LMFDB - Embedded newform 256.11.c.l.255.5

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5593 x^{10} + 23492833 x^{8} - 43148127888 x^{6} + 59505890201856 x^{4} + \cdots + 42\!\cdots\!00 $ x^12 - 5593x^10 + 23492833x^8 - 43148127888x^6 + 59505890201856x^4 - 1615088885760000*x^2 + 42998169600000000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{96}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.5
Root			$-4.51244 - 2.60526i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.l.255.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-67.2765i q^{3} -2403.74 q^{5} -7852.70i q^{7} +54522.9 q^{9} +O(q^{10})$	$q-67.2765i q^{3} -2403.74 q^{5} -7852.70i q^{7} +54522.9 q^{9} -85571.9i q^{11} -246121. q^{13} +161715. i q^{15} -415026. q^{17} -1.25701e6i q^{19} -528302. q^{21} +6.80701e6i q^{23} -3.98765e6 q^{25} -7.64071e6i q^{27} -1.00526e7 q^{29} +1.32161e7i q^{31} -5.75698e6 q^{33} +1.88759e7i q^{35} +4.66892e7 q^{37} +1.65581e7i q^{39} -1.07287e8 q^{41} -2.32758e8i q^{43} -1.31059e8 q^{45} -2.86171e8i q^{47} +2.20810e8 q^{49} +2.79215e7i q^{51} +4.45998e8 q^{53} +2.05693e8i q^{55} -8.45674e7 q^{57} -1.24812e9i q^{59} -1.59522e9 q^{61} -4.28152e8i q^{63} +5.91610e8 q^{65} +1.62268e9i q^{67} +4.57952e8 q^{69} +1.59139e9i q^{71} -1.15372e9 q^{73} +2.68275e8i q^{75} -6.71971e8 q^{77} +6.77895e8i q^{79} +2.70548e9 q^{81} +2.18151e9i q^{83} +9.97616e8 q^{85} +6.76303e8i q^{87} +1.03842e9 q^{89} +1.93271e9i q^{91} +8.89132e8 q^{93} +3.02153e9i q^{95} -1.30759e10 q^{97} -4.66563e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 375204 q^{9}+O(q^{10}) $ 12 * q - 375204 * q^9	$ 12 q - 375204 q^{9} - 1639752 q^{17} + 28703892 q^{25} + 358214544 q^{33} - 496702488 q^{41} + 1385670732 q^{49} - 5733526800 q^{57} + 8895346752 q^{65} - 12161297592 q^{73} - 7708157604 q^{81} + 5871785160 q^{89} + 30465806904 q^{97}+O(q^{100}) $ 12 * q - 375204 * q^9 - 1639752 * q^17 + 28703892 * q^25 + 358214544 * q^33 - 496702488 * q^41 + 1385670732 * q^49 - 5733526800 * q^57 + 8895346752 * q^65 - 12161297592 * q^73 - 7708157604 * q^81 + 5871785160 * q^89 + 30465806904 * 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
$3$	− 67.2765i	− 0.276858i	−0.990372	−	0.138429i	$-0.955795\pi$
$3$	− 67.2765i	− 0.276858i	0.990372	−	0.138429i	$-0.0442052\pi$
$4$	0	0
$5$	−2403.74	−0.769197	−0.384599	−	0.923084i	$-0.625660\pi$
$5$	−2403.74	−0.769197	−0.384599	+	0.923084i	$0.625660\pi$
$6$	0	0
$7$	− 7852.70i	− 0.467228i	−0.972329	−	0.233614i	$-0.924945\pi$
$7$	− 7852.70i	− 0.467228i	0.972329	−	0.233614i	$-0.0750552\pi$
$8$	0	0
$9$	54522.9	0.923350
$10$	0	0
$11$	− 85571.9i	− 0.531334i	−0.964065	−	0.265667i	$-0.914408\pi$
$11$	− 85571.9i	− 0.531334i	0.964065	−	0.265667i	$-0.0855922\pi$
$12$	0	0
$13$	−246121.	−0.662874	−0.331437	−	0.943477i	$-0.607533\pi$
$13$	−246121.	−0.662874	−0.331437	+	0.943477i	$0.607533\pi$
$14$	0	0
$15$	161715.i	0.212958i
$16$	0	0
$17$	−415026.	−0.292302	−0.146151	−	0.989262i	$-0.546688\pi$
$17$	−415026.	−0.292302	−0.146151	+	0.989262i	$0.546688\pi$
$18$	0	0
$19$	− 1.25701e6i	− 0.507659i	−0.967249	−	0.253829i	$-0.918310\pi$
$19$	− 1.25701e6i	− 0.507659i	0.967249	−	0.253829i	$-0.0816901\pi$
$20$	0	0
$21$	−528302.	−0.129356
$22$	0	0
$23$	6.80701e6i	1.05759i	0.848750	+	0.528795i	$0.177356\pi$
$23$	6.80701e6i	1.05759i	−0.848750	+	0.528795i	$0.822644\pi$
$24$	0	0
$25$	−3.98765e6	−0.408336
$26$	0	0
$27$	− 7.64071e6i	− 0.532494i
$28$	0	0
$29$	−1.00526e7	−0.490104	−0.245052	−	0.969510i	$-0.578805\pi$
$29$	−1.00526e7	−0.490104	−0.245052	+	0.969510i	$0.578805\pi$
$30$	0	0
$31$	1.32161e7i	0.461631i	0.972998	+	0.230815i	$0.0741393\pi$
$31$	1.32161e7i	0.461631i	−0.972998	+	0.230815i	$0.925861\pi$
$32$	0	0
$33$	−5.75698e6	−0.147104
$34$	0	0
$35$	1.88759e7i	0.359390i
$36$	0	0
$37$	4.66892e7	0.673299	0.336649	−	0.941630i	$-0.390706\pi$
$37$	4.66892e7	0.673299	0.336649	+	0.941630i	$0.390706\pi$
$38$	0	0
$39$	1.65581e7i	0.183522i
$40$	0	0
$41$	−1.07287e8	−0.926037	−0.463019	−	0.886349i	$-0.653234\pi$
$41$	−1.07287e8	−0.926037	−0.463019	+	0.886349i	$0.653234\pi$
$42$	0	0
$43$	− 2.32758e8i	− 1.58330i	−0.610976	−	0.791649i	$-0.709223\pi$
$43$	− 2.32758e8i	− 1.58330i	0.610976	−	0.791649i	$-0.290777\pi$
$44$	0	0
$45$	−1.31059e8	−0.710238
$46$	0	0
$47$	− 2.86171e8i	− 1.24778i	−0.781514	−	0.623888i	$-0.785552\pi$
$47$	− 2.86171e8i	− 1.24778i	0.781514	−	0.623888i	$-0.214448\pi$
$48$	0	0
$49$	2.20810e8	0.781698
$50$	0	0
$51$	2.79215e7i	0.0809260i
$52$	0	0
$53$	4.45998e8	1.06648	0.533241	−	0.845963i	$-0.320974\pi$
$53$	4.45998e8	1.06648	0.533241	+	0.845963i	$0.320974\pi$
$54$	0	0
$55$	2.05693e8i	0.408701i
$56$	0	0
$57$	−8.45674e7	−0.140549
$58$	0	0
$59$	− 1.24812e9i	− 1.74581i	−0.487890	−	0.872905i	$-0.662233\pi$
$59$	− 1.24812e9i	− 1.74581i	0.487890	−	0.872905i	$-0.337767\pi$
$60$	0	0
$61$	−1.59522e9	−1.88874	−0.944371	−	0.328882i	$-0.893328\pi$
$61$	−1.59522e9	−1.88874	−0.944371	+	0.328882i	$0.893328\pi$
$62$	0	0
$63$	− 4.28152e8i	− 0.431415i
$64$	0	0
$65$	5.91610e8	0.509881
$66$	0	0
$67$	1.62268e9i	1.20187i	0.799297	+	0.600937i	$0.205206\pi$
$67$	1.62268e9i	1.20187i	−0.799297	+	0.600937i	$0.794794\pi$
$68$	0	0
$69$	4.57952e8	0.292802
$70$	0	0
$71$	1.59139e9i	0.882033i	0.897499	+	0.441016i	$0.145382\pi$
$71$	1.59139e9i	0.882033i	−0.897499	+	0.441016i	$0.854618\pi$
$72$	0	0
$73$	−1.15372e9	−0.556527	−0.278264	−	0.960505i	$-0.589759\pi$
$73$	−1.15372e9	−0.556527	−0.278264	+	0.960505i	$0.589759\pi$
$74$	0	0
$75$	2.68275e8i	0.113051i
$76$	0	0
$77$	−6.71971e8	−0.248254
$78$	0	0
$79$	6.77895e8i	0.220306i	0.993915	+	0.110153i	$0.0351341\pi$
$79$	6.77895e8i	0.220306i	−0.993915	+	0.110153i	$0.964866\pi$
$80$	0	0
$81$	2.70548e9	0.775924
$82$	0	0
$83$	2.18151e9i	0.553816i	0.960896	+	0.276908i	$0.0893098\pi$
$83$	2.18151e9i	0.553816i	−0.960896	+	0.276908i	$0.910690\pi$
$84$	0	0
$85$	9.97616e8	0.224838
$86$	0	0
$87$	6.76303e8i	0.135689i
$88$	0	0
$89$	1.03842e9	0.185961	0.0929803	−	0.995668i	$-0.470361\pi$
$89$	1.03842e9	0.185961	0.0929803	+	0.995668i	$0.470361\pi$
$90$	0	0
$91$	1.93271e9i	0.309713i
$92$	0	0
$93$	8.89132e8	0.127806
$94$	0	0
$95$	3.02153e9i	0.390490i
$96$	0	0
$97$	−1.30759e10	−1.52269	−0.761345	−	0.648347i	$-0.775461\pi$
$97$	−1.30759e10	−1.52269	−0.761345	+	0.648347i	$0.775461\pi$
$98$	0	0
$99$	− 4.66563e9i	− 0.490608i
$100$	0	0
$101$	−1.26500e10	−1.20360	−0.601802	−	0.798645i	$-0.705550\pi$
$101$	−1.26500e10	−1.20360	−0.601802	+	0.798645i	$0.705550\pi$
$102$	0	0
$103$	− 9.17592e9i	− 0.791523i	−0.918353	−	0.395762i	$-0.870481\pi$
$103$	− 9.17592e9i	− 0.791523i	0.918353	−	0.395762i	$-0.129519\pi$
$104$	0	0
$105$	1.26990e9	0.0995001
$106$	0	0
$107$	1.90151e10i	1.35575i	0.735176	+	0.677876i	$0.237099\pi$
$107$	1.90151e10i	1.35575i	−0.735176	+	0.677876i	$0.762901\pi$
$108$	0	0
$109$	7.59779e9	0.493804	0.246902	−	0.969040i	$-0.420587\pi$
$109$	7.59779e9	0.493804	0.246902	+	0.969040i	$0.420587\pi$
$110$	0	0
$111$	− 3.14108e9i	− 0.186408i
$112$	0	0
$113$	6.21835e9	0.337507	0.168754	−	0.985658i	$-0.446026\pi$
$113$	6.21835e9	0.337507	0.168754	+	0.985658i	$0.446026\pi$
$114$	0	0
$115$	− 1.63623e10i	− 0.813495i
$116$	0	0
$117$	−1.34192e10	−0.612065
$118$	0	0
$119$	3.25908e9i	0.136571i
$120$	0	0
$121$	1.86149e10	0.717684
$122$	0	0
$123$	7.21790e9i	0.256381i
$124$	0	0
$125$	3.30593e10	1.08329
$126$	0	0
$127$	5.60359e10i	1.69609i	0.529927	+	0.848043i	$0.322219\pi$
$127$	5.60359e10i	1.69609i	−0.529927	+	0.848043i	$0.677781\pi$
$128$	0	0
$129$	−1.56592e10	−0.438349
$130$	0	0
$131$	− 2.21078e10i	− 0.573044i	−0.958074	−	0.286522i	$-0.907501\pi$
$131$	− 2.21078e10i	− 0.573044i	0.958074	−	0.286522i	$-0.0924992\pi$
$132$	0	0
$133$	−9.87095e9	−0.237192
$134$	0	0
$135$	1.83663e10i	0.409593i
$136$	0	0
$137$	3.12575e9	0.0647665	0.0323833	−	0.999476i	$-0.489690\pi$
$137$	3.12575e9	0.0647665	0.0323833	+	0.999476i	$0.489690\pi$
$138$	0	0
$139$	− 4.44487e10i	− 0.856614i	−0.903633	−	0.428307i	$-0.859110\pi$
$139$	− 4.44487e10i	− 0.856614i	0.903633	−	0.428307i	$-0.140890\pi$
$140$	0	0
$141$	−1.92526e10	−0.345457
$142$	0	0
$143$	2.10610e10i	0.352208i
$144$	0	0
$145$	2.41639e10	0.376987
$146$	0	0
$147$	− 1.48553e10i	− 0.216419i
$148$	0	0
$149$	8.82547e10	1.20173	0.600864	−	0.799351i	$-0.294823\pi$
$149$	8.82547e10	1.20173	0.600864	+	0.799351i	$0.294823\pi$
$150$	0	0
$151$	9.18512e10i	1.17004i	0.811020	+	0.585019i	$0.198913\pi$
$151$	9.18512e10i	1.17004i	−0.811020	+	0.585019i	$0.801087\pi$
$152$	0	0
$153$	−2.26284e10	−0.269897
$154$	0	0
$155$	− 3.17681e10i	− 0.355085i
$156$	0	0
$157$	5.44799e9	0.0571135	0.0285567	−	0.999592i	$-0.490909\pi$
$157$	5.44799e9	0.0571135	0.0285567	+	0.999592i	$0.490909\pi$
$158$	0	0
$159$	− 3.00052e10i	− 0.295264i
$160$	0	0
$161$	5.34534e10	0.494135
$162$	0	0
$163$	8.22199e10i	0.714560i	0.933997	+	0.357280i	$0.116296\pi$
$163$	8.22199e10i	0.714560i	−0.933997	+	0.357280i	$0.883704\pi$
$164$	0	0
$165$	1.38383e10	0.113152
$166$	0	0
$167$	9.99313e10i	0.769342i	0.923054	+	0.384671i	$0.125685\pi$
$167$	9.99313e10i	0.769342i	−0.923054	+	0.384671i	$0.874315\pi$
$168$	0	0
$169$	−7.72832e10	−0.560598
$170$	0	0
$171$	− 6.85360e10i	− 0.468746i
$172$	0	0
$173$	−1.90118e11	−1.22686	−0.613428	−	0.789751i	$-0.710210\pi$
$173$	−1.90118e11	−1.22686	−0.613428	+	0.789751i	$0.710210\pi$
$174$	0	0
$175$	3.13138e10i	0.190786i
$176$	0	0
$177$	−8.39692e10	−0.483341
$178$	0	0
$179$	3.09582e11i	1.68465i	0.538966	+	0.842327i	$0.318815\pi$
$179$	3.09582e11i	1.68465i	−0.538966	+	0.842327i	$0.681185\pi$
$180$	0	0
$181$	2.28902e11	1.17830	0.589151	−	0.808023i	$-0.299462\pi$
$181$	2.28902e11	1.17830	0.589151	+	0.808023i	$0.299462\pi$
$182$	0	0
$183$	1.07321e11i	0.522913i
$184$	0	0
$185$	−1.12229e11	−0.517899
$186$	0	0
$187$	3.55146e10i	0.155310i
$188$	0	0
$189$	−6.00002e10	−0.248796
$190$	0	0
$191$	3.17009e11i	1.24711i	0.781779	+	0.623556i	$0.214313\pi$
$191$	3.17009e11i	1.24711i	−0.781779	+	0.623556i	$0.785687\pi$
$192$	0	0
$193$	2.12710e11	0.794330	0.397165	−	0.917747i	$-0.369994\pi$
$193$	2.12710e11	0.794330	0.397165	+	0.917747i	$0.369994\pi$
$194$	0	0
$195$	− 3.98014e10i	− 0.141165i
$196$	0	0
$197$	−2.50809e11	−0.845303	−0.422651	−	0.906292i	$-0.638901\pi$
$197$	−2.50809e11	−0.845303	−0.422651	+	0.906292i	$0.638901\pi$
$198$	0	0
$199$	4.35441e11i	1.39529i	0.716445	+	0.697644i	$0.245768\pi$
$199$	4.35441e11i	1.39529i	−0.716445	+	0.697644i	$0.754232\pi$
$200$	0	0
$201$	1.09168e11	0.332748
$202$	0	0
$203$	7.89400e10i	0.228990i
$204$	0	0
$205$	2.57891e11	0.712305
$206$	0	0
$207$	3.71138e11i	0.976525i
$208$	0	0
$209$	−1.07565e11	−0.269736
$210$	0	0
$211$	− 2.21194e11i	− 0.528885i	−0.964401	−	0.264443i	$-0.914812\pi$
$211$	− 2.21194e11i	− 0.528885i	0.964401	−	0.264443i	$-0.0851880\pi$
$212$	0	0
$213$	1.07063e11	0.244198
$214$	0	0
$215$	5.59491e11i	1.21787i
$216$	0	0
$217$	1.03782e11	0.215687
$218$	0	0
$219$	7.76183e10i	0.154079i
$220$	0	0
$221$	1.02147e11	0.193759
$222$	0	0
$223$	7.83724e11i	1.42115i	0.703623	+	0.710573i	$0.251564\pi$
$223$	7.83724e11i	1.42115i	−0.703623	+	0.710573i	$0.748436\pi$
$224$	0	0
$225$	−2.17418e11	−0.377037
$226$	0	0
$227$	− 4.84719e11i	− 0.804195i	−0.915597	−	0.402097i	$-0.868281\pi$
$227$	− 4.84719e11i	− 0.804195i	0.915597	−	0.402097i	$-0.131719\pi$
$228$	0	0
$229$	−7.58821e11	−1.20493	−0.602465	−	0.798145i	$-0.705815\pi$
$229$	−7.58821e11	−1.20493	−0.602465	+	0.798145i	$0.705815\pi$
$230$	0	0
$231$	4.52078e10i	0.0687311i
$232$	0	0
$233$	6.37909e11	0.928922	0.464461	−	0.885594i	$-0.346248\pi$
$233$	6.37909e11	0.928922	0.464461	+	0.885594i	$0.346248\pi$
$234$	0	0
$235$	6.87882e11i	0.959786i
$236$	0	0
$237$	4.56064e10	0.0609935
$238$	0	0
$239$	1.23185e12i	1.57967i	0.613317	+	0.789837i	$0.289835\pi$
$239$	1.23185e12i	1.57967i	−0.613317	+	0.789837i	$0.710165\pi$
$240$	0	0
$241$	−4.41347e11	−0.542869	−0.271434	−	0.962457i	$-0.587498\pi$
$241$	−4.41347e11	−0.542869	−0.271434	+	0.962457i	$0.587498\pi$
$242$	0	0
$243$	− 6.33192e11i	− 0.747315i
$244$	0	0
$245$	−5.30771e11	−0.601280
$246$	0	0
$247$	3.09377e11i	0.336514i
$248$	0	0
$249$	1.46764e11	0.153328
$250$	0	0
$251$	− 4.41739e11i	− 0.443402i	−0.975115	−	0.221701i	$-0.928839\pi$
$251$	− 4.41739e11i	− 0.443402i	0.975115	−	0.221701i	$-0.0711608\pi$
$252$	0	0
$253$	5.82489e11	0.561934
$254$	0	0
$255$	− 6.71161e10i	− 0.0622480i
$256$	0	0
$257$	3.91352e11	0.349062	0.174531	−	0.984652i	$-0.444159\pi$
$257$	3.91352e11	0.349062	0.174531	+	0.984652i	$0.444159\pi$
$258$	0	0
$259$	− 3.66636e11i	− 0.314584i
$260$	0	0
$261$	−5.48097e11	−0.452538
$262$	0	0
$263$	1.34183e12i	1.06640i	0.845989	+	0.533200i	$0.179011\pi$
$263$	1.34183e12i	1.06640i	−0.845989	+	0.533200i	$0.820989\pi$
$264$	0	0
$265$	−1.07206e12	−0.820336
$266$	0	0
$267$	− 6.98609e10i	− 0.0514847i
$268$	0	0
$269$	−3.63597e11	−0.258142	−0.129071	−	0.991635i	$-0.541200\pi$
$269$	−3.63597e11	−0.258142	−0.129071	+	0.991635i	$0.541200\pi$
$270$	0	0
$271$	1.72087e12i	1.17734i	0.808373	+	0.588671i	$0.200349\pi$
$271$	1.72087e12i	1.17734i	−0.808373	+	0.588671i	$0.799651\pi$
$272$	0	0
$273$	1.30026e11	0.0857466
$274$	0	0
$275$	3.41231e11i	0.216963i
$276$	0	0
$277$	1.02381e12	0.627801	0.313901	−	0.949456i	$-0.398364\pi$
$277$	1.02381e12	0.627801	0.313901	+	0.949456i	$0.398364\pi$
$278$	0	0
$279$	7.20580e11i	0.426247i
$280$	0	0
$281$	−1.09516e11	−0.0625092	−0.0312546	−	0.999511i	$-0.509950\pi$
$281$	−1.09516e11	−0.0625092	−0.0312546	+	0.999511i	$0.509950\pi$
$282$	0	0
$283$	− 4.32354e11i	− 0.238181i	−0.992883	−	0.119090i	$-0.962002\pi$
$283$	− 4.32354e11i	− 0.238181i	0.992883	−	0.119090i	$-0.0379978\pi$
$284$	0	0
$285$	2.03278e11	0.108110
$286$	0	0
$287$	8.42494e11i	0.432670i
$288$	0	0
$289$	−1.84375e12	−0.914560
$290$	0	0
$291$	8.79698e11i	0.421569i
$292$	0	0
$293$	3.50518e12	1.62320	0.811600	−	0.584213i	$-0.198597\pi$
$293$	3.50518e12	1.62320	0.811600	+	0.584213i	$0.198597\pi$
$294$	0	0
$295$	3.00016e12i	1.34287i
$296$	0	0
$297$	−6.53831e11	−0.282933
$298$	0	0
$299$	− 1.67534e12i	− 0.701049i
$300$	0	0
$301$	−1.82778e12	−0.739761
$302$	0	0
$303$	8.51047e11i	0.333227i
$304$	0	0
$305$	3.83451e12	1.45282
$306$	0	0
$307$	4.91271e12i	1.80148i	0.434360	+	0.900739i	$0.356975\pi$
$307$	4.91271e12i	1.80148i	−0.434360	+	0.900739i	$0.643025\pi$
$308$	0	0
$309$	−6.17324e11	−0.219139
$310$	0	0
$311$	4.17826e12i	1.43613i	0.695976	+	0.718065i	$0.254972\pi$
$311$	4.17826e12i	1.43613i	−0.695976	+	0.718065i	$0.745028\pi$
$312$	0	0
$313$	1.26738e12	0.421878	0.210939	−	0.977499i	$-0.432348\pi$
$313$	1.26738e12	0.421878	0.210939	+	0.977499i	$0.432348\pi$
$314$	0	0
$315$	1.02917e12i	0.331843i
$316$	0	0
$317$	−2.67146e12	−0.834550	−0.417275	−	0.908780i	$-0.637015\pi$
$317$	−2.67146e12	−0.834550	−0.417275	+	0.908780i	$0.637015\pi$
$318$	0	0
$319$	8.60220e11i	0.260409i
$320$	0	0
$321$	1.27927e12	0.375351
$322$	0	0
$323$	5.21694e11i	0.148389i
$324$	0	0
$325$	9.81443e11	0.270675
$326$	0	0
$327$	− 5.11152e11i	− 0.136713i
$328$	0	0
$329$	−2.24722e12	−0.582996
$330$	0	0
$331$	2.31898e12i	0.583655i	0.956471	+	0.291827i	$0.0942633\pi$
$331$	2.31898e12i	0.583655i	−0.956471	+	0.291827i	$0.905737\pi$
$332$	0	0
$333$	2.54563e12	0.621690
$334$	0	0
$335$	− 3.90050e12i	− 0.924478i
$336$	0	0
$337$	−3.16148e11	−0.0727346	−0.0363673	−	0.999338i	$-0.511579\pi$
$337$	−3.16148e11	−0.0727346	−0.0363673	+	0.999338i	$0.511579\pi$
$338$	0	0
$339$	− 4.18349e11i	− 0.0934415i
$340$	0	0
$341$	1.13093e12	0.245280
$342$	0	0
$343$	− 3.95215e12i	− 0.832459i
$344$	0	0
$345$	−1.10080e12	−0.225223
$346$	0	0
$347$	− 5.59993e12i	− 1.11310i	−0.830813	−	0.556551i	$-0.812124\pi$
$347$	− 5.59993e12i	− 1.11310i	0.830813	−	0.556551i	$-0.187876\pi$
$348$	0	0
$349$	−9.19554e10	−0.0177603	−0.00888015	−	0.999961i	$-0.502827\pi$
$349$	−9.19554e10	−0.0177603	−0.00888015	+	0.999961i	$0.502827\pi$
$350$	0	0
$351$	1.88054e12i	0.352977i
$352$	0	0
$353$	−9.63420e11	−0.175769	−0.0878845	−	0.996131i	$-0.528011\pi$
$353$	−9.63420e11	−0.175769	−0.0878845	+	0.996131i	$0.528011\pi$
$354$	0	0
$355$	− 3.82529e12i	− 0.678457i
$356$	0	0
$357$	2.19259e11	0.0378109
$358$	0	0
$359$	− 4.12918e12i	− 0.692455i	−0.938151	−	0.346228i	$-0.887462\pi$
$359$	− 4.12918e12i	− 0.692455i	0.938151	−	0.346228i	$-0.112538\pi$
$360$	0	0
$361$	4.55098e12	0.742283
$362$	0	0
$363$	− 1.25234e12i	− 0.198696i
$364$	0	0
$365$	2.77325e12	0.428079
$366$	0	0
$367$	4.45639e11i	0.0669350i	0.999440	+	0.0334675i	$0.0106550\pi$
$367$	4.45639e11i	0.0669350i	−0.999440	+	0.0334675i	$0.989345\pi$
$368$	0	0
$369$	−5.84960e12	−0.855056
$370$	0	0
$371$	− 3.50229e12i	− 0.498290i
$372$	0	0
$373$	−1.13753e13	−1.57550	−0.787751	−	0.615993i	$-0.788755\pi$
$373$	−1.13753e13	−1.57550	−0.787751	+	0.615993i	$0.788755\pi$
$374$	0	0
$375$	− 2.22411e12i	− 0.299917i
$376$	0	0
$377$	2.47415e12	0.324877
$378$	0	0
$379$	1.18502e13i	1.51541i	0.652598	+	0.757704i	$0.273679\pi$
$379$	1.18502e13i	1.51541i	−0.652598	+	0.757704i	$0.726321\pi$
$380$	0	0
$381$	3.76990e12	0.469575
$382$	0	0
$383$	− 2.99098e12i	− 0.362927i	−0.983398	−	0.181463i	$-0.941917\pi$
$383$	− 2.99098e12i	− 0.362927i	0.983398	−	0.181463i	$-0.0580834\pi$
$384$	0	0
$385$	1.61524e12	0.190957
$386$	0	0
$387$	− 1.26907e13i	− 1.46194i
$388$	0	0
$389$	2.61735e12	0.293842	0.146921	−	0.989148i	$-0.453064\pi$
$389$	2.61735e12	0.293842	0.146921	+	0.989148i	$0.453064\pi$
$390$	0	0
$391$	− 2.82509e12i	− 0.309135i
$392$	0	0
$393$	−1.48733e12	−0.158652
$394$	0	0
$395$	− 1.62948e12i	− 0.169459i
$396$	0	0
$397$	8.00203e12	0.811423	0.405712	−	0.914001i	$-0.367024\pi$
$397$	8.00203e12	0.811423	0.405712	+	0.914001i	$0.367024\pi$
$398$	0	0
$399$	6.64082e11i	0.0656685i
$400$	0	0
$401$	−5.29421e12	−0.510599	−0.255299	−	0.966862i	$-0.582174\pi$
$401$	−5.29421e12	−0.510599	−0.255299	+	0.966862i	$0.582174\pi$
$402$	0	0
$403$	− 3.25275e12i	− 0.306003i
$404$	0	0
$405$	−6.50328e12	−0.596839
$406$	0	0
$407$	− 3.99528e12i	− 0.357747i
$408$	0	0
$409$	−1.01399e12	−0.0885970	−0.0442985	−	0.999018i	$-0.514105\pi$
$409$	−1.01399e12	−0.0885970	−0.0442985	+	0.999018i	$0.514105\pi$
$410$	0	0
$411$	− 2.10289e11i	− 0.0179311i
$412$	0	0
$413$	−9.80113e12	−0.815691
$414$	0	0
$415$	− 5.24377e12i	− 0.425994i
$416$	0	0
$417$	−2.99035e12	−0.237160
$418$	0	0
$419$	1.31420e13i	1.01763i	0.860875	+	0.508816i	$0.169917\pi$
$419$	1.31420e13i	1.01763i	−0.860875	+	0.508816i	$0.830083\pi$
$420$	0	0
$421$	−8.51558e12	−0.643879	−0.321939	−	0.946760i	$-0.604335\pi$
$421$	−8.51558e12	−0.643879	−0.321939	+	0.946760i	$0.604335\pi$
$422$	0	0
$423$	− 1.56029e13i	− 1.15213i
$424$	0	0
$425$	1.65498e12	0.119357
$426$	0	0
$427$	1.25268e13i	0.882473i
$428$	0	0
$429$	1.41691e12	0.0975115
$430$	0	0
$431$	− 1.61752e13i	− 1.08759i	−0.839220	−	0.543793i	$-0.816988\pi$
$431$	− 1.61752e13i	− 1.08759i	0.839220	−	0.543793i	$-0.183012\pi$
$432$	0	0
$433$	−2.17538e12	−0.142921	−0.0714603	−	0.997443i	$-0.522766\pi$
$433$	−2.17538e12	−0.142921	−0.0714603	+	0.997443i	$0.522766\pi$
$434$	0	0
$435$	− 1.62566e12i	− 0.104372i
$436$	0	0
$437$	8.55650e12	0.536895
$438$	0	0
$439$	− 1.70488e13i	− 1.04561i	−0.852451	−	0.522807i	$-0.824885\pi$
$439$	− 1.70488e13i	− 1.04561i	0.852451	−	0.522807i	$-0.175115\pi$
$440$	0	0
$441$	1.20392e13	0.721781
$442$	0	0
$443$	− 2.26420e13i	− 1.32708i	−0.748142	−	0.663539i	$-0.769054\pi$
$443$	− 2.26420e13i	− 1.32708i	0.748142	−	0.663539i	$-0.230946\pi$
$444$	0	0
$445$	−2.49608e12	−0.143040
$446$	0	0
$447$	− 5.93746e12i	− 0.332708i
$448$	0	0
$449$	3.17899e13	1.74204	0.871018	−	0.491251i	$-0.163460\pi$
$449$	3.17899e13	1.74204	0.871018	+	0.491251i	$0.163460\pi$
$450$	0	0
$451$	9.18077e12i	0.492035i
$452$	0	0
$453$	6.17942e12	0.323934
$454$	0	0
$455$	− 4.64574e12i	− 0.238231i
$456$	0	0
$457$	−1.65624e13	−0.830885	−0.415443	−	0.909619i	$-0.636373\pi$
$457$	−1.65624e13	−0.830885	−0.415443	+	0.909619i	$0.636373\pi$
$458$	0	0
$459$	3.17110e12i	0.155649i
$460$	0	0
$461$	−3.46530e13	−1.66432	−0.832159	−	0.554536i	$-0.812896\pi$
$461$	−3.46530e13	−1.66432	−0.832159	+	0.554536i	$0.812896\pi$
$462$	0	0
$463$	− 3.86726e13i	− 1.81760i	−0.417232	−	0.908800i	$-0.637000\pi$
$463$	− 3.86726e13i	− 1.81760i	0.417232	−	0.908800i	$-0.363000\pi$
$464$	0	0
$465$	−2.13724e12	−0.0983081
$466$	0	0
$467$	2.82852e13i	1.27343i	0.771100	+	0.636714i	$0.219707\pi$
$467$	2.82852e13i	1.27343i	−0.771100	+	0.636714i	$0.780293\pi$
$468$	0	0
$469$	1.27424e13	0.561549
$470$	0	0
$471$	− 3.66522e11i	− 0.0158123i
$472$	0	0
$473$	−1.99176e13	−0.841261
$474$	0	0
$475$	5.01253e12i	0.207295i
$476$	0	0
$477$	2.43171e13	0.984736
$478$	0	0
$479$	− 4.30642e13i	− 1.70781i	−0.520430	−	0.853904i	$-0.674228\pi$
$479$	− 4.30642e13i	− 1.70781i	0.520430	−	0.853904i	$-0.325772\pi$
$480$	0	0
$481$	−1.14912e13	−0.446312
$482$	0	0
$483$	− 3.59616e12i	− 0.136805i
$484$	0	0
$485$	3.14310e13	1.17125
$486$	0	0
$487$	− 1.99664e13i	− 0.728878i	−0.931227	−	0.364439i	$-0.881261\pi$
$487$	− 1.99664e13i	− 0.728878i	0.931227	−	0.364439i	$-0.118739\pi$
$488$	0	0
$489$	5.53146e12	0.197832
$490$	0	0
$491$	− 2.65452e13i	− 0.930206i	−0.885257	−	0.465103i	$-0.846017\pi$
$491$	− 2.65452e13i	− 0.930206i	0.885257	−	0.465103i	$-0.153983\pi$
$492$	0	0
$493$	4.17209e12	0.143258
$494$	0	0
$495$	1.12150e13i	0.377374i
$496$	0	0
$497$	1.24967e13	0.412110
$498$	0	0
$499$	1.21982e13i	0.394269i	0.980376	+	0.197135i	$0.0631636\pi$
$499$	1.21982e13i	0.394269i	−0.980376	+	0.197135i	$0.936836\pi$
$500$	0	0
$501$	6.72302e12	0.212998
$502$	0	0
$503$	− 3.29508e12i	− 0.102335i	−0.998690	−	0.0511677i	$-0.983706\pi$
$503$	− 3.29508e12i	− 0.102335i	0.998690	−	0.0511677i	$-0.0162943\pi$
$504$	0	0
$505$	3.04073e13	0.925809
$506$	0	0
$507$	5.19934e12i	0.155206i
$508$	0	0
$509$	−3.67173e13	−1.07469	−0.537344	−	0.843363i	$-0.680572\pi$
$509$	−3.67173e13	−1.07469	−0.537344	+	0.843363i	$0.680572\pi$
$510$	0	0
$511$	9.05982e12i	0.260025i
$512$	0	0
$513$	−9.60448e12	−0.270325
$514$	0	0
$515$	2.20566e13i	0.608838i
$516$	0	0
$517$	−2.44882e13	−0.662987
$518$	0	0
$519$	1.27905e13i	0.339665i
$520$	0	0
$521$	−5.37359e13	−1.39983	−0.699916	−	0.714225i	$-0.746779\pi$
$521$	−5.37359e13	−1.39983	−0.699916	+	0.714225i	$0.746779\pi$
$522$	0	0
$523$	7.41830e11i	0.0189582i	0.999955	+	0.00947908i	$0.00301733\pi$
$523$	7.41830e11i	0.0189582i	−0.999955	+	0.00947908i	$0.996983\pi$
$524$	0	0
$525$	2.10668e12	0.0528205
$526$	0	0
$527$	− 5.48503e12i	− 0.134935i
$528$	0	0
$529$	−4.90888e12	−0.118496
$530$	0	0
$531$	− 6.80512e13i	− 1.61199i
$532$	0	0
$533$	2.64056e13	0.613846
$534$	0	0
$535$	− 4.57075e13i	− 1.04284i
$536$	0	0
$537$	2.08276e13	0.466410
$538$	0	0
$539$	− 1.88952e13i	− 0.415343i
$540$	0	0
$541$	6.12187e13	1.32099	0.660493	−	0.750832i	$-0.270347\pi$
$541$	6.12187e13	1.32099	0.660493	+	0.750832i	$0.270347\pi$
$542$	0	0
$543$	− 1.53997e13i	− 0.326222i
$544$	0	0
$545$	−1.82631e13	−0.379833
$546$	0	0
$547$	− 4.83858e13i	− 0.988055i	−0.869446	−	0.494027i	$-0.835524\pi$
$547$	− 4.83858e13i	− 0.988055i	0.869446	−	0.494027i	$-0.164476\pi$
$548$	0	0
$549$	−8.69762e13	−1.74397
$550$	0	0
$551$	1.26362e13i	0.248806i
$552$	0	0
$553$	5.32331e12	0.102933
$554$	0	0
$555$	7.55035e12i	0.143384i
$556$	0	0
$557$	2.62639e13	0.489872	0.244936	−	0.969539i	$-0.421233\pi$
$557$	2.62639e13	0.489872	0.244936	+	0.969539i	$0.421233\pi$
$558$	0	0
$559$	5.72866e13i	1.04953i
$560$	0	0
$561$	2.38930e12	0.0429988
$562$	0	0
$563$	− 1.67074e13i	− 0.295370i	−0.989034	−	0.147685i	$-0.952818\pi$
$563$	− 1.67074e13i	− 0.295370i	0.989034	−	0.147685i	$-0.0471821\pi$
$564$	0	0
$565$	−1.49473e13	−0.259610
$566$	0	0
$567$	− 2.12453e13i	− 0.362534i
$568$	0	0
$569$	2.53618e13	0.425225	0.212612	−	0.977137i	$-0.431803\pi$
$569$	2.53618e13	0.425225	0.212612	+	0.977137i	$0.431803\pi$
$570$	0	0
$571$	7.40453e13i	1.21988i	0.792448	+	0.609940i	$0.208806\pi$
$571$	7.40453e13i	1.21988i	−0.792448	+	0.609940i	$0.791194\pi$
$572$	0	0
$573$	2.13273e13	0.345273
$574$	0	0
$575$	− 2.71440e13i	− 0.431851i
$576$	0	0
$577$	−1.34147e13	−0.209750	−0.104875	−	0.994485i	$-0.533444\pi$
$577$	−1.34147e13	−0.209750	−0.104875	+	0.994485i	$0.533444\pi$
$578$	0	0
$579$	− 1.43104e13i	− 0.219916i
$580$	0	0
$581$	1.71307e13	0.258758
$582$	0	0
$583$	− 3.81649e13i	− 0.566659i
$584$	0	0
$585$	3.22563e13	0.470798
$586$	0	0
$587$	− 3.36017e13i	− 0.482137i	−0.970508	−	0.241069i	$-0.922502\pi$
$587$	− 3.36017e13i	− 0.482137i	0.970508	−	0.241069i	$-0.0774979\pi$
$588$	0	0
$589$	1.66128e13	0.234351
$590$	0	0
$591$	1.68736e13i	0.234029i
$592$	0	0
$593$	−4.00968e12	−0.0546809	−0.0273405	−	0.999626i	$-0.508704\pi$
$593$	−4.00968e12	−0.0546809	−0.0273405	+	0.999626i	$0.508704\pi$
$594$	0	0
$595$	− 7.83398e12i	− 0.105050i
$596$	0	0
$597$	2.92949e13	0.386296
$598$	0	0
$599$	3.83398e13i	0.497182i	0.968609	+	0.248591i	$0.0799675\pi$
$599$	3.83398e13i	0.497182i	−0.968609	+	0.248591i	$0.920032\pi$
$600$	0	0
$601$	1.21658e14	1.55156	0.775781	−	0.631003i	$-0.217356\pi$
$601$	1.21658e14	1.55156	0.775781	+	0.631003i	$0.217356\pi$
$602$	0	0
$603$	8.84732e13i	1.10975i
$604$	0	0
$605$	−4.47453e13	−0.552040
$606$	0	0
$607$	8.13665e13i	0.987421i	0.869626	+	0.493711i	$0.164360\pi$
$607$	8.13665e13i	0.987421i	−0.869626	+	0.493711i	$0.835640\pi$
$608$	0	0
$609$	5.31081e12	0.0633978
$610$	0	0
$611$	7.04326e13i	0.827119i
$612$	0	0
$613$	−7.05978e13	−0.815622	−0.407811	−	0.913066i	$-0.633708\pi$
$613$	−7.05978e13	−0.815622	−0.407811	+	0.913066i	$0.633708\pi$
$614$	0	0
$615$	− 1.73500e13i	− 0.197207i
$616$	0	0
$617$	−1.34578e14	−1.50504	−0.752519	−	0.658571i	$-0.771161\pi$
$617$	−1.34578e14	−1.50504	−0.752519	+	0.658571i	$0.771161\pi$
$618$	0	0
$619$	7.56040e13i	0.831939i	0.909379	+	0.415969i	$0.136558\pi$
$619$	7.56040e13i	0.831939i	−0.909379	+	0.415969i	$0.863442\pi$
$620$	0	0
$621$	5.20104e13	0.563161
$622$	0	0
$623$	− 8.15437e12i	− 0.0868860i
$624$	0	0
$625$	−4.05242e13	−0.424927
$626$	0	0
$627$	7.23659e12i	0.0746787i
$628$	0	0
$629$	−1.93772e13	−0.196806
$630$	0	0
$631$	1.79955e14i	1.79894i	0.436980	+	0.899471i	$0.356048\pi$
$631$	1.79955e14i	1.79894i	−0.436980	+	0.899471i	$0.643952\pi$
$632$	0	0
$633$	−1.48812e13	−0.146426
$634$	0	0
$635$	− 1.34696e14i	− 1.30463i
$636$	0	0
$637$	−5.43460e13	−0.518167
$638$	0	0
$639$	8.67671e13i	0.814425i
$640$	0	0
$641$	1.06457e14	0.983745	0.491873	−	0.870667i	$-0.336313\pi$
$641$	1.06457e14	0.983745	0.491873	+	0.870667i	$0.336313\pi$
$642$	0	0
$643$	5.16793e13i	0.470177i	0.971974	+	0.235089i	$0.0755380\pi$
$643$	5.16793e13i	0.470177i	−0.971974	+	0.235089i	$0.924462\pi$
$644$	0	0
$645$	3.76406e13	0.337177
$646$	0	0
$647$	2.00791e14i	1.77101i	0.464626	+	0.885507i	$0.346189\pi$
$647$	2.00791e14i	1.77101i	−0.464626	+	0.885507i	$0.653811\pi$
$648$	0	0
$649$	−1.06804e14	−0.927609
$650$	0	0
$651$	− 6.98209e12i	− 0.0597146i
$652$	0	0
$653$	−2.19540e14	−1.84904	−0.924521	−	0.381131i	$-0.875535\pi$
$653$	−2.19540e14	−1.84904	−0.924521	+	0.381131i	$0.875535\pi$
$654$	0	0
$655$	5.31413e13i	0.440784i
$656$	0	0
$657$	−6.29042e13	−0.513869
$658$	0	0
$659$	1.06436e14i	0.856370i	0.903691	+	0.428185i	$0.140847\pi$
$659$	1.06436e14i	0.856370i	−0.903691	+	0.428185i	$0.859153\pi$
$660$	0	0
$661$	2.34125e13	0.185541	0.0927704	−	0.995688i	$-0.470428\pi$
$661$	2.34125e13	0.185541	0.0927704	+	0.995688i	$0.470428\pi$
$662$	0	0
$663$	− 6.87206e12i	− 0.0536437i
$664$	0	0
$665$	2.37272e13	0.182448
$666$	0	0
$667$	− 6.84281e13i	− 0.518329i
$668$	0	0
$669$	5.27262e13	0.393455
$670$	0	0
$671$	1.36506e14i	1.00355i
$672$	0	0
$673$	1.51762e14	1.09923	0.549614	−	0.835418i	$-0.314775\pi$
$673$	1.51762e14	1.09923	0.549614	+	0.835418i	$0.314775\pi$
$674$	0	0
$675$	3.04685e13i	0.217436i
$676$	0	0
$677$	1.89286e14	1.33099	0.665497	−	0.746401i	$-0.268220\pi$
$677$	1.89286e14	1.33099	0.665497	+	0.746401i	$0.268220\pi$
$678$	0	0
$679$	1.02681e14i	0.711444i
$680$	0	0
$681$	−3.26102e13	−0.222648
$682$	0	0
$683$	− 1.25529e14i	− 0.844579i	−0.906461	−	0.422289i	$-0.861227\pi$
$683$	− 1.25529e14i	− 0.844579i	0.906461	−	0.422289i	$-0.138773\pi$
$684$	0	0
$685$	−7.51348e12	−0.0498183
$686$	0	0
$687$	5.10508e13i	0.333594i
$688$	0	0
$689$	−1.09769e14	−0.706944
$690$	0	0
$691$	− 8.55273e13i	− 0.542893i	−0.962453	−	0.271447i	$-0.912498\pi$
$691$	− 8.55273e13i	− 0.542893i	0.962453	−	0.271447i	$-0.0875021\pi$
$692$	0	0
$693$	−3.66378e13	−0.229226
$694$	0	0
$695$	1.06843e14i	0.658905i
$696$	0	0
$697$	4.45270e13	0.270682
$698$	0	0
$699$	− 4.29162e13i	− 0.257179i
$700$	0	0
$701$	−1.22613e14	−0.724346	−0.362173	−	0.932111i	$-0.617965\pi$
$701$	−1.22613e14	−0.724346	−0.362173	+	0.932111i	$0.617965\pi$
$702$	0	0
$703$	− 5.86889e13i	− 0.341806i
$704$	0	0
$705$	4.62783e13	0.265724
$706$	0	0
$707$	9.93367e13i	0.562357i
$708$	0	0
$709$	5.22840e13	0.291835	0.145918	−	0.989297i	$-0.453387\pi$
$709$	5.22840e13	0.291835	0.145918	+	0.989297i	$0.453387\pi$
$710$	0	0
$711$	3.69608e13i	0.203420i
$712$	0	0
$713$	−8.99621e13	−0.488216
$714$	0	0
$715$	− 5.06252e13i	− 0.270917i
$716$	0	0
$717$	8.28743e13	0.437345
$718$	0	0
$719$	− 3.26146e13i	− 0.169734i	−0.996392	−	0.0848668i	$-0.972954\pi$
$719$	− 3.26146e13i	− 0.169734i	0.996392	−	0.0848668i	$-0.0270465\pi$
$720$	0	0
$721$	−7.20558e13	−0.369822
$722$	0	0
$723$	2.96923e13i	0.150297i
$724$	0	0
$725$	4.00863e13	0.200127
$726$	0	0
$727$	1.28649e14i	0.633483i	0.948512	+	0.316741i	$0.102589\pi$
$727$	1.28649e14i	0.633483i	−0.948512	+	0.316741i	$0.897411\pi$
$728$	0	0
$729$	1.17157e14	0.569024
$730$	0	0
$731$	9.66008e13i	0.462801i
$732$	0	0
$733$	8.13109e13	0.384263	0.192132	−	0.981369i	$-0.438460\pi$
$733$	8.13109e13	0.384263	0.192132	+	0.981369i	$0.438460\pi$
$734$	0	0
$735$	3.57084e13i	0.166469i
$736$	0	0
$737$	1.38856e14	0.638597
$738$	0	0
$739$	− 2.41369e14i	− 1.09511i	−0.836768	−	0.547557i	$-0.815558\pi$
$739$	− 2.41369e14i	− 1.09511i	0.836768	−	0.547557i	$-0.184442\pi$
$740$	0	0
$741$	2.08138e13	0.0931665
$742$	0	0
$743$	1.38769e13i	0.0612841i	0.999530	+	0.0306421i	$0.00975520\pi$
$743$	1.38769e13i	0.0612841i	−0.999530	+	0.0306421i	$0.990245\pi$
$744$	0	0
$745$	−2.12141e14	−0.924366
$746$	0	0
$747$	1.18942e14i	0.511366i
$748$	0	0
$749$	1.49320e14	0.633445
$750$	0	0
$751$	2.57750e14i	1.07894i	0.842004	+	0.539471i	$0.181376\pi$
$751$	2.57750e14i	1.07894i	−0.842004	+	0.539471i	$0.818624\pi$
$752$	0	0
$753$	−2.97186e13	−0.122759
$754$	0	0
$755$	− 2.20786e14i	− 0.899990i
$756$	0	0
$757$	−3.68750e14	−1.48338	−0.741690	−	0.670743i	$-0.765975\pi$
$757$	−3.68750e14	−1.48338	−0.741690	+	0.670743i	$0.765975\pi$
$758$	0	0
$759$	− 3.91878e13i	− 0.155576i
$760$	0	0
$761$	4.30053e14	1.68500	0.842498	−	0.538699i	$-0.181084\pi$
$761$	4.30053e14	1.68500	0.842498	+	0.538699i	$0.181084\pi$
$762$	0	0
$763$	− 5.96631e13i	− 0.230719i
$764$	0	0
$765$	5.43929e13	0.207604
$766$	0	0
$767$	3.07189e14i	1.15725i
$768$	0	0
$769$	−1.56784e14	−0.583000	−0.291500	−	0.956571i	$-0.594154\pi$
$769$	−1.56784e14	−0.583000	−0.291500	+	0.956571i	$0.594154\pi$
$770$	0	0
$771$	− 2.63288e13i	− 0.0966405i
$772$	0	0
$773$	2.56775e13	0.0930368	0.0465184	−	0.998917i	$-0.485187\pi$
$773$	2.56775e13	0.0930368	0.0465184	+	0.998917i	$0.485187\pi$
$774$	0	0
$775$	− 5.27012e13i	− 0.188500i
$776$	0	0
$777$	−2.46660e13	−0.0870950
$778$	0	0
$779$	1.34861e14i	0.470111i
$780$	0	0
$781$	1.36178e14	0.468654
$782$	0	0
$783$	7.68090e13i	0.260978i
$784$	0	0
$785$	−1.30956e13	−0.0439315
$786$	0	0
$787$	− 6.91213e13i	− 0.228949i	−0.993426	−	0.114474i	$-0.963482\pi$
$787$	− 6.91213e13i	− 0.228949i	0.993426	−	0.114474i	$-0.0365184\pi$
$788$	0	0
$789$	9.02738e13	0.295241
$790$	0	0
$791$	− 4.88308e13i	− 0.157693i
$792$	0	0
$793$	3.92617e14	1.25200
$794$	0	0
$795$	7.21247e13i	0.227116i
$796$	0	0
$797$	2.08751e14	0.649136	0.324568	−	0.945862i	$-0.394781\pi$
$797$	2.08751e14	0.649136	0.324568	+	0.945862i	$0.394781\pi$
$798$	0	0
$799$	1.18769e14i	0.364727i
$800$	0	0
$801$	5.66174e13	0.171707
$802$	0	0
$803$	9.87261e13i	0.295702i
$804$	0	0
$805$	−1.28488e14	−0.380088
$806$	0	0
$807$	2.44615e13i	0.0714687i
$808$	0	0
$809$	−5.13867e14	−1.48289	−0.741444	−	0.671015i	$-0.765859\pi$
$809$	−5.13867e14	−1.48289	−0.741444	+	0.671015i	$0.765859\pi$
$810$	0	0
$811$	9.61782e13i	0.274140i	0.990561	+	0.137070i	$0.0437685\pi$
$811$	9.61782e13i	0.274140i	−0.990561	+	0.137070i	$0.956231\pi$
$812$	0	0
$813$	1.15774e14	0.325956
$814$	0	0
$815$	− 1.97635e14i	− 0.549638i
$816$	0	0
$817$	−2.92580e14	−0.803775
$818$	0	0
$819$	1.05377e14i	0.285974i
$820$	0	0
$821$	−5.48607e14	−1.47077	−0.735386	−	0.677649i	$-0.762999\pi$
$821$	−5.48607e14	−1.47077	−0.735386	+	0.677649i	$0.762999\pi$
$822$	0	0
$823$	− 1.12101e14i	− 0.296900i	−0.988920	−	0.148450i	$-0.952572\pi$
$823$	− 1.12101e14i	− 0.296900i	0.988920	−	0.148450i	$-0.0474285\pi$
$824$	0	0
$825$	2.29568e13	0.0600678
$826$	0	0
$827$	6.89974e14i	1.78363i	0.452398	+	0.891816i	$0.350569\pi$
$827$	6.89974e14i	1.78363i	−0.452398	+	0.891816i	$0.649431\pi$
$828$	0	0
$829$	−3.85717e14	−0.985136	−0.492568	−	0.870274i	$-0.663942\pi$
$829$	−3.85717e14	−0.985136	−0.492568	+	0.870274i	$0.663942\pi$
$830$	0	0
$831$	− 6.88786e13i	− 0.173812i
$832$	0	0
$833$	−9.16421e13	−0.228492
$834$	0	0
$835$	− 2.40209e14i	− 0.591776i
$836$	0	0
$837$	1.00980e14	0.245816
$838$	0	0
$839$	− 1.42443e14i	− 0.342634i	−0.985216	−	0.171317i	$-0.945198\pi$
$839$	− 1.42443e14i	− 0.342634i	0.985216	−	0.171317i	$-0.0548022\pi$
$840$	0	0
$841$	−3.19652e14	−0.759798
$842$	0	0
$843$	7.36782e12i	0.0173062i
$844$	0	0
$845$	1.85769e14	0.431210
$846$	0	0
$847$	− 1.46177e14i	− 0.335322i
$848$	0	0
$849$	−2.90872e13	−0.0659422
$850$	0	0
$851$	3.17814e14i	0.712074i
$852$	0	0
$853$	7.55633e14	1.67327	0.836634	−	0.547762i	$-0.184520\pi$
$853$	7.55633e14	1.67327	0.836634	+	0.547762i	$0.184520\pi$
$854$	0	0
$855$	1.64743e14i	0.360558i
$856$	0	0
$857$	6.90918e14	1.49459	0.747295	−	0.664492i	$-0.231352\pi$
$857$	6.90918e14	1.49459	0.747295	+	0.664492i	$0.231352\pi$
$858$	0	0
$859$	− 5.21002e14i	− 1.11397i	−0.830522	−	0.556985i	$-0.811958\pi$
$859$	− 5.21002e14i	− 1.11397i	0.830522	−	0.556985i	$-0.188042\pi$
$860$	0	0
$861$	5.66800e13	0.119788
$862$	0	0
$863$	− 7.35612e14i	− 1.53672i	−0.640018	−	0.768360i	$-0.721073\pi$
$863$	− 7.35612e14i	− 1.53672i	0.640018	−	0.768360i	$-0.278927\pi$
$864$	0	0
$865$	4.56995e14	0.943694
$866$	0	0
$867$	1.24041e14i	0.253203i
$868$	0	0
$869$	5.80088e13	0.117056
$870$	0	0
$871$	− 3.99375e14i	− 0.796691i
$872$	0	0
$873$	−7.12934e14	−1.40598
$874$	0	0
$875$	− 2.59605e14i	− 0.506142i
$876$	0	0
$877$	−3.90902e14	−0.753476	−0.376738	−	0.926320i	$-0.622954\pi$
$877$	−3.90902e14	−0.753476	−0.376738	+	0.926320i	$0.622954\pi$
$878$	0	0
$879$	− 2.35816e14i	− 0.449396i
$880$	0	0
$881$	3.52174e13	0.0663556	0.0331778	−	0.999449i	$-0.489437\pi$
$881$	3.52174e13	0.0663556	0.0331778	+	0.999449i	$0.489437\pi$
$882$	0	0
$883$	7.34548e14i	1.36841i	0.729289	+	0.684206i	$0.239851\pi$
$883$	7.34548e14i	1.36841i	−0.729289	+	0.684206i	$0.760149\pi$
$884$	0	0
$885$	2.01840e14	0.371785
$886$	0	0
$887$	− 8.32025e14i	− 1.51537i	−0.652621	−	0.757684i	$-0.726331\pi$
$887$	− 8.32025e14i	− 1.51537i	0.652621	−	0.757684i	$-0.273669\pi$
$888$	0	0
$889$	4.40033e14	0.792459
$890$	0	0
$891$	− 2.31513e14i	− 0.412275i
$892$	0	0
$893$	−3.59721e14	−0.633445
$894$	0	0
$895$	− 7.44156e14i	− 1.29583i
$896$	0	0
$897$	−1.12711e14	−0.194091
$898$	0	0
$899$	− 1.32856e14i	− 0.226247i
$900$	0	0
$901$	−1.85101e14	−0.311735
$902$	0	0
$903$	1.22967e14i	0.204809i
$904$	0	0
$905$	−5.50221e14	−0.906346
$906$	0	0
$907$	− 5.56529e14i	− 0.906674i	−0.891339	−	0.453337i	$-0.850233\pi$
$907$	− 5.56529e14i	− 0.906674i	0.891339	−	0.453337i	$-0.149767\pi$
$908$	0	0
$909$	−6.89714e14	−1.11135
$910$	0	0
$911$	1.25125e14i	0.199412i	0.995017	+	0.0997059i	$0.0317902\pi$
$911$	1.25125e14i	0.199412i	−0.995017	+	0.0997059i	$0.968210\pi$
$912$	0	0
$913$	1.86676e14	0.294262
$914$	0	0
$915$	− 2.57972e14i	− 0.402223i
$916$	0	0
$917$	−1.73606e14	−0.267742
$918$	0	0
$919$	− 1.50249e14i	− 0.229211i	−0.993411	−	0.114605i	$-0.963440\pi$
$919$	− 1.50249e14i	− 0.229211i	0.993411	−	0.114605i	$-0.0365604\pi$
$920$	0	0
$921$	3.30510e14	0.498753
$922$	0	0
$923$	− 3.91673e14i	− 0.584677i
$924$	0	0
$925$	−1.86180e14	−0.274932
$926$	0	0
$927$	− 5.00298e14i	− 0.730853i
$928$	0	0
$929$	6.09509e14	0.880849	0.440425	−	0.897790i	$-0.354828\pi$
$929$	6.09509e14	0.880849	0.440425	+	0.897790i	$0.354828\pi$
$930$	0	0
$931$	− 2.77561e14i	− 0.396836i
$932$	0	0
$933$	2.81099e14	0.397604
$934$	0	0
$935$	− 8.53680e13i	− 0.119464i
$936$	0	0
$937$	−1.04183e15	−1.44245	−0.721224	−	0.692702i	$-0.756420\pi$
$937$	−1.04183e15	−1.44245	−0.721224	+	0.692702i	$0.756420\pi$
$938$	0	0
$939$	− 8.52651e13i	− 0.116800i
$940$	0	0
$941$	1.96166e14	0.265874	0.132937	−	0.991125i	$-0.457559\pi$
$941$	1.96166e14	0.265874	0.132937	+	0.991125i	$0.457559\pi$
$942$	0	0
$943$	− 7.30305e14i	− 0.979367i
$944$	0	0
$945$	1.44225e14	0.191373
$946$	0	0
$947$	− 4.29827e14i	− 0.564344i	−0.959364	−	0.282172i	$-0.908945\pi$
$947$	− 4.29827e14i	− 0.564344i	0.959364	−	0.282172i	$-0.0910549\pi$
$948$	0	0
$949$	2.83954e14	0.368908
$950$	0	0
$951$	1.79726e14i	0.231052i
$952$	0	0
$953$	8.03824e14	1.02258	0.511289	−	0.859409i	$-0.329168\pi$
$953$	8.03824e14	1.02258	0.511289	+	0.859409i	$0.329168\pi$
$954$	0	0
$955$	− 7.62008e14i	− 0.959275i
$956$	0	0
$957$	5.78726e13	0.0720963
$958$	0	0
$959$	− 2.45455e13i	− 0.0302607i
$960$	0	0
$961$	6.44963e14	0.786897
$962$	0	0
$963$	1.03676e15i	1.25183i
$964$	0	0
$965$	−5.11299e14	−0.610996
$966$	0	0
$967$	− 3.08534e14i	− 0.364898i	−0.983215	−	0.182449i	$-0.941598\pi$
$967$	− 3.08534e14i	− 0.364898i	0.983215	−	0.182449i	$-0.0584024\pi$
$968$	0	0
$969$	3.50977e13	0.0410828
$970$	0	0
$971$	1.24518e15i	1.44257i	0.692639	+	0.721284i	$0.256448\pi$
$971$	1.24518e15i	1.44257i	−0.692639	+	0.721284i	$0.743552\pi$
$972$	0	0
$973$	−3.49042e14	−0.400234
$974$	0	0
$975$	− 6.60280e13i	− 0.0749385i
$976$	0	0
$977$	−1.62098e15	−1.82098	−0.910489	−	0.413532i	$-0.864295\pi$
$977$	−1.62098e15	−1.82098	−0.910489	+	0.413532i	$0.864295\pi$
$978$	0	0
$979$	− 8.88592e13i	− 0.0988073i
$980$	0	0
$981$	4.14253e14	0.455954
$982$	0	0
$983$	− 1.07722e15i	− 1.17365i	−0.809715	−	0.586823i	$-0.800379\pi$
$983$	− 1.07722e15i	− 1.17365i	0.809715	−	0.586823i	$-0.199621\pi$
$984$	0	0
$985$	6.02881e14	0.650205
$986$	0	0
$987$	1.51185e14i	0.161407i
$988$	0	0
$989$	1.58439e15	1.67448
$990$	0	0
$991$	2.67540e14i	0.279912i	0.990158	+	0.139956i	$0.0446961\pi$
$991$	2.67540e14i	0.279912i	−0.990158	+	0.139956i	$0.955304\pi$
$992$	0	0
$993$	1.56012e14	0.161589
$994$	0	0
$995$	− 1.04669e15i	− 1.07325i
$996$	0	0
$997$	−3.60761e14	−0.366222	−0.183111	−	0.983092i	$-0.558617\pi$
$997$	−3.60761e14	−0.366222	−0.183111	+	0.983092i	$0.558617\pi$
$998$	0	0
$999$	− 3.56739e14i	− 0.358528i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.l.255.5		12
4.3	odd	2	inner	256.11.c.l.255.7		12
8.3	odd	2	inner	256.11.c.l.255.6		12
8.5	even	2	inner	256.11.c.l.255.8		12
16.3	odd	4		64.11.d.b.31.6	yes	12
16.5	even	4		64.11.d.b.31.5	✓	12
16.11	odd	4		64.11.d.b.31.7	yes	12
16.13	even	4		64.11.d.b.31.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.11.d.b.31.5	✓	12	16.5	even	4
64.11.d.b.31.6	yes	12	16.3	odd	4
64.11.d.b.31.7	yes	12	16.11	odd	4
64.11.d.b.31.8	yes	12	16.13	even	4
256.11.c.l.255.5		12	1.1	even	1	trivial
256.11.c.l.255.6		12	8.3	odd	2	inner
256.11.c.l.255.7		12	4.3	odd	2	inner
256.11.c.l.255.8		12	8.5	even	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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-67.2765i q^{3} -2403.74 q^{5} -7852.70i q^{7} +54522.9 q^{9} +O(q^{10})\)	\(q-67.2765i q^{3} -2403.74 q^{5} -7852.70i q^{7} +54522.9 q^{9} -85571.9i q^{11} -246121. q^{13} +161715. i q^{15} -415026. q^{17} -1.25701e6i q^{19} -528302. q^{21} +6.80701e6i q^{23} -3.98765e6 q^{25} -7.64071e6i q^{27} -1.00526e7 q^{29} +1.32161e7i q^{31} -5.75698e6 q^{33} +1.88759e7i q^{35} +4.66892e7 q^{37} +1.65581e7i q^{39} -1.07287e8 q^{41} -2.32758e8i q^{43} -1.31059e8 q^{45} -2.86171e8i q^{47} +2.20810e8 q^{49} +2.79215e7i q^{51} +4.45998e8 q^{53} +2.05693e8i q^{55} -8.45674e7 q^{57} -1.24812e9i q^{59} -1.59522e9 q^{61} -4.28152e8i q^{63} +5.91610e8 q^{65} +1.62268e9i q^{67} +4.57952e8 q^{69} +1.59139e9i q^{71} -1.15372e9 q^{73} +2.68275e8i q^{75} -6.71971e8 q^{77} +6.77895e8i q^{79} +2.70548e9 q^{81} +2.18151e9i q^{83} +9.97616e8 q^{85} +6.76303e8i q^{87} +1.03842e9 q^{89} +1.93271e9i q^{91} +8.89132e8 q^{93} +3.02153e9i q^{95} -1.30759e10 q^{97} -4.66563e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 375204 q^{9}+O(q^{10}) \) 12 * q - 375204 * q^9	\( 12 q - 375204 q^{9} - 1639752 q^{17} + 28703892 q^{25} + 358214544 q^{33} - 496702488 q^{41} + 1385670732 q^{49} - 5733526800 q^{57} + 8895346752 q^{65} - 12161297592 q^{73} - 7708157604 q^{81} + 5871785160 q^{89} + 30465806904 q^{97}+O(q^{100}) \) 12 * q - 375204 * q^9 - 1639752 * q^17 + 28703892 * q^25 + 358214544 * q^33 - 496702488 * q^41 + 1385670732 * q^49 - 5733526800 * q^57 + 8895346752 * q^65 - 12161297592 * q^73 - 7708157604 * q^81 + 5871785160 * q^89 + 30465806904 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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