LMFDB - Embedded newform 5292.2.l.j.3313.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5292,2,Mod(361,5292)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5292.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5292, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.l (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			3313.1
Character	$\chi$	$=$	5292.3313
Dual form			5292.2.l.j.361.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.89246 q^{5} -4.37557 q^{11} +(-0.792201 - 1.37213i) q^{13} +(-2.66678 - 4.61900i) q^{17} +(2.32161 - 4.02115i) q^{19} -0.367799 q^{23} +10.1513 q^{25} +(5.08750 - 8.81180i) q^{29} +(1.14776 - 1.98798i) q^{31} +(-5.44017 + 9.42265i) q^{37} +(0.690443 + 1.19588i) q^{41} +(3.81699 - 6.61122i) q^{43} +(-3.80432 - 6.58928i) q^{47} +(-0.462847 - 0.801674i) q^{53} +17.0317 q^{55} +(-0.460475 + 0.797565i) q^{59} +(-3.27780 - 5.67731i) q^{61} +(3.08361 + 5.34097i) q^{65} +(-7.50420 + 12.9976i) q^{67} +4.91059 q^{71} +(-3.78353 - 6.55327i) q^{73} +(-0.987715 - 1.71077i) q^{79} +(0.253011 - 0.438227i) q^{83} +(10.3803 + 17.9793i) q^{85} +(-6.10987 + 10.5826i) q^{89} +(-9.03679 + 15.6522i) q^{95} +(4.45315 - 7.71308i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{11} - 16 q^{23} + 24 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) $ 24 * q - 8 * q^11 - 16 * q^23 + 24 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Character values

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

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

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
$5$	−3.89246			−1.74076			−0.870381	−	0.492378i	$-0.836128\pi$
$5$	−3.89246			−1.74076			−0.870381	+	0.492378i	$0.836128\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.37557			−1.31928			−0.659642	−	0.751580i	$-0.729292\pi$
$11$	−4.37557			−1.31928			−0.659642	+	0.751580i	$0.729292\pi$
$12$		0			0
$13$	−0.792201	−	1.37213i	−0.219717	−	0.380561i	0.735004	−	0.678062i	$-0.237180\pi$
$13$	−0.792201	−	1.37213i	−0.219717	−	0.380561i	−0.954721	+	0.297501i	$0.903847\pi$
$14$		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
$23$	−0.367799			−0.0766914			−0.0383457	−	0.999265i	$-0.512209\pi$
$23$	−0.367799			−0.0766914			−0.0383457	+	0.999265i	$0.512209\pi$
$24$		0			0
$25$	10.1513			2.03025
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.08750	−	8.81180i	0.944724	−	1.63631i	0.188422	−	0.982088i	$-0.439663\pi$
$29$	5.08750	−	8.81180i	0.944724	−	1.63631i	0.756302	−	0.654222i	$-0.227004\pi$
$30$		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
$37$	−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	0.914891	−	0.403702i	$-0.132277\pi$
$41$	0.690443	+	1.19588i	0.107829	+	0.186766i	−0.807061	+	0.590467i	$0.798943\pi$
$42$		0			0
$43$	3.81699	−	6.61122i	0.582086	−	1.00820i	−0.413146	−	0.910665i	$-0.635570\pi$
$43$	3.81699	−	6.61122i	0.582086	−	1.00820i	0.995232	−	0.0975372i	$-0.0310965\pi$
$44$		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
$53$	−0.462847	−	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
$59$	−0.460475	+	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
$73$	−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
$79$	−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	−0.851806	−	0.523858i	$-0.824492\pi$
$83$	0.253011	−	0.438227i	0.0277715	−	0.0481017i	0.879577	+	0.475756i	$0.157826\pi$
$84$		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
$95$	−9.03679	+	15.6522i	−0.927155	+	1.60588i
$96$		0			0
$97$	4.45315	−	7.71308i	0.452149	−	0.783145i	−0.546370	−	0.837544i	$-0.683991\pi$
$97$	4.45315	−	7.71308i	0.452149	−	0.783145i	0.998519	+	0.0543987i	$0.0173242\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11.0236			−1.09689			−0.548445	−	0.836187i	$-0.684780\pi$
$101$	−11.0236			−1.09689			−0.548445	+	0.836187i	$0.684780\pi$
$102$		0			0
$103$	2.73085			0.269079			0.134540	−	0.990908i	$-0.457044\pi$
$103$	2.73085			0.269079			0.134540	+	0.990908i	$0.457044\pi$
$104$		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	0.995961	+	0.0897875i	$0.0286188\pi$
$113$	6.12019	+	10.6005i	0.575739	+	0.997209i	−0.420222	+	0.907421i	$0.638048\pi$
$114$		0			0
$115$	1.43164			0.133502
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.14559			0.740509
$122$		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
$131$	−5.08683			−0.444438			−0.222219	−	0.974997i	$-0.571330\pi$
$131$	−5.08683			−0.444438			−0.222219	+	0.974997i	$0.571330\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.3442			−1.14007			−0.570034	−	0.821621i	$-0.693070\pi$
$137$	−13.3442			−1.14007			−0.570034	+	0.821621i	$0.693070\pi$
$138$		0			0
$139$	−4.85642	−	8.41157i	−0.411916	−	0.713460i	0.583183	−	0.812341i	$-0.301807\pi$
$139$	−4.85642	−	8.41157i	−0.411916	−	0.713460i	−0.995099	+	0.0988809i	$0.968474\pi$
$140$		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
$149$	−13.8423			−1.13401			−0.567004	−	0.823715i	$-0.691898\pi$
$149$	−13.8423			−1.13401			−0.567004	+	0.823715i	$0.691898\pi$
$150$		0			0
$151$	22.8759			1.86162			0.930809	−	0.365506i	$-0.119104\pi$
$151$	22.8759			1.86162			0.930809	+	0.365506i	$0.119104\pi$
$152$		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
$163$	−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	0.988566	−	0.150791i	$-0.0481821\pi$
$167$	4.69996	+	8.14057i	0.363694	+	0.629936i	−0.624872	+	0.780727i	$0.714849\pi$
$168$		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
$179$	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
$185$	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
$199$	−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
$205$	−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	0.601715	−	0.798711i	$-0.294484\pi$
$211$	−5.67737	−	9.83349i	−0.390846	−	0.676965i	−0.992561	+	0.121745i	$0.961151\pi$
$212$		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
$221$	−4.22525	+	7.31835i	−0.284221	+	0.492285i
$222$		0			0
$223$	−13.3206	+	23.0719i	−0.892011	+	1.54501i	−0.0545504	+	0.998511i	$0.517373\pi$
$223$	−13.3206	+	23.0719i	−0.892011	+	1.54501i	−0.837461	+	0.546498i	$0.815961\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.6027			1.10196			0.550981	−	0.834518i	$-0.314254\pi$
$227$	16.6027			1.10196			0.550981	+	0.834518i	$0.314254\pi$
$228$		0			0
$229$	−14.5014			−0.958282			−0.479141	−	0.877738i	$-0.659052\pi$
$229$	−14.5014			−0.958282			−0.479141	+	0.877738i	$0.659052\pi$
$230$		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	0.989426	+	0.145040i	$0.0463312\pi$
$239$	9.58994	+	16.6103i	0.620322	+	1.07443i	−0.369104	+	0.929388i	$0.620335\pi$
$240$		0			0
$241$	23.3569			1.50455			0.752276	−	0.658848i	$-0.228956\pi$
$241$	23.3569			1.50455			0.752276	+	0.658848i	$0.228956\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.35673			−0.468098
$248$		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
$257$	6.80877			0.424719			0.212360	−	0.977192i	$-0.431885\pi$
$257$	6.80877			0.424719			0.212360	+	0.977192i	$0.431885\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.60001			0.283649			0.141824	−	0.989892i	$-0.454703\pi$
$263$	4.60001			0.283649			0.141824	+	0.989892i	$0.454703\pi$
$264$		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
$275$	−44.4176			−2.67848
$276$		0			0
$277$	−11.2697			−0.677129			−0.338564	−	0.940943i	$-0.609941\pi$
$277$	−11.2697			−0.677129			−0.338564	+	0.940943i	$0.609941\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.50741	+	13.0032i	−0.447854	+	0.775707i	−0.998246	−	0.0592000i	$-0.981145\pi$
$281$	−7.50741	+	13.0032i	−0.447854	+	0.775707i	0.550392	+	0.834907i	$0.314478\pi$
$282$		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
$289$	−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	0.998374	+	0.0570099i	$0.0181567\pi$
$293$	9.38981	+	16.2636i	0.548559	+	0.950132i	−0.449815	+	0.893122i	$0.648510\pi$
$294$		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
$305$	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
$311$	−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
$325$	−8.04184	−	13.9289i	−0.446081	−	0.772635i
$326$		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	0.946671	−	0.322203i	$-0.104423\pi$
$337$	3.56686	+	6.17799i	0.194299	+	0.336537i	−0.752371	+	0.658739i	$0.771090\pi$
$338$		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
$347$	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	−0.972722	−	0.231973i	$-0.925482\pi$
$349$	−5.33296	+	9.23696i	−0.285467	+	0.494443i	0.687256	+	0.726416i	$0.258815\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.8426			0.683545			0.341772	−	0.939783i	$-0.388973\pi$
$353$	12.8426			0.683545			0.341772	+	0.939783i	$0.388973\pi$
$354$		0			0
$355$	−19.1143			−1.01448
$356$		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
$367$	−16.9839			−0.886554			−0.443277	−	0.896385i	$-0.646184\pi$
$367$	−16.9839			−0.886554			−0.443277	+	0.896385i	$0.646184\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−8.77005			−0.454096			−0.227048	−	0.973884i	$-0.572907\pi$
$373$	−8.77005			−0.454096			−0.227048	+	0.973884i	$0.572907\pi$
$374$		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
$383$	9.00720			0.460246			0.230123	−	0.973162i	$-0.426087\pi$
$383$	9.00720			0.460246			0.230123	+	0.973162i	$0.426087\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.78781			0.496262			0.248131	−	0.968727i	$-0.420184\pi$
$389$	9.78781			0.496262			0.248131	+	0.968727i	$0.420184\pi$
$390$		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
$401$	23.4828			1.17267			0.586336	−	0.810068i	$-0.300570\pi$
$401$	23.4828			1.17267			0.586336	+	0.810068i	$0.300570\pi$
$402$		0			0
$403$	−3.63703			−0.181173
$404$		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
$415$	−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	0.946673	−	0.322197i	$-0.104421\pi$
$419$	3.97733	+	6.88894i	0.194305	+	0.336547i	−0.752367	+	0.658744i	$0.771088\pi$
$420$		0			0
$421$	1.30584	−	2.26178i	0.0636426	−	0.110232i	−0.832448	−	0.554102i	$-0.813062\pi$
$421$	1.30584	−	2.26178i	0.0636426	−	0.110232i	0.896091	+	0.443870i	$0.146395\pi$
$422$		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
$431$	−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
$437$	−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
$451$	−3.02108	−	5.23267i	−0.142257	−	0.246397i
$452$		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	−0.823391	−	0.567475i	$-0.807920\pi$
$461$	1.71236	−	2.96589i	0.0797524	−	0.138135i	0.903143	+	0.429340i	$0.141254\pi$
$462$		0			0
$463$	2.38499	+	4.13092i	0.110840	+	0.191980i	0.916109	−	0.400929i	$-0.131313\pi$
$463$	2.38499	+	4.13092i	0.110840	+	0.191980i	−0.805269	+	0.592909i	$0.797979\pi$
$464$		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
$473$	−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
$479$	−16.8506			−0.769922			−0.384961	−	0.922933i	$-0.625785\pi$
$479$	−16.8506			−0.769922			−0.384961	+	0.922933i	$0.625785\pi$
$480$		0			0
$481$	17.2388			0.786023
$482$		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	0.889217	−	0.457485i	$-0.151250\pi$
$491$	1.07281	+	1.85816i	0.0484153	+	0.0838577i	−0.840802	+	0.541342i	$0.817916\pi$
$492$		0			0
$493$	−54.2689			−2.44415
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−27.7154			−1.24071			−0.620357	−	0.784320i	$-0.713012\pi$
$499$	−27.7154			−1.24071			−0.620357	+	0.784320i	$0.713012\pi$
$500$		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
$509$	−29.3689			−1.30175			−0.650876	−	0.759184i	$-0.725598\pi$
$509$	−29.3689			−1.30175			−0.650876	+	0.759184i	$0.725598\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.6298			−0.468403
$516$		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
$527$	−12.2433			−0.533327
$528$		0			0
$529$	−22.8647			−0.994118
$530$		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
$541$	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	−0.697738	−	0.716353i	$-0.745810\pi$
$547$	6.35012	−	10.9987i	0.271512	−	0.470272i	0.969249	+	0.246082i	$0.0791432\pi$
$548$		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
$557$	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
$563$	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
$577$	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
$583$	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	−0.813900	−	0.581005i	$-0.802660\pi$
$587$	2.33110	−	4.03758i	0.0962146	−	0.166649i	0.910115	+	0.414356i	$0.135993\pi$
$588$		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
$599$	−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	−0.945142	−	0.326661i	$-0.894077\pi$
$601$	−4.64993	+	8.05391i	−0.189674	+	0.328526i	0.755467	+	0.655186i	$0.227410\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−31.7064			−1.28905
$606$		0			0
$607$	20.4968			0.831938			0.415969	−	0.909379i	$-0.363442\pi$
$607$	20.4968			0.831938			0.415969	+	0.909379i	$0.363442\pi$
$608$		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	0.966421	−	0.256964i	$-0.0827221\pi$
$617$	6.47499	+	11.2150i	0.260673	+	0.451499i	−0.705748	+	0.708463i	$0.749389\pi$
$618$		0			0
$619$	−35.3980			−1.42277			−0.711383	−	0.702804i	$-0.751931\pi$
$619$	−35.3980			−1.42277			−0.711383	+	0.702804i	$0.751931\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	27.2920			1.09168
$626$		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
$635$	51.3824			2.03905
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.239269			0.00945056			0.00472528	−	0.999989i	$-0.498496\pi$
$641$	0.239269			0.00945056			0.00472528	+	0.999989i	$0.498496\pi$
$642$		0			0
$643$	−4.57211	−	7.91913i	−0.180307	−	0.312300i	0.761678	−	0.647955i	$-0.224376\pi$
$643$	−4.57211	−	7.91913i	−0.180307	−	0.312300i	−0.941985	+	0.335655i	$0.891042\pi$
$644$		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
$653$	4.55775			0.178359			0.0891793	−	0.996016i	$-0.471576\pi$
$653$	4.55775			0.178359			0.0891793	+	0.996016i	$0.471576\pi$
$654$		0			0
$655$	19.8003			0.773662
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16.4631	−	28.5149i	0.641311	−	1.11078i	−0.343829	−	0.939032i	$-0.611724\pi$
$659$	16.4631	−	28.5149i	0.641311	−	1.11078i	0.985140	−	0.171751i	$-0.0549425\pi$
$660$		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
$667$	−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	−0.998725	−	0.0504780i	$-0.983926\pi$
$673$	−11.8205	+	20.4737i	−0.455647	+	0.789204i	0.543078	+	0.839682i	$0.317259\pi$
$674$		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
$683$	−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
$689$	−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
$703$	25.2599	+	43.7515i	0.952697	+	1.65012i
$704$		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
$719$	−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
$725$	51.6446	−	89.4510i	1.91803	−	3.32213i
$726$		0			0
$727$	10.9453	−	18.9578i	0.405938	−	0.703105i	−0.588492	−	0.808503i	$-0.700278\pi$
$727$	10.9453	−	18.9578i	0.405938	−	0.703105i	0.994430	+	0.105398i	$0.0336117\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−40.7163			−1.50595
$732$		0			0
$733$	−8.68831			−0.320910			−0.160455	−	0.987043i	$-0.551296\pi$
$733$	−8.68831			−0.320910			−0.160455	+	0.987043i	$0.551296\pi$
$734$		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	−0.926726	−	0.375737i	$-0.877389\pi$
$743$	−21.5001	−	37.2392i	−0.788761	−	1.36617i	0.137965	−	0.990437i	$-0.455944\pi$
$744$		0			0
$745$	53.8807			1.97404
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.7274			−0.610391			−0.305196	−	0.952290i	$-0.598722\pi$
$751$	−16.7274			−0.610391			−0.305196	+	0.952290i	$0.598722\pi$
$752$		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
$761$	51.6484			1.87225			0.936127	−	0.351663i	$-0.114384\pi$
$761$	51.6484			1.87225			0.936127	+	0.351663i	$0.114384\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.45915			0.0526870
$768$		0			0
$769$	15.3910	+	26.6580i	0.555014	+	0.961313i	0.997902	+	0.0647361i	$0.0206205\pi$
$769$	15.3910	+	26.6580i	0.555014	+	0.961313i	−0.442888	+	0.896577i	$0.646046\pi$
$770$		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
$779$	6.41177			0.229725
$780$		0			0
$781$	−21.4866			−0.768852
$782$		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
$793$	−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	0.886497	+	0.462735i	$0.153132\pi$
$797$	23.8268	+	41.2692i	0.843988	+	1.46183i	−0.0425084	+	0.999096i	$0.513535\pi$
$798$		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
$809$	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
$815$	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
$829$	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
$835$	−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	−0.522372	−	0.852718i	$-0.674953\pi$
$839$	13.8249	−	23.9455i	0.477290	−	0.826690i	0.999661	+	0.0260281i	$0.00828595\pi$
$840$		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
$851$	2.00089	−	3.46564i	0.0685896	−	0.118801i
$852$		0			0
$853$	−22.0459	+	38.1847i	−0.754839	+	1.30742i	0.190616	+	0.981665i	$0.438952\pi$
$853$	−22.0459	+	38.1847i	−0.754839	+	1.30742i	−0.945454	+	0.325754i	$0.894382\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	26.7676			0.914363			0.457182	−	0.889373i	$-0.348859\pi$
$857$	26.7676			0.914363			0.457182	+	0.889373i	$0.348859\pi$
$858$		0			0
$859$	−20.1901			−0.688879			−0.344439	−	0.938809i	$-0.611931\pi$
$859$	−20.1901			−0.688879			−0.344439	+	0.938809i	$0.611931\pi$
$860$		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
$871$	23.7793			0.805731
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	29.2455			0.987549			0.493774	−	0.869590i	$-0.335617\pi$
$877$	29.2455			0.987549			0.493774	+	0.869590i	$0.335617\pi$
$878$		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
$887$	9.92253			0.333166			0.166583	−	0.986027i	$-0.446727\pi$
$887$	9.92253			0.333166			0.166583	+	0.986027i	$0.446727\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−35.3287			−1.18223
$894$		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
$905$	−5.29328			−0.175955
$906$		0			0
$907$	−24.5775			−0.816081			−0.408040	−	0.912964i	$-0.633788\pi$
$907$	−24.5775			−0.816081			−0.408040	+	0.912964i	$0.633788\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.73496	+	16.8614i	−0.322534	+	0.558645i	−0.981010	−	0.193957i	$-0.937868\pi$
$911$	−9.73496	+	16.8614i	−0.322534	+	0.558645i	0.658476	+	0.752601i	$0.271201\pi$
$912$		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
$919$	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
$929$	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
$935$	−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
$941$	−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
$955$	34.8005	+	60.2762i	1.12612	+	1.95049i
$956$		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	0.982139	−	0.188156i	$-0.0602511\pi$
$967$	10.2035	+	17.6729i	0.328121	+	0.568323i	−0.654018	+	0.756479i	$0.726918\pi$
$968$		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
$977$	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
$983$	1.50696			0.0480646			0.0240323	−	0.999711i	$-0.492350\pi$
$983$	1.50696			0.0480646			0.0240323	+	0.999711i	$0.492350\pi$
$984$		0			0
$985$	24.7750			0.789397
$986$		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
$997$	−12.5240			−0.396638			−0.198319	−	0.980138i	$-0.563548\pi$
$997$	−12.5240			−0.396638			−0.198319	+	0.980138i	$0.563548\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.l.j.3313.1		24
3.2	odd	2		1764.2.l.j.961.2		24
7.2	even	3		5292.2.j.i.3529.12		24
7.3	odd	6		5292.2.i.j.2125.1		24
7.4	even	3		5292.2.i.j.2125.12		24
7.5	odd	6		5292.2.j.i.3529.1		24
7.6	odd	2	inner	5292.2.l.j.3313.12		24
9.4	even	3		5292.2.i.j.1549.12		24
9.5	odd	6		1764.2.i.j.373.7		24
21.2	odd	6		1764.2.j.i.1177.10	yes	24
21.5	even	6		1764.2.j.i.1177.3	yes	24
21.11	odd	6		1764.2.i.j.1537.7		24
21.17	even	6		1764.2.i.j.1537.6		24
21.20	even	2		1764.2.l.j.961.11		24
63.4	even	3	inner	5292.2.l.j.361.1		24
63.5	even	6		1764.2.j.i.589.3	✓	24
63.13	odd	6		5292.2.i.j.1549.1		24
63.23	odd	6		1764.2.j.i.589.10	yes	24
63.31	odd	6	inner	5292.2.l.j.361.12		24
63.32	odd	6		1764.2.l.j.949.2		24
63.40	odd	6		5292.2.j.i.1765.1		24
63.41	even	6		1764.2.i.j.373.6		24
63.58	even	3		5292.2.j.i.1765.12		24
63.59	even	6		1764.2.l.j.949.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.j.373.6		24	63.41	even	6
1764.2.i.j.373.7		24	9.5	odd	6
1764.2.i.j.1537.6		24	21.17	even	6
1764.2.i.j.1537.7		24	21.11	odd	6
1764.2.j.i.589.3	✓	24	63.5	even	6
1764.2.j.i.589.10	yes	24	63.23	odd	6
1764.2.j.i.1177.3	yes	24	21.5	even	6
1764.2.j.i.1177.10	yes	24	21.2	odd	6
1764.2.l.j.949.2		24	63.32	odd	6
1764.2.l.j.949.11		24	63.59	even	6
1764.2.l.j.961.2		24	3.2	odd	2
1764.2.l.j.961.11		24	21.20	even	2
5292.2.i.j.1549.1		24	63.13	odd	6
5292.2.i.j.1549.12		24	9.4	even	3
5292.2.i.j.2125.1		24	7.3	odd	6
5292.2.i.j.2125.12		24	7.4	even	3
5292.2.j.i.1765.1		24	63.40	odd	6
5292.2.j.i.1765.12		24	63.58	even	3
5292.2.j.i.3529.1		24	7.5	odd	6
5292.2.j.i.3529.12		24	7.2	even	3
5292.2.l.j.361.1		24	63.4	even	3	inner
5292.2.l.j.361.12		24	63.31	odd	6	inner
5292.2.l.j.3313.1		24	1.1	even	1	trivial
5292.2.l.j.3313.12		24	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-3.89246 q^{5} -4.37557 q^{11} +(-0.792201 - 1.37213i) q^{13} +(-2.66678 - 4.61900i) q^{17} +(2.32161 - 4.02115i) q^{19} -0.367799 q^{23} +10.1513 q^{25} +(5.08750 - 8.81180i) q^{29} +(1.14776 - 1.98798i) q^{31} +(-5.44017 + 9.42265i) q^{37} +(0.690443 + 1.19588i) q^{41} +(3.81699 - 6.61122i) q^{43} +(-3.80432 - 6.58928i) q^{47} +(-0.462847 - 0.801674i) q^{53} +17.0317 q^{55} +(-0.460475 + 0.797565i) q^{59} +(-3.27780 - 5.67731i) q^{61} +(3.08361 + 5.34097i) q^{65} +(-7.50420 + 12.9976i) q^{67} +4.91059 q^{71} +(-3.78353 - 6.55327i) q^{73} +(-0.987715 - 1.71077i) q^{79} +(0.253011 - 0.438227i) q^{83} +(10.3803 + 17.9793i) q^{85} +(-6.10987 + 10.5826i) q^{89} +(-9.03679 + 15.6522i) q^{95} +(4.45315 - 7.71308i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{11} - 16 q^{23} + 24 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} + 36 q^{65} + 12 q^{67} - 48 q^{71} + 12 q^{79} + 12 q^{85} - 32 q^{95}+O(q^{100}) \) 24 * q - 8 * q^11 - 16 * q^23 + 24 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 + 36 * q^65 + 12 * q^67 - 48 * q^71 + 12 * q^79 + 12 * q^85 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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