
$101$	−1.55069e10	−1.47543	−0.737713	−	0.675114i	$-0.764094\pi$
$101$	−1.55069e10	−1.47543	−0.737713	+	0.675114i	$0.764094\pi$
$102$	0	0
$103$	1.83831e10i	1.58574i	0.609390	+	0.792870i	$0.291414\pi$
$103$	1.83831e10i	1.58574i	−0.609390	+	0.792870i	$0.708586\pi$
$104$	0	0
$105$	−9.10497e8	−0.0713399
$106$	0	0
$107$	− 9.93370e9i	− 0.708259i	−0.935196	−	0.354129i	$-0.884777\pi$
$107$	− 9.93370e9i	− 0.708259i	0.935196	−	0.354129i	$-0.115223\pi$
$108$	0	0
$109$	2.00812e10	1.30514	0.652569	−	0.757729i	$-0.273691\pi$
$109$	2.00812e10	1.30514	0.652569	+	0.757729i	$0.273691\pi$
$110$	0	0
$111$	− 2.46522e10i	− 1.46299i
$112$	0	0
$113$	−1.06765e9	−0.0579478	−0.0289739	−	0.999580i	$-0.509224\pi$
$113$	−1.06765e9	−0.0579478	−0.0289739	+	0.999580i	$0.509224\pi$
$114$	0	0
$115$	− 2.60549e9i	− 0.129539i
$116$	0	0
$117$	−4.39695e10	−2.00550
$118$	0	0
$119$	8.56345e9i	0.358851i
$120$	0	0
$121$	2.19584e10	0.846593
$122$	0	0
$123$	7.11016e9i	0.252554i
$124$	0	0
$125$	6.24288e9	0.204567
$126$	0	0
$127$	− 1.22872e10i	− 0.371907i	−0.982559	−	0.185954i	$-0.940463\pi$
$127$	− 1.22872e10i	− 0.371907i	0.982559	−	0.185954i	$-0.0595374\pi$
$128$	0	0
$129$	1.45742e10	0.407978
$130$	0	0
$131$	3.16990e10i	0.821654i	0.911713	+	0.410827i	$0.134760\pi$
$131$	3.16990e10i	0.821654i	−0.911713	+	0.410827i	$0.865240\pi$
$132$	0	0
$133$	−1.09092e10	−0.262141
$134$	0	0
$135$	1.83930e9i	0.0410189i
$136$	0	0
$137$	−3.44061e10	−0.712907	−0.356454	−	0.934313i	$-0.616014\pi$
$137$	−3.44061e10	−0.712907	−0.356454	+	0.934313i	$0.616014\pi$
$138$	0	0
$139$	9.80927e10i	1.89044i	0.326435	+	0.945220i	$0.394153\pi$
$139$	9.80927e10i	1.89044i	−0.326435	+	0.945220i	$0.605847\pi$
$140$	0	0
$141$	1.14676e11	2.05768
$142$	0	0
$143$	3.71292e10i	0.620919i
$144$	0	0
$145$	2.41998e9	0.0377547
$146$	0	0
$147$	8.13529e10i	1.18519i
$148$	0	0
$149$	−9.60238e10	−1.30752	−0.653759	−	0.756703i	$-0.726809\pi$
$149$	−9.60238e10	−1.30752	−0.653759	+	0.756703i	$0.726809\pi$
$150$	0	0
$151$	− 1.23142e11i	− 1.56863i	−0.620364	−	0.784314i	$-0.713015\pi$
$151$	− 1.23142e11i	− 1.56863i	0.620364	−	0.784314i	$-0.286985\pi$
$152$	0	0
$153$	8.25648e10	0.984777
$154$	0	0
$155$	1.20220e10i	0.134375i
$156$	0	0
$157$	4.55265e10	0.477272	0.238636	−	0.971109i	$-0.423300\pi$
$157$	4.55265e10	0.477272	0.238636	+	0.971109i	$0.423300\pi$
$158$	0	0
$159$	1.09509e11i	1.07762i
$160$	0	0
$161$	6.28216e10	0.580737
$162$	0	0
$163$	8.79982e10i	0.764779i	0.924001	+	0.382389i	$0.124899\pi$
$163$	8.79982e10i	0.764779i	−0.924001	+	0.382389i	$0.875101\pi$
$164$	0	0
$165$	7.41291e9	0.0606135
$166$	0	0
$167$	− 5.48586e10i	− 0.422340i	−0.977449	−	0.211170i	$-0.932273\pi$
$167$	− 5.48586e10i	− 0.422340i	0.977449	−	0.211170i	$-0.0677274\pi$
$168$	0	0
$169$	2.08605e11	1.51318
$170$	0	0
$171$	1.05182e11i	0.719381i
$172$	0	0
$173$	−1.77260e11	−1.14388	−0.571940	−	0.820295i	$-0.693809\pi$
$173$	−1.77260e11	−1.14388	−0.571940	+	0.820295i	$0.693809\pi$
$174$	0	0
$175$	7.48617e10i	0.456110i
$176$	0	0
$177$	3.56678e11	2.05310
$178$	0	0
$179$	− 7.41051e9i	− 0.0403258i	−0.999797	−	0.0201629i	$-0.993582\pi$
$179$	− 7.41051e9i	− 0.0403258i	0.999797	−	0.0201629i	$-0.00641849\pi$
$180$	0	0
$181$	1.02815e11	0.529254	0.264627	−	0.964351i	$-0.414751\pi$
$181$	1.02815e11	0.529254	0.264627	+	0.964351i	$0.414751\pi$
$182$	0	0
$183$	2.17850e11i	1.06146i
$184$	0	0
$185$	2.16604e10	0.0999557
$186$	0	0
$187$	− 6.97203e10i	− 0.304895i
$188$	0	0
$189$	−4.43478e10	−0.183892
$190$	0	0
$191$	− 1.98687e11i	− 0.781632i	−0.920469	−	0.390816i	$-0.872193\pi$
$191$	− 1.98687e11i	− 0.781632i	0.920469	−	0.390816i	$-0.127807\pi$
$192$	0	0
$193$	4.26188e10	0.159153	0.0795765	−	0.996829i	$-0.474643\pi$
$193$	4.26188e10	0.159153	0.0795765	+	0.996829i	$0.474643\pi$
$194$	0	0
$195$	− 6.91722e10i	− 0.245334i
$196$	0	0
$197$	−1.96198e10	−0.0661245	−0.0330623	−	0.999453i	$-0.510526\pi$
$197$	−1.96198e10	−0.0661245	−0.0330623	+	0.999453i	$0.510526\pi$
$198$	0	0
$199$	2.17989e11i	0.698504i	0.937029	+	0.349252i	$0.113564\pi$
$199$	2.17989e11i	0.698504i	−0.937029	+	0.349252i	$0.886436\pi$
$200$	0	0
$201$	5.38080e11	1.64009
$202$	0	0
$203$	5.83486e10i	0.169259i
$204$	0	0
$205$	−6.24727e9	−0.0172552
$206$	0	0
$207$	− 6.05696e11i	− 1.59369i
$208$	0	0
$209$	8.88185e10	0.222727
$210$	0	0
$211$	− 4.22653e11i	− 1.01058i	−0.862949	−	0.505291i	$-0.831385\pi$
$211$	− 4.22653e11i	− 1.01058i	0.862949	−	0.505291i	$-0.168615\pi$
$212$	0	0
$213$	1.01100e12	2.30597
$214$	0	0
$215$	1.28055e10i	0.0278743i
$216$	0	0
$217$	−2.89864e11	−0.602416
$218$	0	0
$219$	− 1.03251e12i	− 2.04963i
$220$	0	0
$221$	−6.50581e11	−1.23407
$222$	0	0
$223$	4.19246e10i	0.0760229i	0.999277	+	0.0380114i	$0.0121023\pi$
$223$	4.19246e10i	0.0760229i	−0.999277	+	0.0380114i	$0.987898\pi$
$224$	0	0
$225$	7.21782e11	1.25168
$226$	0	0
$227$	− 1.64197e11i	− 0.272418i	−0.990680	−	0.136209i	$-0.956508\pi$
$227$	− 1.64197e11i	− 0.272418i	0.990680	−	0.136209i	$-0.0434919\pi$
$228$	0	0
$229$	3.25978e11	0.517620	0.258810	−	0.965928i	$-0.416670\pi$
$229$	3.25978e11	0.517620	0.258810	+	0.965928i	$0.416670\pi$
$230$	0	0
$231$	1.78734e11i	0.271737i
$232$	0	0
$233$	9.79445e11	1.42627	0.713133	−	0.701029i	$-0.247276\pi$
$233$	9.79445e11	1.42627	0.713133	+	0.701029i	$0.247276\pi$
$234$	0	0
$235$	1.00759e11i	0.140587i
$236$	0	0
$237$	−1.37678e12	−1.84130
$238$	0	0
$239$	7.64743e11i	0.980677i	0.871532	+	0.490339i	$0.163127\pi$
$239$	7.64743e11i	0.980677i	−0.871532	+	0.490339i	$0.836873\pi$
$240$	0	0
$241$	8.88804e11	1.09325	0.546627	−	0.837376i	$-0.315912\pi$
$241$	8.88804e11	1.09325	0.546627	+	0.837376i	$0.315912\pi$
$242$	0	0
$243$	− 1.18559e12i	− 1.39928i
$244$	0	0
$245$	−7.14800e10	−0.0809756
$246$	0	0
$247$	− 8.28793e11i	− 0.901491i
$248$	0	0
$249$	2.30744e12	2.41064
$250$	0	0
$251$	− 1.37436e12i	− 1.37953i	−0.724031	−	0.689767i	$-0.757713\pi$
$251$	− 1.37436e12i	− 1.37953i	0.724031	−	0.689767i	$-0.242287\pi$
$252$	0	0
$253$	−5.11469e11	−0.493420
$254$	0	0
$255$	1.29890e11i	0.120469i
$256$	0	0
$257$	1.43315e12	1.27828	0.639140	−	0.769091i	$-0.279291\pi$
$257$	1.43315e12	1.27828	0.639140	+	0.769091i	$0.279291\pi$
$258$	0	0
$259$	5.22258e11i	0.448112i
$260$	0	0
$261$	5.62570e11	0.464488
$262$	0	0
$263$	− 2.26828e12i	− 1.80267i	−0.433118	−	0.901337i	$-0.642587\pi$
$263$	− 2.26828e12i	− 1.80267i	0.433118	−	0.901337i	$-0.357413\pi$
$264$	0	0
$265$	−9.62193e10	−0.0736263
$266$	0	0
$267$	2.24967e12i	1.65792i
$268$	0	0
$269$	1.43734e12	1.02047	0.510233	−	0.860036i	$-0.329559\pi$
$269$	1.43734e12	1.02047	0.510233	+	0.860036i	$0.329559\pi$
$270$	0	0
$271$	2.30664e12i	1.57809i	0.614333	+	0.789047i	$0.289425\pi$
$271$	2.30664e12i	1.57809i	−0.614333	+	0.789047i	$0.710575\pi$
$272$	0	0
$273$	1.66783e12	1.09986
$274$	0	0
$275$	− 6.09495e11i	− 0.387531i
$276$	0	0
$277$	3.74193e11	0.229455	0.114727	−	0.993397i	$-0.463401\pi$
$277$	3.74193e11	0.229455	0.114727	+	0.993397i	$0.463401\pi$
$278$	0	0
$279$	2.79474e12i	1.65318i
$280$	0	0
$281$	6.14599e11	0.350801	0.175400	−	0.984497i	$-0.443878\pi$
$281$	6.14599e11	0.350801	0.175400	+	0.984497i	$0.443878\pi$
$282$	0	0
$283$	− 1.91403e12i	− 1.05442i	−0.849734	−	0.527212i	$-0.823237\pi$
$283$	− 1.91403e12i	− 1.05442i	0.849734	−	0.527212i	$-0.176763\pi$
$284$	0	0
$285$	−1.65470e11	−0.0880026
$286$	0	0
$287$	− 1.50629e11i	− 0.0773571i
$288$	0	0
$289$	−7.94348e11	−0.394023
$290$	0	0
$291$	3.56205e12i	1.70701i
$292$	0	0
$293$	2.33021e12	1.07909	0.539543	−	0.841958i	$-0.318597\pi$
$293$	2.33021e12	1.07909	0.539543	+	0.841958i	$0.318597\pi$
$294$	0	0
$295$	3.13392e11i	0.140274i
$296$	0	0
$297$	3.61062e11	0.156243
$298$	0	0
$299$	4.77267e12i	1.99713i
$300$	0	0
$301$	−3.08757e11	−0.124964
$302$	0	0
$303$	− 5.67114e12i	− 2.22053i
$304$	0	0
$305$	−1.91412e11	−0.0725219
$306$	0	0
$307$	1.57987e12i	0.579335i	0.957127	+	0.289667i	$0.0935447\pi$
$307$	1.57987e12i	0.579335i	−0.957127	+	0.289667i	$0.906455\pi$
$308$	0	0
$309$	−6.72301e12	−2.38656
$310$	0	0
$311$	− 8.60168e11i	− 0.295653i	−0.989013	−	0.147826i	$-0.952772\pi$
$311$	− 8.60168e11i	− 0.295653i	0.989013	−	0.147826i	$-0.0472276\pi$
$312$	0	0
$313$	3.45680e12	1.15067	0.575337	−	0.817916i	$-0.304871\pi$
$313$	3.45680e12	1.15067	0.575337	+	0.817916i	$0.304871\pi$
$314$	0	0
$315$	− 1.85975e11i	− 0.0599656i
$316$	0	0
$317$	3.02410e12	0.944713	0.472356	−	0.881408i	$-0.343404\pi$
$317$	3.02410e12	0.944713	0.472356	+	0.881408i	$0.343404\pi$
$318$	0	0
$319$	− 4.75052e11i	− 0.143809i
$320$	0	0
$321$	3.63293e12	1.06594
$322$	0	0
$323$	1.55629e12i	0.442667i
$324$	0	0
$325$	−5.68738e12	−1.56854
$326$	0	0
$327$	7.34403e12i	1.96425i
$328$	0	0
$329$	−2.42943e12	−0.630266
$330$	0	0
$331$	5.70877e12i	1.43682i	0.695620	+	0.718410i	$0.255130\pi$
$331$	5.70877e12i	1.43682i	−0.695620	+	0.718410i	$0.744870\pi$
$332$	0	0
$333$	5.03537e12	1.22973
$334$	0	0
$335$	4.72779e11i	0.112056i
$336$	0	0
$337$	5.41546e11	0.124591	0.0622953	−	0.998058i	$-0.480158\pi$
$337$	5.41546e11	0.124591	0.0622953	+	0.998058i	$0.480158\pi$
$338$	0	0
$339$	− 3.90458e11i	− 0.0872120i
$340$	0	0
$341$	2.35996e12	0.511839
$342$	0	0
$343$	− 3.91202e12i	− 0.824006i
$344$	0	0
$345$	9.52874e11	0.194957
$346$	0	0
$347$	− 6.56530e12i	− 1.30499i	−0.757793	−	0.652495i	$-0.773723\pi$
$347$	− 6.56530e12i	− 1.30499i	0.757793	−	0.652495i	$-0.226277\pi$
$348$	0	0
$349$	4.35199e12	0.840545	0.420273	−	0.907398i	$-0.361934\pi$
$349$	4.35199e12	0.840545	0.420273	+	0.907398i	$0.361934\pi$
$350$	0	0
$351$	− 3.36918e12i	− 0.632396i
$352$	0	0
$353$	−1.41355e12	−0.257893	−0.128946	−	0.991652i	$-0.541159\pi$
$353$	−1.41355e12	−0.257893	−0.128946	+	0.991652i	$0.541159\pi$
$354$	0	0
$355$	8.88307e11i	0.157551i
$356$	0	0
$357$	−3.13180e12	−0.540074
$358$	0	0
$359$	− 6.94913e12i	− 1.16535i	−0.812704	−	0.582677i	$-0.802005\pi$
$359$	− 6.94913e12i	− 1.16535i	0.812704	−	0.582677i	$-0.197995\pi$
$360$	0	0
$361$	4.14847e12	0.676631
$362$	0	0
$363$	8.03058e12i	1.27413i
$364$	0	0
$365$	9.07208e11	0.140037
$366$	0	0
$367$	9.04638e12i	1.35877i	0.733784	+	0.679383i	$0.237752\pi$
$367$	9.04638e12i	1.35877i	−0.733784	+	0.679383i	$0.762248\pi$
$368$	0	0
$369$	−1.45230e12	−0.212287
$370$	0	0
$371$	− 2.31996e12i	− 0.330074i
$372$	0	0
$373$	8.90817e11	0.123380	0.0616899	−	0.998095i	$-0.480351\pi$
$373$	8.90817e11	0.123380	0.0616899	+	0.998095i	$0.480351\pi$
$374$	0	0
$375$	2.28313e12i	0.307875i
$376$	0	0
$377$	−4.43285e12	−0.582072
$378$	0	0
$379$	− 1.30273e13i	− 1.66593i	−0.553326	−	0.832965i	$-0.686641\pi$
$379$	− 1.30273e13i	− 1.66593i	0.553326	−	0.832965i	$-0.313359\pi$
$380$	0	0
$381$	4.49365e12	0.559724
$382$	0	0
$383$	1.26377e13i	1.53347i	0.641964	+	0.766735i	$0.278120\pi$
$383$	1.26377e13i	1.53347i	−0.641964	+	0.766735i	$0.721880\pi$
$384$	0	0
$385$	−1.57043e11	−0.0185659
$386$	0	0
$387$	2.97689e12i	0.342932i
$388$	0	0
$389$	−1.36861e13	−1.53650	−0.768248	−	0.640153i	$-0.778871\pi$
$389$	−1.36861e13	−1.53650	−0.768248	+	0.640153i	$0.778871\pi$
$390$	0	0
$391$	− 8.96201e12i	− 0.980667i
$392$	0	0
$393$	−1.15929e13	−1.23660
$394$	0	0
$395$	− 1.20970e12i	− 0.125803i
$396$	0	0
$397$	−1.19462e12	−0.121137	−0.0605684	−	0.998164i	$-0.519291\pi$
$397$	−1.19462e12	−0.121137	−0.0605684	+	0.998164i	$0.519291\pi$
$398$	0	0
$399$	− 3.98969e12i	− 0.394525i
$400$	0	0
$401$	−8.25527e12	−0.796176	−0.398088	−	0.917347i	$-0.630326\pi$
$401$	−8.25527e12	−0.796176	−0.398088	+	0.917347i	$0.630326\pi$
$402$	0	0
$403$	− 2.20215e13i	− 2.07168i
$404$	0	0
$405$	7.44733e11	0.0683480
$406$	0	0
$407$	− 4.25202e12i	− 0.380736i
$408$	0	0
$409$	−1.91617e13	−1.67424	−0.837119	−	0.547021i	$-0.815762\pi$
$409$	−1.91617e13	−1.67424	−0.837119	+	0.547021i	$0.815762\pi$
$410$	0	0
$411$	− 1.25829e13i	− 1.07293i
$412$	0	0
$413$	−7.55626e12	−0.628864
$414$	0	0
$415$	2.02741e12i	0.164703i
$416$	0	0
$417$	−3.58742e13	−2.84513
$418$	0	0
$419$	− 9.61416e12i	− 0.744460i	−0.928141	−	0.372230i	$-0.878593\pi$
$419$	− 9.61416e12i	− 0.744460i	0.928141	−	0.372230i	$-0.121407\pi$
$420$	0	0
$421$	1.17339e13	0.887220	0.443610	−	0.896220i	$-0.353697\pi$
$421$	1.17339e13	0.887220	0.443610	+	0.896220i	$0.353697\pi$
$422$	0	0
$423$	2.34234e13i	1.72961i
$424$	0	0
$425$	1.06796e13	0.770214
$426$	0	0
$427$	− 4.61517e12i	− 0.325123i
$428$	0	0
$429$	−1.35788e13	−0.934490
$430$	0	0
$431$	− 1.31196e13i	− 0.882131i	−0.897475	−	0.441066i	$-0.854601\pi$
$431$	− 1.31196e13i	− 0.882131i	0.897475	−	0.441066i	$-0.145399\pi$
$432$	0	0
$433$	1.02778e13	0.675247	0.337623	−	0.941281i	$-0.390377\pi$
$433$	1.02778e13	0.675247	0.337623	+	0.941281i	$0.390377\pi$
$434$	0	0
$435$	8.85028e11i	0.0568213i
$436$	0	0
$437$	1.14169e13	0.716379
$438$	0	0
$439$	− 1.92239e12i	− 0.117901i	−0.998261	−	0.0589506i	$-0.981225\pi$
$439$	− 1.92239e12i	− 0.117901i	0.998261	−	0.0589506i	$-0.0187754\pi$
$440$	0	0
$441$	−1.66169e13	−0.996224
$442$	0	0
$443$	− 5.09935e12i	− 0.298880i	−0.988771	−	0.149440i	$-0.952253\pi$
$443$	− 5.09935e12i	− 0.298880i	0.988771	−	0.149440i	$-0.0477471\pi$
$444$	0	0
$445$	−1.97665e12	−0.113274
$446$	0	0
$447$	− 3.51176e13i	− 1.96783i
$448$	0	0
$449$	6.65675e12	0.364780	0.182390	−	0.983226i	$-0.441617\pi$
$449$	6.65675e12	0.364780	0.182390	+	0.983226i	$0.441617\pi$
$450$	0	0
$451$	1.22637e12i	0.0657260i
$452$	0	0
$453$	4.50350e13	2.36080
$454$	0	0
$455$	1.46542e12i	0.0751458i
$456$	0	0
$457$	−1.18960e13	−0.596789	−0.298394	−	0.954443i	$-0.596451\pi$
$457$	−1.18960e13	−0.596789	−0.298394	+	0.954443i	$0.596451\pi$
$458$	0	0
$459$	6.32657e12i	0.310531i
$460$	0	0
$461$	−9.72223e12	−0.466940	−0.233470	−	0.972364i	$-0.575008\pi$
$461$	−9.72223e12	−0.466940	−0.233470	+	0.972364i	$0.575008\pi$
$462$	0	0
$463$	3.32203e12i	0.156134i	0.996948	+	0.0780672i	$0.0248749\pi$
$463$	3.32203e12i	0.156134i	−0.996948	+	0.0780672i	$0.975125\pi$
$464$	0	0
$465$	−4.39665e12	−0.202235
$466$	0	0
$467$	1.04989e13i	0.472671i	0.971672	+	0.236335i	$0.0759464\pi$
$467$	1.04989e13i	0.472671i	−0.971672	+	0.236335i	$0.924054\pi$
$468$	0	0
$469$	−1.13993e13	−0.502358
$470$	0	0
$471$	1.66498e13i	0.718299i
$472$	0	0
$473$	2.51377e12	0.106175
$474$	0	0
$475$	1.36051e13i	0.562643i
$476$	0	0
$477$	−2.23680e13	−0.905807
$478$	0	0
$479$	1.87912e13i	0.745205i	0.927991	+	0.372603i	$0.121535\pi$
$479$	1.87912e13i	0.745205i	−0.927991	+	0.372603i	$0.878465\pi$
$480$	0	0
$481$	−3.96769e13	−1.54104
$482$	0	0
$483$	2.29750e13i	0.874015i
$484$	0	0
$485$	−3.12976e12	−0.116628
$486$	0	0
$487$	3.24005e12i	0.118279i	0.998250	+	0.0591393i	$0.0188356\pi$
$487$	3.24005e12i	0.118279i	−0.998250	+	0.0591393i	$0.981164\pi$
$488$	0	0
$489$	−3.21825e13	−1.15100
$490$	0	0
$491$	− 5.91770e12i	− 0.207370i	−0.994610	−	0.103685i	$-0.966937\pi$
$491$	− 5.91770e12i	− 0.207370i	0.994610	−	0.103685i	$-0.0330633\pi$
$492$	0	0
$493$	8.32391e12	0.285820
$494$	0	0
$495$	1.51414e12i	0.0509495i
$496$	0	0
$497$	−2.14182e13	−0.706318
$498$	0	0
$499$	1.00820e13i	0.325869i	0.986637	+	0.162935i	$0.0520960\pi$
$499$	1.00820e13i	0.325869i	−0.986637	+	0.162935i	$0.947904\pi$
$500$	0	0
$501$	2.00628e13	0.635627
$502$	0	0
$503$	− 1.73241e13i	− 0.538035i	−0.963135	−	0.269017i	$-0.913301\pi$
$503$	− 1.73241e13i	− 0.538035i	0.963135	−	0.269017i	$-0.0866989\pi$
$504$	0	0
$505$	4.98289e12	0.151714
$506$	0	0
$507$	7.62905e13i	2.27736i
$508$	0	0
$509$	−6.47322e13	−1.89466	−0.947330	−	0.320258i	$-0.896230\pi$
$509$	−6.47322e13	−1.89466	−0.947330	+	0.320258i	$0.896230\pi$
$510$	0	0
$511$	2.18739e13i	0.627801i
$512$	0	0
$513$	−8.05959e12	−0.226843
$514$	0	0
$515$	− 5.90711e12i	− 0.163057i
$516$	0	0
$517$	1.97794e13	0.535502
$518$	0	0
$519$	− 6.48271e13i	− 1.72155i
$520$	0	0
$521$	−7.27568e13	−1.89533	−0.947665	−	0.319267i	$-0.896563\pi$
$521$	−7.27568e13	−1.89533	−0.947665	+	0.319267i	$0.896563\pi$
$522$	0	0
$523$	− 1.32565e13i	− 0.338783i	−0.985549	−	0.169392i	$-0.945820\pi$
$523$	− 1.32565e13i	− 0.338783i	0.985549	−	0.169392i	$-0.0541803\pi$
$524$	0	0
$525$	−2.73782e13	−0.686450
$526$	0	0
$527$	4.13515e13i	1.01728i
$528$	0	0
$529$	−2.43189e13	−0.587037
$530$	0	0
$531$	7.28540e13i	1.72576i
$532$	0	0
$533$	1.14436e13	0.266027
$534$	0	0
$535$	3.19204e12i	0.0728281i
$536$	0	0
$537$	2.71015e12	0.0606908
$538$	0	0
$539$	1.40318e13i	0.308440i
$540$	0	0
$541$	6.40632e13	1.38236	0.691182	−	0.722681i	$-0.257090\pi$
$541$	6.40632e13	1.38236	0.691182	+	0.722681i	$0.257090\pi$
$542$	0	0
$543$	3.76013e13i	0.796533i
$544$	0	0
$545$	−6.45276e12	−0.134203
$546$	0	0
$547$	− 1.65431e13i	− 0.337816i	−0.985632	−	0.168908i	$-0.945976\pi$
$547$	− 1.65431e13i	− 0.337816i	0.985632	−	0.168908i	$-0.0540240\pi$
$548$	0	0
$549$	−4.44973e13	−0.892220
$550$	0	0
$551$	1.06041e13i	0.208792i
$552$	0	0
$553$	2.91673e13	0.563989
$554$	0	0
$555$	7.92158e12i	0.150434i
$556$	0	0
$557$	6.86489e13	1.28043	0.640217	−	0.768194i	$-0.278844\pi$
$557$	6.86489e13	1.28043	0.640217	+	0.768194i	$0.278844\pi$
$558$	0	0
$559$	− 2.34568e13i	− 0.429744i
$560$	0	0
$561$	2.54979e13	0.458871
$562$	0	0
$563$	− 1.28930e13i	− 0.227935i	−0.993484	−	0.113968i	$-0.963644\pi$
$563$	− 1.28930e13i	− 0.227935i	0.993484	−	0.113968i	$-0.0363560\pi$
$564$	0	0
$565$	3.43073e11	0.00595859
$566$	0	0
$567$	1.79564e13i	0.306411i
$568$	0	0
$569$	6.92565e13	1.16118	0.580590	−	0.814196i	$-0.302822\pi$
$569$	6.92565e13	1.16118	0.580590	+	0.814196i	$0.302822\pi$
$570$	0	0
$571$	9.54657e12i	0.157278i	0.996903	+	0.0786388i	$0.0250574\pi$
$571$	9.54657e12i	0.157278i	−0.996903	+	0.0786388i	$0.974943\pi$
$572$	0	0
$573$	7.26633e13	1.17636
$574$	0	0
$575$	− 7.83459e13i	− 1.24646i
$576$	0	0
$577$	2.42121e13	0.378576	0.189288	−	0.981922i	$-0.439382\pi$
$577$	2.42121e13	0.378576	0.189288	+	0.981922i	$0.439382\pi$
$578$	0	0
$579$	1.55864e13i	0.239527i
$580$	0	0
$581$	−4.88833e13	−0.738380
$582$	0	0
$583$	1.88882e13i	0.280446i
$584$	0	0
$585$	1.41289e13	0.206219
$586$	0	0
$587$	− 1.17806e14i	− 1.69035i	−0.534488	−	0.845176i	$-0.679496\pi$
$587$	− 1.17806e14i	− 1.69035i	0.534488	−	0.845176i	$-0.320504\pi$
$588$	0	0
$589$	−5.26788e13	−0.743121
$590$	0	0
$591$	− 7.17529e12i	− 0.0995181i
$592$	0	0
$593$	1.00583e14	1.37167	0.685837	−	0.727755i	$-0.259436\pi$
$593$	1.00583e14	1.37167	0.685837	+	0.727755i	$0.259436\pi$
$594$	0	0
$595$	− 2.75173e12i	− 0.0368995i
$596$	0	0
$597$	−7.97223e13	−1.05126
$598$	0	0
$599$	7.06299e13i	0.915913i	0.888975	+	0.457957i	$0.151419\pi$
$599$	7.06299e13i	0.915913i	−0.888975	+	0.457957i	$0.848581\pi$
$600$	0	0
$601$	−1.05996e13	−0.135182	−0.0675910	−	0.997713i	$-0.521531\pi$
$601$	−1.05996e13	−0.135182	−0.0675910	+	0.997713i	$0.521531\pi$
$602$	0	0
$603$	1.09907e14i	1.37860i
$604$	0	0
$605$	−7.05600e12	−0.0870525
$606$	0	0
$607$	− 1.42090e14i	− 1.72433i	−0.506631	−	0.862163i	$-0.669110\pi$
$607$	− 1.42090e14i	− 1.72433i	0.506631	−	0.862163i	$-0.330890\pi$
$608$	0	0
$609$	−2.13391e13	−0.254736
$610$	0	0
$611$	− 1.84568e14i	− 2.16746i
$612$	0	0
$613$	−1.38880e14	−1.60450	−0.802248	−	0.596990i	$-0.796363\pi$
$613$	−1.38880e14	−1.60450	−0.802248	+	0.596990i	$0.796363\pi$
$614$	0	0
$615$	− 2.28474e12i	− 0.0259693i
$616$	0	0
$617$	−2.34158e13	−0.261868	−0.130934	−	0.991391i	$-0.541798\pi$
$617$	−2.34158e13	−0.261868	−0.130934	+	0.991391i	$0.541798\pi$
$618$	0	0
$619$	− 1.63335e14i	− 1.79732i	−0.438647	−	0.898659i	$-0.644542\pi$
$619$	− 1.63335e14i	− 1.79732i	0.438647	−	0.898659i	$-0.355458\pi$
$620$	0	0
$621$	4.64118e13	0.502540
$622$	0	0
$623$	− 4.76595e13i	− 0.507820i
$624$	0	0
$625$	9.23530e13	0.968392
$626$	0	0
$627$	3.24825e13i	0.335206i
$628$	0	0
$629$	7.45044e13	0.756708
$630$	0	0
$631$	− 6.94338e12i	− 0.0694103i	−0.999398	−	0.0347051i	$-0.988951\pi$
$631$	− 6.94338e12i	− 0.0694103i	0.999398	−	0.0347051i	$-0.0110492\pi$
$632$	0	0
$633$	1.54572e14	1.52094
$634$	0	0
$635$	3.94830e12i	0.0382421i
$636$	0	0
$637$	1.30935e14	1.24842
$638$	0	0
$639$	2.06504e14i	1.93831i
$640$	0	0
$641$	−1.69119e14	−1.56279	−0.781396	−	0.624036i	$-0.785492\pi$
$641$	−1.69119e14	−1.56279	−0.781396	+	0.624036i	$0.785492\pi$
$642$	0	0
$643$	− 8.68138e13i	− 0.789830i	−0.918718	−	0.394915i	$-0.870774\pi$
$643$	− 8.68138e13i	− 0.789830i	0.918718	−	0.394915i	$-0.129226\pi$
$644$	0	0
$645$	−4.68320e12	−0.0419512
$646$	0	0
$647$	1.61862e14i	1.42766i	0.700321	+	0.713828i	$0.253040\pi$
$647$	1.61862e14i	1.42766i	−0.700321	+	0.713828i	$0.746960\pi$
$648$	0	0
$649$	6.15201e13	0.534311
$650$	0	0
$651$	− 1.06009e14i	− 0.906642i
$652$	0	0
$653$	1.84565e14	1.55448	0.777238	−	0.629206i	$-0.216620\pi$
$653$	1.84565e14	1.55448	0.777238	+	0.629206i	$0.216620\pi$
$654$	0	0
$655$	− 1.01860e13i	− 0.0844882i
$656$	0	0
$657$	2.10898e14	1.72284
$658$	0	0
$659$	− 5.57893e13i	− 0.448873i	−0.974489	−	0.224437i	$-0.927946\pi$
$659$	− 5.57893e13i	− 0.448873i	0.974489	−	0.224437i	$-0.0720542\pi$
$660$	0	0
$661$	−1.73221e14	−1.37276	−0.686380	−	0.727244i	$-0.740801\pi$
$661$	−1.73221e14	−1.37276	−0.686380	+	0.727244i	$0.740801\pi$
$662$	0	0
$663$	− 2.37929e14i	− 1.85729i
$664$	0	0
$665$	3.50550e12	0.0269552
$666$	0	0
$667$	− 6.10643e13i	− 0.462549i
$668$	0	0
$669$	−1.53326e13	−0.114415
$670$	0	0
$671$	3.75749e13i	0.276239i
$672$	0	0
$673$	7.69685e13	0.557491	0.278745	−	0.960365i	$-0.410081\pi$
$673$	7.69685e13	0.557491	0.278745	+	0.960365i	$0.410081\pi$
$674$	0	0
$675$	5.53069e13i	0.394694i
$676$	0	0
$677$	1.66170e14	1.16845	0.584225	−	0.811592i	$-0.301399\pi$
$677$	1.66170e14	1.16845	0.584225	+	0.811592i	$0.301399\pi$
$678$	0	0
$679$	− 7.54624e13i	− 0.522856i
$680$	0	0
$681$	6.00498e13	0.409992
$682$	0	0
$683$	− 1.39393e14i	− 0.937862i	−0.883235	−	0.468931i	$-0.844639\pi$
$683$	− 1.39393e14i	− 0.937862i	0.883235	−	0.468931i	$-0.155361\pi$
$684$	0	0
$685$	1.10559e13	0.0733061
$686$	0	0
$687$	1.19216e14i	0.779024i
$688$	0	0
$689$	1.76252e14	1.13511
$690$	0	0
$691$	− 9.88768e13i	− 0.627631i	−0.949484	−	0.313815i	$-0.898393\pi$
$691$	− 9.88768e13i	− 0.627631i	0.949484	−	0.313815i	$-0.101607\pi$
$692$	0	0
$693$	−3.65077e13	−0.228412
$694$	0	0
$695$	− 3.15205e13i	− 0.194388i
$696$	0	0
$697$	−2.14885e13	−0.130630
$698$	0	0
$699$	3.58200e14i	2.14654i
$700$	0	0
$701$	−2.47882e14	−1.46438	−0.732191	−	0.681099i	$-0.761502\pi$
$701$	−2.47882e14	−1.46438	−0.732191	+	0.681099i	$0.761502\pi$
$702$	0	0
$703$	9.49132e13i	0.552777i
$704$	0	0
$705$	−3.68494e13	−0.211585
$706$	0	0
$707$	1.20144e14i	0.680148i
$708$	0	0
$709$	1.10096e13	0.0614525	0.0307263	−	0.999528i	$-0.490218\pi$
$709$	1.10096e13	0.0614525	0.0307263	+	0.999528i	$0.490218\pi$
$710$	0	0
$711$	− 2.81218e14i	− 1.54773i
$712$	0	0
$713$	3.03355e14	1.64628
$714$	0	0
$715$	− 1.19309e13i	− 0.0638472i
$716$	0	0
$717$	−2.79680e14	−1.47593
$718$	0	0
$719$	1.00155e14i	0.521227i	0.965443	+	0.260613i	$0.0839248\pi$
$719$	1.00155e14i	0.521227i	−0.965443	+	0.260613i	$0.916075\pi$
$720$	0	0
$721$	1.42428e14	0.731001
$722$	0	0
$723$	3.25051e14i	1.64536i
$724$	0	0
$725$	7.27676e13	0.363286
$726$	0	0
$727$	3.47263e14i	1.70996i	0.518659	+	0.854981i	$0.326432\pi$
$727$	3.47263e14i	1.70996i	−0.518659	+	0.854981i	$0.673568\pi$
$728$	0	0
$729$	2.96738e14	1.44124
$730$	0	0
$731$	4.40466e13i	0.211021i
$732$	0	0
$733$	−9.28015e12	−0.0438566	−0.0219283	−	0.999760i	$-0.506981\pi$
$733$	−9.28015e12	−0.0438566	−0.0219283	+	0.999760i	$0.506981\pi$
$734$	0	0
$735$	− 2.61415e13i	− 0.121869i
$736$	0	0
$737$	9.28085e13	0.426825
$738$	0	0
$739$	− 1.29156e14i	− 0.585994i	−0.956113	−	0.292997i	$-0.905347\pi$
$739$	− 1.29156e14i	− 0.585994i	0.956113	−	0.292997i	$-0.0946526\pi$
$740$	0	0
$741$	3.03104e14	1.35675
$742$	0	0
$743$	6.73037e13i	0.297232i	0.988895	+	0.148616i	$0.0474818\pi$
$743$	6.73037e13i	0.297232i	−0.988895	+	0.148616i	$0.952518\pi$
$744$	0	0
$745$	3.08557e13	0.134448
$746$	0	0
$747$	4.71310e14i	2.02630i
$748$	0	0
$749$	−7.69639e13	−0.326496
$750$	0	0
$751$	− 3.31551e14i	− 1.38788i	−0.720034	−	0.693939i	$-0.755874\pi$
$751$	− 3.31551e14i	− 1.38788i	0.720034	−	0.693939i	$-0.244126\pi$
$752$	0	0
$753$	5.02628e14	2.07621
$754$	0	0
$755$	3.95696e13i	0.161297i
$756$	0	0
$757$	−1.16522e14	−0.468737	−0.234369	−	0.972148i	$-0.575302\pi$
$757$	−1.16522e14	−0.468737	−0.234369	+	0.972148i	$0.575302\pi$
$758$	0	0
$759$	− 1.87053e14i	− 0.742602i
$760$	0	0
$761$	−1.62448e14	−0.636489	−0.318244	−	0.948009i	$-0.603093\pi$
$761$	−1.62448e14	−0.636489	−0.318244	+	0.948009i	$0.603093\pi$
$762$	0	0
$763$	− 1.55584e14i	− 0.601648i
$764$	0	0
$765$	−2.65309e13	−0.101262
$766$	0	0
$767$	− 5.74063e14i	− 2.16263i
$768$	0	0
$769$	−3.73550e14	−1.38905	−0.694523	−	0.719471i	$-0.744385\pi$
$769$	−3.73550e14	−1.38905	−0.694523	+	0.719471i	$0.744385\pi$
$770$	0	0
$771$	5.24128e14i	1.92382i
$772$	0	0
$773$	5.24708e13	0.190117	0.0950583	−	0.995472i	$-0.469696\pi$
$773$	5.24708e13	0.190117	0.0950583	+	0.995472i	$0.469696\pi$
$774$	0	0
$775$	3.61495e14i	1.29299i
$776$	0	0
$777$	−1.90999e14	−0.674413
$778$	0	0
$779$	− 2.73748e13i	− 0.0954253i
$780$	0	0
$781$	1.74378e14	0.600119
$782$	0	0
$783$	4.31073e13i	0.146468i
$784$	0	0
$785$	−1.46292e13	−0.0490764
$786$	0	0
$787$	− 1.31071e14i	− 0.434144i	−0.976156	−	0.217072i	$-0.930349\pi$
$787$	− 1.31071e14i	− 0.434144i	0.976156	−	0.217072i	$-0.0696507\pi$
$788$	0	0
$789$	8.29549e14	2.71304
$790$	0	0
$791$	8.27190e12i	0.0267130i
$792$	0	0
$793$	3.50623e14	1.11808
$794$	0	0
$795$	− 3.51891e13i	− 0.110808i
$796$	0	0
$797$	−3.90800e14	−1.21524	−0.607622	−	0.794227i	$-0.707876\pi$
$797$	−3.90800e14	−1.21524	−0.607622	+	0.794227i	$0.707876\pi$
$798$	0	0
$799$	3.46577e14i	1.06431i
$800$	0	0
$801$	−4.59511e14	−1.39358
$802$	0	0
$803$	− 1.78089e14i	− 0.533407i
$804$	0	0
$805$	−2.01867e13	−0.0597154
$806$	0	0
$807$	5.25661e14i	1.53581i
$808$	0	0
$809$	−6.23564e14	−1.79944	−0.899722	−	0.436464i	$-0.856231\pi$
$809$	−6.23564e14	−1.79944	−0.899722	+	0.436464i	$0.856231\pi$
$810$	0	0
$811$	3.75304e14i	1.06974i	0.844934	+	0.534871i	$0.179640\pi$
$811$	3.75304e14i	1.06974i	−0.844934	+	0.534871i	$0.820360\pi$
$812$	0	0
$813$	−8.43578e14	−2.37505
$814$	0	0
$815$	− 2.82768e13i	− 0.0786399i
$816$	0	0
$817$	−5.61122e13	−0.154151
$818$	0	0
$819$	3.40665e14i	0.924502i
$820$	0	0
$821$	1.31658e14	0.352965	0.176482	−	0.984304i	$-0.443528\pi$
$821$	1.31658e14	0.352965	0.176482	+	0.984304i	$0.443528\pi$
$822$	0	0
$823$	− 8.25554e13i	− 0.218649i	−0.994006	−	0.109324i	$-0.965131\pi$
$823$	− 8.25554e13i	− 0.218649i	0.994006	−	0.109324i	$-0.0348687\pi$
$824$	0	0
$825$	2.22903e14	0.583238
$826$	0	0
$827$	− 1.52417e14i	− 0.394010i	−0.980403	−	0.197005i	$-0.936879\pi$
$827$	− 1.52417e14i	− 0.394010i	0.980403	−	0.197005i	$-0.0631215\pi$
$828$	0	0
$829$	5.41291e14	1.38248	0.691239	−	0.722627i	$-0.257065\pi$
$829$	5.41291e14	1.38248	0.691239	+	0.722627i	$0.257065\pi$
$830$	0	0
$831$	1.36849e14i	0.345332i
$832$	0	0
$833$	−2.45867e14	−0.613021
$834$	0	0
$835$	1.76280e13i	0.0434280i
$836$	0	0
$837$	−2.14148e14	−0.521300
$838$	0	0
$839$	− 5.94342e14i	− 1.42964i	−0.699309	−	0.714819i	$-0.746509\pi$
$839$	− 5.94342e14i	− 1.42964i	0.699309	−	0.714819i	$-0.253491\pi$
$840$	0	0
$841$	−3.63991e14	−0.865188
$842$	0	0
$843$	2.24770e14i	0.527958i
$844$	0	0
$845$	−6.70320e13	−0.155596
$846$	0	0
$847$	− 1.70129e14i	− 0.390266i
$848$	0	0
$849$	6.99993e14	1.58692
$850$	0	0
$851$	− 5.46565e14i	− 1.22460i
$852$	0	0
$853$	−3.37514e14	−0.747388	−0.373694	−	0.927552i	$-0.621909\pi$
$853$	−3.37514e14	−0.747388	−0.373694	+	0.927552i	$0.621909\pi$
$854$	0	0
$855$	− 3.37984e13i	− 0.0739717i
$856$	0	0
$857$	6.29027e14	1.36071	0.680354	−	0.732883i	$-0.261826\pi$
$857$	6.29027e14	1.36071	0.680354	+	0.732883i	$0.261826\pi$
$858$	0	0
$859$	− 8.57840e14i	− 1.83417i	−0.398687	−	0.917087i	$-0.630534\pi$
$859$	− 8.57840e14i	− 1.83417i	0.398687	−	0.917087i	$-0.369466\pi$
$860$	0	0
$861$	5.50878e13	0.116423
$862$	0	0
$863$	4.08250e14i	0.852848i	0.904523	+	0.426424i	$0.140227\pi$
$863$	4.08250e14i	0.852848i	−0.904523	+	0.426424i	$0.859773\pi$
$864$	0	0
$865$	5.69597e13	0.117622
$866$	0	0
$867$	− 2.90507e14i	− 0.593009i
$868$	0	0
$869$	−2.37469e14	−0.479190
$870$	0	0
$871$	− 8.66024e14i	− 1.72758i
$872$	0	0
$873$	−7.27574e14	−1.43485
$874$	0	0
$875$	− 4.83683e13i	− 0.0943020i
$876$	0	0
$877$	5.57624e14	1.07484	0.537419	−	0.843315i	$-0.319399\pi$
$877$	5.57624e14	1.07484	0.537419	+	0.843315i	$0.319399\pi$
$878$	0	0
$879$	8.52197e14i	1.62404i
$880$	0	0
$881$	−7.15353e12	−0.0134785	−0.00673924	−	0.999977i	$-0.502145\pi$
$881$	−7.15353e12	−0.0134785	−0.00673924	+	0.999977i	$0.502145\pi$
$882$	0	0
$883$	− 6.72385e14i	− 1.25261i	−0.779580	−	0.626303i	$-0.784567\pi$
$883$	− 6.72385e14i	− 1.25261i	0.779580	−	0.626303i	$-0.215433\pi$
$884$	0	0
$885$	−1.14613e14	−0.211114
$886$	0	0
$887$	5.29577e14i	0.964519i	0.876028	+	0.482260i	$0.160184\pi$
$887$	5.29577e14i	0.964519i	−0.876028	+	0.482260i	$0.839816\pi$
$888$	0	0
$889$	−9.51984e13	−0.171443
$890$	0	0
$891$	− 1.46194e14i	− 0.260340i
$892$	0	0
$893$	−4.41514e14	−0.777477
$894$	0	0
$895$	2.38125e12i	0.00414658i
$896$	0	0
$897$	−1.74545e15	−3.00570
$898$	0	0
$899$	2.81756e14i	0.479816i
$900$	0	0
$901$	−3.30962e14	−0.557383
$902$	0	0
$903$	− 1.12918e14i	− 0.188072i
$904$	0	0
$905$	−3.30380e13	−0.0544216
$906$	0	0
$907$	− 3.28289e14i	− 0.534836i	−0.963581	−	0.267418i	$-0.913830\pi$
$907$	− 3.28289e14i	− 0.534836i	0.963581	−	0.267418i	$-0.0861704\pi$
$908$	0	0
$909$	1.15837e15	1.86650
$910$	0	0
$911$	1.79388e14i	0.285891i	0.989731	+	0.142946i	$0.0456574\pi$
$911$	1.79388e14i	0.285891i	−0.989731	+	0.142946i	$0.954343\pi$
$912$	0	0
$913$	3.97989e14	0.627360
$914$	0	0
$915$	− 7.00026e13i	− 0.109146i
$916$	0	0
$917$	2.45596e14	0.378769
$918$	0	0
$919$	− 6.61852e14i	− 1.00968i	−0.863213	−	0.504839i	$-0.831552\pi$
$919$	− 6.61852e14i	− 1.00968i	0.863213	−	0.504839i	$-0.168448\pi$
$920$	0	0
$921$	−5.77786e14	−0.871904
$922$	0	0
$923$	− 1.62718e15i	− 2.42899i
$924$	0	0
$925$	6.51318e14	0.961799
$926$	0	0
$927$	− 1.37322e15i	− 2.00605i
$928$	0	0
$929$	−8.76009e14	−1.26599	−0.632995	−	0.774156i	$-0.718175\pi$
$929$	−8.76009e14	−1.26599	−0.632995	+	0.774156i	$0.718175\pi$
$930$	0	0
$931$	− 3.13217e14i	− 0.447813i
$932$	0	0
$933$	3.14579e14	0.444960
$934$	0	0
$935$	2.24035e13i	0.0313515i
$936$	0	0
$937$	−7.21798e14	−0.999350	−0.499675	−	0.866213i	$-0.666547\pi$
$937$	−7.21798e14	−0.999350	−0.499675	+	0.866213i	$0.666547\pi$
$938$	0	0
$939$	1.26421e15i	1.73178i
$940$	0	0
$941$	−2.91967e14	−0.395718	−0.197859	−	0.980231i	$-0.563399\pi$
$941$	−2.91967e14	−0.395718	−0.197859	+	0.980231i	$0.563399\pi$
$942$	0	0
$943$	1.57640e14i	0.211401i
$944$	0	0
$945$	1.42505e13	0.0189091
$946$	0	0
$947$	− 9.33521e14i	− 1.22567i	−0.790210	−	0.612836i	$-0.790029\pi$
$947$	− 9.33521e14i	− 1.22567i	0.790210	−	0.612836i	$-0.209971\pi$
$948$	0	0
$949$	−1.66180e15	−2.15898
$950$	0	0
$951$	1.10597e15i	1.42180i
$952$	0	0
$953$	−9.04625e14	−1.15081	−0.575406	−	0.817868i	$-0.695156\pi$
$953$	−9.04625e14	−1.15081	−0.575406	+	0.817868i	$0.695156\pi$
$954$	0	0
$955$	6.38449e13i	0.0803728i
$956$	0	0
$957$	1.73735e14	0.216435
$958$	0	0
$959$	2.66571e14i	0.328639i
$960$	0	0
$961$	−5.80081e14	−0.707736
$962$	0	0
$963$	7.42050e14i	0.895987i
$964$	0	0
$965$	−1.36949e13	−0.0163652
$966$	0	0
$967$	1.40789e15i	1.66508i	0.553961	+	0.832542i	$0.313116\pi$
$967$	1.40789e15i	1.66508i	−0.553961	+	0.832542i	$0.686884\pi$
$968$	0	0
$969$	−5.69162e14	−0.666219
$970$	0	0
$971$	4.11514e14i	0.476748i	0.971173	+	0.238374i	$0.0766144\pi$
$971$	4.11514e14i	0.476748i	−0.971173	+	0.238374i	$0.923386\pi$
$972$	0	0
$973$	7.59999e14	0.871463
$974$	0	0
$975$	− 2.07998e15i	− 2.36067i
$976$	0	0
$977$	5.70909e13	0.0641349	0.0320674	−	0.999486i	$-0.489791\pi$
$977$	5.70909e13	0.0641349	0.0320674	+	0.999486i	$0.489791\pi$
$978$	0	0
$979$	3.88025e14i	0.431466i
$980$	0	0
$981$	−1.50007e15	−1.65107
$982$	0	0
$983$	− 6.78831e14i	− 0.739595i	−0.929112	−	0.369798i	$-0.879427\pi$
$983$	− 6.78831e14i	− 0.739595i	0.929112	−	0.369798i	$-0.120573\pi$
$984$	0	0
$985$	6.30450e12	0.00679938
$986$	0	0
$987$	− 8.88483e14i	− 0.948557i
$988$	0	0
$989$	3.23127e14	0.341500
$990$	0	0
$991$	3.33077e14i	0.348478i	0.984703	+	0.174239i	$0.0557466\pi$
$991$	3.33077e14i	0.348478i	−0.984703	+	0.174239i	$0.944253\pi$
$992$	0	0
$993$	−2.08780e15	−2.16243
$994$	0	0
$995$	− 7.00473e13i	− 0.0718250i
$996$	0	0
$997$	−1.33627e15	−1.35649	−0.678246	−	0.734835i	$-0.737260\pi$
$997$	−1.33627e15	−1.35649	−0.678246	+	0.734835i	$0.737260\pi$
$998$	0	0
$999$	3.85838e14i	0.387773i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.e.255.4		4
4.3	odd	2	inner	256.11.c.e.255.1		4
8.3	odd	2		256.11.c.h.255.4		4
8.5	even	2		256.11.c.h.255.1		4
16.3	odd	4		128.11.d.e.63.8	yes	8
16.5	even	4		128.11.d.e.63.7	yes	8
16.11	odd	4		128.11.d.e.63.1	✓	8
16.13	even	4		128.11.d.e.63.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.11.d.e.63.1	✓	8	16.11	odd	4
128.11.d.e.63.2	yes	8	16.13	even	4
128.11.d.e.63.7	yes	8	16.5	even	4
128.11.d.e.63.8	yes	8	16.3	odd	4
256.11.c.e.255.1		4	4.3	odd	2	inner
256.11.c.e.255.4		4	1.1	even	1	trivial
256.11.c.h.255.1		4	8.5	even	2
256.11.c.h.255.4		4	8.3	odd	2

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)

\(f(q)\)	\(=\)	\(q+365.717i q^{3} -321.334 q^{5} -7747.76i q^{7} -74700.3 q^{9} +O(q^{10})\)	\(q+365.717i q^{3} -321.334 q^{5} -7747.76i q^{7} -74700.3 q^{9} +63079.2i q^{11} +588612. q^{13} -117518. i q^{15} -1.10528e6 q^{17} -1.40805e6i q^{19} +2.83349e6 q^{21} +8.10835e6i q^{23} -9.66237e6 q^{25} -5.72395e6i q^{27} -7.53103e6 q^{29} -3.74127e7i q^{31} -2.30692e7 q^{33} +2.48962e6i q^{35} -6.74077e7 q^{37} +2.15266e8i q^{39} +1.94417e7 q^{41} -3.98511e7i q^{43} +2.40038e7 q^{45} -3.13565e8i q^{47} +2.22447e8 q^{49} -4.04221e8i q^{51} +2.99437e8 q^{53} -2.02695e7i q^{55} +5.14948e8 q^{57} -9.75283e8i q^{59} +5.95678e8 q^{61} +5.78760e8i q^{63} -1.89141e8 q^{65} -1.47130e9i q^{67} -2.96537e9 q^{69} -2.76443e9i q^{71} -2.82326e9 q^{73} -3.53370e9i q^{75} +4.88723e8 q^{77} +3.76461e9i q^{79} -2.31763e9 q^{81} -6.30934e9i q^{83} +3.55164e8 q^{85} -2.75423e9i q^{87} +6.15139e9 q^{89} -4.56042e9i q^{91} +1.36825e10 q^{93} +4.52454e8i q^{95} +9.73991e9 q^{97} -4.71204e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5600 q^{5} - 57180 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more