LMFDB - Embedded newform 256.4.b.i.129.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,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, 4, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
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:	$15.1044889615$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.4.b.i.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.92820i q^{3} +15.8564i q^{5} +17.8564 q^{7} +2.71281 q^{9} +O(q^{10})$	$q+4.92820i q^{3} +15.8564i q^{5} +17.8564 q^{7} +2.71281 q^{9} +52.9282i q^{11} -8.43078i q^{13} -78.1436 q^{15} +129.138 q^{17} -50.4974i q^{19} +88.0000i q^{21} -128.708 q^{23} -126.426 q^{25} +146.431i q^{27} -111.282i q^{29} -302.851 q^{31} -260.841 q^{33} +283.138i q^{35} -182.995i q^{37} +41.5486 q^{39} +94.5744 q^{41} +184.641i q^{43} +43.0155i q^{45} +296.841 q^{47} -24.1487 q^{49} +636.420i q^{51} -102.995i q^{53} -839.251 q^{55} +248.862 q^{57} -93.3693i q^{59} +338.974i q^{61} +48.4411 q^{63} +133.682 q^{65} -489.041i q^{67} -634.297i q^{69} +86.9845 q^{71} +154.267 q^{73} -623.051i q^{75} +945.108i q^{77} -449.415 q^{79} -648.395 q^{81} -383.933i q^{83} +2047.67i q^{85} +548.420 q^{87} +517.672 q^{89} -150.543i q^{91} -1492.51i q^{93} +800.708 q^{95} +1739.39 q^{97} +143.584i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7} - 100 q^{9}+O(q^{10}) $ 4 * q + 16 * q^7 - 100 * q^9	$ 4 q + 16 q^{7} - 100 q^{9} - 368 q^{15} + 184 q^{17} - 16 q^{23} - 284 q^{25} - 768 q^{31} + 176 q^{33} - 1552 q^{39} + 600 q^{41} - 32 q^{47} - 540 q^{49} - 752 q^{55} + 1328 q^{57} + 1136 q^{63} - 1904 q^{65} - 816 q^{71} - 824 q^{73} - 800 q^{79} - 44 q^{81} + 1584 q^{87} - 1144 q^{89} + 2704 q^{95} + 4408 q^{97}+O(q^{100}) $ 4 * q + 16 * q^7 - 100 * q^9 - 368 * q^15 + 184 * q^17 - 16 * q^23 - 284 * q^25 - 768 * q^31 + 176 * q^33 - 1552 * q^39 + 600 * q^41 - 32 * q^47 - 540 * q^49 - 752 * q^55 + 1328 * q^57 + 1136 * q^63 - 1904 * q^65 - 816 * q^71 - 824 * q^73 - 800 * q^79 - 44 * q^81 + 1584 * q^87 - 1144 * q^89 + 2704 * q^95 + 4408 * 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
$3$	4.92820i	0.948433i	0.880408	+	0.474217i	$0.157269\pi$
$3$	4.92820i	0.948433i	−0.880408	+	0.474217i	$0.842731\pi$
$4$	0	0
$5$	15.8564i	1.41824i	0.705088	+	0.709120i	$0.250908\pi$
$5$	15.8564i	1.41824i	−0.705088	+	0.709120i	$0.749092\pi$
$6$	0	0
$7$	17.8564	0.964155	0.482078	−	0.876128i	$-0.339882\pi$
$7$	17.8564	0.964155	0.482078	+	0.876128i	$0.339882\pi$
$8$	0	0
$9$	2.71281	0.100475
$10$	0	0
$11$	52.9282i	1.45077i	0.688344	+	0.725384i	$0.258338\pi$
$11$	52.9282i	1.45077i	−0.688344	+	0.725384i	$0.741662\pi$
$12$	0	0
$13$	− 8.43078i	− 0.179868i	−0.995948	−	0.0899338i	$-0.971334\pi$
$13$	− 8.43078i	− 0.179868i	0.995948	−	0.0899338i	$-0.0286655\pi$
$14$	0	0
$15$	−78.1436	−1.34511
$16$	0	0
$17$	129.138	1.84239	0.921196	−	0.389098i	$-0.127213\pi$
$17$	129.138	1.84239	0.921196	+	0.389098i	$0.127213\pi$
$18$	0	0
$19$	− 50.4974i	− 0.609732i	−0.952395	−	0.304866i	$-0.901388\pi$
$19$	− 50.4974i	− 0.609732i	0.952395	−	0.304866i	$-0.0986116\pi$
$20$	0	0
$21$	88.0000i	0.914437i
$22$	0	0
$23$	−128.708	−1.16684	−0.583422	−	0.812169i	$-0.698287\pi$
$23$	−128.708	−1.16684	−0.583422	+	0.812169i	$0.698287\pi$
$24$	0	0
$25$	−126.426	−1.01141
$26$	0	0
$27$	146.431i	1.04373i
$28$	0	0
$29$	− 111.282i	− 0.712571i	−0.934377	−	0.356285i	$-0.884043\pi$
$29$	− 111.282i	− 0.712571i	0.934377	−	0.356285i	$-0.115957\pi$
$30$	0	0
$31$	−302.851	−1.75464	−0.877318	−	0.479910i	$-0.840669\pi$
$31$	−302.851	−1.75464	−0.877318	+	0.479910i	$0.840669\pi$
$32$	0	0
$33$	−260.841	−1.37596
$34$	0	0
$35$	283.138i	1.36740i
$36$	0	0
$37$	− 182.995i	− 0.813086i	−0.913632	−	0.406543i	$-0.866734\pi$
$37$	− 182.995i	− 0.813086i	0.913632	−	0.406543i	$-0.133266\pi$
$38$	0	0
$39$	41.5486	0.170592
$40$	0	0
$41$	94.5744	0.360245	0.180122	−	0.983644i	$-0.442351\pi$
$41$	94.5744	0.360245	0.180122	+	0.983644i	$0.442351\pi$
$42$	0	0
$43$	184.641i	0.654825i	0.944881	+	0.327413i	$0.106177\pi$
$43$	184.641i	0.654825i	−0.944881	+	0.327413i	$0.893823\pi$
$44$	0	0
$45$	43.0155i	0.142497i
$46$	0	0
$47$	296.841	0.921249	0.460624	−	0.887595i	$-0.347625\pi$
$47$	296.841	0.921249	0.460624	+	0.887595i	$0.347625\pi$
$48$	0	0
$49$	−24.1487	−0.0704045
$50$	0	0
$51$	636.420i	1.74739i
$52$	0	0
$53$	− 102.995i	− 0.266933i	−0.991053	−	0.133466i	$-0.957389\pi$
$53$	− 102.995i	− 0.266933i	0.991053	−	0.133466i	$-0.0426108\pi$
$54$	0	0
$55$	−839.251	−2.05754
$56$	0	0
$57$	248.862	0.578290
$58$	0	0
$59$	− 93.3693i	− 0.206028i	−0.994680	−	0.103014i	$-0.967151\pi$
$59$	− 93.3693i	− 0.206028i	0.994680	−	0.103014i	$-0.0328486\pi$
$60$	0	0
$61$	338.974i	0.711495i	0.934582	+	0.355748i	$0.115774\pi$
$61$	338.974i	0.711495i	−0.934582	+	0.355748i	$0.884226\pi$
$62$	0	0
$63$	48.4411	0.0968731
$64$	0	0
$65$	133.682	0.255095
$66$	0	0
$67$	− 489.041i	− 0.891729i	−0.895100	−	0.445865i	$-0.852896\pi$
$67$	− 489.041i	− 0.891729i	0.895100	−	0.445865i	$-0.147104\pi$
$68$	0	0
$69$	− 634.297i	− 1.10667i
$70$	0	0
$71$	86.9845	0.145397	0.0726983	−	0.997354i	$-0.476839\pi$
$71$	86.9845	0.145397	0.0726983	+	0.997354i	$0.476839\pi$
$72$	0	0
$73$	154.267	0.247336	0.123668	−	0.992324i	$-0.460534\pi$
$73$	154.267	0.247336	0.123668	+	0.992324i	$0.460534\pi$
$74$	0	0
$75$	− 623.051i	− 0.959250i
$76$	0	0
$77$	945.108i	1.39877i
$78$	0	0
$79$	−449.415	−0.640040	−0.320020	−	0.947411i	$-0.603690\pi$
$79$	−449.415	−0.640040	−0.320020	+	0.947411i	$0.603690\pi$
$80$	0	0
$81$	−648.395	−0.889430
$82$	0	0
$83$	− 383.933i	− 0.507737i	−0.967239	−	0.253868i	$-0.918297\pi$
$83$	− 383.933i	− 0.507737i	0.967239	−	0.253868i	$-0.0817030\pi$
$84$	0	0
$85$	2047.67i	2.61295i
$86$	0	0
$87$	548.420	0.675826
$88$	0	0
$89$	517.672	0.616551	0.308276	−	0.951297i	$-0.400248\pi$
$89$	517.672	0.616551	0.308276	+	0.951297i	$0.400248\pi$
$90$	0	0
$91$	− 150.543i	− 0.173420i
$92$	0	0
$93$	− 1492.51i	− 1.66415i
$94$	0	0
$95$	800.708	0.864746
$96$	0	0
$97$	1739.39	1.82071	0.910355	−	0.413829i	$-0.135809\pi$
$97$	1739.39	1.82071	0.910355	+	0.413829i	$0.135809\pi$
$98$	0	0
$99$	143.584i	0.145765i
$100$	0	0
$101$	− 1158.99i	− 1.14182i	−0.821011	−	0.570912i	$-0.806590\pi$
$101$	− 1158.99i	− 1.14182i	0.821011	−	0.570912i	$-0.193410\pi$
$102$	0	0
$103$	1635.01	1.56410	0.782048	−	0.623218i	$-0.214175\pi$
$103$	1635.01	1.56410	0.782048	+	0.623218i	$0.214175\pi$
$104$	0	0
$105$	−1395.36	−1.29689
$106$	0	0
$107$	282.589i	0.255317i	0.991818	+	0.127659i	$0.0407462\pi$
$107$	282.589i	0.255317i	−0.991818	+	0.127659i	$0.959254\pi$
$108$	0	0
$109$	− 1004.98i	− 0.883120i	−0.897232	−	0.441560i	$-0.854425\pi$
$109$	− 1004.98i	− 0.883120i	0.897232	−	0.441560i	$-0.145575\pi$
$110$	0	0
$111$	901.836	0.771157
$112$	0	0
$113$	−567.108	−0.472115	−0.236057	−	0.971739i	$-0.575855\pi$
$113$	−567.108	−0.472115	−0.236057	+	0.971739i	$0.575855\pi$
$114$	0	0
$115$	− 2040.84i	− 1.65486i
$116$	0	0
$117$	− 22.8711i	− 0.0180721i
$118$	0	0
$119$	2305.95	1.77635
$120$	0	0
$121$	−1470.39	−1.10473
$122$	0	0
$123$	466.082i	0.341668i
$124$	0	0
$125$	− 22.6053i	− 0.0161750i
$126$	0	0
$127$	−496.616	−0.346988	−0.173494	−	0.984835i	$-0.555506\pi$
$127$	−496.616	−0.346988	−0.173494	+	0.984835i	$0.555506\pi$
$128$	0	0
$129$	−909.948	−0.621058
$130$	0	0
$131$	− 1658.72i	− 1.10628i	−0.833087	−	0.553142i	$-0.813429\pi$
$131$	− 1658.72i	− 1.10628i	0.833087	−	0.553142i	$-0.186571\pi$
$132$	0	0
$133$	− 901.703i	− 0.587876i
$134$	0	0
$135$	−2321.87	−1.48025
$136$	0	0
$137$	432.872	0.269947	0.134974	−	0.990849i	$-0.456905\pi$
$137$	432.872	0.269947	0.134974	+	0.990849i	$0.456905\pi$
$138$	0	0
$139$	− 146.148i	− 0.0891809i	−0.999005	−	0.0445904i	$-0.985802\pi$
$139$	− 146.148i	− 0.0891809i	0.999005	−	0.0445904i	$-0.0141983\pi$
$140$	0	0
$141$	1462.89i	0.873743i
$142$	0	0
$143$	446.226	0.260946
$144$	0	0
$145$	1764.53	1.01060
$146$	0	0
$147$	− 119.010i	− 0.0667740i
$148$	0	0
$149$	3556.29i	1.95532i	0.210198	+	0.977659i	$0.432589\pi$
$149$	3556.29i	1.95532i	−0.210198	+	0.977659i	$0.567411\pi$
$150$	0	0
$151$	1320.32	0.711562	0.355781	−	0.934569i	$-0.384215\pi$
$151$	1320.32	0.711562	0.355781	+	0.934569i	$0.384215\pi$
$152$	0	0
$153$	350.328	0.185114
$154$	0	0
$155$	− 4802.13i	− 2.48849i
$156$	0	0
$157$	− 2040.43i	− 1.03722i	−0.855010	−	0.518612i	$-0.826449\pi$
$157$	− 2040.43i	− 1.03722i	0.855010	−	0.518612i	$-0.173551\pi$
$158$	0	0
$159$	507.580	0.253168
$160$	0	0
$161$	−2298.26	−1.12502
$162$	0	0
$163$	2392.21i	1.14952i	0.818321	+	0.574762i	$0.194905\pi$
$163$	2392.21i	1.14952i	−0.818321	+	0.574762i	$0.805095\pi$
$164$	0	0
$165$	− 4136.00i	− 1.95144i
$166$	0	0
$167$	−2068.34	−0.958400	−0.479200	−	0.877706i	$-0.659073\pi$
$167$	−2068.34	−0.958400	−0.479200	+	0.877706i	$0.659073\pi$
$168$	0	0
$169$	2125.92	0.967648
$170$	0	0
$171$	− 136.990i	− 0.0612625i
$172$	0	0
$173$	2441.87i	1.07313i	0.843859	+	0.536566i	$0.180279\pi$
$173$	2441.87i	1.07313i	−0.843859	+	0.536566i	$0.819721\pi$
$174$	0	0
$175$	−2257.51	−0.975152
$176$	0	0
$177$	460.143	0.195404
$178$	0	0
$179$	2430.84i	1.01502i	0.861645	+	0.507512i	$0.169435\pi$
$179$	2430.84i	1.01502i	−0.861645	+	0.507512i	$0.830565\pi$
$180$	0	0
$181$	1928.96i	0.792148i	0.918219	+	0.396074i	$0.129628\pi$
$181$	1928.96i	0.792148i	−0.918219	+	0.396074i	$0.870372\pi$
$182$	0	0
$183$	−1670.53	−0.674806
$184$	0	0
$185$	2901.64	1.15315
$186$	0	0
$187$	6835.07i	2.67288i
$188$	0	0
$189$	2614.73i	1.00631i
$190$	0	0
$191$	4087.34	1.54843	0.774214	−	0.632924i	$-0.218145\pi$
$191$	4087.34	1.54843	0.774214	+	0.632924i	$0.218145\pi$
$192$	0	0
$193$	−1156.52	−0.431339	−0.215669	−	0.976466i	$-0.569193\pi$
$193$	−1156.52	−0.431339	−0.215669	+	0.976466i	$0.569193\pi$
$194$	0	0
$195$	658.811i	0.241941i
$196$	0	0
$197$	− 2658.44i	− 0.961452i	−0.876871	−	0.480726i	$-0.840373\pi$
$197$	− 2658.44i	− 0.961452i	0.876871	−	0.480726i	$-0.159627\pi$
$198$	0	0
$199$	742.369	0.264448	0.132224	−	0.991220i	$-0.457788\pi$
$199$	742.369	0.264448	0.132224	+	0.991220i	$0.457788\pi$
$200$	0	0
$201$	2410.09	0.845745
$202$	0	0
$203$	− 1987.10i	− 0.687029i
$204$	0	0
$205$	1499.61i	0.510914i
$206$	0	0
$207$	−349.160	−0.117238
$208$	0	0
$209$	2672.74	0.884580
$210$	0	0
$211$	3838.30i	1.25232i	0.779694	+	0.626160i	$0.215374\pi$
$211$	3838.30i	1.25232i	−0.779694	+	0.626160i	$0.784626\pi$
$212$	0	0
$213$	428.677i	0.137899i
$214$	0	0
$215$	−2927.74	−0.928700
$216$	0	0
$217$	−5407.84	−1.69174
$218$	0	0
$219$	760.257i	0.234582i
$220$	0	0
$221$	− 1088.74i	− 0.331387i
$222$	0	0
$223$	−3418.34	−1.02650	−0.513249	−	0.858240i	$-0.671558\pi$
$223$	−3418.34	−1.02650	−0.513249	+	0.858240i	$0.671558\pi$
$224$	0	0
$225$	−342.969	−0.101620
$226$	0	0
$227$	− 515.729i	− 0.150793i	−0.997154	−	0.0753967i	$-0.975978\pi$
$227$	− 515.729i	− 0.150793i	0.997154	−	0.0753967i	$-0.0240223\pi$
$228$	0	0
$229$	1595.26i	0.460340i	0.973150	+	0.230170i	$0.0739282\pi$
$229$	1595.26i	0.460340i	−0.973150	+	0.230170i	$0.926072\pi$
$230$	0	0
$231$	−4657.68	−1.32664
$232$	0	0
$233$	−5251.27	−1.47649	−0.738245	−	0.674533i	$-0.764345\pi$
$233$	−5251.27	−1.47649	−0.738245	+	0.674533i	$0.764345\pi$
$234$	0	0
$235$	4706.83i	1.30655i
$236$	0	0
$237$	− 2214.81i	− 0.607035i
$238$	0	0
$239$	−2029.27	−0.549216	−0.274608	−	0.961556i	$-0.588548\pi$
$239$	−2029.27	−0.549216	−0.274608	+	0.961556i	$0.588548\pi$
$240$	0	0
$241$	−4615.72	−1.23371	−0.616856	−	0.787076i	$-0.711594\pi$
$241$	−4615.72	−1.23371	−0.616856	+	0.787076i	$0.711594\pi$
$242$	0	0
$243$	758.210i	0.200161i
$244$	0	0
$245$	− 382.912i	− 0.0998505i
$246$	0	0
$247$	−425.733	−0.109671
$248$	0	0
$249$	1892.10	0.481554
$250$	0	0
$251$	− 2702.52i	− 0.679607i	−0.940496	−	0.339804i	$-0.889639\pi$
$251$	− 2702.52i	− 0.679607i	0.940496	−	0.339804i	$-0.110361\pi$
$252$	0	0
$253$	− 6812.27i	− 1.69282i
$254$	0	0
$255$	−10091.3	−2.47821
$256$	0	0
$257$	991.785	0.240723	0.120362	−	0.992730i	$-0.461595\pi$
$257$	991.785	0.240723	0.120362	+	0.992730i	$0.461595\pi$
$258$	0	0
$259$	− 3267.63i	− 0.783941i
$260$	0	0
$261$	− 301.887i	− 0.0715952i
$262$	0	0
$263$	7430.74	1.74220	0.871101	−	0.491105i	$-0.163407\pi$
$263$	7430.74	1.74220	0.871101	+	0.491105i	$0.163407\pi$
$264$	0	0
$265$	1633.13	0.378575
$266$	0	0
$267$	2551.19i	0.584758i
$268$	0	0
$269$	− 3281.50i	− 0.743778i	−0.928277	−	0.371889i	$-0.878710\pi$
$269$	− 3281.50i	− 0.743778i	0.928277	−	0.371889i	$-0.121290\pi$
$270$	0	0
$271$	−221.518	−0.0496542	−0.0248271	−	0.999692i	$-0.507904\pi$
$271$	−221.518	−0.0496542	−0.0248271	+	0.999692i	$0.507904\pi$
$272$	0	0
$273$	741.909	0.164477
$274$	0	0
$275$	− 6691.48i	− 1.46731i
$276$	0	0
$277$	1106.28i	0.239963i	0.992776	+	0.119982i	$0.0382835\pi$
$277$	1106.28i	0.239963i	−0.992776	+	0.119982i	$0.961716\pi$
$278$	0	0
$279$	−821.579	−0.176296
$280$	0	0
$281$	2372.52	0.503676	0.251838	−	0.967769i	$-0.418965\pi$
$281$	2372.52	0.503676	0.251838	+	0.967769i	$0.418965\pi$
$282$	0	0
$283$	2549.89i	0.535602i	0.963474	+	0.267801i	$0.0862970\pi$
$283$	2549.89i	0.535602i	−0.963474	+	0.267801i	$0.913703\pi$
$284$	0	0
$285$	3946.05i	0.820154i
$286$	0	0
$287$	1688.76	0.347332
$288$	0	0
$289$	11763.7	2.39441
$290$	0	0
$291$	8572.09i	1.72682i
$292$	0	0
$293$	130.319i	0.0259840i	0.999916	+	0.0129920i	$0.00413560\pi$
$293$	130.319i	0.0259840i	−0.999916	+	0.0129920i	$0.995864\pi$
$294$	0	0
$295$	1480.50	0.292197
$296$	0	0
$297$	−7750.32	−1.51421
$298$	0	0
$299$	1085.11i	0.209877i
$300$	0	0
$301$	3297.03i	0.631353i
$302$	0	0
$303$	5711.76	1.08294
$304$	0	0
$305$	−5374.91	−1.00907
$306$	0	0
$307$	62.1182i	0.0115481i	0.999983	+	0.00577406i	$0.00183795\pi$
$307$	62.1182i	0.0115481i	−0.999983	+	0.00577406i	$0.998162\pi$
$308$	0	0
$309$	8057.64i	1.48344i
$310$	0	0
$311$	593.837	0.108275	0.0541373	−	0.998533i	$-0.482759\pi$
$311$	593.837	0.108275	0.0541373	+	0.998533i	$0.482759\pi$
$312$	0	0
$313$	−3669.69	−0.662694	−0.331347	−	0.943509i	$-0.607503\pi$
$313$	−3669.69	−0.662694	−0.331347	+	0.943509i	$0.607503\pi$
$314$	0	0
$315$	768.102i	0.137389i
$316$	0	0
$317$	− 374.339i	− 0.0663249i	−0.999450	−	0.0331625i	$-0.989442\pi$
$317$	− 374.339i	− 0.0663249i	0.999450	−	0.0331625i	$-0.0105579\pi$
$318$	0	0
$319$	5889.96	1.03378
$320$	0	0
$321$	−1392.66	−0.242151
$322$	0	0
$323$	− 6521.16i	− 1.12337i
$324$	0	0
$325$	1065.87i	0.181919i
$326$	0	0
$327$	4952.77	0.837580
$328$	0	0
$329$	5300.51	0.888227
$330$	0	0
$331$	− 5871.13i	− 0.974945i	−0.873138	−	0.487472i	$-0.837919\pi$
$331$	− 5871.13i	− 0.974945i	0.873138	−	0.487472i	$-0.162081\pi$
$332$	0	0
$333$	− 496.431i	− 0.0816944i
$334$	0	0
$335$	7754.43	1.26469
$336$	0	0
$337$	1047.83	0.169373	0.0846865	−	0.996408i	$-0.473011\pi$
$337$	1047.83	0.169373	0.0846865	+	0.996408i	$0.473011\pi$
$338$	0	0
$339$	− 2794.82i	− 0.447769i
$340$	0	0
$341$	− 16029.4i	− 2.54557i
$342$	0	0
$343$	−6555.96	−1.03204
$344$	0	0
$345$	10057.7	1.56953
$346$	0	0
$347$	− 3921.88i	− 0.606737i	−0.952873	−	0.303368i	$-0.901889\pi$
$347$	− 3921.88i	− 0.606737i	0.952873	−	0.303368i	$-0.0981113\pi$
$348$	0	0
$349$	11604.9i	1.77993i	0.456027	+	0.889966i	$0.349272\pi$
$349$	11604.9i	1.77993i	−0.456027	+	0.889966i	$0.650728\pi$
$350$	0	0
$351$	1234.53	0.187733
$352$	0	0
$353$	−1530.22	−0.230724	−0.115362	−	0.993324i	$-0.536803\pi$
$353$	−1530.22	−0.230724	−0.115362	+	0.993324i	$0.536803\pi$
$354$	0	0
$355$	1379.26i	0.206207i
$356$	0	0
$357$	11364.2i	1.68475i
$358$	0	0
$359$	10764.0	1.58246	0.791231	−	0.611518i	$-0.209441\pi$
$359$	10764.0	1.58246	0.791231	+	0.611518i	$0.209441\pi$
$360$	0	0
$361$	4309.01	0.628227
$362$	0	0
$363$	− 7246.40i	− 1.04776i
$364$	0	0
$365$	2446.11i	0.350782i
$366$	0	0
$367$	−10713.0	−1.52374	−0.761869	−	0.647731i	$-0.775718\pi$
$367$	−10713.0	−1.52374	−0.761869	+	0.647731i	$0.775718\pi$
$368$	0	0
$369$	256.563	0.0361954
$370$	0	0
$371$	− 1839.12i	− 0.257365i
$372$	0	0
$373$	− 9186.32i	− 1.27520i	−0.770368	−	0.637600i	$-0.779927\pi$
$373$	− 9186.32i	− 1.27520i	0.770368	−	0.637600i	$-0.220073\pi$
$374$	0	0
$375$	111.404	0.0153409
$376$	0	0
$377$	−938.194	−0.128168
$378$	0	0
$379$	− 14201.0i	− 1.92468i	−0.271842	−	0.962342i	$-0.587633\pi$
$379$	− 14201.0i	− 1.92468i	0.271842	−	0.962342i	$-0.412367\pi$
$380$	0	0
$381$	− 2447.42i	− 0.329095i
$382$	0	0
$383$	9691.24	1.29295	0.646474	−	0.762936i	$-0.276243\pi$
$383$	9691.24	1.29295	0.646474	+	0.762936i	$0.276243\pi$
$384$	0	0
$385$	−14986.0	−1.98379
$386$	0	0
$387$	500.897i	0.0657933i
$388$	0	0
$389$	6358.79i	0.828801i	0.910095	+	0.414400i	$0.136009\pi$
$389$	6358.79i	0.828801i	−0.910095	+	0.414400i	$0.863991\pi$
$390$	0	0
$391$	−16621.1	−2.14978
$392$	0	0
$393$	8174.52	1.04924
$394$	0	0
$395$	− 7126.11i	− 0.907731i
$396$	0	0
$397$	− 13034.2i	− 1.64778i	−0.566752	−	0.823888i	$-0.691800\pi$
$397$	− 13034.2i	− 1.64778i	0.566752	−	0.823888i	$-0.308200\pi$
$398$	0	0
$399$	4443.77	0.557561
$400$	0	0
$401$	6705.06	0.834999	0.417499	−	0.908677i	$-0.362907\pi$
$401$	6705.06	0.834999	0.417499	+	0.908677i	$0.362907\pi$
$402$	0	0
$403$	2553.27i	0.315602i
$404$	0	0
$405$	− 10281.2i	− 1.26143i
$406$	0	0
$407$	9685.59	1.17960
$408$	0	0
$409$	−10967.8	−1.32598	−0.662988	−	0.748630i	$-0.730712\pi$
$409$	−10967.8	−1.32598	−0.662988	+	0.748630i	$0.730712\pi$
$410$	0	0
$411$	2133.28i	0.256027i
$412$	0	0
$413$	− 1667.24i	− 0.198643i
$414$	0	0
$415$	6087.80	0.720093
$416$	0	0
$417$	720.249	0.0845821
$418$	0	0
$419$	− 13686.0i	− 1.59571i	−0.602849	−	0.797856i	$-0.705968\pi$
$419$	− 13686.0i	− 1.59571i	0.602849	−	0.797856i	$-0.294032\pi$
$420$	0	0
$421$	− 8552.92i	− 0.990128i	−0.868857	−	0.495064i	$-0.835145\pi$
$421$	− 8552.92i	− 0.990128i	0.868857	−	0.495064i	$-0.164855\pi$
$422$	0	0
$423$	805.274	0.0925621
$424$	0	0
$425$	−16326.4	−1.86340
$426$	0	0
$427$	6052.86i	0.685992i
$428$	0	0
$429$	2199.09i	0.247490i
$430$	0	0
$431$	12187.6	1.36208	0.681039	−	0.732247i	$-0.261528\pi$
$431$	12187.6	1.36208	0.681039	+	0.732247i	$0.261528\pi$
$432$	0	0
$433$	−8906.35	−0.988480	−0.494240	−	0.869326i	$-0.664554\pi$
$433$	−8906.35	−0.988480	−0.494240	+	0.869326i	$0.664554\pi$
$434$	0	0
$435$	8695.98i	0.958483i
$436$	0	0
$437$	6499.41i	0.711462i
$438$	0	0
$439$	4655.21	0.506107	0.253054	−	0.967452i	$-0.418565\pi$
$439$	4655.21	0.506107	0.253054	+	0.967452i	$0.418565\pi$
$440$	0	0
$441$	−65.5110	−0.00707386
$442$	0	0
$443$	10008.8i	1.07343i	0.843763	+	0.536716i	$0.180335\pi$
$443$	10008.8i	1.07343i	−0.843763	+	0.536716i	$0.819665\pi$
$444$	0	0
$445$	8208.41i	0.874418i
$446$	0	0
$447$	−17526.1	−1.85449
$448$	0	0
$449$	−3373.71	−0.354600	−0.177300	−	0.984157i	$-0.556736\pi$
$449$	−3373.71	−0.354600	−0.177300	+	0.984157i	$0.556736\pi$
$450$	0	0
$451$	5005.65i	0.522632i
$452$	0	0
$453$	6506.79i	0.674869i
$454$	0	0
$455$	2387.08	0.245952
$456$	0	0
$457$	13600.4	1.39212	0.696060	−	0.717984i	$-0.254935\pi$
$457$	13600.4	1.39212	0.696060	+	0.717984i	$0.254935\pi$
$458$	0	0
$459$	18909.8i	1.92295i
$460$	0	0
$461$	3559.45i	0.359609i	0.983702	+	0.179805i	$0.0575466\pi$
$461$	3559.45i	0.359609i	−0.983702	+	0.179805i	$0.942453\pi$
$462$	0	0
$463$	−8019.14	−0.804926	−0.402463	−	0.915436i	$-0.631846\pi$
$463$	−8019.14	−0.804926	−0.402463	+	0.915436i	$0.631846\pi$
$464$	0	0
$465$	23665.9	2.36017
$466$	0	0
$467$	2066.14i	0.204731i	0.994747	+	0.102365i	$0.0326411\pi$
$467$	2066.14i	0.204731i	−0.994747	+	0.102365i	$0.967359\pi$
$468$	0	0
$469$	− 8732.51i	− 0.859765i
$470$	0	0
$471$	10055.7	0.983737
$472$	0	0
$473$	−9772.72	−0.950000
$474$	0	0
$475$	6384.17i	0.616686i
$476$	0	0
$477$	− 279.406i	− 0.0268199i
$478$	0	0
$479$	−3679.26	−0.350960	−0.175480	−	0.984483i	$-0.556148\pi$
$479$	−3679.26	−0.350960	−0.175480	+	0.984483i	$0.556148\pi$
$480$	0	0
$481$	−1542.79	−0.146248
$482$	0	0
$483$	− 11326.3i	− 1.06700i
$484$	0	0
$485$	27580.5i	2.58220i
$486$	0	0
$487$	−10884.4	−1.01277	−0.506386	−	0.862307i	$-0.669019\pi$
$487$	−10884.4	−1.01277	−0.506386	+	0.862307i	$0.669019\pi$
$488$	0	0
$489$	−11789.3	−1.09025
$490$	0	0
$491$	− 14548.9i	− 1.33723i	−0.743608	−	0.668616i	$-0.766887\pi$
$491$	− 14548.9i	− 1.33723i	0.743608	−	0.668616i	$-0.233113\pi$
$492$	0	0
$493$	− 14370.8i	− 1.31284i
$494$	0	0
$495$	−2276.73	−0.206730
$496$	0	0
$497$	1553.23	0.140185
$498$	0	0
$499$	− 8461.92i	− 0.759134i	−0.925164	−	0.379567i	$-0.876073\pi$
$499$	− 8461.92i	− 0.759134i	0.925164	−	0.379567i	$-0.123927\pi$
$500$	0	0
$501$	− 10193.2i	− 0.908978i
$502$	0	0
$503$	983.702	0.0871990	0.0435995	−	0.999049i	$-0.486117\pi$
$503$	983.702	0.0871990	0.0435995	+	0.999049i	$0.486117\pi$
$504$	0	0
$505$	18377.5	1.61938
$506$	0	0
$507$	10477.0i	0.917749i
$508$	0	0
$509$	− 14472.5i	− 1.26028i	−0.776481	−	0.630140i	$-0.782997\pi$
$509$	− 14472.5i	− 1.26028i	0.776481	−	0.630140i	$-0.217003\pi$
$510$	0	0
$511$	2754.65	0.238470
$512$	0	0
$513$	7394.38	0.636393
$514$	0	0
$515$	25925.3i	2.21826i
$516$	0	0
$517$	15711.3i	1.33652i
$518$	0	0
$519$	−12034.0	−1.01779
$520$	0	0
$521$	10761.8	0.904959	0.452480	−	0.891775i	$-0.350540\pi$
$521$	10761.8	0.904959	0.452480	+	0.891775i	$0.350540\pi$
$522$	0	0
$523$	− 1516.67i	− 0.126806i	−0.997988	−	0.0634029i	$-0.979805\pi$
$523$	− 1516.67i	− 0.126806i	0.997988	−	0.0634029i	$-0.0201953\pi$
$524$	0	0
$525$	− 11125.5i	− 0.924866i
$526$	0	0
$527$	−39109.7	−3.23273
$528$	0	0
$529$	4398.66	0.361524
$530$	0	0
$531$	− 253.293i	− 0.0207006i
$532$	0	0
$533$	− 797.336i	− 0.0647963i
$534$	0	0
$535$	−4480.85	−0.362101
$536$	0	0
$537$	−11979.7	−0.962682
$538$	0	0
$539$	− 1278.15i	− 0.102141i
$540$	0	0
$541$	22541.9i	1.79141i	0.444652	+	0.895704i	$0.353327\pi$
$541$	22541.9i	1.79141i	−0.444652	+	0.895704i	$0.646673\pi$
$542$	0	0
$543$	−9506.33	−0.751299
$544$	0	0
$545$	15935.4	1.25248
$546$	0	0
$547$	10202.0i	0.797448i	0.917071	+	0.398724i	$0.130547\pi$
$547$	10202.0i	0.797448i	−0.917071	+	0.398724i	$0.869453\pi$
$548$	0	0
$549$	919.574i	0.0714872i
$550$	0	0
$551$	−5619.46	−0.434477
$552$	0	0
$553$	−8024.94	−0.617098
$554$	0	0
$555$	14299.9i	1.09369i
$556$	0	0
$557$	7139.06i	0.543073i	0.962428	+	0.271536i	$0.0875317\pi$
$557$	7139.06i	0.543073i	−0.962428	+	0.271536i	$0.912468\pi$
$558$	0	0
$559$	1556.67	0.117782
$560$	0	0
$561$	−33684.6	−2.53505
$562$	0	0
$563$	− 19558.0i	− 1.46407i	−0.681268	−	0.732034i	$-0.738571\pi$
$563$	− 19558.0i	− 1.46407i	0.681268	−	0.732034i	$-0.261429\pi$
$564$	0	0
$565$	− 8992.29i	− 0.669572i
$566$	0	0
$567$	−11578.0	−0.857549
$568$	0	0
$569$	5835.86	0.429969	0.214984	−	0.976617i	$-0.431030\pi$
$569$	5835.86	0.429969	0.214984	+	0.976617i	$0.431030\pi$
$570$	0	0
$571$	− 9820.65i	− 0.719757i	−0.932999	−	0.359879i	$-0.882818\pi$
$571$	− 9820.65i	− 0.719757i	0.932999	−	0.359879i	$-0.117182\pi$
$572$	0	0
$573$	20143.3i	1.46858i
$574$	0	0
$575$	16271.9	1.18015
$576$	0	0
$577$	451.231	0.0325563	0.0162782	−	0.999868i	$-0.494818\pi$
$577$	451.231	0.0325563	0.0162782	+	0.999868i	$0.494818\pi$
$578$	0	0
$579$	− 5699.58i	− 0.409096i
$580$	0	0
$581$	− 6855.67i	− 0.489537i
$582$	0	0
$583$	5451.33	0.387257
$584$	0	0
$585$	362.654	0.0256306
$586$	0	0
$587$	24775.7i	1.74208i	0.491209	+	0.871042i	$0.336555\pi$
$587$	24775.7i	1.74208i	−0.491209	+	0.871042i	$0.663445\pi$
$588$	0	0
$589$	15293.2i	1.06986i
$590$	0	0
$591$	13101.3	0.911873
$592$	0	0
$593$	−4207.64	−0.291378	−0.145689	−	0.989330i	$-0.546540\pi$
$593$	−4207.64	−0.291378	−0.145689	+	0.989330i	$0.546540\pi$
$594$	0	0
$595$	36564.1i	2.51929i
$596$	0	0
$597$	3658.54i	0.250811i
$598$	0	0
$599$	−1916.32	−0.130715	−0.0653577	−	0.997862i	$-0.520819\pi$
$599$	−1916.32	−0.130715	−0.0653577	+	0.997862i	$0.520819\pi$
$600$	0	0
$601$	19361.1	1.31407	0.657034	−	0.753861i	$-0.271811\pi$
$601$	19361.1	1.31407	0.657034	+	0.753861i	$0.271811\pi$
$602$	0	0
$603$	− 1326.68i	− 0.0895961i
$604$	0	0
$605$	− 23315.2i	− 1.56677i
$606$	0	0
$607$	6097.02	0.407694	0.203847	−	0.979003i	$-0.434655\pi$
$607$	6097.02	0.407694	0.203847	+	0.979003i	$0.434655\pi$
$608$	0	0
$609$	9792.82	0.651601
$610$	0	0
$611$	− 2502.60i	− 0.165703i
$612$	0	0
$613$	− 13837.7i	− 0.911743i	−0.890046	−	0.455872i	$-0.849328\pi$
$613$	− 13837.7i	− 0.911743i	0.890046	−	0.455872i	$-0.150672\pi$
$614$	0	0
$615$	−7390.38	−0.484567
$616$	0	0
$617$	−11959.1	−0.780318	−0.390159	−	0.920747i	$-0.627580\pi$
$617$	−11959.1	−0.780318	−0.390159	+	0.920747i	$0.627580\pi$
$618$	0	0
$619$	− 8385.02i	− 0.544463i	−0.962232	−	0.272231i	$-0.912238\pi$
$619$	− 8385.02i	− 0.544463i	0.962232	−	0.272231i	$-0.0877616\pi$
$620$	0	0
$621$	− 18846.8i	− 1.21787i
$622$	0	0
$623$	9243.75	0.594451
$624$	0	0
$625$	−15444.8	−0.988465
$626$	0	0
$627$	13171.8i	0.838965i
$628$	0	0
$629$	− 23631.7i	− 1.49802i
$630$	0	0
$631$	−13592.1	−0.857513	−0.428757	−	0.903420i	$-0.641048\pi$
$631$	−13592.1	−0.857513	−0.428757	+	0.903420i	$0.641048\pi$
$632$	0	0
$633$	−18915.9	−1.18774
$634$	0	0
$635$	− 7874.54i	− 0.492113i
$636$	0	0
$637$	203.593i	0.0126635i
$638$	0	0
$639$	235.973	0.0146087
$640$	0	0
$641$	−18123.4	−1.11674	−0.558371	−	0.829591i	$-0.688573\pi$
$641$	−18123.4	−1.11674	−0.558371	+	0.829591i	$0.688573\pi$
$642$	0	0
$643$	25042.1i	1.53587i	0.640528	+	0.767935i	$0.278716\pi$
$643$	25042.1i	1.53587i	−0.640528	+	0.767935i	$0.721284\pi$
$644$	0	0
$645$	− 14428.5i	− 0.880809i
$646$	0	0
$647$	6022.08	0.365923	0.182962	−	0.983120i	$-0.441432\pi$
$647$	6022.08	0.365923	0.182962	+	0.983120i	$0.441432\pi$
$648$	0	0
$649$	4941.87	0.298899
$650$	0	0
$651$	− 26650.9i	− 1.60450i
$652$	0	0
$653$	14093.8i	0.844617i	0.906452	+	0.422308i	$0.138780\pi$
$653$	14093.8i	0.844617i	−0.906452	+	0.422308i	$0.861220\pi$
$654$	0	0
$655$	26301.4	1.56898
$656$	0	0
$657$	418.496	0.0248510
$658$	0	0
$659$	− 17240.7i	− 1.01912i	−0.860434	−	0.509562i	$-0.829807\pi$
$659$	− 17240.7i	− 1.01912i	0.860434	−	0.509562i	$-0.170193\pi$
$660$	0	0
$661$	20209.5i	1.18920i	0.804023	+	0.594598i	$0.202689\pi$
$661$	20209.5i	1.18920i	−0.804023	+	0.594598i	$0.797311\pi$
$662$	0	0
$663$	5365.52	0.314298
$664$	0	0
$665$	14297.8	0.833749
$666$	0	0
$667$	14322.8i	0.831459i
$668$	0	0
$669$	− 16846.3i	− 0.973564i
$670$	0	0
$671$	−17941.3	−1.03221
$672$	0	0
$673$	6838.55	0.391689	0.195845	−	0.980635i	$-0.437255\pi$
$673$	6838.55	0.391689	0.195845	+	0.980635i	$0.437255\pi$
$674$	0	0
$675$	− 18512.6i	− 1.05563i
$676$	0	0
$677$	− 187.425i	− 0.0106401i	−0.999986	−	0.00532003i	$-0.998307\pi$
$677$	− 187.425i	− 0.0106401i	0.999986	−	0.00532003i	$-0.00169343\pi$
$678$	0	0
$679$	31059.3	1.75545
$680$	0	0
$681$	2541.62	0.143018
$682$	0	0
$683$	− 23476.0i	− 1.31520i	−0.753367	−	0.657601i	$-0.771571\pi$
$683$	− 23476.0i	− 1.31520i	0.753367	−	0.657601i	$-0.228429\pi$
$684$	0	0
$685$	6863.79i	0.382850i
$686$	0	0
$687$	−7861.77	−0.436602
$688$	0	0
$689$	−868.327	−0.0480125
$690$	0	0
$691$	− 6309.92i	− 0.347382i	−0.984800	−	0.173691i	$-0.944431\pi$
$691$	− 6309.92i	− 0.347382i	0.984800	−	0.173691i	$-0.0555694\pi$
$692$	0	0
$693$	2563.90i	0.140540i
$694$	0	0
$695$	2317.39	0.126480
$696$	0	0
$697$	12213.2	0.663712
$698$	0	0
$699$	− 25879.3i	− 1.40035i
$700$	0	0
$701$	24920.4i	1.34270i	0.741142	+	0.671348i	$0.234284\pi$
$701$	24920.4i	1.34270i	−0.741142	+	0.671348i	$0.765716\pi$
$702$	0	0
$703$	−9240.77	−0.495764
$704$	0	0
$705$	−23196.2	−1.23918
$706$	0	0
$707$	− 20695.5i	− 1.10090i
$708$	0	0
$709$	14478.9i	0.766949i	0.923551	+	0.383474i	$0.125273\pi$
$709$	14478.9i	0.766949i	−0.923551	+	0.383474i	$0.874727\pi$
$710$	0	0
$711$	−1219.18	−0.0643077
$712$	0	0
$713$	38979.3	2.04738
$714$	0	0
$715$	7075.54i	0.370084i
$716$	0	0
$717$	− 10000.7i	− 0.520895i
$718$	0	0
$719$	6411.30	0.332547	0.166273	−	0.986080i	$-0.446827\pi$
$719$	6411.30	0.332547	0.166273	+	0.986080i	$0.446827\pi$
$720$	0	0
$721$	29195.3	1.50803
$722$	0	0
$723$	− 22747.2i	− 1.17009i
$724$	0	0
$725$	14068.9i	0.720698i
$726$	0	0
$727$	−20626.3	−1.05225	−0.526126	−	0.850407i	$-0.676356\pi$
$727$	−20626.3	−1.05225	−0.526126	+	0.850407i	$0.676356\pi$
$728$	0	0
$729$	−21243.3	−1.07927
$730$	0	0
$731$	23844.3i	1.20645i
$732$	0	0
$733$	− 5732.29i	− 0.288850i	−0.989516	−	0.144425i	$-0.953867\pi$
$733$	− 5732.29i	− 0.288850i	0.989516	−	0.144425i	$-0.0461332\pi$
$734$	0	0
$735$	1887.07	0.0947015
$736$	0	0
$737$	25884.1	1.29369
$738$	0	0
$739$	− 36336.0i	− 1.80872i	−0.426773	−	0.904359i	$-0.640350\pi$
$739$	− 36336.0i	− 1.80872i	0.426773	−	0.904359i	$-0.359650\pi$
$740$	0	0
$741$	− 2098.10i	− 0.104016i
$742$	0	0
$743$	4363.83	0.215469	0.107734	−	0.994180i	$-0.465640\pi$
$743$	4363.83	0.215469	0.107734	+	0.994180i	$0.465640\pi$
$744$	0	0
$745$	−56389.9	−2.77311
$746$	0	0
$747$	− 1041.54i	− 0.0510146i
$748$	0	0
$749$	5046.03i	0.246166i
$750$	0	0
$751$	16162.7	0.785334	0.392667	−	0.919681i	$-0.371552\pi$
$751$	16162.7	0.785334	0.392667	+	0.919681i	$0.371552\pi$
$752$	0	0
$753$	13318.6	0.644562
$754$	0	0
$755$	20935.5i	1.00917i
$756$	0	0
$757$	4411.92i	0.211828i	0.994375	+	0.105914i	$0.0337769\pi$
$757$	4411.92i	0.211828i	−0.994375	+	0.105914i	$0.966223\pi$
$758$	0	0
$759$	33572.2	1.60553
$760$	0	0
$761$	−28281.8	−1.34719	−0.673597	−	0.739099i	$-0.735251\pi$
$761$	−28281.8	−1.34719	−0.673597	+	0.739099i	$0.735251\pi$
$762$	0	0
$763$	− 17945.4i	− 0.851465i
$764$	0	0
$765$	5554.95i	0.262535i
$766$	0	0
$767$	−787.176	−0.0370577
$768$	0	0
$769$	−15016.8	−0.704186	−0.352093	−	0.935965i	$-0.614530\pi$
$769$	−15016.8	−0.704186	−0.352093	+	0.935965i	$0.614530\pi$
$770$	0	0
$771$	4887.72i	0.228310i
$772$	0	0
$773$	8674.11i	0.403604i	0.979426	+	0.201802i	$0.0646798\pi$
$773$	8674.11i	0.403604i	−0.979426	+	0.201802i	$0.935320\pi$
$774$	0	0
$775$	38288.2	1.77465
$776$	0	0
$777$	16103.5	0.743515
$778$	0	0
$779$	− 4775.76i	− 0.219653i
$780$	0	0
$781$	4603.94i	0.210937i
$782$	0	0
$783$	16295.1	0.743729
$784$	0	0
$785$	32353.9	1.47103
$786$	0	0
$787$	− 23660.2i	− 1.07166i	−0.844327	−	0.535829i	$-0.819999\pi$
$787$	− 23660.2i	− 1.07166i	0.844327	−	0.535829i	$-0.180001\pi$
$788$	0	0
$789$	36620.2i	1.65236i
$790$	0	0
$791$	−10126.5	−0.455192
$792$	0	0
$793$	2857.82	0.127975
$794$	0	0
$795$	8048.39i	0.359053i
$796$	0	0
$797$	− 35432.5i	− 1.57476i	−0.616468	−	0.787380i	$-0.711437\pi$
$797$	− 35432.5i	− 1.57476i	0.616468	−	0.787380i	$-0.288563\pi$
$798$	0	0
$799$	38333.6	1.69730
$800$	0	0
$801$	1404.35	0.0619477
$802$	0	0
$803$	8165.05i	0.358827i
$804$	0	0
$805$	− 36442.1i	− 1.59555i
$806$	0	0
$807$	16171.9	0.705424
$808$	0	0
$809$	14005.5	0.608664	0.304332	−	0.952566i	$-0.401567\pi$
$809$	14005.5	0.608664	0.304332	+	0.952566i	$0.401567\pi$
$810$	0	0
$811$	22580.8i	0.977706i	0.872366	+	0.488853i	$0.162584\pi$
$811$	22580.8i	0.977706i	−0.872366	+	0.488853i	$0.837416\pi$
$812$	0	0
$813$	− 1091.69i	− 0.0470937i
$814$	0	0
$815$	−37931.8	−1.63030
$816$	0	0
$817$	9323.90	0.399268
$818$	0	0
$819$	− 408.396i	− 0.0174243i
$820$	0	0
$821$	21308.9i	0.905827i	0.891554	+	0.452914i	$0.149615\pi$
$821$	21308.9i	0.905827i	−0.891554	+	0.452914i	$0.850385\pi$
$822$	0	0
$823$	13571.4	0.574810	0.287405	−	0.957809i	$-0.407207\pi$
$823$	13571.4	0.574810	0.287405	+	0.957809i	$0.407207\pi$
$824$	0	0
$825$	32977.0	1.39165
$826$	0	0
$827$	19295.2i	0.811318i	0.914025	+	0.405659i	$0.132958\pi$
$827$	19295.2i	0.811318i	−0.914025	+	0.405659i	$0.867042\pi$
$828$	0	0
$829$	− 28752.7i	− 1.20461i	−0.798265	−	0.602306i	$-0.794249\pi$
$829$	− 28752.7i	− 1.20461i	0.798265	−	0.602306i	$-0.205751\pi$
$830$	0	0
$831$	−5451.96	−0.227589
$832$	0	0
$833$	−3118.53	−0.129713
$834$	0	0
$835$	− 32796.4i	− 1.35924i
$836$	0	0
$837$	− 44346.7i	− 1.83136i
$838$	0	0
$839$	−566.927	−0.0233284	−0.0116642	−	0.999932i	$-0.503713\pi$
$839$	−566.927	−0.0233284	−0.0116642	+	0.999932i	$0.503713\pi$
$840$	0	0
$841$	12005.3	0.492243
$842$	0	0
$843$	11692.3i	0.477703i
$844$	0	0
$845$	33709.5i	1.37236i
$846$	0	0
$847$	−26256.0	−1.06513
$848$	0	0
$849$	−12566.4	−0.507983
$850$	0	0
$851$	23552.8i	0.948744i
$852$	0	0
$853$	34698.1i	1.39278i	0.717665	+	0.696389i	$0.245211\pi$
$853$	34698.1i	1.39278i	−0.717665	+	0.696389i	$0.754789\pi$
$854$	0	0
$855$	2172.17	0.0868850
$856$	0	0
$857$	−37362.0	−1.48922	−0.744610	−	0.667500i	$-0.767364\pi$
$857$	−37362.0	−1.48922	−0.744610	+	0.667500i	$0.767364\pi$
$858$	0	0
$859$	− 29572.2i	− 1.17461i	−0.809366	−	0.587305i	$-0.800189\pi$
$859$	− 29572.2i	− 1.17461i	0.809366	−	0.587305i	$-0.199811\pi$
$860$	0	0
$861$	8322.54i	0.329421i
$862$	0	0
$863$	11636.5	0.458994	0.229497	−	0.973309i	$-0.426292\pi$
$863$	11636.5	0.458994	0.229497	+	0.973309i	$0.426292\pi$
$864$	0	0
$865$	−38719.2	−1.52196
$866$	0	0
$867$	57974.1i	2.27094i
$868$	0	0
$869$	− 23786.7i	− 0.928550i
$870$	0	0
$871$	−4123.00	−0.160393
$872$	0	0
$873$	4718.65	0.182935
$874$	0	0
$875$	− 403.649i	− 0.0155952i
$876$	0	0
$877$	− 19878.2i	− 0.765380i	−0.923877	−	0.382690i	$-0.874998\pi$
$877$	− 19878.2i	− 0.765380i	0.923877	−	0.382690i	$-0.125002\pi$
$878$	0	0
$879$	−642.238	−0.0246441
$880$	0	0
$881$	−36460.3	−1.39430	−0.697150	−	0.716926i	$-0.745549\pi$
$881$	−36460.3	−1.39430	−0.697150	+	0.716926i	$0.745549\pi$
$882$	0	0
$883$	− 28119.9i	− 1.07170i	−0.844314	−	0.535849i	$-0.819991\pi$
$883$	− 28119.9i	− 1.07170i	0.844314	−	0.535849i	$-0.180009\pi$
$884$	0	0
$885$	7296.21i	0.277129i
$886$	0	0
$887$	−41888.4	−1.58565	−0.792826	−	0.609448i	$-0.791391\pi$
$887$	−41888.4	−1.58565	−0.792826	+	0.609448i	$0.791391\pi$
$888$	0	0
$889$	−8867.77	−0.334551
$890$	0	0
$891$	− 34318.4i	− 1.29036i
$892$	0	0
$893$	− 14989.7i	− 0.561715i
$894$	0	0
$895$	−38544.3	−1.43955
$896$	0	0
$897$	−5347.62	−0.199055
$898$	0	0
$899$	33701.9i	1.25030i
$900$	0	0
$901$	− 13300.6i	− 0.491795i
$902$	0	0
$903$	−16248.4	−0.598796
$904$	0	0
$905$	−30586.4	−1.12346
$906$	0	0
$907$	− 12575.3i	− 0.460371i	−0.973147	−	0.230185i	$-0.926067\pi$
$907$	− 12575.3i	− 0.460371i	0.973147	−	0.230185i	$-0.0739333\pi$
$908$	0	0
$909$	− 3144.14i	− 0.114724i
$910$	0	0
$911$	39814.7	1.44799	0.723995	−	0.689805i	$-0.242304\pi$
$911$	39814.7	1.44799	0.723995	+	0.689805i	$0.242304\pi$
$912$	0	0
$913$	20320.9	0.736609
$914$	0	0
$915$	− 26488.7i	− 0.957036i
$916$	0	0
$917$	− 29618.8i	− 1.06663i
$918$	0	0
$919$	8987.93	0.322616	0.161308	−	0.986904i	$-0.448429\pi$
$919$	8987.93	0.322616	0.161308	+	0.986904i	$0.448429\pi$
$920$	0	0
$921$	−306.131	−0.0109526
$922$	0	0
$923$	− 733.348i	− 0.0261521i
$924$	0	0
$925$	23135.2i	0.822359i
$926$	0	0
$927$	4435.46	0.157152
$928$	0	0
$929$	234.252	0.00827293	0.00413647	−	0.999991i	$-0.498683\pi$
$929$	234.252	0.00827293	0.00413647	+	0.999991i	$0.498683\pi$
$930$	0	0
$931$	1219.45i	0.0429279i
$932$	0	0
$933$	2926.55i	0.102691i
$934$	0	0
$935$	−108380.	−3.79079
$936$	0	0
$937$	−17624.6	−0.614483	−0.307241	−	0.951632i	$-0.599406\pi$
$937$	−17624.6	−0.614483	−0.307241	+	0.951632i	$0.599406\pi$
$938$	0	0
$939$	− 18085.0i	− 0.628521i
$940$	0	0
$941$	30532.1i	1.05772i	0.848708	+	0.528861i	$0.177381\pi$
$941$	30532.1i	1.05772i	−0.848708	+	0.528861i	$0.822619\pi$
$942$	0	0
$943$	−12172.4	−0.420349
$944$	0	0
$945$	−41460.2	−1.42720
$946$	0	0
$947$	42092.8i	1.44438i	0.691692	+	0.722192i	$0.256865\pi$
$947$	42092.8i	1.44438i	−0.691692	+	0.722192i	$0.743135\pi$
$948$	0	0
$949$	− 1300.59i	− 0.0444877i
$950$	0	0
$951$	1844.82	0.0629048
$952$	0	0
$953$	48641.0	1.65334	0.826672	−	0.562683i	$-0.190231\pi$
$953$	48641.0	1.65334	0.826672	+	0.562683i	$0.190231\pi$
$954$	0	0
$955$	64810.6i	2.19604i
$956$	0	0
$957$	29026.9i	0.980467i
$958$	0	0
$959$	7729.54	0.260271
$960$	0	0
$961$	61927.9	2.07874
$962$	0	0
$963$	766.612i	0.0256529i
$964$	0	0
$965$	− 18338.3i	− 0.611742i
$966$	0	0
$967$	−53000.5	−1.76254	−0.881272	−	0.472609i	$-0.843312\pi$
$967$	−53000.5	−1.76254	−0.881272	+	0.472609i	$0.843312\pi$
$968$	0	0
$969$	32137.6	1.06544
$970$	0	0
$971$	26286.3i	0.868761i	0.900729	+	0.434381i	$0.143033\pi$
$971$	26286.3i	0.868761i	−0.900729	+	0.434381i	$0.856967\pi$
$972$	0	0
$973$	− 2609.68i	− 0.0859842i
$974$	0	0
$975$	−5252.81	−0.172538
$976$	0	0
$977$	−30508.0	−0.999014	−0.499507	−	0.866310i	$-0.666486\pi$
$977$	−30508.0	−0.999014	−0.499507	+	0.866310i	$0.666486\pi$
$978$	0	0
$979$	27399.4i	0.894473i
$980$	0	0
$981$	− 2726.34i	− 0.0887311i
$982$	0	0
$983$	37044.3	1.20196	0.600982	−	0.799263i	$-0.294776\pi$
$983$	37044.3	1.20196	0.600982	+	0.799263i	$0.294776\pi$
$984$	0	0
$985$	42153.3	1.36357
$986$	0	0
$987$	26122.0i	0.842424i
$988$	0	0
$989$	− 23764.7i	− 0.764079i
$990$	0	0
$991$	−47803.5	−1.53232	−0.766160	−	0.642650i	$-0.777835\pi$
$991$	−47803.5	−1.53232	−0.766160	+	0.642650i	$0.777835\pi$
$992$	0	0
$993$	28934.1	0.924670
$994$	0	0
$995$	11771.3i	0.375051i
$996$	0	0
$997$	− 13478.1i	− 0.428140i	−0.976818	−	0.214070i	$-0.931328\pi$
$997$	− 13478.1i	− 0.428140i	0.976818	−	0.214070i	$-0.0686720\pi$
$998$	0	0
$999$	26796.1	0.848639

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.4.b.i.129.3		4
4.3	odd	2		256.4.b.h.129.2		4
8.3	odd	2		256.4.b.h.129.3		4
8.5	even	2	inner	256.4.b.i.129.2		4
16.3	odd	4		128.4.a.f.1.2	yes	2
16.5	even	4		128.4.a.e.1.2	✓	2
16.11	odd	4		128.4.a.g.1.1	yes	2
16.13	even	4		128.4.a.h.1.1	yes	2
48.5	odd	4		1152.4.a.s.1.2		2
48.11	even	4		1152.4.a.t.1.2		2
48.29	odd	4		1152.4.a.q.1.1		2
48.35	even	4		1152.4.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.a.e.1.2	✓	2	16.5	even	4
128.4.a.f.1.2	yes	2	16.3	odd	4
128.4.a.g.1.1	yes	2	16.11	odd	4
128.4.a.h.1.1	yes	2	16.13	even	4
256.4.b.h.129.2		4	4.3	odd	2
256.4.b.h.129.3		4	8.3	odd	2
256.4.b.i.129.2		4	8.5	even	2	inner
256.4.b.i.129.3		4	1.1	even	1	trivial
1152.4.a.q.1.1		2	48.29	odd	4
1152.4.a.r.1.1		2	48.35	even	4
1152.4.a.s.1.2		2	48.5	odd	4
1152.4.a.t.1.2		2	48.11	even	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.92820i q^{3} +15.8564i q^{5} +17.8564 q^{7} +2.71281 q^{9} +O(q^{10})\)	\(q+4.92820i q^{3} +15.8564i q^{5} +17.8564 q^{7} +2.71281 q^{9} +52.9282i q^{11} -8.43078i q^{13} -78.1436 q^{15} +129.138 q^{17} -50.4974i q^{19} +88.0000i q^{21} -128.708 q^{23} -126.426 q^{25} +146.431i q^{27} -111.282i q^{29} -302.851 q^{31} -260.841 q^{33} +283.138i q^{35} -182.995i q^{37} +41.5486 q^{39} +94.5744 q^{41} +184.641i q^{43} +43.0155i q^{45} +296.841 q^{47} -24.1487 q^{49} +636.420i q^{51} -102.995i q^{53} -839.251 q^{55} +248.862 q^{57} -93.3693i q^{59} +338.974i q^{61} +48.4411 q^{63} +133.682 q^{65} -489.041i q^{67} -634.297i q^{69} +86.9845 q^{71} +154.267 q^{73} -623.051i q^{75} +945.108i q^{77} -449.415 q^{79} -648.395 q^{81} -383.933i q^{83} +2047.67i q^{85} +548.420 q^{87} +517.672 q^{89} -150.543i q^{91} -1492.51i q^{93} +800.708 q^{95} +1739.39 q^{97} +143.584i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7} - 100 q^{9}+O(q^{10}) \) 4 * q + 16 * q^7 - 100 * q^9	\( 4 q + 16 q^{7} - 100 q^{9} - 368 q^{15} + 184 q^{17} - 16 q^{23} - 284 q^{25} - 768 q^{31} + 176 q^{33} - 1552 q^{39} + 600 q^{41} - 32 q^{47} - 540 q^{49} - 752 q^{55} + 1328 q^{57} + 1136 q^{63} - 1904 q^{65} - 816 q^{71} - 824 q^{73} - 800 q^{79} - 44 q^{81} + 1584 q^{87} - 1144 q^{89} + 2704 q^{95} + 4408 q^{97}+O(q^{100}) \) 4 * q + 16 * q^7 - 100 * q^9 - 368 * q^15 + 184 * q^17 - 16 * q^23 - 284 * q^25 - 768 * q^31 + 176 * q^33 - 1552 * q^39 + 600 * q^41 - 32 * q^47 - 540 * q^49 - 752 * q^55 + 1328 * q^57 + 1136 * q^63 - 1904 * q^65 - 816 * q^71 - 824 * q^73 - 800 * q^79 - 44 * q^81 + 1584 * q^87 - 1144 * q^89 + 2704 * q^95 + 4408 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more