LMFDB - Embedded newform 5292.2.bm.c.2285.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(2285,5292)]

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, 5, 1]))

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

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

chi := DirichletCharacter("5292.2285");

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.bm (of order $6$, 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:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2285.5
Character	$\chi$	$=$	5292.2285
Dual form			5292.2.bm.c.4625.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62774 q^{5} +O(q^{10})$	$q-2.62774 q^{5} -5.98739i q^{11} +(-2.54231 - 1.46780i) q^{13} +(0.0167322 - 0.0289811i) q^{17} +(-7.07369 + 4.08400i) q^{19} +8.13607i q^{23} +1.90501 q^{25} +(0.949006 - 0.547909i) q^{29} +(-7.01724 + 4.05141i) q^{31} +(-0.105898 - 0.183421i) q^{37} +(2.23087 - 3.86399i) q^{41} +(3.78001 + 6.54717i) q^{43} +(3.53805 - 6.12809i) q^{47} +(-1.59204 - 0.919166i) q^{53} +15.7333i q^{55} +(-2.36844 - 4.10225i) q^{59} +(10.3773 + 5.99136i) q^{61} +(6.68053 + 3.85700i) q^{65} +(-3.81147 - 6.60166i) q^{67} -3.52461i q^{71} +(-2.24398 - 1.29556i) q^{73} +(3.58952 - 6.21723i) q^{79} +(0.932743 + 1.61556i) q^{83} +(-0.0439680 + 0.0761548i) q^{85} +(3.56125 + 6.16826i) q^{89} +(18.5878 - 10.7317i) q^{95} +(-13.2520 + 7.65104i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 48 q^{25} - 48 q^{53} - 72 q^{65} - 24 q^{79} - 24 q^{85} + 96 q^{95}+O(q^{100}) $ 48 * q + 48 * q^25 - 48 * q^53 - 72 * q^65 - 24 * q^79 - 24 * q^85 + 96 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	−2.62774			−1.17516			−0.587580	−	0.809166i	$-0.699919\pi$
$5$	−2.62774			−1.17516			−0.587580	+	0.809166i	$0.699919\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.98739i		−	1.80527i	−0.430411	−	0.902633i	$-0.641631\pi$
$11$		−	5.98739i		−	1.80527i	0.430411	−	0.902633i	$-0.358369\pi$
$12$		0			0
$13$	−2.54231	−	1.46780i	−0.705110	−	0.407095i	0.104138	−	0.994563i	$-0.466792\pi$
$13$	−2.54231	−	1.46780i	−0.705110	−	0.407095i	−0.809248	+	0.587467i	$0.800125\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.0167322	−	0.0289811i	0.00405817	−	0.00702895i	−0.863989	−	0.503510i	$-0.832042\pi$
$17$	0.0167322	−	0.0289811i	0.00405817	−	0.00702895i	0.868047	+	0.496481i	$0.165375\pi$
$18$		0			0
$19$	−7.07369	+	4.08400i	−1.62282	+	0.936933i	−0.636653	+	0.771150i	$0.719682\pi$
$19$	−7.07369	+	4.08400i	−1.62282	+	0.936933i	−0.986162	+	0.165783i	$0.946985\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.13607i			1.69649i	0.529605	+	0.848244i	$0.322340\pi$
$23$			8.13607i			1.69649i	−0.529605	+	0.848244i	$0.677660\pi$
$24$		0			0
$25$	1.90501			0.381003
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.949006	−	0.547909i	0.176226	−	0.101744i	−0.409292	−	0.912403i	$-0.634224\pi$
$29$	0.949006	−	0.547909i	0.176226	−	0.101744i	0.585518	+	0.810659i	$0.300891\pi$
$30$		0			0
$31$	−7.01724	+	4.05141i	−1.26033	+	0.727654i	−0.973139	−	0.230218i	$-0.926056\pi$
$31$	−7.01724	+	4.05141i	−1.26033	+	0.727654i	−0.287195	+	0.957872i	$0.592723\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.105898	−	0.183421i	−0.0174095	−	0.0301542i	0.857189	−	0.515001i	$-0.172209\pi$
$37$	−0.105898	−	0.183421i	−0.0174095	−	0.0301542i	−0.874599	+	0.484847i	$0.838875\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.23087	−	3.86399i	0.348404	−	0.603453i	−0.637562	−	0.770399i	$-0.720057\pi$
$41$	2.23087	−	3.86399i	0.348404	−	0.603453i	0.985966	+	0.166946i	$0.0533904\pi$
$42$		0			0
$43$	3.78001	+	6.54717i	0.576446	+	0.998434i	0.995883	+	0.0906496i	$0.0288943\pi$
$43$	3.78001	+	6.54717i	0.576446	+	0.998434i	−0.419437	+	0.907785i	$0.637772\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.53805	−	6.12809i	0.516078	−	0.893873i	−0.483748	−	0.875207i	$-0.660725\pi$
$47$	3.53805	−	6.12809i	0.516078	−	0.893873i	0.999826	−	0.0186658i	$-0.00594185\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.59204	−	0.919166i	−0.218684	−	0.126257i	0.386657	−	0.922224i	$-0.373630\pi$
$53$	−1.59204	−	0.919166i	−0.218684	−	0.126257i	−0.605341	+	0.795967i	$0.706963\pi$
$54$		0			0
$55$			15.7333i			2.12148i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.36844	−	4.10225i	−0.308344	−	0.534068i	0.669656	−	0.742671i	$-0.266442\pi$
$59$	−2.36844	−	4.10225i	−0.308344	−	0.534068i	−0.978000	+	0.208603i	$0.933108\pi$
$60$		0			0
$61$	10.3773	+	5.99136i	1.32868	+	0.767115i	0.985096	−	0.172006i	$-0.0550248\pi$
$61$	10.3773	+	5.99136i	1.32868	+	0.767115i	0.343586	+	0.939121i	$0.388358\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.68053	+	3.85700i	0.828618	+	0.478403i
$66$		0			0
$67$	−3.81147	−	6.60166i	−0.465646	−	0.806522i	0.533585	−	0.845747i	$-0.320844\pi$
$67$	−3.81147	−	6.60166i	−0.465646	−	0.806522i	−0.999230	+	0.0392248i	$0.987511\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.52461i		−	0.418294i	−0.977884	−	0.209147i	$-0.932931\pi$
$71$		−	3.52461i		−	0.418294i	0.977884	−	0.209147i	$-0.0670687\pi$
$72$		0			0
$73$	−2.24398	−	1.29556i	−0.262638	−	0.151634i	0.362899	−	0.931828i	$-0.381787\pi$
$73$	−2.24398	−	1.29556i	−0.262638	−	0.151634i	−0.625537	+	0.780194i	$0.715120\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.58952	−	6.21723i	0.403852	−	0.699493i	−0.590335	−	0.807159i	$-0.701004\pi$
$79$	3.58952	−	6.21723i	0.403852	−	0.699493i	0.994187	+	0.107666i	$0.0343376\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.932743	+	1.61556i	0.102382	+	0.177331i	0.912666	−	0.408707i	$-0.134020\pi$
$83$	0.932743	+	1.61556i	0.102382	+	0.177331i	−0.810284	+	0.586038i	$0.800687\pi$
$84$		0			0
$85$	−0.0439680	+	0.0761548i	−0.00476900	+	0.00826014i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.56125	+	6.16826i	0.377492	+	0.653835i	0.990697	−	0.136089i	$-0.0434534\pi$
$89$	3.56125	+	6.16826i	0.377492	+	0.653835i	−0.613205	+	0.789924i	$0.710120\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	18.5878	−	10.7317i	1.90707	−	1.10105i
$96$		0			0
$97$	−13.2520	+	7.65104i	−1.34554	+	0.776845i	−0.987613	−	0.156907i	$-0.949848\pi$
$97$	−13.2520	+	7.65104i	−1.34554	+	0.776845i	−0.357922	+	0.933752i	$0.616514\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.77515			0.674153			0.337077	−	0.941477i	$-0.390562\pi$
$101$	6.77515			0.674153			0.337077	+	0.941477i	$0.390562\pi$
$102$		0			0
$103$		−	4.88220i		−	0.481058i	−0.970642	−	0.240529i	$-0.922679\pi$
$103$		−	4.88220i		−	0.481058i	0.970642	−	0.240529i	$-0.0773209\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.8238	−	7.40383i	1.23972	−	0.715756i	0.270687	−	0.962667i	$-0.412749\pi$
$107$	12.8238	−	7.40383i	1.23972	−	0.715756i	0.969038	+	0.246912i	$0.0794158\pi$
$108$		0			0
$109$	5.82144	−	10.0830i	0.557593	−	0.965780i	−0.440104	−	0.897947i	$-0.645058\pi$
$109$	5.82144	−	10.0830i	0.557593	−	0.965780i	0.997697	−	0.0678327i	$-0.0216084\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.24947	+	0.721383i	0.117540	+	0.0678620i	0.557618	−	0.830098i	$-0.311716\pi$
$113$	1.24947	+	0.721383i	0.117540	+	0.0678620i	−0.440077	+	0.897960i	$0.645049\pi$
$114$		0			0
$115$		−	21.3795i		−	1.99365i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−24.8489			−2.25899
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.13282			0.727421
$126$		0			0
$127$	8.25511			0.732523			0.366261	−	0.930512i	$-0.380638\pi$
$127$	8.25511			0.732523			0.366261	+	0.930512i	$0.380638\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	13.2891			1.16107			0.580536	−	0.814235i	$-0.302843\pi$
$131$	13.2891			1.16107			0.580536	+	0.814235i	$0.302843\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.68989i		−	0.571556i	−0.958296	−	0.285778i	$-0.907748\pi$
$137$		−	6.68989i		−	0.571556i	0.958296	−	0.285778i	$-0.0922520\pi$
$138$		0			0
$139$	5.65156	+	3.26293i	0.479359	+	0.276758i	0.720149	−	0.693819i	$-0.244073\pi$
$139$	5.65156	+	3.26293i	0.479359	+	0.276758i	−0.240790	+	0.970577i	$0.577407\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.78831	+	15.2218i	−0.734916	+	1.27291i
$144$		0			0
$145$	−2.49374	+	1.43976i	−0.207094	+	0.119566i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			5.41089i			0.443277i	0.975129	+	0.221639i	$0.0711405\pi$
$149$			5.41089i			0.443277i	−0.975129	+	0.221639i	$0.928859\pi$
$150$		0			0
$151$	20.0715			1.63340			0.816699	−	0.577064i	$-0.195802\pi$
$151$	20.0715			1.63340			0.816699	+	0.577064i	$0.195802\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	18.4395	−	10.6460i	1.48110	−	0.855111i
$156$		0			0
$157$	1.13796	−	0.657000i	0.0908188	−	0.0524343i	−0.453903	−	0.891051i	$-0.649969\pi$
$157$	1.13796	−	0.657000i	0.0908188	−	0.0524343i	0.544722	+	0.838617i	$0.316635\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.30283	+	14.3809i	0.650328	+	1.12640i	0.983043	+	0.183374i	$0.0587018\pi$
$163$	8.30283	+	14.3809i	0.650328	+	1.12640i	−0.332716	+	0.943027i	$0.607965\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.18145	+	10.7066i	−0.478335	+	0.828501i	−0.999691	−	0.0248384i	$-0.992093\pi$
$167$	−6.18145	+	10.7066i	−0.478335	+	0.828501i	0.521356	+	0.853339i	$0.325426\pi$
$168$		0			0
$169$	−2.19111	−	3.79511i	−0.168547	−	0.291931i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.18456	−	8.97991i	0.394174	−	0.682730i	−0.598821	−	0.800883i	$-0.704364\pi$
$173$	5.18456	−	8.97991i	0.394174	−	0.682730i	0.992995	+	0.118153i	$0.0376972\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−19.5936	−	11.3124i	−1.46450	−	0.845528i	−0.465284	−	0.885162i	$-0.654048\pi$
$179$	−19.5936	−	11.3124i	−1.46450	−	0.845528i	−0.999214	+	0.0396332i	$0.987381\pi$
$180$		0			0
$181$			6.15765i			0.457695i	0.973462	+	0.228847i	$0.0734956\pi$
$181$			6.15765i			0.457695i	−0.973462	+	0.228847i	$0.926504\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.278273	+	0.481982i	0.0204590	+	0.0354360i
$186$		0			0
$187$	−0.173521	−	0.100182i	−0.0126891	−	0.00732607i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.9110	+	11.4956i	1.44071	+	0.831794i	0.997897	−	0.0648193i	$-0.0206471\pi$
$191$	19.9110	+	11.4956i	1.44071	+	0.831794i	0.442813	+	0.896614i	$0.353980\pi$
$192$		0			0
$193$	8.61512	+	14.9218i	0.620130	+	1.07410i	0.989461	+	0.144799i	$0.0462535\pi$
$193$	8.61512	+	14.9218i	0.620130	+	1.07410i	−0.369331	+	0.929298i	$0.620413\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.31094i			0.307141i	0.988138	+	0.153571i	$0.0490773\pi$
$197$			4.31094i			0.307141i	−0.988138	+	0.153571i	$0.950923\pi$
$198$		0			0
$199$	6.35934	+	3.67156i	0.450801	+	0.260270i	0.708169	−	0.706043i	$-0.249522\pi$
$199$	6.35934	+	3.67156i	0.450801	+	0.260270i	−0.257367	+	0.966314i	$0.582855\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−5.86216	+	10.1536i	−0.409431	+	0.709155i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.4525	+	42.3529i	1.69141	+	2.92961i
$210$		0			0
$211$	−6.79668	+	11.7722i	−0.467903	+	0.810431i	−0.999327	−	0.0366744i	$-0.988324\pi$
$211$	−6.79668	+	11.7722i	−0.467903	+	0.810431i	0.531425	+	0.847106i	$0.321657\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.93288	−	17.2043i	−0.677417	−	1.17332i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.0850771	+	0.0491193i	−0.00572291	+	0.00330412i
$222$		0			0
$223$	−22.6248	+	13.0624i	−1.51507	+	0.874725i	−0.515225	+	0.857055i	$0.672292\pi$
$223$	−22.6248	+	13.0624i	−1.51507	+	0.874725i	−0.999844	+	0.0176705i	$0.994375\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.16204			0.209872			0.104936	−	0.994479i	$-0.466536\pi$
$227$	3.16204			0.209872			0.104936	+	0.994479i	$0.466536\pi$
$228$		0			0
$229$		−	13.6178i		−	0.899892i	−0.893056	−	0.449946i	$-0.851443\pi$
$229$		−	13.6178i		−	0.899892i	0.893056	−	0.449946i	$-0.148557\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−23.3765	+	13.4964i	−1.53145	+	0.884182i	−0.532152	+	0.846649i	$0.678617\pi$
$233$	−23.3765	+	13.4964i	−1.53145	+	0.884182i	−0.999295	+	0.0375332i	$0.988050\pi$
$234$		0			0
$235$	−9.29708	+	16.1030i	−0.606475	+	1.05044i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.46107	+	4.30765i	0.482617	+	0.278639i	0.721506	−	0.692408i	$-0.243450\pi$
$239$	7.46107	+	4.30765i	0.482617	+	0.278639i	−0.238890	+	0.971047i	$0.576783\pi$
$240$		0			0
$241$			23.2461i			1.49741i	0.662902	+	0.748706i	$0.269324\pi$
$241$			23.2461i			1.49741i	−0.662902	+	0.748706i	$0.730676\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	23.9780			1.52568
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.09095			−0.573816			−0.286908	−	0.957958i	$-0.592627\pi$
$251$	−9.09095			−0.573816			−0.286908	+	0.957958i	$0.592627\pi$
$252$		0			0
$253$	48.7139			3.06261
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.7392			−0.919403			−0.459702	−	0.888073i	$-0.652044\pi$
$257$	−14.7392			−0.919403			−0.459702	+	0.888073i	$0.652044\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			27.9001i			1.72040i	0.509959	+	0.860198i	$0.329660\pi$
$263$			27.9001i			1.72040i	−0.509959	+	0.860198i	$0.670340\pi$
$264$		0			0
$265$	4.18347	+	2.41533i	0.256989	+	0.148372i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.57488	−	11.3880i	0.400878	−	0.694341i	−0.592954	−	0.805236i	$-0.702039\pi$
$269$	6.57488	−	11.3880i	0.400878	−	0.694341i	0.993832	+	0.110896i	$0.0353719\pi$
$270$		0			0
$271$	15.4231	−	8.90451i	0.936885	−	0.540911i	0.0479022	−	0.998852i	$-0.484746\pi$
$271$	15.4231	−	8.90451i	0.936885	−	0.540911i	0.888982	+	0.457941i	$0.151413\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	11.4061i		−	0.687812i
$276$		0			0
$277$	17.7887			1.06882			0.534410	−	0.845225i	$-0.320534\pi$
$277$	17.7887			1.06882			0.534410	+	0.845225i	$0.320534\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	18.3680	−	10.6048i	1.09574	−	0.632628i	0.160644	−	0.987012i	$-0.448643\pi$
$281$	18.3680	−	10.6048i	1.09574	−	0.632628i	0.935100	+	0.354384i	$0.115309\pi$
$282$		0			0
$283$	16.2504	−	9.38216i	0.965985	−	0.557711i	0.0679748	−	0.997687i	$-0.478346\pi$
$283$	16.2504	−	9.38216i	0.965985	−	0.557711i	0.898010	+	0.439976i	$0.145013\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.49944	+	14.7215i	0.499967	+	0.865968i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.40500	+	11.0938i	−0.374184	+	0.648106i	−0.990205	−	0.139624i	$-0.955411\pi$
$293$	−6.40500	+	11.0938i	−0.374184	+	0.648106i	0.616020	+	0.787730i	$0.288744\pi$
$294$		0			0
$295$	6.22364	+	10.7797i	0.362354	+	0.627616i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.9422	−	20.6844i	0.690633	−	1.19621i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−27.2689	−	15.7437i	−1.56142	−	0.901484i
$306$		0			0
$307$			3.97281i			0.226740i	0.993553	+	0.113370i	$0.0361646\pi$
$307$			3.97281i			0.226740i	−0.993553	+	0.113370i	$0.963835\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.293203	+	0.507843i	0.0166260	+	0.0287971i	0.874219	−	0.485532i	$-0.161374\pi$
$311$	0.293203	+	0.507843i	0.0166260	+	0.0287971i	−0.857593	+	0.514329i	$0.828041\pi$
$312$		0			0
$313$	0.860061	+	0.496556i	0.0486135	+	0.0280670i	0.524110	−	0.851651i	$-0.324398\pi$
$313$	0.860061	+	0.496556i	0.0486135	+	0.0280670i	−0.475496	+	0.879718i	$0.657731\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−20.6022	−	11.8947i	−1.15714	−	0.668073i	−0.206520	−	0.978442i	$-0.566214\pi$
$317$	−20.6022	−	11.8947i	−1.15714	−	0.668073i	−0.950616	+	0.310369i	$0.899547\pi$
$318$		0			0
$319$	−3.28054	−	5.68207i	−0.183675	−	0.318135i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.273338i			0.0152089i
$324$		0			0
$325$	−4.84314	−	2.79619i	−0.268649	−	0.155105i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−13.5486	+	23.4668i	−0.744697	+	1.28985i	0.205640	+	0.978628i	$0.434072\pi$
$331$	−13.5486	+	23.4668i	−0.744697	+	1.28985i	−0.950336	+	0.311224i	$0.899261\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.0156	+	17.3475i	0.547208	+	0.947793i
$336$		0			0
$337$	0.618503	−	1.07128i	0.0336920	−	0.0583562i	−0.848688	−	0.528894i	$-0.822607\pi$
$337$	0.618503	−	1.07128i	0.0336920	−	0.0583562i	0.882380	+	0.470538i	$0.155940\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	24.2574	+	42.0150i	1.31361	+	2.27524i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.6468	−	6.72430i	0.625234	−	0.360979i	−0.153670	−	0.988122i	$-0.549109\pi$
$347$	11.6468	−	6.72430i	0.625234	−	0.360979i	0.778904	+	0.627143i	$0.215776\pi$
$348$		0			0
$349$	12.4728	−	7.20115i	0.667652	−	0.385469i	−0.127535	−	0.991834i	$-0.540706\pi$
$349$	12.4728	−	7.20115i	0.667652	−	0.385469i	0.795186	+	0.606365i	$0.207373\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.55329			0.402021			0.201011	−	0.979589i	$-0.435577\pi$
$353$	7.55329			0.402021			0.201011	+	0.979589i	$0.435577\pi$
$354$		0			0
$355$			9.26175i			0.491563i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.8285	−	12.0253i	1.09928	−	0.634672i	0.163251	−	0.986585i	$-0.447802\pi$
$359$	20.8285	−	12.0253i	1.09928	−	0.634672i	0.936033	+	0.351913i	$0.114469\pi$
$360$		0			0
$361$	23.8580	−	41.3233i	1.25569	−	2.17491i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.89659	+	3.40440i	0.308642	+	0.178194i
$366$		0			0
$367$			7.25219i			0.378561i	0.981923	+	0.189281i	$0.0606156\pi$
$367$			7.25219i			0.378561i	−0.981923	+	0.189281i	$0.939384\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	12.3063			0.637196			0.318598	−	0.947890i	$-0.396788\pi$
$373$	12.3063			0.637196			0.318598	+	0.947890i	$0.396788\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.21689			−0.165678
$378$		0			0
$379$	−20.4289			−1.04936			−0.524680	−	0.851299i	$-0.675815\pi$
$379$	−20.4289			−1.04936			−0.524680	+	0.851299i	$0.675815\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−35.2920			−1.80334			−0.901668	−	0.432429i	$-0.857657\pi$
$383$	−35.2920			−1.80334			−0.901668	+	0.432429i	$0.857657\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	22.2667i		−	1.12897i	−0.825444	−	0.564484i	$-0.809075\pi$
$389$		−	22.2667i		−	1.12897i	0.825444	−	0.564484i	$-0.190925\pi$
$390$		0			0
$391$	0.235792	+	0.136135i	0.0119245	+	0.00688463i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.43232	+	16.3373i	−0.474592	+	0.822017i
$396$		0			0
$397$	16.8558	−	9.73170i	0.845969	−	0.488420i	−0.0133200	−	0.999911i	$-0.504240\pi$
$397$	16.8558	−	9.73170i	0.845969	−	0.488420i	0.859289	+	0.511491i	$0.170907\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			8.60311i			0.429619i	0.976656	+	0.214809i	$0.0689130\pi$
$401$			8.60311i			0.429619i	−0.976656	+	0.214809i	$0.931087\pi$
$402$		0			0
$403$	23.7867			1.18490
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.09821	+	0.634053i	−0.0544364	+	0.0314289i
$408$		0			0
$409$	16.7571	−	9.67472i	0.828585	−	0.478384i	−0.0247826	−	0.999693i	$-0.507889\pi$
$409$	16.7571	−	9.67472i	0.828585	−	0.478384i	0.853368	+	0.521309i	$0.174556\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.45101	−	4.24527i	−0.120315	−	0.208392i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.81222	−	15.2632i	0.430505	−	0.745657i	−0.566412	−	0.824122i	$-0.691669\pi$
$419$	8.81222	−	15.2632i	0.430505	−	0.745657i	0.996917	+	0.0784657i	$0.0250021\pi$
$420$		0			0
$421$	−5.77040	−	9.99463i	−0.281232	−	0.487109i	0.690456	−	0.723374i	$-0.257410\pi$
$421$	−5.77040	−	9.99463i	−0.281232	−	0.487109i	−0.971689	+	0.236266i	$0.924076\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.0318752	−	0.0552094i	0.00154617	−	0.00267805i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.90849	+	5.72067i	0.477275	+	0.275555i	0.719280	−	0.694720i	$-0.244472\pi$
$431$	9.90849	+	5.72067i	0.477275	+	0.275555i	−0.242005	+	0.970275i	$0.577805\pi$
$432$		0			0
$433$		−	6.25525i		−	0.300608i	−0.988640	−	0.150304i	$-0.951975\pi$
$433$		−	6.25525i		−	0.300608i	0.988640	−	0.150304i	$-0.0480253\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−33.2277	−	57.5521i	−1.58950	−	2.75309i
$438$		0			0
$439$	13.2543	+	7.65239i	0.632595	+	0.365229i	0.781756	−	0.623584i	$-0.214324\pi$
$439$	13.2543	+	7.65239i	0.632595	+	0.365229i	−0.149162	+	0.988813i	$0.547657\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.4838	−	11.2490i	−0.925705	−	0.534456i	−0.0402540	−	0.999189i	$-0.512817\pi$
$443$	−19.4838	−	11.2490i	−0.925705	−	0.534456i	−0.885450	+	0.464734i	$0.846150\pi$
$444$		0			0
$445$	−9.35803	−	16.2086i	−0.443613	−	0.768361i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	25.9709i		−	1.22564i	−0.790222	−	0.612821i	$-0.790035\pi$
$449$		−	25.9709i		−	1.22564i	0.790222	−	0.612821i	$-0.209965\pi$
$450$		0			0
$451$	−23.1352	−	13.3571i	−1.08939	−	0.628962i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.0222019	+	0.0384547i	−0.00103856	+	0.00179884i	−0.866544	−	0.499100i	$-0.833664\pi$
$457$	−0.0222019	+	0.0384547i	−0.00103856	+	0.00179884i	0.865506	+	0.500899i	$0.166997\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.46783	+	2.54236i	0.0683636	+	0.118409i	0.898181	−	0.439626i	$-0.144889\pi$
$461$	1.46783	+	2.54236i	0.0683636	+	0.118409i	−0.829817	+	0.558035i	$0.811556\pi$
$462$		0			0
$463$	−19.2017	+	33.2583i	−0.892378	+	1.54564i	−0.0553609	+	0.998466i	$0.517631\pi$
$463$	−19.2017	+	33.2583i	−0.892378	+	1.54564i	−0.837017	+	0.547177i	$0.815702\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.63913	+	14.9634i	0.399771	+	0.692424i	0.993697	−	0.112095i	$-0.0357562\pi$
$467$	8.63913	+	14.9634i	0.399771	+	0.692424i	−0.593926	+	0.804520i	$0.702423\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	39.2005	−	22.6324i	1.80244	−	1.04064i
$474$		0			0
$475$	−13.4755	+	7.78007i	−0.618297	+	0.356974i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−36.9142			−1.68665			−0.843326	−	0.537403i	$-0.819405\pi$
$479$	−36.9142			−1.68665			−0.843326	+	0.537403i	$0.819405\pi$
$480$		0			0
$481$			0.621750i			0.0283494i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	34.8228	−	20.1049i	1.58122	−	0.912918i
$486$		0			0
$487$	−2.03199	+	3.51951i	−0.0920783	+	0.159484i	−0.908385	−	0.418134i	$-0.862684\pi$
$487$	−2.03199	+	3.51951i	−0.0920783	+	0.159484i	0.816307	+	0.577618i	$0.196018\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.30460	−	2.48526i	−0.194264	−	0.112158i	0.399713	−	0.916640i	$-0.369110\pi$
$491$	−4.30460	−	2.48526i	−0.194264	−	0.112158i	−0.593977	+	0.804482i	$0.702443\pi$
$492$		0			0
$493$		−	0.0366710i		−	0.00165158i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	9.20669			0.412148			0.206074	−	0.978536i	$-0.433931\pi$
$499$	9.20669			0.412148			0.206074	+	0.978536i	$0.433931\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.16946			0.0521436			0.0260718	−	0.999660i	$-0.491700\pi$
$503$	1.16946			0.0521436			0.0260718	+	0.999660i	$0.491700\pi$
$504$		0			0
$505$	−17.8033			−0.792238
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.72412			−0.431014			−0.215507	−	0.976502i	$-0.569140\pi$
$509$	−9.72412			−0.431014			−0.215507	+	0.976502i	$0.569140\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			12.8292i			0.565320i
$516$		0			0
$517$	−36.6912	−	21.1837i	−1.61368	−	0.931658i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.23317	+	10.7962i	−0.273080	+	0.472989i	−0.969649	−	0.244501i	$-0.921376\pi$
$521$	−6.23317	+	10.7962i	−0.273080	+	0.472989i	0.696569	+	0.717490i	$0.254709\pi$
$522$		0			0
$523$	15.6222	−	9.01951i	0.683113	−	0.394395i	−0.117914	−	0.993024i	$-0.537621\pi$
$523$	15.6222	−	9.01951i	0.683113	−	0.394395i	0.801027	+	0.598628i	$0.204287\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			0.271157i			0.0118118i
$528$		0			0
$529$	−43.1957			−1.87807
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−11.3431	+	6.54897i	−0.491326	+	0.283667i
$534$		0			0
$535$	−33.6976	+	19.4553i	−1.45688	+	0.841128i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.87785	−	3.25253i	−0.0807349	−	0.139837i	0.822831	−	0.568286i	$-0.192393\pi$
$541$	−1.87785	−	3.25253i	−0.0807349	−	0.139837i	−0.903566	+	0.428449i	$0.859060\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.2972	+	26.4956i	−0.655262	+	1.13495i
$546$		0			0
$547$	5.05062	+	8.74793i	0.215949	+	0.374034i	0.953566	−	0.301185i	$-0.0973822\pi$
$547$	5.05062	+	8.74793i	0.215949	+	0.374034i	−0.737617	+	0.675219i	$0.764049\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.47531	+	7.75147i	−0.190655	+	0.330224i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.7817	−	9.11158i	−0.668693	−	0.386070i	0.126888	−	0.991917i	$-0.459501\pi$
$557$	−15.7817	−	9.11158i	−0.668693	−	0.386070i	−0.795581	+	0.605847i	$0.792834\pi$
$558$		0			0
$559$		−	22.1933i		−	0.938674i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.34691	+	14.4573i	0.351780	+	0.609301i	0.986561	−	0.163391i	$-0.0522431\pi$
$563$	8.34691	+	14.4573i	0.351780	+	0.609301i	−0.634781	+	0.772692i	$0.718910\pi$
$564$		0			0
$565$	−3.28329	−	1.89561i	−0.138129	−	0.0797487i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.8711	+	12.6273i	0.916886	+	0.529365i	0.882640	−	0.470049i	$-0.155764\pi$
$569$	21.8711	+	12.6273i	0.916886	+	0.529365i	0.0342459	+	0.999413i	$0.489097\pi$
$570$		0			0
$571$	2.88981	+	5.00529i	0.120935	+	0.209465i	0.920137	−	0.391598i	$-0.128078\pi$
$571$	2.88981	+	5.00529i	0.120935	+	0.209465i	−0.799202	+	0.601063i	$0.794744\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			15.4993i			0.646367i
$576$		0			0
$577$	28.6539	+	16.5433i	1.19288	+	0.688708i	0.958958	−	0.283548i	$-0.0915115\pi$
$577$	28.6539	+	16.5433i	1.19288	+	0.688708i	0.233919	+	0.972256i	$0.424845\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−5.50340	+	9.53218i	−0.227928	+	0.394782i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.1186	+	24.4541i	0.582737	+	1.00933i	0.995153	+	0.0983341i	$0.0313514\pi$
$587$	14.1186	+	24.4541i	0.582737	+	1.00933i	−0.412417	+	0.910995i	$0.635315\pi$
$588$		0			0
$589$	33.0919	−	57.3168i	1.36353	−	2.36170i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9.29769	−	16.1041i	−0.381810	−	0.661315i	0.609511	−	0.792778i	$-0.291366\pi$
$593$	−9.29769	−	16.1041i	−0.381810	−	0.661315i	−0.991321	+	0.131463i	$0.958033\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.77276	−	4.48760i	0.317586	−	0.183358i	−0.332730	−	0.943022i	$-0.607970\pi$
$599$	7.77276	−	4.48760i	0.317586	−	0.183358i	0.650316	+	0.759664i	$0.274636\pi$
$600$		0			0
$601$	−4.71245	+	2.72073i	−0.192225	+	0.110981i	−0.593024	−	0.805185i	$-0.702066\pi$
$601$	−4.71245	+	2.72073i	−0.192225	+	0.110981i	0.400799	+	0.916166i	$0.368733\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	65.2963			2.65467
$606$		0			0
$607$			36.9781i			1.50089i	0.660930	+	0.750447i	$0.270162\pi$
$607$			36.9781i			1.50089i	−0.660930	+	0.750447i	$0.729838\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−17.9897	+	10.3863i	−0.727783	+	0.420186i
$612$		0			0
$613$	5.93439	−	10.2787i	0.239688	−	0.415151i	−0.720937	−	0.693001i	$-0.756288\pi$
$613$	5.93439	−	10.2787i	0.239688	−	0.415151i	0.960625	+	0.277849i	$0.0896216\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.2845	−	9.40188i	−0.655591	−	0.378506i	0.135004	−	0.990845i	$-0.456895\pi$
$617$	−16.2845	−	9.40188i	−0.655591	−	0.378506i	−0.790595	+	0.612339i	$0.790229\pi$
$618$		0			0
$619$		−	11.8810i		−	0.477536i	−0.971077	−	0.238768i	$-0.923256\pi$
$619$		−	11.8810i		−	0.477536i	0.971077	−	0.238768i	$-0.0767436\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−30.8960			−1.23584
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.00708765			−0.000282603
$630$		0			0
$631$	−28.3350			−1.12800			−0.563998	−	0.825776i	$-0.690738\pi$
$631$	−28.3350			−1.12800			−0.563998	+	0.825776i	$0.690738\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−21.6923			−0.860832
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			28.3633i			1.12028i	0.828396	+	0.560142i	$0.189254\pi$
$641$			28.3633i			1.12028i	−0.828396	+	0.560142i	$0.810746\pi$
$642$		0			0
$643$	16.0912	+	9.29024i	0.634573	+	0.366371i	0.782521	−	0.622624i	$-0.213933\pi$
$643$	16.0912	+	9.29024i	0.634573	+	0.366371i	−0.147948	+	0.988995i	$0.547267\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.4482	+	37.1494i	−0.843216	+	1.46049i	0.0439448	+	0.999034i	$0.486007\pi$
$647$	−21.4482	+	37.1494i	−0.843216	+	1.46049i	−0.887161	+	0.461460i	$0.847326\pi$
$648$		0			0
$649$	−24.5618	+	14.1808i	−0.964135	+	0.556644i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	3.68027i		−	0.144020i	−0.997404	−	0.0720101i	$-0.977059\pi$
$653$		−	3.68027i		−	0.144020i	0.997404	−	0.0720101i	$-0.0229414\pi$
$654$		0			0
$655$	−34.9202			−1.36445
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.7955	−	7.96483i	0.537396	−	0.310266i	−0.206627	−	0.978420i	$-0.566249\pi$
$659$	13.7955	−	7.96483i	0.537396	−	0.310266i	0.744023	+	0.668154i	$0.232915\pi$
$660$		0			0
$661$	−1.14378	+	0.660360i	−0.0444878	+	0.0256850i	−0.522079	−	0.852897i	$-0.674843\pi$
$661$	−1.14378	+	0.660360i	−0.0444878	+	0.0256850i	0.477591	+	0.878582i	$0.341510\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.45783	+	7.72118i	0.172608	+	0.298965i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	35.8726	−	62.1332i	1.38485	−	2.39863i
$672$		0			0
$673$	21.7987	+	37.7565i	0.840280	+	1.45541i	0.889658	+	0.456627i	$0.150943\pi$
$673$	21.7987	+	37.7565i	0.840280	+	1.45541i	−0.0493788	+	0.998780i	$0.515724\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.4677	−	25.0588i	0.556039	−	0.963088i	−0.441782	−	0.897122i	$-0.645654\pi$
$677$	14.4677	−	25.0588i	0.556039	−	0.963088i	0.997822	−	0.0659663i	$-0.0210130\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.90755	+	1.10132i	0.0729903	+	0.0421410i	0.536051	−	0.844186i	$-0.319915\pi$
$683$	1.90755	+	1.10132i	0.0729903	+	0.0421410i	−0.463061	+	0.886327i	$0.653249\pi$
$684$		0			0
$685$			17.5793i			0.671670i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.69831	+	4.67361i	0.102797	+	0.178050i
$690$		0			0
$691$	−25.8896	−	14.9474i	−0.984888	−	0.568625i	−0.0811456	−	0.996702i	$-0.525858\pi$
$691$	−25.8896	−	14.9474i	−0.984888	−	0.568625i	−0.903742	+	0.428077i	$0.859191\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14.8508	−	8.57413i	−0.563324	−	0.325235i
$696$		0			0
$697$	−0.0746551	−	0.129306i	−0.00282776	−	0.00489783i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			45.4119i			1.71518i	0.514332	+	0.857591i	$0.328040\pi$
$701$			45.4119i			1.71518i	−0.514332	+	0.857591i	$0.671960\pi$
$702$		0			0
$703$	1.49818	+	0.864975i	0.0565049	+	0.0326231i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.92591	−	13.7281i	0.297664	−	0.515569i	−0.677937	−	0.735120i	$-0.737126\pi$
$709$	7.92591	−	13.7281i	0.297664	−	0.515569i	0.975601	+	0.219551i	$0.0704592\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−32.9626	−	57.0928i	−1.23446	−	2.13814i
$714$		0			0
$715$	23.0934	−	39.9989i	0.863644	−	1.49588i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.30154	+	5.71844i	0.123127	+	0.213262i	0.920999	−	0.389565i	$-0.127375\pi$
$719$	3.30154	+	5.71844i	0.123127	+	0.213262i	−0.797872	+	0.602826i	$0.794041\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.80787	−	1.04377i	0.0671426	−	0.0387648i
$726$		0			0
$727$	31.2086	−	18.0183i	1.15746	−	0.668261i	0.206767	−	0.978390i	$-0.433706\pi$
$727$	31.2086	−	18.0183i	1.15746	−	0.668261i	0.950694	+	0.310129i	$0.100372\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.252992			0.00935726
$732$		0			0
$733$			30.0583i			1.11023i	0.831774	+	0.555114i	$0.187325\pi$
$733$			30.0583i			1.11023i	−0.831774	+	0.555114i	$0.812675\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−39.5268	+	22.8208i	−1.45599	+	0.840614i
$738$		0			0
$739$	−4.83576	+	8.37577i	−0.177886	+	0.308108i	−0.941156	−	0.337972i	$-0.890259\pi$
$739$	−4.83576	+	8.37577i	−0.177886	+	0.308108i	0.763270	+	0.646079i	$0.223593\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−11.0205	−	6.36269i	−0.404303	−	0.233424i	0.284036	−	0.958814i	$-0.408326\pi$
$743$	−11.0205	−	6.36269i	−0.404303	−	0.233424i	−0.688339	+	0.725389i	$0.741660\pi$
$744$		0			0
$745$		−	14.2184i		−	0.520922i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	21.3741			0.779950			0.389975	−	0.920825i	$-0.372484\pi$
$751$	21.3741			0.779950			0.389975	+	0.920825i	$0.372484\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−52.7428			−1.91951
$756$		0			0
$757$	−7.21065			−0.262076			−0.131038	−	0.991377i	$-0.541831\pi$
$757$	−7.21065			−0.262076			−0.131038	+	0.991377i	$0.541831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.3240			−0.809245			−0.404623	−	0.914484i	$-0.632597\pi$
$761$	−22.3240			−0.809245			−0.404623	+	0.914484i	$0.632597\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			13.9056i			0.502102i
$768$		0			0
$769$	−13.3202	−	7.69042i	−0.480338	−	0.277324i	0.240219	−	0.970719i	$-0.422781\pi$
$769$	−13.3202	−	7.69042i	−0.480338	−	0.277324i	−0.720558	+	0.693395i	$0.756114\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.17527	−	10.6959i	0.222109	−	0.384704i	−0.733339	−	0.679863i	$-0.762039\pi$
$773$	6.17527	−	10.6959i	0.222109	−	0.384704i	0.955448	+	0.295159i	$0.0953726\pi$
$774$		0			0
$775$	−13.3680	+	7.71799i	−0.480191	+	0.277238i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			36.4435i			1.30572i
$780$		0			0
$781$	−21.1032			−0.755132
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.99025	+	1.72642i	−0.106727	+	0.0616187i
$786$		0			0
$787$	−12.5825	+	7.26452i	−0.448518	+	0.258952i	−0.707204	−	0.707009i	$-0.750044\pi$
$787$	−12.5825	+	7.26452i	−0.448518	+	0.258952i	0.258686	+	0.965961i	$0.416710\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−17.5883	−	30.4638i	−0.624578	−	1.08180i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.0040	−	38.1120i	0.779420	−	1.35000i	−0.152856	−	0.988248i	$-0.548847\pi$
$797$	22.0040	−	38.1120i	0.779420	−	1.35000i	0.932276	−	0.361747i	$-0.117819\pi$
$798$		0			0
$799$	−0.118399	−	0.205073i	−0.00418866	−	0.00725497i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.75703	+	13.4356i	−0.273740	+	0.474131i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−41.4554	−	23.9343i	−1.45749	−	0.841484i	−0.458606	−	0.888640i	$-0.651651\pi$
$809$	−41.4554	−	23.9343i	−1.45749	−	0.841484i	−0.998888	+	0.0471551i	$0.984985\pi$
$810$		0			0
$811$			17.4775i			0.613720i	0.951755	+	0.306860i	$0.0992783\pi$
$811$			17.4775i			0.613720i	−0.951755	+	0.306860i	$0.900722\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21.8177	−	37.7893i	−0.764240	−	1.32370i
$816$		0			0
$817$	−53.4772	−	30.8751i	−1.87093	−	1.08018i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.40574	−	3.69836i	−0.223562	−	0.129074i	0.384037	−	0.923318i	$-0.374534\pi$
$821$	−6.40574	−	3.69836i	−0.223562	−	0.129074i	−0.607598	+	0.794244i	$0.707867\pi$
$822$		0			0
$823$	21.4718	+	37.1903i	0.748460	+	1.29637i	0.948560	+	0.316596i	$0.102540\pi$
$823$	21.4718	+	37.1903i	0.748460	+	1.29637i	−0.200100	+	0.979775i	$0.564127\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			43.0042i			1.49540i	0.664035	+	0.747701i	$0.268843\pi$
$827$			43.0042i			1.49540i	−0.664035	+	0.747701i	$0.731157\pi$
$828$		0			0
$829$	10.0780	+	5.81851i	0.350022	+	0.202085i	0.664695	−	0.747115i	$-0.268561\pi$
$829$	10.0780	+	5.81851i	0.350022	+	0.202085i	−0.314673	+	0.949200i	$0.601895\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	16.2432	−	28.1341i	0.562121	−	0.973621i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.936892	+	1.62274i	0.0323451	+	0.0560234i	0.881745	−	0.471727i	$-0.156369\pi$
$839$	0.936892	+	1.62274i	0.0323451	+	0.0560234i	−0.849400	+	0.527750i	$0.823036\pi$
$840$		0			0
$841$	−13.8996	+	24.0748i	−0.479296	+	0.830166i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.75766	+	9.97255i	0.198069	+	0.343066i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.49233	−	0.861595i	0.0511563	−	0.0295351i
$852$		0			0
$853$	34.9301	−	20.1669i	1.19598	−	0.690501i	0.236325	−	0.971674i	$-0.424057\pi$
$853$	34.9301	−	20.1669i	1.19598	−	0.690501i	0.959657	+	0.281173i	$0.0907236\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	32.2681			1.10226			0.551129	−	0.834420i	$-0.314197\pi$
$857$	32.2681			1.10226			0.551129	+	0.834420i	$0.314197\pi$
$858$		0			0
$859$		−	38.9025i		−	1.32734i	−0.748027	−	0.663668i	$-0.768999\pi$
$859$		−	38.9025i		−	1.32734i	0.748027	−	0.663668i	$-0.231001\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33.1319	−	19.1287i	1.12782	−	0.651148i	0.184435	−	0.982845i	$-0.440954\pi$
$863$	33.1319	−	19.1287i	1.12782	−	0.651148i	0.943386	+	0.331697i	$0.107621\pi$
$864$		0			0
$865$	−13.6237	+	23.5969i	−0.463218	+	0.802318i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−37.2250	−	21.4919i	−1.26277	−	0.729061i
$870$		0			0
$871$			22.3780i			0.758249i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	9.52303			0.321570			0.160785	−	0.986989i	$-0.448597\pi$
$877$	9.52303			0.321570			0.160785	+	0.986989i	$0.448597\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.5770			−0.558493			−0.279247	−	0.960219i	$-0.590085\pi$
$881$	−16.5770			−0.558493			−0.279247	+	0.960219i	$0.590085\pi$
$882$		0			0
$883$	−34.9830			−1.17727			−0.588636	−	0.808398i	$-0.700335\pi$
$883$	−34.9830			−1.17727			−0.588636	+	0.808398i	$0.700335\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	34.7590			1.16709			0.583547	−	0.812079i	$-0.301664\pi$
$887$	34.7590			1.16709			0.583547	+	0.812079i	$0.301664\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			57.7976i			1.93412i
$894$		0			0
$895$	51.4870	+	29.7260i	1.72102	+	0.993632i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.43960	+	7.68962i	−0.148069	+	0.256463i
$900$		0			0
$901$	−0.0532769	+	0.0307594i	−0.00177491	+	0.00102474i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	16.1807i		−	0.537865i
$906$		0			0
$907$	31.8232			1.05667			0.528336	−	0.849035i	$-0.322816\pi$
$907$	31.8232			1.05667			0.528336	+	0.849035i	$0.322816\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.51591	+	4.33931i	−0.249013	+	0.143768i	−0.619312	−	0.785145i	$-0.712589\pi$
$911$	−7.51591	+	4.33931i	−0.249013	+	0.143768i	0.370299	+	0.928913i	$0.379255\pi$
$912$		0			0
$913$	9.67298	−	5.58470i	0.320129	−	0.184827i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−10.3922	−	17.9999i	−0.342808	−	0.593760i	0.642145	−	0.766583i	$-0.278045\pi$
$919$	−10.3922	−	17.9999i	−0.342808	−	0.593760i	−0.984953	+	0.172823i	$0.944711\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.17343	+	8.96064i	−0.170286	+	0.294943i
$924$		0			0
$925$	−0.201737	−	0.349419i	−0.00663308	−	0.0114888i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.5361	−	35.5695i	0.673767	−	1.16700i	−0.303060	−	0.952971i	$-0.598008\pi$
$929$	20.5361	−	35.5695i	0.673767	−	1.16700i	0.976828	−	0.214028i	$-0.0686583\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.455968	+	0.263254i	0.0149118	+	0.00860931i
$936$		0			0
$937$			8.38277i			0.273853i	0.990581	+	0.136927i	$0.0437225\pi$
$937$			8.38277i			0.273853i	−0.990581	+	0.136927i	$0.956278\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.99670	−	8.65453i	−0.162888	−	0.282130i	0.773015	−	0.634387i	$-0.218747\pi$
$941$	−4.99670	−	8.65453i	−0.162888	−	0.282130i	−0.935903	+	0.352257i	$0.885414\pi$
$942$		0			0
$943$	31.4377	+	18.1506i	1.02375	+	0.591063i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.3659	+	11.7582i	0.661801	+	0.382091i	0.792963	−	0.609270i	$-0.208537\pi$
$947$	20.3659	+	11.7582i	0.661801	+	0.382091i	−0.131162	+	0.991361i	$0.541871\pi$
$948$		0			0
$949$	3.80326	+	6.58744i	0.123459	+	0.213837i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	34.7300i		−	1.12501i	−0.826793	−	0.562507i	$-0.809837\pi$
$953$		−	34.7300i		−	1.12501i	0.826793	−	0.562507i	$-0.190163\pi$
$954$		0			0
$955$	−52.3210	−	30.2075i	−1.69307	−	0.977492i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	17.3278	−	30.0126i	0.558962	−	0.968150i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.6383	−	39.2107i	−0.728753	−	1.26224i
$966$		0			0
$967$	8.98645	−	15.5650i	0.288985	−	0.500536i	−0.684583	−	0.728935i	$-0.740016\pi$
$967$	8.98645	−	15.5650i	0.288985	−	0.500536i	0.973568	+	0.228399i	$0.0733489\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.9001	+	36.2001i	0.670717	+	1.16172i	0.977701	+	0.210001i	$0.0673468\pi$
$971$	20.9001	+	36.2001i	0.670717	+	1.16172i	−0.306984	+	0.951715i	$0.599320\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−38.6499	+	22.3145i	−1.23652	+	0.713905i	−0.968381	−	0.249475i	$-0.919742\pi$
$977$	−38.6499	+	22.3145i	−1.23652	+	0.713905i	−0.268139	+	0.963380i	$0.586409\pi$
$978$		0			0
$979$	36.9318	−	21.3226i	1.18035	−	0.681473i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	57.1202			1.82185			0.910925	−	0.412571i	$-0.135369\pi$
$983$	57.1202			1.82185			0.910925	+	0.412571i	$0.135369\pi$
$984$		0			0
$985$		−	11.3280i		−	0.360941i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−53.2683	+	30.7544i	−1.69383	+	0.977935i
$990$		0			0
$991$	−18.6791	+	32.3532i	−0.593362	+	1.02773i	0.400413	+	0.916335i	$0.368867\pi$
$991$	−18.6791	+	32.3532i	−0.593362	+	1.02773i	−0.993776	+	0.111399i	$0.964467\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.7107	−	9.64792i	−0.529764	−	0.305859i
$996$		0			0
$997$		−	39.3210i		−	1.24531i	−0.782497	−	0.622654i	$-0.786054\pi$
$997$		−	39.3210i		−	1.24531i	0.782497	−	0.622654i	$-0.213946\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.bm.c.2285.5		48
3.2	odd	2		1764.2.bm.c.1697.7		48
7.2	even	3		5292.2.w.c.1097.20		48
7.3	odd	6		5292.2.x.c.881.5		48
7.4	even	3		5292.2.x.c.881.20		48
7.5	odd	6		5292.2.w.c.1097.5		48
7.6	odd	2	inner	5292.2.bm.c.2285.20		48
9.2	odd	6		5292.2.w.c.521.5		48
9.7	even	3		1764.2.w.c.1109.16		48
21.2	odd	6		1764.2.w.c.509.9		48
21.5	even	6		1764.2.w.c.509.16		48
21.11	odd	6		1764.2.x.c.293.23	yes	48
21.17	even	6		1764.2.x.c.293.2	✓	48
21.20	even	2		1764.2.bm.c.1697.18		48
63.2	odd	6	inner	5292.2.bm.c.4625.20		48
63.11	odd	6		5292.2.x.c.4409.5		48
63.16	even	3		1764.2.bm.c.1685.18		48
63.20	even	6		5292.2.w.c.521.20		48
63.25	even	3		1764.2.x.c.1469.2	yes	48
63.34	odd	6		1764.2.w.c.1109.9		48
63.38	even	6		5292.2.x.c.4409.20		48
63.47	even	6	inner	5292.2.bm.c.4625.5		48
63.52	odd	6		1764.2.x.c.1469.23	yes	48
63.61	odd	6		1764.2.bm.c.1685.7		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.w.c.509.9		48	21.2	odd	6
1764.2.w.c.509.16		48	21.5	even	6
1764.2.w.c.1109.9		48	63.34	odd	6
1764.2.w.c.1109.16		48	9.7	even	3
1764.2.x.c.293.2	✓	48	21.17	even	6
1764.2.x.c.293.23	yes	48	21.11	odd	6
1764.2.x.c.1469.2	yes	48	63.25	even	3
1764.2.x.c.1469.23	yes	48	63.52	odd	6
1764.2.bm.c.1685.7		48	63.61	odd	6
1764.2.bm.c.1685.18		48	63.16	even	3
1764.2.bm.c.1697.7		48	3.2	odd	2
1764.2.bm.c.1697.18		48	21.20	even	2
5292.2.w.c.521.5		48	9.2	odd	6
5292.2.w.c.521.20		48	63.20	even	6
5292.2.w.c.1097.5		48	7.5	odd	6
5292.2.w.c.1097.20		48	7.2	even	3
5292.2.x.c.881.5		48	7.3	odd	6
5292.2.x.c.881.20		48	7.4	even	3
5292.2.x.c.4409.5		48	63.11	odd	6
5292.2.x.c.4409.20		48	63.38	even	6
5292.2.bm.c.2285.5		48	1.1	even	1	trivial
5292.2.bm.c.2285.20		48	7.6	odd	2	inner
5292.2.bm.c.4625.5		48	63.47	even	6	inner
5292.2.bm.c.4625.20		48	63.2	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.bm (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.62774 q^{5} +O(q^{10})\)	\(q-2.62774 q^{5} -5.98739i q^{11} +(-2.54231 - 1.46780i) q^{13} +(0.0167322 - 0.0289811i) q^{17} +(-7.07369 + 4.08400i) q^{19} +8.13607i q^{23} +1.90501 q^{25} +(0.949006 - 0.547909i) q^{29} +(-7.01724 + 4.05141i) q^{31} +(-0.105898 - 0.183421i) q^{37} +(2.23087 - 3.86399i) q^{41} +(3.78001 + 6.54717i) q^{43} +(3.53805 - 6.12809i) q^{47} +(-1.59204 - 0.919166i) q^{53} +15.7333i q^{55} +(-2.36844 - 4.10225i) q^{59} +(10.3773 + 5.99136i) q^{61} +(6.68053 + 3.85700i) q^{65} +(-3.81147 - 6.60166i) q^{67} -3.52461i q^{71} +(-2.24398 - 1.29556i) q^{73} +(3.58952 - 6.21723i) q^{79} +(0.932743 + 1.61556i) q^{83} +(-0.0439680 + 0.0761548i) q^{85} +(3.56125 + 6.16826i) q^{89} +(18.5878 - 10.7317i) q^{95} +(-13.2520 + 7.65104i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 48 q^{25} - 48 q^{53} - 72 q^{65} - 24 q^{79} - 24 q^{85} + 96 q^{95}+O(q^{100}) \) 48 * q + 48 * q^25 - 48 * q^53 - 72 * q^65 - 24 * q^79 - 24 * q^85 + 96 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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