LMFDB - Embedded newform 2940.2.f.b.1469.29

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,2,Mod(1469,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2940.1469"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.4760181943$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1469.29
Character	$\chi$	$=$	2940.1469
Dual form			2940.2.f.b.1469.31

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.726948 - 1.57212i) q^{3} +(-2.19795 + 0.411111i) q^{5} +(-1.94309 - 2.28569i) q^{9} +1.72528i q^{11} -1.24079 q^{13} +(-0.951482 + 3.75429i) q^{15} -4.75664i q^{17} +4.23622i q^{19} -2.60815 q^{23} +(4.66198 - 1.80720i) q^{25} +(-5.00590 + 1.39319i) q^{27} +2.97676i q^{29} +0.179474i q^{31} +(2.71235 + 1.25419i) q^{33} +5.41835i q^{37} +(-0.901986 + 1.95066i) q^{39} -1.41531 q^{41} +8.03664i q^{43} +(5.21050 + 4.22501i) q^{45} -7.87495i q^{47} +(-7.47799 - 3.45783i) q^{51} +6.64074 q^{53} +(-0.709284 - 3.79209i) q^{55} +(6.65983 + 3.07951i) q^{57} -2.07100 q^{59} +9.34523i q^{61} +(2.72718 - 0.510101i) q^{65} +9.46664i q^{67} +(-1.89599 + 4.10031i) q^{69} +8.56168i q^{71} +15.6657 q^{73} +(0.547880 - 8.64291i) q^{75} +0.643921 q^{79} +(-1.44878 + 8.88263i) q^{81} +4.73726i q^{83} +(1.95551 + 10.4549i) q^{85} +(4.67981 + 2.16395i) q^{87} -8.25733 q^{89} +(0.282154 + 0.130468i) q^{93} +(-1.74156 - 9.31101i) q^{95} -16.8182 q^{97} +(3.94347 - 3.35239i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 8 q^{9} - 16 q^{15} + 16 q^{25} + 56 q^{39} + 8 q^{51} + 48 q^{79} - 24 q^{81} + 32 q^{85} + 88 q^{99}+O(q^{100}) $ 48 * q - 8 * q^9 - 16 * q^15 + 16 * q^25 + 56 * q^39 + 8 * q^51 + 48 * q^79 - 24 * q^81 + 32 * q^85 + 88 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	0.726948	−	1.57212i	0.419704	−	0.907661i
$3$	0.726948	−	1.57212i	0.419704	−	0.907661i
$4$		0			0
$5$	−2.19795	+	0.411111i	−0.982953	+	0.183854i
$5$	−2.19795	+	0.411111i	−0.982953	+	0.183854i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.94309	−	2.28569i	−0.647698	−	0.761897i
$10$		0			0
$11$			1.72528i			0.520193i	0.965583	+	0.260096i	$0.0837543\pi$
$11$			1.72528i			0.520193i	−0.965583	+	0.260096i	$0.916246\pi$
$12$		0			0
$13$	−1.24079			−0.344132			−0.172066	−	0.985085i	$-0.555044\pi$
$13$	−1.24079			−0.344132			−0.172066	+	0.985085i	$0.555044\pi$
$14$		0			0
$15$	−0.951482	+	3.75429i	−0.245672	+	0.969353i
$16$		0			0
$17$		−	4.75664i		−	1.15365i	−0.816866	−	0.576827i	$-0.804291\pi$
$17$		−	4.75664i		−	1.15365i	0.816866	−	0.576827i	$-0.195709\pi$
$18$		0			0
$19$			4.23622i			0.971857i	0.873999	+	0.485928i	$0.161518\pi$
$19$			4.23622i			0.971857i	−0.873999	+	0.485928i	$0.838482\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.60815			−0.543836			−0.271918	−	0.962320i	$-0.587658\pi$
$23$	−2.60815			−0.543836			−0.271918	+	0.962320i	$0.587658\pi$
$24$		0			0
$25$	4.66198	−	1.80720i	0.932395	−	0.361441i
$26$		0			0
$27$	−5.00590	+	1.39319i	−0.963386	+	0.268119i
$28$		0			0
$29$			2.97676i			0.552770i	0.961047	+	0.276385i	$0.0891365\pi$
$29$			2.97676i			0.552770i	−0.961047	+	0.276385i	$0.910864\pi$
$30$		0			0
$31$			0.179474i			0.0322345i	0.999870	+	0.0161172i	$0.00513050\pi$
$31$			0.179474i			0.0322345i	−0.999870	+	0.0161172i	$0.994870\pi$
$32$		0			0
$33$	2.71235	+	1.25419i	0.472159	+	0.218327i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			5.41835i			0.890771i	0.895339	+	0.445385i	$0.146933\pi$
$37$			5.41835i			0.890771i	−0.895339	+	0.445385i	$0.853067\pi$
$38$		0			0
$39$	−0.901986	+	1.95066i	−0.144433	+	0.312355i
$40$		0			0
$41$	−1.41531			−0.221034			−0.110517	−	0.993874i	$-0.535251\pi$
$41$	−1.41531			−0.221034			−0.110517	+	0.993874i	$0.535251\pi$
$42$		0			0
$43$			8.03664i			1.22558i	0.790247	+	0.612788i	$0.209952\pi$
$43$			8.03664i			1.22558i	−0.790247	+	0.612788i	$0.790048\pi$
$44$		0			0
$45$	5.21050	+	4.22501i	0.776735	+	0.629827i
$46$		0			0
$47$		−	7.87495i		−	1.14868i	−0.818617	−	0.574340i	$-0.805259\pi$
$47$		−	7.87495i		−	1.14868i	0.818617	−	0.574340i	$-0.194741\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−7.47799	−	3.45783i	−1.04713	−	0.484193i
$52$		0			0
$53$	6.64074			0.912175			0.456088	−	0.889935i	$-0.349250\pi$
$53$	6.64074			0.912175			0.456088	+	0.889935i	$0.349250\pi$
$54$		0			0
$55$	−0.709284	−	3.79209i	−0.0956398	−	0.511325i
$56$		0			0
$57$	6.65983	+	3.07951i	0.882117	+	0.407892i
$58$		0			0
$59$	−2.07100			−0.269621			−0.134810	−	0.990871i	$-0.543043\pi$
$59$	−2.07100			−0.269621			−0.134810	+	0.990871i	$0.543043\pi$
$60$		0			0
$61$			9.34523i			1.19653i	0.801297	+	0.598267i	$0.204144\pi$
$61$			9.34523i			1.19653i	−0.801297	+	0.598267i	$0.795856\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.72718	−	0.510101i	0.338266	−	0.0632702i
$66$		0			0
$67$			9.46664i			1.15653i	0.815848	+	0.578267i	$0.196271\pi$
$67$			9.46664i			1.15653i	−0.815848	+	0.578267i	$0.803729\pi$
$68$		0			0
$69$	−1.89599	+	4.10031i	−0.228250	+	0.493619i
$70$		0			0
$71$			8.56168i			1.01608i	0.861332	+	0.508042i	$0.169631\pi$
$71$			8.56168i			1.01608i	−0.861332	+	0.508042i	$0.830369\pi$
$72$		0			0
$73$	15.6657			1.83353			0.916764	−	0.399430i	$-0.130792\pi$
$73$	15.6657			1.83353			0.916764	+	0.399430i	$0.130792\pi$
$74$		0			0
$75$	0.547880	−	8.64291i	0.0632637	−	0.997997i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.643921			0.0724467			0.0362234	−	0.999344i	$-0.488467\pi$
$79$	0.643921			0.0724467			0.0362234	+	0.999344i	$0.488467\pi$
$80$		0			0
$81$	−1.44878	+	8.88263i	−0.160975	+	0.986958i
$82$		0			0
$83$			4.73726i			0.519982i	0.965611	+	0.259991i	$0.0837196\pi$
$83$			4.73726i			0.519982i	−0.965611	+	0.259991i	$0.916280\pi$
$84$		0			0
$85$	1.95551	+	10.4549i	0.212105	+	1.13399i
$86$		0			0
$87$	4.67981	+	2.16395i	0.501728	+	0.232000i
$88$		0			0
$89$	−8.25733			−0.875275			−0.437637	−	0.899152i	$-0.644185\pi$
$89$	−8.25733			−0.875275			−0.437637	+	0.899152i	$0.644185\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.282154	+	0.130468i	0.0292580	+	0.0135289i
$94$		0			0
$95$	−1.74156	−	9.31101i	−0.178680	−	0.955290i
$96$		0			0
$97$	−16.8182			−1.70763			−0.853813	−	0.520579i	$-0.825716\pi$
$97$	−16.8182			−1.70763			−0.853813	+	0.520579i	$0.825716\pi$
$98$		0			0
$99$	3.94347	−	3.35239i	0.396333	−	0.336928i
$100$		0			0
$101$	−15.7374			−1.56593			−0.782967	−	0.622063i	$-0.786295\pi$
$101$	−15.7374			−1.56593			−0.782967	+	0.622063i	$0.786295\pi$
$102$		0			0
$103$	14.8011			1.45840			0.729200	−	0.684301i	$-0.239893\pi$
$103$	14.8011			1.45840			0.729200	+	0.684301i	$0.239893\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.61940			0.833269			0.416634	−	0.909074i	$-0.363210\pi$
$107$	8.61940			0.833269			0.416634	+	0.909074i	$0.363210\pi$
$108$		0			0
$109$	6.54267			0.626674			0.313337	−	0.949642i	$-0.398553\pi$
$109$	6.54267			0.626674			0.313337	+	0.949642i	$0.398553\pi$
$110$		0			0
$111$	8.51827	+	3.93886i	0.808518	+	0.373860i
$112$		0			0
$113$	19.3171			1.81720			0.908598	−	0.417671i	$-0.137154\pi$
$113$	19.3171			1.81720			0.908598	+	0.417671i	$0.137154\pi$
$114$		0			0
$115$	5.73257	−	1.07224i	0.534565	−	0.0999867i
$116$		0			0
$117$	2.41096	+	2.83605i	0.222893	+	0.262193i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.02339			0.729400
$122$		0			0
$123$	−1.02885	+	2.22502i	−0.0927686	+	0.200624i
$124$		0			0
$125$	−9.50383	+	5.88874i	−0.850048	+	0.526705i
$126$		0			0
$127$			15.8653i			1.40782i	0.710289	+	0.703910i	$0.248564\pi$
$127$			15.8653i			1.40782i	−0.710289	+	0.703910i	$0.751436\pi$
$128$		0			0
$129$	12.6345	+	5.84222i	1.11241	+	0.514379i
$130$		0			0
$131$	14.5600			1.27211			0.636056	−	0.771643i	$-0.280565\pi$
$131$	14.5600			1.27211			0.636056	+	0.771643i	$0.280565\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	10.4300	−	5.12014i	0.897668	−	0.440671i
$136$		0			0
$137$	−10.7801			−0.921002			−0.460501	−	0.887659i	$-0.652330\pi$
$137$	−10.7801			−0.921002			−0.460501	+	0.887659i	$0.652330\pi$
$138$		0			0
$139$		−	17.5395i		−	1.48768i	−0.668358	−	0.743840i	$-0.733002\pi$
$139$		−	17.5395i		−	1.48768i	0.668358	−	0.743840i	$-0.266998\pi$
$140$		0			0
$141$	−12.3803	−	5.72468i	−1.04261	−	0.482105i
$142$		0			0
$143$		−	2.14071i		−	0.179015i
$144$		0			0
$145$	−1.22378	−	6.54277i	−0.101629	−	0.543347i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			17.8727i			1.46419i	0.681204	+	0.732094i	$0.261457\pi$
$149$			17.8727i			1.46419i	−0.681204	+	0.732094i	$0.738543\pi$
$150$		0			0
$151$	−20.7055			−1.68499			−0.842495	−	0.538704i	$-0.818914\pi$
$151$	−20.7055			−1.68499			−0.842495	+	0.538704i	$0.818914\pi$
$152$		0			0
$153$	−10.8722	+	9.24260i	−0.878967	+	0.747220i
$154$		0			0
$155$	−0.0737838	−	0.394475i	−0.00592645	−	0.0316850i
$156$		0			0
$157$	17.1190			1.36625			0.683123	−	0.730303i	$-0.260621\pi$
$157$	17.1190			1.36625			0.683123	+	0.730303i	$0.260621\pi$
$158$		0			0
$159$	4.82747	−	10.4400i	0.382843	−	0.827946i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			12.1625i			0.952640i	0.879272	+	0.476320i	$0.158030\pi$
$163$			12.1625i			0.952640i	−0.879272	+	0.476320i	$0.841970\pi$
$164$		0			0
$165$	−6.47721	−	1.64158i	−0.504250	−	0.127797i
$166$		0			0
$167$			2.24481i			0.173709i	0.996221	+	0.0868543i	$0.0276815\pi$
$167$			2.24481i			0.173709i	−0.996221	+	0.0868543i	$0.972319\pi$
$168$		0			0
$169$	−11.4605			−0.881573
$170$		0			0
$171$	9.68271	−	8.23138i	0.740455	−	0.629469i
$172$		0			0
$173$			19.2816i			1.46595i	0.680253	+	0.732977i	$0.261870\pi$
$173$			19.2816i			1.46595i	−0.680253	+	0.732977i	$0.738130\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.50551	+	3.25585i	−0.113161	+	0.244725i
$178$		0			0
$179$		−	22.3831i		−	1.67299i	−0.547972	−	0.836497i	$-0.684600\pi$
$179$		−	22.3831i		−	1.67299i	0.547972	−	0.836497i	$-0.315400\pi$
$180$		0			0
$181$			8.41345i			0.625367i	0.949857	+	0.312683i	$0.101228\pi$
$181$			8.41345i			0.625367i	−0.949857	+	0.312683i	$0.898772\pi$
$182$		0			0
$183$	14.6918	+	6.79350i	1.08605	+	0.502190i
$184$		0			0
$185$	−2.22754	−	11.9093i	−0.163772	−	0.875586i
$186$		0			0
$187$	8.20656			0.600123
$188$		0			0
$189$		0			0
$190$		0			0
$191$			21.4075i			1.54899i	0.632580	+	0.774495i	$0.281996\pi$
$191$			21.4075i			1.54899i	−0.632580	+	0.774495i	$0.718004\pi$
$192$		0			0
$193$		−	8.89422i		−	0.640220i	−0.947380	−	0.320110i	$-0.896280\pi$
$193$		−	8.89422i		−	0.640220i	0.947380	−	0.320110i	$-0.103720\pi$
$194$		0			0
$195$	1.18058	−	4.65827i	0.0845434	−	0.333585i
$196$		0			0
$197$	−8.70586			−0.620267			−0.310133	−	0.950693i	$-0.600374\pi$
$197$	−8.70586			−0.620267			−0.310133	+	0.950693i	$0.600374\pi$
$198$		0			0
$199$			12.4368i			0.881624i	0.897599	+	0.440812i	$0.145309\pi$
$199$			12.4368i			0.881624i	−0.897599	+	0.440812i	$0.854691\pi$
$200$		0			0
$201$	14.8826	+	6.88175i	1.04974	+	0.485401i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.11077	−	0.581848i	0.217266	−	0.0406380i
$206$		0			0
$207$	5.06787	+	5.96142i	0.352241	+	0.414347i
$208$		0			0
$209$	−7.30869			−0.505553
$210$		0			0
$211$	−7.93329			−0.546150			−0.273075	−	0.961993i	$-0.588041\pi$
$211$	−7.93329			−0.546150			−0.273075	+	0.961993i	$0.588041\pi$
$212$		0			0
$213$	13.4599	+	6.22390i	0.922261	+	0.426454i
$214$		0			0
$215$	−3.30395	−	17.6641i	−0.225328	−	1.20468i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	11.3881	−	24.6282i	0.769538	−	1.66422i
$220$		0			0
$221$			5.90197i			0.397009i
$222$		0			0
$223$	−2.75970			−0.184803			−0.0924014	−	0.995722i	$-0.529454\pi$
$223$	−2.75970			−0.184803			−0.0924014	+	0.995722i	$0.529454\pi$
$224$		0			0
$225$	−13.1894	−	7.14427i	−0.879291	−	0.476285i
$226$		0			0
$227$		−	6.93266i		−	0.460137i	−0.973174	−	0.230068i	$-0.926105\pi$
$227$		−	6.93266i		−	0.460137i	0.973174	−	0.230068i	$-0.0738950\pi$
$228$		0			0
$229$		−	17.0420i		−	1.12617i	−0.826400	−	0.563084i	$-0.809615\pi$
$229$		−	17.0420i		−	1.12617i	0.826400	−	0.563084i	$-0.190385\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.3673			−1.39982			−0.699911	−	0.714230i	$-0.746777\pi$
$233$	−21.3673			−1.39982			−0.699911	+	0.714230i	$0.746777\pi$
$234$		0			0
$235$	3.23748	+	17.3087i	0.211190	+	1.12910i
$236$		0			0
$237$	0.468097	−	1.01232i	0.0304062	−	0.0657571i
$238$		0			0
$239$			1.52427i			0.0985968i	0.998784	+	0.0492984i	$0.0156985\pi$
$239$			1.52427i			0.0985968i	−0.998784	+	0.0492984i	$0.984301\pi$
$240$		0			0
$241$		−	17.2920i		−	1.11388i	−0.830554	−	0.556939i	$-0.811976\pi$
$241$		−	17.2920i		−	1.11388i	0.830554	−	0.556939i	$-0.188024\pi$
$242$		0			0
$243$	12.9113	+	8.73485i	0.828262	+	0.560341i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	5.25624i		−	0.334447i
$248$		0			0
$249$	7.44752	+	3.44374i	0.471967	+	0.218238i
$250$		0			0
$251$	12.1085			0.764281			0.382140	−	0.924104i	$-0.375187\pi$
$251$	12.1085			0.764281			0.382140	+	0.924104i	$0.375187\pi$
$252$		0			0
$253$		−	4.49979i		−	0.282899i
$254$		0			0
$255$	17.8578	+	4.52586i	1.11830	+	0.283420i
$256$		0			0
$257$		−	16.5309i		−	1.03117i	−0.856839	−	0.515583i	$-0.827575\pi$
$257$		−	16.5309i		−	1.03117i	0.856839	−	0.515583i	$-0.172425\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.80395	−	5.78412i	0.421154	−	0.358028i
$262$		0			0
$263$	24.3013			1.49848			0.749241	−	0.662297i	$-0.230418\pi$
$263$	24.3013			1.49848			0.749241	+	0.662297i	$0.230418\pi$
$264$		0			0
$265$	−14.5960	+	2.73008i	−0.896626	+	0.167708i
$266$		0			0
$267$	−6.00265	+	12.9815i	−0.367356	+	0.794453i
$268$		0			0
$269$	−5.91108			−0.360405			−0.180202	−	0.983630i	$-0.557675\pi$
$269$	−5.91108			−0.360405			−0.180202	+	0.983630i	$0.557675\pi$
$270$		0			0
$271$		−	0.135959i		−	0.00825892i	−0.999991	−	0.00412946i	$-0.998686\pi$
$271$		−	0.135959i		−	0.00825892i	0.999991	−	0.00412946i	$-0.00131445\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.11794	+	8.04323i	0.188019	+	0.485025i
$276$		0			0
$277$		−	17.5681i		−	1.05557i	−0.849379	−	0.527784i	$-0.823023\pi$
$277$		−	17.5681i		−	1.05557i	0.849379	−	0.527784i	$-0.176977\pi$
$278$		0			0
$279$	0.410222	−	0.348735i	0.0245594	−	0.0208782i
$280$		0			0
$281$			8.12449i			0.484667i	0.970193	+	0.242333i	$0.0779128\pi$
$281$			8.12449i			0.484667i	−0.970193	+	0.242333i	$0.922087\pi$
$282$		0			0
$283$	−16.6384			−0.989051			−0.494526	−	0.869163i	$-0.664658\pi$
$283$	−16.6384			−0.989051			−0.494526	+	0.869163i	$0.664658\pi$
$284$		0			0
$285$	−15.9040	−	4.03069i	−0.942072	−	0.238757i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.62563			−0.330919
$290$		0			0
$291$	−12.2259	+	26.4401i	−0.716697	+	1.54995i
$292$		0			0
$293$			4.87957i			0.285067i	0.989790	+	0.142534i	$0.0455249\pi$
$293$			4.87957i			0.285067i	−0.989790	+	0.142534i	$0.954475\pi$
$294$		0			0
$295$	4.55195	−	0.851410i	0.265025	−	0.0495710i
$296$		0			0
$297$	−2.40364	−	8.63660i	−0.139474	−	0.501146i
$298$		0			0
$299$	3.23615			0.187151
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−11.4403	+	24.7411i	−0.657228	+	1.42134i
$304$		0			0
$305$	−3.84193	−	20.5404i	−0.219988	−	1.17614i
$306$		0			0
$307$	18.5719			1.05995			0.529977	−	0.848012i	$-0.322200\pi$
$307$	18.5719			1.05995			0.529977	+	0.848012i	$0.322200\pi$
$308$		0			0
$309$	10.7597	−	23.2691i	0.612095	−	1.32373i
$310$		0			0
$311$	−28.8269			−1.63462			−0.817312	−	0.576195i	$-0.804537\pi$
$311$	−28.8269			−1.63462			−0.817312	+	0.576195i	$0.804537\pi$
$312$		0			0
$313$	5.31901			0.300648			0.150324	−	0.988637i	$-0.451968\pi$
$313$	5.31901			0.300648			0.150324	+	0.988637i	$0.451968\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.959456			0.0538884			0.0269442	−	0.999637i	$-0.491422\pi$
$317$	0.959456			0.0538884			0.0269442	+	0.999637i	$0.491422\pi$
$318$		0			0
$319$	−5.13575			−0.287547
$320$		0			0
$321$	6.26585	−	13.5507i	0.349726	−	0.756326i
$322$		0			0
$323$	20.1502			1.12119
$324$		0			0
$325$	−5.78451	+	2.24235i	−0.320867	+	0.124383i
$326$		0			0
$327$	4.75618	−	10.2858i	0.263017	−	0.568807i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−17.8535			−0.981319			−0.490659	−	0.871351i	$-0.663244\pi$
$331$	−17.8535			−0.981319			−0.490659	+	0.871351i	$0.663244\pi$
$332$		0			0
$333$	12.3847	−	10.5284i	0.678676	−	0.576950i
$334$		0			0
$335$	−3.89184	−	20.8072i	−0.212634	−	1.13682i
$336$		0			0
$337$			1.47586i			0.0803951i	0.999192	+	0.0401975i	$0.0127987\pi$
$337$			1.47586i			0.0803951i	−0.999192	+	0.0401975i	$0.987201\pi$
$338$		0			0
$339$	14.0425	−	30.3687i	0.762684	−	1.64940i
$340$		0			0
$341$	−0.309644			−0.0167681
$342$		0			0
$343$		0			0
$344$		0			0
$345$	2.48160	−	9.79173i	0.133605	−	0.527169i
$346$		0			0
$347$	20.7568			1.11428			0.557141	−	0.830418i	$-0.311898\pi$
$347$	20.7568			1.11428			0.557141	+	0.830418i	$0.311898\pi$
$348$		0			0
$349$			12.0460i			0.644806i	0.946603	+	0.322403i	$0.104491\pi$
$349$			12.0460i			0.644806i	−0.946603	+	0.322403i	$0.895509\pi$
$350$		0			0
$351$	6.21125	−	1.72865i	0.331532	−	0.0922683i
$352$		0			0
$353$			5.44001i			0.289542i	0.989465	+	0.144771i	$0.0462446\pi$
$353$			5.44001i			0.289542i	−0.989465	+	0.144771i	$0.953755\pi$
$354$		0			0
$355$	−3.51980	−	18.8182i	−0.186812	−	0.998764i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			8.90499i			0.469987i	0.971997	+	0.234994i	$0.0755069\pi$
$359$			8.90499i			0.469987i	−0.971997	+	0.234994i	$0.924493\pi$
$360$		0			0
$361$	1.05440			0.0554947
$362$		0			0
$363$	5.83259	−	12.6137i	0.306132	−	0.662048i
$364$		0			0
$365$	−34.4324	+	6.44033i	−1.80227	+	0.337102i
$366$		0			0
$367$	−16.7979			−0.876843			−0.438422	−	0.898769i	$-0.644462\pi$
$367$	−16.7979			−0.876843			−0.438422	+	0.898769i	$0.644462\pi$
$368$		0			0
$369$	2.75007	+	3.23495i	0.143163	+	0.168405i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			4.38678i			0.227139i	0.993530	+	0.113569i	$0.0362285\pi$
$373$			4.38678i			0.227139i	−0.993530	+	0.113569i	$0.963772\pi$
$374$		0			0
$375$	2.34898	+	19.2219i	0.121301	+	0.992616i
$376$		0			0
$377$		−	3.69352i		−	0.190226i
$378$		0			0
$379$	−12.8658			−0.660873			−0.330436	−	0.943828i	$-0.607196\pi$
$379$	−12.8658			−0.660873			−0.330436	+	0.943828i	$0.607196\pi$
$380$		0			0
$381$	24.9421	+	11.5333i	1.27782	+	0.590867i
$382$		0			0
$383$			33.0774i			1.69018i	0.534628	+	0.845088i	$0.320452\pi$
$383$			33.0774i			1.69018i	−0.534628	+	0.845088i	$0.679548\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	18.3693	−	15.6160i	0.933764	−	0.793803i
$388$		0			0
$389$		−	0.922630i		−	0.0467792i	−0.999726	−	0.0233896i	$-0.992554\pi$
$389$		−	0.922630i		−	0.0467792i	0.999726	−	0.0233896i	$-0.00744582\pi$
$390$		0			0
$391$			12.4060i			0.627399i
$392$		0			0
$393$	10.5844	−	22.8900i	0.533910	−	1.15465i
$394$		0			0
$395$	−1.41531	+	0.264723i	−0.0712118	+	0.0133197i
$396$		0			0
$397$	−22.2990			−1.11916			−0.559578	−	0.828778i	$-0.689037\pi$
$397$	−22.2990			−1.11916			−0.559578	+	0.828778i	$0.689037\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.60434i			0.179992i	0.995942	+	0.0899961i	$0.0286855\pi$
$401$			3.60434i			0.179992i	−0.995942	+	0.0899961i	$0.971315\pi$
$402$		0			0
$403$		−	0.222689i		−	0.0110929i
$404$		0			0
$405$	−0.467408	−	20.1192i	−0.0232257	−	0.999730i
$406$		0			0
$407$	−9.34819			−0.463372
$408$		0			0
$409$			25.9838i			1.28481i	0.766363	+	0.642407i	$0.222064\pi$
$409$			25.9838i			1.28481i	−0.766363	+	0.642407i	$0.777936\pi$
$410$		0			0
$411$	−7.83654	+	16.9475i	−0.386548	+	0.835958i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.94754	−	10.4123i	−0.0956010	−	0.511118i
$416$		0			0
$417$	−27.5741	−	12.7503i	−1.35031	−	0.624385i
$418$		0			0
$419$	−11.7471			−0.573882			−0.286941	−	0.957948i	$-0.592638\pi$
$419$	−11.7471			−0.573882			−0.286941	+	0.957948i	$0.592638\pi$
$420$		0			0
$421$	10.1171			0.493077			0.246538	−	0.969133i	$-0.420707\pi$
$421$	10.1171			0.493077			0.246538	+	0.969133i	$0.420707\pi$
$422$		0			0
$423$	−17.9997	+	15.3018i	−0.875176	+	0.743997i
$424$		0			0
$425$	−8.59622	−	22.1753i	−0.416978	−	1.07566i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−3.36544	−	1.55618i	−0.162485	−	0.0751332i
$430$		0			0
$431$		−	13.8264i		−	0.665994i	−0.942928	−	0.332997i	$-0.891940\pi$
$431$		−	13.8264i		−	0.665994i	0.942928	−	0.332997i	$-0.108060\pi$
$432$		0			0
$433$	−4.69553			−0.225653			−0.112826	−	0.993615i	$-0.535990\pi$
$433$	−4.69553			−0.225653			−0.112826	+	0.993615i	$0.535990\pi$
$434$		0			0
$435$	−11.1756	−	2.83233i	−0.535829	−	0.135800i
$436$		0			0
$437$		−	11.0487i		−	0.528530i
$438$		0			0
$439$			39.2207i			1.87190i	0.352133	+	0.935950i	$0.385457\pi$
$439$			39.2207i			1.87190i	−0.352133	+	0.935950i	$0.614543\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−31.7445			−1.50823			−0.754114	−	0.656743i	$-0.771934\pi$
$443$	−31.7445			−1.50823			−0.754114	+	0.656743i	$0.771934\pi$
$444$		0			0
$445$	18.1492	−	3.39468i	0.860354	−	0.160923i
$446$		0			0
$447$	28.0979	+	12.9925i	1.32899	+	0.614525i
$448$		0			0
$449$			31.3461i			1.47931i	0.672985	+	0.739656i	$0.265012\pi$
$449$			31.3461i			1.47931i	−0.672985	+	0.739656i	$0.734988\pi$
$450$		0			0
$451$		−	2.44180i		−	0.114980i
$452$		0			0
$453$	−15.0518	+	32.5514i	−0.707196	+	1.52940i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	26.5180i		−	1.24046i	−0.784420	−	0.620229i	$-0.787039\pi$
$457$		−	26.5180i		−	1.24046i	0.784420	−	0.620229i	$-0.212961\pi$
$458$		0			0
$459$	6.62689	+	23.8113i	0.309317	+	1.11141i
$460$		0			0
$461$	−15.1409			−0.705184			−0.352592	−	0.935777i	$-0.614700\pi$
$461$	−15.1409			−0.705184			−0.352592	+	0.935777i	$0.614700\pi$
$462$		0			0
$463$		−	36.2691i		−	1.68557i	−0.538252	−	0.842784i	$-0.680915\pi$
$463$		−	36.2691i		−	1.68557i	0.538252	−	0.842784i	$-0.319085\pi$
$464$		0			0
$465$	−0.673797	−	0.170766i	−0.0312466	−	0.00791909i
$466$		0			0
$467$			7.78888i			0.360426i	0.983628	+	0.180213i	$0.0576788\pi$
$467$			7.78888i			0.360426i	−0.983628	+	0.180213i	$0.942321\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	12.4446	−	26.9131i	0.573418	−	1.24009i
$472$		0			0
$473$	−13.8655			−0.637536
$474$		0			0
$475$	7.65572	+	19.7492i	0.351269	+	0.906154i
$476$		0			0
$477$	−12.9036	−	15.1787i	−0.590814	−	0.694984i
$478$		0			0
$479$	−33.7914			−1.54397			−0.771984	−	0.635642i	$-0.780735\pi$
$479$	−33.7914			−1.54397			−0.771984	+	0.635642i	$0.780735\pi$
$480$		0			0
$481$		−	6.72300i		−	0.306543i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	36.9655	−	6.91414i	1.67852	−	0.313955i
$486$		0			0
$487$			10.8580i			0.492022i	0.969267	+	0.246011i	$0.0791199\pi$
$487$			10.8580i			0.492022i	−0.969267	+	0.246011i	$0.920880\pi$
$488$		0			0
$489$	19.1208	+	8.84150i	0.864674	+	0.399826i
$490$		0			0
$491$			8.39794i			0.378994i	0.981881	+	0.189497i	$0.0606857\pi$
$491$			8.39794i			0.378994i	−0.981881	+	0.189497i	$0.939314\pi$
$492$		0			0
$493$	14.1594			0.637706
$494$		0			0
$495$	−7.28934	+	8.98959i	−0.327632	+	0.404052i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−14.3785			−0.643668			−0.321834	−	0.946796i	$-0.604299\pi$
$499$	−14.3785			−0.643668			−0.321834	+	0.946796i	$0.604299\pi$
$500$		0			0
$501$	3.52910	+	1.63186i	0.157669	+	0.0729061i
$502$		0			0
$503$		−	2.96439i		−	0.132175i	−0.997814	−	0.0660877i	$-0.978948\pi$
$503$		−	2.96439i		−	0.132175i	0.997814	−	0.0660877i	$-0.0210517\pi$
$504$		0			0
$505$	34.5901	−	6.46984i	1.53924	−	0.287904i
$506$		0			0
$507$	−8.33115	+	18.0172i	−0.369999	+	0.800170i
$508$		0			0
$509$	−30.5783			−1.35536			−0.677680	−	0.735357i	$-0.737015\pi$
$509$	−30.5783			−1.35536			−0.677680	+	0.735357i	$0.737015\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−5.90186	−	21.2061i	−0.260573	−	0.936273i
$514$		0			0
$515$	−32.5322	+	6.08491i	−1.43354	+	0.268133i
$516$		0			0
$517$	13.5865			0.597535
$518$		0			0
$519$	30.3129	+	14.0167i	1.33059	+	0.615266i
$520$		0			0
$521$	−26.6701			−1.16844			−0.584220	−	0.811596i	$-0.698599\pi$
$521$	−26.6701			−1.16844			−0.584220	+	0.811596i	$0.698599\pi$
$522$		0			0
$523$	20.2062			0.883556			0.441778	−	0.897124i	$-0.354348\pi$
$523$	20.2062			0.883556			0.441778	+	0.897124i	$0.354348\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.853693			0.0371875
$528$		0			0
$529$	−16.1976			−0.704243
$530$		0			0
$531$	4.02414	+	4.73366i	0.174633	+	0.205424i
$532$		0			0
$533$	1.75609			0.0760647
$534$		0			0
$535$	−18.9450	+	3.54353i	−0.819064	+	0.153200i
$536$		0			0
$537$	−35.1889	−	16.2714i	−1.51851	−	0.702161i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−15.2721			−0.656597			−0.328298	−	0.944574i	$-0.606475\pi$
$541$	−15.2721			−0.656597			−0.328298	+	0.944574i	$0.606475\pi$
$542$		0			0
$543$	13.2269	+	6.11614i	0.567621	+	0.262469i
$544$		0			0
$545$	−14.3805	+	2.68976i	−0.615991	+	0.115217i
$546$		0			0
$547$		−	15.1601i		−	0.648199i	−0.946023	−	0.324099i	$-0.894939\pi$
$547$		−	15.1601i		−	0.648199i	0.946023	−	0.324099i	$-0.105061\pi$
$548$		0			0
$549$	21.3603	−	18.1587i	0.911637	−	0.774993i
$550$		0			0
$551$	−12.6102			−0.537213
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−20.3420	−	5.15546i	−0.863471	−	0.218837i
$556$		0			0
$557$	−22.1748			−0.939574			−0.469787	−	0.882780i	$-0.655669\pi$
$557$	−22.1748			−0.939574			−0.469787	+	0.882780i	$0.655669\pi$
$558$		0			0
$559$		−	9.97175i		−	0.421760i
$560$		0			0
$561$	5.96574	−	12.9017i	0.251874	−	0.544708i
$562$		0			0
$563$			36.1336i			1.52285i	0.648252	+	0.761425i	$0.275500\pi$
$563$			36.1336i			1.52285i	−0.648252	+	0.761425i	$0.724500\pi$
$564$		0			0
$565$	−42.4580	+	7.94146i	−1.78622	+	0.334100i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			9.23544i			0.387170i	0.981084	+	0.193585i	$0.0620115\pi$
$569$			9.23544i			0.387170i	−0.981084	+	0.193585i	$0.937989\pi$
$570$		0			0
$571$	35.7525			1.49620			0.748098	−	0.663589i	$-0.230967\pi$
$571$	35.7525			1.49620			0.748098	+	0.663589i	$0.230967\pi$
$572$		0			0
$573$	33.6550	+	15.5621i	1.40596	+	0.650117i
$574$		0			0
$575$	−12.1591	+	4.71345i	−0.507070	+	0.196564i
$576$		0			0
$577$	−27.1169			−1.12889			−0.564447	−	0.825470i	$-0.690910\pi$
$577$	−27.1169			−1.12889			−0.564447	+	0.825470i	$0.690910\pi$
$578$		0			0
$579$	−13.9827	−	6.46563i	−0.581103	−	0.268702i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			11.4572i			0.474507i
$584$		0			0
$585$	−6.46511	−	5.24233i	−0.267299	−	0.216744i
$586$		0			0
$587$		−	7.14942i		−	0.295088i	−0.989055	−	0.147544i	$-0.952863\pi$
$587$		−	7.14942i		−	0.295088i	0.989055	−	0.147544i	$-0.0471368\pi$
$588$		0			0
$589$	−0.760292			−0.0313273
$590$		0			0
$591$	−6.32870	+	13.6866i	−0.260328	+	0.562992i
$592$		0			0
$593$			41.2878i			1.69549i	0.530406	+	0.847744i	$0.322040\pi$
$593$			41.2878i			1.69549i	−0.530406	+	0.847744i	$0.677960\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19.5521	+	9.04093i	0.800216	+	0.370021i
$598$		0			0
$599$		−	11.4425i		−	0.467527i	−0.972293	−	0.233763i	$-0.924896\pi$
$599$		−	11.4425i		−	0.467527i	0.972293	−	0.233763i	$-0.0751041\pi$
$600$		0			0
$601$			12.5356i			0.511340i	0.966764	+	0.255670i	$0.0822960\pi$
$601$			12.5356i			0.511340i	−0.966764	+	0.255670i	$0.917704\pi$
$602$		0			0
$603$	21.6378	−	18.3946i	0.881160	−	0.749084i
$604$		0			0
$605$	−17.6350	+	3.29851i	−0.716966	+	0.134103i
$606$		0			0
$607$	−2.89316			−0.117430			−0.0587148	−	0.998275i	$-0.518700\pi$
$607$	−2.89316			−0.117430			−0.0587148	+	0.998275i	$0.518700\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			9.77112i			0.395297i
$612$		0			0
$613$		−	24.5127i		−	0.990058i	−0.868877	−	0.495029i	$-0.835158\pi$
$613$		−	24.5127i		−	0.990058i	0.868877	−	0.495029i	$-0.164842\pi$
$614$		0			0
$615$	1.34664	−	5.31347i	0.0543017	−	0.214260i
$616$		0			0
$617$	−14.0471			−0.565514			−0.282757	−	0.959192i	$-0.591249\pi$
$617$	−14.0471			−0.565514			−0.282757	+	0.959192i	$0.591249\pi$
$618$		0			0
$619$		−	3.19919i		−	0.128586i	−0.997931	−	0.0642932i	$-0.979521\pi$
$619$		−	3.19919i		−	0.128586i	0.997931	−	0.0642932i	$-0.0204793\pi$
$620$		0			0
$621$	13.0561	−	3.63364i	0.523924	−	0.145813i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	18.4680	−	16.8503i	0.738721	−	0.674011i
$626$		0			0
$627$	−5.31304	+	11.4901i	−0.212182	+	0.458871i
$628$		0			0
$629$	25.7731			1.02764
$630$		0			0
$631$	16.6389			0.662384			0.331192	−	0.943563i	$-0.392549\pi$
$631$	16.6389			0.662384			0.331192	+	0.943563i	$0.392549\pi$
$632$		0			0
$633$	−5.76709	+	12.4720i	−0.229221	+	0.495719i
$634$		0			0
$635$	−6.52241	−	34.8712i	−0.258834	−	1.38382i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	19.5694	−	16.6361i	0.774152	−	0.658116i
$640$		0			0
$641$		−	17.2353i		−	0.680755i	−0.940289	−	0.340378i	$-0.889445\pi$
$641$		−	17.2353i		−	0.680755i	0.940289	−	0.340378i	$-0.110555\pi$
$642$		0			0
$643$	37.6573			1.48506			0.742530	−	0.669813i	$-0.233626\pi$
$643$	37.6573			1.48506			0.742530	+	0.669813i	$0.233626\pi$
$644$		0			0
$645$	−30.1719	−	7.64672i	−1.18802	−	0.301089i
$646$		0			0
$647$			33.4209i			1.31391i	0.753929	+	0.656956i	$0.228156\pi$
$647$			33.4209i			1.31391i	−0.753929	+	0.656956i	$0.771844\pi$
$648$		0			0
$649$		−	3.57306i		−	0.140255i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.8570			−0.503135			−0.251567	−	0.967840i	$-0.580946\pi$
$653$	−12.8570			−0.503135			−0.251567	+	0.967840i	$0.580946\pi$
$654$		0			0
$655$	−32.0022	+	5.98578i	−1.25043	+	0.233884i
$656$		0			0
$657$	−30.4398	−	35.8069i	−1.18757	−	1.39696i
$658$		0			0
$659$		−	22.3060i		−	0.868919i	−0.900691	−	0.434460i	$-0.856939\pi$
$659$		−	22.3060i		−	0.868919i	0.900691	−	0.434460i	$-0.143061\pi$
$660$		0			0
$661$		−	12.5102i		−	0.486589i	−0.969953	−	0.243294i	$-0.921772\pi$
$661$		−	12.5102i		−	0.486589i	0.969953	−	0.243294i	$-0.0782281\pi$
$662$		0			0
$663$	9.27858	+	4.29042i	0.360350	+	0.166626i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	7.76382i		−	0.300616i
$668$		0			0
$669$	−2.00615	+	4.33856i	−0.0775624	+	0.167738i
$670$		0			0
$671$	−16.1232			−0.622429
$672$		0			0
$673$		−	49.4108i		−	1.90464i	−0.305094	−	0.952322i	$-0.598688\pi$
$673$		−	49.4108i		−	1.90464i	0.305094	−	0.952322i	$-0.401312\pi$
$674$		0			0
$675$	−20.8196	+	15.5417i	−0.801347	+	0.598200i
$676$		0			0
$677$			3.44410i			0.132367i	0.997807	+	0.0661837i	$0.0210824\pi$
$677$			3.44410i			0.132367i	−0.997807	+	0.0661837i	$0.978918\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−10.8989	−	5.03968i	−0.417648	−	0.193121i
$682$		0			0
$683$	46.5764			1.78220			0.891098	−	0.453811i	$-0.149936\pi$
$683$	46.5764			1.78220			0.891098	+	0.453811i	$0.149936\pi$
$684$		0			0
$685$	23.6940	−	4.43180i	0.905303	−	0.169330i
$686$		0			0
$687$	−26.7920	−	12.3887i	−1.02218	−	0.472657i
$688$		0			0
$689$	−8.23973			−0.313909
$690$		0			0
$691$		−	10.6259i		−	0.404226i	−0.979362	−	0.202113i	$-0.935219\pi$
$691$		−	10.6259i		−	0.404226i	0.979362	−	0.202113i	$-0.0647809\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.21068	+	38.5509i	0.273517	+	1.46232i
$696$		0			0
$697$			6.73210i			0.254997i
$698$		0			0
$699$	−15.5330	+	33.5919i	−0.587510	+	1.27056i
$700$		0			0
$701$			23.3919i			0.883500i	0.897138	+	0.441750i	$0.145642\pi$
$701$			23.3919i			0.883500i	−0.897138	+	0.441750i	$0.854358\pi$
$702$		0			0
$703$	−22.9533			−0.865701
$704$		0			0
$705$	29.5648	+	7.49287i	1.11348	+	0.282198i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−8.50206			−0.319302			−0.159651	−	0.987174i	$-0.551037\pi$
$709$	−8.50206			−0.319302			−0.159651	+	0.987174i	$0.551037\pi$
$710$		0			0
$711$	−1.25120	−	1.47180i	−0.0469236	−	0.0551970i
$712$		0			0
$713$		−	0.468094i		−	0.0175303i
$714$		0			0
$715$	0.880068	+	4.70517i	0.0329127	+	0.175963i
$716$		0			0
$717$	2.39633	+	1.10806i	0.0894925	+	0.0413814i
$718$		0			0
$719$	40.5970			1.51401			0.757006	−	0.653408i	$-0.226661\pi$
$719$	40.5970			1.51401			0.757006	+	0.653408i	$0.226661\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−27.1851	−	12.5704i	−1.01102	−	0.467498i
$724$		0			0
$725$	5.37961	+	13.8776i	0.199794	+	0.515400i
$726$		0			0
$727$	15.8178			0.586651			0.293325	−	0.956013i	$-0.405238\pi$
$727$	15.8178			0.586651			0.293325	+	0.956013i	$0.405238\pi$
$728$		0			0
$729$	23.1181	−	13.9483i	0.856224	−	0.516604i
$730$		0			0
$731$	38.2274			1.41389
$732$		0			0
$733$	−20.1829			−0.745473			−0.372736	−	0.927937i	$-0.621580\pi$
$733$	−20.1829			−0.745473			−0.372736	+	0.927937i	$0.621580\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.3326			−0.601620
$738$		0			0
$739$	20.6458			0.759469			0.379734	−	0.925096i	$-0.376015\pi$
$739$	20.6458			0.759469			0.379734	+	0.925096i	$0.376015\pi$
$740$		0			0
$741$	−8.26342	−	3.82102i	−0.303564	−	0.140369i
$742$		0			0
$743$	31.8671			1.16909			0.584545	−	0.811361i	$-0.301273\pi$
$743$	31.8671			1.16909			0.584545	+	0.811361i	$0.301273\pi$
$744$		0			0
$745$	−7.34766	−	39.2833i	−0.269198	−	1.43923i
$746$		0			0
$747$	10.8279	−	9.20494i	0.396173	−	0.336791i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	35.0974			1.28072			0.640360	−	0.768075i	$-0.278785\pi$
$751$	35.0974			1.28072			0.640360	+	0.768075i	$0.278785\pi$
$752$		0			0
$753$	8.80224	−	19.0359i	0.320771	−	0.693708i
$754$		0			0
$755$	45.5097	−	8.51226i	1.65627	−	0.309793i
$756$		0			0
$757$		−	37.1926i		−	1.35179i	−0.736999	−	0.675894i	$-0.763758\pi$
$757$		−	37.1926i		−	1.35179i	0.736999	−	0.675894i	$-0.236242\pi$
$758$		0			0
$759$	−7.07419	−	3.27111i	−0.256777	−	0.118734i
$760$		0			0
$761$	−26.4360			−0.958305			−0.479152	−	0.877732i	$-0.659056\pi$
$761$	−26.4360			−0.958305			−0.479152	+	0.877732i	$0.659056\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	20.0969	−	24.7845i	0.726604	−	0.896084i
$766$		0			0
$767$	2.56966			0.0927852
$768$		0			0
$769$		−	47.0058i		−	1.69507i	−0.530738	−	0.847536i	$-0.678085\pi$
$769$		−	47.0058i		−	1.69507i	0.530738	−	0.847536i	$-0.321915\pi$
$770$		0			0
$771$	−25.9884	−	12.0171i	−0.935950	−	0.432784i
$772$		0			0
$773$		−	20.9970i		−	0.755210i	−0.925967	−	0.377605i	$-0.876748\pi$
$773$		−	20.9970i		−	0.755210i	0.925967	−	0.377605i	$-0.123252\pi$
$774$		0			0
$775$	0.324346	+	0.836703i	0.0116509	+	0.0300553i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	5.99555i		−	0.214813i
$780$		0			0
$781$	−14.7713			−0.528560
$782$		0			0
$783$	−4.14718	−	14.9013i	−0.148208	−	0.532531i
$784$		0			0
$785$	−37.6267	+	7.03782i	−1.34296	+	0.251190i
$786$		0			0
$787$	−34.2223			−1.21989			−0.609946	−	0.792443i	$-0.708809\pi$
$787$	−34.2223			−1.21989			−0.609946	+	0.792443i	$0.708809\pi$
$788$		0			0
$789$	17.6658	−	38.2045i	0.628919	−	1.36011i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	11.5954i		−	0.411766i
$794$		0			0
$795$	−6.31854	+	24.9312i	−0.224096	+	0.884220i
$796$		0			0
$797$		−	32.2568i		−	1.14259i	−0.820744	−	0.571296i	$-0.806441\pi$
$797$		−	32.2568i		−	1.14259i	0.820744	−	0.571296i	$-0.193559\pi$
$798$		0			0
$799$	−37.4583			−1.32518
$800$		0			0
$801$	16.0448	+	18.8737i	0.566914	+	0.666870i
$802$		0			0
$803$			27.0277i			0.953787i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−4.29704	+	9.29289i	−0.151263	+	0.327125i
$808$		0			0
$809$		−	9.29300i		−	0.326725i	−0.986566	−	0.163362i	$-0.947766\pi$
$809$		−	9.29300i		−	0.326725i	0.986566	−	0.163362i	$-0.0522339\pi$
$810$		0			0
$811$		−	20.6750i		−	0.725996i	−0.931790	−	0.362998i	$-0.881753\pi$
$811$		−	20.6750i		−	0.725996i	0.931790	−	0.362998i	$-0.118247\pi$
$812$		0			0
$813$	−0.213743	−	0.0988351i	−0.00749630	−	0.00346630i
$814$		0			0
$815$	−5.00014	−	26.7326i	−0.175147	−	0.936401i
$816$		0			0
$817$	−34.0450			−1.19108
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	7.54303i		−	0.263253i	−0.991299	−	0.131627i	$-0.957980\pi$
$821$		−	7.54303i		−	0.263253i	0.991299	−	0.131627i	$-0.0420200\pi$
$822$		0			0
$823$			17.4639i			0.608755i	0.952551	+	0.304377i	$0.0984484\pi$
$823$			17.4639i			0.608755i	−0.952551	+	0.304377i	$0.901552\pi$
$824$		0			0
$825$	14.9115	+	0.945248i	0.519151	+	0.0329093i
$826$		0			0
$827$	−38.3714			−1.33430			−0.667152	−	0.744921i	$-0.732487\pi$
$827$	−38.3714			−1.33430			−0.667152	+	0.744921i	$0.732487\pi$
$828$		0			0
$829$			24.0625i			0.835725i	0.908510	+	0.417863i	$0.137221\pi$
$829$			24.0625i			0.835725i	−0.908510	+	0.417863i	$0.862779\pi$
$830$		0			0
$831$	−27.6191	−	12.7711i	−0.958098	−	0.443025i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.922866	−	4.93398i	−0.0319371	−	0.170747i
$836$		0			0
$837$	−0.250041	−	0.898429i	−0.00864268	−	0.0310542i
$838$		0			0
$839$	−44.6178			−1.54038			−0.770188	−	0.637817i	$-0.779838\pi$
$839$	−44.6178			−1.54038			−0.770188	+	0.637817i	$0.779838\pi$
$840$		0			0
$841$	20.1389			0.694445
$842$		0			0
$843$	12.7726	+	5.90608i	0.439913	+	0.203416i
$844$		0			0
$845$	25.1895	−	4.71152i	0.866545	−	0.162081i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−12.0953	+	26.1575i	−0.415108	+	0.897724i
$850$		0			0
$851$		−	14.1318i		−	0.484433i
$852$		0			0
$853$	18.8961			0.646991			0.323495	−	0.946230i	$-0.395142\pi$
$853$	18.8961			0.646991			0.323495	+	0.946230i	$0.395142\pi$
$854$		0			0
$855$	−17.8981	+	22.0728i	−0.612102	+	0.754875i
$856$		0			0
$857$		−	5.73649i		−	0.195955i	−0.995189	−	0.0979775i	$-0.968763\pi$
$857$		−	5.73649i		−	0.195955i	0.995189	−	0.0979775i	$-0.0312373\pi$
$858$		0			0
$859$			7.17954i			0.244963i	0.992471	+	0.122481i	$0.0390852\pi$
$859$			7.17954i			0.244963i	−0.992471	+	0.122481i	$0.960915\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37.5538			−1.27835			−0.639173	−	0.769063i	$-0.720723\pi$
$863$	−37.5538			−1.27835			−0.639173	+	0.769063i	$0.720723\pi$
$864$		0			0
$865$	−7.92689	−	42.3800i	−0.269522	−	1.44096i
$866$		0			0
$867$	−4.08954	+	8.84414i	−0.138888	+	0.300363i
$868$		0			0
$869$			1.11095i			0.0376863i
$870$		0			0
$871$		−	11.7461i		−	0.398000i
$872$		0			0
$873$	32.6793	+	38.4412i	1.10603	+	1.30104i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			40.3087i			1.36113i	0.732689	+	0.680564i	$0.238265\pi$
$877$			40.3087i			1.36113i	−0.732689	+	0.680564i	$0.761735\pi$
$878$		0			0
$879$	7.67124	+	3.54719i	0.258745	+	0.119644i
$880$		0			0
$881$	20.5873			0.693603			0.346802	−	0.937938i	$-0.387268\pi$
$881$	20.5873			0.693603			0.346802	+	0.937938i	$0.387268\pi$
$882$		0			0
$883$		−	13.6651i		−	0.459868i	−0.973206	−	0.229934i	$-0.926149\pi$
$883$		−	13.6651i		−	0.459868i	0.973206	−	0.229934i	$-0.0738511\pi$
$884$		0			0
$885$	1.97052	−	7.77512i	0.0662382	−	0.261358i
$886$		0			0
$887$		−	10.5807i		−	0.355266i	−0.984097	−	0.177633i	$-0.943156\pi$
$887$		−	10.5807i		−	0.355266i	0.984097	−	0.177633i	$-0.0568440\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−15.3251	−	2.49955i	−0.513409	−	0.0837381i
$892$		0			0
$893$	33.3601			1.11635
$894$		0			0
$895$	9.20195	+	49.1970i	0.307587	+	1.64448i
$896$		0			0
$897$	2.35251	−	5.08760i	0.0785481	−	0.169870i
$898$		0			0
$899$	−0.534251			−0.0178183
$900$		0			0
$901$		−	31.5876i		−	1.05234i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.45886	−	18.4923i	−0.114976	−	0.614706i
$906$		0			0
$907$			4.44682i			0.147654i	0.997271	+	0.0738271i	$0.0235213\pi$
$907$			4.44682i			0.147654i	−0.997271	+	0.0738271i	$0.976479\pi$
$908$		0			0
$909$	30.5793	+	35.9709i	1.01425	+	1.19308i
$910$		0			0
$911$		−	56.2126i		−	1.86241i	−0.364501	−	0.931203i	$-0.618760\pi$
$911$		−	56.2126i		−	1.86241i	0.364501	−	0.931203i	$-0.381240\pi$
$912$		0			0
$913$	−8.17312			−0.270491
$914$		0			0
$915$	−35.0847	−	8.89182i	−1.15986	−	0.293954i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	43.3160			1.42886			0.714432	−	0.699705i	$-0.246685\pi$
$919$	43.3160			1.42886			0.714432	+	0.699705i	$0.246685\pi$
$920$		0			0
$921$	13.5008	−	29.1972i	0.444867	−	0.962079i
$922$		0			0
$923$		−	10.6232i		−	0.349667i
$924$		0			0
$925$	9.79206	+	25.2602i	0.321961	+	0.830550i
$926$		0			0
$927$	−28.7600	−	33.8308i	−0.944602	−	1.11115i
$928$		0			0
$929$	−2.02666			−0.0664925			−0.0332462	−	0.999447i	$-0.510585\pi$
$929$	−2.02666			−0.0664925			−0.0332462	+	0.999447i	$0.510585\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−20.9557	+	45.3193i	−0.686058	+	1.48369i
$934$		0			0
$935$	−18.0376	+	3.37381i	−0.589893	+	0.110335i
$936$		0			0
$937$	−41.1052			−1.34285			−0.671424	−	0.741074i	$-0.734317\pi$
$937$	−41.1052			−1.34285			−0.671424	+	0.741074i	$0.734317\pi$
$938$		0			0
$939$	3.86664	−	8.36210i	0.126183	−	0.272887i
$940$		0			0
$941$	15.1589			0.494167			0.247083	−	0.968994i	$-0.420528\pi$
$941$	15.1589			0.494167			0.247083	+	0.968994i	$0.420528\pi$
$942$		0			0
$943$	3.69132			0.120206
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27.2201			0.884535			0.442267	−	0.896883i	$-0.354174\pi$
$947$	27.2201			0.884535			0.442267	+	0.896883i	$0.354174\pi$
$948$		0			0
$949$	−19.4377			−0.630975
$950$		0			0
$951$	0.697475	−	1.50838i	0.0226172	−	0.0489125i
$952$		0			0
$953$	−0.378966			−0.0122759			−0.00613795	−	0.999981i	$-0.501954\pi$
$953$	−0.378966			−0.0122759			−0.00613795	+	0.999981i	$0.501954\pi$
$954$		0			0
$955$	−8.80085	−	47.0526i	−0.284789	−	1.52259i
$956$		0			0
$957$	−3.73342	+	8.07400i	−0.120684	+	0.260995i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	30.9678			0.998961
$962$		0			0
$963$	−16.7483	−	19.7013i	−0.539706	−	0.634865i
$964$		0			0
$965$	3.65651	+	19.5490i	0.117707	+	0.629306i
$966$		0			0
$967$			15.2235i			0.489554i	0.969579	+	0.244777i	$0.0787147\pi$
$967$			15.2235i			0.489554i	−0.969579	+	0.244777i	$0.921285\pi$
$968$		0			0
$969$	14.6481	−	31.6784i	0.470566	−	1.01766i
$970$		0			0
$971$	28.0624			0.900566			0.450283	−	0.892886i	$-0.351323\pi$
$971$	28.0624			0.900566			0.450283	+	0.892886i	$0.351323\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−0.679801	+	10.7240i	−0.0217711	+	0.343443i
$976$		0			0
$977$	−34.2912			−1.09707			−0.548536	−	0.836127i	$-0.684814\pi$
$977$	−34.2912			−1.09707			−0.548536	+	0.836127i	$0.684814\pi$
$978$		0			0
$979$		−	14.2462i		−	0.455312i
$980$		0			0
$981$	−12.7130	−	14.9545i	−0.405895	−	0.477461i
$982$		0			0
$983$			9.35033i			0.298229i	0.988820	+	0.149115i	$0.0476424\pi$
$983$			9.35033i			0.298229i	−0.988820	+	0.149115i	$0.952358\pi$
$984$		0			0
$985$	19.1350	−	3.57907i	0.609693	−	0.114039i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	20.9607i		−	0.666513i
$990$		0			0
$991$	21.5334			0.684032			0.342016	−	0.939694i	$-0.388890\pi$
$991$	21.5334			0.684032			0.342016	+	0.939694i	$0.388890\pi$
$992$		0			0
$993$	−12.9786	+	28.0678i	−0.411863	+	0.890705i
$994$		0			0
$995$	−5.11292	−	27.3356i	−0.162091	−	0.866595i
$996$		0			0
$997$	39.6965			1.25720			0.628600	−	0.777729i	$-0.283628\pi$
$997$	39.6965			1.25720			0.628600	+	0.777729i	$0.283628\pi$
$998$		0			0
$999$	−7.54877	−	27.1237i	−0.238833	−	0.858156i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.f.b.1469.29	yes	48
3.2	odd	2	inner	2940.2.f.b.1469.32	yes	48
5.4	even	2	inner	2940.2.f.b.1469.19	yes	48
7.6	odd	2	inner	2940.2.f.b.1469.20	yes	48
15.14	odd	2	inner	2940.2.f.b.1469.18	yes	48
21.20	even	2	inner	2940.2.f.b.1469.17	✓	48
35.34	odd	2	inner	2940.2.f.b.1469.30	yes	48
105.104	even	2	inner	2940.2.f.b.1469.31	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2940.2.f.b.1469.17	✓	48	21.20	even	2	inner
2940.2.f.b.1469.18	yes	48	15.14	odd	2	inner
2940.2.f.b.1469.19	yes	48	5.4	even	2	inner
2940.2.f.b.1469.20	yes	48	7.6	odd	2	inner
2940.2.f.b.1469.29	yes	48	1.1	even	1	trivial
2940.2.f.b.1469.30	yes	48	35.34	odd	2	inner
2940.2.f.b.1469.31	yes	48	105.104	even	2	inner
2940.2.f.b.1469.32	yes	48	3.2	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:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.726948 - 1.57212i) q^{3} +(-2.19795 + 0.411111i) q^{5} +(-1.94309 - 2.28569i) q^{9} +1.72528i q^{11} -1.24079 q^{13} +(-0.951482 + 3.75429i) q^{15} -4.75664i q^{17} +4.23622i q^{19} -2.60815 q^{23} +(4.66198 - 1.80720i) q^{25} +(-5.00590 + 1.39319i) q^{27} +2.97676i q^{29} +0.179474i q^{31} +(2.71235 + 1.25419i) q^{33} +5.41835i q^{37} +(-0.901986 + 1.95066i) q^{39} -1.41531 q^{41} +8.03664i q^{43} +(5.21050 + 4.22501i) q^{45} -7.87495i q^{47} +(-7.47799 - 3.45783i) q^{51} +6.64074 q^{53} +(-0.709284 - 3.79209i) q^{55} +(6.65983 + 3.07951i) q^{57} -2.07100 q^{59} +9.34523i q^{61} +(2.72718 - 0.510101i) q^{65} +9.46664i q^{67} +(-1.89599 + 4.10031i) q^{69} +8.56168i q^{71} +15.6657 q^{73} +(0.547880 - 8.64291i) q^{75} +0.643921 q^{79} +(-1.44878 + 8.88263i) q^{81} +4.73726i q^{83} +(1.95551 + 10.4549i) q^{85} +(4.67981 + 2.16395i) q^{87} -8.25733 q^{89} +(0.282154 + 0.130468i) q^{93} +(-1.74156 - 9.31101i) q^{95} -16.8182 q^{97} +(3.94347 - 3.35239i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{9} - 16 q^{15} + 16 q^{25} + 56 q^{39} + 8 q^{51} + 48 q^{79} - 24 q^{81} + 32 q^{85} + 88 q^{99}+O(q^{100}) \) 48 * q - 8 * q^9 - 16 * q^15 + 16 * q^25 + 56 * q^39 + 8 * q^51 + 48 * q^79 - 24 * q^81 + 32 * q^85 + 88 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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