LMFDB - Embedded newform 5292.2.j.i.1765.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1765,5292)]

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, 0]))

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

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

chi := DirichletCharacter("5292.1765");

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.j (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			1765.12
Character	$\chi$	$=$	5292.1765
Dual form			5292.2.j.i.3529.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.94623 - 3.37097i) q^{5} +O(q^{10})$	$q+(1.94623 - 3.37097i) q^{5} +(2.18778 + 3.78935i) q^{11} +(-0.792201 + 1.37213i) q^{13} +5.33356 q^{17} -4.64323 q^{19} +(0.183900 - 0.318523i) q^{23} +(-5.07564 - 8.79126i) q^{25} +(5.08750 + 8.81180i) q^{29} +(1.14776 - 1.98798i) q^{31} +10.8803 q^{37} +(0.690443 - 1.19588i) q^{41} +(3.81699 + 6.61122i) q^{43} +(-3.80432 - 6.58928i) q^{47} +0.925693 q^{53} +17.0317 q^{55} +(-0.460475 + 0.797565i) q^{59} +(-3.27780 - 5.67731i) q^{61} +(3.08361 + 5.34097i) q^{65} +(-7.50420 + 12.9976i) q^{67} +4.91059 q^{71} +7.56707 q^{73} +(-0.987715 - 1.71077i) q^{79} +(0.253011 + 0.438227i) q^{83} +(10.3803 - 17.9793i) q^{85} +12.2197 q^{89} +(-9.03679 + 15.6522i) q^{95} +(4.45315 + 7.71308i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} + 24 q^{37} - 32 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) $ 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 + 24 * q^37 - 32 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

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	0.00877856	−	0.999961i	$-0.497206\pi$
$5$	1.94623	−	3.37097i	0.870381	−	1.50754i	0.861603	−	0.507583i	$-0.169461\pi$
$6$		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	0.980708	+	0.195476i	$0.0626253\pi$
$11$	2.18778	+	3.78935i	0.659642	+	1.14253i	−0.321067	+	0.947057i	$0.604041\pi$
$12$		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
$17$	5.33356			1.29358			0.646789	−	0.762669i	$-0.276111\pi$
$17$	5.33356			1.29358			0.646789	+	0.762669i	$0.276111\pi$
$18$		0			0
$19$	−4.64323			−1.06523			−0.532614	−	0.846358i	$-0.678790\pi$
$19$	−4.64323			−1.06523			−0.532614	+	0.846358i	$0.678790\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.183900	−	0.318523i	0.0383457	−	0.0664167i	−0.846216	−	0.532841i	$-0.821125\pi$
$23$	0.183900	−	0.318523i	0.0383457	−	0.0664167i	0.884561	+	0.466424i	$0.154458\pi$
$24$		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
$31$	1.14776	−	1.98798i	0.206144	−	0.357052i	−0.744353	−	0.667787i	$-0.767242\pi$
$31$	1.14776	−	1.98798i	0.206144	−	0.357052i	0.950497	+	0.310735i	$0.100575\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10.8803			1.78872			0.894359	−	0.447351i	$-0.147632\pi$
$37$	10.8803			1.78872			0.894359	+	0.447351i	$0.147632\pi$
$38$		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
$47$	−3.80432	−	6.58928i	−0.554918	−	0.961145i	−0.997910	−	0.0646200i	$-0.979416\pi$
$47$	−3.80432	−	6.58928i	−0.554918	−	0.961145i	0.442992	−	0.896525i	$-0.353917\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.925693			0.127154			0.0635769	−	0.997977i	$-0.479749\pi$
$53$	0.925693			0.127154			0.0635769	+	0.997977i	$0.479749\pi$
$54$		0			0
$55$	17.0317			2.29656
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.460475	+	0.797565i	−0.0599487	+	0.103834i	−0.894442	−	0.447184i	$-0.852427\pi$
$59$	−0.460475	+	0.797565i	−0.0599487	+	0.103834i	0.834493	+	0.551018i	$0.185760\pi$
$60$		0			0
$61$	−3.27780	−	5.67731i	−0.419679	−	0.726905i	0.576228	−	0.817289i	$-0.304524\pi$
$61$	−3.27780	−	5.67731i	−0.419679	−	0.726905i	−0.995907	+	0.0903836i	$0.971191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.08361	+	5.34097i	0.382475	+	0.662466i
$66$		0			0
$67$	−7.50420	+	12.9976i	−0.916783	+	1.58792i	−0.112514	+	0.993650i	$0.535890\pi$
$67$	−7.50420	+	12.9976i	−0.916783	+	1.58792i	−0.804269	+	0.594265i	$0.797443\pi$
$68$		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
$73$	7.56707			0.885658			0.442829	−	0.896606i	$-0.353975\pi$
$73$	7.56707			0.885658			0.442829	+	0.896606i	$0.353975\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.987715	−	1.71077i	−0.111127	−	0.192477i	0.805098	−	0.593142i	$-0.202113\pi$
$79$	−0.987715	−	1.71077i	−0.111127	−	0.192477i	−0.916225	+	0.400665i	$0.868779\pi$
$80$		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
$89$	12.2197			1.29529			0.647645	−	0.761942i	$-0.275754\pi$
$89$	12.2197			1.29529			0.647645	+	0.761942i	$0.275754\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.03679	+	15.6522i	−0.927155	+	1.60588i
$96$		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	0.998381	+	0.0568740i	$0.0181133\pi$
$101$	5.51180	+	9.54672i	0.548445	+	0.949935i	−0.449936	+	0.893061i	$0.648553\pi$
$102$		0			0
$103$	−1.36543	+	2.36499i	−0.134540	+	0.233029i	−0.925421	−	0.378939i	$-0.876289\pi$
$103$	−1.36543	+	2.36499i	−0.134540	+	0.233029i	0.790882	+	0.611969i	$0.209622\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.7921			1.04331			0.521655	−	0.853156i	$-0.325315\pi$
$107$	10.7921			1.04331			0.521655	+	0.853156i	$0.325315\pi$
$108$		0			0
$109$	−9.98374			−0.956269			−0.478134	−	0.878287i	$-0.658687\pi$
$109$	−9.98374			−0.956269			−0.478134	+	0.878287i	$0.658687\pi$
$110$		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	−0.733262	−	0.679946i	$-0.762003\pi$
$131$	2.54342	−	4.40532i	0.222219	−	0.384895i	0.955482	+	0.295051i	$0.0953366\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.67208	+	11.5564i	0.570034	+	0.987328i	0.996562	+	0.0828538i	$0.0264034\pi$
$137$	6.67208	+	11.5564i	0.570034	+	0.987328i	−0.426527	+	0.904475i	$0.640263\pi$
$138$		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
$143$	−6.93265			−0.579738
$144$		0			0
$145$	39.6058			3.28908
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.92116	−	11.9878i	0.567004	−	0.982079i	−0.429856	−	0.902897i	$-0.641436\pi$
$149$	6.92116	−	11.9878i	0.567004	−	0.982079i	0.996860	−	0.0791820i	$-0.0252308\pi$
$150$		0			0
$151$	−11.4380	−	19.8112i	−0.930809	−	1.61221i	−0.781942	−	0.623352i	$-0.785771\pi$
$151$	−11.4380	−	19.8112i	−0.930809	−	1.61221i	−0.148867	−	0.988857i	$-0.547563\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.46762	−	7.73815i	−0.358848	−	0.621543i
$156$		0			0
$157$	−5.78991	+	10.0284i	−0.462085	+	0.800355i	−0.999065	−	0.0432404i	$-0.986232\pi$
$157$	−5.78991	+	10.0284i	−0.462085	+	0.800355i	0.536980	+	0.843595i	$0.319565\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3.41155			0.267213			0.133607	−	0.991034i	$-0.457344\pi$
$163$	3.41155			0.267213			0.133607	+	0.991034i	$0.457344\pi$
$164$		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
$173$	−3.20256	−	5.54700i	−0.243486	−	0.421730i	0.718219	−	0.695817i	$-0.244958\pi$
$173$	−3.20256	−	5.54700i	−0.243486	−	0.421730i	−0.961705	+	0.274087i	$0.911624\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−17.0808			−1.27668			−0.638338	−	0.769756i	$-0.720378\pi$
$179$	−17.0808			−1.27668			−0.638338	+	0.769756i	$0.720378\pi$
$180$		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
$185$	21.1757	−	36.6773i	1.55687	−	2.69657i
$186$		0			0
$187$	11.6687	+	20.2107i	0.853298	+	1.47796i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.94048	−	15.4854i	−0.646911	−	1.12048i	−0.983857	−	0.178958i	$-0.942727\pi$
$191$	−8.94048	−	15.4854i	−0.646911	−	1.12048i	0.336946	−	0.941524i	$-0.390606\pi$
$192$		0			0
$193$	6.50664	−	11.2698i	0.468358	−	0.811220i	−0.530988	−	0.847380i	$-0.678179\pi$
$193$	6.50664	−	11.2698i	0.468358	−	0.811220i	0.999346	+	0.0361591i	$0.0115123\pi$
$194$		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
$199$	23.1529			1.64126			0.820631	−	0.571458i	$-0.193622\pi$
$199$	23.1529			1.64126			0.820631	+	0.571458i	$0.193622\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.68753	−	4.65493i	−0.187705	−	0.325114i
$206$		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
$215$	29.7150			2.02655
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.22525	+	7.31835i	−0.284221	+	0.492285i
$222$		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	−0.998204	−	0.0599042i	$-0.980920\pi$
$227$	−8.30136	−	14.3784i	−0.550981	−	0.954326i	0.447224	−	0.894422i	$-0.352413\pi$
$228$		0			0
$229$	7.25072	−	12.5586i	0.479141	−	0.829897i	−0.520573	−	0.853817i	$-0.674282\pi$
$229$	7.25072	−	12.5586i	0.479141	−	0.829897i	0.999714	+	0.0239205i	$0.00761485\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.9804			−1.04691			−0.523456	−	0.852053i	$-0.675357\pi$
$233$	−15.9804			−1.04691			−0.523456	+	0.852053i	$0.675357\pi$
$234$		0			0
$235$	−29.6164			−1.93196
$236$		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	−0.946717	−	0.322067i	$-0.895623\pi$
$241$	−11.6785	−	20.2277i	−0.752276	−	1.30298i	0.194441	−	0.980914i	$-0.437711\pi$
$242$		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	−0.952453	−	0.304687i	$-0.901448\pi$
$257$	−3.40438	+	5.89657i	−0.212360	+	0.367818i	0.740093	+	0.672505i	$0.234782\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.30000	−	3.98372i	−0.141824	−	0.245647i	0.786359	−	0.617769i	$-0.211963\pi$
$263$	−2.30000	−	3.98372i	−0.141824	−	0.245647i	−0.928184	+	0.372122i	$0.878630\pi$
$264$		0			0
$265$	1.80161	−	3.12049i	0.110672	−	0.191690i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.50179			0.518363			0.259182	−	0.965829i	$-0.416547\pi$
$269$	8.50179			0.518363			0.259182	+	0.965829i	$0.416547\pi$
$270$		0			0
$271$	6.58381			0.399938			0.199969	−	0.979802i	$-0.435916\pi$
$271$	6.58381			0.399938			0.199969	+	0.979802i	$0.435916\pi$
$272$		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	0.984163	−	0.177266i	$-0.0567254\pi$
$277$	5.63483	+	9.75982i	0.338564	+	0.586411i	−0.645599	+	0.763677i	$0.723392\pi$
$278$		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
$283$	−7.33657	+	12.7073i	−0.436114	+	0.755371i	−0.997386	−	0.0722602i	$-0.976979\pi$
$283$	−7.33657	+	12.7073i	−0.436114	+	0.755371i	0.561272	+	0.827631i	$0.310312\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	11.4469			0.673344
$290$		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
$299$	0.291371	+	0.504669i	0.0168504	+	0.0291858i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−25.5174			−1.46112
$306$		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
$311$	−6.79681	+	11.7724i	−0.385412	+	0.667553i	−0.991826	−	0.127596i	$-0.959274\pi$
$311$	−6.79681	+	11.7724i	−0.385412	+	0.667553i	0.606414	+	0.795149i	$0.292607\pi$
$312$		0			0
$313$	6.93222	+	12.0070i	0.391832	+	0.678673i	0.992691	−	0.120682i	$-0.0385080\pi$
$313$	6.93222	+	12.0070i	0.391832	+	0.678673i	−0.600859	+	0.799355i	$0.705175\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.5428	+	19.9927i	0.648309	+	1.12290i	0.983527	+	0.180764i	$0.0578569\pi$
$317$	11.5428	+	19.9927i	0.648309	+	1.12290i	−0.335217	+	0.942141i	$0.608810\pi$
$318$		0			0
$319$	−22.2607	+	38.5566i	−1.24636	+	2.15876i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.7649			−1.37796
$324$		0			0
$325$	16.0837			0.892163
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.82647	+	13.5558i	0.430182	+	0.745097i	0.996889	−	0.0788227i	$-0.0251161\pi$
$331$	7.82647	+	13.5558i	0.430182	+	0.745097i	−0.566707	+	0.823920i	$0.691783\pi$
$332$		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
$341$	10.0442			0.543925
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.77827	−	4.81211i	0.149146	−	0.258328i	−0.781766	−	0.623571i	$-0.785681\pi$
$347$	2.77827	−	4.81211i	0.149146	−	0.258328i	0.930912	+	0.365244i	$0.119014\pi$
$348$		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	0.642989	−	0.765875i	$-0.277694\pi$
$353$	−6.42132	−	11.1221i	−0.341772	−	0.591967i	−0.984762	+	0.173908i	$0.944361\pi$
$354$		0			0
$355$	9.55715	−	16.5535i	0.507241	−	0.878567i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.6835			−1.35552			−0.677761	−	0.735282i	$-0.737050\pi$
$359$	−25.6835			−1.35552			−0.677761	+	0.735282i	$0.737050\pi$
$360$		0			0
$361$	2.55954			0.134713
$362$		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	0.997930	−	0.0643031i	$-0.0204824\pi$
$367$	8.49197	+	14.7085i	0.443277	+	0.767778i	−0.554653	+	0.832082i	$0.687149\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4.38503	−	7.59509i	0.227048	−	0.393259i	−0.729884	−	0.683571i	$-0.760426\pi$
$373$	4.38503	−	7.59509i	0.227048	−	0.393259i	0.956932	+	0.290312i	$0.0937592\pi$
$374$		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	−0.957844	−	0.287288i	$-0.907246\pi$
$383$	−4.50360	+	7.80046i	−0.230123	+	0.398585i	0.727721	+	0.685873i	$0.240580\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.89390	−	8.47649i	−0.248131	−	0.429775i	0.714876	−	0.699251i	$-0.246483\pi$
$389$	−4.89390	−	8.47649i	−0.248131	−	0.429775i	−0.963007	+	0.269476i	$0.913150\pi$
$390$		0			0
$391$	0.980839	−	1.69886i	0.0496032	−	0.0859152i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.68929			−0.386890
$396$		0			0
$397$	13.9186			0.698554			0.349277	−	0.937020i	$-0.386427\pi$
$397$	13.9186			0.698554			0.349277	+	0.937020i	$0.386427\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.7414	+	20.3367i	−0.586336	+	1.01556i	0.408371	+	0.912816i	$0.366097\pi$
$401$	−11.7414	+	20.3367i	−0.586336	+	1.01556i	−0.994707	+	0.102748i	$0.967236\pi$
$402$		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
$409$	6.81225	−	11.7992i	0.336844	−	0.583431i	−0.646993	−	0.762496i	$-0.723974\pi$
$409$	6.81225	−	11.7992i	0.336844	−	0.583431i	0.983837	+	0.179064i	$0.0573071\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.96967			0.0966873
$416$		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
$425$	−27.0712	−	46.8887i	−1.31315	−	2.27444i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.58213			0.0762086			0.0381043	−	0.999274i	$-0.487868\pi$
$431$	1.58213			0.0762086			0.0381043	+	0.999274i	$0.487868\pi$
$432$		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
$437$	−0.853887	+	1.47898i	−0.0408470	+	0.0707490i
$438$		0			0
$439$	12.4806	+	21.6170i	0.595665	+	1.03172i	0.993453	+	0.114244i	$0.0364446\pi$
$439$	12.4806	+	21.6170i	0.595665	+	1.03172i	−0.397788	+	0.917477i	$0.630222\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.542263	+	0.939227i	0.0257637	+	0.0446240i	0.878620	−	0.477522i	$-0.158465\pi$
$443$	0.542263	+	0.939227i	0.0257637	+	0.0446240i	−0.852856	+	0.522146i	$0.825132\pi$
$444$		0			0
$445$	23.7825	−	41.1924i	1.12740	−	1.95271i
$446$		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
$451$	6.04216			0.284514
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	18.1513	+	31.4390i	0.849083	+	1.47065i	0.882028	+	0.471197i	$0.156178\pi$
$457$	18.1513	+	31.4390i	0.849083	+	1.47065i	−0.0329453	+	0.999457i	$0.510489\pi$
$458$		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
$467$	20.0979			0.930021			0.465010	−	0.885305i	$-0.346051\pi$
$467$	20.0979			0.930021			0.465010	+	0.885305i	$0.346051\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16.7015	+	28.9279i	−0.767936	+	1.33010i
$474$		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	0.991764	−	0.128080i	$-0.0408816\pi$
$479$	8.42528	+	14.5930i	0.384961	+	0.666772i	−0.606803	+	0.794852i	$0.707548\pi$
$480$		0			0
$481$	−8.61941	+	14.9293i	−0.393011	+	0.680716i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	34.6675			1.57417
$486$		0			0
$487$	23.9831			1.08678			0.543389	−	0.839481i	$-0.317141\pi$
$487$	23.9831			1.08678			0.543389	+	0.839481i	$0.317141\pi$
$488$		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	−0.369062	−	0.929405i	$-0.620321\pi$
$499$	13.8577	−	24.0023i	0.620357	−	1.07449i	0.989419	−	0.145085i	$-0.0463455\pi$
$500$		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	−0.332034	−	0.943267i	$-0.607735\pi$
$509$	14.6844	−	25.4342i	0.650876	−	1.12735i	0.982911	−	0.184083i	$-0.0589316\pi$
$510$		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
$521$	−16.5078			−0.723219			−0.361610	−	0.932330i	$-0.617773\pi$
$521$	−16.5078			−0.723219			−0.361610	+	0.932330i	$0.617773\pi$
$522$		0			0
$523$	−44.6952			−1.95439			−0.977193	−	0.212352i	$-0.931888\pi$
$523$	−44.6952			−1.95439			−0.977193	+	0.212352i	$0.931888\pi$
$524$		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
$535$	21.0039	−	36.3798i	0.908078	−	1.57284i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−20.0737			−0.863037			−0.431519	−	0.902104i	$-0.642022\pi$
$541$	−20.0737			−0.863037			−0.431519	+	0.902104i	$0.642022\pi$
$542$		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
$551$	−23.6224	−	40.9152i	−1.00635	−	1.74305i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.6949			−0.622642			−0.311321	−	0.950305i	$-0.600771\pi$
$557$	−14.6949			−0.622642			−0.311321	+	0.950305i	$0.600771\pi$
$558$		0			0
$559$	−12.0953			−0.511576
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.3930	−	23.1974i	0.564448	−	0.977653i	−0.432653	−	0.901561i	$-0.642422\pi$
$563$	13.3930	−	23.1974i	0.564448	−	0.977653i	0.997101	−	0.0760922i	$-0.0242443\pi$
$564$		0			0
$565$	−23.8226	−	41.2620i	−1.00222	−	1.73590i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.24168	+	5.61475i	0.135898	+	0.235383i	0.925940	−	0.377670i	$-0.123275\pi$
$569$	3.24168	+	5.61475i	0.135898	+	0.235383i	−0.790042	+	0.613053i	$0.789941\pi$
$570$		0			0
$571$	−7.81632	+	13.5383i	−0.327103	+	0.566559i	−0.981936	−	0.189215i	$-0.939406\pi$
$571$	−7.81632	+	13.5383i	−0.327103	+	0.566559i	0.654833	+	0.755774i	$0.272739\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.73363			−0.155703
$576$		0			0
$577$	−29.1600			−1.21395			−0.606974	−	0.794722i	$-0.707617\pi$
$577$	−29.1600			−1.21395			−0.606974	+	0.794722i	$0.707617\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.02522	+	3.50778i	0.0838759	+	0.145277i
$584$		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
$593$	−31.6644			−1.30030			−0.650150	−	0.759805i	$-0.725294\pi$
$593$	−31.6644			−1.30030			−0.650150	+	0.759805i	$0.725294\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5.15268	+	8.92470i	−0.210533	+	0.364653i	−0.951881	−	0.306467i	$-0.900853\pi$
$599$	−5.15268	+	8.92470i	−0.210533	+	0.364653i	0.741349	+	0.671120i	$0.234187\pi$
$600$		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	−0.995530	−	0.0944495i	$-0.969891\pi$
$607$	−10.2484	+	17.7507i	−0.415969	+	0.720480i	0.579561	+	0.814929i	$0.303224\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.0551			0.487699
$612$		0			0
$613$	−14.3448			−0.579381			−0.289691	−	0.957120i	$-0.593552\pi$
$613$	−14.3448			−0.579381			−0.289691	+	0.957120i	$0.593552\pi$
$614$		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	0.964338	+	0.264674i	$0.0852642\pi$
$619$	17.6990	+	30.6556i	0.711383	+	1.23215i	−0.252955	+	0.967478i	$0.581402\pi$
$620$		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	0.863653	−	0.504087i	$-0.168171\pi$
$641$	−0.119634	−	0.207213i	−0.00472528	−	0.00818442i	−0.868378	+	0.495902i	$0.834837\pi$
$642$		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
$647$	−6.31214			−0.248156			−0.124078	−	0.992272i	$-0.539597\pi$
$647$	−6.31214			−0.248156			−0.124078	+	0.992272i	$0.539597\pi$
$648$		0			0
$649$	−4.02968			−0.158179
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.27888	+	3.94713i	−0.0891793	+	0.154463i	−0.907164	−	0.420776i	$-0.861758\pi$
$653$	−2.27888	+	3.94713i	−0.0891793	+	0.154463i	0.817985	+	0.575239i	$0.195091\pi$
$654$		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
$661$	−0.270668	+	0.468811i	−0.0105278	+	0.0182346i	−0.871241	−	0.490855i	$-0.836684\pi$
$661$	−0.270668	+	0.468811i	−0.0105278	+	0.0182346i	0.860714	+	0.509090i	$0.170018\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	3.74235			0.144905
$668$		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
$677$	−1.36494	−	2.36415i	−0.0524591	−	0.0908618i	0.838603	−	0.544742i	$-0.183373\pi$
$677$	−1.36494	−	2.36415i	−0.0524591	−	0.0908618i	−0.891062	+	0.453881i	$0.850039\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	31.9279			1.22169			0.610844	−	0.791751i	$-0.290830\pi$
$683$	31.9279			1.22169			0.610844	+	0.791751i	$0.290830\pi$
$684$		0			0
$685$	51.9417			1.98459
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.733335	+	1.27017i	−0.0279378	+	0.0483897i
$690$		0			0
$691$	1.19103	+	2.06292i	0.0453089	+	0.0784773i	0.887790	−	0.460248i	$-0.152239\pi$
$691$	1.19103	+	2.06292i	0.0453089	+	0.0784773i	−0.842482	+	0.538725i	$0.818906\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.9034	+	32.7417i	0.717048	+	1.24196i
$696$		0			0
$697$	3.68252	−	6.37831i	0.139485	−	0.241596i
$698$		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
$703$	−50.5199			−1.90539
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−20.1600	−	34.9182i	−0.757126	−	1.31138i	−0.944310	−	0.329056i	$-0.893269\pi$
$709$	−20.1600	−	34.9182i	−0.757126	−	1.31138i	0.187184	−	0.982325i	$-0.440064\pi$
$710$		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
$719$	34.0893			1.27132			0.635658	−	0.771971i	$-0.280729\pi$
$719$	34.0893			1.27132			0.635658	+	0.771971i	$0.280729\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	51.6446	−	89.4510i	1.91803	−	3.32213i
$726$		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	−0.774577	−	0.632480i	$-0.782037\pi$
$733$	4.34416	−	7.52430i	0.160455	−	0.277916i	0.935032	+	0.354563i	$0.115370\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−65.6702			−2.41899
$738$		0			0
$739$	6.69385			0.246237			0.123119	−	0.992392i	$-0.460710\pi$
$739$	6.69385			0.246237			0.123119	+	0.992392i	$0.460710\pi$
$740$		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	−0.672109	−	0.740452i	$-0.734612\pi$
$751$	8.36369	−	14.4863i	0.305196	−	0.528614i	0.977305	+	0.211838i	$0.0679448\pi$
$752$		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	−0.163514	+	0.986541i	$0.552283\pi$
$761$	−25.8242	+	44.7288i	−0.936127	+	1.62142i	−0.772613	+	0.634878i	$0.781050\pi$
$762$		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
$773$	−27.0517			−0.972982			−0.486491	−	0.873686i	$-0.661723\pi$
$773$	−27.0517			−0.972982			−0.486491	+	0.873686i	$0.661723\pi$
$774$		0			0
$775$	−23.3025			−0.837050
$776$		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
$787$	17.5997	−	30.4837i	0.627363	−	1.08662i	−0.360716	−	0.932676i	$-0.617468\pi$
$787$	17.5997	−	30.4837i	0.627363	−	1.08662i	0.988079	−	0.153949i	$-0.0491992\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	10.3867			0.368842
$794$		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
$803$	16.5551	+	28.6743i	0.584217	+	1.01189i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−19.7542			−0.694521			−0.347261	−	0.937769i	$-0.612888\pi$
$809$	−19.7542			−0.694521			−0.347261	+	0.937769i	$0.612888\pi$
$810$		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
$815$	6.63967	−	11.5002i	0.232577	−	0.402836i
$816$		0			0
$817$	−17.7231	−	30.6974i	−0.620054	−	1.07397i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.00013	+	8.66048i	0.174506	+	0.302253i	0.939990	−	0.341202i	$-0.110834\pi$
$821$	5.00013	+	8.66048i	0.174506	+	0.302253i	−0.765484	+	0.643455i	$0.777501\pi$
$822$		0			0
$823$	17.5138	−	30.3348i	0.610493	−	1.05741i	−0.380664	−	0.924713i	$-0.624305\pi$
$823$	17.5138	−	30.3348i	0.610493	−	1.05741i	0.991157	−	0.132692i	$-0.0423621\pi$
$824$		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
$829$	−12.4417			−0.432117			−0.216058	−	0.976380i	$-0.569320\pi$
$829$	−12.4417			−0.432117			−0.216058	+	0.976380i	$0.569320\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−18.2944	−	31.6869i	−0.633105	−	1.09657i
$836$		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
$845$	40.8307			1.40462
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.00089	−	3.46564i	0.0685896	−	0.118801i
$852$		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	0.541629	−	0.840618i	$-0.317808\pi$
$857$	−13.3838	−	23.1814i	−0.457182	−	0.791862i	−0.998811	+	0.0487557i	$0.984474\pi$
$858$		0			0
$859$	10.0951	−	17.4852i	0.344439	−	0.596587i	−0.640812	−	0.767698i	$-0.721402\pi$
$859$	10.0951	−	17.4852i	0.344439	−	0.596587i	0.985252	+	0.171111i	$0.0547357\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.49075			−0.323069			−0.161534	−	0.986867i	$-0.551644\pi$
$863$	−9.49075			−0.323069			−0.161534	+	0.986867i	$0.551644\pi$
$864$		0			0
$865$	−24.9317			−0.847703
$866$		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	−0.999974	−	0.00717380i	$-0.997716\pi$
$877$	−14.6227	+	25.3273i	−0.493774	+	0.855242i	0.506200	+	0.862416i	$0.331050\pi$
$878$		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	−0.937216	−	0.348749i	$-0.886607\pi$
$887$	−4.96127	+	8.59316i	−0.166583	+	0.288530i	0.770633	+	0.637279i	$0.219940\pi$
$888$		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
$899$	23.3569			0.778997
$900$		0			0
$901$	4.93724			0.164483
$902$		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	0.994670	−	0.103109i	$-0.0328789\pi$
$907$	12.2887	+	21.2847i	0.408040	+	0.706747i	−0.586630	+	0.809855i	$0.699546\pi$
$908$		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
$913$	−1.10707	+	1.91749i	−0.0366385	+	0.0634598i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−13.2398			−0.436742			−0.218371	−	0.975866i	$-0.570074\pi$
$919$	−13.2398			−0.436742			−0.218371	+	0.975866i	$0.570074\pi$
$920$		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
$929$	15.8682	+	27.4845i	0.520618	+	0.901737i	0.999713	+	0.0239734i	$0.00763170\pi$
$929$	15.8682	+	27.4845i	0.520618	+	0.901737i	−0.479095	+	0.877763i	$0.659035\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	90.8398			2.97078
$936$		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
$941$	−19.8286	+	34.3441i	−0.646394	+	1.11959i	0.337584	+	0.941295i	$0.390390\pi$
$941$	−19.8286	+	34.3441i	−0.646394	+	1.11959i	−0.983978	+	0.178291i	$0.942943\pi$
$942$		0			0
$943$	−0.253944	−	0.439845i	−0.00826957	−	0.0143233i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30.5172	−	52.8573i	−0.991675	−	1.71763i	−0.607350	−	0.794434i	$-0.707768\pi$
$947$	−30.5172	−	52.8573i	−0.991675	−	1.71763i	−0.384325	−	0.923198i	$-0.625566\pi$
$948$		0			0
$949$	−5.99464	+	10.3830i	−0.194594	+	0.337047i
$950$		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
$955$	−69.6010			−2.25224
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	12.8653	+	22.2833i	0.415009	+	0.718817i
$962$		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
$971$	−1.17880			−0.0378296			−0.0189148	−	0.999821i	$-0.506021\pi$
$971$	−1.17880			−0.0378296			−0.0189148	+	0.999821i	$0.506021\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.10487	−	7.10984i	0.131326	−	0.227464i	−0.792862	−	0.609402i	$-0.791410\pi$
$977$	4.10487	−	7.10984i	0.131326	−	0.227464i	0.924188	+	0.381938i	$0.124743\pi$
$978$		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	0.853759	−	0.520668i	$-0.174317\pi$
$983$	−0.753481	−	1.30507i	−0.0240323	−	0.0416252i	−0.877791	+	0.479043i	$0.840984\pi$
$984$		0			0
$985$	−12.3875	+	21.4558i	−0.394698	+	0.683638i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.80777			0.0892820
$990$		0			0
$991$	32.4459			1.03068			0.515339	−	0.856987i	$-0.327666\pi$
$991$	32.4459			1.03068			0.515339	+	0.856987i	$0.327666\pi$
$992$		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	0.947983	−	0.318320i	$-0.103119\pi$
$997$	6.26198	+	10.8461i	0.198319	+	0.343498i	−0.749665	+	0.661818i	$0.769785\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.j.373.6		24	21.17	even	6
1764.2.i.j.373.7		24	21.11	odd	6
1764.2.i.j.1537.6		24	63.47	even	6
1764.2.i.j.1537.7		24	63.2	odd	6
1764.2.j.i.589.3	✓	24	21.20	even	2
1764.2.j.i.589.10	yes	24	3.2	odd	2
1764.2.j.i.1177.3	yes	24	63.20	even	6
1764.2.j.i.1177.10	yes	24	9.2	odd	6
1764.2.l.j.949.2		24	21.2	odd	6
1764.2.l.j.949.11		24	21.5	even	6
1764.2.l.j.961.2		24	63.11	odd	6
1764.2.l.j.961.11		24	63.38	even	6
5292.2.i.j.1549.1		24	7.3	odd	6
5292.2.i.j.1549.12		24	7.4	even	3
5292.2.i.j.2125.1		24	63.61	odd	6
5292.2.i.j.2125.12		24	63.16	even	3
5292.2.j.i.1765.1		24	7.6	odd	2	inner
5292.2.j.i.1765.12		24	1.1	even	1	trivial
5292.2.j.i.3529.1		24	63.34	odd	6	inner
5292.2.j.i.3529.12		24	9.7	even	3	inner
5292.2.l.j.361.1		24	7.2	even	3
5292.2.l.j.361.12		24	7.5	odd	6
5292.2.l.j.3313.1		24	63.25	even	3
5292.2.l.j.3313.12		24	63.52	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.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.94623 - 3.37097i) q^{5} +O(q^{10})\)	\(q+(1.94623 - 3.37097i) q^{5} +(2.18778 + 3.78935i) q^{11} +(-0.792201 + 1.37213i) q^{13} +5.33356 q^{17} -4.64323 q^{19} +(0.183900 - 0.318523i) q^{23} +(-5.07564 - 8.79126i) q^{25} +(5.08750 + 8.81180i) q^{29} +(1.14776 - 1.98798i) q^{31} +10.8803 q^{37} +(0.690443 - 1.19588i) q^{41} +(3.81699 + 6.61122i) q^{43} +(-3.80432 - 6.58928i) q^{47} +0.925693 q^{53} +17.0317 q^{55} +(-0.460475 + 0.797565i) q^{59} +(-3.27780 - 5.67731i) q^{61} +(3.08361 + 5.34097i) q^{65} +(-7.50420 + 12.9976i) q^{67} +4.91059 q^{71} +7.56707 q^{73} +(-0.987715 - 1.71077i) q^{79} +(0.253011 + 0.438227i) q^{83} +(10.3803 - 17.9793i) q^{85} +12.2197 q^{89} +(-9.03679 + 15.6522i) q^{95} +(4.45315 + 7.71308i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} + 24 q^{37} - 32 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) \) 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 + 24 * q^37 - 32 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more