LMFDB - Embedded newform 256.5.c.k.255.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,5,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, 5, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 5 $
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:	$26.4627105495$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - x^{2} + 2x + 27 $ x^4 - 2x^3 - x^2 + 2x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.3
Root			$-1.30278 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.5.c.k.255.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.54118i q^{3} -16.8444 q^{5} +40.7922i q^{7} +60.3776 q^{9} +O(q^{10})$	$q+4.54118i q^{3} -16.8444 q^{5} +40.7922i q^{7} +60.3776 q^{9} +104.133i q^{11} -13.4668 q^{13} -76.4936i q^{15} +73.3776 q^{17} -502.501i q^{19} -185.245 q^{21} +973.921i q^{23} -341.266 q^{25} +642.022i q^{27} -1256.49 q^{29} -1086.12i q^{31} -472.888 q^{33} -687.120i q^{35} -1953.73 q^{37} -61.1551i q^{39} +194.266 q^{41} +1808.79i q^{43} -1017.03 q^{45} -627.155i q^{47} +737.000 q^{49} +333.221i q^{51} -3011.33 q^{53} -1754.06i q^{55} +2281.95 q^{57} -5696.96i q^{59} -6076.93 q^{61} +2462.93i q^{63} +226.840 q^{65} +3518.77i q^{67} -4422.76 q^{69} -3340.00i q^{71} -4810.79 q^{73} -1549.75i q^{75} -4247.82 q^{77} +5466.15i q^{79} +1975.05 q^{81} +3945.61i q^{83} -1236.00 q^{85} -5705.94i q^{87} +700.314 q^{89} -549.339i q^{91} +4932.25 q^{93} +8464.34i q^{95} -12767.9 q^{97} +6287.32i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 48 q^{5} - 220 q^{9}+O(q^{10}) $ 4 * q + 48 * q^5 - 220 * q^9	$ 4 q + 48 q^{5} - 220 q^{9} - 400 q^{13} - 168 q^{17} - 1664 q^{21} + 1404 q^{25} - 1680 q^{29} + 416 q^{33} - 5392 q^{37} - 1992 q^{41} - 15952 q^{45} + 2948 q^{49} - 13776 q^{53} - 4256 q^{57} - 9424 q^{61} - 14784 q^{65} - 16768 q^{69} + 9832 q^{73} - 4992 q^{77} + 21284 q^{81} - 15328 q^{85} + 13416 q^{89} - 24576 q^{93} - 10920 q^{97}+O(q^{100}) $ 4 * q + 48 * q^5 - 220 * q^9 - 400 * q^13 - 168 * q^17 - 1664 * q^21 + 1404 * q^25 - 1680 * q^29 + 416 * q^33 - 5392 * q^37 - 1992 * q^41 - 15952 * q^45 + 2948 * q^49 - 13776 * q^53 - 4256 * q^57 - 9424 * q^61 - 14784 * q^65 - 16768 * q^69 + 9832 * q^73 - 4992 * q^77 + 21284 * q^81 - 15328 * q^85 + 13416 * q^89 - 24576 * q^93 - 10920 * 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
$3$	4.54118i	0.504576i	0.967652	+	0.252288i	$0.0811830\pi$
$3$	4.54118i	0.504576i	−0.967652	+	0.252288i	$0.918817\pi$
$4$	0	0
$5$	−16.8444	−0.673776	−0.336888	−	0.941545i	$-0.609374\pi$
$5$	−16.8444	−0.673776	−0.336888	+	0.941545i	$0.609374\pi$
$6$	0	0
$7$	40.7922i	0.832493i	0.909252	+	0.416246i	$0.136655\pi$
$7$	40.7922i	0.832493i	−0.909252	+	0.416246i	$0.863345\pi$
$8$	0	0
$9$	60.3776	0.745403
$10$	0	0
$11$	104.133i	0.860605i	0.902685	+	0.430303i	$0.141593\pi$
$11$	104.133i	0.860605i	−0.902685	+	0.430303i	$0.858407\pi$
$12$	0	0
$13$	−13.4668	−0.0796850	−0.0398425	−	0.999206i	$-0.512686\pi$
$13$	−13.4668	−0.0796850	−0.0398425	+	0.999206i	$0.512686\pi$
$14$	0	0
$15$	− 76.4936i	− 0.339971i
$16$	0	0
$17$	73.3776	0.253902	0.126951	−	0.991909i	$-0.459481\pi$
$17$	73.3776	0.253902	0.126951	+	0.991909i	$0.459481\pi$
$18$	0	0
$19$	− 502.501i	− 1.39197i	−0.718056	−	0.695985i	$-0.754968\pi$
$19$	− 502.501i	− 1.39197i	0.718056	−	0.695985i	$-0.245032\pi$
$20$	0	0
$21$	−185.245	−0.420056
$22$	0	0
$23$	973.921i	1.84106i	0.390671	+	0.920530i	$0.372243\pi$
$23$	973.921i	1.84106i	−0.390671	+	0.920530i	$0.627757\pi$
$24$	0	0
$25$	−341.266	−0.546025
$26$	0	0
$27$	642.022i	0.880689i
$28$	0	0
$29$	−1256.49	−1.49404	−0.747020	−	0.664801i	$-0.768516\pi$
$29$	−1256.49	−1.49404	−0.747020	+	0.664801i	$0.768516\pi$
$30$	0	0
$31$	− 1086.12i	− 1.13019i	−0.825025	−	0.565097i	$-0.808839\pi$
$31$	− 1086.12i	− 1.13019i	0.825025	−	0.565097i	$-0.191161\pi$
$32$	0	0
$33$	−472.888	−0.434241
$34$	0	0
$35$	− 687.120i	− 0.560914i
$36$	0	0
$37$	−1953.73	−1.42712	−0.713562	−	0.700592i	$-0.752919\pi$
$37$	−1953.73	−1.42712	−0.713562	+	0.700592i	$0.752919\pi$
$38$	0	0
$39$	− 61.1551i	− 0.0402072i
$40$	0	0
$41$	194.266	0.115566	0.0577828	−	0.998329i	$-0.481597\pi$
$41$	194.266	0.115566	0.0577828	+	0.998329i	$0.481597\pi$
$42$	0	0
$43$	1808.79i	0.978254i	0.872212	+	0.489127i	$0.162685\pi$
$43$	1808.79i	0.978254i	−0.872212	+	0.489127i	$0.837315\pi$
$44$	0	0
$45$	−1017.03	−0.502235
$46$	0	0
$47$	− 627.155i	− 0.283909i	−0.989873	−	0.141954i	$-0.954661\pi$
$47$	− 627.155i	− 0.283909i	0.989873	−	0.141954i	$-0.0453386\pi$
$48$	0	0
$49$	737.000	0.306955
$50$	0	0
$51$	333.221i	0.128113i
$52$	0	0
$53$	−3011.33	−1.07203	−0.536015	−	0.844208i	$-0.680071\pi$
$53$	−3011.33	−1.07203	−0.536015	+	0.844208i	$0.680071\pi$
$54$	0	0
$55$	− 1754.06i	− 0.579855i
$56$	0	0
$57$	2281.95	0.702355
$58$	0	0
$59$	− 5696.96i	− 1.63659i	−0.574800	−	0.818294i	$-0.694920\pi$
$59$	− 5696.96i	− 1.63659i	0.574800	−	0.818294i	$-0.305080\pi$
$60$	0	0
$61$	−6076.93	−1.63314	−0.816572	−	0.577244i	$-0.804128\pi$
$61$	−6076.93	−1.63314	−0.816572	+	0.577244i	$0.804128\pi$
$62$	0	0
$63$	2462.93i	0.620543i
$64$	0	0
$65$	226.840	0.0536899
$66$	0	0
$67$	3518.77i	0.783864i	0.919994	+	0.391932i	$0.128193\pi$
$67$	3518.77i	0.783864i	−0.919994	+	0.391932i	$0.871807\pi$
$68$	0	0
$69$	−4422.76	−0.928955
$70$	0	0
$71$	− 3340.00i	− 0.662567i	−0.943531	−	0.331283i	$-0.892518\pi$
$71$	− 3340.00i	− 0.662567i	0.943531	−	0.331283i	$-0.107482\pi$
$72$	0	0
$73$	−4810.79	−0.902757	−0.451378	−	0.892333i	$-0.649068\pi$
$73$	−4810.79	−0.902757	−0.451378	+	0.892333i	$0.649068\pi$
$74$	0	0
$75$	− 1549.75i	− 0.275511i
$76$	0	0
$77$	−4247.82	−0.716448
$78$	0	0
$79$	5466.15i	0.875845i	0.899013	+	0.437923i	$0.144286\pi$
$79$	5466.15i	0.875845i	−0.899013	+	0.437923i	$0.855714\pi$
$80$	0	0
$81$	1975.05	0.301029
$82$	0	0
$83$	3945.61i	0.572741i	0.958119	+	0.286370i	$0.0924488\pi$
$83$	3945.61i	0.572741i	−0.958119	+	0.286370i	$0.907551\pi$
$84$	0	0
$85$	−1236.00	−0.171073
$86$	0	0
$87$	− 5705.94i	− 0.753857i
$88$	0	0
$89$	700.314	0.0884124	0.0442062	−	0.999022i	$-0.485924\pi$
$89$	700.314	0.0884124	0.0442062	+	0.999022i	$0.485924\pi$
$90$	0	0
$91$	− 549.339i	− 0.0663372i
$92$	0	0
$93$	4932.25	0.570269
$94$	0	0
$95$	8464.34i	0.937877i
$96$	0	0
$97$	−12767.9	−1.35698	−0.678492	−	0.734608i	$-0.737366\pi$
$97$	−12767.9	−1.35698	−0.678492	+	0.734608i	$0.737366\pi$
$98$	0	0
$99$	6287.32i	0.641498i
$100$	0	0
$101$	4197.02	0.411432	0.205716	−	0.978612i	$-0.434048\pi$
$101$	4197.02	0.411432	0.205716	+	0.978612i	$0.434048\pi$
$102$	0	0
$103$	13201.4i	1.24436i	0.782875	+	0.622179i	$0.213752\pi$
$103$	13201.4i	1.24436i	−0.782875	+	0.622179i	$0.786248\pi$
$104$	0	0
$105$	3120.34	0.283024
$106$	0	0
$107$	179.932i	0.0157160i	0.999969	+	0.00785800i	$0.00250130\pi$
$107$	179.932i	0.0157160i	−0.999969	+	0.00785800i	$0.997499\pi$
$108$	0	0
$109$	1774.81	0.149382	0.0746909	−	0.997207i	$-0.476203\pi$
$109$	1774.81	0.149382	0.0746909	+	0.997207i	$0.476203\pi$
$110$	0	0
$111$	− 8872.26i	− 0.720093i
$112$	0	0
$113$	16994.7	1.33093	0.665467	−	0.746427i	$-0.268232\pi$
$113$	16994.7	1.33093	0.665467	+	0.746427i	$0.268232\pi$
$114$	0	0
$115$	− 16405.1i	− 1.24046i
$116$	0	0
$117$	−813.092	−0.0593975
$118$	0	0
$119$	2993.23i	0.211372i
$120$	0	0
$121$	3797.27	0.259359
$122$	0	0
$123$	882.197i	0.0583117i
$124$	0	0
$125$	16276.2	1.04168
$126$	0	0
$127$	− 22216.9i	− 1.37745i	−0.725024	−	0.688724i	$-0.758171\pi$
$127$	− 22216.9i	− 1.37745i	0.725024	−	0.688724i	$-0.241829\pi$
$128$	0	0
$129$	−8214.06	−0.493604
$130$	0	0
$131$	− 15326.7i	− 0.893113i	−0.894755	−	0.446557i	$-0.852650\pi$
$131$	− 15326.7i	− 0.893113i	0.894755	−	0.446557i	$-0.147350\pi$
$132$	0	0
$133$	20498.1	1.15881
$134$	0	0
$135$	− 10814.5i	− 0.593387i
$136$	0	0
$137$	−4816.22	−0.256605	−0.128303	−	0.991735i	$-0.540953\pi$
$137$	−4816.22	−0.256605	−0.128303	+	0.991735i	$0.540953\pi$
$138$	0	0
$139$	26354.8i	1.36405i	0.731330	+	0.682024i	$0.238900\pi$
$139$	26354.8i	1.36405i	−0.731330	+	0.682024i	$0.761100\pi$
$140$	0	0
$141$	2848.02	0.143254
$142$	0	0
$143$	− 1402.34i	− 0.0685773i
$144$	0	0
$145$	21164.8	1.00665
$146$	0	0
$147$	3346.85i	0.154882i
$148$	0	0
$149$	35333.0	1.59150	0.795752	−	0.605622i	$-0.207076\pi$
$149$	35333.0	1.59150	0.795752	+	0.605622i	$0.207076\pi$
$150$	0	0
$151$	35811.1i	1.57059i	0.619120	+	0.785296i	$0.287489\pi$
$151$	35811.1i	1.57059i	−0.619120	+	0.785296i	$0.712511\pi$
$152$	0	0
$153$	4430.37	0.189259
$154$	0	0
$155$	18295.0i	0.761498i
$156$	0	0
$157$	27036.6	1.09687	0.548433	−	0.836195i	$-0.315225\pi$
$157$	27036.6	1.09687	0.548433	+	0.836195i	$0.315225\pi$
$158$	0	0
$159$	− 13675.0i	− 0.540921i
$160$	0	0
$161$	−39728.3	−1.53267
$162$	0	0
$163$	− 15689.0i	− 0.590499i	−0.955420	−	0.295250i	$-0.904597\pi$
$163$	− 15689.0i	− 0.590499i	0.955420	−	0.295250i	$-0.0954028\pi$
$164$	0	0
$165$	7965.52	0.292581
$166$	0	0
$167$	− 10137.3i	− 0.363487i	−0.983346	−	0.181743i	$-0.941826\pi$
$167$	− 10137.3i	− 0.363487i	0.983346	−	0.181743i	$-0.0581740\pi$
$168$	0	0
$169$	−28379.6	−0.993650
$170$	0	0
$171$	− 30339.8i	− 1.03758i
$172$	0	0
$173$	39217.1	1.31034	0.655169	−	0.755483i	$-0.272598\pi$
$173$	39217.1	1.31034	0.655169	+	0.755483i	$0.272598\pi$
$174$	0	0
$175$	− 13921.0i	− 0.454562i
$176$	0	0
$177$	25871.0	0.825783
$178$	0	0
$179$	35042.8i	1.09369i	0.837235	+	0.546843i	$0.184171\pi$
$179$	35042.8i	1.09369i	−0.837235	+	0.546843i	$0.815829\pi$
$180$	0	0
$181$	−14059.3	−0.429147	−0.214574	−	0.976708i	$-0.568836\pi$
$181$	−14059.3	−0.429147	−0.214574	+	0.976708i	$0.568836\pi$
$182$	0	0
$183$	− 27596.5i	− 0.824045i
$184$	0	0
$185$	32909.5	0.961562
$186$	0	0
$187$	7641.05i	0.218509i
$188$	0	0
$189$	−26189.5	−0.733167
$190$	0	0
$191$	45804.6i	1.25558i	0.778385	+	0.627788i	$0.216039\pi$
$191$	45804.6i	1.25558i	−0.778385	+	0.627788i	$0.783961\pi$
$192$	0	0
$193$	−40765.7	−1.09441	−0.547206	−	0.836998i	$-0.684308\pi$
$193$	−40765.7	−1.09441	−0.547206	+	0.836998i	$0.684308\pi$
$194$	0	0
$195$	1030.12i	0.0270906i
$196$	0	0
$197$	8098.75	0.208682	0.104341	−	0.994542i	$-0.466727\pi$
$197$	8098.75	0.208682	0.104341	+	0.994542i	$0.466727\pi$
$198$	0	0
$199$	12023.2i	0.303609i	0.988411	+	0.151805i	$0.0485085\pi$
$199$	12023.2i	0.303609i	−0.988411	+	0.151805i	$0.951492\pi$
$200$	0	0
$201$	−15979.4	−0.395519
$202$	0	0
$203$	− 51254.9i	− 1.24378i
$204$	0	0
$205$	−3272.29	−0.0778654
$206$	0	0
$207$	58803.1i	1.37233i
$208$	0	0
$209$	52327.1	1.19794
$210$	0	0
$211$	− 32994.3i	− 0.741094i	−0.928814	−	0.370547i	$-0.879170\pi$
$211$	− 32994.3i	− 0.741094i	0.928814	−	0.370547i	$-0.120830\pi$
$212$	0	0
$213$	15167.6	0.334315
$214$	0	0
$215$	− 30468.0i	− 0.659125i
$216$	0	0
$217$	44305.0	0.940878
$218$	0	0
$219$	− 21846.7i	− 0.455510i
$220$	0	0
$221$	−988.160	−0.0202322
$222$	0	0
$223$	8325.83i	0.167424i	0.996490	+	0.0837120i	$0.0266776\pi$
$223$	8325.83i	0.167424i	−0.996490	+	0.0837120i	$0.973322\pi$
$224$	0	0
$225$	−20604.8	−0.407009
$226$	0	0
$227$	98932.8i	1.91994i	0.280096	+	0.959972i	$0.409634\pi$
$227$	98932.8i	1.91994i	−0.280096	+	0.959972i	$0.590366\pi$
$228$	0	0
$229$	−39028.4	−0.744235	−0.372117	−	0.928186i	$-0.621368\pi$
$229$	−39028.4	−0.744235	−0.372117	+	0.928186i	$0.621368\pi$
$230$	0	0
$231$	− 19290.1i	− 0.361502i
$232$	0	0
$233$	−47062.2	−0.866882	−0.433441	−	0.901182i	$-0.642701\pi$
$233$	−47062.2	−0.866882	−0.433441	+	0.901182i	$0.642701\pi$
$234$	0	0
$235$	10564.0i	0.191291i
$236$	0	0
$237$	−24822.8	−0.441930
$238$	0	0
$239$	18051.8i	0.316028i	0.987437	+	0.158014i	$0.0505091\pi$
$239$	18051.8i	0.316028i	−0.987437	+	0.158014i	$0.949491\pi$
$240$	0	0
$241$	87375.0	1.50437	0.752183	−	0.658955i	$-0.229001\pi$
$241$	87375.0	1.50437	0.752183	+	0.658955i	$0.229001\pi$
$242$	0	0
$243$	60972.8i	1.03258i
$244$	0	0
$245$	−12414.3	−0.206819
$246$	0	0
$247$	6767.07i	0.110919i
$248$	0	0
$249$	−17917.8	−0.288991
$250$	0	0
$251$	− 14707.9i	− 0.233456i	−0.993164	−	0.116728i	$-0.962760\pi$
$251$	− 14707.9i	− 0.233456i	0.993164	−	0.116728i	$-0.0372405\pi$
$252$	0	0
$253$	−101418.	−1.58443
$254$	0	0
$255$	− 5612.92i	− 0.0863194i
$256$	0	0
$257$	−105989.	−1.60470	−0.802351	−	0.596852i	$-0.796418\pi$
$257$	−105989.	−1.60470	−0.802351	+	0.596852i	$0.796418\pi$
$258$	0	0
$259$	− 79697.0i	− 1.18807i
$260$	0	0
$261$	−75863.8	−1.11366
$262$	0	0
$263$	2401.32i	0.0347166i	0.999849	+	0.0173583i	$0.00552560\pi$
$263$	2401.32i	0.0347166i	−0.999849	+	0.0173583i	$0.994474\pi$
$264$	0	0
$265$	50724.1	0.722309
$266$	0	0
$267$	3180.26i	0.0446108i
$268$	0	0
$269$	−17257.4	−0.238490	−0.119245	−	0.992865i	$-0.538047\pi$
$269$	−17257.4	−0.238490	−0.119245	+	0.992865i	$0.538047\pi$
$270$	0	0
$271$	− 119967.i	− 1.63351i	−0.576982	−	0.816757i	$-0.695770\pi$
$271$	− 119967.i	− 1.63351i	0.576982	−	0.816757i	$-0.304230\pi$
$272$	0	0
$273$	2494.65	0.0334722
$274$	0	0
$275$	− 35537.1i	− 0.469912i
$276$	0	0
$277$	−101645.	−1.32473	−0.662363	−	0.749183i	$-0.730446\pi$
$277$	−101645.	−1.32473	−0.662363	+	0.749183i	$0.730446\pi$
$278$	0	0
$279$	− 65577.1i	− 0.842450i
$280$	0	0
$281$	−12028.5	−0.152335	−0.0761675	−	0.997095i	$-0.524268\pi$
$281$	−12028.5	−0.152335	−0.0761675	+	0.997095i	$0.524268\pi$
$282$	0	0
$283$	98095.9i	1.22484i	0.790534	+	0.612418i	$0.209803\pi$
$283$	98095.9i	1.22484i	−0.790534	+	0.612418i	$0.790197\pi$
$284$	0	0
$285$	−38438.1	−0.473230
$286$	0	0
$287$	7924.52i	0.0962076i
$288$	0	0
$289$	−78136.7	−0.935534
$290$	0	0
$291$	− 57981.2i	− 0.684701i
$292$	0	0
$293$	−2060.54	−0.0240019	−0.0120010	−	0.999928i	$-0.503820\pi$
$293$	−2060.54	−0.0240019	−0.0120010	+	0.999928i	$0.503820\pi$
$294$	0	0
$295$	95962.0i	1.10269i
$296$	0	0
$297$	−66855.8	−0.757925
$298$	0	0
$299$	− 13115.6i	− 0.146705i
$300$	0	0
$301$	−73784.5	−0.814390
$302$	0	0
$303$	19059.5i	0.207599i
$304$	0	0
$305$	102362.	1.10037
$306$	0	0
$307$	− 76322.7i	− 0.809798i	−0.914361	−	0.404899i	$-0.867307\pi$
$307$	− 76322.7i	− 0.809798i	0.914361	−	0.404899i	$-0.132693\pi$
$308$	0	0
$309$	−59949.9	−0.627873
$310$	0	0
$311$	93007.1i	0.961602i	0.876830	+	0.480801i	$0.159654\pi$
$311$	93007.1i	0.961602i	−0.876830	+	0.480801i	$0.840346\pi$
$312$	0	0
$313$	44008.6	0.449210	0.224605	−	0.974450i	$-0.427891\pi$
$313$	44008.6	0.449210	0.224605	+	0.974450i	$0.427891\pi$
$314$	0	0
$315$	− 41486.7i	− 0.418107i
$316$	0	0
$317$	110474.	1.09936	0.549680	−	0.835375i	$-0.314750\pi$
$317$	110474.	1.09936	0.549680	+	0.835375i	$0.314750\pi$
$318$	0	0
$319$	− 130842.i	− 1.28578i
$320$	0	0
$321$	−817.107	−0.00792992
$322$	0	0
$323$	− 36872.4i	− 0.353424i
$324$	0	0
$325$	4595.75	0.0435100
$326$	0	0
$327$	8059.72i	0.0753745i
$328$	0	0
$329$	25583.0	0.236352
$330$	0	0
$331$	210881.i	1.92478i	0.271673	+	0.962390i	$0.412423\pi$
$331$	210881.i	1.92478i	−0.271673	+	0.962390i	$0.587577\pi$
$332$	0	0
$333$	−117962.	−1.06378
$334$	0	0
$335$	− 59271.5i	− 0.528149i
$336$	0	0
$337$	46756.0	0.411697	0.205848	−	0.978584i	$-0.434005\pi$
$337$	46756.0	0.411697	0.205848	+	0.978584i	$0.434005\pi$
$338$	0	0
$339$	77176.1i	0.671558i
$340$	0	0
$341$	113101.	0.972650
$342$	0	0
$343$	128006.i	1.08803i
$344$	0	0
$345$	74498.7	0.625908
$346$	0	0
$347$	− 41355.2i	− 0.343456i	−0.985144	−	0.171728i	$-0.945065\pi$
$347$	− 41355.2i	− 0.343456i	0.985144	−	0.171728i	$-0.0549350\pi$
$348$	0	0
$349$	−60146.7	−0.493812	−0.246906	−	0.969039i	$-0.579414\pi$
$349$	−60146.7	−0.493812	−0.246906	+	0.969039i	$0.579414\pi$
$350$	0	0
$351$	− 8645.96i	− 0.0701777i
$352$	0	0
$353$	−12988.7	−0.104236	−0.0521178	−	0.998641i	$-0.516597\pi$
$353$	−12988.7	−0.104236	−0.0521178	+	0.998641i	$0.516597\pi$
$354$	0	0
$355$	56260.3i	0.446422i
$356$	0	0
$357$	−13592.8	−0.106653
$358$	0	0
$359$	− 191555.i	− 1.48630i	−0.669127	−	0.743148i	$-0.733332\pi$
$359$	− 191555.i	− 1.48630i	0.669127	−	0.743148i	$-0.266668\pi$
$360$	0	0
$361$	−122187.	−0.937582
$362$	0	0
$363$	17244.1i	0.130866i
$364$	0	0
$365$	81034.9	0.608256
$366$	0	0
$367$	163572.i	1.21444i	0.794532	+	0.607222i	$0.207716\pi$
$367$	163572.i	1.21444i	−0.794532	+	0.607222i	$0.792284\pi$
$368$	0	0
$369$	11729.3	0.0861430
$370$	0	0
$371$	− 122839.i	− 0.892458i
$372$	0	0
$373$	−55059.2	−0.395742	−0.197871	−	0.980228i	$-0.563403\pi$
$373$	−55059.2	−0.395742	−0.197871	+	0.980228i	$0.563403\pi$
$374$	0	0
$375$	73913.1i	0.525605i
$376$	0	0
$377$	16920.8	0.119053
$378$	0	0
$379$	− 108892.i	− 0.758083i	−0.925380	−	0.379041i	$-0.876254\pi$
$379$	− 108892.i	− 0.758083i	0.925380	−	0.379041i	$-0.123746\pi$
$380$	0	0
$381$	100891.	0.695027
$382$	0	0
$383$	169594.i	1.15615i	0.815985	+	0.578073i	$0.196195\pi$
$383$	169594.i	1.15615i	−0.815985	+	0.578073i	$0.803805\pi$
$384$	0	0
$385$	71552.0	0.482726
$386$	0	0
$387$	109211.i	0.729194i
$388$	0	0
$389$	183067.	1.20979	0.604896	−	0.796304i	$-0.293215\pi$
$389$	183067.	1.20979	0.604896	+	0.796304i	$0.293215\pi$
$390$	0	0
$391$	71464.0i	0.467449i
$392$	0	0
$393$	69601.4	0.450644
$394$	0	0
$395$	− 92074.1i	− 0.590124i
$396$	0	0
$397$	145471.	0.922984	0.461492	−	0.887144i	$-0.347314\pi$
$397$	145471.	0.922984	0.461492	+	0.887144i	$0.347314\pi$
$398$	0	0
$399$	93085.7i	0.584706i
$400$	0	0
$401$	−188314.	−1.17110	−0.585550	−	0.810636i	$-0.699121\pi$
$401$	−188314.	−1.17110	−0.585550	+	0.810636i	$0.699121\pi$
$402$	0	0
$403$	14626.5i	0.0900595i
$404$	0	0
$405$	−33268.5	−0.202826
$406$	0	0
$407$	− 203448.i	− 1.22819i
$408$	0	0
$409$	−88454.1	−0.528776	−0.264388	−	0.964416i	$-0.585170\pi$
$409$	−88454.1	−0.528776	−0.264388	+	0.964416i	$0.585170\pi$
$410$	0	0
$411$	− 21871.4i	− 0.129477i
$412$	0	0
$413$	232391.	1.36245
$414$	0	0
$415$	− 66461.5i	− 0.385899i
$416$	0	0
$417$	−119682.	−0.688266
$418$	0	0
$419$	146392.i	0.833851i	0.908941	+	0.416925i	$0.136892\pi$
$419$	146392.i	0.833851i	−0.908941	+	0.416925i	$0.863108\pi$
$420$	0	0
$421$	127042.	0.716774	0.358387	−	0.933573i	$-0.383327\pi$
$421$	127042.	0.716774	0.358387	+	0.933573i	$0.383327\pi$
$422$	0	0
$423$	− 37866.1i	− 0.211626i
$424$	0	0
$425$	−25041.3	−0.138637
$426$	0	0
$427$	− 247891.i	− 1.35958i
$428$	0	0
$429$	6368.28	0.0346025
$430$	0	0
$431$	− 113658.i	− 0.611850i	−0.952056	−	0.305925i	$-0.901034\pi$
$431$	− 113658.i	− 0.611850i	0.952056	−	0.305925i	$-0.0989656\pi$
$432$	0	0
$433$	−9580.62	−0.0510996	−0.0255498	−	0.999674i	$-0.508134\pi$
$433$	−9580.62	−0.0510996	−0.0255498	+	0.999674i	$0.508134\pi$
$434$	0	0
$435$	96113.3i	0.507931i
$436$	0	0
$437$	489397.	2.56270
$438$	0	0
$439$	− 310714.i	− 1.61225i	−0.591747	−	0.806123i	$-0.701562\pi$
$439$	− 310714.i	− 1.61225i	0.591747	−	0.806123i	$-0.298438\pi$
$440$	0	0
$441$	44498.3	0.228805
$442$	0	0
$443$	− 89033.9i	− 0.453678i	−0.973932	−	0.226839i	$-0.927161\pi$
$443$	− 89033.9i	− 0.453678i	0.973932	−	0.226839i	$-0.0728392\pi$
$444$	0	0
$445$	−11796.4	−0.0595702
$446$	0	0
$447$	160454.i	0.803035i
$448$	0	0
$449$	282195.	1.39977	0.699885	−	0.714256i	$-0.253235\pi$
$449$	282195.	1.39977	0.699885	+	0.714256i	$0.253235\pi$
$450$	0	0
$451$	20229.5i	0.0994564i
$452$	0	0
$453$	−162625.	−0.792484
$454$	0	0
$455$	9253.28i	0.0446965i
$456$	0	0
$457$	100299.	0.480246	0.240123	−	0.970742i	$-0.422812\pi$
$457$	100299.	0.480246	0.240123	+	0.970742i	$0.422812\pi$
$458$	0	0
$459$	47110.1i	0.223608i
$460$	0	0
$461$	309619.	1.45689	0.728443	−	0.685106i	$-0.240244\pi$
$461$	309619.	1.45689	0.728443	+	0.685106i	$0.240244\pi$
$462$	0	0
$463$	15604.3i	0.0727917i	0.999337	+	0.0363959i	$0.0115877\pi$
$463$	15604.3i	0.0727917i	−0.999337	+	0.0363959i	$0.988412\pi$
$464$	0	0
$465$	−83080.9	−0.384234
$466$	0	0
$467$	137144.i	0.628844i	0.949283	+	0.314422i	$0.101811\pi$
$467$	137144.i	0.628844i	−0.949283	+	0.314422i	$0.898189\pi$
$468$	0	0
$469$	−143538.	−0.652561
$470$	0	0
$471$	122778.i	0.553452i
$472$	0	0
$473$	−188355.	−0.841891
$474$	0	0
$475$	171487.i	0.760051i
$476$	0	0
$477$	−181817.	−0.799095
$478$	0	0
$479$	16795.8i	0.0732031i	0.999330	+	0.0366016i	$0.0116532\pi$
$479$	16795.8i	0.0732031i	−0.999330	+	0.0366016i	$0.988347\pi$
$480$	0	0
$481$	26310.5	0.113720
$482$	0	0
$483$	− 180414.i	− 0.773349i
$484$	0	0
$485$	215067.	0.914303
$486$	0	0
$487$	− 119105.i	− 0.502194i	−0.967962	−	0.251097i	$-0.919209\pi$
$487$	− 119105.i	− 0.502194i	0.967962	−	0.251097i	$-0.0807913\pi$
$488$	0	0
$489$	71246.5	0.297952
$490$	0	0
$491$	379148.i	1.57270i	0.617783	+	0.786349i	$0.288031\pi$
$491$	379148.i	1.57270i	−0.617783	+	0.786349i	$0.711969\pi$
$492$	0	0
$493$	−92198.1	−0.379340
$494$	0	0
$495$	− 105906.i	− 0.432226i
$496$	0	0
$497$	136246.	0.551582
$498$	0	0
$499$	54844.8i	0.220260i	0.993917	+	0.110130i	$0.0351266\pi$
$499$	54844.8i	0.220260i	−0.993917	+	0.110130i	$0.964873\pi$
$500$	0	0
$501$	46035.3	0.183407
$502$	0	0
$503$	146931.i	0.580732i	0.956916	+	0.290366i	$0.0937771\pi$
$503$	146931.i	0.580732i	−0.956916	+	0.290366i	$0.906223\pi$
$504$	0	0
$505$	−70696.4	−0.277213
$506$	0	0
$507$	− 128877.i	− 0.501372i
$508$	0	0
$509$	156866.	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$509$	156866.	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$510$	0	0
$511$	− 196243.i	− 0.751539i
$512$	0	0
$513$	322617.	1.22589
$514$	0	0
$515$	− 222370.i	− 0.838419i
$516$	0	0
$517$	65307.6	0.244333
$518$	0	0
$519$	178092.i	0.661165i
$520$	0	0
$521$	433939.	1.59865	0.799325	−	0.600899i	$-0.205191\pi$
$521$	433939.	1.59865	0.799325	+	0.600899i	$0.205191\pi$
$522$	0	0
$523$	− 244305.i	− 0.893161i	−0.894744	−	0.446580i	$-0.852642\pi$
$523$	− 244305.i	− 0.893161i	0.894744	−	0.446580i	$-0.147358\pi$
$524$	0	0
$525$	63217.7	0.229361
$526$	0	0
$527$	− 79696.6i	− 0.286958i
$528$	0	0
$529$	−668681.	−2.38950
$530$	0	0
$531$	− 343969.i	− 1.21992i
$532$	0	0
$533$	−2616.13	−0.00920885
$534$	0	0
$535$	− 3030.86i	− 0.0105891i
$536$	0	0
$537$	−159136.	−0.551848
$538$	0	0
$539$	76746.2i	0.264167i
$540$	0	0
$541$	−263893.	−0.901642	−0.450821	−	0.892614i	$-0.648869\pi$
$541$	−263893.	−0.901642	−0.450821	+	0.892614i	$0.648869\pi$
$542$	0	0
$543$	− 63845.8i	− 0.216537i
$544$	0	0
$545$	−29895.5	−0.100650
$546$	0	0
$547$	− 5684.58i	− 0.0189987i	−0.999955	−	0.00949935i	$-0.996976\pi$
$547$	− 5684.58i	− 0.0189987i	0.999955	−	0.00949935i	$-0.00302378\pi$
$548$	0	0
$549$	−366911.	−1.21735
$550$	0	0
$551$	631387.i	2.07966i
$552$	0	0
$553$	−222976.	−0.729135
$554$	0	0
$555$	149448.i	0.485181i
$556$	0	0
$557$	229010.	0.738150	0.369075	−	0.929400i	$-0.379675\pi$
$557$	229010.	0.738150	0.369075	+	0.929400i	$0.379675\pi$
$558$	0	0
$559$	− 24358.6i	− 0.0779522i
$560$	0	0
$561$	−34699.4	−0.110255
$562$	0	0
$563$	8586.54i	0.0270895i	0.999908	+	0.0135448i	$0.00431156\pi$
$563$	8586.54i	0.0270895i	−0.999908	+	0.0135448i	$0.995688\pi$
$564$	0	0
$565$	−286266.	−0.896752
$566$	0	0
$567$	80566.5i	0.250604i
$568$	0	0
$569$	−58381.6	−0.180323	−0.0901616	−	0.995927i	$-0.528738\pi$
$569$	−58381.6	−0.180323	−0.0901616	+	0.995927i	$0.528738\pi$
$570$	0	0
$571$	512966.i	1.57332i	0.617389	+	0.786658i	$0.288191\pi$
$571$	512966.i	1.57332i	−0.617389	+	0.786658i	$0.711809\pi$
$572$	0	0
$573$	−208007.	−0.633533
$574$	0	0
$575$	− 332366.i	− 1.00527i
$576$	0	0
$577$	457649.	1.37461	0.687307	−	0.726367i	$-0.258793\pi$
$577$	457649.	1.37461	0.687307	+	0.726367i	$0.258793\pi$
$578$	0	0
$579$	− 185125.i	− 0.552214i
$580$	0	0
$581$	−160950.	−0.476803
$582$	0	0
$583$	− 313580.i	− 0.922595i
$584$	0	0
$585$	13696.1	0.0400206
$586$	0	0
$587$	− 76953.3i	− 0.223332i	−0.993746	−	0.111666i	$-0.964381\pi$
$587$	− 76953.3i	− 0.223332i	0.993746	−	0.111666i	$-0.0356187\pi$
$588$	0	0
$589$	−545775.	−1.57320
$590$	0	0
$591$	36777.9i	0.105296i
$592$	0	0
$593$	−559786.	−1.59189	−0.795944	−	0.605370i	$-0.793025\pi$
$593$	−559786.	−1.59189	−0.795944	+	0.605370i	$0.793025\pi$
$594$	0	0
$595$	− 50419.2i	− 0.142417i
$596$	0	0
$597$	−54599.8	−0.153194
$598$	0	0
$599$	209237.i	0.583157i	0.956547	+	0.291579i	$0.0941805\pi$
$599$	209237.i	0.583157i	−0.956547	+	0.291579i	$0.905820\pi$
$600$	0	0
$601$	−2016.35	−0.00558236	−0.00279118	−	0.999996i	$-0.500888\pi$
$601$	−2016.35	−0.00558236	−0.00279118	+	0.999996i	$0.500888\pi$
$602$	0	0
$603$	212455.i	0.584295i
$604$	0	0
$605$	−63962.8	−0.174750
$606$	0	0
$607$	648350.i	1.75967i	0.475275	+	0.879837i	$0.342349\pi$
$607$	648350.i	1.75967i	−0.475275	+	0.879837i	$0.657651\pi$
$608$	0	0
$609$	232758.	0.627581
$610$	0	0
$611$	8445.75i	0.0226233i
$612$	0	0
$613$	−303258.	−0.807033	−0.403516	−	0.914972i	$-0.632212\pi$
$613$	−303258.	−0.807033	−0.403516	+	0.914972i	$0.632212\pi$
$614$	0	0
$615$	− 14860.1i	− 0.0392890i
$616$	0	0
$617$	−174055.	−0.457210	−0.228605	−	0.973519i	$-0.573416\pi$
$617$	−174055.	−0.457210	−0.228605	+	0.973519i	$0.573416\pi$
$618$	0	0
$619$	− 577628.i	− 1.50753i	−0.657142	−	0.753767i	$-0.728235\pi$
$619$	− 577628.i	− 1.50753i	0.657142	−	0.753767i	$-0.271765\pi$
$620$	0	0
$621$	−625279.	−1.62140
$622$	0	0
$623$	28567.3i	0.0736027i
$624$	0	0
$625$	−60871.5	−0.155831
$626$	0	0
$627$	237627.i	0.604450i
$628$	0	0
$629$	−143360.	−0.362349
$630$	0	0
$631$	166511.i	0.418200i	0.977894	+	0.209100i	$0.0670533\pi$
$631$	166511.i	0.418200i	−0.977894	+	0.209100i	$0.932947\pi$
$632$	0	0
$633$	149833.	0.373939
$634$	0	0
$635$	374230.i	0.928092i
$636$	0	0
$637$	−9925.01	−0.0244598
$638$	0	0
$639$	− 201661.i	− 0.493879i
$640$	0	0
$641$	515858.	1.25549	0.627746	−	0.778418i	$-0.283978\pi$
$641$	515858.	1.25549	0.627746	+	0.778418i	$0.283978\pi$
$642$	0	0
$643$	585939.i	1.41720i	0.705611	+	0.708599i	$0.250673\pi$
$643$	585939.i	1.41720i	−0.705611	+	0.708599i	$0.749327\pi$
$644$	0	0
$645$	138361.	0.332579
$646$	0	0
$647$	− 734055.i	− 1.75356i	−0.480895	−	0.876778i	$-0.659688\pi$
$647$	− 734055.i	− 1.75356i	0.480895	−	0.876778i	$-0.340312\pi$
$648$	0	0
$649$	593243.	1.40846
$650$	0	0
$651$	201197.i	0.474745i
$652$	0	0
$653$	−78218.9	−0.183436	−0.0917181	−	0.995785i	$-0.529236\pi$
$653$	−78218.9	−0.183436	−0.0917181	+	0.995785i	$0.529236\pi$
$654$	0	0
$655$	258169.i	0.601759i
$656$	0	0
$657$	−290464.	−0.672918
$658$	0	0
$659$	− 4105.91i	− 0.00945450i	−0.999989	−	0.00472725i	$-0.998495\pi$
$659$	− 4105.91i	− 0.00945450i	0.999989	−	0.00472725i	$-0.00150474\pi$
$660$	0	0
$661$	−478700.	−1.09562	−0.547811	−	0.836602i	$-0.684539\pi$
$661$	−478700.	−1.09562	−0.547811	+	0.836602i	$0.684539\pi$
$662$	0	0
$663$	− 4487.42i	− 0.0102087i
$664$	0	0
$665$	−345279.	−0.780776
$666$	0	0
$667$	− 1.22372e6i	− 2.75062i
$668$	0	0
$669$	−37809.1	−0.0844782
$670$	0	0
$671$	− 632810.i	− 1.40549i
$672$	0	0
$673$	292887.	0.646651	0.323326	−	0.946288i	$-0.395199\pi$
$673$	292887.	0.646651	0.323326	+	0.946288i	$0.395199\pi$
$674$	0	0
$675$	− 219100.i	− 0.480878i
$676$	0	0
$677$	358940.	0.783149	0.391575	−	0.920146i	$-0.371930\pi$
$677$	358940.	0.783149	0.391575	+	0.920146i	$0.371930\pi$
$678$	0	0
$679$	− 520828.i	− 1.12968i
$680$	0	0
$681$	−449272.	−0.968758
$682$	0	0
$683$	− 166893.i	− 0.357765i	−0.983870	−	0.178883i	$-0.942752\pi$
$683$	− 166893.i	− 0.357765i	0.983870	−	0.178883i	$-0.0572482\pi$
$684$	0	0
$685$	81126.4	0.172895
$686$	0	0
$687$	− 177235.i	− 0.375523i
$688$	0	0
$689$	40552.9	0.0854248
$690$	0	0
$691$	102959.i	0.215629i	0.994171	+	0.107814i	$0.0343853\pi$
$691$	102959.i	0.215629i	−0.994171	+	0.107814i	$0.965615\pi$
$692$	0	0
$693$	−256473.	−0.534042
$694$	0	0
$695$	− 443930.i	− 0.919063i
$696$	0	0
$697$	14254.8	0.0293423
$698$	0	0
$699$	− 213718.i	− 0.437408i
$700$	0	0
$701$	196855.	0.400599	0.200299	−	0.979735i	$-0.435808\pi$
$701$	196855.	0.400599	0.200299	+	0.979735i	$0.435808\pi$
$702$	0	0
$703$	981753.i	1.98651i
$704$	0	0
$705$	−47973.3	−0.0965209
$706$	0	0
$707$	171206.i	0.342515i
$708$	0	0
$709$	−621291.	−1.23595	−0.617977	−	0.786196i	$-0.712048\pi$
$709$	−621291.	−1.23595	−0.617977	+	0.786196i	$0.712048\pi$
$710$	0	0
$711$	330033.i	0.652857i
$712$	0	0
$713$	1.05779e6	2.08075
$714$	0	0
$715$	23621.6i	0.0462058i
$716$	0	0
$717$	−81976.7	−0.159460
$718$	0	0
$719$	699397.i	1.35290i	0.736488	+	0.676450i	$0.236483\pi$
$719$	699397.i	1.35290i	−0.736488	+	0.676450i	$0.763517\pi$
$720$	0	0
$721$	−538513.	−1.03592
$722$	0	0
$723$	396786.i	0.759067i
$724$	0	0
$725$	428796.	0.815784
$726$	0	0
$727$	− 526955.i	− 0.997022i	−0.866883	−	0.498511i	$-0.833880\pi$
$727$	− 526955.i	− 0.997022i	0.866883	−	0.498511i	$-0.166120\pi$
$728$	0	0
$729$	−116910.	−0.219987
$730$	0	0
$731$	132725.i	0.248381i
$732$	0	0
$733$	−13466.6	−0.0250640	−0.0125320	−	0.999921i	$-0.503989\pi$
$733$	−13466.6	−0.0250640	−0.0125320	+	0.999921i	$0.503989\pi$
$734$	0	0
$735$	− 56375.8i	− 0.104356i
$736$	0	0
$737$	−366420.	−0.674597
$738$	0	0
$739$	352928.i	0.646245i	0.946357	+	0.323122i	$0.104733\pi$
$739$	352928.i	0.646245i	−0.946357	+	0.323122i	$0.895267\pi$
$740$	0	0
$741$	−30730.5	−0.0559672
$742$	0	0
$743$	539226.i	0.976772i	0.872628	+	0.488386i	$0.162414\pi$
$743$	539226.i	0.976772i	−0.872628	+	0.488386i	$0.837586\pi$
$744$	0	0
$745$	−595164.	−1.07232
$746$	0	0
$747$	238227.i	0.426923i
$748$	0	0
$749$	−7339.83	−0.0130835
$750$	0	0
$751$	− 301606.i	− 0.534762i	−0.963591	−	0.267381i	$-0.913842\pi$
$751$	− 301606.i	− 0.534762i	0.963591	−	0.267381i	$-0.0861582\pi$
$752$	0	0
$753$	66791.4	0.117796
$754$	0	0
$755$	− 603217.i	− 1.05823i
$756$	0	0
$757$	402304.	0.702042	0.351021	−	0.936368i	$-0.385835\pi$
$757$	402304.	0.702042	0.351021	+	0.936368i	$0.385835\pi$
$758$	0	0
$759$	− 460556.i	− 0.799464i
$760$	0	0
$761$	−622961.	−1.07570	−0.537851	−	0.843040i	$-0.680764\pi$
$761$	−622961.	−1.07570	−0.537851	+	0.843040i	$0.680764\pi$
$762$	0	0
$763$	72398.1i	0.124359i
$764$	0	0
$765$	−74627.0	−0.127518
$766$	0	0
$767$	76719.7i	0.130412i
$768$	0	0
$769$	563590.	0.953039	0.476520	−	0.879164i	$-0.341898\pi$
$769$	563590.	0.953039	0.476520	+	0.879164i	$0.341898\pi$
$770$	0	0
$771$	− 481316.i	− 0.809694i
$772$	0	0
$773$	510328.	0.854063	0.427032	−	0.904237i	$-0.359559\pi$
$773$	510328.	0.854063	0.427032	+	0.904237i	$0.359559\pi$
$774$	0	0
$775$	370654.i	0.617114i
$776$	0	0
$777$	361919.	0.599472
$778$	0	0
$779$	− 97618.9i	− 0.160864i
$780$	0	0
$781$	347805.	0.570208
$782$	0	0
$783$	− 806693.i	− 1.31578i
$784$	0	0
$785$	−455416.	−0.739042
$786$	0	0
$787$	970831.i	1.56745i	0.621107	+	0.783726i	$0.286683\pi$
$787$	970831.i	1.56745i	−0.621107	+	0.783726i	$0.713317\pi$
$788$	0	0
$789$	−10904.8	−0.0175172
$790$	0	0
$791$	693250.i	1.10799i
$792$	0	0
$793$	81836.6	0.130137
$794$	0	0
$795$	230348.i	0.364460i
$796$	0	0
$797$	1.12132e6	1.76528	0.882639	−	0.470051i	$-0.155765\pi$
$797$	1.12132e6	1.76528	0.882639	+	0.470051i	$0.155765\pi$
$798$	0	0
$799$	− 46019.1i	− 0.0720850i
$800$	0	0
$801$	42283.3	0.0659028
$802$	0	0
$803$	− 500963.i	− 0.776917i
$804$	0	0
$805$	669200.	1.03268
$806$	0	0
$807$	− 78368.9i	− 0.120336i
$808$	0	0
$809$	−533014.	−0.814406	−0.407203	−	0.913338i	$-0.633496\pi$
$809$	−533014.	−0.814406	−0.407203	+	0.913338i	$0.633496\pi$
$810$	0	0
$811$	736257.i	1.11941i	0.828693	+	0.559703i	$0.189085\pi$
$811$	736257.i	1.11941i	−0.828693	+	0.559703i	$0.810915\pi$
$812$	0	0
$813$	544792.	0.824232
$814$	0	0
$815$	264271.i	0.397864i
$816$	0	0
$817$	908921.	1.36170
$818$	0	0
$819$	− 33167.8i	− 0.0494480i
$820$	0	0
$821$	221216.	0.328194	0.164097	−	0.986444i	$-0.447529\pi$
$821$	221216.	0.328194	0.164097	+	0.986444i	$0.447529\pi$
$822$	0	0
$823$	427766.i	0.631548i	0.948834	+	0.315774i	$0.102264\pi$
$823$	427766.i	0.631548i	−0.948834	+	0.315774i	$0.897736\pi$
$824$	0	0
$825$	161381.	0.237106
$826$	0	0
$827$	− 87673.4i	− 0.128191i	−0.997944	−	0.0640954i	$-0.979584\pi$
$827$	− 87673.4i	− 0.128191i	0.997944	−	0.0640954i	$-0.0204162\pi$
$828$	0	0
$829$	−442564.	−0.643971	−0.321986	−	0.946745i	$-0.604350\pi$
$829$	−442564.	−0.643971	−0.321986	+	0.946745i	$0.604350\pi$
$830$	0	0
$831$	− 461588.i	− 0.668425i
$832$	0	0
$833$	54079.3	0.0779366
$834$	0	0
$835$	170757.i	0.244909i
$836$	0	0
$837$	697310.	0.995349
$838$	0	0
$839$	705591.i	1.00237i	0.865339	+	0.501186i	$0.167103\pi$
$839$	705591.i	1.00237i	−0.865339	+	0.501186i	$0.832897\pi$
$840$	0	0
$841$	871481.	1.23216
$842$	0	0
$843$	− 54623.8i	− 0.0768646i
$844$	0	0
$845$	478038.	0.669498
$846$	0	0
$847$	154899.i	0.215914i
$848$	0	0
$849$	−445472.	−0.618023
$850$	0	0
$851$	− 1.90278e6i	− 2.62742i
$852$	0	0
$853$	1.11225e6	1.52864	0.764320	−	0.644837i	$-0.223075\pi$
$853$	1.11225e6	1.52864	0.764320	+	0.644837i	$0.223075\pi$
$854$	0	0
$855$	511057.i	0.699096i
$856$	0	0
$857$	119068.	0.162119	0.0810597	−	0.996709i	$-0.474170\pi$
$857$	119068.	0.162119	0.0810597	+	0.996709i	$0.474170\pi$
$858$	0	0
$859$	− 78675.9i	− 0.106624i	−0.998578	−	0.0533120i	$-0.983022\pi$
$859$	− 78675.9i	− 0.106624i	0.998578	−	0.0533120i	$-0.0169778\pi$
$860$	0	0
$861$	−35986.7	−0.0485440
$862$	0	0
$863$	− 92999.5i	− 0.124870i	−0.998049	−	0.0624351i	$-0.980113\pi$
$863$	− 92999.5i	− 0.124870i	0.998049	−	0.0624351i	$-0.0198866\pi$
$864$	0	0
$865$	−660589.	−0.882874
$866$	0	0
$867$	− 354833.i	− 0.472048i
$868$	0	0
$869$	−569208.	−0.753757
$870$	0	0
$871$	− 47386.4i	− 0.0624622i
$872$	0	0
$873$	−770893.	−1.01150
$874$	0	0
$875$	663940.i	0.867187i
$876$	0	0
$877$	861315.	1.11986	0.559929	−	0.828541i	$-0.310829\pi$
$877$	861315.	1.11986	0.559929	+	0.828541i	$0.310829\pi$
$878$	0	0
$879$	− 9357.30i	− 0.0121108i
$880$	0	0
$881$	1.03692e6	1.33596	0.667981	−	0.744178i	$-0.267159\pi$
$881$	1.03692e6	1.33596	0.667981	+	0.744178i	$0.267159\pi$
$882$	0	0
$883$	− 1.07531e6i	− 1.37915i	−0.724212	−	0.689577i	$-0.757797\pi$
$883$	− 1.07531e6i	− 1.37915i	0.724212	−	0.689577i	$-0.242203\pi$
$884$	0	0
$885$	−435781.	−0.556393
$886$	0	0
$887$	97396.5i	0.123793i	0.998083	+	0.0618965i	$0.0197149\pi$
$887$	97396.5i	0.123793i	−0.998083	+	0.0618965i	$0.980285\pi$
$888$	0	0
$889$	906273.	1.14672
$890$	0	0
$891$	205668.i	0.259067i
$892$	0	0
$893$	−315146.	−0.395193
$894$	0	0
$895$	− 590276.i	− 0.736900i
$896$	0	0
$897$	59560.2	0.0740238
$898$	0	0
$899$	1.36469e6i	1.68855i
$900$	0	0
$901$	−220965.	−0.272191
$902$	0	0
$903$	− 335069.i	− 0.410922i
$904$	0	0
$905$	236820.	0.289149
$906$	0	0
$907$	172583.i	0.209790i	0.994483	+	0.104895i	$0.0334506\pi$
$907$	172583.i	0.209790i	−0.994483	+	0.104895i	$0.966549\pi$
$908$	0	0
$909$	253406.	0.306683
$910$	0	0
$911$	− 1.07491e6i	− 1.29519i	−0.761983	−	0.647597i	$-0.775774\pi$
$911$	− 1.07491e6i	− 1.29519i	0.761983	−	0.647597i	$-0.224226\pi$
$912$	0	0
$913$	−410869.	−0.492904
$914$	0	0
$915$	464846.i	0.555222i
$916$	0	0
$917$	625210.	0.743510
$918$	0	0
$919$	− 763121.i	− 0.903571i	−0.892127	−	0.451785i	$-0.850787\pi$
$919$	− 763121.i	− 0.903571i	0.892127	−	0.451785i	$-0.149213\pi$
$920$	0	0
$921$	346595.	0.408605
$922$	0	0
$923$	44979.0i	0.0527966i
$924$	0	0
$925$	666742.	0.779246
$926$	0	0
$927$	797069.i	0.927548i
$928$	0	0
$929$	−742596.	−0.860441	−0.430220	−	0.902724i	$-0.641564\pi$
$929$	−742596.	−0.860441	−0.430220	+	0.902724i	$0.641564\pi$
$930$	0	0
$931$	− 370344.i	− 0.427273i
$932$	0	0
$933$	−422362.	−0.485201
$934$	0	0
$935$	− 128709.i	− 0.147226i
$936$	0	0
$937$	−1.57493e6	−1.79383	−0.896915	−	0.442204i	$-0.854197\pi$
$937$	−1.57493e6	−1.79383	−0.896915	+	0.442204i	$0.854197\pi$
$938$	0	0
$939$	199851.i	0.226661i
$940$	0	0
$941$	43641.7	0.0492859	0.0246429	−	0.999696i	$-0.492155\pi$
$941$	43641.7	0.0492859	0.0246429	+	0.999696i	$0.492155\pi$
$942$	0	0
$943$	189200.i	0.212763i
$944$	0	0
$945$	441146.	0.493991
$946$	0	0
$947$	1.56638e6i	1.74661i	0.487170	+	0.873307i	$0.338029\pi$
$947$	1.56638e6i	1.74661i	−0.487170	+	0.873307i	$0.661971\pi$
$948$	0	0
$949$	64785.8	0.0719362
$950$	0	0
$951$	501681.i	0.554711i
$952$	0	0
$953$	−280777.	−0.309154	−0.154577	−	0.987981i	$-0.549402\pi$
$953$	−280777.	−0.309154	−0.154577	+	0.987981i	$0.549402\pi$
$954$	0	0
$955$	− 771552.i	− 0.845977i
$956$	0	0
$957$	594178.	0.648773
$958$	0	0
$959$	− 196464.i	− 0.213622i
$960$	0	0
$961$	−256127.	−0.277337
$962$	0	0
$963$	10863.9i	0.0117148i
$964$	0	0
$965$	686675.	0.737389
$966$	0	0
$967$	− 273606.i	− 0.292599i	−0.989240	−	0.146299i	$-0.953264\pi$
$967$	− 273606.i	− 0.292599i	0.989240	−	0.146299i	$-0.0467363\pi$
$968$	0	0
$969$	167444.	0.178329
$970$	0	0
$971$	− 813827.i	− 0.863164i	−0.902074	−	0.431582i	$-0.857955\pi$
$971$	− 813827.i	− 0.863164i	0.902074	−	0.431582i	$-0.142045\pi$
$972$	0	0
$973$	−1.07507e6	−1.13556
$974$	0	0
$975$	20870.1i	0.0219541i
$976$	0	0
$977$	195689.	0.205011	0.102506	−	0.994732i	$-0.467314\pi$
$977$	195689.	0.205011	0.102506	+	0.994732i	$0.467314\pi$
$978$	0	0
$979$	72926.0i	0.0760881i
$980$	0	0
$981$	107159.	0.111350
$982$	0	0
$983$	− 1.01172e6i	− 1.04701i	−0.852022	−	0.523506i	$-0.824624\pi$
$983$	− 1.01172e6i	− 1.04701i	0.852022	−	0.523506i	$-0.175376\pi$
$984$	0	0
$985$	−136419.	−0.140605
$986$	0	0
$987$	116177.i	0.119258i
$988$	0	0
$989$	−1.76162e6	−1.80103
$990$	0	0
$991$	− 419686.i	− 0.427344i	−0.976905	−	0.213672i	$-0.931458\pi$
$991$	− 419686.i	− 0.427344i	0.976905	−	0.213672i	$-0.0685424\pi$
$992$	0	0
$993$	−957648.	−0.971198
$994$	0	0
$995$	− 202524.i	− 0.204565i
$996$	0	0
$997$	499410.	0.502420	0.251210	−	0.967933i	$-0.419172\pi$
$997$	499410.	0.502420	0.251210	+	0.967933i	$0.419172\pi$
$998$	0	0
$999$	− 1.25434e6i	− 1.25685i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.5.c.k.255.3		4
4.3	odd	2	inner	256.5.c.k.255.2		4
8.3	odd	2		256.5.c.g.255.3		4
8.5	even	2		256.5.c.g.255.2		4
16.3	odd	4		128.5.d.d.63.6	yes	8
16.5	even	4		128.5.d.d.63.5	yes	8
16.11	odd	4		128.5.d.d.63.3	✓	8
16.13	even	4		128.5.d.d.63.4	yes	8
48.5	odd	4		1152.5.b.l.703.5		8
48.11	even	4		1152.5.b.l.703.6		8
48.29	odd	4		1152.5.b.l.703.3		8
48.35	even	4		1152.5.b.l.703.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.d.d.63.3	✓	8	16.11	odd	4
128.5.d.d.63.4	yes	8	16.13	even	4
128.5.d.d.63.5	yes	8	16.5	even	4
128.5.d.d.63.6	yes	8	16.3	odd	4
256.5.c.g.255.2		4	8.5	even	2
256.5.c.g.255.3		4	8.3	odd	2
256.5.c.k.255.2		4	4.3	odd	2	inner
256.5.c.k.255.3		4	1.1	even	1	trivial
1152.5.b.l.703.3		8	48.29	odd	4
1152.5.b.l.703.4		8	48.35	even	4
1152.5.b.l.703.5		8	48.5	odd	4
1152.5.b.l.703.6		8	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.54118i q^{3} -16.8444 q^{5} +40.7922i q^{7} +60.3776 q^{9} +O(q^{10})\)	\(q+4.54118i q^{3} -16.8444 q^{5} +40.7922i q^{7} +60.3776 q^{9} +104.133i q^{11} -13.4668 q^{13} -76.4936i q^{15} +73.3776 q^{17} -502.501i q^{19} -185.245 q^{21} +973.921i q^{23} -341.266 q^{25} +642.022i q^{27} -1256.49 q^{29} -1086.12i q^{31} -472.888 q^{33} -687.120i q^{35} -1953.73 q^{37} -61.1551i q^{39} +194.266 q^{41} +1808.79i q^{43} -1017.03 q^{45} -627.155i q^{47} +737.000 q^{49} +333.221i q^{51} -3011.33 q^{53} -1754.06i q^{55} +2281.95 q^{57} -5696.96i q^{59} -6076.93 q^{61} +2462.93i q^{63} +226.840 q^{65} +3518.77i q^{67} -4422.76 q^{69} -3340.00i q^{71} -4810.79 q^{73} -1549.75i q^{75} -4247.82 q^{77} +5466.15i q^{79} +1975.05 q^{81} +3945.61i q^{83} -1236.00 q^{85} -5705.94i q^{87} +700.314 q^{89} -549.339i q^{91} +4932.25 q^{93} +8464.34i q^{95} -12767.9 q^{97} +6287.32i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 48 q^{5} - 220 q^{9}+O(q^{10}) \) 4 * q + 48 * q^5 - 220 * q^9	\( 4 q + 48 q^{5} - 220 q^{9} - 400 q^{13} - 168 q^{17} - 1664 q^{21} + 1404 q^{25} - 1680 q^{29} + 416 q^{33} - 5392 q^{37} - 1992 q^{41} - 15952 q^{45} + 2948 q^{49} - 13776 q^{53} - 4256 q^{57} - 9424 q^{61} - 14784 q^{65} - 16768 q^{69} + 9832 q^{73} - 4992 q^{77} + 21284 q^{81} - 15328 q^{85} + 13416 q^{89} - 24576 q^{93} - 10920 q^{97}+O(q^{100}) \) 4 * q + 48 * q^5 - 220 * q^9 - 400 * q^13 - 168 * q^17 - 1664 * q^21 + 1404 * q^25 - 1680 * q^29 + 416 * q^33 - 5392 * q^37 - 1992 * q^41 - 15952 * q^45 + 2948 * q^49 - 13776 * q^53 - 4256 * q^57 - 9424 * q^61 - 14784 * q^65 - 16768 * q^69 + 9832 * q^73 - 4992 * q^77 + 21284 * q^81 - 15328 * q^85 + 13416 * q^89 - 24576 * q^93 - 10920 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more