LMFDB - Embedded newform 1152.4.l.a.863.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,4,Mod(287,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.287"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			863.13
Character	$\chi$	$=$	1152.863
Dual form			1152.4.l.a.287.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.40838 + 2.40838i) q^{5} +11.7205 q^{7} +(34.7409 - 34.7409i) q^{11} +(3.17950 + 3.17950i) q^{13} -98.0797i q^{17} +(-15.9562 + 15.9562i) q^{19} +69.6819i q^{23} -113.399i q^{25} +(-15.9649 + 15.9649i) q^{29} -121.295i q^{31} +(28.2275 + 28.2275i) q^{35} +(37.0567 - 37.0567i) q^{37} -59.3202 q^{41} +(-241.737 - 241.737i) q^{43} -395.106 q^{47} -205.629 q^{49} +(458.194 + 458.194i) q^{53} +167.339 q^{55} +(257.629 - 257.629i) q^{59} +(373.295 + 373.295i) q^{61} +15.3149i q^{65} +(-648.397 + 648.397i) q^{67} -787.139i q^{71} -1074.96i q^{73} +(407.181 - 407.181i) q^{77} -382.146i q^{79} +(491.315 + 491.315i) q^{83} +(236.213 - 236.213i) q^{85} -624.448 q^{89} +(37.2653 + 37.2653i) q^{91} -76.8575 q^{95} +1665.45 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{19} + 864 q^{43} + 2352 q^{49} + 576 q^{55} - 1824 q^{61} + 816 q^{67} + 480 q^{85} - 3600 q^{91}+O(q^{100}) $ 48 * q - 48 * q^19 + 864 * q^43 + 2352 * q^49 + 576 * q^55 - 1824 * q^61 + 816 * q^67 + 480 * q^85 - 3600 * q^91

Character values

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

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	2.40838	+	2.40838i	0.215412	+	0.215412i	0.806562	−	0.591150i	$-0.201326\pi$
$5$	2.40838	+	2.40838i	0.215412	+	0.215412i	−0.591150	+	0.806562i	$0.701326\pi$
$6$		0			0
$7$	11.7205			0.632848			0.316424	−	0.948618i	$-0.397518\pi$
$7$	11.7205			0.632848			0.316424	+	0.948618i	$0.397518\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	34.7409	−	34.7409i	0.952252	−	0.952252i	−0.0466586	−	0.998911i	$-0.514857\pi$
$11$	34.7409	−	34.7409i	0.952252	−	0.952252i	0.998911	+	0.0466586i	$0.0148573\pi$
$12$		0			0
$13$	3.17950	+	3.17950i	0.0678333	+	0.0678333i	0.740210	−	0.672376i	$-0.234726\pi$
$13$	3.17950	+	3.17950i	0.0678333	+	0.0678333i	−0.672376	+	0.740210i	$0.734726\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	98.0797i		−	1.39928i	−0.714494	−	0.699642i	$-0.753343\pi$
$17$		−	98.0797i		−	1.39928i	0.714494	−	0.699642i	$-0.246657\pi$
$18$		0			0
$19$	−15.9562	+	15.9562i	−0.192664	+	0.192664i	−0.796846	−	0.604182i	$-0.793500\pi$
$19$	−15.9562	+	15.9562i	−0.192664	+	0.192664i	0.604182	+	0.796846i	$0.293500\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			69.6819i			0.631725i	0.948805	+	0.315863i	$0.102294\pi$
$23$			69.6819i			0.631725i	−0.948805	+	0.315863i	$0.897706\pi$
$24$		0			0
$25$		−	113.399i		−	0.907195i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−15.9649	+	15.9649i	−0.102228	+	0.102228i	−0.756371	−	0.654143i	$-0.773029\pi$
$29$	−15.9649	+	15.9649i	−0.102228	+	0.102228i	0.654143	+	0.756371i	$0.273029\pi$
$30$		0			0
$31$		−	121.295i		−	0.702748i	−0.936235	−	0.351374i	$-0.885715\pi$
$31$		−	121.295i		−	0.702748i	0.936235	−	0.351374i	$-0.114285\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	28.2275	+	28.2275i	0.136323	+	0.136323i
$36$		0			0
$37$	37.0567	−	37.0567i	0.164651	−	0.164651i	−0.619973	−	0.784623i	$-0.712856\pi$
$37$	37.0567	−	37.0567i	0.164651	−	0.164651i	0.784623	+	0.619973i	$0.212856\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−59.3202			−0.225958			−0.112979	−	0.993597i	$-0.536039\pi$
$41$	−59.3202			−0.225958			−0.112979	+	0.993597i	$0.536039\pi$
$42$		0			0
$43$	−241.737	−	241.737i	−0.857314	−	0.857314i	0.133707	−	0.991021i	$-0.457312\pi$
$43$	−241.737	−	241.737i	−0.857314	−	0.857314i	−0.991021	+	0.133707i	$0.957312\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−395.106			−1.22622			−0.613108	−	0.789999i	$-0.710081\pi$
$47$	−395.106			−1.22622			−0.613108	+	0.789999i	$0.710081\pi$
$48$		0			0
$49$	−205.629			−0.599503
$50$		0			0
$51$		0			0
$52$		0			0
$53$	458.194	+	458.194i	1.18751	+	1.18751i	0.977755	+	0.209750i	$0.0672650\pi$
$53$	458.194	+	458.194i	1.18751	+	1.18751i	0.209750	+	0.977755i	$0.432735\pi$
$54$		0			0
$55$	167.339			0.410254
$56$		0			0
$57$		0			0
$58$		0			0
$59$	257.629	−	257.629i	0.568483	−	0.568483i	−0.363220	−	0.931703i	$-0.618323\pi$
$59$	257.629	−	257.629i	0.568483	−	0.568483i	0.931703	+	0.363220i	$0.118323\pi$
$60$		0			0
$61$	373.295	+	373.295i	0.783533	+	0.783533i	0.980425	−	0.196892i	$-0.0630848\pi$
$61$	373.295	+	373.295i	0.783533	+	0.783533i	−0.196892	+	0.980425i	$0.563085\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			15.3149i			0.0292243i
$66$		0			0
$67$	−648.397	+	648.397i	−1.18230	+	1.18230i	−0.203157	+	0.979146i	$0.565120\pi$
$67$	−648.397	+	648.397i	−1.18230	+	1.18230i	−0.979146	+	0.203157i	$0.934880\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	787.139i		−	1.31572i	−0.753140	−	0.657861i	$-0.771462\pi$
$71$		−	787.139i		−	1.31572i	0.753140	−	0.657861i	$-0.228538\pi$
$72$		0			0
$73$		−	1074.96i		−	1.72349i	−0.507340	−	0.861746i	$-0.669371\pi$
$73$		−	1074.96i		−	1.72349i	0.507340	−	0.861746i	$-0.330629\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	407.181	−	407.181i	0.602631	−	0.602631i
$78$		0			0
$79$		−	382.146i		−	0.544238i	−0.962264	−	0.272119i	$-0.912276\pi$
$79$		−	382.146i		−	0.544238i	0.962264	−	0.272119i	$-0.0877245\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	491.315	+	491.315i	0.649744	+	0.649744i	0.952931	−	0.303187i	$-0.0980506\pi$
$83$	491.315	+	491.315i	0.649744	+	0.649744i	−0.303187	+	0.952931i	$0.598051\pi$
$84$		0			0
$85$	236.213	−	236.213i	0.301423	−	0.301423i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−624.448			−0.743723			−0.371862	−	0.928288i	$-0.621280\pi$
$89$	−624.448			−0.743723			−0.371862	+	0.928288i	$0.621280\pi$
$90$		0			0
$91$	37.2653	+	37.2653i	0.0429282	+	0.0429282i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−76.8575			−0.0830043
$96$		0			0
$97$	1665.45			1.74331			0.871654	−	0.490122i	$-0.163048\pi$
$97$	1665.45			1.74331			0.871654	+	0.490122i	$0.163048\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	625.952	+	625.952i	0.616678	+	0.616678i	0.944678	−	0.327999i	$-0.106374\pi$
$101$	625.952	+	625.952i	0.616678	+	0.616678i	−0.327999	+	0.944678i	$0.606374\pi$
$102$		0			0
$103$	−1641.52			−1.57033			−0.785163	−	0.619289i	$-0.787421\pi$
$103$	−1641.52			−1.57033			−0.785163	+	0.619289i	$0.787421\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	956.649	−	956.649i	0.864325	−	0.864325i	−0.127512	−	0.991837i	$-0.540699\pi$
$107$	956.649	−	956.649i	0.864325	−	0.864325i	0.991837	+	0.127512i	$0.0406991\pi$
$108$		0			0
$109$	−1221.44	−	1221.44i	−1.07333	−	1.07333i	−0.997089	−	0.0762403i	$-0.975708\pi$
$109$	−1221.44	−	1221.44i	−1.07333	−	1.07333i	−0.0762403	−	0.997089i	$-0.524292\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			360.983i			0.300517i	0.988647	+	0.150258i	$0.0480105\pi$
$113$			360.983i			0.300517i	−0.988647	+	0.150258i	$0.951989\pi$
$114$		0			0
$115$	−167.821	+	167.821i	−0.136081	+	0.136081i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	1149.54i		−	0.885534i
$120$		0			0
$121$		−	1082.86i		−	0.813569i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	574.157	−	574.157i	0.410833	−	0.410833i
$126$		0			0
$127$		−	1713.36i		−	1.19713i	−0.801073	−	0.598566i	$-0.795737\pi$
$127$		−	1713.36i		−	1.19713i	0.801073	−	0.598566i	$-0.204263\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−191.155	−	191.155i	−0.127491	−	0.127491i	0.640482	−	0.767973i	$-0.278735\pi$
$131$	−191.155	−	191.155i	−0.127491	−	0.127491i	−0.767973	+	0.640482i	$0.778735\pi$
$132$		0			0
$133$	−187.015	+	187.015i	−0.121927	+	0.121927i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2130.50			1.32862			0.664311	−	0.747456i	$-0.268725\pi$
$137$	2130.50			1.32862			0.664311	+	0.747456i	$0.268725\pi$
$138$		0			0
$139$	−423.046	−	423.046i	−0.258146	−	0.258146i	0.566154	−	0.824300i	$-0.308431\pi$
$139$	−423.046	−	423.046i	−0.258146	−	0.258146i	−0.824300	+	0.566154i	$0.808431\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	220.917			0.129189
$144$		0			0
$145$	−76.8994			−0.0440424
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2020.44	+	2020.44i	1.11088	+	1.11088i	0.993032	+	0.117844i	$0.0375982\pi$
$149$	2020.44	+	2020.44i	1.11088	+	1.11088i	0.117844	+	0.993032i	$0.462402\pi$
$150$		0			0
$151$	−2605.57			−1.40423			−0.702115	−	0.712064i	$-0.747761\pi$
$151$	−2605.57			−1.40423			−0.702115	+	0.712064i	$0.747761\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	292.124	−	292.124i	0.151381	−	0.151381i
$156$		0			0
$157$	−815.970	−	815.970i	−0.414787	−	0.414787i	0.468615	−	0.883402i	$-0.344753\pi$
$157$	−815.970	−	815.970i	−0.414787	−	0.414787i	−0.883402	+	0.468615i	$0.844753\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			816.708i			0.399786i
$162$		0			0
$163$	268.825	−	268.825i	0.129178	−	0.129178i	−0.639562	−	0.768740i	$-0.720884\pi$
$163$	268.825	−	268.825i	0.129178	−	0.129178i	0.768740	+	0.639562i	$0.220884\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1537.53i		−	0.712440i	−0.934402	−	0.356220i	$-0.884065\pi$
$167$		−	1537.53i		−	0.712440i	0.934402	−	0.356220i	$-0.115935\pi$
$168$		0			0
$169$		−	2176.78i		−	0.990797i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	598.582	−	598.582i	0.263060	−	0.263060i	−0.563236	−	0.826296i	$-0.690444\pi$
$173$	598.582	−	598.582i	0.263060	−	0.263060i	0.826296	+	0.563236i	$0.190444\pi$
$174$		0			0
$175$		−	1329.10i		−	0.574117i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−285.351	−	285.351i	−0.119152	−	0.119152i	0.645017	−	0.764168i	$-0.276850\pi$
$179$	−285.351	−	285.351i	−0.119152	−	0.119152i	−0.764168	+	0.645017i	$0.776850\pi$
$180$		0			0
$181$	2919.93	−	2919.93i	1.19910	−	1.19910i	0.224660	−	0.974437i	$-0.427873\pi$
$181$	2919.93	−	2919.93i	1.19910	−	1.19910i	0.974437	−	0.224660i	$-0.0721270\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	178.493			0.0709356
$186$		0			0
$187$	−3407.38	−	3407.38i	−1.33247	−	1.33247i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1085.25			0.411129			0.205564	−	0.978644i	$-0.434097\pi$
$191$	1085.25			0.411129			0.205564	+	0.978644i	$0.434097\pi$
$192$		0			0
$193$	1150.76			0.429191			0.214595	−	0.976703i	$-0.431157\pi$
$193$	1150.76			0.429191			0.214595	+	0.976703i	$0.431157\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2302.47	−	2302.47i	−0.832713	−	0.832713i	0.155174	−	0.987887i	$-0.450406\pi$
$197$	−2302.47	−	2302.47i	−0.832713	−	0.832713i	−0.987887	+	0.155174i	$0.950406\pi$
$198$		0			0
$199$	−1300.97			−0.463432			−0.231716	−	0.972783i	$-0.574434\pi$
$199$	−1300.97			−0.463432			−0.231716	+	0.972783i	$0.574434\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−187.117	+	187.117i	−0.0646949	+	0.0646949i
$204$		0			0
$205$	−142.866	−	142.866i	−0.0486740	−	0.0486740i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1108.67i			0.366929i
$210$		0			0
$211$	−1702.90	+	1702.90i	−0.555604	+	0.555604i	−0.928053	−	0.372448i	$-0.878518\pi$
$211$	−1702.90	+	1702.90i	−0.555604	+	0.555604i	0.372448	+	0.928053i	$0.378518\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1164.39i		−	0.369352i
$216$		0			0
$217$		−	1421.64i		−	0.444733i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	311.844	−	311.844i	0.0949180	−	0.0949180i
$222$		0			0
$223$			5316.70i			1.59656i	0.602288	+	0.798279i	$0.294256\pi$
$223$			5316.70i			1.59656i	−0.602288	+	0.798279i	$0.705744\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3132.85	−	3132.85i	−0.916013	−	0.916013i	0.0807239	−	0.996737i	$-0.474277\pi$
$227$	−3132.85	−	3132.85i	−0.916013	−	0.916013i	−0.996737	+	0.0807239i	$0.974277\pi$
$228$		0			0
$229$	4530.55	−	4530.55i	1.30737	−	1.30737i	0.384059	−	0.923308i	$-0.374526\pi$
$229$	4530.55	−	4530.55i	1.30737	−	1.30737i	0.923308	−	0.384059i	$-0.125474\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5381.47			1.51310			0.756549	−	0.653937i	$-0.226884\pi$
$233$	5381.47			1.51310			0.756549	+	0.653937i	$0.226884\pi$
$234$		0			0
$235$	−951.566	−	951.566i	−0.264142	−	0.264142i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4955.44			1.34117			0.670587	−	0.741831i	$-0.266042\pi$
$239$	4955.44			1.34117			0.670587	+	0.741831i	$0.266042\pi$
$240$		0			0
$241$	−589.625			−0.157598			−0.0787989	−	0.996891i	$-0.525109\pi$
$241$	−589.625			−0.157598			−0.0787989	+	0.996891i	$0.525109\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−495.234	−	495.234i	−0.129140	−	0.129140i
$246$		0			0
$247$	−101.466			−0.0261381
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2418.09	−	2418.09i	0.608082	−	0.608082i	−0.334362	−	0.942445i	$-0.608521\pi$
$251$	2418.09	−	2418.09i	0.608082	−	0.608082i	0.942445	+	0.334362i	$0.108521\pi$
$252$		0			0
$253$	2420.81	+	2420.81i	0.601562	+	0.601562i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	976.607i		−	0.237039i	−0.992952	−	0.118520i	$-0.962185\pi$
$257$		−	976.607i		−	0.237039i	0.992952	−	0.118520i	$-0.0378148\pi$
$258$		0			0
$259$	434.323	−	434.323i	0.104199	−	0.104199i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1820.40i			0.426809i	0.976964	+	0.213404i	$0.0684552\pi$
$263$			1820.40i			0.426809i	−0.976964	+	0.213404i	$0.931545\pi$
$264$		0			0
$265$			2207.01i			0.511606i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4795.34	+	4795.34i	−1.08690	+	1.08690i	−0.0910580	+	0.995846i	$0.529025\pi$
$269$	−4795.34	+	4795.34i	−1.08690	+	1.08690i	−0.995846	+	0.0910580i	$0.970975\pi$
$270$		0			0
$271$		−	1294.24i		−	0.290108i	−0.989424	−	0.145054i	$-0.953664\pi$
$271$		−	1294.24i		−	0.290108i	0.989424	−	0.145054i	$-0.0463356\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3939.60	−	3939.60i	−0.863879	−	0.863879i
$276$		0			0
$277$	−701.252	+	701.252i	−0.152109	+	0.152109i	−0.779059	−	0.626950i	$-0.784303\pi$
$277$	−701.252	+	701.252i	−0.152109	+	0.152109i	0.626950	+	0.779059i	$0.284303\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5755.44			1.22185			0.610927	−	0.791687i	$-0.290797\pi$
$281$	5755.44			1.22185			0.610927	+	0.791687i	$0.290797\pi$
$282$		0			0
$283$	−3908.33	−	3908.33i	−0.820939	−	0.820939i	0.165303	−	0.986243i	$-0.447140\pi$
$283$	−3908.33	−	3908.33i	−0.820939	−	0.820939i	−0.986243	+	0.165303i	$0.947140\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−695.264			−0.142997
$288$		0			0
$289$	−4706.62			−0.957993
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3513.72	+	3513.72i	0.700592	+	0.700592i	0.964538	−	0.263946i	$-0.0850239\pi$
$293$	3513.72	+	3513.72i	0.700592	+	0.700592i	−0.263946	+	0.964538i	$0.585024\pi$
$294$		0			0
$295$	1240.94			0.244916
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−221.553	+	221.553i	−0.0428520	+	0.0428520i
$300$		0			0
$301$	−2833.28	−	2833.28i	−0.542550	−	0.542550i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1798.07i			0.337565i
$306$		0			0
$307$	2537.12	−	2537.12i	0.471666	−	0.471666i	−0.430788	−	0.902453i	$-0.641764\pi$
$307$	2537.12	−	2537.12i	0.471666	−	0.471666i	0.902453	+	0.430788i	$0.141764\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			7063.50i			1.28789i	0.765071	+	0.643946i	$0.222704\pi$
$311$			7063.50i			1.28789i	−0.765071	+	0.643946i	$0.777296\pi$
$312$		0			0
$313$		−	579.160i		−	0.104588i	−0.998632	−	0.0522940i	$-0.983347\pi$
$313$		−	579.160i		−	0.104588i	0.998632	−	0.0522940i	$-0.0166533\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	340.991	−	340.991i	0.0604164	−	0.0604164i	−0.676253	−	0.736669i	$-0.736397\pi$
$317$	340.991	−	340.991i	0.0604164	−	0.0604164i	0.736669	+	0.676253i	$0.236397\pi$
$318$		0			0
$319$			1109.27i			0.194694i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1564.98	+	1564.98i	0.269591	+	0.269591i
$324$		0			0
$325$	360.553	−	360.553i	0.0615381	−	0.0615381i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4630.85			−0.776009
$330$		0			0
$331$	2509.44	+	2509.44i	0.416711	+	0.416711i	0.884068	−	0.467358i	$-0.154794\pi$
$331$	2509.44	+	2509.44i	0.416711	+	0.416711i	−0.467358	+	0.884068i	$0.654794\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3123.18			−0.509365
$336$		0			0
$337$	1805.84			0.291899			0.145950	−	0.989292i	$-0.453376\pi$
$337$	1805.84			0.291899			0.145950	+	0.989292i	$0.453376\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4213.89	−	4213.89i	−0.669193	−	0.669193i
$342$		0			0
$343$	−6430.22			−1.01224
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3463.32	+	3463.32i	−0.535795	+	0.535795i	−0.922291	−	0.386496i	$-0.873685\pi$
$347$	−3463.32	+	3463.32i	−0.535795	+	0.535795i	0.386496	+	0.922291i	$0.373685\pi$
$348$		0			0
$349$	−5797.35	−	5797.35i	−0.889183	−	0.889183i	0.105261	−	0.994445i	$-0.466432\pi$
$349$	−5797.35	−	5797.35i	−0.889183	−	0.889183i	−0.994445	+	0.105261i	$0.966432\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2793.32i		−	0.421172i	−0.977575	−	0.210586i	$-0.932463\pi$
$353$		−	2793.32i		−	0.421172i	0.977575	−	0.210586i	$-0.0675372\pi$
$354$		0			0
$355$	1895.73	−	1895.73i	0.283423	−	0.283423i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6896.95i		−	1.01395i	−0.861962	−	0.506974i	$-0.830764\pi$
$359$		−	6896.95i		−	1.01395i	0.861962	−	0.506974i	$-0.169236\pi$
$360$		0			0
$361$			6349.80i			0.925761i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2588.92	−	2588.92i	0.371261	−	0.371261i
$366$		0			0
$367$		−	1724.22i		−	0.245242i	−0.992454	−	0.122621i	$-0.960870\pi$
$367$		−	1724.22i		−	0.245242i	0.992454	−	0.122621i	$-0.0391299\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5370.27	+	5370.27i	0.751511	+	0.751511i
$372$		0			0
$373$	−5937.62	+	5937.62i	−0.824231	+	0.824231i	−0.986712	−	0.162481i	$-0.948051\pi$
$373$	−5937.62	+	5937.62i	−0.824231	+	0.824231i	0.162481	+	0.986712i	$0.448051\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−101.521			−0.0138689
$378$		0			0
$379$	8009.60	+	8009.60i	1.08556	+	1.08556i	0.995980	+	0.0895751i	$0.0285509\pi$
$379$	8009.60	+	8009.60i	1.08556	+	1.08556i	0.0895751	+	0.995980i	$0.471449\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12372.3			1.65064			0.825319	−	0.564667i	$-0.190995\pi$
$383$	12372.3			1.65064			0.825319	+	0.564667i	$0.190995\pi$
$384$		0			0
$385$	1961.30			0.259628
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3950.83	−	3950.83i	−0.514948	−	0.514948i	0.401090	−	0.916039i	$-0.368631\pi$
$389$	−3950.83	−	3950.83i	−0.514948	−	0.514948i	−0.916039	+	0.401090i	$0.868631\pi$
$390$		0			0
$391$	6834.38			0.883962
$392$		0			0
$393$		0			0
$394$		0			0
$395$	920.355	−	920.355i	0.117236	−	0.117236i
$396$		0			0
$397$	9644.39	+	9644.39i	1.21924	+	1.21924i	0.967899	+	0.251341i	$0.0808715\pi$
$397$	9644.39	+	9644.39i	1.21924	+	1.21924i	0.251341	+	0.967899i	$0.419129\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3647.28i			0.454206i	0.973871	+	0.227103i	$0.0729254\pi$
$401$			3647.28i			0.454206i	−0.973871	+	0.227103i	$0.927075\pi$
$402$		0			0
$403$	385.656	−	385.656i	0.0476697	−	0.0476697i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2574.76i		−	0.313578i
$408$		0			0
$409$		−	860.719i		−	0.104058i	−0.998646	−	0.0520291i	$-0.983431\pi$
$409$		−	860.719i		−	0.104058i	0.998646	−	0.0520291i	$-0.0165689\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3019.55	−	3019.55i	0.359763	−	0.359763i
$414$		0			0
$415$			2366.55i			0.279926i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1158.11	−	1158.11i	−0.135029	−	0.135029i	0.636362	−	0.771391i	$-0.280439\pi$
$419$	−1158.11	−	1158.11i	−0.135029	−	0.135029i	−0.771391	+	0.636362i	$0.780439\pi$
$420$		0			0
$421$	−2410.72	+	2410.72i	−0.279076	+	0.279076i	−0.832740	−	0.553664i	$-0.813229\pi$
$421$	−2410.72	+	2410.72i	−0.279076	+	0.279076i	0.553664	+	0.832740i	$0.313229\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11122.2			−1.26942
$426$		0			0
$427$	4375.21	+	4375.21i	0.495858	+	0.495858i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10544.5			−1.17844			−0.589221	−	0.807972i	$-0.700565\pi$
$431$	−10544.5			−1.17844			−0.589221	+	0.807972i	$0.700565\pi$
$432$		0			0
$433$	1205.31			0.133773			0.0668864	−	0.997761i	$-0.478693\pi$
$433$	1205.31			0.133773			0.0668864	+	0.997761i	$0.478693\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1111.86	−	1111.86i	−0.121711	−	0.121711i
$438$		0			0
$439$	16643.3			1.80943			0.904716	−	0.426015i	$-0.140083\pi$
$439$	16643.3			1.80943			0.904716	+	0.426015i	$0.140083\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7621.94	+	7621.94i	−0.817448	+	0.817448i	−0.985738	−	0.168290i	$-0.946176\pi$
$443$	−7621.94	+	7621.94i	−0.817448	+	0.817448i	0.168290	+	0.985738i	$0.446176\pi$
$444$		0			0
$445$	−1503.91	−	1503.91i	−0.160207	−	0.160207i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1505.28i			0.158215i	0.996866	+	0.0791077i	$0.0252071\pi$
$449$			1505.28i			0.158215i	−0.996866	+	0.0791077i	$0.974793\pi$
$450$		0			0
$451$	−2060.84	+	2060.84i	−0.215169	+	0.215169i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			179.498i			0.0184945i
$456$		0			0
$457$			6918.12i			0.708131i	0.935221	+	0.354065i	$0.115201\pi$
$457$			6918.12i			0.708131i	−0.935221	+	0.354065i	$0.884799\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3348.19	−	3348.19i	0.338266	−	0.338266i	−0.517448	−	0.855715i	$-0.673118\pi$
$461$	3348.19	−	3348.19i	0.338266	−	0.338266i	0.855715	+	0.517448i	$0.173118\pi$
$462$		0			0
$463$			2122.03i			0.213000i	0.994313	+	0.106500i	$0.0339644\pi$
$463$			2122.03i			0.213000i	−0.994313	+	0.106500i	$0.966036\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12230.3	+	12230.3i	1.21189	+	1.21189i	0.970405	+	0.241482i	$0.0776334\pi$
$467$	12230.3	+	12230.3i	1.21189	+	1.21189i	0.241482	+	0.970405i	$0.422367\pi$
$468$		0			0
$469$	−7599.55	+	7599.55i	−0.748219	+	0.748219i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16796.3			−1.63276
$474$		0			0
$475$	1809.43	+	1809.43i	0.174784	+	0.174784i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−20589.9			−1.96404			−0.982020	−	0.188779i	$-0.939547\pi$
$479$	−20589.9			−1.96404			−0.982020	+	0.188779i	$0.939547\pi$
$480$		0			0
$481$	235.643			0.0223376
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4011.04	+	4011.04i	0.375530	+	0.375530i
$486$		0			0
$487$	−8672.88			−0.806994			−0.403497	−	0.914981i	$-0.632205\pi$
$487$	−8672.88			−0.806994			−0.403497	+	0.914981i	$0.632205\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1815.32	+	1815.32i	−0.166852	+	0.166852i	−0.785594	−	0.618742i	$-0.787643\pi$
$491$	−1815.32	+	1815.32i	−0.166852	+	0.166852i	0.618742	+	0.785594i	$0.287643\pi$
$492$		0			0
$493$	1565.84	+	1565.84i	0.143046	+	0.143046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	9225.68i		−	0.832652i
$498$		0			0
$499$	−2716.12	+	2716.12i	−0.243668	+	0.243668i	−0.818366	−	0.574698i	$-0.805120\pi$
$499$	−2716.12	+	2716.12i	−0.243668	+	0.243668i	0.574698	+	0.818366i	$0.305120\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			17354.3i			1.53835i	0.639040	+	0.769173i	$0.279332\pi$
$503$			17354.3i			1.53835i	−0.639040	+	0.769173i	$0.720668\pi$
$504$		0			0
$505$			3015.06i			0.265680i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8211.14	+	8211.14i	−0.715034	+	0.715034i	−0.967584	−	0.252550i	$-0.918731\pi$
$509$	−8211.14	+	8211.14i	−0.715034	+	0.715034i	0.252550	+	0.967584i	$0.418731\pi$
$510$		0			0
$511$		−	12599.1i		−	1.09071i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3953.41	−	3953.41i	−0.338268	−	0.338268i
$516$		0			0
$517$	−13726.3	+	13726.3i	−1.16767	+	1.16767i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9350.18			−0.786255			−0.393127	−	0.919484i	$-0.628607\pi$
$521$	−9350.18			−0.786255			−0.393127	+	0.919484i	$0.628607\pi$
$522$		0			0
$523$	11818.7	+	11818.7i	0.988140	+	0.988140i	0.999930	−	0.0117907i	$-0.00375319\pi$
$523$	11818.7	+	11818.7i	0.988140	+	0.988140i	−0.0117907	+	0.999930i	$0.503753\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11896.5			−0.983343
$528$		0			0
$529$	7311.43			0.600923
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−188.608	−	188.608i	−0.0153275	−	0.0153275i
$534$		0			0
$535$	4607.96			0.372373
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7143.75	+	7143.75i	−0.570878	+	0.570878i
$540$		0			0
$541$	10764.1	+	10764.1i	0.855422	+	0.855422i	0.990795	−	0.135373i	$-0.0432232\pi$
$541$	10764.1	+	10764.1i	0.855422	+	0.855422i	−0.135373	+	0.990795i	$0.543223\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	5883.40i		−	0.462417i
$546$		0			0
$547$	2093.77	−	2093.77i	0.163662	−	0.163662i	−0.620525	−	0.784187i	$-0.713080\pi$
$547$	2093.77	−	2093.77i	0.163662	−	0.163662i	0.784187	+	0.620525i	$0.213080\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	509.481i		−	0.0393913i
$552$		0			0
$553$		−	4478.95i		−	0.344420i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5245.63	+	5245.63i	−0.399038	+	0.399038i	−0.877894	−	0.478855i	$-0.841052\pi$
$557$	−5245.63	+	5245.63i	−0.399038	+	0.399038i	0.478855	+	0.877894i	$0.341052\pi$
$558$		0			0
$559$		−	1537.20i		−	0.116309i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7593.93	+	7593.93i	0.568465	+	0.568465i	0.931698	−	0.363233i	$-0.118327\pi$
$563$	7593.93	+	7593.93i	0.568465	+	0.568465i	−0.363233	+	0.931698i	$0.618327\pi$
$564$		0			0
$565$	−869.384	+	869.384i	−0.0647350	+	0.0647350i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3416.36			0.251707			0.125853	−	0.992049i	$-0.459833\pi$
$569$	3416.36			0.251707			0.125853	+	0.992049i	$0.459833\pi$
$570$		0			0
$571$	1823.75	+	1823.75i	0.133663	+	0.133663i	0.770773	−	0.637110i	$-0.219870\pi$
$571$	1823.75	+	1823.75i	0.133663	+	0.133663i	−0.637110	+	0.770773i	$0.719870\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7901.88			0.573098
$576$		0			0
$577$	−23143.8			−1.66982			−0.834912	−	0.550383i	$-0.814482\pi$
$577$	−23143.8			−1.66982			−0.834912	+	0.550383i	$0.814482\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5758.46	+	5758.46i	0.411190	+	0.411190i
$582$		0			0
$583$	31836.1			2.26161
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16398.7	−	16398.7i	1.15306	−	1.15306i	0.167122	−	0.985936i	$-0.446553\pi$
$587$	16398.7	−	16398.7i	1.15306	−	1.15306i	0.985936	−	0.167122i	$-0.0534473\pi$
$588$		0			0
$589$	1935.41	+	1935.41i	0.135394	+	0.135394i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	13256.2i		−	0.917989i	−0.888439	−	0.458994i	$-0.848210\pi$
$593$		−	13256.2i		−	0.917989i	0.888439	−	0.458994i	$-0.151790\pi$
$594$		0			0
$595$	2768.54	−	2768.54i	0.190755	−	0.190755i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			9973.40i			0.680304i	0.940370	+	0.340152i	$0.110479\pi$
$599$			9973.40i			0.680304i	−0.940370	+	0.340152i	$0.889521\pi$
$600$		0			0
$601$			22549.7i			1.53048i	0.643744	+	0.765241i	$0.277380\pi$
$601$			22549.7i			1.53048i	−0.643744	+	0.765241i	$0.722620\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2607.94	−	2607.94i	0.175253	−	0.175253i
$606$		0			0
$607$			21635.2i			1.44670i	0.690482	+	0.723350i	$0.257398\pi$
$607$			21635.2i			1.44670i	−0.690482	+	0.723350i	$0.742602\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1256.24	−	1256.24i	−0.0831783	−	0.0831783i
$612$		0			0
$613$	−2977.25	+	2977.25i	−0.196166	+	0.196166i	−0.798354	−	0.602188i	$-0.794296\pi$
$613$	−2977.25	+	2977.25i	−0.196166	+	0.196166i	0.602188	+	0.798354i	$0.294296\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6649.18			−0.433851			−0.216925	−	0.976188i	$-0.569603\pi$
$617$	−6649.18			−0.433851			−0.216925	+	0.976188i	$0.569603\pi$
$618$		0			0
$619$	−2920.44	−	2920.44i	−0.189632	−	0.189632i	0.605905	−	0.795537i	$-0.292811\pi$
$619$	−2920.44	−	2920.44i	−0.189632	−	0.189632i	−0.795537	+	0.605905i	$0.792811\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7318.86			−0.470664
$624$		0			0
$625$	−11409.3			−0.730198
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3634.50	−	3634.50i	−0.230393	−	0.230393i
$630$		0			0
$631$	3289.01			0.207501			0.103751	−	0.994603i	$-0.466916\pi$
$631$	3289.01			0.207501			0.103751	+	0.994603i	$0.466916\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4126.42	−	4126.42i	0.257877	−	0.257877i
$636$		0			0
$637$	−653.798	−	653.798i	−0.0406663	−	0.0406663i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	9161.75i		−	0.564536i	−0.959336	−	0.282268i	$-0.908913\pi$
$641$		−	9161.75i		−	0.564536i	0.959336	−	0.282268i	$-0.0910867\pi$
$642$		0			0
$643$	6165.52	−	6165.52i	0.378141	−	0.378141i	−0.492290	−	0.870431i	$-0.663840\pi$
$643$	6165.52	−	6165.52i	0.378141	−	0.378141i	0.870431	+	0.492290i	$0.163840\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			6481.62i			0.393846i	0.980419	+	0.196923i	$0.0630950\pi$
$647$			6481.62i			0.393846i	−0.980419	+	0.196923i	$0.936905\pi$
$648$		0			0
$649$		−	17900.5i		−	1.08268i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	772.442	−	772.442i	0.0462910	−	0.0462910i	−0.683582	−	0.729873i	$-0.739579\pi$
$653$	772.442	−	772.442i	0.0462910	−	0.0462910i	0.729873	+	0.683582i	$0.239579\pi$
$654$		0			0
$655$		−	920.751i		−	0.0549263i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1416.83	−	1416.83i	−0.0837510	−	0.0837510i	0.663990	−	0.747741i	$-0.268862\pi$
$659$	−1416.83	−	1416.83i	−0.0837510	−	0.0837510i	−0.747741	+	0.663990i	$0.768862\pi$
$660$		0			0
$661$	7384.66	−	7384.66i	0.434539	−	0.434539i	−0.455630	−	0.890169i	$-0.650586\pi$
$661$	7384.66	−	7384.66i	0.434539	−	0.434539i	0.890169	+	0.455630i	$0.150586\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−900.809			−0.0525291
$666$		0			0
$667$	−1112.47	−	1112.47i	−0.0645801	−	0.0645801i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	25937.2			1.49224
$672$		0			0
$673$	8695.15			0.498029			0.249014	−	0.968500i	$-0.419893\pi$
$673$	8695.15			0.498029			0.249014	+	0.968500i	$0.419893\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19700.8	−	19700.8i	−1.11841	−	1.11841i	−0.991975	−	0.126436i	$-0.959646\pi$
$677$	−19700.8	−	19700.8i	−1.11841	−	1.11841i	−0.126436	−	0.991975i	$-0.540354\pi$
$678$		0			0
$679$	19519.9			1.10325
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6263.20	+	6263.20i	−0.350885	+	0.350885i	−0.860439	−	0.509554i	$-0.829811\pi$
$683$	−6263.20	+	6263.20i	−0.350885	+	0.350885i	0.509554	+	0.860439i	$0.329811\pi$
$684$		0			0
$685$	5131.07	+	5131.07i	0.286202	+	0.286202i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			2913.65i			0.161105i
$690$		0			0
$691$	−21784.7	+	21784.7i	−1.19932	+	1.19932i	−0.224945	+	0.974371i	$0.572220\pi$
$691$	−21784.7	+	21784.7i	−1.19932	+	1.19932i	−0.974371	+	0.224945i	$0.927780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	2037.71i		−	0.111216i
$696$		0			0
$697$			5818.11i			0.316179i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6306.61	+	6306.61i	−0.339797	+	0.339797i	−0.856291	−	0.516494i	$-0.827237\pi$
$701$	−6306.61	+	6306.61i	−0.339797	+	0.339797i	0.516494	+	0.856291i	$0.327237\pi$
$702$		0			0
$703$			1182.57i			0.0634445i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7336.48	+	7336.48i	0.390264	+	0.390264i
$708$		0			0
$709$	−6829.13	+	6829.13i	−0.361740	+	0.361740i	−0.864453	−	0.502713i	$-0.832335\pi$
$709$	−6829.13	+	6829.13i	−0.361740	+	0.361740i	0.502713	+	0.864453i	$0.332335\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8452.05			0.443944
$714$		0			0
$715$	532.053	+	532.053i	0.0278289	+	0.0278289i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3688.19			−0.191302			−0.0956511	−	0.995415i	$-0.530493\pi$
$719$	−3688.19			−0.191302			−0.0956511	+	0.995415i	$0.530493\pi$
$720$		0			0
$721$	−19239.4			−0.993779
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1810.41	+	1810.41i	0.0927408	+	0.0927408i
$726$		0			0
$727$	−28693.9			−1.46382			−0.731911	−	0.681400i	$-0.761371\pi$
$727$	−28693.9			−1.46382			−0.731911	+	0.681400i	$0.761371\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−23709.5	+	23709.5i	−1.19963	+	1.19963i
$732$		0			0
$733$	4890.55	+	4890.55i	0.246434	+	0.246434i	0.819506	−	0.573071i	$-0.194248\pi$
$733$	4890.55	+	4890.55i	0.246434	+	0.246434i	−0.573071	+	0.819506i	$0.694248\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			45051.8i			2.25170i
$738$		0			0
$739$	23911.6	−	23911.6i	1.19026	−	1.19026i	0.213269	−	0.976993i	$-0.431589\pi$
$739$	23911.6	−	23911.6i	1.19026	−	1.19026i	0.976993	−	0.213269i	$-0.0684111\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	14426.5i		−	0.712323i	−0.934424	−	0.356162i	$-0.884085\pi$
$743$		−	14426.5i		−	0.712323i	0.934424	−	0.356162i	$-0.115915\pi$
$744$		0			0
$745$			9731.96i			0.478593i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11212.4	−	11212.4i	0.546987	−	0.546987i
$750$		0			0
$751$		−	17255.7i		−	0.838439i	−0.907885	−	0.419220i	$-0.862304\pi$
$751$		−	17255.7i		−	0.838439i	0.907885	−	0.419220i	$-0.137696\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6275.22	−	6275.22i	−0.302488	−	0.302488i
$756$		0			0
$757$	12874.2	−	12874.2i	0.618124	−	0.618124i	−0.326926	−	0.945050i	$-0.606013\pi$
$757$	12874.2	−	12874.2i	0.618124	−	0.618124i	0.945050	+	0.326926i	$0.106013\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4694.94			0.223642			0.111821	−	0.993728i	$-0.464332\pi$
$761$	4694.94			0.223642			0.111821	+	0.993728i	$0.464332\pi$
$762$		0			0
$763$	−14315.9	−	14315.9i	−0.679255	−	0.679255i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1638.26			0.0771242
$768$		0			0
$769$	27268.7			1.27872			0.639359	−	0.768908i	$-0.279200\pi$
$769$	27268.7			1.27872			0.639359	+	0.768908i	$0.279200\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10010.2	+	10010.2i	0.465772	+	0.465772i	0.900542	−	0.434769i	$-0.143170\pi$
$773$	10010.2	+	10010.2i	0.465772	+	0.465772i	−0.434769	+	0.900542i	$0.643170\pi$
$774$		0			0
$775$	−13754.8			−0.637529
$776$		0			0
$777$		0			0
$778$		0			0
$779$	946.527	−	946.527i	0.0435338	−	0.0435338i
$780$		0			0
$781$	−27345.9	−	27345.9i	−1.25290	−	1.25290i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	3930.34i		−	0.178700i
$786$		0			0
$787$	1703.37	−	1703.37i	0.0771521	−	0.0771521i	−0.667478	−	0.744630i	$-0.732626\pi$
$787$	1703.37	−	1703.37i	0.0771521	−	0.0771521i	0.744630	+	0.667478i	$0.232626\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4230.90i			0.190182i
$792$		0			0
$793$			2373.78i			0.106299i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13767.6	+	13767.6i	−0.611886	+	0.611886i	−0.943437	−	0.331551i	$-0.892428\pi$
$797$	−13767.6	+	13767.6i	−0.611886	+	0.611886i	0.331551	+	0.943437i	$0.392428\pi$
$798$		0			0
$799$			38751.9i			1.71582i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−37345.2	−	37345.2i	−1.64120	−	1.64120i
$804$		0			0
$805$	−1966.95	+	1966.95i	−0.0861189	+	0.0861189i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	34122.6			1.48292			0.741462	−	0.670995i	$-0.234133\pi$
$809$	34122.6			1.48292			0.741462	+	0.670995i	$0.234133\pi$
$810$		0			0
$811$	19921.8	+	19921.8i	0.862576	+	0.862576i	0.991637	−	0.129061i	$-0.0411962\pi$
$811$	19921.8	+	19921.8i	0.862576	+	0.862576i	−0.129061	+	0.991637i	$0.541196\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1294.87			0.0556529
$816$		0			0
$817$	7714.42			0.330347
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3546.97	−	3546.97i	−0.150780	−	0.150780i	0.627687	−	0.778466i	$-0.284002\pi$
$821$	−3546.97	−	3546.97i	−0.150780	−	0.150780i	−0.778466	+	0.627687i	$0.784002\pi$
$822$		0			0
$823$	−33931.8			−1.43717			−0.718583	−	0.695442i	$-0.755209\pi$
$823$	−33931.8			−1.43717			−0.718583	+	0.695442i	$0.755209\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5818.78	−	5818.78i	0.244666	−	0.244666i	−0.574111	−	0.818777i	$-0.694652\pi$
$827$	5818.78	−	5818.78i	0.244666	−	0.244666i	0.818777	+	0.574111i	$0.194652\pi$
$828$		0			0
$829$	−7895.08	−	7895.08i	−0.330769	−	0.330769i	0.522110	−	0.852878i	$-0.325145\pi$
$829$	−7895.08	−	7895.08i	−0.330769	−	0.330769i	−0.852878	+	0.522110i	$0.825145\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			20168.1i			0.838874i
$834$		0			0
$835$	3702.95	−	3702.95i	0.153468	−	0.153468i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	23123.0i		−	0.951485i	−0.879585	−	0.475743i	$-0.842179\pi$
$839$		−	23123.0i		−	0.951485i	0.879585	−	0.475743i	$-0.157821\pi$
$840$		0			0
$841$			23879.2i			0.979099i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5242.52	−	5242.52i	0.213430	−	0.213430i
$846$		0			0
$847$		−	12691.7i		−	0.514866i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2582.18	+	2582.18i	0.104014	+	0.104014i
$852$		0			0
$853$	10555.7	−	10555.7i	0.423707	−	0.423707i	−0.462771	−	0.886478i	$-0.653145\pi$
$853$	10555.7	−	10555.7i	0.423707	−	0.423707i	0.886478	+	0.462771i	$0.153145\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14835.9			0.591349			0.295674	−	0.955289i	$-0.404456\pi$
$857$	14835.9			0.591349			0.295674	+	0.955289i	$0.404456\pi$
$858$		0			0
$859$	−21200.1	−	21200.1i	−0.842069	−	0.842069i	0.147059	−	0.989128i	$-0.453019\pi$
$859$	−21200.1	−	21200.1i	−0.842069	−	0.842069i	−0.989128	+	0.147059i	$0.953019\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	38729.6			1.52766			0.763830	−	0.645418i	$-0.223317\pi$
$863$	38729.6			1.52766			0.763830	+	0.645418i	$0.223317\pi$
$864$		0			0
$865$	2883.23			0.113333
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13276.1	−	13276.1i	−0.518252	−	0.518252i
$870$		0			0
$871$	−4123.15			−0.160399
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6729.42	−	6729.42i	0.259995	−	0.259995i
$876$		0			0
$877$	−3724.13	−	3724.13i	−0.143392	−	0.143392i	0.631766	−	0.775159i	$-0.282330\pi$
$877$	−3724.13	−	3724.13i	−0.143392	−	0.143392i	−0.775159	+	0.631766i	$0.782330\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	10798.5i		−	0.412951i	−0.978452	−	0.206475i	$-0.933801\pi$
$881$		−	10798.5i		−	0.412951i	0.978452	−	0.206475i	$-0.0661993\pi$
$882$		0			0
$883$	3539.05	−	3539.05i	0.134879	−	0.134879i	−0.636444	−	0.771323i	$-0.719595\pi$
$883$	3539.05	−	3539.05i	0.134879	−	0.134879i	0.771323	+	0.636444i	$0.219595\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			22802.5i			0.863172i	0.902072	+	0.431586i	$0.142046\pi$
$887$			22802.5i			0.863172i	−0.902072	+	0.431586i	$0.857954\pi$
$888$		0			0
$889$		−	20081.4i		−	0.757604i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6304.40	−	6304.40i	0.236247	−	0.236247i
$894$		0			0
$895$		−	1374.47i		−	0.0513334i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1936.46	+	1936.46i	0.0718406	+	0.0718406i
$900$		0			0
$901$	44939.5	−	44939.5i	1.66166	−	1.66166i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	14064.6			0.516600
$906$		0			0
$907$	−6649.86	−	6649.86i	−0.243445	−	0.243445i	0.574829	−	0.818274i	$-0.305069\pi$
$907$	−6649.86	−	6649.86i	−0.243445	−	0.243445i	−0.818274	+	0.574829i	$0.805069\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2309.48			−0.0839917			−0.0419958	−	0.999118i	$-0.513372\pi$
$911$	−2309.48			−0.0839917			−0.0419958	+	0.999118i	$0.513372\pi$
$912$		0			0
$913$	34137.4			1.23744
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2240.44	−	2240.44i	−0.0806825	−	0.0806825i
$918$		0			0
$919$	−48567.3			−1.74329			−0.871646	−	0.490135i	$-0.836947\pi$
$919$	−48567.3			−1.74329			−0.871646	+	0.490135i	$0.836947\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2502.70	−	2502.70i	0.0892498	−	0.0892498i
$924$		0			0
$925$	−4202.20	−	4202.20i	−0.149370	−	0.149370i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	55572.0i		−	1.96260i	−0.192477	−	0.981302i	$-0.561652\pi$
$929$		−	55572.0i		−	1.96260i	0.192477	−	0.981302i	$-0.438348\pi$
$930$		0			0
$931$	3281.07	−	3281.07i	0.115502	−	0.115502i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	16412.5i		−	0.574061i
$936$		0			0
$937$		−	46921.8i		−	1.63593i	−0.575268	−	0.817965i	$-0.695102\pi$
$937$		−	46921.8i		−	1.63593i	0.575268	−	0.817965i	$-0.304898\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15833.3	+	15833.3i	−0.548514	+	0.548514i	−0.926011	−	0.377497i	$-0.876785\pi$
$941$	−15833.3	+	15833.3i	−0.548514	+	0.548514i	0.377497	+	0.926011i	$0.376785\pi$
$942$		0			0
$943$		−	4133.55i		−	0.142743i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4616.34	−	4616.34i	−0.158406	−	0.158406i	0.623454	−	0.781860i	$-0.285729\pi$
$947$	−4616.34	−	4616.34i	−0.158406	−	0.158406i	−0.781860	+	0.623454i	$0.785729\pi$
$948$		0			0
$949$	3417.84	−	3417.84i	0.116910	−	0.116910i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7702.58			−0.261816			−0.130908	−	0.991394i	$-0.541789\pi$
$953$	−7702.58			−0.261816			−0.130908	+	0.991394i	$0.541789\pi$
$954$		0			0
$955$	2613.69	+	2613.69i	0.0885622	+	0.0885622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24970.6			0.840817
$960$		0			0
$961$	15078.6			0.506145
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2771.48	+	2771.48i	0.0924530	+	0.0924530i
$966$		0			0
$967$	3885.39			0.129210			0.0646048	−	0.997911i	$-0.479421\pi$
$967$	3885.39			0.129210			0.0646048	+	0.997911i	$0.479421\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27634.4	−	27634.4i	0.913315	−	0.913315i	−0.0832167	−	0.996531i	$-0.526519\pi$
$971$	27634.4	−	27634.4i	0.913315	−	0.913315i	0.996531	+	0.0832167i	$0.0265194\pi$
$972$		0			0
$973$	−4958.31	−	4958.31i	−0.163367	−	0.163367i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			59115.6i			1.93580i	0.251336	+	0.967900i	$0.419130\pi$
$977$			59115.6i			1.93580i	−0.251336	+	0.967900i	$0.580870\pi$
$978$		0			0
$979$	−21693.9	+	21693.9i	−0.708212	+	0.708212i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	13689.3i		−	0.444172i	−0.975027	−	0.222086i	$-0.928713\pi$
$983$		−	13689.3i		−	0.444172i	0.975027	−	0.222086i	$-0.0712866\pi$
$984$		0			0
$985$		−	11090.5i		−	0.358753i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16844.7	−	16844.7i	0.541587	−	0.541587i
$990$		0			0
$991$		−	37624.8i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$991$		−	37624.8i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3133.22	−	3133.22i	−0.0998290	−	0.0998290i
$996$		0			0
$997$	−11609.6	+	11609.6i	−0.368786	+	0.368786i	−0.867034	−	0.498248i	$-0.833977\pi$
$997$	−11609.6	+	11609.6i	−0.368786	+	0.368786i	0.498248	+	0.867034i	$0.333977\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.l.a.863.13		48
3.2	odd	2	inner	1152.4.l.a.863.12		48
4.3	odd	2		1152.4.l.b.863.13		48
8.3	odd	2		144.4.l.a.35.12	✓	48
8.5	even	2		576.4.l.a.431.12		48
12.11	even	2		1152.4.l.b.863.12		48
16.3	odd	4		576.4.l.a.143.13		48
16.5	even	4		1152.4.l.b.287.12		48
16.11	odd	4	inner	1152.4.l.a.287.12		48
16.13	even	4		144.4.l.a.107.13	yes	48
24.5	odd	2		576.4.l.a.431.13		48
24.11	even	2		144.4.l.a.35.13	yes	48
48.5	odd	4		1152.4.l.b.287.13		48
48.11	even	4	inner	1152.4.l.a.287.13		48
48.29	odd	4		144.4.l.a.107.12	yes	48
48.35	even	4		576.4.l.a.143.12		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.12	✓	48	8.3	odd	2
144.4.l.a.35.13	yes	48	24.11	even	2
144.4.l.a.107.12	yes	48	48.29	odd	4
144.4.l.a.107.13	yes	48	16.13	even	4
576.4.l.a.143.12		48	48.35	even	4
576.4.l.a.143.13		48	16.3	odd	4
576.4.l.a.431.12		48	8.5	even	2
576.4.l.a.431.13		48	24.5	odd	2
1152.4.l.a.287.12		48	16.11	odd	4	inner
1152.4.l.a.287.13		48	48.11	even	4	inner
1152.4.l.a.863.12		48	3.2	odd	2	inner
1152.4.l.a.863.13		48	1.1	even	1	trivial
1152.4.l.b.287.12		48	16.5	even	4
1152.4.l.b.287.13		48	48.5	odd	4
1152.4.l.b.863.12		48	12.11	even	2
1152.4.l.b.863.13		48	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.40838 + 2.40838i) q^{5} +11.7205 q^{7} +(34.7409 - 34.7409i) q^{11} +(3.17950 + 3.17950i) q^{13} -98.0797i q^{17} +(-15.9562 + 15.9562i) q^{19} +69.6819i q^{23} -113.399i q^{25} +(-15.9649 + 15.9649i) q^{29} -121.295i q^{31} +(28.2275 + 28.2275i) q^{35} +(37.0567 - 37.0567i) q^{37} -59.3202 q^{41} +(-241.737 - 241.737i) q^{43} -395.106 q^{47} -205.629 q^{49} +(458.194 + 458.194i) q^{53} +167.339 q^{55} +(257.629 - 257.629i) q^{59} +(373.295 + 373.295i) q^{61} +15.3149i q^{65} +(-648.397 + 648.397i) q^{67} -787.139i q^{71} -1074.96i q^{73} +(407.181 - 407.181i) q^{77} -382.146i q^{79} +(491.315 + 491.315i) q^{83} +(236.213 - 236.213i) q^{85} -624.448 q^{89} +(37.2653 + 37.2653i) q^{91} -76.8575 q^{95} +1665.45 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{19} + 864 q^{43} + 2352 q^{49} + 576 q^{55} - 1824 q^{61} + 816 q^{67} + 480 q^{85} - 3600 q^{91}+O(q^{100}) \) 48 * q - 48 * q^19 + 864 * q^43 + 2352 * q^49 + 576 * q^55 - 1824 * q^61 + 816 * q^67 + 480 * q^85 - 3600 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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