LMFDB - Embedded newform 256.8.b.l.129.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(129,256)]

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([0, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.b (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:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.50765497344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 101x^{4} + 2512x^{2} + 36 $ x^6 + 101x^4 + 2512x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.5
Root			$-6.65206i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.8.b.l.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+51.2165i q^{3} -145.477i q^{5} +192.521 q^{7} -436.129 q^{9} +O(q^{10})$	$q+51.2165i q^{3} -145.477i q^{5} +192.521 q^{7} -436.129 q^{9} +5081.41i q^{11} -9384.12i q^{13} +7450.81 q^{15} -17222.6 q^{17} -17236.8i q^{19} +9860.23i q^{21} -18418.0 q^{23} +56961.5 q^{25} +89673.5i q^{27} -58384.4i q^{29} +99913.3 q^{31} -260252. q^{33} -28007.3i q^{35} -484307. i q^{37} +480622. q^{39} -569425. q^{41} -18958.6i q^{43} +63446.6i q^{45} -1.22220e6 q^{47} -786479. q^{49} -882079. i q^{51} -1.64310e6i q^{53} +739227. q^{55} +882809. q^{57} +63612.4i q^{59} +1.39333e6i q^{61} -83963.7 q^{63} -1.36517e6 q^{65} -3.02319e6i q^{67} -943304. i q^{69} +3.28271e6 q^{71} +4.52034e6 q^{73} +2.91737e6i q^{75} +978276. i q^{77} +3.18742e6 q^{79} -5.54657e6 q^{81} -7.31201e6i q^{83} +2.50548e6i q^{85} +2.99024e6 q^{87} +3.55637e6 q^{89} -1.80664e6i q^{91} +5.11721e6i q^{93} -2.50756e6 q^{95} +1.36320e7 q^{97} -2.21615e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 24 q^{7} + 106 q^{9}+O(q^{10}) $ 6 * q + 24 * q^7 + 106 * q^9	$ 6 q + 24 q^{7} + 106 q^{9} - 8232 q^{15} + 1076 q^{17} + 86632 q^{23} + 81206 q^{25} - 313856 q^{31} - 367000 q^{33} + 1243368 q^{39} + 1819396 q^{41} - 1705904 q^{47} - 3273066 q^{49} + 7411352 q^{55} + 6490536 q^{57} - 5088856 q^{63} - 9549544 q^{65} + 19882936 q^{71} + 11588940 q^{73} - 2009008 q^{79} - 12889762 q^{81} + 36376264 q^{87} + 9662700 q^{89} + 30260056 q^{95} - 10501388 q^{97}+O(q^{100}) $ 6 * q + 24 * q^7 + 106 * q^9 - 8232 * q^15 + 1076 * q^17 + 86632 * q^23 + 81206 * q^25 - 313856 * q^31 - 367000 * q^33 + 1243368 * q^39 + 1819396 * q^41 - 1705904 * q^47 - 3273066 * q^49 + 7411352 * q^55 + 6490536 * q^57 - 5088856 * q^63 - 9549544 * q^65 + 19882936 * q^71 + 11588940 * q^73 - 2009008 * q^79 - 12889762 * q^81 + 36376264 * q^87 + 9662700 * q^89 + 30260056 * q^95 - 10501388 * 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
$3$	51.2165i	1.09518i	0.836747	+	0.547590i	$0.184455\pi$
$3$	51.2165i	1.09518i	−0.836747	+	0.547590i	$0.815545\pi$
$4$	0	0
$5$	− 145.477i	− 0.520473i	−0.965545	−	0.260237i	$-0.916199\pi$
$5$	− 145.477i	− 0.520473i	0.965545	−	0.260237i	$-0.0838006\pi$
$6$	0	0
$7$	192.521	0.212146	0.106073	−	0.994358i	$-0.466172\pi$
$7$	192.521	0.212146	0.106073	+	0.994358i	$0.466172\pi$
$8$	0	0
$9$	−436.129	−0.199419
$10$	0	0
$11$	5081.41i	1.15109i	0.817770	+	0.575546i	$0.195210\pi$
$11$	5081.41i	1.15109i	−0.817770	+	0.575546i	$0.804790\pi$
$12$	0	0
$13$	− 9384.12i	− 1.18465i	−0.805697	−	0.592327i	$-0.798209\pi$
$13$	− 9384.12i	− 1.18465i	0.805697	−	0.592327i	$-0.201791\pi$
$14$	0	0
$15$	7450.81	0.570012
$16$	0	0
$17$	−17222.6	−0.850211	−0.425105	−	0.905144i	$-0.639763\pi$
$17$	−17222.6	−0.850211	−0.425105	+	0.905144i	$0.639763\pi$
$18$	0	0
$19$	− 17236.8i	− 0.576527i	−0.957551	−	0.288263i	$-0.906922\pi$
$19$	− 17236.8i	− 0.576527i	0.957551	−	0.288263i	$-0.0930779\pi$
$20$	0	0
$21$	9860.23i	0.232337i
$22$	0	0
$23$	−18418.0	−0.315642	−0.157821	−	0.987468i	$-0.550447\pi$
$23$	−18418.0	−0.315642	−0.157821	+	0.987468i	$0.550447\pi$
$24$	0	0
$25$	56961.5	0.729107
$26$	0	0
$27$	89673.5i	0.876780i
$28$	0	0
$29$	− 58384.4i	− 0.444533i	−0.974986	−	0.222266i	$-0.928655\pi$
$29$	− 58384.4i	− 0.444533i	0.974986	−	0.222266i	$-0.0713455\pi$
$30$	0	0
$31$	99913.3	0.602362	0.301181	−	0.953567i	$-0.402619\pi$
$31$	99913.3	0.602362	0.301181	+	0.953567i	$0.402619\pi$
$32$	0	0
$33$	−260252.	−1.26065
$34$	0	0
$35$	− 28007.3i	− 0.110416i
$36$	0	0
$37$	− 484307.i	− 1.57186i	−0.618314	−	0.785931i	$-0.712184\pi$
$37$	− 484307.i	− 1.57186i	0.618314	−	0.785931i	$-0.287816\pi$
$38$	0	0
$39$	480622.	1.29741
$40$	0	0
$41$	−569425.	−1.29031	−0.645153	−	0.764053i	$-0.723206\pi$
$41$	−569425.	−1.29031	−0.645153	+	0.764053i	$0.723206\pi$
$42$	0	0
$43$	− 18958.6i	− 0.0363636i	−0.999835	−	0.0181818i	$-0.994212\pi$
$43$	− 18958.6i	− 0.0363636i	0.999835	−	0.0181818i	$-0.00578776\pi$
$44$	0	0
$45$	63446.6i	0.103792i
$46$	0	0
$47$	−1.22220e6	−1.71712	−0.858558	−	0.512717i	$-0.828639\pi$
$47$	−1.22220e6	−1.71712	−0.858558	+	0.512717i	$0.828639\pi$
$48$	0	0
$49$	−786479.	−0.954994
$50$	0	0
$51$	− 882079.i	− 0.931133i
$52$	0	0
$53$	− 1.64310e6i	− 1.51600i	−0.652257	−	0.757998i	$-0.726178\pi$
$53$	− 1.64310e6i	− 1.51600i	0.652257	−	0.757998i	$-0.273822\pi$
$54$	0	0
$55$	739227.	0.599113
$56$	0	0
$57$	882809.	0.631400
$58$	0	0
$59$	63612.4i	0.0403236i	0.999797	+	0.0201618i	$0.00641814\pi$
$59$	63612.4i	0.0403236i	−0.999797	+	0.0201618i	$0.993582\pi$
$60$	0	0
$61$	1.39333e6i	0.785960i	0.919547	+	0.392980i	$0.128556\pi$
$61$	1.39333e6i	0.785960i	−0.919547	+	0.392980i	$0.871444\pi$
$62$	0	0
$63$	−83963.7	−0.0423058
$64$	0	0
$65$	−1.36517e6	−0.616581
$66$	0	0
$67$	− 3.02319e6i	− 1.22802i	−0.789300	−	0.614008i	$-0.789556\pi$
$67$	− 3.02319e6i	− 1.22802i	0.789300	−	0.614008i	$-0.210444\pi$
$68$	0	0
$69$	− 943304.i	− 0.345685i
$70$	0	0
$71$	3.28271e6	1.08850	0.544250	−	0.838923i	$-0.316814\pi$
$71$	3.28271e6	1.08850	0.544250	+	0.838923i	$0.316814\pi$
$72$	0	0
$73$	4.52034e6	1.36001	0.680003	−	0.733210i	$-0.261979\pi$
$73$	4.52034e6	1.36001	0.680003	+	0.733210i	$0.261979\pi$
$74$	0	0
$75$	2.91737e6i	0.798504i
$76$	0	0
$77$	978276.i	0.244199i
$78$	0	0
$79$	3.18742e6	0.727352	0.363676	−	0.931525i	$-0.381521\pi$
$79$	3.18742e6	0.727352	0.363676	+	0.931525i	$0.381521\pi$
$80$	0	0
$81$	−5.54657e6	−1.15965
$82$	0	0
$83$	− 7.31201e6i	− 1.40367i	−0.712342	−	0.701833i	$-0.752365\pi$
$83$	− 7.31201e6i	− 1.40367i	0.712342	−	0.701833i	$-0.247635\pi$
$84$	0	0
$85$	2.50548e6i	0.442512i
$86$	0	0
$87$	2.99024e6	0.486843
$88$	0	0
$89$	3.55637e6	0.534739	0.267369	−	0.963594i	$-0.413846\pi$
$89$	3.55637e6	0.534739	0.267369	+	0.963594i	$0.413846\pi$
$90$	0	0
$91$	− 1.80664e6i	− 0.251319i
$92$	0	0
$93$	5.11721e6i	0.659695i
$94$	0	0
$95$	−2.50756e6	−0.300067
$96$	0	0
$97$	1.36320e7	1.51656	0.758281	−	0.651928i	$-0.226040\pi$
$97$	1.36320e7	1.51656	0.758281	+	0.651928i	$0.226040\pi$
$98$	0	0
$99$	− 2.21615e6i	− 0.229549i
$100$	0	0
$101$	2.80336e6i	0.270741i	0.990795	+	0.135371i	$0.0432225\pi$
$101$	2.80336e6i	0.270741i	−0.990795	+	0.135371i	$0.956778\pi$
$102$	0	0
$103$	1.87446e7	1.69023	0.845116	−	0.534582i	$-0.179531\pi$
$103$	1.87446e7	1.69023	0.845116	+	0.534582i	$0.179531\pi$
$104$	0	0
$105$	1.43443e6	0.120926
$106$	0	0
$107$	− 1.61604e7i	− 1.27529i	−0.770331	−	0.637644i	$-0.779909\pi$
$107$	− 1.61604e7i	− 1.27529i	0.770331	−	0.637644i	$-0.220091\pi$
$108$	0	0
$109$	1.51664e7i	1.12173i	0.827906	+	0.560867i	$0.189532\pi$
$109$	1.51664e7i	1.12173i	−0.827906	+	0.560867i	$0.810468\pi$
$110$	0	0
$111$	2.48045e7	1.72147
$112$	0	0
$113$	7.70281e6	0.502197	0.251099	−	0.967962i	$-0.419208\pi$
$113$	7.70281e6	0.502197	0.251099	+	0.967962i	$0.419208\pi$
$114$	0	0
$115$	2.67939e6i	0.164283i
$116$	0	0
$117$	4.09268e6i	0.236242i
$118$	0	0
$119$	−3.31570e6	−0.180368
$120$	0	0
$121$	−6.33357e6	−0.325012
$122$	0	0
$123$	− 2.91639e7i	− 1.41312i
$124$	0	0
$125$	− 1.96519e7i	− 0.899954i
$126$	0	0
$127$	7.30616e6	0.316502	0.158251	−	0.987399i	$-0.449415\pi$
$127$	7.30616e6	0.316502	0.158251	+	0.987399i	$0.449415\pi$
$128$	0	0
$129$	970993.	0.0398246
$130$	0	0
$131$	− 1.64805e7i	− 0.640504i	−0.947332	−	0.320252i	$-0.896232\pi$
$131$	− 1.64805e7i	− 0.640504i	0.947332	−	0.320252i	$-0.103768\pi$
$132$	0	0
$133$	− 3.31844e6i	− 0.122308i
$134$	0	0
$135$	1.30454e7	0.456341
$136$	0	0
$137$	−4.74046e7	−1.57506	−0.787532	−	0.616273i	$-0.788642\pi$
$137$	−4.74046e7	−1.57506	−0.787532	+	0.616273i	$0.788642\pi$
$138$	0	0
$139$	− 4.60622e7i	− 1.45476i	−0.686233	−	0.727382i	$-0.740737\pi$
$139$	− 4.60622e7i	− 1.45476i	0.686233	−	0.727382i	$-0.259263\pi$
$140$	0	0
$141$	− 6.25968e7i	− 1.88055i
$142$	0	0
$143$	4.76846e7	1.36365
$144$	0	0
$145$	−8.49357e6	−0.231368
$146$	0	0
$147$	− 4.02807e7i	− 1.04589i
$148$	0	0
$149$	1.97163e7i	0.488285i	0.969739	+	0.244142i	$0.0785064\pi$
$149$	1.97163e7i	0.488285i	−0.969739	+	0.244142i	$0.921494\pi$
$150$	0	0
$151$	−5.53585e7	−1.30847	−0.654237	−	0.756289i	$-0.727010\pi$
$151$	−5.53585e7	−1.30847	−0.654237	+	0.756289i	$0.727010\pi$
$152$	0	0
$153$	7.51125e6	0.169548
$154$	0	0
$155$	− 1.45351e7i	− 0.313513i
$156$	0	0
$157$	− 3.33244e7i	− 0.687248i	−0.939107	−	0.343624i	$-0.888345\pi$
$157$	− 3.33244e7i	− 0.687248i	0.939107	−	0.343624i	$-0.111655\pi$
$158$	0	0
$159$	8.41537e7	1.66029
$160$	0	0
$161$	−3.54584e6	−0.0669620
$162$	0	0
$163$	− 3.57147e6i	− 0.0645938i	−0.999478	−	0.0322969i	$-0.989718\pi$
$163$	− 3.57147e6i	− 0.0645938i	0.999478	−	0.0322969i	$-0.0102822\pi$
$164$	0	0
$165$	3.78606e7i	0.656136i
$166$	0	0
$167$	1.16844e8	1.94133	0.970667	−	0.240429i	$-0.0772882\pi$
$167$	1.16844e8	1.94133	0.970667	+	0.240429i	$0.0772882\pi$
$168$	0	0
$169$	−2.53132e7	−0.403407
$170$	0	0
$171$	7.51747e6i	0.114970i
$172$	0	0
$173$	2.83019e7i	0.415579i	0.978174	+	0.207790i	$0.0666270\pi$
$173$	2.83019e7i	0.415579i	−0.978174	+	0.207790i	$0.933373\pi$
$174$	0	0
$175$	1.09663e7	0.154677
$176$	0	0
$177$	−3.25800e6	−0.0441616
$178$	0	0
$179$	7.62530e7i	0.993737i	0.867826	+	0.496868i	$0.165517\pi$
$179$	7.62530e7i	0.993737i	−0.867826	+	0.496868i	$0.834483\pi$
$180$	0	0
$181$	− 5.39175e7i	− 0.675857i	−0.941172	−	0.337929i	$-0.890274\pi$
$181$	− 5.39175e7i	− 0.675857i	0.941172	−	0.337929i	$-0.109726\pi$
$182$	0	0
$183$	−7.13616e7	−0.860768
$184$	0	0
$185$	−7.04554e7	−0.818113
$186$	0	0
$187$	− 8.75149e7i	− 0.978670i
$188$	0	0
$189$	1.72640e7i	0.186005i
$190$	0	0
$191$	−6.12934e7	−0.636498	−0.318249	−	0.948007i	$-0.603095\pi$
$191$	−6.12934e7	−0.636498	−0.318249	+	0.948007i	$0.603095\pi$
$192$	0	0
$193$	−6.41428e6	−0.0642239	−0.0321120	−	0.999484i	$-0.510223\pi$
$193$	−6.41428e6	−0.0642239	−0.0321120	+	0.999484i	$0.510223\pi$
$194$	0	0
$195$	− 6.99193e7i	− 0.675267i
$196$	0	0
$197$	− 1.33319e8i	− 1.24240i	−0.783653	−	0.621198i	$-0.786646\pi$
$197$	− 1.33319e8i	− 1.24240i	0.783653	−	0.621198i	$-0.213354\pi$
$198$	0	0
$199$	1.76933e8	1.59156	0.795780	−	0.605585i	$-0.207061\pi$
$199$	1.76933e8	1.59156	0.795780	+	0.605585i	$0.207061\pi$
$200$	0	0
$201$	1.54837e8	1.34490
$202$	0	0
$203$	− 1.12402e7i	− 0.0943057i
$204$	0	0
$205$	8.28380e7i	0.671570i
$206$	0	0
$207$	8.03261e6	0.0629449
$208$	0	0
$209$	8.75873e7	0.663635
$210$	0	0
$211$	− 1.12921e8i	− 0.827532i	−0.910383	−	0.413766i	$-0.864213\pi$
$211$	− 1.12921e8i	− 0.827532i	0.910383	−	0.413766i	$-0.135787\pi$
$212$	0	0
$213$	1.68129e8i	1.19210i
$214$	0	0
$215$	−2.75803e6	−0.0189263
$216$	0	0
$217$	1.92354e7	0.127788
$218$	0	0
$219$	2.31516e8i	1.48945i
$220$	0	0
$221$	1.61619e8i	1.00721i
$222$	0	0
$223$	−3.09574e8	−1.86938	−0.934689	−	0.355465i	$-0.884322\pi$
$223$	−3.09574e8	−1.86938	−0.934689	+	0.355465i	$0.884322\pi$
$224$	0	0
$225$	−2.48425e7	−0.145398
$226$	0	0
$227$	8.20252e7i	0.465433i	0.972545	+	0.232716i	$0.0747614\pi$
$227$	8.20252e7i	0.465433i	−0.972545	+	0.232716i	$0.925239\pi$
$228$	0	0
$229$	− 8.46442e7i	− 0.465772i	−0.972504	−	0.232886i	$-0.925183\pi$
$229$	− 8.46442e7i	− 0.465772i	0.972504	−	0.232886i	$-0.0748169\pi$
$230$	0	0
$231$	−5.01039e7	−0.267442
$232$	0	0
$233$	2.29590e8	1.18907	0.594535	−	0.804069i	$-0.297336\pi$
$233$	2.29590e8	1.18907	0.594535	+	0.804069i	$0.297336\pi$
$234$	0	0
$235$	1.77802e8i	0.893713i
$236$	0	0
$237$	1.63249e8i	0.796582i
$238$	0	0
$239$	4.29560e7	0.203531	0.101766	−	0.994808i	$-0.467551\pi$
$239$	4.29560e7	0.203531	0.101766	+	0.994808i	$0.467551\pi$
$240$	0	0
$241$	−1.01479e8	−0.466999	−0.233499	−	0.972357i	$-0.575018\pi$
$241$	−1.01479e8	−0.466999	−0.233499	+	0.972357i	$0.575018\pi$
$242$	0	0
$243$	− 8.79601e7i	− 0.393246i
$244$	0	0
$245$	1.14414e8i	0.497049i
$246$	0	0
$247$	−1.61752e8	−0.682985
$248$	0	0
$249$	3.74496e8	1.53727
$250$	0	0
$251$	1.44611e8i	0.577223i	0.957446	+	0.288612i	$0.0931936\pi$
$251$	1.44611e8i	0.577223i	−0.957446	+	0.288612i	$0.906806\pi$
$252$	0	0
$253$	− 9.35893e7i	− 0.363333i
$254$	0	0
$255$	−1.28322e8	−0.484630
$256$	0	0
$257$	−2.87026e8	−1.05477	−0.527383	−	0.849628i	$-0.676827\pi$
$257$	−2.87026e8	−1.05477	−0.527383	+	0.849628i	$0.676827\pi$
$258$	0	0
$259$	− 9.32390e7i	− 0.333464i
$260$	0	0
$261$	2.54631e7i	0.0886481i
$262$	0	0
$263$	−2.41676e8	−0.819197	−0.409598	−	0.912266i	$-0.634331\pi$
$263$	−2.41676e8	−0.819197	−0.409598	+	0.912266i	$0.634331\pi$
$264$	0	0
$265$	−2.39033e8	−0.789035
$266$	0	0
$267$	1.82145e8i	0.585635i
$268$	0	0
$269$	− 8.32948e7i	− 0.260907i	−0.991454	−	0.130453i	$-0.958357\pi$
$269$	− 8.32948e7i	− 0.260907i	0.991454	−	0.130453i	$-0.0416433\pi$
$270$	0	0
$271$	−3.13948e8	−0.958221	−0.479110	−	0.877755i	$-0.659041\pi$
$271$	−3.13948e8	−0.958221	−0.479110	+	0.877755i	$0.659041\pi$
$272$	0	0
$273$	9.25295e7	0.275240
$274$	0	0
$275$	2.89445e8i	0.839269i
$276$	0	0
$277$	− 4.19831e8i	− 1.18685i	−0.804890	−	0.593425i	$-0.797776\pi$
$277$	− 4.19831e8i	− 1.18685i	0.804890	−	0.593425i	$-0.202224\pi$
$278$	0	0
$279$	−4.35751e7	−0.120122
$280$	0	0
$281$	−3.77038e8	−1.01371	−0.506854	−	0.862032i	$-0.669192\pi$
$281$	−3.77038e8	−1.01371	−0.506854	+	0.862032i	$0.669192\pi$
$282$	0	0
$283$	7.46059e7i	0.195668i	0.995203	+	0.0978341i	$0.0311915\pi$
$283$	7.46059e7i	0.195668i	−0.995203	+	0.0978341i	$0.968809\pi$
$284$	0	0
$285$	− 1.28428e8i	− 0.328627i
$286$	0	0
$287$	−1.09626e8	−0.273733
$288$	0	0
$289$	−1.13722e8	−0.277142
$290$	0	0
$291$	6.98186e8i	1.66091i
$292$	0	0
$293$	− 4.39379e8i	− 1.02048i	−0.860033	−	0.510238i	$-0.829557\pi$
$293$	− 4.39379e8i	− 1.02048i	0.860033	−	0.510238i	$-0.170443\pi$
$294$	0	0
$295$	9.25413e6	0.0209874
$296$	0	0
$297$	−4.55668e8	−1.00925
$298$	0	0
$299$	1.72837e8i	0.373927i
$300$	0	0
$301$	− 3.64992e6i	− 0.00771437i
$302$	0	0
$303$	−1.43578e8	−0.296510
$304$	0	0
$305$	2.02698e8	0.409071
$306$	0	0
$307$	5.09082e8i	1.00416i	0.864821	+	0.502081i	$0.167432\pi$
$307$	5.09082e8i	1.00416i	−0.864821	+	0.502081i	$0.832568\pi$
$308$	0	0
$309$	9.60034e8i	1.85111i
$310$	0	0
$311$	−1.36937e8	−0.258143	−0.129072	−	0.991635i	$-0.541200\pi$
$311$	−1.36937e8	−0.258143	−0.129072	+	0.991635i	$0.541200\pi$
$312$	0	0
$313$	−1.83418e8	−0.338094	−0.169047	−	0.985608i	$-0.554069\pi$
$313$	−1.83418e8	−0.338094	−0.169047	+	0.985608i	$0.554069\pi$
$314$	0	0
$315$	1.22148e7i	0.0220190i
$316$	0	0
$317$	− 7.69406e8i	− 1.35659i	−0.734790	−	0.678294i	$-0.762719\pi$
$317$	− 7.69406e8i	− 1.35659i	0.734790	−	0.678294i	$-0.237281\pi$
$318$	0	0
$319$	2.96675e8	0.511698
$320$	0	0
$321$	8.27678e8	1.39667
$322$	0	0
$323$	2.96862e8i	0.490169i
$324$	0	0
$325$	− 5.34534e8i	− 0.863740i
$326$	0	0
$327$	−7.76770e8	−1.22850
$328$	0	0
$329$	−2.35298e8	−0.364278
$330$	0	0
$331$	− 1.19278e8i	− 0.180785i	−0.995906	−	0.0903926i	$-0.971188\pi$
$331$	− 1.19278e8i	− 0.180785i	0.995906	−	0.0903926i	$-0.0288122\pi$
$332$	0	0
$333$	2.11220e8i	0.313459i
$334$	0	0
$335$	−4.39804e8	−0.639150
$336$	0	0
$337$	−5.92823e8	−0.843763	−0.421882	−	0.906651i	$-0.638630\pi$
$337$	−5.92823e8	−0.843763	−0.421882	+	0.906651i	$0.638630\pi$
$338$	0	0
$339$	3.94511e8i	0.549996i
$340$	0	0
$341$	5.07701e8i	0.693374i
$342$	0	0
$343$	−3.09962e8	−0.414743
$344$	0	0
$345$	−1.37229e8	−0.179920
$346$	0	0
$347$	− 7.05723e8i	− 0.906736i	−0.891323	−	0.453368i	$-0.850222\pi$
$347$	− 7.05723e8i	− 0.906736i	0.891323	−	0.453368i	$-0.149778\pi$
$348$	0	0
$349$	− 3.96668e8i	− 0.499503i	−0.968310	−	0.249751i	$-0.919651\pi$
$349$	− 3.96668e8i	− 0.499503i	0.968310	−	0.249751i	$-0.0803489\pi$
$350$	0	0
$351$	8.41507e8	1.03868
$352$	0	0
$353$	1.96190e7	0.0237392	0.0118696	−	0.999930i	$-0.496222\pi$
$353$	1.96190e7	0.0237392	0.0118696	+	0.999930i	$0.496222\pi$
$354$	0	0
$355$	− 4.77558e8i	− 0.566535i
$356$	0	0
$357$	− 1.69818e8i	− 0.197536i
$358$	0	0
$359$	−1.71544e9	−1.95680	−0.978398	−	0.206732i	$-0.933717\pi$
$359$	−1.71544e9	−1.95680	−0.978398	+	0.206732i	$0.933717\pi$
$360$	0	0
$361$	5.96764e8	0.667617
$362$	0	0
$363$	− 3.24383e8i	− 0.355947i
$364$	0	0
$365$	− 6.57604e8i	− 0.707847i
$366$	0	0
$367$	−1.54817e8	−0.163489	−0.0817444	−	0.996653i	$-0.526049\pi$
$367$	−1.54817e8	−0.163489	−0.0817444	+	0.996653i	$0.526049\pi$
$368$	0	0
$369$	2.48342e8	0.257311
$370$	0	0
$371$	− 3.16330e8i	− 0.321612i
$372$	0	0
$373$	1.92398e9i	1.91964i	0.280613	+	0.959821i	$0.409462\pi$
$373$	1.92398e9i	1.91964i	−0.280613	+	0.959821i	$0.590538\pi$
$374$	0	0
$375$	1.00650e9	0.985612
$376$	0	0
$377$	−5.47886e8	−0.526618
$378$	0	0
$379$	1.57142e9i	1.48270i	0.671116	+	0.741352i	$0.265815\pi$
$379$	1.57142e9i	1.48270i	−0.671116	+	0.741352i	$0.734185\pi$
$380$	0	0
$381$	3.74196e8i	0.346626i
$382$	0	0
$383$	8.75347e8	0.796131	0.398065	−	0.917357i	$-0.369682\pi$
$383$	8.75347e8	0.796131	0.398065	+	0.917357i	$0.369682\pi$
$384$	0	0
$385$	1.42316e8	0.127099
$386$	0	0
$387$	8.26839e6i	0.00725157i
$388$	0	0
$389$	− 1.37884e8i	− 0.118765i	−0.998235	−	0.0593827i	$-0.981087\pi$
$389$	− 1.37884e8i	− 0.118765i	0.998235	−	0.0593827i	$-0.0189132\pi$
$390$	0	0
$391$	3.17205e8	0.268362
$392$	0	0
$393$	8.44075e8	0.701467
$394$	0	0
$395$	− 4.63696e8i	− 0.378568i
$396$	0	0
$397$	1.71487e9i	1.37551i	0.725941	+	0.687757i	$0.241405\pi$
$397$	1.71487e9i	1.37551i	−0.725941	+	0.687757i	$0.758595\pi$
$398$	0	0
$399$	1.69959e8	0.133949
$400$	0	0
$401$	4.96564e8	0.384565	0.192283	−	0.981340i	$-0.438411\pi$
$401$	4.96564e8	0.384565	0.192283	+	0.981340i	$0.438411\pi$
$402$	0	0
$403$	− 9.37599e8i	− 0.713591i
$404$	0	0
$405$	8.06898e8i	0.603567i
$406$	0	0
$407$	2.46096e9	1.80936
$408$	0	0
$409$	1.96571e9	1.42065	0.710325	−	0.703874i	$-0.248548\pi$
$409$	1.96571e9	1.42065	0.710325	+	0.703874i	$0.248548\pi$
$410$	0	0
$411$	− 2.42790e9i	− 1.72498i
$412$	0	0
$413$	1.22467e7i	0.00855448i
$414$	0	0
$415$	−1.06373e9	−0.730571
$416$	0	0
$417$	2.35914e9	1.59323
$418$	0	0
$419$	− 2.31255e8i	− 0.153583i	−0.997047	−	0.0767913i	$-0.975532\pi$
$419$	− 2.31255e8i	− 0.153583i	0.997047	−	0.0767913i	$-0.0244675\pi$
$420$	0	0
$421$	− 2.73476e9i	− 1.78621i	−0.449851	−	0.893103i	$-0.648523\pi$
$421$	− 2.73476e9i	− 1.78621i	0.449851	−	0.893103i	$-0.351477\pi$
$422$	0	0
$423$	5.33036e8	0.342425
$424$	0	0
$425$	−9.81023e8	−0.619895
$426$	0	0
$427$	2.68245e8i	0.166738i
$428$	0	0
$429$	2.44224e9i	1.49344i
$430$	0	0
$431$	6.59396e8	0.396712	0.198356	−	0.980130i	$-0.436440\pi$
$431$	6.59396e8	0.396712	0.198356	+	0.980130i	$0.436440\pi$
$432$	0	0
$433$	−1.84454e9	−1.09189	−0.545946	−	0.837820i	$-0.683830\pi$
$433$	−1.84454e9	−1.09189	−0.545946	+	0.837820i	$0.683830\pi$
$434$	0	0
$435$	− 4.35011e8i	− 0.253389i
$436$	0	0
$437$	3.17467e8i	0.181976i
$438$	0	0
$439$	2.34127e9	1.32077	0.660384	−	0.750928i	$-0.270394\pi$
$439$	2.34127e9	1.32077	0.660384	+	0.750928i	$0.270394\pi$
$440$	0	0
$441$	3.43006e8	0.190444
$442$	0	0
$443$	1.69165e9i	0.924483i	0.886754	+	0.462241i	$0.152955\pi$
$443$	1.69165e9i	0.924483i	−0.886754	+	0.462241i	$0.847045\pi$
$444$	0	0
$445$	− 5.17369e8i	− 0.278317i
$446$	0	0
$447$	−1.00980e9	−0.534760
$448$	0	0
$449$	−3.05472e9	−1.59261	−0.796304	−	0.604897i	$-0.793214\pi$
$449$	−3.05472e9	−1.59261	−0.796304	+	0.604897i	$0.793214\pi$
$450$	0	0
$451$	− 2.89348e9i	− 1.48526i
$452$	0	0
$453$	− 2.83527e9i	− 1.43301i
$454$	0	0
$455$	−2.62823e8	−0.130805
$456$	0	0
$457$	−3.90086e8	−0.191185	−0.0955924	−	0.995421i	$-0.530475\pi$
$457$	−3.90086e8	−0.191185	−0.0955924	+	0.995421i	$0.530475\pi$
$458$	0	0
$459$	− 1.54441e9i	− 0.745448i
$460$	0	0
$461$	− 1.36426e9i	− 0.648550i	−0.945963	−	0.324275i	$-0.894880\pi$
$461$	− 1.36426e9i	− 0.648550i	0.945963	−	0.324275i	$-0.105120\pi$
$462$	0	0
$463$	1.33810e9	0.626551	0.313276	−	0.949662i	$-0.398574\pi$
$463$	1.33810e9	0.626551	0.313276	+	0.949662i	$0.398574\pi$
$464$	0	0
$465$	7.44435e8	0.343354
$466$	0	0
$467$	− 1.60450e8i	− 0.0729005i	−0.999335	−	0.0364503i	$-0.988395\pi$
$467$	− 1.60450e8i	− 0.0729005i	0.999335	−	0.0364503i	$-0.0116051\pi$
$468$	0	0
$469$	− 5.82027e8i	− 0.260518i
$470$	0	0
$471$	1.70676e9	0.752660
$472$	0	0
$473$	9.63364e7	0.0418578
$474$	0	0
$475$	− 9.81835e8i	− 0.420350i
$476$	0	0
$477$	7.16602e8i	0.302318i
$478$	0	0
$479$	2.56265e9	1.06541	0.532704	−	0.846302i	$-0.321176\pi$
$479$	2.56265e9	1.06541	0.532704	+	0.846302i	$0.321176\pi$
$480$	0	0
$481$	−4.54479e9	−1.86211
$482$	0	0
$483$	− 1.81605e8i	− 0.0733355i
$484$	0	0
$485$	− 1.98315e9i	− 0.789330i
$486$	0	0
$487$	4.63335e9	1.81779	0.908895	−	0.417024i	$-0.136927\pi$
$487$	4.63335e9	1.81779	0.908895	+	0.417024i	$0.136927\pi$
$488$	0	0
$489$	1.82918e8	0.0707418
$490$	0	0
$491$	1.84615e9i	0.703854i	0.936028	+	0.351927i	$0.114473\pi$
$491$	1.84615e9i	0.703854i	−0.936028	+	0.351927i	$0.885527\pi$
$492$	0	0
$493$	1.00553e9i	0.377947i
$494$	0	0
$495$	−3.22398e8	−0.119474
$496$	0	0
$497$	6.31989e8	0.230920
$498$	0	0
$499$	− 5.12844e9i	− 1.84771i	−0.382746	−	0.923854i	$-0.625022\pi$
$499$	− 5.12844e9i	− 1.84771i	0.382746	−	0.923854i	$-0.374978\pi$
$500$	0	0
$501$	5.98436e9i	2.12611i
$502$	0	0
$503$	−2.60493e9	−0.912657	−0.456328	−	0.889811i	$-0.650836\pi$
$503$	−2.60493e9	−0.912657	−0.456328	+	0.889811i	$0.650836\pi$
$504$	0	0
$505$	4.07824e8	0.140914
$506$	0	0
$507$	− 1.29645e9i	− 0.441803i
$508$	0	0
$509$	− 5.90571e9i	− 1.98500i	−0.122261	−	0.992498i	$-0.539015\pi$
$509$	− 5.90571e9i	− 1.98500i	0.122261	−	0.992498i	$-0.460985\pi$
$510$	0	0
$511$	8.70257e8	0.288519
$512$	0	0
$513$	1.54569e9	0.505487
$514$	0	0
$515$	− 2.72691e9i	− 0.879721i
$516$	0	0
$517$	− 6.21050e9i	− 1.97656i
$518$	0	0
$519$	−1.44952e9	−0.455134
$520$	0	0
$521$	1.11569e9	0.345629	0.172815	−	0.984954i	$-0.444714\pi$
$521$	1.11569e9	0.345629	0.172815	+	0.984954i	$0.444714\pi$
$522$	0	0
$523$	4.12256e8i	0.126012i	0.998013	+	0.0630059i	$0.0200687\pi$
$523$	4.12256e8i	0.126012i	−0.998013	+	0.0630059i	$0.979931\pi$
$524$	0	0
$525$	5.61653e8i	0.169399i
$526$	0	0
$527$	−1.72076e9	−0.512135
$528$	0	0
$529$	−3.06560e9	−0.900370
$530$	0	0
$531$	− 2.77432e7i	− 0.00804129i
$532$	0	0
$533$	5.34355e9i	1.52857i
$534$	0	0
$535$	−2.35096e9	−0.663753
$536$	0	0
$537$	−3.90541e9	−1.08832
$538$	0	0
$539$	− 3.99642e9i	− 1.09929i
$540$	0	0
$541$	− 7.94914e8i	− 0.215839i	−0.994160	−	0.107919i	$-0.965581\pi$
$541$	− 7.94914e8i	− 0.215839i	0.994160	−	0.107919i	$-0.0344189\pi$
$542$	0	0
$543$	2.76147e9	0.740185
$544$	0	0
$545$	2.20636e9	0.583833
$546$	0	0
$547$	5.28117e9i	1.37967i	0.723967	+	0.689834i	$0.242317\pi$
$547$	5.28117e9i	1.37967i	−0.723967	+	0.689834i	$0.757683\pi$
$548$	0	0
$549$	− 6.07672e8i	− 0.156735i
$550$	0	0
$551$	−1.00636e9	−0.256285
$552$	0	0
$553$	6.13644e8	0.154305
$554$	0	0
$555$	− 3.60848e9i	− 0.895980i
$556$	0	0
$557$	5.63862e9i	1.38254i	0.722594	+	0.691272i	$0.242950\pi$
$557$	5.63862e9i	1.38254i	−0.722594	+	0.691272i	$0.757050\pi$
$558$	0	0
$559$	−1.77910e8	−0.0430783
$560$	0	0
$561$	4.48221e9	1.07182
$562$	0	0
$563$	7.06564e9i	1.66868i	0.551252	+	0.834338i	$0.314150\pi$
$563$	7.06564e9i	1.66868i	−0.551252	+	0.834338i	$0.685850\pi$
$564$	0	0
$565$	− 1.12058e9i	− 0.261380i
$566$	0	0
$567$	−1.06783e9	−0.246015
$568$	0	0
$569$	−4.40034e9	−1.00137	−0.500684	−	0.865630i	$-0.666918\pi$
$569$	−4.40034e9	−1.00137	−0.500684	+	0.865630i	$0.666918\pi$
$570$	0	0
$571$	4.60968e9i	1.03620i	0.855319	+	0.518101i	$0.173361\pi$
$571$	4.60968e9i	1.03620i	−0.855319	+	0.518101i	$0.826639\pi$
$572$	0	0
$573$	− 3.13923e9i	− 0.697079i
$574$	0	0
$575$	−1.04912e9	−0.230137
$576$	0	0
$577$	2.61376e9	0.566436	0.283218	−	0.959056i	$-0.408598\pi$
$577$	2.61376e9	0.566436	0.283218	+	0.959056i	$0.408598\pi$
$578$	0	0
$579$	− 3.28517e8i	− 0.0703368i
$580$	0	0
$581$	− 1.40771e9i	− 0.297781i
$582$	0	0
$583$	8.34925e9	1.74505
$584$	0	0
$585$	5.95390e8	0.122958
$586$	0	0
$587$	− 6.25292e9i	− 1.27600i	−0.770038	−	0.637998i	$-0.779763\pi$
$587$	− 6.25292e9i	− 1.27600i	0.770038	−	0.637998i	$-0.220237\pi$
$588$	0	0
$589$	− 1.72219e9i	− 0.347278i
$590$	0	0
$591$	6.82813e9	1.36065
$592$	0	0
$593$	6.05058e9	1.19153	0.595765	−	0.803158i	$-0.296849\pi$
$593$	6.05058e9	1.19153	0.595765	+	0.803158i	$0.296849\pi$
$594$	0	0
$595$	4.82357e8i	0.0938770i
$596$	0	0
$597$	9.06189e9i	1.74304i
$598$	0	0
$599$	−9.42695e9	−1.79216	−0.896081	−	0.443891i	$-0.853598\pi$
$599$	−9.42695e9	−1.79216	−0.896081	+	0.443891i	$0.853598\pi$
$600$	0	0
$601$	1.75834e9	0.330401	0.165200	−	0.986260i	$-0.447173\pi$
$601$	1.75834e9	0.330401	0.165200	+	0.986260i	$0.447173\pi$
$602$	0	0
$603$	1.31850e9i	0.244889i
$604$	0	0
$605$	9.21387e8i	0.169160i
$606$	0	0
$607$	−7.23873e9	−1.31372	−0.656859	−	0.754013i	$-0.728115\pi$
$607$	−7.23873e9	−1.31372	−0.656859	+	0.754013i	$0.728115\pi$
$608$	0	0
$609$	5.75683e8	0.103282
$610$	0	0
$611$	1.14693e10i	2.03419i
$612$	0	0
$613$	1.25590e8i	0.0220213i	0.999939	+	0.0110107i	$0.00350487\pi$
$613$	1.25590e8i	0.0220213i	−0.999939	+	0.0110107i	$0.996495\pi$
$614$	0	0
$615$	−4.24267e9	−0.735490
$616$	0	0
$617$	−1.80957e9	−0.310153	−0.155077	−	0.987902i	$-0.549562\pi$
$617$	−1.80957e9	−0.310153	−0.155077	+	0.987902i	$0.549562\pi$
$618$	0	0
$619$	9.74012e9i	1.65062i	0.564680	+	0.825310i	$0.309000\pi$
$619$	9.74012e9i	1.65062i	−0.564680	+	0.825310i	$0.691000\pi$
$620$	0	0
$621$	− 1.65160e9i	− 0.276749i
$622$	0	0
$623$	6.84674e8	0.113442
$624$	0	0
$625$	1.59122e9	0.260705
$626$	0	0
$627$	4.48592e9i	0.726800i
$628$	0	0
$629$	8.34100e9i	1.33641i
$630$	0	0
$631$	−3.98157e9	−0.630887	−0.315444	−	0.948944i	$-0.602153\pi$
$631$	−3.98157e9	−0.630887	−0.315444	+	0.948944i	$0.602153\pi$
$632$	0	0
$633$	5.78340e9	0.906296
$634$	0	0
$635$	− 1.06288e9i	− 0.164731i
$636$	0	0
$637$	7.38041e9i	1.13134i
$638$	0	0
$639$	−1.43168e9	−0.217067
$640$	0	0
$641$	3.99235e9	0.598723	0.299361	−	0.954140i	$-0.403226\pi$
$641$	3.99235e9	0.598723	0.299361	+	0.954140i	$0.403226\pi$
$642$	0	0
$643$	− 1.07939e10i	− 1.60118i	−0.599213	−	0.800589i	$-0.704520\pi$
$643$	− 1.07939e10i	− 1.60118i	0.599213	−	0.800589i	$-0.295480\pi$
$644$	0	0
$645$	− 1.41257e8i	− 0.0207277i
$646$	0	0
$647$	−2.19085e9	−0.318015	−0.159008	−	0.987277i	$-0.550829\pi$
$647$	−2.19085e9	−0.318015	−0.159008	+	0.987277i	$0.550829\pi$
$648$	0	0
$649$	−3.23241e8	−0.0464162
$650$	0	0
$651$	9.85168e8i	0.139951i
$652$	0	0
$653$	1.30043e9i	0.182763i	0.995816	+	0.0913817i	$0.0291283\pi$
$653$	1.30043e9i	0.182763i	−0.995816	+	0.0913817i	$0.970872\pi$
$654$	0	0
$655$	−2.39754e9	−0.333365
$656$	0	0
$657$	−1.97145e9	−0.271210
$658$	0	0
$659$	− 1.49570e9i	− 0.203585i	−0.994806	−	0.101793i	$-0.967542\pi$
$659$	− 1.49570e9i	− 0.203585i	0.994806	−	0.101793i	$-0.0324578\pi$
$660$	0	0
$661$	4.45984e9i	0.600640i	0.953839	+	0.300320i	$0.0970934\pi$
$661$	4.45984e9i	0.600640i	−0.953839	+	0.300320i	$0.902907\pi$
$662$	0	0
$663$	−8.27753e9	−1.10307
$664$	0	0
$665$	−4.82756e8	−0.0636578
$666$	0	0
$667$	1.07532e9i	0.140313i
$668$	0	0
$669$	− 1.58553e10i	− 2.04731i
$670$	0	0
$671$	−7.08010e9	−0.904712
$672$	0	0
$673$	−6.93326e9	−0.876768	−0.438384	−	0.898788i	$-0.644449\pi$
$673$	−6.93326e9	−0.876768	−0.438384	+	0.898788i	$0.644449\pi$
$674$	0	0
$675$	5.10794e9i	0.639267i
$676$	0	0
$677$	− 1.17696e10i	− 1.45781i	−0.684613	−	0.728907i	$-0.740029\pi$
$677$	− 1.17696e10i	− 1.45781i	0.684613	−	0.728907i	$-0.259971\pi$
$678$	0	0
$679$	2.62445e9	0.321732
$680$	0	0
$681$	−4.20104e9	−0.509732
$682$	0	0
$683$	1.18271e10i	1.42039i	0.704006	+	0.710194i	$0.251393\pi$
$683$	1.18271e10i	1.42039i	−0.704006	+	0.710194i	$0.748607\pi$
$684$	0	0
$685$	6.89627e9i	0.819779i
$686$	0	0
$687$	4.33518e9	0.510104
$688$	0	0
$689$	−1.54190e10	−1.79593
$690$	0	0
$691$	5.64930e9i	0.651361i	0.945480	+	0.325680i	$0.105593\pi$
$691$	5.64930e9i	0.651361i	−0.945480	+	0.325680i	$0.894407\pi$
$692$	0	0
$693$	− 4.26654e8i	− 0.0486978i
$694$	0	0
$695$	−6.70098e9	−0.757166
$696$	0	0
$697$	9.80695e9	1.09703
$698$	0	0
$699$	1.17588e10i	1.30225i
$700$	0	0
$701$	− 5.07474e9i	− 0.556417i	−0.960521	−	0.278208i	$-0.910259\pi$
$701$	− 5.07474e9i	− 0.556417i	0.960521	−	0.278208i	$-0.0897406\pi$
$702$	0	0
$703$	−8.34790e9	−0.906220
$704$	0	0
$705$	−9.10637e9	−0.978777
$706$	0	0
$707$	5.39705e8i	0.0574365i
$708$	0	0
$709$	− 1.79283e10i	− 1.88919i	−0.328232	−	0.944597i	$-0.606453\pi$
$709$	− 1.79283e10i	− 1.88919i	0.328232	−	0.944597i	$-0.393547\pi$
$710$	0	0
$711$	−1.39013e9	−0.145048
$712$	0	0
$713$	−1.84020e9	−0.190131
$714$	0	0
$715$	− 6.93700e9i	− 0.709742i
$716$	0	0
$717$	2.20005e9i	0.222903i
$718$	0	0
$719$	−1.11920e10	−1.12294	−0.561468	−	0.827498i	$-0.689763\pi$
$719$	−1.11920e10	−1.12294	−0.561468	+	0.827498i	$0.689763\pi$
$720$	0	0
$721$	3.60873e9	0.358575
$722$	0	0
$723$	− 5.19739e9i	− 0.511448i
$724$	0	0
$725$	− 3.32566e9i	− 0.324112i
$726$	0	0
$727$	1.52463e10	1.47161	0.735805	−	0.677193i	$-0.236804\pi$
$727$	1.52463e10	1.47161	0.735805	+	0.677193i	$0.236804\pi$
$728$	0	0
$729$	−7.62535e9	−0.728976
$730$	0	0
$731$	3.26516e8i	0.0309167i
$732$	0	0
$733$	1.33611e10i	1.25308i	0.779389	+	0.626541i	$0.215530\pi$
$733$	1.33611e10i	1.25308i	−0.779389	+	0.626541i	$0.784470\pi$
$734$	0	0
$735$	−5.85990e9	−0.544358
$736$	0	0
$737$	1.53621e10	1.41356
$738$	0	0
$739$	1.12829e9i	0.102841i	0.998677	+	0.0514203i	$0.0163748\pi$
$739$	1.12829e9i	0.102841i	−0.998677	+	0.0514203i	$0.983625\pi$
$740$	0	0
$741$	− 8.28439e9i	− 0.747991i
$742$	0	0
$743$	1.70680e7	0.00152659	0.000763295	−	1.00000i	$-0.499757\pi$
$743$	1.70680e7	0.00152659	0.000763295		1.00000i	$0.499757\pi$
$744$	0	0
$745$	2.86826e9	0.254139
$746$	0	0
$747$	3.18898e9i	0.279917i
$748$	0	0
$749$	− 3.11120e9i	− 0.270547i
$750$	0	0
$751$	−1.70601e10	−1.46974	−0.734871	−	0.678207i	$-0.762757\pi$
$751$	−1.70601e10	−1.46974	−0.734871	+	0.678207i	$0.762757\pi$
$752$	0	0
$753$	−7.40648e9	−0.632163
$754$	0	0
$755$	8.05338e9i	0.681026i
$756$	0	0
$757$	− 1.22318e9i	− 0.102484i	−0.998686	−	0.0512418i	$-0.983682\pi$
$757$	− 1.22318e9i	− 0.102484i	0.998686	−	0.0512418i	$-0.0163179\pi$
$758$	0	0
$759$	4.79332e9	0.397915
$760$	0	0
$761$	−7.82094e9	−0.643299	−0.321649	−	0.946859i	$-0.604237\pi$
$761$	−7.82094e9	−0.643299	−0.321649	+	0.946859i	$0.604237\pi$
$762$	0	0
$763$	2.91984e9i	0.237971i
$764$	0	0
$765$	− 1.09271e9i	− 0.0882452i
$766$	0	0
$767$	5.96946e8	0.0477696
$768$	0	0
$769$	−8.17810e9	−0.648500	−0.324250	−	0.945971i	$-0.605112\pi$
$769$	−8.17810e9	−0.648500	−0.324250	+	0.945971i	$0.605112\pi$
$770$	0	0
$771$	− 1.47005e10i	− 1.15516i
$772$	0	0
$773$	1.07643e10i	0.838218i	0.907936	+	0.419109i	$0.137657\pi$
$773$	1.07643e10i	0.838218i	−0.907936	+	0.419109i	$0.862343\pi$
$774$	0	0
$775$	5.69122e9	0.439187
$776$	0	0
$777$	4.77537e9	0.365203
$778$	0	0
$779$	9.81507e9i	0.743896i
$780$	0	0
$781$	1.66808e10i	1.25296i
$782$	0	0
$783$	5.23553e9	0.389758
$784$	0	0
$785$	−4.84792e9	−0.357694
$786$	0	0
$787$	− 1.74535e10i	− 1.27635i	−0.769890	−	0.638177i	$-0.779689\pi$
$787$	− 1.74535e10i	− 1.27635i	0.769890	−	0.638177i	$-0.220311\pi$
$788$	0	0
$789$	− 1.23778e10i	− 0.897168i
$790$	0	0
$791$	1.48295e9	0.106539
$792$	0	0
$793$	1.30752e10	0.931091
$794$	0	0
$795$	− 1.22424e10i	− 0.864135i
$796$	0	0
$797$	− 1.19624e8i	− 0.00836976i	−0.999991	−	0.00418488i	$-0.998668\pi$
$797$	− 1.19624e8i	− 0.00836976i	0.999991	−	0.00418488i	$-0.00133209\pi$
$798$	0	0
$799$	2.10494e10	1.45991
$800$	0	0
$801$	−1.55103e9	−0.106637
$802$	0	0
$803$	2.29697e10i	1.56549i
$804$	0	0
$805$	5.15837e8i	0.0348520i
$806$	0	0
$807$	4.26607e9	0.285740
$808$	0	0
$809$	3.68783e9	0.244879	0.122439	−	0.992476i	$-0.460928\pi$
$809$	3.68783e9	0.244879	0.122439	+	0.992476i	$0.460928\pi$
$810$	0	0
$811$	− 4.94079e9i	− 0.325254i	−0.986688	−	0.162627i	$-0.948003\pi$
$811$	− 4.94079e9i	− 0.325254i	0.986688	−	0.162627i	$-0.0519968\pi$
$812$	0	0
$813$	− 1.60793e10i	− 1.04942i
$814$	0	0
$815$	−5.19567e8	−0.0336194
$816$	0	0
$817$	−3.26786e8	−0.0209646
$818$	0	0
$819$	7.87925e8i	0.0501177i
$820$	0	0
$821$	9.73545e9i	0.613981i	0.951713	+	0.306990i	$0.0993219\pi$
$821$	9.73545e9i	0.613981i	−0.951713	+	0.306990i	$0.900678\pi$
$822$	0	0
$823$	8.04183e9	0.502869	0.251435	−	0.967874i	$-0.419098\pi$
$823$	8.04183e9	0.502869	0.251435	+	0.967874i	$0.419098\pi$
$824$	0	0
$825$	−1.48243e10	−0.919151
$826$	0	0
$827$	2.48354e10i	1.52687i	0.645885	+	0.763435i	$0.276488\pi$
$827$	2.48354e10i	1.52687i	−0.645885	+	0.763435i	$0.723512\pi$
$828$	0	0
$829$	− 9.35472e8i	− 0.0570282i	−0.999593	−	0.0285141i	$-0.990922\pi$
$829$	− 9.35472e8i	− 0.0570282i	0.999593	−	0.0285141i	$-0.00907756\pi$
$830$	0	0
$831$	2.15023e10	1.29981
$832$	0	0
$833$	1.35452e10	0.811946
$834$	0	0
$835$	− 1.69981e10i	− 1.01041i
$836$	0	0
$837$	8.95958e9i	0.528139i
$838$	0	0
$839$	−9.51816e9	−0.556399	−0.278199	−	0.960523i	$-0.589738\pi$
$839$	−9.51816e9	−0.556399	−0.278199	+	0.960523i	$0.589738\pi$
$840$	0	0
$841$	1.38411e10	0.802391
$842$	0	0
$843$	− 1.93106e10i	− 1.11019i
$844$	0	0
$845$	3.68248e9i	0.209962i
$846$	0	0
$847$	−1.21934e9	−0.0689499
$848$	0	0
$849$	−3.82105e9	−0.214292
$850$	0	0
$851$	8.91995e9i	0.496146i
$852$	0	0
$853$	1.66989e10i	0.921228i	0.887601	+	0.460614i	$0.152371\pi$
$853$	1.66989e10i	0.921228i	−0.887601	+	0.460614i	$0.847629\pi$
$854$	0	0
$855$	1.09362e9	0.0598389
$856$	0	0
$857$	1.60352e10	0.870243	0.435121	−	0.900372i	$-0.356705\pi$
$857$	1.60352e10	0.870243	0.435121	+	0.900372i	$0.356705\pi$
$858$	0	0
$859$	− 2.41808e10i	− 1.30165i	−0.759227	−	0.650826i	$-0.774423\pi$
$859$	− 2.41808e10i	− 1.30165i	0.759227	−	0.650826i	$-0.225577\pi$
$860$	0	0
$861$	− 5.61465e9i	− 0.299786i
$862$	0	0
$863$	6.03439e9	0.319592	0.159796	−	0.987150i	$-0.448916\pi$
$863$	6.03439e9	0.319592	0.159796	+	0.987150i	$0.448916\pi$
$864$	0	0
$865$	4.11727e9	0.216298
$866$	0	0
$867$	− 5.82445e9i	− 0.303520i
$868$	0	0
$869$	1.61966e10i	0.837249i
$870$	0	0
$871$	−2.83700e10	−1.45478
$872$	0	0
$873$	−5.94533e9	−0.302431
$874$	0	0
$875$	− 3.78340e9i	− 0.190921i
$876$	0	0
$877$	− 3.87528e9i	− 0.194001i	−0.995284	−	0.0970007i	$-0.969075\pi$
$877$	− 3.87528e9i	− 0.194001i	0.995284	−	0.0970007i	$-0.0309249\pi$
$878$	0	0
$879$	2.25035e10	1.11761
$880$	0	0
$881$	3.37191e10	1.66135	0.830673	−	0.556760i	$-0.187956\pi$
$881$	3.37191e10	1.66135	0.830673	+	0.556760i	$0.187956\pi$
$882$	0	0
$883$	− 3.25227e10i	− 1.58973i	−0.606784	−	0.794867i	$-0.707541\pi$
$883$	− 3.25227e10i	− 1.58973i	0.606784	−	0.794867i	$-0.292459\pi$
$884$	0	0
$885$	4.73964e8i	0.0229850i
$886$	0	0
$887$	1.17993e10	0.567704	0.283852	−	0.958868i	$-0.408388\pi$
$887$	1.17993e10	0.567704	0.283852	+	0.958868i	$0.408388\pi$
$888$	0	0
$889$	1.40659e9	0.0671444
$890$	0	0
$891$	− 2.81844e10i	− 1.33486i
$892$	0	0
$893$	2.10668e10i	0.989963i
$894$	0	0
$895$	1.10930e10	0.517214
$896$	0	0
$897$	−8.85208e9	−0.409517
$898$	0	0
$899$	− 5.83338e9i	− 0.267770i
$900$	0	0
$901$	2.82983e10i	1.28892i
$902$	0	0
$903$	1.86936e8	0.00844862
$904$	0	0
$905$	−7.84375e9	−0.351766
$906$	0	0
$907$	− 1.60172e10i	− 0.712788i	−0.934336	−	0.356394i	$-0.884006\pi$
$907$	− 1.60172e10i	− 0.712788i	0.934336	−	0.356394i	$-0.115994\pi$
$908$	0	0
$909$	− 1.22263e9i	− 0.0539908i
$910$	0	0
$911$	8.42391e9	0.369147	0.184574	−	0.982819i	$-0.440910\pi$
$911$	8.42391e9	0.369147	0.184574	+	0.982819i	$0.440910\pi$
$912$	0	0
$913$	3.71554e10	1.61575
$914$	0	0
$915$	1.03815e10i	0.448007i
$916$	0	0
$917$	− 3.17284e9i	− 0.135880i
$918$	0	0
$919$	4.03951e10	1.71682	0.858408	−	0.512967i	$-0.171454\pi$
$919$	4.03951e10	1.71682	0.858408	+	0.512967i	$0.171454\pi$
$920$	0	0
$921$	−2.60734e10	−1.09974
$922$	0	0
$923$	− 3.08053e10i	− 1.28950i
$924$	0	0
$925$	− 2.75868e10i	− 1.14606i
$926$	0	0
$927$	−8.17507e9	−0.337064
$928$	0	0
$929$	6.24836e9	0.255688	0.127844	−	0.991794i	$-0.459194\pi$
$929$	6.24836e9	0.255688	0.127844	+	0.991794i	$0.459194\pi$
$930$	0	0
$931$	1.35564e10i	0.550580i
$932$	0	0
$933$	− 7.01346e9i	− 0.282713i
$934$	0	0
$935$	−1.27314e10	−0.509372
$936$	0	0
$937$	−3.33715e10	−1.32522	−0.662608	−	0.748966i	$-0.730551\pi$
$937$	−3.33715e10	−1.32522	−0.662608	+	0.748966i	$0.730551\pi$
$938$	0	0
$939$	− 9.39403e9i	− 0.370273i
$940$	0	0
$941$	− 1.63777e10i	− 0.640749i	−0.947291	−	0.320375i	$-0.896191\pi$
$941$	− 1.63777e10i	− 0.640749i	0.947291	−	0.320375i	$-0.103809\pi$
$942$	0	0
$943$	1.04877e10	0.407275
$944$	0	0
$945$	2.51151e9	0.0968107
$946$	0	0
$947$	9.07481e9i	0.347226i	0.984814	+	0.173613i	$0.0555442\pi$
$947$	9.07481e9i	0.347226i	−0.984814	+	0.173613i	$0.944456\pi$
$948$	0	0
$949$	− 4.24194e10i	− 1.61114i
$950$	0	0
$951$	3.94063e10	1.48571
$952$	0	0
$953$	1.87187e10	0.700569	0.350284	−	0.936643i	$-0.386085\pi$
$953$	1.87187e10	0.700569	0.350284	+	0.936643i	$0.386085\pi$
$954$	0	0
$955$	8.91676e9i	0.331280i
$956$	0	0
$957$	1.51947e10i	0.560401i
$958$	0	0
$959$	−9.12636e9	−0.334143
$960$	0	0
$961$	−1.75299e10	−0.637160
$962$	0	0
$963$	7.04800e9i	0.254316i
$964$	0	0
$965$	9.33128e8i	0.0334269i
$966$	0	0
$967$	−2.29498e10	−0.816182	−0.408091	−	0.912941i	$-0.633805\pi$
$967$	−2.29498e10	−0.816182	−0.408091	+	0.912941i	$0.633805\pi$
$968$	0	0
$969$	−1.52042e10	−0.536823
$970$	0	0
$971$	− 4.95632e9i	− 0.173737i	−0.996220	−	0.0868684i	$-0.972314\pi$
$971$	− 4.95632e9i	− 0.173737i	0.996220	−	0.0868684i	$-0.0276860\pi$
$972$	0	0
$973$	− 8.86791e9i	− 0.308622i
$974$	0	0
$975$	2.73769e10	0.945951
$976$	0	0
$977$	−3.14703e10	−1.07962	−0.539809	−	0.841787i	$-0.681504\pi$
$977$	−3.14703e10	−1.07962	−0.539809	+	0.841787i	$0.681504\pi$
$978$	0	0
$979$	1.80714e10i	0.615533i
$980$	0	0
$981$	− 6.61450e9i	− 0.223695i
$982$	0	0
$983$	−2.75291e10	−0.924389	−0.462195	−	0.886778i	$-0.652938\pi$
$983$	−2.75291e10	−0.924389	−0.462195	+	0.886778i	$0.652938\pi$
$984$	0	0
$985$	−1.93948e10	−0.646635
$986$	0	0
$987$	− 1.20512e10i	− 0.398950i
$988$	0	0
$989$	3.49179e8i	0.0114779i
$990$	0	0
$991$	5.41201e10	1.76645	0.883224	−	0.468951i	$-0.155368\pi$
$991$	5.41201e10	1.76645	0.883224	+	0.468951i	$0.155368\pi$
$992$	0	0
$993$	6.10901e9	0.197992
$994$	0	0
$995$	− 2.57396e10i	− 0.828365i
$996$	0	0
$997$	1.70649e10i	0.545343i	0.962107	+	0.272672i	$0.0879072\pi$
$997$	1.70649e10i	0.545343i	−0.962107	+	0.272672i	$0.912093\pi$
$998$	0	0
$999$	4.34295e10	1.37818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.b.l.129.5		6
4.3	odd	2		256.8.b.k.129.2		6
8.3	odd	2		256.8.b.k.129.5		6
8.5	even	2	inner	256.8.b.l.129.2		6
16.3	odd	4		128.8.a.b.1.3	yes	3
16.5	even	4		128.8.a.a.1.3	✓	3
16.11	odd	4		128.8.a.c.1.1	yes	3
16.13	even	4		128.8.a.d.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.8.a.a.1.3	✓	3	16.5	even	4
128.8.a.b.1.3	yes	3	16.3	odd	4
128.8.a.c.1.1	yes	3	16.11	odd	4
128.8.a.d.1.1	yes	3	16.13	even	4
256.8.b.k.129.2		6	4.3	odd	2
256.8.b.k.129.5		6	8.3	odd	2
256.8.b.l.129.2		6	8.5	even	2	inner
256.8.b.l.129.5		6	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+51.2165i q^{3} -145.477i q^{5} +192.521 q^{7} -436.129 q^{9} +O(q^{10})\)	\(q+51.2165i q^{3} -145.477i q^{5} +192.521 q^{7} -436.129 q^{9} +5081.41i q^{11} -9384.12i q^{13} +7450.81 q^{15} -17222.6 q^{17} -17236.8i q^{19} +9860.23i q^{21} -18418.0 q^{23} +56961.5 q^{25} +89673.5i q^{27} -58384.4i q^{29} +99913.3 q^{31} -260252. q^{33} -28007.3i q^{35} -484307. i q^{37} +480622. q^{39} -569425. q^{41} -18958.6i q^{43} +63446.6i q^{45} -1.22220e6 q^{47} -786479. q^{49} -882079. i q^{51} -1.64310e6i q^{53} +739227. q^{55} +882809. q^{57} +63612.4i q^{59} +1.39333e6i q^{61} -83963.7 q^{63} -1.36517e6 q^{65} -3.02319e6i q^{67} -943304. i q^{69} +3.28271e6 q^{71} +4.52034e6 q^{73} +2.91737e6i q^{75} +978276. i q^{77} +3.18742e6 q^{79} -5.54657e6 q^{81} -7.31201e6i q^{83} +2.50548e6i q^{85} +2.99024e6 q^{87} +3.55637e6 q^{89} -1.80664e6i q^{91} +5.11721e6i q^{93} -2.50756e6 q^{95} +1.36320e7 q^{97} -2.21615e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24 q^{7} + 106 q^{9}+O(q^{10}) \) 6 * q + 24 * q^7 + 106 * q^9	\( 6 q + 24 q^{7} + 106 q^{9} - 8232 q^{15} + 1076 q^{17} + 86632 q^{23} + 81206 q^{25} - 313856 q^{31} - 367000 q^{33} + 1243368 q^{39} + 1819396 q^{41} - 1705904 q^{47} - 3273066 q^{49} + 7411352 q^{55} + 6490536 q^{57} - 5088856 q^{63} - 9549544 q^{65} + 19882936 q^{71} + 11588940 q^{73} - 2009008 q^{79} - 12889762 q^{81} + 36376264 q^{87} + 9662700 q^{89} + 30260056 q^{95} - 10501388 q^{97}+O(q^{100}) \) 6 * q + 24 * q^7 + 106 * q^9 - 8232 * q^15 + 1076 * q^17 + 86632 * q^23 + 81206 * q^25 - 313856 * q^31 - 367000 * q^33 + 1243368 * q^39 + 1819396 * q^41 - 1705904 * q^47 - 3273066 * q^49 + 7411352 * q^55 + 6490536 * q^57 - 5088856 * q^63 - 9549544 * q^65 + 19882936 * q^71 + 11588940 * q^73 - 2009008 * q^79 - 12889762 * q^81 + 36376264 * q^87 + 9662700 * q^89 + 30260056 * q^95 - 10501388 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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