LMFDB - Embedded newform 5292.2.i.j.1549.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1549,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("5292.1549");

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.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.12
Character	$\chi$	$=$	5292.1549
Dual form			5292.2.i.j.2125.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.94623 + 3.37097i) q^{5} +O(q^{10})$	$q+(1.94623 + 3.37097i) q^{5} +(2.18778 - 3.78935i) q^{11} +(-0.792201 + 1.37213i) q^{13} +(-2.66678 - 4.61900i) q^{17} +(2.32161 - 4.02115i) q^{19} +(0.183900 + 0.318523i) q^{23} +(-5.07564 + 8.79126i) q^{25} +(5.08750 + 8.81180i) q^{29} -2.29552 q^{31} +(-5.44017 + 9.42265i) q^{37} +(0.690443 - 1.19588i) q^{41} +(3.81699 + 6.61122i) q^{43} +7.60865 q^{47} +(-0.462847 - 0.801674i) q^{53} +17.0317 q^{55} +0.920949 q^{59} +6.55560 q^{61} -6.16722 q^{65} +15.0084 q^{67} +4.91059 q^{71} +(-3.78353 - 6.55327i) q^{73} +1.97543 q^{79} +(0.253011 + 0.438227i) q^{83} +(10.3803 - 17.9793i) q^{85} +(-6.10987 + 10.5826i) q^{89} +18.0736 q^{95} +(4.45315 + 7.71308i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} - 72 q^{65} - 24 q^{67} - 48 q^{71} - 24 q^{79} + 12 q^{85} + 64 q^{95}+O(q^{100}) $ 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 - 72 * q^65 - 24 * q^67 - 48 * q^71 - 24 * q^79 + 12 * q^85 + 64 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.94623	+	3.37097i	0.870381	+	1.50754i	0.861603	+	0.507583i	$0.169461\pi$
$5$	1.94623	+	3.37097i	0.870381	+	1.50754i	0.00877856	+	0.999961i	$0.497206\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.18778	−	3.78935i	0.659642	−	1.14253i	−0.321067	−	0.947057i	$-0.604041\pi$
$11$	2.18778	−	3.78935i	0.659642	−	1.14253i	0.980708	−	0.195476i	$-0.0626253\pi$
$12$		0			0
$13$	−0.792201	+	1.37213i	−0.219717	+	0.380561i	−0.954721	−	0.297501i	$-0.903847\pi$
$13$	−0.792201	+	1.37213i	−0.219717	+	0.380561i	0.735004	+	0.678062i	$0.237180\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.66678	−	4.61900i	−0.646789	−	1.12027i	−0.983885	−	0.178801i	$-0.942778\pi$
$17$	−2.66678	−	4.61900i	−0.646789	−	1.12027i	0.337096	−	0.941470i	$-0.390555\pi$
$18$		0			0
$19$	2.32161	−	4.02115i	0.532614	−	0.922515i	−0.466660	−	0.884437i	$-0.654543\pi$
$19$	2.32161	−	4.02115i	0.532614	−	0.922515i	0.999275	−	0.0380786i	$-0.0121237\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.183900	+	0.318523i	0.0383457	+	0.0664167i	0.884561	−	0.466424i	$-0.154458\pi$
$23$	0.183900	+	0.318523i	0.0383457	+	0.0664167i	−0.846216	+	0.532841i	$0.821125\pi$
$24$		0			0
$25$	−5.07564	+	8.79126i	−1.01513	+	1.75825i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.08750	+	8.81180i	0.944724	+	1.63631i	0.756302	+	0.654222i	$0.227004\pi$
$29$	5.08750	+	8.81180i	0.944724	+	1.63631i	0.188422	+	0.982088i	$0.439663\pi$
$30$		0			0
$31$	−2.29552			−0.412288			−0.206144	−	0.978522i	$-0.566092\pi$
$31$	−2.29552			−0.412288			−0.206144	+	0.978522i	$0.566092\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.44017	+	9.42265i	−0.894359	+	1.54907i	−0.0597623	+	0.998213i	$0.519034\pi$
$37$	−5.44017	+	9.42265i	−0.894359	+	1.54907i	−0.834596	+	0.550862i	$0.814299\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.690443	−	1.19588i	0.107829	−	0.186766i	−0.807061	−	0.590467i	$-0.798943\pi$
$41$	0.690443	−	1.19588i	0.107829	−	0.186766i	0.914891	+	0.403702i	$0.132277\pi$
$42$		0			0
$43$	3.81699	+	6.61122i	0.582086	+	1.00820i	0.995232	+	0.0975372i	$0.0310965\pi$
$43$	3.81699	+	6.61122i	0.582086	+	1.00820i	−0.413146	+	0.910665i	$0.635570\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.60865			1.10984			0.554918	−	0.831905i	$-0.312750\pi$
$47$	7.60865			1.10984			0.554918	+	0.831905i	$0.312750\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.462847	−	0.801674i	−0.0635769	−	0.110118i	0.832485	−	0.554048i	$-0.186917\pi$
$53$	−0.462847	−	0.801674i	−0.0635769	−	0.110118i	−0.896062	+	0.443929i	$0.853584\pi$
$54$		0			0
$55$	17.0317			2.29656
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.920949			0.119897			0.0599487	−	0.998201i	$-0.480906\pi$
$59$	0.920949			0.119897			0.0599487	+	0.998201i	$0.480906\pi$
$60$		0			0
$61$	6.55560			0.839358			0.419679	−	0.907673i	$-0.362143\pi$
$61$	6.55560			0.839358			0.419679	+	0.907673i	$0.362143\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.16722			−0.764950
$66$		0			0
$67$	15.0084			1.83357			0.916783	−	0.399385i	$-0.130776\pi$
$67$	15.0084			1.83357			0.916783	+	0.399385i	$0.130776\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.91059			0.582780			0.291390	−	0.956604i	$-0.405882\pi$
$71$	4.91059			0.582780			0.291390	+	0.956604i	$0.405882\pi$
$72$		0			0
$73$	−3.78353	−	6.55327i	−0.442829	−	0.767003i	0.555069	−	0.831804i	$-0.312692\pi$
$73$	−3.78353	−	6.55327i	−0.442829	−	0.767003i	−0.997898	+	0.0648016i	$0.979359\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.97543			0.222253			0.111127	−	0.993806i	$-0.464554\pi$
$79$	1.97543			0.222253			0.111127	+	0.993806i	$0.464554\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.253011	+	0.438227i	0.0277715	+	0.0481017i	0.879577	−	0.475756i	$-0.157826\pi$
$83$	0.253011	+	0.438227i	0.0277715	+	0.0481017i	−0.851806	+	0.523858i	$0.824492\pi$
$84$		0			0
$85$	10.3803	−	17.9793i	1.12591	−	1.95013i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.10987	+	10.5826i	−0.647645	+	1.12175i	0.336039	+	0.941848i	$0.390913\pi$
$89$	−6.10987	+	10.5826i	−0.647645	+	1.12175i	−0.983684	+	0.179906i	$0.942421\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	18.0736			1.85431
$96$		0			0
$97$	4.45315	+	7.71308i	0.452149	+	0.783145i	0.998519	−	0.0543987i	$-0.0173242\pi$
$97$	4.45315	+	7.71308i	0.452149	+	0.783145i	−0.546370	+	0.837544i	$0.683991\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.51180	−	9.54672i	0.548445	−	0.949935i	−0.449936	−	0.893061i	$-0.648553\pi$
$101$	5.51180	−	9.54672i	0.548445	−	0.949935i	0.998381	−	0.0568740i	$-0.0181133\pi$
$102$		0			0
$103$	−1.36543	−	2.36499i	−0.134540	−	0.233029i	0.790882	−	0.611969i	$-0.209622\pi$
$103$	−1.36543	−	2.36499i	−0.134540	−	0.233029i	−0.925421	+	0.378939i	$0.876289\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.39605	+	9.34623i	−0.521655	+	0.903534i	0.478027	+	0.878345i	$0.341352\pi$
$107$	−5.39605	+	9.34623i	−0.521655	+	0.903534i	−0.999683	+	0.0251887i	$0.991981\pi$
$108$		0			0
$109$	4.99187	+	8.64617i	0.478134	+	0.828153i	0.999686	−	0.0250668i	$-0.00797984\pi$
$109$	4.99187	+	8.64617i	0.478134	+	0.828153i	−0.521551	+	0.853220i	$0.674647\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.12019	−	10.6005i	0.575739	−	0.997209i	−0.420222	−	0.907421i	$-0.638048\pi$
$113$	6.12019	−	10.6005i	0.575739	−	0.997209i	0.995961	−	0.0897875i	$-0.0286188\pi$
$114$		0			0
$115$	−0.715822	+	1.23984i	−0.0667508	+	0.115616i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.07280	−	7.05429i	−0.370254	−	0.641299i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−20.0511			−1.79343
$126$		0			0
$127$	−13.2005			−1.17135			−0.585677	−	0.810544i	$-0.699171\pi$
$127$	−13.2005			−1.17135			−0.585677	+	0.810544i	$0.699171\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.54342	+	4.40532i	0.222219	+	0.384895i	0.955482	−	0.295051i	$-0.0953366\pi$
$131$	2.54342	+	4.40532i	0.222219	+	0.384895i	−0.733262	+	0.679946i	$0.762003\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.67208	−	11.5564i	0.570034	−	0.987328i	−0.426527	−	0.904475i	$-0.640263\pi$
$137$	6.67208	−	11.5564i	0.570034	−	0.987328i	0.996562	−	0.0828538i	$-0.0264034\pi$
$138$		0			0
$139$	−4.85642	+	8.41157i	−0.411916	+	0.713460i	−0.995099	−	0.0988809i	$-0.968474\pi$
$139$	−4.85642	+	8.41157i	−0.411916	+	0.713460i	0.583183	+	0.812341i	$0.301807\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.46633	+	6.00386i	0.289869	+	0.502068i
$144$		0			0
$145$	−19.8029	+	34.2996i	−1.64454	+	2.84843i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.92116	+	11.9878i	0.567004	+	0.982079i	0.996860	+	0.0791820i	$0.0252308\pi$
$149$	6.92116	+	11.9878i	0.567004	+	0.982079i	−0.429856	+	0.902897i	$0.641436\pi$
$150$		0			0
$151$	−11.4380	+	19.8112i	−0.930809	+	1.61221i	−0.148867	+	0.988857i	$0.547563\pi$
$151$	−11.4380	+	19.8112i	−0.930809	+	1.61221i	−0.781942	+	0.623352i	$0.785771\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.46762	−	7.73815i	−0.358848	−	0.621543i
$156$		0			0
$157$	11.5798			0.924170			0.462085	−	0.886836i	$-0.347101\pi$
$157$	11.5798			0.924170			0.462085	+	0.886836i	$0.347101\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.70577	+	2.95449i	−0.133607	+	0.231413i	−0.925064	−	0.379811i	$-0.875989\pi$
$163$	−1.70577	+	2.95449i	−0.133607	+	0.231413i	0.791458	+	0.611224i	$0.209323\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.69996	−	8.14057i	0.363694	−	0.629936i	−0.624872	−	0.780727i	$-0.714849\pi$
$167$	4.69996	−	8.14057i	0.363694	−	0.629936i	0.988566	+	0.150791i	$0.0481821\pi$
$168$		0			0
$169$	5.24484	+	9.08432i	0.403449	+	0.698794i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.40512			0.486972			0.243486	−	0.969904i	$-0.421709\pi$
$173$	6.40512			0.486972			0.243486	+	0.969904i	$0.421709\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.54038	+	14.7924i	0.638338	+	1.10563i	0.985797	+	0.167938i	$0.0537110\pi$
$179$	8.54038	+	14.7924i	0.638338	+	1.10563i	−0.347460	+	0.937695i	$0.612956\pi$
$180$		0			0
$181$	1.35988			0.101079			0.0505395	−	0.998722i	$-0.483906\pi$
$181$	1.35988			0.101079			0.0505395	+	0.998722i	$0.483906\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−42.3513			−3.11373
$186$		0			0
$187$	−23.3374			−1.70660
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17.8810			1.29382			0.646911	−	0.762566i	$-0.276061\pi$
$191$	17.8810			1.29382			0.646911	+	0.762566i	$0.276061\pi$
$192$		0			0
$193$	−13.0133			−0.936717			−0.468358	−	0.883539i	$-0.655154\pi$
$193$	−13.0133			−0.936717			−0.468358	+	0.883539i	$0.655154\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.36486			−0.453478			−0.226739	−	0.973956i	$-0.572806\pi$
$197$	−6.36486			−0.453478			−0.226739	+	0.973956i	$0.572806\pi$
$198$		0			0
$199$	−11.5764	−	20.0510i	−0.820631	−	1.42137i	−0.905213	−	0.424958i	$-0.860289\pi$
$199$	−11.5764	−	20.0510i	−0.820631	−	1.42137i	0.0845818	−	0.996417i	$-0.473045\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	5.37505			0.375410
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.1584	−	17.5948i	−0.702669	−	1.21706i
$210$		0			0
$211$	−5.67737	+	9.83349i	−0.390846	+	0.676965i	−0.992561	−	0.121745i	$-0.961151\pi$
$211$	−5.67737	+	9.83349i	−0.390846	+	0.676965i	0.601715	+	0.798711i	$0.294484\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14.8575	+	25.7339i	−1.01327	+	1.75504i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.45050			0.568442
$222$		0			0
$223$	−13.3206	−	23.0719i	−0.892011	−	1.54501i	−0.837461	−	0.546498i	$-0.815961\pi$
$223$	−13.3206	−	23.0719i	−0.892011	−	1.54501i	−0.0545504	−	0.998511i	$-0.517373\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.30136	+	14.3784i	−0.550981	+	0.954326i	0.447224	+	0.894422i	$0.352413\pi$
$227$	−8.30136	+	14.3784i	−0.550981	+	0.954326i	−0.998204	+	0.0599042i	$0.980920\pi$
$228$		0			0
$229$	7.25072	+	12.5586i	0.479141	+	0.829897i	0.999714	−	0.0239205i	$-0.00761485\pi$
$229$	7.25072	+	12.5586i	0.479141	+	0.829897i	−0.520573	+	0.853817i	$0.674282\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.99020	−	13.8394i	0.523456	−	0.906652i	−0.476172	−	0.879352i	$-0.657976\pi$
$233$	7.99020	−	13.8394i	0.523456	−	0.906652i	0.999627	−	0.0272993i	$-0.00869073\pi$
$234$		0			0
$235$	14.8082	+	25.6485i	0.965980	+	1.67313i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.58994	−	16.6103i	0.620322	−	1.07443i	−0.369104	−	0.929388i	$-0.620335\pi$
$239$	9.58994	−	16.6103i	0.620322	−	1.07443i	0.989426	−	0.145040i	$-0.0463312\pi$
$240$		0			0
$241$	−11.6785	+	20.2277i	−0.752276	+	1.30298i	0.194441	+	0.980914i	$0.437711\pi$
$241$	−11.6785	+	20.2277i	−0.752276	+	1.30298i	−0.946717	+	0.322067i	$0.895623\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.67837	+	6.37112i	0.234049	+	0.405384i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	30.4619			1.92274			0.961371	−	0.275257i	$-0.0887631\pi$
$251$	30.4619			1.92274			0.961371	+	0.275257i	$0.0887631\pi$
$252$		0			0
$253$	1.60933			0.101178
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.40438	−	5.89657i	−0.212360	−	0.367818i	0.740093	−	0.672505i	$-0.234782\pi$
$257$	−3.40438	−	5.89657i	−0.212360	−	0.367818i	−0.952453	+	0.304687i	$0.901448\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.30000	+	3.98372i	−0.141824	+	0.245647i	−0.928184	−	0.372122i	$-0.878630\pi$
$263$	−2.30000	+	3.98372i	−0.141824	+	0.245647i	0.786359	+	0.617769i	$0.211963\pi$
$264$		0			0
$265$	1.80161	−	3.12049i	0.110672	−	0.191690i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.25090	−	7.36277i	−0.259182	−	0.448916i	0.706841	−	0.707372i	$-0.250120\pi$
$269$	−4.25090	−	7.36277i	−0.259182	−	0.448916i	−0.966023	+	0.258456i	$0.916786\pi$
$270$		0			0
$271$	−3.29191	+	5.70175i	−0.199969	+	0.346357i	−0.948518	−	0.316723i	$-0.897417\pi$
$271$	−3.29191	+	5.70175i	−0.199969	+	0.346357i	0.748549	+	0.663079i	$0.230751\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	22.2088	+	38.4668i	1.33924	+	2.31963i
$276$		0			0
$277$	5.63483	−	9.75982i	0.338564	−	0.586411i	−0.645599	−	0.763677i	$-0.723392\pi$
$277$	5.63483	−	9.75982i	0.338564	−	0.586411i	0.984163	+	0.177266i	$0.0567254\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.50741	−	13.0032i	−0.447854	−	0.775707i	0.550392	−	0.834907i	$-0.314478\pi$
$281$	−7.50741	−	13.0032i	−0.447854	−	0.775707i	−0.998246	+	0.0592000i	$0.981145\pi$
$282$		0			0
$283$	14.6731			0.872227			0.436114	−	0.899892i	$-0.356355\pi$
$283$	14.6731			0.872227			0.436114	+	0.899892i	$0.356355\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.72343	+	9.91327i	−0.336672	+	0.583133i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.38981	−	16.2636i	0.548559	−	0.950132i	−0.449815	−	0.893122i	$-0.648510\pi$
$293$	9.38981	−	16.2636i	0.548559	−	0.950132i	0.998374	−	0.0570099i	$-0.0181567\pi$
$294$		0			0
$295$	1.79238	+	3.10449i	0.104356	+	0.180751i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.582741			−0.0337008
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.7587	+	22.0987i	0.730562	+	1.26537i
$306$		0			0
$307$	−28.9425			−1.65184			−0.825919	−	0.563789i	$-0.809343\pi$
$307$	−28.9425			−1.65184			−0.825919	+	0.563789i	$0.809343\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.5936			0.770824			0.385412	−	0.922745i	$-0.374059\pi$
$311$	13.5936			0.770824			0.385412	+	0.922745i	$0.374059\pi$
$312$		0			0
$313$	−13.8644			−0.783665			−0.391832	−	0.920037i	$-0.628159\pi$
$313$	−13.8644			−0.783665			−0.391832	+	0.920037i	$0.628159\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−23.0856			−1.29662			−0.648309	−	0.761377i	$-0.724524\pi$
$317$	−23.0856			−1.29662			−0.648309	+	0.761377i	$0.724524\pi$
$318$		0			0
$319$	44.5214			2.49272
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.7649			−1.37796
$324$		0			0
$325$	−8.04184	−	13.9289i	−0.446081	−	0.772635i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.6529			−0.860364			−0.430182	−	0.902742i	$-0.641551\pi$
$331$	−15.6529			−0.860364			−0.430182	+	0.902742i	$0.641551\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	29.2098	+	50.5929i	1.59590	+	2.76418i
$336$		0			0
$337$	3.56686	−	6.17799i	0.194299	−	0.336537i	−0.752371	−	0.658739i	$-0.771090\pi$
$337$	3.56686	−	6.17799i	0.194299	−	0.336537i	0.946671	+	0.322203i	$0.104423\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5.02211	+	8.69855i	−0.271962	+	0.471053i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.55655			−0.298291			−0.149146	−	0.988815i	$-0.547652\pi$
$347$	−5.55655			−0.298291			−0.149146	+	0.988815i	$0.547652\pi$
$348$		0			0
$349$	−5.33296	−	9.23696i	−0.285467	−	0.494443i	0.687256	−	0.726416i	$-0.258815\pi$
$349$	−5.33296	−	9.23696i	−0.285467	−	0.494443i	−0.972722	+	0.231973i	$0.925482\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.42132	+	11.1221i	−0.341772	+	0.591967i	−0.984762	−	0.173908i	$-0.944361\pi$
$353$	−6.42132	+	11.1221i	−0.341772	+	0.591967i	0.642989	+	0.765875i	$0.277694\pi$
$354$		0			0
$355$	9.55715	+	16.5535i	0.507241	+	0.878567i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.8417	−	22.2426i	0.677761	−	1.17392i	−0.297892	−	0.954600i	$-0.596284\pi$
$359$	12.8417	−	22.2426i	0.677761	−	1.17392i	0.975653	−	0.219318i	$-0.0703831\pi$
$360$		0			0
$361$	−1.27977	−	2.21663i	−0.0673563	−	0.116665i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.7273	−	25.5084i	0.770861	−	1.33517i
$366$		0			0
$367$	8.49197	−	14.7085i	0.443277	−	0.767778i	−0.554653	−	0.832082i	$-0.687149\pi$
$367$	8.49197	−	14.7085i	0.443277	−	0.767778i	0.997930	+	0.0643031i	$0.0204824\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4.38503	+	7.59509i	0.227048	+	0.393259i	0.956932	−	0.290312i	$-0.0937592\pi$
$373$	4.38503	+	7.59509i	0.227048	+	0.393259i	−0.729884	+	0.683571i	$0.760426\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−16.1213			−0.830288
$378$		0			0
$379$	11.7002			0.601001			0.300500	−	0.953782i	$-0.402846\pi$
$379$	11.7002			0.601001			0.300500	+	0.953782i	$0.402846\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.50360	−	7.80046i	−0.230123	−	0.398585i	0.727721	−	0.685873i	$-0.240580\pi$
$383$	−4.50360	−	7.80046i	−0.230123	−	0.398585i	−0.957844	+	0.287288i	$0.907246\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.89390	+	8.47649i	−0.248131	+	0.429775i	−0.963007	−	0.269476i	$-0.913150\pi$
$389$	−4.89390	+	8.47649i	−0.248131	+	0.429775i	0.714876	+	0.699251i	$0.246483\pi$
$390$		0			0
$391$	0.980839	−	1.69886i	0.0496032	−	0.0859152i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.84465	+	6.65912i	0.193445	+	0.335057i
$396$		0			0
$397$	−6.95929	+	12.0538i	−0.349277	+	0.604965i	−0.986121	−	0.166027i	$-0.946906\pi$
$397$	−6.95929	+	12.0538i	−0.349277	+	0.604965i	0.636844	+	0.770992i	$0.280239\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.7414	−	20.3367i	−0.586336	−	1.01556i	−0.994707	−	0.102748i	$-0.967236\pi$
$401$	−11.7414	−	20.3367i	−0.586336	−	1.01556i	0.408371	−	0.912816i	$-0.366097\pi$
$402$		0			0
$403$	1.81852	−	3.14976i	0.0905867	−	0.156901i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.8038	+	41.2295i	1.17991	+	2.04367i
$408$		0			0
$409$	−13.6245			−0.673689			−0.336844	−	0.941560i	$-0.609360\pi$
$409$	−13.6245			−0.673689			−0.336844	+	0.941560i	$0.609360\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.984835	+	1.70578i	−0.0483436	+	0.0837336i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.97733	−	6.88894i	0.194305	−	0.336547i	−0.752367	−	0.658744i	$-0.771088\pi$
$419$	3.97733	−	6.88894i	0.194305	−	0.336547i	0.946673	+	0.322197i	$0.104421\pi$
$420$		0			0
$421$	1.30584	+	2.26178i	0.0636426	+	0.110232i	0.896091	−	0.443870i	$-0.146395\pi$
$421$	1.30584	+	2.26178i	0.0636426	+	0.110232i	−0.832448	+	0.554102i	$0.813062\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	54.1424			2.62629
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.791065	−	1.37017i	−0.0381043	−	0.0659985i	0.846344	−	0.532636i	$-0.178799\pi$
$431$	−0.791065	−	1.37017i	−0.0381043	−	0.0659985i	−0.884449	+	0.466638i	$0.845465\pi$
$432$		0			0
$433$	5.17110			0.248507			0.124254	−	0.992250i	$-0.460346\pi$
$433$	5.17110			0.248507			0.124254	+	0.992250i	$0.460346\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.70777			0.0816939
$438$		0			0
$439$	−24.9611			−1.19133			−0.595665	−	0.803233i	$-0.703111\pi$
$439$	−24.9611			−1.19133			−0.595665	+	0.803233i	$0.703111\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.08453			−0.0515274			−0.0257637	−	0.999668i	$-0.508202\pi$
$443$	−1.08453			−0.0515274			−0.0257637	+	0.999668i	$0.508202\pi$
$444$		0			0
$445$	−47.5649			−2.25479
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.23372			−0.199802			−0.0999008	−	0.994997i	$-0.531853\pi$
$449$	−4.23372			−0.199802			−0.0999008	+	0.994997i	$0.531853\pi$
$450$		0			0
$451$	−3.02108	−	5.23267i	−0.142257	−	0.246397i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−36.3026			−1.69817			−0.849083	−	0.528260i	$-0.822845\pi$
$457$	−36.3026			−1.69817			−0.849083	+	0.528260i	$0.822845\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.71236	+	2.96589i	0.0797524	+	0.138135i	0.903143	−	0.429340i	$-0.141254\pi$
$461$	1.71236	+	2.96589i	0.0797524	+	0.138135i	−0.823391	+	0.567475i	$0.807920\pi$
$462$		0			0
$463$	2.38499	−	4.13092i	0.110840	−	0.191980i	−0.805269	−	0.592909i	$-0.797979\pi$
$463$	2.38499	−	4.13092i	0.110840	−	0.191980i	0.916109	+	0.400929i	$0.131313\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.0490	+	17.4053i	−0.465010	+	0.805422i	−0.999202	−	0.0399417i	$-0.987283\pi$
$467$	−10.0490	+	17.4053i	−0.465010	+	0.805422i	0.534192	+	0.845363i	$0.320616\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	33.4030			1.53587
$474$		0			0
$475$	23.5673	+	40.8198i	1.08134	+	1.87294i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.42528	−	14.5930i	0.384961	−	0.666772i	−0.606803	−	0.794852i	$-0.707548\pi$
$479$	8.42528	−	14.5930i	0.384961	−	0.666772i	0.991764	+	0.128080i	$0.0408816\pi$
$480$		0			0
$481$	−8.61941	−	14.9293i	−0.393011	−	0.680716i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17.3337	+	30.0229i	−0.787084	+	1.36327i
$486$		0			0
$487$	−11.9916	−	20.7700i	−0.543389	−	0.941178i	−0.998706	−	0.0508486i	$-0.983807\pi$
$487$	−11.9916	−	20.7700i	−0.543389	−	0.941178i	0.455317	−	0.890329i	$-0.349526\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.07281	−	1.85816i	0.0484153	−	0.0838577i	−0.840802	−	0.541342i	$-0.817916\pi$
$491$	1.07281	−	1.85816i	0.0484153	−	0.0838577i	0.889217	+	0.457485i	$0.151250\pi$
$492$		0			0
$493$	27.1345	−	46.9983i	1.22207	−	2.11670i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	13.8577	+	24.0023i	0.620357	+	1.07449i	0.989419	+	0.145085i	$0.0463455\pi$
$499$	13.8577	+	24.0023i	0.620357	+	1.07449i	−0.369062	+	0.929405i	$0.620321\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−22.4265			−0.999949			−0.499974	−	0.866040i	$-0.666657\pi$
$503$	−22.4265			−0.999949			−0.499974	+	0.866040i	$0.666657\pi$
$504$		0			0
$505$	42.9090			1.90943
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.6844	+	25.4342i	0.650876	+	1.12735i	0.982911	+	0.184083i	$0.0589316\pi$
$509$	14.6844	+	25.4342i	0.650876	+	1.12735i	−0.332034	+	0.943267i	$0.607735\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.31488	−	9.20563i	0.234201	−	0.405649i
$516$		0			0
$517$	16.6461	−	28.8318i	0.732093	−	1.26802i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.25389	+	14.2962i	0.361610	+	0.626326i	0.988226	−	0.153002i	$-0.0488940\pi$
$521$	8.25389	+	14.2962i	0.361610	+	0.626326i	−0.626616	+	0.779328i	$0.715561\pi$
$522$		0			0
$523$	22.3476	−	38.7072i	0.977193	−	1.69255i	0.304695	−	0.952450i	$-0.401446\pi$
$523$	22.3476	−	38.7072i	0.977193	−	1.69255i	0.672499	−	0.740098i	$-0.265221\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.12166	+	10.6030i	0.266664	+	0.461875i
$528$		0			0
$529$	11.4324	−	19.8014i	0.497059	−	0.860932i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.09394	+	1.89476i	0.0473838	+	0.0820711i
$534$		0			0
$535$	−42.0078			−1.81616
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	10.0369	−	17.3844i	0.431519	−	0.747412i	−0.565486	−	0.824758i	$-0.691311\pi$
$541$	10.0369	−	17.3844i	0.431519	−	0.747412i	0.997004	+	0.0773460i	$0.0246446\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.4307	+	33.6549i	−0.832319	+	1.44162i
$546$		0			0
$547$	6.35012	+	10.9987i	0.271512	+	0.470272i	0.969249	−	0.246082i	$-0.0791432\pi$
$547$	6.35012	+	10.9987i	0.271512	+	0.470272i	−0.697738	+	0.716353i	$0.745810\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	47.2448			2.01270
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.34743	+	12.7261i	0.311321	+	0.539223i	0.978649	−	0.205541i	$-0.0658953\pi$
$557$	7.34743	+	12.7261i	0.311321	+	0.539223i	−0.667328	+	0.744764i	$0.732562\pi$
$558$		0			0
$559$	−12.0953			−0.511576
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−26.7860			−1.12890			−0.564448	−	0.825469i	$-0.690911\pi$
$563$	−26.7860			−1.12890			−0.564448	+	0.825469i	$0.690911\pi$
$564$		0			0
$565$	47.6452			2.00445
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.48336			−0.271797			−0.135898	−	0.990723i	$-0.543392\pi$
$569$	−6.48336			−0.271797			−0.135898	+	0.990723i	$0.543392\pi$
$570$		0			0
$571$	15.6326			0.654205			0.327103	−	0.944989i	$-0.393928\pi$
$571$	15.6326			0.654205			0.327103	+	0.944989i	$0.393928\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.73363			−0.155703
$576$		0			0
$577$	14.5800	+	25.2533i	0.606974	+	1.05131i	0.991736	+	0.128294i	$0.0409500\pi$
$577$	14.5800	+	25.2533i	0.606974	+	1.05131i	−0.384763	+	0.923016i	$0.625717\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.05043			−0.167752
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.33110	+	4.03758i	0.0962146	+	0.166649i	0.910115	−	0.414356i	$-0.135993\pi$
$587$	2.33110	+	4.03758i	0.0962146	+	0.166649i	−0.813900	+	0.581005i	$0.802660\pi$
$588$		0			0
$589$	−5.32932	+	9.23065i	−0.219591	+	0.380342i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.8322	−	27.4222i	0.650150	−	1.12609i	−0.332936	−	0.942950i	$-0.608039\pi$
$593$	15.8322	−	27.4222i	0.650150	−	1.12609i	0.983086	−	0.183144i	$-0.0586275\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.3054			0.421065			0.210533	−	0.977587i	$-0.432480\pi$
$599$	10.3054			0.421065			0.210533	+	0.977587i	$0.432480\pi$
$600$		0			0
$601$	−4.64993	−	8.05391i	−0.189674	−	0.328526i	0.755467	−	0.655186i	$-0.227410\pi$
$601$	−4.64993	−	8.05391i	−0.189674	−	0.328526i	−0.945142	+	0.326661i	$0.894077\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.8532	−	27.4586i	0.644525	−	1.11635i
$606$		0			0
$607$	−10.2484	−	17.7507i	−0.415969	−	0.720480i	0.579561	−	0.814929i	$-0.303224\pi$
$607$	−10.2484	−	17.7507i	−0.415969	−	0.720480i	−0.995530	+	0.0944495i	$0.969891\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.02757	+	10.4401i	−0.243850	+	0.422360i
$612$		0			0
$613$	7.17240	+	12.4230i	0.289691	+	0.501759i	0.973736	−	0.227681i	$-0.0731143\pi$
$613$	7.17240	+	12.4230i	0.289691	+	0.501759i	−0.684045	+	0.729440i	$0.739781\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.47499	−	11.2150i	0.260673	−	0.451499i	−0.705748	−	0.708463i	$-0.749389\pi$
$617$	6.47499	−	11.2150i	0.260673	−	0.451499i	0.966421	+	0.256964i	$0.0827221\pi$
$618$		0			0
$619$	17.6990	−	30.6556i	0.711383	−	1.23215i	−0.252955	−	0.967478i	$-0.581402\pi$
$619$	17.6990	−	30.6556i	0.711383	−	1.23215i	0.964338	−	0.264674i	$-0.0852642\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−13.6460	−	23.6355i	−0.545839	−	0.945422i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	58.0310			2.31385
$630$		0			0
$631$	33.1936			1.32141			0.660707	−	0.750644i	$-0.270256\pi$
$631$	33.1936			1.32141			0.660707	+	0.750644i	$0.270256\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−25.6912	−	44.4985i	−1.01952	−	1.76587i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.119634	+	0.207213i	−0.00472528	+	0.00818442i	−0.868378	−	0.495902i	$-0.834837\pi$
$641$	−0.119634	+	0.207213i	−0.00472528	+	0.00818442i	0.863653	+	0.504087i	$0.168171\pi$
$642$		0			0
$643$	−4.57211	+	7.91913i	−0.180307	+	0.312300i	−0.941985	−	0.335655i	$-0.891042\pi$
$643$	−4.57211	+	7.91913i	−0.180307	+	0.312300i	0.761678	+	0.647955i	$0.224376\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.15607	+	5.46648i	0.124078	+	0.214909i	0.921372	−	0.388682i	$-0.127069\pi$
$647$	3.15607	+	5.46648i	0.124078	+	0.214909i	−0.797294	+	0.603591i	$0.793736\pi$
$648$		0			0
$649$	2.01484	−	3.48980i	0.0790893	−	0.136987i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.27888	−	3.94713i	−0.0891793	−	0.154463i	0.817985	−	0.575239i	$-0.195091\pi$
$653$	−2.27888	−	3.94713i	−0.0891793	−	0.154463i	−0.907164	+	0.420776i	$0.861758\pi$
$654$		0			0
$655$	−9.90015	+	17.1476i	−0.386831	+	0.670011i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16.4631	+	28.5149i	0.641311	+	1.11078i	0.985140	+	0.171751i	$0.0549425\pi$
$659$	16.4631	+	28.5149i	0.641311	+	1.11078i	−0.343829	+	0.939032i	$0.611724\pi$
$660$		0			0
$661$	0.541337			0.0210556			0.0105278	−	0.999945i	$-0.496649\pi$
$661$	0.541337			0.0210556			0.0105278	+	0.999945i	$0.496649\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.87118	+	3.24097i	−0.0724523	+	0.125491i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.3422	−	24.8415i	0.553676	−	0.958994i
$672$		0			0
$673$	−11.8205	−	20.4737i	−0.455647	−	0.789204i	0.543078	−	0.839682i	$-0.317259\pi$
$673$	−11.8205	−	20.4737i	−0.455647	−	0.789204i	−0.998725	+	0.0504780i	$0.983926\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.72989			0.104918			0.0524591	−	0.998623i	$-0.483294\pi$
$677$	2.72989			0.104918			0.0524591	+	0.998623i	$0.483294\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.9640	−	27.6504i	−0.610844	−	1.05801i	−0.991098	−	0.133131i	$-0.957497\pi$
$683$	−15.9640	−	27.6504i	−0.610844	−	1.05801i	0.380254	−	0.924882i	$-0.375836\pi$
$684$		0			0
$685$	51.9417			1.98459
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.46667			0.0558756
$690$		0			0
$691$	−2.38206			−0.0906178			−0.0453089	−	0.998973i	$-0.514427\pi$
$691$	−2.38206			−0.0906178			−0.0453089	+	0.998973i	$0.514427\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−37.8069			−1.43410
$696$		0			0
$697$	−7.36504			−0.278971
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−50.3767			−1.90270			−0.951350	−	0.308111i	$-0.900303\pi$
$701$	−50.3767			−1.90270			−0.951350	+	0.308111i	$0.900303\pi$
$702$		0			0
$703$	25.2599	+	43.7515i	0.952697	+	1.65012i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	40.3201			1.51425			0.757126	−	0.653268i	$-0.226603\pi$
$709$	40.3201			1.51425			0.757126	+	0.653268i	$0.226603\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.422146	−	0.731178i	−0.0158095	−	0.0273828i
$714$		0			0
$715$	−13.4926	+	23.3698i	−0.504593	+	0.873981i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.0446	+	29.5222i	−0.635658	+	1.10099i	0.350718	+	0.936481i	$0.385938\pi$
$719$	−17.0446	+	29.5222i	−0.635658	+	1.10099i	−0.986375	+	0.164510i	$0.947396\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−103.289			−3.83606
$726$		0			0
$727$	10.9453	+	18.9578i	0.405938	+	0.703105i	0.994430	−	0.105398i	$-0.0336117\pi$
$727$	10.9453	+	18.9578i	0.405938	+	0.703105i	−0.588492	+	0.808503i	$0.700278\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	20.3581	−	35.2613i	0.752973	−	1.30419i
$732$		0			0
$733$	4.34416	+	7.52430i	0.160455	+	0.277916i	0.935032	−	0.354563i	$-0.115370\pi$
$733$	4.34416	+	7.52430i	0.160455	+	0.277916i	−0.774577	+	0.632480i	$0.782037\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	32.8351	−	56.8721i	1.20950	−	2.09491i
$738$		0			0
$739$	−3.34692	−	5.79704i	−0.123119	−	0.213248i	0.797877	−	0.602820i	$-0.205956\pi$
$739$	−3.34692	−	5.79704i	−0.123119	−	0.213248i	−0.920996	+	0.389572i	$0.872623\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.5001	+	37.2392i	−0.788761	+	1.36617i	0.137965	+	0.990437i	$0.455944\pi$
$743$	−21.5001	+	37.2392i	−0.788761	+	1.36617i	−0.926726	+	0.375737i	$0.877389\pi$
$744$		0			0
$745$	−26.9404	+	46.6621i	−0.987019	+	1.70957i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8.36369	+	14.4863i	0.305196	+	0.528614i	0.977305	−	0.211838i	$-0.0679448\pi$
$751$	8.36369	+	14.4863i	0.305196	+	0.528614i	−0.672109	+	0.740452i	$0.734612\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−89.0438			−3.24064
$756$		0			0
$757$	−4.68561			−0.170301			−0.0851507	−	0.996368i	$-0.527137\pi$
$757$	−4.68561			−0.170301			−0.0851507	+	0.996368i	$0.527137\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.8242	−	44.7288i	−0.936127	−	1.62142i	−0.772613	−	0.634878i	$-0.781050\pi$
$761$	−25.8242	−	44.7288i	−0.936127	−	1.62142i	−0.163514	−	0.986541i	$-0.552283\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.729576	+	1.26366i	−0.0263435	+	0.0456282i
$768$		0			0
$769$	15.3910	−	26.6580i	0.555014	−	0.961313i	−0.442888	−	0.896577i	$-0.646046\pi$
$769$	15.3910	−	26.6580i	0.555014	−	0.961313i	0.997902	−	0.0647361i	$-0.0206205\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.5259	+	23.4275i	0.486491	+	0.842627i	0.999879	−	0.0155292i	$-0.00494329\pi$
$773$	13.5259	+	23.4275i	0.486491	+	0.842627i	−0.513388	+	0.858156i	$0.671610\pi$
$774$		0			0
$775$	11.6512	−	20.1805i	0.418525	−	0.724907i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.20588	−	5.55275i	−0.114863	−	0.198948i
$780$		0			0
$781$	10.7433	−	18.6080i	0.384426	−	0.665845i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.5370	+	39.0352i	0.804380	+	1.39323i
$786$		0			0
$787$	−35.1995			−1.25473			−0.627363	−	0.778727i	$-0.715866\pi$
$787$	−35.1995			−1.25473			−0.627363	+	0.778727i	$0.715866\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−5.19335	+	8.99514i	−0.184421	+	0.319427i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	23.8268	−	41.2692i	0.843988	−	1.46183i	−0.0425084	−	0.999096i	$-0.513535\pi$
$797$	23.8268	−	41.2692i	0.843988	−	1.46183i	0.886497	−	0.462735i	$-0.153132\pi$
$798$		0			0
$799$	−20.2906	−	35.1443i	−0.717829	−	1.24332i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−33.1102			−1.16843
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.87711	+	17.1077i	0.347261	+	0.601473i	0.985762	−	0.168148i	$-0.0537786\pi$
$809$	9.87711	+	17.1077i	0.347261	+	0.601473i	−0.638501	+	0.769621i	$0.720445\pi$
$810$		0			0
$811$	−49.2424			−1.72913			−0.864567	−	0.502518i	$-0.832407\pi$
$811$	−49.2424			−1.72913			−0.864567	+	0.502518i	$0.832407\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.2793			−0.465155
$816$		0			0
$817$	35.4463			1.24011
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.0003			−0.349012			−0.174506	−	0.984656i	$-0.555833\pi$
$821$	−10.0003			−0.349012			−0.174506	+	0.984656i	$0.555833\pi$
$822$		0			0
$823$	−35.0276			−1.22099			−0.610493	−	0.792021i	$-0.709029\pi$
$823$	−35.0276			−1.22099			−0.610493	+	0.792021i	$0.709029\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.7079			0.789631			0.394816	−	0.918760i	$-0.370808\pi$
$827$	22.7079			0.789631			0.394816	+	0.918760i	$0.370808\pi$
$828$		0			0
$829$	6.22083	+	10.7748i	0.216058	+	0.374224i	0.953599	−	0.301078i	$-0.0973465\pi$
$829$	6.22083	+	10.7748i	0.216058	+	0.374224i	−0.737541	+	0.675302i	$0.764013\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	36.5888			1.26621
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13.8249	+	23.9455i	0.477290	+	0.826690i	0.999661	−	0.0260281i	$-0.00828595\pi$
$839$	13.8249	+	23.9455i	0.477290	+	0.826690i	−0.522372	+	0.852718i	$0.674953\pi$
$840$		0			0
$841$	−37.2652	+	64.5453i	−1.28501	+	2.22570i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−20.4153	+	35.3604i	−0.702309	+	1.21643i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.00178			−0.137179
$852$		0			0
$853$	−22.0459	−	38.1847i	−0.754839	−	1.30742i	−0.945454	−	0.325754i	$-0.894382\pi$
$853$	−22.0459	−	38.1847i	−0.754839	−	1.30742i	0.190616	−	0.981665i	$-0.438952\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13.3838	+	23.1814i	−0.457182	+	0.791862i	−0.998811	−	0.0487557i	$-0.984474\pi$
$857$	−13.3838	+	23.1814i	−0.457182	+	0.791862i	0.541629	+	0.840618i	$0.317808\pi$
$858$		0			0
$859$	10.0951	+	17.4852i	0.344439	+	0.596587i	0.985252	−	0.171111i	$-0.0547357\pi$
$859$	10.0951	+	17.4852i	0.344439	+	0.596587i	−0.640812	+	0.767698i	$0.721402\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.74538	−	8.21923i	0.161534	−	0.279786i	−0.773885	−	0.633327i	$-0.781689\pi$
$863$	4.74538	−	8.21923i	0.161534	−	0.279786i	0.935419	+	0.353541i	$0.115022\pi$
$864$		0			0
$865$	12.4659	+	21.5915i	0.423852	+	0.734133i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.32182	−	7.48560i	0.146608	−	0.253932i
$870$		0			0
$871$	−11.8897	+	20.5935i	−0.402866	+	0.697784i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.6227	−	25.3273i	−0.493774	−	0.855242i	0.506200	−	0.862416i	$-0.331050\pi$
$877$	−14.6227	−	25.3273i	−0.493774	−	0.855242i	−0.999974	+	0.00717380i	$0.997716\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37.2768			−1.25589			−0.627944	−	0.778259i	$-0.716103\pi$
$881$	−37.2768			−1.25589			−0.627944	+	0.778259i	$0.716103\pi$
$882$		0			0
$883$	−56.9436			−1.91630			−0.958152	−	0.286260i	$-0.907588\pi$
$883$	−56.9436			−1.91630			−0.958152	+	0.286260i	$0.907588\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.96127	−	8.59316i	−0.166583	−	0.288530i	0.770633	−	0.637279i	$-0.219940\pi$
$887$	−4.96127	−	8.59316i	−0.166583	−	0.288530i	−0.937216	+	0.348749i	$0.886607\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	17.6643	−	30.5955i	0.591114	−	1.02384i
$894$		0			0
$895$	−33.2431	+	57.5787i	−1.11119	+	1.92465i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.6785	−	20.2277i	−0.389499	−	0.674632i
$900$		0			0
$901$	−2.46862	+	4.27577i	−0.0822416	+	0.142447i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.64664	+	4.58412i	0.0879774	+	0.152381i
$906$		0			0
$907$	12.2887	−	21.2847i	0.408040	−	0.706747i	−0.586630	−	0.809855i	$-0.699546\pi$
$907$	12.2887	−	21.2847i	0.408040	−	0.706747i	0.994670	+	0.103109i	$0.0328789\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.73496	−	16.8614i	−0.322534	−	0.558645i	0.658476	−	0.752601i	$-0.271201\pi$
$911$	−9.73496	−	16.8614i	−0.322534	−	0.558645i	−0.981010	+	0.193957i	$0.937868\pi$
$912$		0			0
$913$	2.21413			0.0732770
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	6.61992	−	11.4660i	0.218371	−	0.378230i	−0.735939	−	0.677048i	$-0.763259\pi$
$919$	6.61992	−	11.4660i	0.218371	−	0.378230i	0.954310	+	0.298818i	$0.0965923\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.89017	+	6.73798i	−0.128047	+	0.221783i
$924$		0			0
$925$	−55.2247	−	95.6519i	−1.81578	−	3.14502i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−31.7363			−1.04124			−0.520618	−	0.853790i	$-0.674298\pi$
$929$	−31.7363			−1.04124			−0.520618	+	0.853790i	$0.674298\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−45.4199	−	78.6696i	−1.48539	−	2.57277i
$936$		0			0
$937$	−13.5019			−0.441087			−0.220543	−	0.975377i	$-0.570783\pi$
$937$	−13.5019			−0.441087			−0.220543	+	0.975377i	$0.570783\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39.6572			1.29279			0.646394	−	0.763004i	$-0.276276\pi$
$941$	39.6572			1.29279			0.646394	+	0.763004i	$0.276276\pi$
$942$		0			0
$943$	0.507889			0.0165391
$944$		0			0
$945$		0			0
$946$		0			0
$947$	61.0344			1.98335			0.991675	−	0.128764i	$-0.0411009\pi$
$947$	61.0344			1.98335			0.991675	+	0.128764i	$0.0411009\pi$
$948$		0			0
$949$	11.9893			0.389188
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.22726			−0.169328			−0.0846638	−	0.996410i	$-0.526982\pi$
$953$	−5.22726			−0.169328			−0.0846638	+	0.996410i	$0.526982\pi$
$954$		0			0
$955$	34.8005	+	60.2762i	1.12612	+	1.95049i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−25.7306			−0.830018
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−25.3269	−	43.8674i	−0.815301	−	1.41214i
$966$		0			0
$967$	10.2035	−	17.6729i	0.328121	−	0.568323i	−0.654018	−	0.756479i	$-0.726918\pi$
$967$	10.2035	−	17.6729i	0.328121	−	0.568323i	0.982139	+	0.188156i	$0.0602511\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.589402	−	1.02087i	0.0189148	−	0.0327614i	−0.856413	−	0.516291i	$-0.827312\pi$
$971$	0.589402	−	1.02087i	0.0189148	−	0.0327614i	0.875328	+	0.483530i	$0.160646\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.20973			−0.262653			−0.131326	−	0.991339i	$-0.541924\pi$
$977$	−8.20973			−0.262653			−0.131326	+	0.991339i	$0.541924\pi$
$978$		0			0
$979$	26.7342	+	46.3049i	0.854427	+	1.47991i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.753481	+	1.30507i	−0.0240323	+	0.0416252i	−0.877791	−	0.479043i	$-0.840984\pi$
$983$	−0.753481	+	1.30507i	−0.0240323	+	0.0416252i	0.853759	+	0.520668i	$0.174317\pi$
$984$		0			0
$985$	−12.3875	−	21.4558i	−0.394698	−	0.683638i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.40389	+	2.43160i	−0.0446410	+	0.0773204i
$990$		0			0
$991$	−16.2229	−	28.0990i	−0.515339	−	0.892593i	−0.999842	−	0.0178030i	$-0.994333\pi$
$991$	−16.2229	−	28.0990i	−0.515339	−	0.892593i	0.484503	−	0.874790i	$-0.339001\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	45.0608	−	78.0476i	1.42852	−	2.47428i
$996$		0			0
$997$	6.26198	−	10.8461i	0.198319	−	0.343498i	−0.749665	−	0.661818i	$-0.769785\pi$
$997$	6.26198	−	10.8461i	0.198319	−	0.343498i	0.947983	+	0.318320i	$0.103119\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.i.j.1549.12		24
3.2	odd	2		1764.2.i.j.373.7		24
7.2	even	3		5292.2.j.i.1765.12		24
7.3	odd	6		5292.2.l.j.361.12		24
7.4	even	3		5292.2.l.j.361.1		24
7.5	odd	6		5292.2.j.i.1765.1		24
7.6	odd	2	inner	5292.2.i.j.1549.1		24
9.2	odd	6		1764.2.l.j.961.2		24
9.7	even	3		5292.2.l.j.3313.1		24
21.2	odd	6		1764.2.j.i.589.10	yes	24
21.5	even	6		1764.2.j.i.589.3	✓	24
21.11	odd	6		1764.2.l.j.949.2		24
21.17	even	6		1764.2.l.j.949.11		24
21.20	even	2		1764.2.i.j.373.6		24
63.2	odd	6		1764.2.j.i.1177.10	yes	24
63.11	odd	6		1764.2.i.j.1537.7		24
63.16	even	3		5292.2.j.i.3529.12		24
63.20	even	6		1764.2.l.j.961.11		24
63.25	even	3	inner	5292.2.i.j.2125.12		24
63.34	odd	6		5292.2.l.j.3313.12		24
63.38	even	6		1764.2.i.j.1537.6		24
63.47	even	6		1764.2.j.i.1177.3	yes	24
63.52	odd	6	inner	5292.2.i.j.2125.1		24
63.61	odd	6		5292.2.j.i.3529.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.j.373.6		24	21.20	even	2
1764.2.i.j.373.7		24	3.2	odd	2
1764.2.i.j.1537.6		24	63.38	even	6
1764.2.i.j.1537.7		24	63.11	odd	6
1764.2.j.i.589.3	✓	24	21.5	even	6
1764.2.j.i.589.10	yes	24	21.2	odd	6
1764.2.j.i.1177.3	yes	24	63.47	even	6
1764.2.j.i.1177.10	yes	24	63.2	odd	6
1764.2.l.j.949.2		24	21.11	odd	6
1764.2.l.j.949.11		24	21.17	even	6
1764.2.l.j.961.2		24	9.2	odd	6
1764.2.l.j.961.11		24	63.20	even	6
5292.2.i.j.1549.1		24	7.6	odd	2	inner
5292.2.i.j.1549.12		24	1.1	even	1	trivial
5292.2.i.j.2125.1		24	63.52	odd	6	inner
5292.2.i.j.2125.12		24	63.25	even	3	inner
5292.2.j.i.1765.1		24	7.5	odd	6
5292.2.j.i.1765.12		24	7.2	even	3
5292.2.j.i.3529.1		24	63.61	odd	6
5292.2.j.i.3529.12		24	63.16	even	3
5292.2.l.j.361.1		24	7.4	even	3
5292.2.l.j.361.12		24	7.3	odd	6
5292.2.l.j.3313.1		24	9.7	even	3
5292.2.l.j.3313.12		24	63.34	odd	6

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.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.94623 + 3.37097i) q^{5} +O(q^{10})\)	\(q+(1.94623 + 3.37097i) q^{5} +(2.18778 - 3.78935i) q^{11} +(-0.792201 + 1.37213i) q^{13} +(-2.66678 - 4.61900i) q^{17} +(2.32161 - 4.02115i) q^{19} +(0.183900 + 0.318523i) q^{23} +(-5.07564 + 8.79126i) q^{25} +(5.08750 + 8.81180i) q^{29} -2.29552 q^{31} +(-5.44017 + 9.42265i) q^{37} +(0.690443 - 1.19588i) q^{41} +(3.81699 + 6.61122i) q^{43} +7.60865 q^{47} +(-0.462847 - 0.801674i) q^{53} +17.0317 q^{55} +0.920949 q^{59} +6.55560 q^{61} -6.16722 q^{65} +15.0084 q^{67} +4.91059 q^{71} +(-3.78353 - 6.55327i) q^{73} +1.97543 q^{79} +(0.253011 + 0.438227i) q^{83} +(10.3803 - 17.9793i) q^{85} +(-6.10987 + 10.5826i) q^{89} +18.0736 q^{95} +(4.45315 + 7.71308i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} - 72 q^{65} - 24 q^{67} - 48 q^{71} - 24 q^{79} + 12 q^{85} + 64 q^{95}+O(q^{100}) \) 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 - 72 * q^65 - 24 * q^67 - 48 * q^71 - 24 * q^79 + 12 * q^85 + 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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