LMFDB - Embedded newform 5292.2.l.j.361.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(361,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5292.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.l (of order $3$, degree $2$, 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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.3
Character	$\chi$	$=$	5292.361
Dual form			5292.2.l.j.3313.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38486 q^{5} +2.25956 q^{11} +(-2.37884 + 4.12027i) q^{13} +(2.15202 - 3.72740i) q^{17} +(-4.29815 - 7.44461i) q^{19} -1.32910 q^{23} +0.687562 q^{25} +(3.87886 + 6.71839i) q^{29} +(0.405320 + 0.702036i) q^{31} +(2.31613 + 4.01166i) q^{37} +(5.00426 - 8.66764i) q^{41} +(-1.74292 - 3.01883i) q^{43} +(2.18338 - 3.78173i) q^{47} +(-5.83934 + 10.1140i) q^{53} -5.38874 q^{55} +(2.40463 + 4.16495i) q^{59} +(-0.575967 + 0.997604i) q^{61} +(5.67319 - 9.82626i) q^{65} +(2.06381 + 3.57463i) q^{67} +4.41593 q^{71} +(-6.05590 + 10.4891i) q^{73} +(4.23312 - 7.33198i) q^{79} +(0.817808 + 1.41648i) q^{83} +(-5.13226 + 8.88934i) q^{85} +(3.17155 + 5.49329i) q^{89} +(10.2505 + 17.7543i) q^{95} +(-5.98278 - 10.3625i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{11} - 16 q^{23} + 24 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) $ 24 * q - 8 * q^11 - 16 * q^23 + 24 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	−2.38486			−1.06654			−0.533271	−	0.845944i	$-0.679037\pi$
$5$	−2.38486			−1.06654			−0.533271	+	0.845944i	$0.679037\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.25956			0.681283			0.340642	−	0.940193i	$-0.389356\pi$
$11$	2.25956			0.681283			0.340642	+	0.940193i	$0.389356\pi$
$12$		0			0
$13$	−2.37884	+	4.12027i	−0.659770	+	1.14276i	0.320904	+	0.947112i	$0.396013\pi$
$13$	−2.37884	+	4.12027i	−0.659770	+	1.14276i	−0.980675	+	0.195644i	$0.937320\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.15202	−	3.72740i	0.521941	−	0.904028i	−0.477733	−	0.878505i	$-0.658541\pi$
$17$	2.15202	−	3.72740i	0.521941	−	0.904028i	0.999674	−	0.0255234i	$-0.00812524\pi$
$18$		0			0
$19$	−4.29815	−	7.44461i	−0.986062	−	1.70791i	−0.637118	−	0.770766i	$-0.719874\pi$
$19$	−4.29815	−	7.44461i	−0.986062	−	1.70791i	−0.348944	−	0.937144i	$-0.613460\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.32910			−0.277137			−0.138568	−	0.990353i	$-0.544250\pi$
$23$	−1.32910			−0.277137			−0.138568	+	0.990353i	$0.544250\pi$
$24$		0			0
$25$	0.687562			0.137512
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.87886	+	6.71839i	0.720287	+	1.24757i	0.960885	+	0.276948i	$0.0893231\pi$
$29$	3.87886	+	6.71839i	0.720287	+	1.24757i	−0.240598	+	0.970625i	$0.577344\pi$
$30$		0			0
$31$	0.405320	+	0.702036i	0.0727977	+	0.126089i	0.900126	−	0.435629i	$-0.143474\pi$
$31$	0.405320	+	0.702036i	0.0727977	+	0.126089i	−0.827329	+	0.561718i	$0.810141\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.31613	+	4.01166i	0.380770	+	0.659513i	0.991172	−	0.132579i	$-0.0423257\pi$
$37$	2.31613	+	4.01166i	0.380770	+	0.659513i	−0.610403	+	0.792091i	$0.708992\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.00426	−	8.66764i	0.781534	−	1.35366i	−0.149513	−	0.988760i	$-0.547771\pi$
$41$	5.00426	−	8.66764i	0.781534	−	1.35366i	0.931048	−	0.364898i	$-0.118896\pi$
$42$		0			0
$43$	−1.74292	−	3.01883i	−0.265793	−	0.460367i	0.701978	−	0.712199i	$-0.252300\pi$
$43$	−1.74292	−	3.01883i	−0.265793	−	0.460367i	−0.967771	+	0.251831i	$0.918967\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.18338	−	3.78173i	0.318479	−	0.551622i	−0.661692	−	0.749776i	$-0.730161\pi$
$47$	2.18338	−	3.78173i	0.318479	−	0.551622i	0.980171	+	0.198154i	$0.0634946\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.83934	+	10.1140i	−0.802094	+	1.38927i	0.116141	+	0.993233i	$0.462948\pi$
$53$	−5.83934	+	10.1140i	−0.802094	+	1.38927i	−0.918235	+	0.396036i	$0.870386\pi$
$54$		0			0
$55$	−5.38874			−0.726617
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.40463	+	4.16495i	0.313056	+	0.542230i	0.979022	−	0.203752i	$-0.0653137\pi$
$59$	2.40463	+	4.16495i	0.313056	+	0.542230i	−0.665966	+	0.745982i	$0.731980\pi$
$60$		0			0
$61$	−0.575967	+	0.997604i	−0.0737450	+	0.127730i	−0.900540	−	0.434774i	$-0.856828\pi$
$61$	−0.575967	+	0.997604i	−0.0737450	+	0.127730i	0.826795	+	0.562504i	$0.190162\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.67319	−	9.82626i	0.703673	−	1.21880i
$66$		0			0
$67$	2.06381	+	3.57463i	0.252135	+	0.436710i	0.964113	−	0.265491i	$-0.0855341\pi$
$67$	2.06381	+	3.57463i	0.252135	+	0.436710i	−0.711979	+	0.702201i	$0.752201\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.41593			0.524074			0.262037	−	0.965058i	$-0.415606\pi$
$71$	4.41593			0.524074			0.262037	+	0.965058i	$0.415606\pi$
$72$		0			0
$73$	−6.05590	+	10.4891i	−0.708790	+	1.22766i	0.256517	+	0.966540i	$0.417425\pi$
$73$	−6.05590	+	10.4891i	−0.708790	+	1.22766i	−0.965306	+	0.261120i	$0.915908\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.23312	−	7.33198i	0.476263	−	0.824913i	−0.523367	−	0.852108i	$-0.675324\pi$
$79$	4.23312	−	7.33198i	0.476263	−	0.824913i	0.999630	+	0.0271950i	$0.00865752\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.817808	+	1.41648i	0.0897661	+	0.155479i	0.907412	−	0.420242i	$-0.138055\pi$
$83$	0.817808	+	1.41648i	0.0897661	+	0.155479i	−0.817646	+	0.575721i	$0.804721\pi$
$84$		0			0
$85$	−5.13226	+	8.88934i	−0.556672	+	0.964184i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.17155	+	5.49329i	0.336184	+	0.582287i	0.983711	−	0.179755i	$-0.0575304\pi$
$89$	3.17155	+	5.49329i	0.336184	+	0.582287i	−0.647528	+	0.762042i	$0.724197\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.2505	+	17.7543i	1.05168	+	1.82156i
$96$		0			0
$97$	−5.98278	−	10.3625i	−0.607459	−	1.05215i	−0.991658	−	0.128900i	$-0.958855\pi$
$97$	−5.98278	−	10.3625i	−0.607459	−	1.05215i	0.384198	−	0.923251i	$-0.374478\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.8470			1.07931			0.539656	−	0.841885i	$-0.318554\pi$
$101$	10.8470			1.07931			0.539656	+	0.841885i	$0.318554\pi$
$102$		0			0
$103$	−18.3445			−1.80754			−0.903769	−	0.428020i	$-0.859211\pi$
$103$	−18.3445			−1.80754			−0.903769	+	0.428020i	$0.859211\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.91706	+	3.32044i	0.185329	+	0.321000i	0.943687	−	0.330838i	$-0.107332\pi$
$107$	1.91706	+	3.32044i	0.185329	+	0.321000i	−0.758358	+	0.651838i	$0.773998\pi$
$108$		0			0
$109$	−1.32248	+	2.29060i	−0.126671	+	0.219400i	−0.922385	−	0.386272i	$-0.873762\pi$
$109$	−1.32248	+	2.29060i	−0.126671	+	0.219400i	0.795714	+	0.605672i	$0.207096\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.64275	−	4.57738i	0.248609	−	0.430603i	−0.714531	−	0.699604i	$-0.753360\pi$
$113$	2.64275	−	4.57738i	0.248609	−	0.430603i	0.963140	+	0.269000i	$0.0866933\pi$
$114$		0			0
$115$	3.16972			0.295578
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.89438			−0.535853
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.2846			0.919880
$126$		0			0
$127$	−7.67115			−0.680704			−0.340352	−	0.940298i	$-0.610546\pi$
$127$	−7.67115			−0.680704			−0.340352	+	0.940298i	$0.610546\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.3801			−1.69325			−0.846625	−	0.532189i	$-0.821369\pi$
$131$	−19.3801			−1.69325			−0.846625	+	0.532189i	$0.821369\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.68997			0.827870			0.413935	−	0.910306i	$-0.364154\pi$
$137$	9.68997			0.827870			0.413935	+	0.910306i	$0.364154\pi$
$138$		0			0
$139$	−3.81197	+	6.60252i	−0.323327	+	0.560018i	−0.981172	−	0.193135i	$-0.938135\pi$
$139$	−3.81197	+	6.60252i	−0.323327	+	0.560018i	0.657846	+	0.753153i	$0.271468\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.37513	+	9.30999i	−0.449491	+	0.778541i
$144$		0			0
$145$	−9.25055	−	16.0224i	−0.768216	−	1.33059i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.09179			0.744829			0.372414	−	0.928067i	$-0.378530\pi$
$149$	9.09179			0.744829			0.372414	+	0.928067i	$0.378530\pi$
$150$		0			0
$151$	−20.2582			−1.64859			−0.824294	−	0.566162i	$-0.808428\pi$
$151$	−20.2582			−1.64859			−0.824294	+	0.566162i	$0.808428\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.966633	−	1.67426i	−0.0776418	−	0.134480i
$156$		0			0
$157$	4.18075	+	7.24127i	0.333660	+	0.577917i	0.983227	−	0.182388i	$-0.0583828\pi$
$157$	4.18075	+	7.24127i	0.333660	+	0.577917i	−0.649566	+	0.760305i	$0.725049\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	9.69195	+	16.7869i	0.759132	+	1.31486i	0.943294	+	0.331959i	$0.107710\pi$
$163$	9.69195	+	16.7869i	0.759132	+	1.31486i	−0.184162	+	0.982896i	$0.558957\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.84158	+	15.3141i	−0.684182	+	1.18504i	0.289511	+	0.957175i	$0.406507\pi$
$167$	−8.84158	+	15.3141i	−0.684182	+	1.18504i	−0.973693	+	0.227864i	$0.926826\pi$
$168$		0			0
$169$	−4.81772	−	8.34454i	−0.370594	−	0.641888i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.5928	+	18.3473i	−0.805356	+	1.39492i	0.110695	+	0.993854i	$0.464692\pi$
$173$	−10.5928	+	18.3473i	−0.805356	+	1.39492i	−0.916051	+	0.401063i	$0.868641\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.49389	−	11.2478i	0.485376	−	0.840697i	−0.514482	−	0.857501i	$-0.672016\pi$
$179$	6.49389	−	11.2478i	0.485376	−	0.840697i	0.999859	+	0.0168043i	$0.00534924\pi$
$180$		0			0
$181$	16.8238			1.25050			0.625251	−	0.780423i	$-0.284996\pi$
$181$	16.8238			1.25050			0.625251	+	0.780423i	$0.284996\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.52365	−	9.56725i	−0.406107	−	0.703398i
$186$		0			0
$187$	4.86262	−	8.42230i	0.355590	−	0.615899i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.58561	−	7.94251i	0.331803	−	0.574700i	−0.651062	−	0.759024i	$-0.725676\pi$
$191$	4.58561	−	7.94251i	0.331803	−	0.574700i	0.982865	+	0.184324i	$0.0590096\pi$
$192$		0			0
$193$	12.8153	+	22.1968i	0.922466	+	1.59776i	0.795586	+	0.605840i	$0.207163\pi$
$193$	12.8153	+	22.1968i	0.922466	+	1.59776i	0.126880	+	0.991918i	$0.459504\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.1036			−1.14734			−0.573668	−	0.819088i	$-0.694480\pi$
$197$	−16.1036			−1.14734			−0.573668	+	0.819088i	$0.694480\pi$
$198$		0			0
$199$	−9.61411	+	16.6521i	−0.681526	+	1.18044i	0.292989	+	0.956116i	$0.405350\pi$
$199$	−9.61411	+	16.6521i	−0.681526	+	1.18044i	−0.974515	+	0.224322i	$0.927983\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−11.9345	+	20.6711i	−0.833540	+	1.44373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.71192	−	16.8215i	−0.671788	−	1.16357i
$210$		0			0
$211$	−12.3251	+	21.3477i	−0.848496	+	1.46964i	0.0340549	+	0.999420i	$0.489158\pi$
$211$	−12.3251	+	21.3477i	−0.848496	+	1.46964i	−0.882551	+	0.470218i	$0.844175\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.15663	+	7.19949i	0.283480	+	0.491001i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.2386	+	17.7338i	0.688723	+	1.19290i
$222$		0			0
$223$	7.41074	+	12.8358i	0.496260	+	0.859547i	0.999991	−	0.00431335i	$-0.00137299\pi$
$223$	7.41074	+	12.8358i	0.496260	+	0.859547i	−0.503731	+	0.863861i	$0.668040\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.1550			1.20499			0.602496	−	0.798122i	$-0.294173\pi$
$227$	18.1550			1.20499			0.602496	+	0.798122i	$0.294173\pi$
$228$		0			0
$229$	9.50691			0.628235			0.314117	−	0.949384i	$-0.398291\pi$
$229$	9.50691			0.628235			0.314117	+	0.949384i	$0.398291\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.1715	+	17.6176i	0.666360	+	1.15417i	0.978915	+	0.204270i	$0.0654820\pi$
$233$	10.1715	+	17.6176i	0.666360	+	1.15417i	−0.312554	+	0.949900i	$0.601185\pi$
$234$		0			0
$235$	−5.20707	+	9.01890i	−0.339671	+	0.588328i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.90544	+	15.4247i	−0.576045	+	0.997739i	0.419882	+	0.907579i	$0.362071\pi$
$239$	−8.90544	+	15.4247i	−0.576045	+	0.997739i	−0.995927	+	0.0901607i	$0.971262\pi$
$240$		0			0
$241$	6.28873			0.405093			0.202546	−	0.979273i	$-0.435078\pi$
$241$	6.28873			0.405093			0.202546	+	0.979273i	$0.435078\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	40.8983			2.60230
$248$		0			0
$249$		0			0
$250$		0			0
$251$	25.1868			1.58978			0.794889	−	0.606755i	$-0.207529\pi$
$251$	25.1868			1.58978			0.794889	+	0.606755i	$0.207529\pi$
$252$		0			0
$253$	−3.00318			−0.188809
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−28.6803			−1.78903			−0.894514	−	0.447040i	$-0.852478\pi$
$257$	−28.6803			−1.78903			−0.894514	+	0.447040i	$0.852478\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	14.4168			0.888977			0.444489	−	0.895784i	$-0.353385\pi$
$263$	14.4168			0.888977			0.444489	+	0.895784i	$0.353385\pi$
$264$		0			0
$265$	13.9260	−	24.1205i	0.855468	−	1.48171i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.44997	+	11.1717i	−0.393262	+	0.681150i	−0.992878	−	0.119138i	$-0.961987\pi$
$269$	−6.44997	+	11.1717i	−0.393262	+	0.681150i	0.599616	+	0.800288i	$0.295320\pi$
$270$		0			0
$271$	9.73110	+	16.8548i	0.591122	+	1.02385i	0.994082	+	0.108636i	$0.0346483\pi$
$271$	9.73110	+	16.8548i	0.591122	+	1.02385i	−0.402959	+	0.915218i	$0.632018\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.55359			0.0936849
$276$		0			0
$277$	15.9124			0.956084			0.478042	−	0.878337i	$-0.341347\pi$
$277$	15.9124			0.956084			0.478042	+	0.878337i	$0.341347\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	13.9998	+	24.2484i	0.835158	+	1.44654i	0.893901	+	0.448263i	$0.147957\pi$
$281$	13.9998	+	24.2484i	0.835158	+	1.44654i	−0.0587432	+	0.998273i	$0.518709\pi$
$282$		0			0
$283$	−2.62345	−	4.54394i	−0.155948	−	0.270109i	0.777456	−	0.628937i	$-0.216510\pi$
$283$	−2.62345	−	4.54394i	−0.155948	−	0.270109i	−0.933404	+	0.358828i	$0.883176\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.762363	−	1.32045i	−0.0448449	−	0.0776736i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.65939	+	2.87415i	−0.0969428	+	0.167910i	−0.910418	−	0.413690i	$-0.864240\pi$
$293$	−1.65939	+	2.87415i	−0.0969428	+	0.167910i	0.813475	+	0.581600i	$0.197573\pi$
$294$		0			0
$295$	−5.73471	−	9.93282i	−0.333888	−	0.578311i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.16171	−	5.47625i	0.182847	−	0.316700i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.37360	−	2.37915i	0.0786521	−	0.136230i
$306$		0			0
$307$	12.4703			0.711715			0.355857	−	0.934540i	$-0.384189\pi$
$307$	12.4703			0.711715			0.355857	+	0.934540i	$0.384189\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.3383	+	19.6385i	0.642936	+	1.11360i	0.984774	+	0.173840i	$0.0556174\pi$
$311$	11.3383	+	19.6385i	0.642936	+	1.11360i	−0.341838	+	0.939759i	$0.611049\pi$
$312$		0			0
$313$	−3.16108	+	5.47515i	−0.178675	+	0.309474i	−0.941427	−	0.337217i	$-0.890514\pi$
$313$	−3.16108	+	5.47515i	−0.178675	+	0.309474i	0.762752	+	0.646691i	$0.223848\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.79041	+	13.4934i	−0.437553	+	0.757864i	−0.997500	−	0.0706643i	$-0.977488\pi$
$317$	−7.79041	+	13.4934i	−0.437553	+	0.757864i	0.559947	+	0.828528i	$0.310821\pi$
$318$		0			0
$319$	8.76453	+	15.1806i	0.490719	+	0.849951i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−36.9987			−2.05867
$324$		0			0
$325$	−1.63560	+	2.83294i	−0.0907266	+	0.157143i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.8634	−	18.8159i	0.597104	−	1.03422i	−0.396142	−	0.918189i	$-0.629651\pi$
$331$	10.8634	−	18.8159i	0.597104	−	1.03422i	0.993246	−	0.116026i	$-0.0370155\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.92190	−	8.52499i	−0.268912	−	0.465770i
$336$		0			0
$337$	4.04329	−	7.00319i	0.220252	−	0.381488i	−0.734632	−	0.678465i	$-0.762645\pi$
$337$	4.04329	−	7.00319i	0.220252	−	0.381488i	0.954884	+	0.296977i	$0.0959786\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.915846	+	1.58629i	0.0495959	+	0.0859026i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.466877	−	0.808654i	−0.0250632	−	0.0434108i	0.853222	−	0.521548i	$-0.174645\pi$
$347$	−0.466877	−	0.808654i	−0.0250632	−	0.0434108i	−0.878285	+	0.478138i	$0.841312\pi$
$348$		0			0
$349$	1.90264	+	3.29548i	0.101846	+	0.176403i	0.912445	−	0.409199i	$-0.134192\pi$
$349$	1.90264	+	3.29548i	0.101846	+	0.176403i	−0.810599	+	0.585601i	$0.800858\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.1195			−0.698280			−0.349140	−	0.937071i	$-0.613526\pi$
$353$	−13.1195			−0.698280			−0.349140	+	0.937071i	$0.613526\pi$
$354$		0			0
$355$	−10.5314			−0.558947
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.7829	−	18.6765i	−0.569100	−	0.985710i	−0.996655	−	0.0817206i	$-0.973958\pi$
$359$	−10.7829	−	18.6765i	−0.569100	−	0.985710i	0.427556	−	0.903989i	$-0.359375\pi$
$360$		0			0
$361$	−27.4481	+	47.5415i	−1.44464	+	2.50218i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.4425	−	25.0151i	0.755954	−	1.30935i
$366$		0			0
$367$	−18.6619			−0.974143			−0.487072	−	0.873362i	$-0.661935\pi$
$367$	−18.6619			−0.974143			−0.487072	+	0.873362i	$0.661935\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−20.1707			−1.04440			−0.522201	−	0.852823i	$-0.674889\pi$
$373$	−20.1707			−1.04440			−0.522201	+	0.852823i	$0.674889\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−36.9087			−1.90090
$378$		0			0
$379$	18.2436			0.937110			0.468555	−	0.883434i	$-0.344775\pi$
$379$	18.2436			0.937110			0.468555	+	0.883434i	$0.344775\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−22.8563			−1.16790			−0.583952	−	0.811788i	$-0.698494\pi$
$383$	−22.8563			−1.16790			−0.583952	+	0.811788i	$0.698494\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.3429			−0.625811			−0.312905	−	0.949784i	$-0.601302\pi$
$389$	−12.3429			−0.625811			−0.312905	+	0.949784i	$0.601302\pi$
$390$		0			0
$391$	−2.86025	+	4.95410i	−0.144649	+	0.250539i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.0954	+	17.4858i	−0.507955	+	0.879804i
$396$		0			0
$397$	−13.1016	−	22.6927i	−0.657551	−	1.13891i	−0.981248	−	0.192751i	$-0.938259\pi$
$397$	−13.1016	−	22.6927i	−0.657551	−	1.13891i	0.323696	−	0.946161i	$-0.395074\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.91118			−0.195315			−0.0976575	−	0.995220i	$-0.531135\pi$
$401$	−3.91118			−0.195315			−0.0976575	+	0.995220i	$0.531135\pi$
$402$		0			0
$403$	−3.85676			−0.192119
$404$		0			0
$405$		0			0
$406$		0			0
$407$	5.23344	+	9.06459i	0.259412	+	0.449315i
$408$		0			0
$409$	1.05065	+	1.81978i	0.0519513	+	0.0899823i	0.890832	−	0.454334i	$-0.150123\pi$
$409$	1.05065	+	1.81978i	0.0519513	+	0.0899823i	−0.838880	+	0.544316i	$0.816789\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.95036	−	3.37812i	−0.0957393	−	0.165825i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.7017	+	20.2679i	−0.571664	+	0.990150i	0.424732	+	0.905319i	$0.360368\pi$
$419$	−11.7017	+	20.2679i	−0.571664	+	0.990150i	−0.996395	+	0.0848311i	$0.972965\pi$
$420$		0			0
$421$	9.78341	+	16.9454i	0.476814	+	0.825866i	0.999647	−	0.0265688i	$-0.00845812\pi$
$421$	9.78341	+	16.9454i	0.476814	+	0.825866i	−0.522833	+	0.852435i	$0.675125\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.47965	−	2.56282i	0.0717733	−	0.124315i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.35438	−	5.80996i	0.161575	−	0.279856i	−0.773859	−	0.633358i	$-0.781676\pi$
$431$	3.35438	−	5.80996i	0.161575	−	0.279856i	0.935434	+	0.353502i	$0.115009\pi$
$432$		0			0
$433$	−9.46607			−0.454910			−0.227455	−	0.973789i	$-0.573041\pi$
$433$	−9.46607			−0.454910			−0.227455	+	0.973789i	$0.573041\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.71267	+	9.89463i	0.273274	+	0.473324i
$438$		0			0
$439$	3.82386	−	6.62312i	0.182503	−	0.316104i	−0.760229	−	0.649655i	$-0.774913\pi$
$439$	3.82386	−	6.62312i	0.182503	−	0.316104i	0.942732	+	0.333550i	$0.108247\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.0763	−	31.3091i	0.858833	−	1.48754i	−0.0142102	−	0.999899i	$-0.504523\pi$
$443$	18.0763	−	31.3091i	0.858833	−	1.48754i	0.873043	−	0.487643i	$-0.162143\pi$
$444$		0			0
$445$	−7.56371	−	13.1007i	−0.358554	−	0.621034i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.7337			−0.648132			−0.324066	−	0.946034i	$-0.605050\pi$
$449$	−13.7337			−0.648132			−0.324066	+	0.946034i	$0.605050\pi$
$450$		0			0
$451$	11.3074	−	19.5851i	0.532446	−	0.922224i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.213109	−	0.369115i	0.00996881	−	0.0172665i	−0.860998	−	0.508608i	$-0.830160\pi$
$457$	0.213109	−	0.369115i	0.00996881	−	0.0172665i	0.870967	+	0.491342i	$0.163493\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.26453	−	3.92228i	−0.105470	−	0.182679i	0.808460	−	0.588551i	$-0.200301\pi$
$461$	−2.26453	−	3.92228i	−0.105470	−	0.182679i	−0.913930	+	0.405872i	$0.866968\pi$
$462$		0			0
$463$	16.5598	−	28.6825i	0.769600	−	1.33299i	−0.168179	−	0.985756i	$-0.553789\pi$
$463$	16.5598	−	28.6825i	0.769600	−	1.33299i	0.937780	−	0.347230i	$-0.112878\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.756660	+	1.31057i	0.0350140	+	0.0606461i	0.883001	−	0.469370i	$-0.155519\pi$
$467$	0.756660	+	1.31057i	0.0350140	+	0.0606461i	−0.847987	+	0.530016i	$0.822186\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.93824	−	6.82123i	−0.181080	−	0.313640i
$474$		0			0
$475$	−2.95524	−	5.11863i	−0.135596	−	0.234859i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	22.2926			1.01858			0.509288	−	0.860596i	$-0.329909\pi$
$479$	22.2926			1.01858			0.509288	+	0.860596i	$0.329909\pi$
$480$		0			0
$481$	−22.0388			−1.00488
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.2681	+	24.7131i	0.647881	+	1.12216i
$486$		0			0
$487$	−2.31676	+	4.01275i	−0.104983	+	0.181835i	−0.913731	−	0.406319i	$-0.866812\pi$
$487$	−2.31676	+	4.01275i	−0.104983	+	0.181835i	0.808749	+	0.588155i	$0.200145\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.5858	−	21.7993i	0.567990	−	0.983788i	−0.428774	−	0.903412i	$-0.641054\pi$
$491$	12.5858	−	21.7993i	0.567990	−	0.983788i	0.996765	−	0.0803766i	$-0.0256123\pi$
$492$		0			0
$493$	33.3895			1.50379
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.27455			−0.101823			−0.0509114	−	0.998703i	$-0.516213\pi$
$499$	−2.27455			−0.101823			−0.0509114	+	0.998703i	$0.516213\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−8.94793			−0.398968			−0.199484	−	0.979901i	$-0.563927\pi$
$503$	−8.94793			−0.398968			−0.199484	+	0.979901i	$0.563927\pi$
$504$		0			0
$505$	−25.8685			−1.15113
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−33.2687			−1.47461			−0.737304	−	0.675561i	$-0.763901\pi$
$509$	−33.2687			−1.47461			−0.737304	+	0.675561i	$0.763901\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	43.7491			1.92782
$516$		0			0
$517$	4.93349	−	8.54505i	0.216975	−	0.375811i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.57396	−	13.1185i	0.331821	−	0.574731i	−0.651048	−	0.759037i	$-0.725670\pi$
$521$	7.57396	−	13.1185i	0.331821	−	0.574731i	0.982869	+	0.184306i	$0.0590036\pi$
$522$		0			0
$523$	−5.23952	−	9.07512i	−0.229108	−	0.396827i	0.728436	−	0.685114i	$-0.240248\pi$
$523$	−5.23952	−	9.07512i	−0.229108	−	0.396827i	−0.957544	+	0.288287i	$0.906914\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.48903			0.151984
$528$		0			0
$529$	−21.2335			−0.923195
$530$		0			0
$531$		0			0
$532$		0			0
$533$	23.8086	+	41.2378i	1.03127	+	1.78621i
$534$		0			0
$535$	−4.57192	−	7.91880i	−0.197661	−	0.342360i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.43874	−	4.22402i	−0.104850	−	0.181605i	0.808827	−	0.588047i	$-0.200103\pi$
$541$	−2.43874	−	4.22402i	−0.104850	−	0.181605i	−0.913677	+	0.406442i	$0.866769\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.15393	−	5.46277i	0.135100	−	0.234000i
$546$		0			0
$547$	9.62179	+	16.6654i	0.411398	+	0.712562i	0.995043	−	0.0994468i	$-0.0317073\pi$
$547$	9.62179	+	16.6654i	0.411398	+	0.712562i	−0.583645	+	0.812009i	$0.698374\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	33.3438	−	57.7532i	1.42049	−	2.46037i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.32388	−	12.6853i	0.310323	−	0.537495i	−0.668109	−	0.744063i	$-0.732896\pi$
$557$	7.32388	−	12.6853i	0.310323	−	0.537495i	0.978432	+	0.206568i	$0.0662295\pi$
$558$		0			0
$559$	16.5845			0.701450
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.85335	−	3.21010i	−0.0781095	−	0.135290i	0.824325	−	0.566117i	$-0.191555\pi$
$563$	−1.85335	−	3.21010i	−0.0781095	−	0.135290i	−0.902434	+	0.430828i	$0.858222\pi$
$564$		0			0
$565$	−6.30259	+	10.9164i	−0.265152	+	0.459257i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.1768	−	26.2871i	0.636246	−	1.10201i	−0.350004	−	0.936748i	$-0.613820\pi$
$569$	15.1768	−	26.2871i	0.636246	−	1.10201i	0.986250	−	0.165262i	$-0.0528470\pi$
$570$		0			0
$571$	5.88458	+	10.1924i	0.246262	+	0.426539i	0.962486	−	0.271332i	$-0.0874642\pi$
$571$	5.88458	+	10.1924i	0.246262	+	0.426539i	−0.716224	+	0.697871i	$0.754131\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.913839			−0.0381097
$576$		0			0
$577$	−1.79640	+	3.11146i	−0.0747852	+	0.129532i	−0.900993	−	0.433834i	$-0.857160\pi$
$577$	−1.79640	+	3.11146i	−0.0747852	+	0.129532i	0.826208	+	0.563366i	$0.190494\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−13.1943	+	22.8533i	−0.546454	+	0.946485i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.62298	−	8.00724i	−0.190811	−	0.330494i	0.754708	−	0.656060i	$-0.227778\pi$
$587$	−4.62298	−	8.00724i	−0.190811	−	0.330494i	−0.945519	+	0.325566i	$0.894445\pi$
$588$		0			0
$589$	3.48425	−	6.03490i	0.143566	−	0.248664i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17.0396	+	29.5135i	0.699735	+	1.21198i	0.968558	+	0.248786i	$0.0800317\pi$
$593$	17.0396	+	29.5135i	0.699735	+	1.21198i	−0.268824	+	0.963189i	$0.586635\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.51340	−	6.08539i	−0.143554	−	0.248642i	0.785279	−	0.619142i	$-0.212520\pi$
$599$	−3.51340	−	6.08539i	−0.143554	−	0.248642i	−0.928832	+	0.370500i	$0.879186\pi$
$600$		0			0
$601$	−2.31218	−	4.00481i	−0.0943158	−	0.163360i	0.815007	−	0.579451i	$-0.196733\pi$
$601$	−2.31218	−	4.00481i	−0.0943158	−	0.163360i	−0.909323	+	0.416091i	$0.863400\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.0573			0.571510
$606$		0			0
$607$	20.2081			0.820222			0.410111	−	0.912036i	$-0.365490\pi$
$607$	20.2081			0.820222			0.410111	+	0.912036i	$0.365490\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.3878	+	17.9922i	0.420246	+	0.727888i
$612$		0			0
$613$	8.91037	−	15.4332i	0.359887	−	0.623342i	−0.628055	−	0.778169i	$-0.716149\pi$
$613$	8.91037	−	15.4332i	0.359887	−	0.623342i	0.987942	+	0.154827i	$0.0494820\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.0671	+	26.0969i	−0.606577	+	1.05062i	0.385223	+	0.922824i	$0.374125\pi$
$617$	−15.0671	+	26.0969i	−0.606577	+	1.05062i	−0.991800	+	0.127799i	$0.959209\pi$
$618$		0			0
$619$	−31.9116			−1.28264			−0.641318	−	0.767275i	$-0.721612\pi$
$619$	−31.9116			−1.28264			−0.641318	+	0.767275i	$0.721612\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−27.9651			−1.11860
$626$		0			0
$627$		0			0
$628$		0			0
$629$	19.9374			0.794957
$630$		0			0
$631$	−10.1430			−0.403788			−0.201894	−	0.979407i	$-0.564710\pi$
$631$	−10.1430			−0.403788			−0.201894	+	0.979407i	$0.564710\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.2946			0.726000
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−43.8512			−1.73202			−0.866009	−	0.500028i	$-0.833323\pi$
$641$	−43.8512			−1.73202			−0.866009	+	0.500028i	$0.833323\pi$
$642$		0			0
$643$	1.10737	−	1.91802i	0.0436704	−	0.0756394i	−0.843364	−	0.537343i	$-0.819428\pi$
$643$	1.10737	−	1.91802i	0.0436704	−	0.0756394i	0.887034	+	0.461703i	$0.152762\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.4633	−	18.1230i	0.411355	−	0.712488i	−0.583683	−	0.811981i	$-0.698389\pi$
$647$	10.4633	−	18.1230i	0.411355	−	0.712488i	0.995038	+	0.0994938i	$0.0317224\pi$
$648$		0			0
$649$	5.43341	+	9.41095i	0.213280	+	0.369412i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.02883			−0.314193			−0.157096	−	0.987583i	$-0.550213\pi$
$653$	−8.02883			−0.314193			−0.157096	+	0.987583i	$0.550213\pi$
$654$		0			0
$655$	46.2189			1.80592
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.25793	−	16.0352i	−0.360638	−	0.624643i	0.627428	−	0.778675i	$-0.284108\pi$
$659$	−9.25793	−	16.0352i	−0.360638	−	0.624643i	−0.988066	+	0.154031i	$0.950774\pi$
$660$		0			0
$661$	10.4273	+	18.0606i	0.405574	+	0.702474i	0.994388	−	0.105794i	$-0.0337385\pi$
$661$	10.4273	+	18.0606i	0.405574	+	0.702474i	−0.588814	+	0.808268i	$0.700405\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−5.15540	−	8.92941i	−0.199618	−	0.345748i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.30143	+	2.25415i	−0.0502412	+	0.0870204i
$672$		0			0
$673$	11.4484	+	19.8292i	0.441303	+	0.764360i	0.997786	−	0.0664992i	$-0.0211830\pi$
$673$	11.4484	+	19.8292i	0.441303	+	0.764360i	−0.556483	+	0.830859i	$0.687850\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	23.3119	−	40.3774i	0.895948	−	1.55183i	0.0633218	−	0.997993i	$-0.479831\pi$
$677$	23.3119	−	40.3774i	0.895948	−	1.55183i	0.832627	−	0.553835i	$-0.186836\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.3102	+	31.7141i	−0.700618	+	1.21351i	0.267631	+	0.963521i	$0.413759\pi$
$683$	−18.3102	+	31.7141i	−0.700618	+	1.21351i	−0.968250	+	0.249985i	$0.919574\pi$
$684$		0			0
$685$	−23.1092			−0.882958
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−27.7816	−	48.1192i	−1.05840	−	1.83320i
$690$		0			0
$691$	13.2586	−	22.9645i	0.504380	−	0.873611i	−0.495607	−	0.868547i	$-0.665055\pi$
$691$	13.2586	−	22.9645i	0.504380	−	0.873611i	0.999987	−	0.00506472i	$-0.00161216\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.09101	−	15.7461i	0.344842	−	0.597283i
$696$		0			0
$697$	−21.5385	−	37.3058i	−0.815830	−	1.41306i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29.9931			1.13282			0.566411	−	0.824123i	$-0.308331\pi$
$701$	29.9931			1.13282			0.566411	+	0.824123i	$0.308331\pi$
$702$		0			0
$703$	19.9101	−	34.4854i	0.750925	−	1.30064i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.48805	+	9.50558i	−0.206108	+	0.356990i	−0.950485	−	0.310770i	$-0.899413\pi$
$709$	−5.48805	+	9.50558i	−0.206108	+	0.356990i	0.744377	+	0.667759i	$0.232746\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.538712	−	0.933076i	−0.0201749	−	0.0349440i
$714$		0			0
$715$	12.8189	−	22.2030i	0.479401	−	0.830346i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.55211	+	11.3486i	0.244352	+	0.423231i	0.961949	−	0.273228i	$-0.0880913\pi$
$719$	6.55211	+	11.3486i	0.244352	+	0.423231i	−0.717597	+	0.696459i	$0.754758\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.66696	+	4.61931i	0.0990483	+	0.171557i
$726$		0			0
$727$	8.96026	+	15.5196i	0.332318	+	0.575591i	0.982966	−	0.183788i	$-0.0588361\pi$
$727$	8.96026	+	15.5196i	0.332318	+	0.575591i	−0.650648	+	0.759379i	$0.725503\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15.0032			−0.554913
$732$		0			0
$733$	−5.91320			−0.218409			−0.109204	−	0.994019i	$-0.534830\pi$
$733$	−5.91320			−0.218409			−0.109204	+	0.994019i	$0.534830\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.66331	+	8.07709i	0.171775	+	0.297523i
$738$		0			0
$739$	14.5887	−	25.2683i	0.536653	−	0.929511i	−0.462428	−	0.886657i	$-0.653022\pi$
$739$	14.5887	−	25.2683i	0.536653	−	0.929511i	0.999081	−	0.0428538i	$-0.0136450\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.8512	−	32.6513i	0.691584	−	1.19786i	−0.279735	−	0.960077i	$-0.590247\pi$
$743$	18.8512	−	32.6513i	0.691584	−	1.19786i	0.971319	−	0.237781i	$-0.0764201\pi$
$744$		0			0
$745$	−21.6827			−0.794391
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	40.6553			1.48353			0.741767	−	0.670658i	$-0.233988\pi$
$751$	40.6553			1.48353			0.741767	+	0.670658i	$0.233988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	48.3130			1.75829
$756$		0			0
$757$	17.6704			0.642241			0.321120	−	0.947038i	$-0.395941\pi$
$757$	17.6704			0.642241			0.321120	+	0.947038i	$0.395941\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.28837			−0.0467033			−0.0233516	−	0.999727i	$-0.507434\pi$
$761$	−1.28837			−0.0467033			−0.0233516	+	0.999727i	$0.507434\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−22.8809			−0.826182
$768$		0			0
$769$	−4.19275	+	7.26205i	−0.151194	+	0.261876i	−0.931667	−	0.363314i	$-0.881645\pi$
$769$	−4.19275	+	7.26205i	−0.151194	+	0.261876i	0.780472	+	0.625190i	$0.214979\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−25.1530	+	43.5663i	−0.904691	+	1.56697i	−0.0833598	+	0.996520i	$0.526565\pi$
$773$	−25.1530	+	43.5663i	−0.904691	+	1.56697i	−0.821331	+	0.570451i	$0.806768\pi$
$774$		0			0
$775$	0.278683	+	0.482693i	0.0100106	+	0.0173388i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−86.0362			−3.08257
$780$		0			0
$781$	9.97806			0.357043
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.97051	−	17.2694i	−0.355863	−	0.616373i
$786$		0			0
$787$	11.1922	+	19.3855i	0.398960	+	0.691020i	0.993598	−	0.112974i	$-0.0360378\pi$
$787$	11.1922	+	19.3855i	0.398960	+	0.691020i	−0.594638	+	0.803994i	$0.702704\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.74026	−	4.74627i	−0.0973095	−	0.168545i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.706182	+	1.22314i	−0.0250143	+	0.0433260i	−0.878262	−	0.478181i	$-0.841296\pi$
$797$	−0.706182	+	1.22314i	−0.0250143	+	0.0433260i	0.853247	+	0.521507i	$0.174630\pi$
$798$		0			0
$799$	−9.39736	−	16.2767i	−0.332455	−	0.575828i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13.6837	+	23.7008i	−0.482887	+	0.836384i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10.0424	−	17.3939i	0.353071	−	0.611537i	−0.633715	−	0.773567i	$-0.718471\pi$
$809$	10.0424	−	17.3939i	0.353071	−	0.611537i	0.986786	+	0.162030i	$0.0518040\pi$
$810$		0			0
$811$	−55.7821			−1.95878			−0.979388	−	0.201988i	$-0.935260\pi$
$811$	−55.7821			−1.95878			−0.979388	+	0.201988i	$0.935260\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−23.1140	−	40.0345i	−0.809646	−	1.40235i
$816$		0			0
$817$	−14.9827	+	25.9507i	−0.524177	+	0.907901i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.4346	+	42.3219i	−0.852772	+	1.47704i	0.0259249	+	0.999664i	$0.491747\pi$
$821$	−24.4346	+	42.3219i	−0.852772	+	1.47704i	−0.878697	+	0.477380i	$0.841586\pi$
$822$		0			0
$823$	0.266319	+	0.461277i	0.00928328	+	0.0160791i	0.870630	−	0.491939i	$-0.163712\pi$
$823$	0.266319	+	0.461277i	0.00928328	+	0.0160791i	−0.861346	+	0.508018i	$0.830378\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10.1951			−0.354518			−0.177259	−	0.984164i	$-0.556723\pi$
$827$	−10.1951			−0.354518			−0.177259	+	0.984164i	$0.556723\pi$
$828$		0			0
$829$	10.6346	−	18.4197i	0.369355	−	0.639742i	−0.620110	−	0.784515i	$-0.712912\pi$
$829$	10.6346	−	18.4197i	0.369355	−	0.639742i	0.989465	+	0.144773i	$0.0462452\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	21.0859	−	36.5219i	0.729709	−	1.26389i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.47717	+	16.4149i	0.327188	+	0.566707i	0.981953	−	0.189126i	$-0.0605655\pi$
$839$	9.47717	+	16.4149i	0.327188	+	0.566707i	−0.654765	+	0.755833i	$0.727232\pi$
$840$		0			0
$841$	−15.5912	+	27.0047i	−0.537626	+	0.931196i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.4896	+	19.9006i	0.395254	+	0.684601i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.07837	−	5.33190i	−0.105525	−	0.182775i
$852$		0			0
$853$	2.75811	+	4.77718i	0.0944358	+	0.163568i	0.909373	−	0.415982i	$-0.136562\pi$
$853$	2.75811	+	4.77718i	0.0944358	+	0.163568i	−0.814937	+	0.579549i	$0.803229\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.8494			0.541406			0.270703	−	0.962663i	$-0.412744\pi$
$857$	15.8494			0.541406			0.270703	+	0.962663i	$0.412744\pi$
$858$		0			0
$859$	21.4009			0.730189			0.365095	−	0.930970i	$-0.381037\pi$
$859$	21.4009			0.730189			0.365095	+	0.930970i	$0.381037\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.01860	−	3.49632i	−0.0687140	−	0.119016i	0.829621	−	0.558326i	$-0.188556\pi$
$863$	−2.01860	−	3.49632i	−0.0687140	−	0.119016i	−0.898335	+	0.439310i	$0.855223\pi$
$864$		0			0
$865$	25.2624	−	43.7557i	0.858946	−	1.48774i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.56500	−	16.5671i	0.324470	−	0.561999i
$870$		0			0
$871$	−19.6379			−0.665404
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25.3810			−0.857057			−0.428529	−	0.903528i	$-0.640968\pi$
$877$	−25.3810			−0.857057			−0.428529	+	0.903528i	$0.640968\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.7755			−0.632561			−0.316281	−	0.948666i	$-0.602434\pi$
$881$	−18.7755			−0.632561			−0.316281	+	0.948666i	$0.602434\pi$
$882$		0			0
$883$	8.52167			0.286777			0.143388	−	0.989666i	$-0.454200\pi$
$883$	8.52167			0.286777			0.143388	+	0.989666i	$0.454200\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.1008			−0.339152			−0.169576	−	0.985517i	$-0.554240\pi$
$887$	−10.1008			−0.339152			−0.169576	+	0.985517i	$0.554240\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−37.5380			−1.25616
$894$		0			0
$895$	−15.4870	+	26.8243i	−0.517674	+	0.896638i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.14437	+	5.44620i	−0.104870	+	0.181641i
$900$		0			0
$901$	25.1327	+	43.5311i	0.837292	+	1.45023i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−40.1224			−1.33371
$906$		0			0
$907$	−21.2176			−0.704519			−0.352260	−	0.935902i	$-0.614587\pi$
$907$	−21.2176			−0.704519			−0.352260	+	0.935902i	$0.614587\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.46609	+	7.73550i	0.147968	+	0.256289i	0.930476	−	0.366352i	$-0.119393\pi$
$911$	4.46609	+	7.73550i	0.147968	+	0.256289i	−0.782508	+	0.622640i	$0.786060\pi$
$912$		0			0
$913$	1.84789	+	3.20063i	0.0611561	+	0.105925i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−17.4564	−	30.2354i	−0.575834	−	0.997374i	−0.995950	−	0.0899041i	$-0.971344\pi$
$919$	−17.4564	−	30.2354i	−0.575834	−	0.997374i	0.420116	−	0.907470i	$-0.361989\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.5048	+	18.1948i	−0.345769	+	0.598889i
$924$		0			0
$925$	1.59248	+	2.75826i	0.0523605	+	0.0906911i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.51692	+	9.55559i	−0.181004	+	0.313509i	−0.942223	−	0.334987i	$-0.891268\pi$
$929$	−5.51692	+	9.55559i	−0.181004	+	0.313509i	0.761218	+	0.648496i	$0.224601\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−11.5967	+	20.0860i	−0.379251	+	0.656883i
$936$		0			0
$937$	−13.6426			−0.445686			−0.222843	−	0.974854i	$-0.571534\pi$
$937$	−13.6426			−0.445686			−0.222843	+	0.974854i	$0.571534\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.0278	+	20.8328i	0.392095	+	0.679129i	0.992726	−	0.120398i	$-0.0384170\pi$
$941$	12.0278	+	20.8328i	0.392095	+	0.679129i	−0.600630	+	0.799527i	$0.705084\pi$
$942$		0			0
$943$	−6.65117	+	11.5202i	−0.216592	+	0.375148i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.69602	−	11.5978i	0.217591	−	0.376879i	−0.736480	−	0.676460i	$-0.763513\pi$
$947$	6.69602	−	11.5978i	0.217591	−	0.376879i	0.954071	+	0.299580i	$0.0968466\pi$
$948$		0			0
$949$	−28.8120	−	49.9038i	−0.935277	−	1.61995i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−36.3444			−1.17731			−0.588655	−	0.808384i	$-0.700342\pi$
$953$	−36.3444			−1.17731			−0.588655	+	0.808384i	$0.700342\pi$
$954$		0			0
$955$	−10.9360	+	18.9418i	−0.353882	+	0.612942i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.1714	−	26.2777i	0.489401	−	0.847667i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−30.5627	−	52.9362i	−0.983849	−	1.70408i
$966$		0			0
$967$	4.82455	−	8.35637i	0.155147	−	0.268723i	−0.777965	−	0.628307i	$-0.783748\pi$
$967$	4.82455	−	8.35637i	0.155147	−	0.268723i	0.933113	+	0.359584i	$0.117082\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21.6888	+	37.5660i	0.696025	+	1.20555i	0.969834	+	0.243767i	$0.0783832\pi$
$971$	21.6888	+	37.5660i	0.696025	+	1.20555i	−0.273809	+	0.961784i	$0.588283\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.60004	+	4.50340i	0.0831826	+	0.144076i	0.904615	−	0.426229i	$-0.140158\pi$
$977$	2.60004	+	4.50340i	0.0831826	+	0.144076i	−0.821433	+	0.570305i	$0.806825\pi$
$978$		0			0
$979$	7.16631	+	12.4124i	0.229036	+	0.396703i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−23.2697			−0.742190			−0.371095	−	0.928595i	$-0.621018\pi$
$983$	−23.2697			−0.742190			−0.371095	+	0.928595i	$0.621018\pi$
$984$		0			0
$985$	38.4049			1.22368
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.31652	+	4.01233i	0.0736610	+	0.127585i
$990$		0			0
$991$	19.5191	−	33.8080i	0.620044	−	1.07395i	−0.369433	−	0.929257i	$-0.620448\pi$
$991$	19.5191	−	33.8080i	0.620044	−	1.07395i	0.989477	−	0.144691i	$-0.0462187\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	22.9283	−	39.7130i	0.726877	−	1.25899i
$996$		0			0
$997$	−24.1809			−0.765818			−0.382909	−	0.923786i	$-0.625078\pi$
$997$	−24.1809			−0.765818			−0.382909	+	0.923786i	$0.625078\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.l.j.361.3		24
3.2	odd	2		1764.2.l.j.949.1		24
7.2	even	3		5292.2.i.j.1549.10		24
7.3	odd	6		5292.2.j.i.1765.3		24
7.4	even	3		5292.2.j.i.1765.10		24
7.5	odd	6		5292.2.i.j.1549.3		24
7.6	odd	2	inner	5292.2.l.j.361.10		24
9.2	odd	6		1764.2.i.j.1537.8		24
9.7	even	3		5292.2.i.j.2125.10		24
21.2	odd	6		1764.2.i.j.373.8		24
21.5	even	6		1764.2.i.j.373.5		24
21.11	odd	6		1764.2.j.i.589.8	yes	24
21.17	even	6		1764.2.j.i.589.5	✓	24
21.20	even	2		1764.2.l.j.949.12		24
63.2	odd	6		1764.2.l.j.961.1		24
63.11	odd	6		1764.2.j.i.1177.8	yes	24
63.16	even	3	inner	5292.2.l.j.3313.3		24
63.20	even	6		1764.2.i.j.1537.5		24
63.25	even	3		5292.2.j.i.3529.10		24
63.34	odd	6		5292.2.i.j.2125.3		24
63.38	even	6		1764.2.j.i.1177.5	yes	24
63.47	even	6		1764.2.l.j.961.12		24
63.52	odd	6		5292.2.j.i.3529.3		24
63.61	odd	6	inner	5292.2.l.j.3313.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.j.373.5		24	21.5	even	6
1764.2.i.j.373.8		24	21.2	odd	6
1764.2.i.j.1537.5		24	63.20	even	6
1764.2.i.j.1537.8		24	9.2	odd	6
1764.2.j.i.589.5	✓	24	21.17	even	6
1764.2.j.i.589.8	yes	24	21.11	odd	6
1764.2.j.i.1177.5	yes	24	63.38	even	6
1764.2.j.i.1177.8	yes	24	63.11	odd	6
1764.2.l.j.949.1		24	3.2	odd	2
1764.2.l.j.949.12		24	21.20	even	2
1764.2.l.j.961.1		24	63.2	odd	6
1764.2.l.j.961.12		24	63.47	even	6
5292.2.i.j.1549.3		24	7.5	odd	6
5292.2.i.j.1549.10		24	7.2	even	3
5292.2.i.j.2125.3		24	63.34	odd	6
5292.2.i.j.2125.10		24	9.7	even	3
5292.2.j.i.1765.3		24	7.3	odd	6
5292.2.j.i.1765.10		24	7.4	even	3
5292.2.j.i.3529.3		24	63.52	odd	6
5292.2.j.i.3529.10		24	63.25	even	3
5292.2.l.j.361.3		24	1.1	even	1	trivial
5292.2.l.j.361.10		24	7.6	odd	2	inner
5292.2.l.j.3313.3		24	63.16	even	3	inner
5292.2.l.j.3313.10		24	63.61	odd	6	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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.38486 q^{5} +2.25956 q^{11} +(-2.37884 + 4.12027i) q^{13} +(2.15202 - 3.72740i) q^{17} +(-4.29815 - 7.44461i) q^{19} -1.32910 q^{23} +0.687562 q^{25} +(3.87886 + 6.71839i) q^{29} +(0.405320 + 0.702036i) q^{31} +(2.31613 + 4.01166i) q^{37} +(5.00426 - 8.66764i) q^{41} +(-1.74292 - 3.01883i) q^{43} +(2.18338 - 3.78173i) q^{47} +(-5.83934 + 10.1140i) q^{53} -5.38874 q^{55} +(2.40463 + 4.16495i) q^{59} +(-0.575967 + 0.997604i) q^{61} +(5.67319 - 9.82626i) q^{65} +(2.06381 + 3.57463i) q^{67} +4.41593 q^{71} +(-6.05590 + 10.4891i) q^{73} +(4.23312 - 7.33198i) q^{79} +(0.817808 + 1.41648i) q^{83} +(-5.13226 + 8.88934i) q^{85} +(3.17155 + 5.49329i) q^{89} +(10.2505 + 17.7543i) q^{95} +(-5.98278 - 10.3625i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{11} - 16 q^{23} + 24 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) \) 24 * q - 8 * q^11 - 16 * q^23 + 24 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more