LMFDB - Embedded newform 1152.5.b.l.703.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,5,Mod(703,1152)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1152.703");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1152.b (of order $2$, degree $1$, 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:	$119.082197473$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1871773696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 31x^{4} + 81 $ x^8 + 31*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{38} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.4
Root			$-1.62831 - 1.62831i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.5.b.l.703.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.8444i q^{5} +40.7922i q^{7} +O(q^{10})$	$q-16.8444i q^{5} +40.7922i q^{7} +104.133 q^{11} -13.4668i q^{13} -73.3776 q^{17} -502.501 q^{19} -973.921i q^{23} +341.266 q^{25} +1256.49i q^{29} +1086.12i q^{31} +687.120 q^{35} +1953.73i q^{37} +194.266 q^{41} -1808.79 q^{43} -627.155i q^{47} +737.000 q^{49} -3011.33i q^{53} -1754.06i q^{55} -5696.96 q^{59} -6076.93i q^{61} -226.840 q^{65} +3518.77 q^{67} +3340.00i q^{71} +4810.79 q^{73} +4247.82i q^{77} -5466.15i q^{79} -3945.61 q^{83} +1236.00i q^{85} +700.314 q^{89} +549.339 q^{91} +8464.34i q^{95} -12767.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 336 q^{17} - 2808 q^{25} - 3984 q^{41} + 5896 q^{49} + 29568 q^{65} - 19664 q^{73} + 26832 q^{89} - 21840 q^{97}+O(q^{100}) $ 8 * q + 336 * q^17 - 2808 * q^25 - 3984 * q^41 + 5896 * q^49 + 29568 * q^65 - 19664 * q^73 + 26832 * q^89 - 21840 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 16.8444i	− 0.673776i	−0.941545	−	0.336888i	$-0.890626\pi$
$5$	− 16.8444i	− 0.673776i	0.941545	−	0.336888i	$-0.109374\pi$
$6$	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
$9$	0	0
$10$	0	0
$11$	104.133	0.860605	0.430303	−	0.902685i	$-0.358407\pi$
$11$	104.133	0.860605	0.430303	+	0.902685i	$0.358407\pi$
$12$	0	0
$13$	− 13.4668i	− 0.0796850i	−0.999206	−	0.0398425i	$-0.987314\pi$
$13$	− 13.4668i	− 0.0796850i	0.999206	−	0.0398425i	$-0.0126856\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−73.3776	−0.253902	−0.126951	−	0.991909i	$-0.540519\pi$
$17$	−73.3776	−0.253902	−0.126951	+	0.991909i	$0.540519\pi$
$18$	0	0
$19$	−502.501	−1.39197	−0.695985	−	0.718056i	$-0.745032\pi$
$19$	−502.501	−1.39197	−0.695985	+	0.718056i	$0.745032\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 973.921i	− 1.84106i	−0.390671	−	0.920530i	$-0.627757\pi$
$23$	− 973.921i	− 1.84106i	0.390671	−	0.920530i	$-0.372243\pi$
$24$	0	0
$25$	341.266	0.546025
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1256.49i	1.49404i	0.664801	+	0.747020i	$0.268516\pi$
$29$	1256.49i	1.49404i	−0.664801	+	0.747020i	$0.731484\pi$
$30$	0	0
$31$	1086.12i	1.13019i	0.825025	+	0.565097i	$0.191161\pi$
$31$	1086.12i	1.13019i	−0.825025	+	0.565097i	$0.808839\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	687.120	0.560914
$36$	0	0
$37$	1953.73i	1.42712i	0.700592	+	0.713562i	$0.252919\pi$
$37$	1953.73i	1.42712i	−0.700592	+	0.713562i	$0.747081\pi$
$38$	0	0
$39$	0	0
$40$	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
$43$	−1808.79	−0.978254	−0.489127	−	0.872212i	$-0.662685\pi$
$43$	−1808.79	−0.978254	−0.489127	+	0.872212i	$0.662685\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	0	0
$53$	− 3011.33i	− 1.07203i	−0.844208	−	0.536015i	$-0.819929\pi$
$53$	− 3011.33i	− 1.07203i	0.844208	−	0.536015i	$-0.180071\pi$
$54$	0	0
$55$	− 1754.06i	− 0.579855i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5696.96	−1.63659	−0.818294	−	0.574800i	$-0.805080\pi$
$59$	−5696.96	−1.63659	−0.818294	+	0.574800i	$0.805080\pi$
$60$	0	0
$61$	− 6076.93i	− 1.63314i	−0.577244	−	0.816572i	$-0.695872\pi$
$61$	− 6076.93i	− 1.63314i	0.577244	−	0.816572i	$-0.304128\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−226.840	−0.0536899
$66$	0	0
$67$	3518.77	0.783864	0.391932	−	0.919994i	$-0.371807\pi$
$67$	3518.77	0.783864	0.391932	+	0.919994i	$0.371807\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3340.00i	0.662567i	0.943531	+	0.331283i	$0.107482\pi$
$71$	3340.00i	0.662567i	−0.943531	+	0.331283i	$0.892518\pi$
$72$	0	0
$73$	4810.79	0.902757	0.451378	−	0.892333i	$-0.350932\pi$
$73$	4810.79	0.902757	0.451378	+	0.892333i	$0.350932\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4247.82i	0.716448i
$78$	0	0
$79$	− 5466.15i	− 0.875845i	−0.899013	−	0.437923i	$-0.855714\pi$
$79$	− 5466.15i	− 0.875845i	0.899013	−	0.437923i	$-0.144286\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3945.61	−0.572741	−0.286370	−	0.958119i	$-0.592449\pi$
$83$	−3945.61	−0.572741	−0.286370	+	0.958119i	$0.592449\pi$
$84$	0	0
$85$	1236.00i	0.171073i
$86$	0	0
$87$	0	0
$88$	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
$91$	549.339	0.0663372
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	4197.02i	0.411432i	0.978612	+	0.205716i	$0.0659524\pi$
$101$	4197.02i	0.411432i	−0.978612	+	0.205716i	$0.934048\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	179.932	0.0157160	0.00785800	−	0.999969i	$-0.497499\pi$
$107$	179.932	0.0157160	0.00785800	+	0.999969i	$0.497499\pi$
$108$	0	0
$109$	1774.81i	0.149382i	0.997207	+	0.0746909i	$0.0237970\pi$
$109$	1774.81i	0.149382i	−0.997207	+	0.0746909i	$0.976203\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16994.7	−1.33093	−0.665467	−	0.746427i	$-0.731768\pi$
$113$	−16994.7	−1.33093	−0.665467	+	0.746427i	$0.731768\pi$
$114$	0	0
$115$	−16405.1	−1.24046
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 2993.23i	− 0.211372i
$120$	0	0
$121$	−3797.27	−0.259359
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 16276.2i	− 1.04168i
$126$	0	0
$127$	22216.9i	1.37745i	0.725024	+	0.688724i	$0.241829\pi$
$127$	22216.9i	1.37745i	−0.725024	+	0.688724i	$0.758171\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15326.7	0.893113	0.446557	−	0.894755i	$-0.352650\pi$
$131$	15326.7	0.893113	0.446557	+	0.894755i	$0.352650\pi$
$132$	0	0
$133$	− 20498.1i	− 1.15881i
$134$	0	0
$135$	0	0
$136$	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
$139$	−26354.8	−1.36405	−0.682024	−	0.731330i	$-0.738900\pi$
$139$	−26354.8	−1.36405	−0.682024	+	0.731330i	$0.738900\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1402.34i	− 0.0685773i
$144$	0	0
$145$	21164.8	1.00665
$146$	0	0
$147$	0	0
$148$	0	0
$149$	35333.0i	1.59150i	0.605622	+	0.795752i	$0.292924\pi$
$149$	35333.0i	1.59150i	−0.605622	+	0.795752i	$0.707076\pi$
$150$	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
$153$	0	0
$154$	0	0
$155$	18295.0	0.761498
$156$	0	0
$157$	27036.6i	1.09687i	0.836195	+	0.548433i	$0.184775\pi$
$157$	27036.6i	1.09687i	−0.836195	+	0.548433i	$0.815225\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	39728.3	1.53267
$162$	0	0
$163$	−15689.0	−0.590499	−0.295250	−	0.955420i	$-0.595403\pi$
$163$	−15689.0	−0.590499	−0.295250	+	0.955420i	$0.595403\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10137.3i	0.363487i	0.983346	+	0.181743i	$0.0581740\pi$
$167$	10137.3i	0.363487i	−0.983346	+	0.181743i	$0.941826\pi$
$168$	0	0
$169$	28379.6	0.993650
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 39217.1i	− 1.31034i	−0.755483	−	0.655169i	$-0.772598\pi$
$173$	− 39217.1i	− 1.31034i	0.755483	−	0.655169i	$-0.227402\pi$
$174$	0	0
$175$	13921.0i	0.454562i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−35042.8	−1.09369	−0.546843	−	0.837235i	$-0.684171\pi$
$179$	−35042.8	−1.09369	−0.546843	+	0.837235i	$0.684171\pi$
$180$	0	0
$181$	14059.3i	0.429147i	0.976708	+	0.214574i	$0.0688362\pi$
$181$	14059.3i	0.429147i	−0.976708	+	0.214574i	$0.931164\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	32909.5	0.961562
$186$	0	0
$187$	−7641.05	−0.218509
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	0	0
$197$	8098.75i	0.208682i	0.994542	+	0.104341i	$0.0332734\pi$
$197$	8098.75i	0.208682i	−0.994542	+	0.104341i	$0.966727\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	−51254.9	−1.24378
$204$	0	0
$205$	− 3272.29i	− 0.0778654i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−52327.1	−1.19794
$210$	0	0
$211$	−32994.3	−0.741094	−0.370547	−	0.928814i	$-0.620830\pi$
$211$	−32994.3	−0.741094	−0.370547	+	0.928814i	$0.620830\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	30468.0i	0.659125i
$216$	0	0
$217$	−44305.0	−0.940878
$218$	0	0
$219$	0	0
$220$	0	0
$221$	988.160i	0.0202322i
$222$	0	0
$223$	− 8325.83i	− 0.167424i	−0.996490	−	0.0837120i	$-0.973322\pi$
$223$	− 8325.83i	− 0.167424i	0.996490	−	0.0837120i	$-0.0266776\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−98932.8	−1.91994	−0.959972	−	0.280096i	$-0.909634\pi$
$227$	−98932.8	−1.91994	−0.959972	+	0.280096i	$0.909634\pi$
$228$	0	0
$229$	39028.4i	0.744235i	0.928186	+	0.372117i	$0.121368\pi$
$229$	39028.4i	0.744235i	−0.928186	+	0.372117i	$0.878632\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	−10564.0	−0.191291
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	− 12414.3i	− 0.206819i
$246$	0	0
$247$	6767.07i	0.110919i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14707.9	−0.233456	−0.116728	−	0.993164i	$-0.537240\pi$
$251$	−14707.9	−0.233456	−0.116728	+	0.993164i	$0.537240\pi$
$252$	0	0
$253$	− 101418.i	− 1.58443i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	105989.	1.60470	0.802351	−	0.596852i	$-0.203582\pi$
$257$	105989.	1.60470	0.802351	+	0.596852i	$0.203582\pi$
$258$	0	0
$259$	−79697.0	−1.18807
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2401.32i	− 0.0347166i	−0.999849	−	0.0173583i	$-0.994474\pi$
$263$	− 2401.32i	− 0.0347166i	0.999849	−	0.0173583i	$-0.00552560\pi$
$264$	0	0
$265$	−50724.1	−0.722309
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17257.4i	0.238490i	0.992865	+	0.119245i	$0.0380474\pi$
$269$	17257.4i	0.238490i	−0.992865	+	0.119245i	$0.961953\pi$
$270$	0	0
$271$	119967.i	1.63351i	0.576982	+	0.816757i	$0.304230\pi$
$271$	119967.i	1.63351i	−0.576982	+	0.816757i	$0.695770\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	35537.1	0.469912
$276$	0	0
$277$	101645.i	1.32473i	0.749183	+	0.662363i	$0.230446\pi$
$277$	101645.i	1.32473i	−0.749183	+	0.662363i	$0.769554\pi$
$278$	0	0
$279$	0	0
$280$	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
$283$	−98095.9	−1.22484	−0.612418	−	0.790534i	$-0.709803\pi$
$283$	−98095.9	−1.22484	−0.612418	+	0.790534i	$0.709803\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7924.52i	0.0962076i
$288$	0	0
$289$	−78136.7	−0.935534
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2060.54i	− 0.0240019i	−0.999928	−	0.0120010i	$-0.996180\pi$
$293$	− 2060.54i	− 0.0240019i	0.999928	−	0.0120010i	$-0.00382012\pi$
$294$	0	0
$295$	95962.0i	1.10269i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13115.6	−0.146705
$300$	0	0
$301$	− 73784.5i	− 0.814390i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−102362.	−1.10037
$306$	0	0
$307$	−76322.7	−0.809798	−0.404899	−	0.914361i	$-0.632693\pi$
$307$	−76322.7	−0.809798	−0.404899	+	0.914361i	$0.632693\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 93007.1i	− 0.961602i	−0.876830	−	0.480801i	$-0.840346\pi$
$311$	− 93007.1i	− 0.961602i	0.876830	−	0.480801i	$-0.159654\pi$
$312$	0	0
$313$	−44008.6	−0.449210	−0.224605	−	0.974450i	$-0.572109\pi$
$313$	−44008.6	−0.449210	−0.224605	+	0.974450i	$0.572109\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 110474.i	− 1.09936i	−0.835375	−	0.549680i	$-0.814750\pi$
$317$	− 110474.i	− 1.09936i	0.835375	−	0.549680i	$-0.185250\pi$
$318$	0	0
$319$	130842.i	1.28578i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	36872.4	0.353424
$324$	0	0
$325$	− 4595.75i	− 0.0435100i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	25583.0	0.236352
$330$	0	0
$331$	−210881.	−1.92478	−0.962390	−	0.271673i	$-0.912423\pi$
$331$	−210881.	−1.92478	−0.962390	+	0.271673i	$0.912423\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	113101.i	0.972650i
$342$	0	0
$343$	128006.i	1.08803i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−41355.2	−0.343456	−0.171728	−	0.985144i	$-0.554935\pi$
$347$	−41355.2	−0.343456	−0.171728	+	0.985144i	$0.554935\pi$
$348$	0	0
$349$	− 60146.7i	− 0.493812i	−0.969039	−	0.246906i	$-0.920586\pi$
$349$	− 60146.7i	− 0.493812i	0.969039	−	0.246906i	$-0.0794138\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12988.7	0.104236	0.0521178	−	0.998641i	$-0.483403\pi$
$353$	12988.7	0.104236	0.0521178	+	0.998641i	$0.483403\pi$
$354$	0	0
$355$	56260.3	0.446422
$356$	0	0
$357$	0	0
$358$	0	0
$359$	191555.i	1.48630i	0.669127	+	0.743148i	$0.266668\pi$
$359$	191555.i	1.48630i	−0.669127	+	0.743148i	$0.733332\pi$
$360$	0	0
$361$	122187.	0.937582
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 81034.9i	− 0.608256i
$366$	0	0
$367$	− 163572.i	− 1.21444i	−0.794532	−	0.607222i	$-0.792284\pi$
$367$	− 163572.i	− 1.21444i	0.794532	−	0.607222i	$-0.207716\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	122839.	0.892458
$372$	0	0
$373$	55059.2i	0.395742i	0.980228	+	0.197871i	$0.0634028\pi$
$373$	55059.2i	0.395742i	−0.980228	+	0.197871i	$0.936597\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16920.8	0.119053
$378$	0	0
$379$	108892.	0.758083	0.379041	−	0.925380i	$-0.376254\pi$
$379$	108892.	0.758083	0.379041	+	0.925380i	$0.376254\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	0	0
$389$	183067.i	1.20979i	0.796304	+	0.604896i	$0.206785\pi$
$389$	183067.i	1.20979i	−0.796304	+	0.604896i	$0.793215\pi$
$390$	0	0
$391$	71464.0i	0.467449i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−92074.1	−0.590124
$396$	0	0
$397$	145471.i	0.922984i	0.887144	+	0.461492i	$0.152686\pi$
$397$	145471.i	0.922984i	−0.887144	+	0.461492i	$0.847314\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	188314.	1.17110	0.585550	−	0.810636i	$-0.300879\pi$
$401$	188314.	1.17110	0.585550	+	0.810636i	$0.300879\pi$
$402$	0	0
$403$	14626.5	0.0900595
$404$	0	0
$405$	0	0
$406$	0	0
$407$	203448.i	1.22819i
$408$	0	0
$409$	88454.1	0.528776	0.264388	−	0.964416i	$-0.414830\pi$
$409$	88454.1	0.528776	0.264388	+	0.964416i	$0.414830\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 232391.i	− 1.36245i
$414$	0	0
$415$	66461.5i	0.385899i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−146392.	−0.833851	−0.416925	−	0.908941i	$-0.636892\pi$
$419$	−146392.	−0.833851	−0.416925	+	0.908941i	$0.636892\pi$
$420$	0	0
$421$	− 127042.i	− 0.716774i	−0.933573	−	0.358387i	$-0.883327\pi$
$421$	− 127042.i	− 0.716774i	0.933573	−	0.358387i	$-0.116673\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−25041.3	−0.138637
$426$	0	0
$427$	247891.	1.35958
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	489397.i	2.56270i
$438$	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
$441$	0	0
$442$	0	0
$443$	−89033.9	−0.453678	−0.226839	−	0.973932i	$-0.572839\pi$
$443$	−89033.9	−0.453678	−0.226839	+	0.973932i	$0.572839\pi$
$444$	0	0
$445$	− 11796.4i	− 0.0595702i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−282195.	−1.39977	−0.699885	−	0.714256i	$-0.746765\pi$
$449$	−282195.	−1.39977	−0.699885	+	0.714256i	$0.746765\pi$
$450$	0	0
$451$	20229.5	0.0994564
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 9253.28i	− 0.0446965i
$456$	0	0
$457$	−100299.	−0.480246	−0.240123	−	0.970742i	$-0.577188\pi$
$457$	−100299.	−0.480246	−0.240123	+	0.970742i	$0.577188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 309619.i	− 1.45689i	−0.685106	−	0.728443i	$-0.740244\pi$
$461$	− 309619.i	− 1.45689i	0.685106	−	0.728443i	$-0.259756\pi$
$462$	0	0
$463$	− 15604.3i	− 0.0727917i	−0.999337	−	0.0363959i	$-0.988412\pi$
$463$	− 15604.3i	− 0.0727917i	0.999337	−	0.0363959i	$-0.0115877\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−137144.	−0.628844	−0.314422	−	0.949283i	$-0.601811\pi$
$467$	−137144.	−0.628844	−0.314422	+	0.949283i	$0.601811\pi$
$468$	0	0
$469$	143538.i	0.652561i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−188355.	−0.841891
$474$	0	0
$475$	−171487.	−0.760051
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	215067.i	0.914303i
$486$	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
$489$	0	0
$490$	0	0
$491$	379148.	1.57270	0.786349	−	0.617783i	$-0.211969\pi$
$491$	379148.	1.57270	0.786349	+	0.617783i	$0.211969\pi$
$492$	0	0
$493$	− 92198.1i	− 0.379340i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−136246.	−0.551582
$498$	0	0
$499$	54844.8	0.220260	0.110130	−	0.993917i	$-0.464873\pi$
$499$	54844.8	0.220260	0.110130	+	0.993917i	$0.464873\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 146931.i	− 0.580732i	−0.956916	−	0.290366i	$-0.906223\pi$
$503$	− 146931.i	− 0.580732i	0.956916	−	0.290366i	$-0.0937771\pi$
$504$	0	0
$505$	70696.4	0.277213
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 156866.i	− 0.605470i	−0.953075	−	0.302735i	$-0.902100\pi$
$509$	− 156866.i	− 0.605470i	0.953075	−	0.302735i	$-0.0978997\pi$
$510$	0	0
$511$	196243.i	0.751539i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	222370.	0.838419
$516$	0	0
$517$	− 65307.6i	− 0.244333i
$518$	0	0
$519$	0	0
$520$	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.	0.893161	0.446580	−	0.894744i	$-0.352642\pi$
$523$	244305.	0.893161	0.446580	+	0.894744i	$0.352642\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 79696.6i	− 0.286958i
$528$	0	0
$529$	−668681.	−2.38950
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2616.13i	− 0.00920885i
$534$	0	0
$535$	− 3030.86i	− 0.0105891i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	76746.2	0.264167
$540$	0	0
$541$	− 263893.i	− 0.901642i	−0.892614	−	0.450821i	$-0.851131\pi$
$541$	− 263893.i	− 0.901642i	0.892614	−	0.450821i	$-0.148869\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	29895.5	0.100650
$546$	0	0
$547$	−5684.58	−0.0189987	−0.00949935	−	0.999955i	$-0.503024\pi$
$547$	−5684.58	−0.0189987	−0.00949935	+	0.999955i	$0.503024\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 631387.i	− 2.07966i
$552$	0	0
$553$	222976.	0.729135
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 229010.i	− 0.738150i	−0.929400	−	0.369075i	$-0.879675\pi$
$557$	− 229010.i	− 0.738150i	0.929400	−	0.369075i	$-0.120325\pi$
$558$	0	0
$559$	24358.6i	0.0779522i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8586.54	−0.0270895	−0.0135448	−	0.999908i	$-0.504312\pi$
$563$	−8586.54	−0.0270895	−0.0135448	+	0.999908i	$0.504312\pi$
$564$	0	0
$565$	286266.i	0.896752i
$566$	0	0
$567$	0	0
$568$	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.	−1.57332	−0.786658	−	0.617389i	$-0.788191\pi$
$571$	−512966.	−1.57332	−0.786658	+	0.617389i	$0.788191\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	− 160950.i	− 0.476803i
$582$	0	0
$583$	− 313580.i	− 0.922595i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−76953.3	−0.223332	−0.111666	−	0.993746i	$-0.535619\pi$
$587$	−76953.3	−0.223332	−0.111666	+	0.993746i	$0.535619\pi$
$588$	0	0
$589$	− 545775.i	− 1.57320i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	559786.	1.59189	0.795944	−	0.605370i	$-0.206975\pi$
$593$	559786.	1.59189	0.795944	+	0.605370i	$0.206975\pi$
$594$	0	0
$595$	−50419.2	−0.142417
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 209237.i	− 0.583157i	−0.956547	−	0.291579i	$-0.905820\pi$
$599$	− 209237.i	− 0.583157i	0.956547	−	0.291579i	$-0.0941805\pi$
$600$	0	0
$601$	2016.35	0.00558236	0.00279118	−	0.999996i	$-0.499112\pi$
$601$	2016.35	0.00558236	0.00279118	+	0.999996i	$0.499112\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	63962.8i	0.174750i
$606$	0	0
$607$	− 648350.i	− 1.75967i	−0.475275	−	0.879837i	$-0.657651\pi$
$607$	− 648350.i	− 1.75967i	0.475275	−	0.879837i	$-0.342349\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8445.75	−0.0226233
$612$	0	0
$613$	303258.i	0.807033i	0.914972	+	0.403516i	$0.132212\pi$
$613$	303258.i	0.807033i	−0.914972	+	0.403516i	$0.867788\pi$
$614$	0	0
$615$	0	0
$616$	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.	1.50753	0.753767	−	0.657142i	$-0.228235\pi$
$619$	577628.	1.50753	0.753767	+	0.657142i	$0.228235\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	28567.3i	0.0736027i
$624$	0	0
$625$	−60871.5	−0.155831
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 143360.i	− 0.362349i
$630$	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
$633$	0	0
$634$	0	0
$635$	374230.	0.928092
$636$	0	0
$637$	− 9925.01i	− 0.0244598i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−515858.	−1.25549	−0.627746	−	0.778418i	$-0.716022\pi$
$641$	−515858.	−1.25549	−0.627746	+	0.778418i	$0.716022\pi$
$642$	0	0
$643$	585939.	1.41720	0.708599	−	0.705611i	$-0.249327\pi$
$643$	585939.	1.41720	0.708599	+	0.705611i	$0.249327\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	734055.i	1.75356i	0.480895	+	0.876778i	$0.340312\pi$
$647$	734055.i	1.75356i	−0.480895	+	0.876778i	$0.659688\pi$
$648$	0	0
$649$	−593243.	−1.40846
$650$	0	0
$651$	0	0
$652$	0	0
$653$	78218.9i	0.183436i	0.995785	+	0.0917181i	$0.0292359\pi$
$653$	78218.9i	0.183436i	−0.995785	+	0.0917181i	$0.970764\pi$
$654$	0	0
$655$	− 258169.i	− 0.601759i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4105.91	0.00945450	0.00472725	−	0.999989i	$-0.498495\pi$
$659$	4105.91	0.00945450	0.00472725	+	0.999989i	$0.498495\pi$
$660$	0	0
$661$	478700.i	1.09562i	0.836602	+	0.547811i	$0.184539\pi$
$661$	478700.i	1.09562i	−0.836602	+	0.547811i	$0.815461\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−345279.	−0.780776
$666$	0	0
$667$	1.22372e6	2.75062
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	358940.i	0.783149i	0.920146	+	0.391575i	$0.128070\pi$
$677$	358940.i	0.783149i	−0.920146	+	0.391575i	$0.871930\pi$
$678$	0	0
$679$	− 520828.i	− 1.12968i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−166893.	−0.357765	−0.178883	−	0.983870i	$-0.557248\pi$
$683$	−166893.	−0.357765	−0.178883	+	0.983870i	$0.557248\pi$
$684$	0	0
$685$	81126.4i	0.172895i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−40552.9	−0.0854248
$690$	0	0
$691$	102959.	0.215629	0.107814	−	0.994171i	$-0.465615\pi$
$691$	102959.	0.215629	0.107814	+	0.994171i	$0.465615\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	443930.i	0.919063i
$696$	0	0
$697$	−14254.8	−0.0293423
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 196855.i	− 0.400599i	−0.979735	−	0.200299i	$-0.935808\pi$
$701$	− 196855.i	− 0.400599i	0.979735	−	0.200299i	$-0.0641915\pi$
$702$	0	0
$703$	− 981753.i	− 1.98651i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−171206.	−0.342515
$708$	0	0
$709$	621291.i	1.23595i	0.786196	+	0.617977i	$0.212048\pi$
$709$	621291.i	1.23595i	−0.786196	+	0.617977i	$0.787952\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.05779e6	2.08075
$714$	0	0
$715$	−23621.6	−0.0462058
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	428796.i	0.815784i
$726$	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
$729$	0	0
$730$	0	0
$731$	132725.	0.248381
$732$	0	0
$733$	− 13466.6i	− 0.0250640i	−0.999921	−	0.0125320i	$-0.996011\pi$
$733$	− 13466.6i	− 0.0250640i	0.999921	−	0.0125320i	$-0.00398917\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	366420.	0.674597
$738$	0	0
$739$	352928.	0.646245	0.323122	−	0.946357i	$-0.395267\pi$
$739$	352928.	0.646245	0.323122	+	0.946357i	$0.395267\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 539226.i	− 0.976772i	−0.872628	−	0.488386i	$-0.837586\pi$
$743$	− 539226.i	− 0.976772i	0.872628	−	0.488386i	$-0.162414\pi$
$744$	0	0
$745$	595164.	1.07232
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7339.83i	0.0130835i
$750$	0	0
$751$	301606.i	0.534762i	0.963591	+	0.267381i	$0.0861582\pi$
$751$	301606.i	0.534762i	−0.963591	+	0.267381i	$0.913842\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	603217.	1.05823
$756$	0	0
$757$	− 402304.i	− 0.702042i	−0.936368	−	0.351021i	$-0.885835\pi$
$757$	− 402304.i	− 0.702042i	0.936368	−	0.351021i	$-0.114165\pi$
$758$	0	0
$759$	0	0
$760$	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
$763$	−72398.1	−0.124359
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	510328.i	0.854063i	0.904237	+	0.427032i	$0.140441\pi$
$773$	510328.i	0.854063i	−0.904237	+	0.427032i	$0.859559\pi$
$774$	0	0
$775$	370654.i	0.617114i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−97618.9	−0.160864
$780$	0	0
$781$	347805.i	0.570208i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	455416.	0.739042
$786$	0	0
$787$	970831.	1.56745	0.783726	−	0.621107i	$-0.213317\pi$
$787$	970831.	1.56745	0.783726	+	0.621107i	$0.213317\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 693250.i	− 1.10799i
$792$	0	0
$793$	−81836.6	−0.130137
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.12132e6i	− 1.76528i	−0.470051	−	0.882639i	$-0.655765\pi$
$797$	− 1.12132e6i	− 1.76528i	0.470051	−	0.882639i	$-0.344235\pi$
$798$	0	0
$799$	46019.1i	0.0720850i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	500963.	0.776917
$804$	0	0
$805$	− 669200.i	− 1.03268i
$806$	0	0
$807$	0	0
$808$	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.	−1.11941	−0.559703	−	0.828693i	$-0.689085\pi$
$811$	−736257.	−1.11941	−0.559703	+	0.828693i	$0.689085\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	264271.i	0.397864i
$816$	0	0
$817$	908921.	1.36170
$818$	0	0
$819$	0	0
$820$	0	0
$821$	221216.i	0.328194i	0.986444	+	0.164097i	$0.0524711\pi$
$821$	221216.i	0.328194i	−0.986444	+	0.164097i	$0.947529\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	−87673.4	−0.128191	−0.0640954	−	0.997944i	$-0.520416\pi$
$827$	−87673.4	−0.128191	−0.0640954	+	0.997944i	$0.520416\pi$
$828$	0	0
$829$	− 442564.i	− 0.643971i	−0.946745	−	0.321986i	$-0.895650\pi$
$829$	− 442564.i	− 0.643971i	0.946745	−	0.321986i	$-0.104350\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−54079.3	−0.0779366
$834$	0	0
$835$	170757.	0.244909
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 705591.i	− 1.00237i	−0.865339	−	0.501186i	$-0.832897\pi$
$839$	− 705591.i	− 1.00237i	0.865339	−	0.501186i	$-0.167103\pi$
$840$	0	0
$841$	−871481.	−1.23216
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 478038.i	− 0.669498i
$846$	0	0
$847$	− 154899.i	− 0.215914i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.90278e6	2.62742
$852$	0	0
$853$	− 1.11225e6i	− 1.52864i	−0.644837	−	0.764320i	$-0.723075\pi$
$853$	− 1.11225e6i	− 1.52864i	0.644837	−	0.764320i	$-0.276925\pi$
$854$	0	0
$855$	0	0
$856$	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
$859$	78675.9	0.106624	0.0533120	−	0.998578i	$-0.483022\pi$
$859$	78675.9	0.106624	0.0533120	+	0.998578i	$0.483022\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	− 569208.i	− 0.753757i
$870$	0	0
$871$	− 47386.4i	− 0.0624622i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	663940.	0.867187
$876$	0	0
$877$	861315.i	1.11986i	0.828541	+	0.559929i	$0.189171\pi$
$877$	861315.i	1.11986i	−0.828541	+	0.559929i	$0.810829\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.03692e6	−1.33596	−0.667981	−	0.744178i	$-0.732841\pi$
$881$	−1.03692e6	−1.33596	−0.667981	+	0.744178i	$0.732841\pi$
$882$	0	0
$883$	−1.07531e6	−1.37915	−0.689577	−	0.724212i	$-0.742203\pi$
$883$	−1.07531e6	−1.37915	−0.689577	+	0.724212i	$0.742203\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 97396.5i	− 0.123793i	−0.998083	−	0.0618965i	$-0.980285\pi$
$887$	− 97396.5i	− 0.123793i	0.998083	−	0.0618965i	$-0.0197149\pi$
$888$	0	0
$889$	−906273.	−1.14672
$890$	0	0
$891$	0	0
$892$	0	0
$893$	315146.i	0.395193i
$894$	0	0
$895$	590276.i	0.736900i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.36469e6	−1.68855
$900$	0	0
$901$	220965.i	0.272191i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	236820.	0.289149
$906$	0	0
$907$	−172583.	−0.209790	−0.104895	−	0.994483i	$-0.533451\pi$
$907$	−172583.	−0.209790	−0.104895	+	0.994483i	$0.533451\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	625210.i	0.743510i
$918$	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
$921$	0	0
$922$	0	0
$923$	44979.0	0.0527966
$924$	0	0
$925$	666742.i	0.779246i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	742596.	0.860441	0.430220	−	0.902724i	$-0.358436\pi$
$929$	742596.	0.860441	0.430220	+	0.902724i	$0.358436\pi$
$930$	0	0
$931$	−370344.	−0.427273
$932$	0	0
$933$	0	0
$934$	0	0
$935$	128709.i	0.147226i
$936$	0	0
$937$	1.57493e6	1.79383	0.896915	−	0.442204i	$-0.145803\pi$
$937$	1.57493e6	1.79383	0.896915	+	0.442204i	$0.145803\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 43641.7i	− 0.0492859i	−0.999696	−	0.0246429i	$-0.992155\pi$
$941$	− 43641.7i	− 0.0492859i	0.999696	−	0.0246429i	$-0.00784489\pi$
$942$	0	0
$943$	− 189200.i	− 0.212763i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.56638e6	−1.74661	−0.873307	−	0.487170i	$-0.838029\pi$
$947$	−1.56638e6	−1.74661	−0.873307	+	0.487170i	$0.838029\pi$
$948$	0	0
$949$	− 64785.8i	− 0.0719362i
$950$	0	0
$951$	0	0
$952$	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.	0.845977
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 196464.i	− 0.213622i
$960$	0	0
$961$	−256127.	−0.277337
$962$	0	0
$963$	0	0
$964$	0	0
$965$	686675.i	0.737389i
$966$	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
$969$	0	0
$970$	0	0
$971$	−813827.	−0.863164	−0.431582	−	0.902074i	$-0.642045\pi$
$971$	−813827.	−0.863164	−0.431582	+	0.902074i	$0.642045\pi$
$972$	0	0
$973$	− 1.07507e6i	− 1.13556i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−195689.	−0.205011	−0.102506	−	0.994732i	$-0.532686\pi$
$977$	−195689.	−0.205011	−0.102506	+	0.994732i	$0.532686\pi$
$978$	0	0
$979$	72926.0	0.0760881
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.01172e6i	1.04701i	0.852022	+	0.523506i	$0.175376\pi$
$983$	1.01172e6i	1.04701i	−0.852022	+	0.523506i	$0.824624\pi$
$984$	0	0
$985$	136419.	0.140605
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.76162e6i	1.80103i
$990$	0	0
$991$	419686.i	0.427344i	0.976905	+	0.213672i	$0.0685424\pi$
$991$	419686.i	0.427344i	−0.976905	+	0.213672i	$0.931458\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	202524.	0.204565
$996$	0	0
$997$	− 499410.i	− 0.502420i	−0.967933	−	0.251210i	$-0.919172\pi$
$997$	− 499410.i	− 0.502420i	0.967933	−	0.251210i	$-0.0808285\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.d.d.63.3	✓	8	24.5	odd	2
128.5.d.d.63.4	yes	8	12.11	even	2
128.5.d.d.63.5	yes	8	24.11	even	2
128.5.d.d.63.6	yes	8	3.2	odd	2
256.5.c.g.255.2		4	48.35	even	4
256.5.c.g.255.3		4	48.29	odd	4
256.5.c.k.255.2		4	48.5	odd	4
256.5.c.k.255.3		4	48.11	even	4
1152.5.b.l.703.3		8	4.3	odd	2	inner
1152.5.b.l.703.4		8	1.1	even	1	trivial
1152.5.b.l.703.5		8	8.3	odd	2	inner
1152.5.b.l.703.6		8	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1152.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.8444i q^{5} +40.7922i q^{7} +O(q^{10})\)	\(q-16.8444i q^{5} +40.7922i q^{7} +104.133 q^{11} -13.4668i q^{13} -73.3776 q^{17} -502.501 q^{19} -973.921i q^{23} +341.266 q^{25} +1256.49i q^{29} +1086.12i q^{31} +687.120 q^{35} +1953.73i q^{37} +194.266 q^{41} -1808.79 q^{43} -627.155i q^{47} +737.000 q^{49} -3011.33i q^{53} -1754.06i q^{55} -5696.96 q^{59} -6076.93i q^{61} -226.840 q^{65} +3518.77 q^{67} +3340.00i q^{71} +4810.79 q^{73} +4247.82i q^{77} -5466.15i q^{79} -3945.61 q^{83} +1236.00i q^{85} +700.314 q^{89} +549.339 q^{91} +8464.34i q^{95} -12767.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 336 q^{17} - 2808 q^{25} - 3984 q^{41} + 5896 q^{49} + 29568 q^{65} - 19664 q^{73} + 26832 q^{89} - 21840 q^{97}+O(q^{100}) \) 8 * q + 336 * q^17 - 2808 * q^25 - 3984 * q^41 + 5896 * q^49 + 29568 * q^65 - 19664 * q^73 + 26832 * q^89 - 21840 * q^97

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