LMFDB - Embedded newform 5292.2.bm.c.4625.20

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			4625.20
Character	$\chi$	$=$	5292.4625
Dual form			5292.2.bm.c.2285.20

$q$-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 $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

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

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	$-0.300081\pi$
$5$	2.62774			1.17516			0.587580	+	0.809166i	$0.300081\pi$
$6$		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	$0.358369\pi$
$11$			5.98739i			1.80527i	−0.430411	+	0.902633i	$0.641631\pi$
$12$		0			0
$13$	2.54231	−	1.46780i	0.705110	−	0.407095i	−0.104138	−	0.994563i	$-0.533208\pi$
$13$	2.54231	−	1.46780i	0.705110	−	0.407095i	0.809248	+	0.587467i	$0.199875\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.0167322	−	0.0289811i	−0.00405817	−	0.00702895i	0.863989	−	0.503510i	$-0.167958\pi$
$17$	−0.0167322	−	0.0289811i	−0.00405817	−	0.00702895i	−0.868047	+	0.496481i	$0.834625\pi$
$18$		0			0
$19$	7.07369	+	4.08400i	1.62282	+	0.936933i	0.986162	+	0.165783i	$0.0530150\pi$
$19$	7.07369	+	4.08400i	1.62282	+	0.936933i	0.636653	+	0.771150i	$0.280318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.13607i		−	1.69649i	−0.529605	−	0.848244i	$-0.677660\pi$
$23$		−	8.13607i		−	1.69649i	0.529605	−	0.848244i	$-0.322340\pi$
$24$		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	0.585518	−	0.810659i	$-0.300891\pi$
$29$	0.949006	+	0.547909i	0.176226	+	0.101744i	−0.409292	+	0.912403i	$0.634224\pi$
$30$		0			0
$31$	7.01724	+	4.05141i	1.26033	+	0.727654i	0.973139	−	0.230218i	$-0.0739439\pi$
$31$	7.01724	+	4.05141i	1.26033	+	0.727654i	0.287195	+	0.957872i	$0.407277\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.105898	+	0.183421i	−0.0174095	+	0.0301542i	−0.874599	−	0.484847i	$-0.838875\pi$
$37$	−0.105898	+	0.183421i	−0.0174095	+	0.0301542i	0.857189	+	0.515001i	$0.172209\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.23087	−	3.86399i	−0.348404	−	0.603453i	0.637562	−	0.770399i	$-0.279943\pi$
$41$	−2.23087	−	3.86399i	−0.348404	−	0.603453i	−0.985966	+	0.166946i	$0.946610\pi$
$42$		0			0
$43$	3.78001	−	6.54717i	0.576446	−	0.998434i	−0.419437	−	0.907785i	$-0.637772\pi$
$43$	3.78001	−	6.54717i	0.576446	−	0.998434i	0.995883	−	0.0906496i	$-0.0288943\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.53805	−	6.12809i	−0.516078	−	0.893873i	−0.999826	−	0.0186658i	$-0.994058\pi$
$47$	−3.53805	−	6.12809i	−0.516078	−	0.893873i	0.483748	−	0.875207i	$-0.339275\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.59204	+	0.919166i	−0.218684	+	0.126257i	−0.605341	−	0.795967i	$-0.706963\pi$
$53$	−1.59204	+	0.919166i	−0.218684	+	0.126257i	0.386657	+	0.922224i	$0.373630\pi$
$54$		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	$-0.733558\pi$
$59$	2.36844	−	4.10225i	0.308344	−	0.534068i	0.978000	+	0.208603i	$0.0668918\pi$
$60$		0			0
$61$	−10.3773	+	5.99136i	−1.32868	+	0.767115i	−0.985096	−	0.172006i	$-0.944975\pi$
$61$	−10.3773	+	5.99136i	−1.32868	+	0.767115i	−0.343586	+	0.939121i	$0.611642\pi$
$62$		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	−0.999230	−	0.0392248i	$-0.987511\pi$
$67$	−3.81147	+	6.60166i	−0.465646	+	0.806522i	0.533585	+	0.845747i	$0.320844\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.52461i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$71$			3.52461i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$72$		0			0
$73$	2.24398	−	1.29556i	0.262638	−	0.151634i	−0.362899	−	0.931828i	$-0.618213\pi$
$73$	2.24398	−	1.29556i	0.262638	−	0.151634i	0.625537	+	0.780194i	$0.284880\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.58952	+	6.21723i	0.403852	+	0.699493i	0.994187	−	0.107666i	$-0.0343376\pi$
$79$	3.58952	+	6.21723i	0.403852	+	0.699493i	−0.590335	+	0.807159i	$0.701004\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.932743	+	1.61556i	−0.102382	+	0.177331i	−0.912666	−	0.408707i	$-0.865980\pi$
$83$	−0.932743	+	1.61556i	−0.102382	+	0.177331i	0.810284	+	0.586038i	$0.199313\pi$
$84$		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	$-0.956547\pi$
$89$	−3.56125	+	6.16826i	−0.377492	+	0.653835i	0.613205	+	0.789924i	$0.289880\pi$
$90$		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	$-0.0501522\pi$
$97$	13.2520	+	7.65104i	1.34554	+	0.776845i	0.357922	+	0.933752i	$0.383486\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.77515			−0.674153			−0.337077	−	0.941477i	$-0.609438\pi$
$101$	−6.77515			−0.674153			−0.337077	+	0.941477i	$0.609438\pi$
$102$		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	0.969038	−	0.246912i	$-0.0794158\pi$
$107$	12.8238	+	7.40383i	1.23972	+	0.715756i	0.270687	+	0.962667i	$0.412749\pi$
$108$		0			0
$109$	5.82144	+	10.0830i	0.557593	+	0.965780i	0.997697	+	0.0678327i	$0.0216084\pi$
$109$	5.82144	+	10.0830i	0.557593	+	0.965780i	−0.440104	+	0.897947i	$0.645058\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.24947	−	0.721383i	0.117540	−	0.0678620i	−0.440077	−	0.897960i	$-0.645049\pi$
$113$	1.24947	−	0.721383i	0.117540	−	0.0678620i	0.557618	+	0.830098i	$0.311716\pi$
$114$		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	$-0.697157\pi$
$131$	−13.2891			−1.16107			−0.580536	+	0.814235i	$0.697157\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.68989i			0.571556i	0.958296	+	0.285778i	$0.0922520\pi$
$137$			6.68989i			0.571556i	−0.958296	+	0.285778i	$0.907748\pi$
$138$		0			0
$139$	−5.65156	+	3.26293i	−0.479359	+	0.276758i	−0.720149	−	0.693819i	$-0.755927\pi$
$139$	−5.65156	+	3.26293i	−0.479359	+	0.276758i	0.240790	+	0.970577i	$0.422593\pi$
$140$		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	$-0.928859\pi$
$149$		−	5.41089i		−	0.443277i	0.975129	−	0.221639i	$-0.0711405\pi$
$150$		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	$-0.350031\pi$
$157$	−1.13796	−	0.657000i	−0.0908188	−	0.0524343i	−0.544722	+	0.838617i	$0.683365\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.30283	−	14.3809i	0.650328	−	1.12640i	−0.332716	−	0.943027i	$-0.607965\pi$
$163$	8.30283	−	14.3809i	0.650328	−	1.12640i	0.983043	−	0.183374i	$-0.0587018\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.18145	+	10.7066i	0.478335	+	0.828501i	0.999691	−	0.0248384i	$-0.00790711\pi$
$167$	6.18145	+	10.7066i	0.478335	+	0.828501i	−0.521356	+	0.853339i	$0.674574\pi$
$168$		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	$-0.295636\pi$
$173$	−5.18456	−	8.97991i	−0.394174	−	0.682730i	−0.992995	+	0.118153i	$0.962303\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−19.5936	+	11.3124i	−1.46450	+	0.845528i	−0.999214	−	0.0396332i	$-0.987381\pi$
$179$	−19.5936	+	11.3124i	−1.46450	+	0.845528i	−0.465284	+	0.885162i	$0.654048\pi$
$180$		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	0.442813	−	0.896614i	$-0.353980\pi$
$191$	19.9110	−	11.4956i	1.44071	−	0.831794i	0.997897	+	0.0648193i	$0.0206471\pi$
$192$		0			0
$193$	8.61512	−	14.9218i	0.620130	−	1.07410i	−0.369331	−	0.929298i	$-0.620413\pi$
$193$	8.61512	−	14.9218i	0.620130	−	1.07410i	0.989461	−	0.144799i	$-0.0462535\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.31094i		−	0.307141i	−0.988138	−	0.153571i	$-0.950923\pi$
$197$		−	4.31094i		−	0.307141i	0.988138	−	0.153571i	$-0.0490773\pi$
$198$		0			0
$199$	−6.35934	+	3.67156i	−0.450801	+	0.260270i	−0.708169	−	0.706043i	$-0.750478\pi$
$199$	−6.35934	+	3.67156i	−0.450801	+	0.260270i	0.257367	+	0.966314i	$0.417145\pi$
$200$		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	0.531425	−	0.847106i	$-0.321657\pi$
$211$	−6.79668	−	11.7722i	−0.467903	−	0.810431i	−0.999327	+	0.0366744i	$0.988324\pi$
$212$		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	0.999844	+	0.0176705i	$0.00562499\pi$
$223$	22.6248	+	13.0624i	1.51507	+	0.874725i	0.515225	+	0.857055i	$0.327708\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.16204			−0.209872			−0.104936	−	0.994479i	$-0.533464\pi$
$227$	−3.16204			−0.209872			−0.104936	+	0.994479i	$0.533464\pi$
$228$		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	−0.999295	−	0.0375332i	$-0.988050\pi$
$233$	−23.3765	−	13.4964i	−1.53145	−	0.884182i	−0.532152	−	0.846649i	$-0.678617\pi$
$234$		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	−0.238890	−	0.971047i	$-0.576783\pi$
$239$	7.46107	−	4.30765i	0.482617	−	0.278639i	0.721506	+	0.692408i	$0.243450\pi$
$240$		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	$-0.407373\pi$
$251$	9.09095			0.573816			0.286908	+	0.957958i	$0.407373\pi$
$252$		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	$-0.347956\pi$
$257$	14.7392			0.919403			0.459702	+	0.888073i	$0.347956\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	27.9001i		−	1.72040i	−0.509959	−	0.860198i	$-0.670340\pi$
$263$		−	27.9001i		−	1.72040i	0.509959	−	0.860198i	$-0.329660\pi$
$264$		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	$-0.297961\pi$
$269$	−6.57488	−	11.3880i	−0.400878	−	0.694341i	−0.993832	+	0.110896i	$0.964628\pi$
$270$		0			0
$271$	−15.4231	−	8.90451i	−0.936885	−	0.540911i	−0.0479022	−	0.998852i	$-0.515254\pi$
$271$	−15.4231	−	8.90451i	−0.936885	−	0.540911i	−0.888982	+	0.457941i	$0.848587\pi$
$272$		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	0.935100	−	0.354384i	$-0.115309\pi$
$281$	18.3680	+	10.6048i	1.09574	+	0.632628i	0.160644	+	0.987012i	$0.448643\pi$
$282$		0			0
$283$	−16.2504	−	9.38216i	−0.965985	−	0.557711i	−0.0679748	−	0.997687i	$-0.521654\pi$
$283$	−16.2504	−	9.38216i	−0.965985	−	0.557711i	−0.898010	+	0.439976i	$0.854987\pi$
$284$		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	$-0.0445894\pi$
$293$	6.40500	+	11.0938i	0.374184	+	0.648106i	−0.616020	+	0.787730i	$0.711256\pi$
$294$		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	$-0.838626\pi$
$311$	−0.293203	+	0.507843i	−0.0166260	+	0.0287971i	0.857593	+	0.514329i	$0.171959\pi$
$312$		0			0
$313$	−0.860061	+	0.496556i	−0.0486135	+	0.0280670i	−0.524110	−	0.851651i	$-0.675602\pi$
$313$	−0.860061	+	0.496556i	−0.0486135	+	0.0280670i	0.475496	+	0.879718i	$0.342269\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−20.6022	+	11.8947i	−1.15714	+	0.668073i	−0.950616	−	0.310369i	$-0.899547\pi$
$317$	−20.6022	+	11.8947i	−1.15714	+	0.668073i	−0.206520	+	0.978442i	$0.566214\pi$
$318$		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	−0.950336	−	0.311224i	$-0.899261\pi$
$331$	−13.5486	−	23.4668i	−0.744697	−	1.28985i	0.205640	−	0.978628i	$-0.434072\pi$
$332$		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	0.882380	−	0.470538i	$-0.155940\pi$
$337$	0.618503	+	1.07128i	0.0336920	+	0.0583562i	−0.848688	+	0.528894i	$0.822607\pi$
$338$		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	0.778904	−	0.627143i	$-0.215776\pi$
$347$	11.6468	+	6.72430i	0.625234	+	0.360979i	−0.153670	+	0.988122i	$0.549109\pi$
$348$		0			0
$349$	−12.4728	−	7.20115i	−0.667652	−	0.385469i	0.127535	−	0.991834i	$-0.459294\pi$
$349$	−12.4728	−	7.20115i	−0.667652	−	0.385469i	−0.795186	+	0.606365i	$0.792627\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.55329			−0.402021			−0.201011	−	0.979589i	$-0.564423\pi$
$353$	−7.55329			−0.402021			−0.201011	+	0.979589i	$0.564423\pi$
$354$		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	0.936033	−	0.351913i	$-0.114469\pi$
$359$	20.8285	+	12.0253i	1.09928	+	0.634672i	0.163251	+	0.986585i	$0.447802\pi$
$360$		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	$-0.142343\pi$
$383$	35.2920			1.80334			0.901668	+	0.432429i	$0.142343\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			22.2667i			1.12897i	0.825444	+	0.564484i	$0.190925\pi$
$389$			22.2667i			1.12897i	−0.825444	+	0.564484i	$0.809075\pi$
$390$		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	$-0.495760\pi$
$397$	−16.8558	−	9.73170i	−0.845969	−	0.488420i	−0.859289	+	0.511491i	$0.829093\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	8.60311i		−	0.429619i	−0.976656	−	0.214809i	$-0.931087\pi$
$401$		−	8.60311i		−	0.429619i	0.976656	−	0.214809i	$-0.0689130\pi$
$402$		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	$-0.492111\pi$
$409$	−16.7571	−	9.67472i	−0.828585	−	0.478384i	−0.853368	+	0.521309i	$0.825444\pi$
$410$		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	$-0.308331\pi$
$419$	−8.81222	−	15.2632i	−0.430505	−	0.745657i	−0.996917	+	0.0784657i	$0.974998\pi$
$420$		0			0
$421$	−5.77040	+	9.99463i	−0.281232	+	0.487109i	−0.971689	−	0.236266i	$-0.924076\pi$
$421$	−5.77040	+	9.99463i	−0.281232	+	0.487109i	0.690456	+	0.723374i	$0.257410\pi$
$422$		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	−0.242005	−	0.970275i	$-0.577805\pi$
$431$	9.90849	−	5.72067i	0.477275	−	0.275555i	0.719280	+	0.694720i	$0.244472\pi$
$432$		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	$-0.785676\pi$
$439$	−13.2543	+	7.65239i	−0.632595	+	0.365229i	0.149162	+	0.988813i	$0.452343\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.4838	+	11.2490i	−0.925705	+	0.534456i	−0.885450	−	0.464734i	$-0.846150\pi$
$443$	−19.4838	+	11.2490i	−0.925705	+	0.534456i	−0.0402540	+	0.999189i	$0.512817\pi$
$444$		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	$0.209965\pi$
$449$			25.9709i			1.22564i	−0.790222	+	0.612821i	$0.790035\pi$
$450$		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	0.865506	−	0.500899i	$-0.166997\pi$
$457$	−0.0222019	−	0.0384547i	−0.00103856	−	0.00179884i	−0.866544	+	0.499100i	$0.833664\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.46783	+	2.54236i	−0.0683636	+	0.118409i	−0.898181	−	0.439626i	$-0.855111\pi$
$461$	−1.46783	+	2.54236i	−0.0683636	+	0.118409i	0.829817	+	0.558035i	$0.188444\pi$
$462$		0			0
$463$	−19.2017	−	33.2583i	−0.892378	−	1.54564i	−0.837017	−	0.547177i	$-0.815702\pi$
$463$	−19.2017	−	33.2583i	−0.892378	−	1.54564i	−0.0553609	−	0.998466i	$-0.517631\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.63913	+	14.9634i	−0.399771	+	0.692424i	−0.993697	−	0.112095i	$-0.964244\pi$
$467$	−8.63913	+	14.9634i	−0.399771	+	0.692424i	0.593926	+	0.804520i	$0.297577\pi$
$468$		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	$-0.180595\pi$
$479$	36.9142			1.68665			0.843326	+	0.537403i	$0.180595\pi$
$480$		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	0.816307	−	0.577618i	$-0.196018\pi$
$487$	−2.03199	−	3.51951i	−0.0920783	−	0.159484i	−0.908385	+	0.418134i	$0.862684\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.30460	+	2.48526i	−0.194264	+	0.112158i	−0.593977	−	0.804482i	$-0.702443\pi$
$491$	−4.30460	+	2.48526i	−0.194264	+	0.112158i	0.399713	+	0.916640i	$0.369110\pi$
$492$		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	$-0.508300\pi$
$503$	−1.16946			−0.0521436			−0.0260718	+	0.999660i	$0.508300\pi$
$504$		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	$-0.430860\pi$
$509$	9.72412			0.431014			0.215507	+	0.976502i	$0.430860\pi$
$510$		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	$-0.0786243\pi$
$521$	6.23317	+	10.7962i	0.273080	+	0.472989i	−0.696569	+	0.717490i	$0.745291\pi$
$522$		0			0
$523$	−15.6222	−	9.01951i	−0.683113	−	0.394395i	0.117914	−	0.993024i	$-0.462379\pi$
$523$	−15.6222	−	9.01951i	−0.683113	−	0.394395i	−0.801027	+	0.598628i	$0.795713\pi$
$524$		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	−0.903566	−	0.428449i	$-0.859060\pi$
$541$	−1.87785	+	3.25253i	−0.0807349	+	0.139837i	0.822831	+	0.568286i	$0.192393\pi$
$542$		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	−0.737617	−	0.675219i	$-0.764049\pi$
$547$	5.05062	−	8.74793i	0.215949	−	0.374034i	0.953566	+	0.301185i	$0.0973822\pi$
$548$		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	−0.795581	−	0.605847i	$-0.792834\pi$
$557$	−15.7817	+	9.11158i	−0.668693	+	0.386070i	0.126888	+	0.991917i	$0.459501\pi$
$558$		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	$-0.947757\pi$
$563$	−8.34691	+	14.4573i	−0.351780	+	0.609301i	0.634781	+	0.772692i	$0.281090\pi$
$564$		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	0.0342459	−	0.999413i	$-0.489097\pi$
$569$	21.8711	−	12.6273i	0.916886	−	0.529365i	0.882640	+	0.470049i	$0.155764\pi$
$570$		0			0
$571$	2.88981	−	5.00529i	0.120935	−	0.209465i	−0.799202	−	0.601063i	$-0.794744\pi$
$571$	2.88981	−	5.00529i	0.120935	−	0.209465i	0.920137	+	0.391598i	$0.128078\pi$
$572$		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	$-0.908488\pi$
$577$	−28.6539	+	16.5433i	−1.19288	+	0.688708i	−0.233919	+	0.972256i	$0.575155\pi$
$578$		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	0.412417	+	0.910995i	$0.364685\pi$
$587$	−14.1186	+	24.4541i	−0.582737	+	1.00933i	−0.995153	+	0.0983341i	$0.968649\pi$
$588$		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	$-0.708634\pi$
$593$	9.29769	−	16.1041i	0.381810	−	0.661315i	0.991321	+	0.131463i	$0.0419674\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.77276	+	4.48760i	0.317586	+	0.183358i	0.650316	−	0.759664i	$-0.274636\pi$
$599$	7.77276	+	4.48760i	0.317586	+	0.183358i	−0.332730	+	0.943022i	$0.607970\pi$
$600$		0			0
$601$	4.71245	+	2.72073i	0.192225	+	0.110981i	0.593024	−	0.805185i	$-0.297934\pi$
$601$	4.71245	+	2.72073i	0.192225	+	0.110981i	−0.400799	+	0.916166i	$0.631267\pi$
$602$		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	0.960625	−	0.277849i	$-0.0896216\pi$
$613$	5.93439	+	10.2787i	0.239688	+	0.415151i	−0.720937	+	0.693001i	$0.756288\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.2845	+	9.40188i	−0.655591	+	0.378506i	−0.790595	−	0.612339i	$-0.790229\pi$
$617$	−16.2845	+	9.40188i	−0.655591	+	0.378506i	0.135004	+	0.990845i	$0.456895\pi$
$618$		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	$-0.810746\pi$
$641$		−	28.3633i		−	1.12028i	0.828396	−	0.560142i	$-0.189254\pi$
$642$		0			0
$643$	−16.0912	+	9.29024i	−0.634573	+	0.366371i	−0.782521	−	0.622624i	$-0.786067\pi$
$643$	−16.0912	+	9.29024i	−0.634573	+	0.366371i	0.147948	+	0.988995i	$0.452733\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.4482	+	37.1494i	0.843216	+	1.46049i	0.887161	+	0.461460i	$0.152674\pi$
$647$	21.4482	+	37.1494i	0.843216	+	1.46049i	−0.0439448	+	0.999034i	$0.513993\pi$
$648$		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	$0.0229414\pi$
$653$			3.68027i			0.144020i	−0.997404	+	0.0720101i	$0.977059\pi$
$654$		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	0.744023	−	0.668154i	$-0.232915\pi$
$659$	13.7955	+	7.96483i	0.537396	+	0.310266i	−0.206627	+	0.978420i	$0.566249\pi$
$660$		0			0
$661$	1.14378	+	0.660360i	0.0444878	+	0.0256850i	0.522079	−	0.852897i	$-0.325157\pi$
$661$	1.14378	+	0.660360i	0.0444878	+	0.0256850i	−0.477591	+	0.878582i	$0.658490\pi$
$662$		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	−0.0493788	−	0.998780i	$-0.515724\pi$
$673$	21.7987	−	37.7565i	0.840280	−	1.45541i	0.889658	−	0.456627i	$-0.150943\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.4677	−	25.0588i	−0.556039	−	0.963088i	−0.997822	−	0.0659663i	$-0.978987\pi$
$677$	−14.4677	−	25.0588i	−0.556039	−	0.963088i	0.441782	−	0.897122i	$-0.354346\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.90755	−	1.10132i	0.0729903	−	0.0421410i	−0.463061	−	0.886327i	$-0.653249\pi$
$683$	1.90755	−	1.10132i	0.0729903	−	0.0421410i	0.536051	+	0.844186i	$0.319915\pi$
$684$		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	$-0.474142\pi$
$691$	25.8896	−	14.9474i	0.984888	−	0.568625i	0.903742	+	0.428077i	$0.140809\pi$
$692$		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	$-0.671960\pi$
$701$		−	45.4119i		−	1.71518i	0.514332	−	0.857591i	$-0.328040\pi$
$702$		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	0.975601	−	0.219551i	$-0.0704592\pi$
$709$	7.92591	+	13.7281i	0.297664	+	0.515569i	−0.677937	+	0.735120i	$0.737126\pi$
$710$		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	$-0.872625\pi$
$719$	−3.30154	+	5.71844i	−0.123127	+	0.213262i	0.797872	+	0.602826i	$0.205959\pi$
$720$		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	$-0.566294\pi$
$727$	−31.2086	−	18.0183i	−1.15746	−	0.668261i	−0.950694	+	0.310129i	$0.899628\pi$
$728$		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	0.763270	−	0.646079i	$-0.223593\pi$
$739$	−4.83576	−	8.37577i	−0.177886	−	0.308108i	−0.941156	+	0.337972i	$0.890259\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−11.0205	+	6.36269i	−0.404303	+	0.233424i	−0.688339	−	0.725389i	$-0.741660\pi$
$743$	−11.0205	+	6.36269i	−0.404303	+	0.233424i	0.284036	+	0.958814i	$0.408326\pi$
$744$		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	$-0.367403\pi$
$761$	22.3240			0.809245			0.404623	+	0.914484i	$0.367403\pi$
$762$		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	$-0.577219\pi$
$769$	13.3202	−	7.69042i	0.480338	−	0.277324i	0.720558	+	0.693395i	$0.243886\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.17527	−	10.6959i	−0.222109	−	0.384704i	0.733339	−	0.679863i	$-0.237961\pi$
$773$	−6.17527	−	10.6959i	−0.222109	−	0.384704i	−0.955448	+	0.295159i	$0.904627\pi$
$774$		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	$-0.249956\pi$
$787$	12.5825	+	7.26452i	0.448518	+	0.258952i	−0.258686	+	0.965961i	$0.583290\pi$
$788$		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	−0.932276	−	0.361747i	$-0.882181\pi$
$797$	−22.0040	−	38.1120i	−0.779420	−	1.35000i	0.152856	−	0.988248i	$-0.451153\pi$
$798$		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	−0.998888	−	0.0471551i	$-0.984985\pi$
$809$	−41.4554	+	23.9343i	−1.45749	+	0.841484i	−0.458606	+	0.888640i	$0.651651\pi$
$810$		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	−0.607598	−	0.794244i	$-0.707867\pi$
$821$	−6.40574	+	3.69836i	−0.223562	+	0.129074i	0.384037	+	0.923318i	$0.374534\pi$
$822$		0			0
$823$	21.4718	−	37.1903i	0.748460	−	1.29637i	−0.200100	−	0.979775i	$-0.564127\pi$
$823$	21.4718	−	37.1903i	0.748460	−	1.29637i	0.948560	−	0.316596i	$-0.102540\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	43.0042i		−	1.49540i	−0.664035	−	0.747701i	$-0.731157\pi$
$827$		−	43.0042i		−	1.49540i	0.664035	−	0.747701i	$-0.268843\pi$
$828$		0			0
$829$	−10.0780	+	5.81851i	−0.350022	+	0.202085i	−0.664695	−	0.747115i	$-0.731439\pi$
$829$	−10.0780	+	5.81851i	−0.350022	+	0.202085i	0.314673	+	0.949200i	$0.398105\pi$
$830$		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	$-0.843631\pi$
$839$	−0.936892	+	1.62274i	−0.0323451	+	0.0560234i	0.849400	+	0.527750i	$0.176964\pi$
$840$		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	$-0.575943\pi$
$853$	−34.9301	−	20.1669i	−1.19598	−	0.690501i	−0.959657	+	0.281173i	$0.909276\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−32.2681			−1.10226			−0.551129	−	0.834420i	$-0.685803\pi$
$857$	−32.2681			−1.10226			−0.551129	+	0.834420i	$0.685803\pi$
$858$		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	0.943386	−	0.331697i	$-0.107621\pi$
$863$	33.1319	+	19.1287i	1.12782	+	0.651148i	0.184435	+	0.982845i	$0.440954\pi$
$864$		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	$-0.409915\pi$
$881$	16.5770			0.558493			0.279247	+	0.960219i	$0.409915\pi$
$882$		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	$-0.698336\pi$
$887$	−34.7590			−1.16709			−0.583547	+	0.812079i	$0.698336\pi$
$888$		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	0.370299	−	0.928913i	$-0.379255\pi$
$911$	−7.51591	−	4.33931i	−0.249013	−	0.143768i	−0.619312	+	0.785145i	$0.712589\pi$
$912$		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	−0.984953	−	0.172823i	$-0.944711\pi$
$919$	−10.3922	+	17.9999i	−0.342808	+	0.593760i	0.642145	+	0.766583i	$0.278045\pi$
$920$		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	−0.976828	−	0.214028i	$-0.931342\pi$
$929$	−20.5361	−	35.5695i	−0.673767	−	1.16700i	0.303060	−	0.952971i	$-0.401992\pi$
$930$		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	$-0.781253\pi$
$941$	4.99670	−	8.65453i	0.162888	−	0.282130i	0.935903	+	0.352257i	$0.114586\pi$
$942$		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	−0.131162	−	0.991361i	$-0.541871\pi$
$947$	20.3659	−	11.7582i	0.661801	−	0.382091i	0.792963	+	0.609270i	$0.208537\pi$
$948$		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	$0.190163\pi$
$953$			34.7300i			1.12501i	−0.826793	+	0.562507i	$0.809837\pi$
$954$		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	0.973568	−	0.228399i	$-0.0733489\pi$
$967$	8.98645	+	15.5650i	0.288985	+	0.500536i	−0.684583	+	0.728935i	$0.740016\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.9001	+	36.2001i	−0.670717	+	1.16172i	0.306984	+	0.951715i	$0.400680\pi$
$971$	−20.9001	+	36.2001i	−0.670717	+	1.16172i	−0.977701	+	0.210001i	$0.932653\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−38.6499	−	22.3145i	−1.23652	−	0.713905i	−0.268139	−	0.963380i	$-0.586409\pi$
$977$	−38.6499	−	22.3145i	−1.23652	−	0.713905i	−0.968381	+	0.249475i	$0.919742\pi$
$978$		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	$-0.864631\pi$
$983$	−57.1202			−1.82185			−0.910925	+	0.412571i	$0.864631\pi$
$984$		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	−0.993776	−	0.111399i	$-0.964467\pi$
$991$	−18.6791	−	32.3532i	−0.593362	−	1.02773i	0.400413	−	0.916335i	$-0.368867\pi$
$992$		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.4625.20		48
3.2	odd	2		1764.2.bm.c.1685.18		48
7.2	even	3		5292.2.x.c.4409.5		48
7.3	odd	6		5292.2.w.c.521.20		48
7.4	even	3		5292.2.w.c.521.5		48
7.5	odd	6		5292.2.x.c.4409.20		48
7.6	odd	2	inner	5292.2.bm.c.4625.5		48
9.4	even	3		1764.2.w.c.509.9		48
9.5	odd	6		5292.2.w.c.1097.20		48
21.2	odd	6		1764.2.x.c.1469.2	yes	48
21.5	even	6		1764.2.x.c.1469.23	yes	48
21.11	odd	6		1764.2.w.c.1109.16		48
21.17	even	6		1764.2.w.c.1109.9		48
21.20	even	2		1764.2.bm.c.1685.7		48
63.4	even	3		1764.2.bm.c.1697.7		48
63.5	even	6		5292.2.x.c.881.5		48
63.13	odd	6		1764.2.w.c.509.16		48
63.23	odd	6		5292.2.x.c.881.20		48
63.31	odd	6		1764.2.bm.c.1697.18		48
63.32	odd	6	inner	5292.2.bm.c.2285.5		48
63.40	odd	6		1764.2.x.c.293.2	✓	48
63.41	even	6		5292.2.w.c.1097.5		48
63.58	even	3		1764.2.x.c.293.23	yes	48
63.59	even	6	inner	5292.2.bm.c.2285.20		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.w.c.509.9		48	9.4	even	3
1764.2.w.c.509.16		48	63.13	odd	6
1764.2.w.c.1109.9		48	21.17	even	6
1764.2.w.c.1109.16		48	21.11	odd	6
1764.2.x.c.293.2	✓	48	63.40	odd	6
1764.2.x.c.293.23	yes	48	63.58	even	3
1764.2.x.c.1469.2	yes	48	21.2	odd	6
1764.2.x.c.1469.23	yes	48	21.5	even	6
1764.2.bm.c.1685.7		48	21.20	even	2
1764.2.bm.c.1685.18		48	3.2	odd	2
1764.2.bm.c.1697.7		48	63.4	even	3
1764.2.bm.c.1697.18		48	63.31	odd	6
5292.2.w.c.521.5		48	7.4	even	3
5292.2.w.c.521.20		48	7.3	odd	6
5292.2.w.c.1097.5		48	63.41	even	6
5292.2.w.c.1097.20		48	9.5	odd	6
5292.2.x.c.881.5		48	63.5	even	6
5292.2.x.c.881.20		48	63.23	odd	6
5292.2.x.c.4409.5		48	7.2	even	3
5292.2.x.c.4409.20		48	7.5	odd	6
5292.2.bm.c.2285.5		48	63.32	odd	6	inner
5292.2.bm.c.2285.20		48	63.59	even	6	inner
5292.2.bm.c.4625.5		48	7.6	odd	2	inner
5292.2.bm.c.4625.20		48	1.1	even	1	trivial

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

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