LMFDB - Embedded newform 4032.2.v.e.1583.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 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("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1583.10
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.e.3599.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0893433 - 0.0893433i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(-0.0893433 - 0.0893433i) q^{5} -1.00000 q^{7} +(-2.42480 + 2.42480i) q^{11} +(-3.86569 - 3.86569i) q^{13} +0.794810i q^{17} +(2.65233 - 2.65233i) q^{19} +3.92175i q^{23} -4.98404i q^{25} +(7.47818 - 7.47818i) q^{29} +5.55554i q^{31} +(0.0893433 + 0.0893433i) q^{35} +(-6.35455 + 6.35455i) q^{37} +6.96091 q^{41} +(1.25316 + 1.25316i) q^{43} -6.48295 q^{47} +1.00000 q^{49} +(-0.620687 - 0.620687i) q^{53} +0.433280 q^{55} +(3.39065 - 3.39065i) q^{59} +(7.51460 + 7.51460i) q^{61} +0.690747i q^{65} +(-2.22915 + 2.22915i) q^{67} +7.95182i q^{71} +12.9376i q^{73} +(2.42480 - 2.42480i) q^{77} +10.1941i q^{79} +(11.2820 + 11.2820i) q^{83} +(0.0710109 - 0.0710109i) q^{85} +5.52785 q^{89} +(3.86569 + 3.86569i) q^{91} -0.473935 q^{95} -4.33616 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{7}+O(q^{10}) $ 40 * q - 40 * q^7	$ 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) $ 40 * q - 40 * q^7 - 24 * q^13 + 32 * q^19 - 8 * q^37 - 32 * q^43 + 40 * q^49 - 48 * q^55 - 24 * q^61 + 64 * q^85 + 24 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	−0.0893433	−	0.0893433i	−0.0399555	−	0.0399555i	0.686847	−	0.726802i	$-0.258994\pi$
$5$	−0.0893433	−	0.0893433i	−0.0399555	−	0.0399555i	−0.726802	+	0.686847i	$0.758994\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.42480	+	2.42480i	−0.731106	+	0.731106i	−0.970839	−	0.239733i	$-0.922940\pi$
$11$	−2.42480	+	2.42480i	−0.731106	+	0.731106i	0.239733	+	0.970839i	$0.422940\pi$
$12$		0			0
$13$	−3.86569	−	3.86569i	−1.07215	−	1.07215i	−0.997186	−	0.0749629i	$-0.976116\pi$
$13$	−3.86569	−	3.86569i	−1.07215	−	1.07215i	−0.0749629	−	0.997186i	$-0.523884\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.794810i			0.192770i	0.995344	+	0.0963848i	$0.0307280\pi$
$17$			0.794810i			0.192770i	−0.995344	+	0.0963848i	$0.969272\pi$
$18$		0			0
$19$	2.65233	−	2.65233i	0.608485	−	0.608485i	−0.334065	−	0.942550i	$-0.608420\pi$
$19$	2.65233	−	2.65233i	0.608485	−	0.608485i	0.942550	+	0.334065i	$0.108420\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.92175i			0.817741i	0.912592	+	0.408871i	$0.134077\pi$
$23$			3.92175i			0.817741i	−0.912592	+	0.408871i	$0.865923\pi$
$24$		0			0
$25$		−	4.98404i		−	0.996807i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.47818	−	7.47818i	1.38866	−	1.38866i	0.560525	−	0.828138i	$-0.310599\pi$
$29$	7.47818	−	7.47818i	1.38866	−	1.38866i	0.828138	−	0.560525i	$-0.189401\pi$
$30$		0			0
$31$			5.55554i			0.997804i	0.866658	+	0.498902i	$0.166263\pi$
$31$			5.55554i			0.997804i	−0.866658	+	0.498902i	$0.833737\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.0893433	+	0.0893433i	0.0151018	+	0.0151018i
$36$		0			0
$37$	−6.35455	+	6.35455i	−1.04468	+	1.04468i	−0.0457270	+	0.998954i	$0.514560\pi$
$37$	−6.35455	+	6.35455i	−1.04468	+	1.04468i	−0.998954	+	0.0457270i	$0.985440\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.96091			1.08711			0.543556	−	0.839373i	$-0.317078\pi$
$41$	6.96091			1.08711			0.543556	+	0.839373i	$0.317078\pi$
$42$		0			0
$43$	1.25316	+	1.25316i	0.191105	+	0.191105i	0.796173	−	0.605069i	$-0.206854\pi$
$43$	1.25316	+	1.25316i	0.191105	+	0.191105i	−0.605069	+	0.796173i	$0.706854\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.48295			−0.945635			−0.472818	−	0.881160i	$-0.656763\pi$
$47$	−6.48295			−0.945635			−0.472818	+	0.881160i	$0.656763\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.620687	−	0.620687i	−0.0852579	−	0.0852579i	0.663192	−	0.748450i	$-0.269201\pi$
$53$	−0.620687	−	0.620687i	−0.0852579	−	0.0852579i	−0.748450	+	0.663192i	$0.769201\pi$
$54$		0			0
$55$	0.433280			0.0584234
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.39065	−	3.39065i	0.441425	−	0.441425i	−0.451066	−	0.892491i	$-0.648956\pi$
$59$	3.39065	−	3.39065i	0.441425	−	0.441425i	0.892491	+	0.451066i	$0.148956\pi$
$60$		0			0
$61$	7.51460	+	7.51460i	0.962146	+	0.962146i	0.999309	−	0.0371630i	$-0.0118321\pi$
$61$	7.51460	+	7.51460i	0.962146	+	0.962146i	−0.0371630	+	0.999309i	$0.511832\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.690747i			0.0856766i
$66$		0			0
$67$	−2.22915	+	2.22915i	−0.272334	+	0.272334i	−0.830039	−	0.557705i	$-0.811682\pi$
$67$	−2.22915	+	2.22915i	−0.272334	+	0.272334i	0.557705	+	0.830039i	$0.311682\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.95182i			0.943707i	0.881677	+	0.471853i	$0.156415\pi$
$71$			7.95182i			0.943707i	−0.881677	+	0.471853i	$0.843585\pi$
$72$		0			0
$73$			12.9376i			1.51423i	0.653283	+	0.757114i	$0.273391\pi$
$73$			12.9376i			1.51423i	−0.653283	+	0.757114i	$0.726609\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.42480	−	2.42480i	0.276332	−	0.276332i
$78$		0			0
$79$			10.1941i			1.14693i	0.819230	+	0.573465i	$0.194401\pi$
$79$			10.1941i			1.14693i	−0.819230	+	0.573465i	$0.805599\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.2820	+	11.2820i	1.23836	+	1.23836i	0.960670	+	0.277694i	$0.0895700\pi$
$83$	11.2820	+	11.2820i	1.23836	+	1.23836i	0.277694	+	0.960670i	$0.410430\pi$
$84$		0			0
$85$	0.0710109	−	0.0710109i	0.00770221	−	0.00770221i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.52785			0.585951			0.292976	−	0.956120i	$-0.405355\pi$
$89$	5.52785			0.585951			0.292976	+	0.956120i	$0.405355\pi$
$90$		0			0
$91$	3.86569	+	3.86569i	0.405234	+	0.405234i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.473935			−0.0486247
$96$		0			0
$97$	−4.33616			−0.440270			−0.220135	−	0.975469i	$-0.570650\pi$
$97$	−4.33616			−0.440270			−0.220135	+	0.975469i	$0.570650\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.66216	+	9.66216i	0.961421	+	0.961421i	0.999283	−	0.0378620i	$-0.0120547\pi$
$101$	9.66216	+	9.66216i	0.961421	+	0.961421i	−0.0378620	+	0.999283i	$0.512055\pi$
$102$		0			0
$103$	−16.4000			−1.61594			−0.807969	−	0.589225i	$-0.799433\pi$
$103$	−16.4000			−1.61594			−0.807969	+	0.589225i	$0.799433\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.48912	+	5.48912i	−0.530653	+	0.530653i	−0.920767	−	0.390114i	$-0.872436\pi$
$107$	−5.48912	+	5.48912i	−0.530653	+	0.530653i	0.390114	+	0.920767i	$0.372436\pi$
$108$		0			0
$109$	−14.6355	−	14.6355i	−1.40183	−	1.40183i	−0.794301	−	0.607524i	$-0.792163\pi$
$109$	−14.6355	−	14.6355i	−1.40183	−	1.40183i	−0.607524	−	0.794301i	$-0.707837\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.28264i			0.779165i	0.920992	+	0.389582i	$0.127381\pi$
$113$			8.28264i			0.779165i	−0.920992	+	0.389582i	$0.872619\pi$
$114$		0			0
$115$	0.350382	−	0.350382i	0.0326733	−	0.0326733i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.794810i		−	0.0728601i
$120$		0			0
$121$		−	0.759336i		−	0.0690305i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.892006	+	0.892006i	−0.0797835	+	0.0797835i
$126$		0			0
$127$			3.39522i			0.301277i	0.988589	+	0.150638i	$0.0481329\pi$
$127$			3.39522i			0.301277i	−0.988589	+	0.150638i	$0.951867\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.73455	+	8.73455i	0.763141	+	0.763141i	0.976889	−	0.213748i	$-0.0685671\pi$
$131$	8.73455	+	8.73455i	0.763141	+	0.763141i	−0.213748	+	0.976889i	$0.568567\pi$
$132$		0			0
$133$	−2.65233	+	2.65233i	−0.229986	+	0.229986i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.84889			−0.414269			−0.207134	−	0.978313i	$-0.566414\pi$
$137$	−4.84889			−0.414269			−0.207134	+	0.978313i	$0.566414\pi$
$138$		0			0
$139$	2.19464	+	2.19464i	0.186147	+	0.186147i	0.794028	−	0.607881i	$-0.207980\pi$
$139$	2.19464	+	2.19464i	0.186147	+	0.186147i	−0.607881	+	0.794028i	$0.707980\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.7471			1.56771
$144$		0			0
$145$	−1.33625			−0.110969
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.83836	+	2.83836i	0.232528	+	0.232528i	0.813747	−	0.581219i	$-0.197424\pi$
$149$	2.83836	+	2.83836i	0.232528	+	0.232528i	−0.581219	+	0.813747i	$0.697424\pi$
$150$		0			0
$151$	18.9821			1.54474			0.772370	−	0.635172i	$-0.219071\pi$
$151$	18.9821			1.54474			0.772370	+	0.635172i	$0.219071\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.496350	−	0.496350i	0.0398678	−	0.0398678i
$156$		0			0
$157$	−12.3221	−	12.3221i	−0.983412	−	0.983412i	0.0164522	−	0.999865i	$-0.494763\pi$
$157$	−12.3221	−	12.3221i	−0.983412	−	0.983412i	−0.999865	+	0.0164522i	$0.994763\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	3.92175i		−	0.309077i
$162$		0			0
$163$	3.50400	−	3.50400i	0.274454	−	0.274454i	−0.556436	−	0.830890i	$-0.687832\pi$
$163$	3.50400	−	3.50400i	0.274454	−	0.274454i	0.830890	+	0.556436i	$0.187832\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	17.4719i		−	1.35202i	−0.736893	−	0.676009i	$-0.763708\pi$
$167$		−	17.4719i		−	1.35202i	0.736893	−	0.676009i	$-0.236292\pi$
$168$		0			0
$169$			16.8871i			1.29901i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.45528	+	1.45528i	−0.110643	+	0.110643i	−0.760261	−	0.649618i	$-0.774929\pi$
$173$	−1.45528	+	1.45528i	−0.110643	+	0.110643i	0.649618	+	0.760261i	$0.274929\pi$
$174$		0			0
$175$			4.98404i			0.376758i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.22288	−	2.22288i	−0.166146	−	0.166146i	0.619137	−	0.785283i	$-0.287482\pi$
$179$	−2.22288	−	2.22288i	−0.166146	−	0.166146i	−0.785283	+	0.619137i	$0.787482\pi$
$180$		0			0
$181$	3.13227	−	3.13227i	0.232820	−	0.232820i	−0.581049	−	0.813869i	$-0.697358\pi$
$181$	3.13227	−	3.13227i	0.232820	−	0.232820i	0.813869	+	0.581049i	$0.197358\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.13547			0.0834816
$186$		0			0
$187$	−1.92726	−	1.92726i	−0.140935	−	0.140935i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.61134			0.261308			0.130654	−	0.991428i	$-0.458292\pi$
$191$	3.61134			0.261308			0.130654	+	0.991428i	$0.458292\pi$
$192$		0			0
$193$	−3.01689			−0.217160			−0.108580	−	0.994088i	$-0.534630\pi$
$193$	−3.01689			−0.217160			−0.108580	+	0.994088i	$0.534630\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.282108	−	0.282108i	−0.0200994	−	0.0200994i	0.696986	−	0.717085i	$-0.254524\pi$
$197$	−0.282108	−	0.282108i	−0.0200994	−	0.0200994i	−0.717085	+	0.696986i	$0.754524\pi$
$198$		0			0
$199$	−3.40015			−0.241030			−0.120515	−	0.992712i	$-0.538455\pi$
$199$	−3.40015			−0.241030			−0.120515	+	0.992712i	$0.538455\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.47818	+	7.47818i	−0.524865	+	0.524865i
$204$		0			0
$205$	−0.621911	−	0.621911i	−0.0434361	−	0.0434361i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			12.8627i			0.889734i
$210$		0			0
$211$	−6.04290	+	6.04290i	−0.416010	+	0.416010i	−0.883826	−	0.467816i	$-0.845041\pi$
$211$	−6.04290	+	6.04290i	−0.416010	+	0.416010i	0.467816	+	0.883826i	$0.345041\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	0.223922i		−	0.0152714i
$216$		0			0
$217$		−	5.55554i		−	0.377135i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.07249	−	3.07249i	0.206678	−	0.206678i
$222$		0			0
$223$			22.2108i			1.48734i	0.668545	+	0.743672i	$0.266918\pi$
$223$			22.2108i			1.48734i	−0.668545	+	0.743672i	$0.733082\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.1859	+	16.1859i	1.07430	+	1.07430i	0.997009	+	0.0772896i	$0.0246266\pi$
$227$	16.1859	+	16.1859i	1.07430	+	1.07430i	0.0772896	+	0.997009i	$0.475373\pi$
$228$		0			0
$229$	−7.75579	+	7.75579i	−0.512517	+	0.512517i	−0.915297	−	0.402780i	$-0.868044\pi$
$229$	−7.75579	+	7.75579i	−0.512517	+	0.512517i	0.402780	+	0.915297i	$0.368044\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.7060			1.22547			0.612734	−	0.790289i	$-0.290070\pi$
$233$	18.7060			1.22547			0.612734	+	0.790289i	$0.290070\pi$
$234$		0			0
$235$	0.579208	+	0.579208i	0.0377834	+	0.0377834i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.3964			−1.31933			−0.659667	−	0.751558i	$-0.729303\pi$
$239$	−20.3964			−1.31933			−0.659667	+	0.751558i	$0.729303\pi$
$240$		0			0
$241$	10.4808			0.675129			0.337565	−	0.941302i	$-0.390397\pi$
$241$	10.4808			0.675129			0.337565	+	0.941302i	$0.390397\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.0893433	−	0.0893433i	−0.00570793	−	0.00570793i
$246$		0			0
$247$	−20.5061			−1.30477
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.2508	−	22.2508i	1.40446	−	1.40446i	0.619320	−	0.785139i	$-0.287408\pi$
$251$	22.2508	−	22.2508i	1.40446	−	1.40446i	0.785139	−	0.619320i	$-0.212592\pi$
$252$		0			0
$253$	−9.50947	−	9.50947i	−0.597855	−	0.597855i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			29.3780i			1.83255i	0.400547	+	0.916276i	$0.368820\pi$
$257$			29.3780i			1.83255i	−0.400547	+	0.916276i	$0.631180\pi$
$258$		0			0
$259$	6.35455	−	6.35455i	0.394852	−	0.394852i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			8.75090i			0.539603i	0.962916	+	0.269802i	$0.0869582\pi$
$263$			8.75090i			0.539603i	−0.962916	+	0.269802i	$0.913042\pi$
$264$		0			0
$265$			0.110908i			0.00681305i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.92219	−	8.92219i	0.543996	−	0.543996i	−0.380702	−	0.924698i	$-0.624318\pi$
$269$	8.92219	−	8.92219i	0.543996	−	0.543996i	0.924698	+	0.380702i	$0.124318\pi$
$270$		0			0
$271$		−	3.36141i		−	0.204191i	−0.994775	−	0.102096i	$-0.967445\pi$
$271$		−	3.36141i		−	0.204191i	0.994775	−	0.102096i	$-0.0325548\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.0853	+	12.0853i	0.728771	+	0.728771i
$276$		0			0
$277$	0.516082	−	0.516082i	0.0310083	−	0.0310083i	−0.691433	−	0.722441i	$-0.743020\pi$
$277$	0.516082	−	0.516082i	0.0310083	−	0.0310083i	0.722441	+	0.691433i	$0.243020\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.0206			−0.776747			−0.388373	−	0.921502i	$-0.626963\pi$
$281$	−13.0206			−0.776747			−0.388373	+	0.921502i	$0.626963\pi$
$282$		0			0
$283$	−7.04669	−	7.04669i	−0.418882	−	0.418882i	0.465936	−	0.884818i	$-0.345718\pi$
$283$	−7.04669	−	7.04669i	−0.418882	−	0.418882i	−0.884818	+	0.465936i	$0.845718\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.96091			−0.410890
$288$		0			0
$289$	16.3683			0.962840
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.61446	+	3.61446i	0.211159	+	0.211159i	0.804760	−	0.593601i	$-0.202294\pi$
$293$	3.61446	+	3.61446i	0.211159	+	0.211159i	−0.593601	+	0.804760i	$0.702294\pi$
$294$		0			0
$295$	−0.605864			−0.0352747
$296$		0			0
$297$		0			0
$298$		0			0
$299$	15.1603	−	15.1603i	0.876741	−	0.876741i
$300$		0			0
$301$	−1.25316	−	1.25316i	−0.0722308	−	0.0722308i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	1.34276i		−	0.0768861i
$306$		0			0
$307$	15.5015	−	15.5015i	0.884715	−	0.884715i	−0.109295	−	0.994009i	$-0.534859\pi$
$307$	15.5015	−	15.5015i	0.884715	−	0.884715i	0.994009	+	0.109295i	$0.0348592\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			17.6216i			0.999227i	0.866248	+	0.499613i	$0.166525\pi$
$311$			17.6216i			0.999227i	−0.866248	+	0.499613i	$0.833475\pi$
$312$		0			0
$313$			18.5910i			1.05083i	0.850847	+	0.525414i	$0.176089\pi$
$313$			18.5910i			1.05083i	−0.850847	+	0.525414i	$0.823911\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.0237	−	13.0237i	0.731486	−	0.731486i	−0.239428	−	0.970914i	$-0.576960\pi$
$317$	13.0237	−	13.0237i	0.731486	−	0.731486i	0.970914	+	0.239428i	$0.0769600\pi$
$318$		0			0
$319$			36.2662i			2.03052i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.10810	+	2.10810i	0.117298	+	0.117298i
$324$		0			0
$325$	−19.2667	+	19.2667i	−1.06873	+	1.06873i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.48295			0.357417
$330$		0			0
$331$	0.844531	+	0.844531i	0.0464196	+	0.0464196i	0.729936	−	0.683516i	$-0.239550\pi$
$331$	0.844531	+	0.844531i	0.0464196	+	0.0464196i	−0.683516	+	0.729936i	$0.739550\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.398319			0.0217625
$336$		0			0
$337$	24.2226			1.31949			0.659746	−	0.751489i	$-0.270664\pi$
$337$	24.2226			1.31949			0.659746	+	0.751489i	$0.270664\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−13.4711	−	13.4711i	−0.729500	−	0.729500i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.75310	−	8.75310i	0.469891	−	0.469891i	−0.431988	−	0.901879i	$-0.642188\pi$
$347$	8.75310	−	8.75310i	0.469891	−	0.469891i	0.901879	+	0.431988i	$0.142188\pi$
$348$		0			0
$349$	−6.87743	−	6.87743i	−0.368140	−	0.368140i	0.498658	−	0.866799i	$-0.333826\pi$
$349$	−6.87743	−	6.87743i	−0.368140	−	0.368140i	−0.866799	+	0.498658i	$0.833826\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			22.8780i			1.21767i	0.793295	+	0.608837i	$0.208364\pi$
$353$			22.8780i			1.21767i	−0.793295	+	0.608837i	$0.791636\pi$
$354$		0			0
$355$	0.710441	−	0.710441i	0.0377063	−	0.0377063i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.82162i		−	0.465587i	−0.972526	−	0.232794i	$-0.925213\pi$
$359$		−	8.82162i		−	0.465587i	0.972526	−	0.232794i	$-0.0747867\pi$
$360$		0			0
$361$			4.93033i			0.259491i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.15588	−	1.15588i	0.0605017	−	0.0605017i
$366$		0			0
$367$			21.0160i			1.09703i	0.836142	+	0.548513i	$0.184806\pi$
$367$			21.0160i			1.09703i	−0.836142	+	0.548513i	$0.815194\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.620687	+	0.620687i	0.0322245	+	0.0322245i
$372$		0			0
$373$	−10.3425	+	10.3425i	−0.535516	+	0.535516i	−0.922209	−	0.386693i	$-0.873617\pi$
$373$	−10.3425	+	10.3425i	−0.535516	+	0.535516i	0.386693	+	0.922209i	$0.373617\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−57.8166			−2.97771
$378$		0			0
$379$	−16.1353	−	16.1353i	−0.828813	−	0.828813i	0.158539	−	0.987353i	$-0.449322\pi$
$379$	−16.1353	−	16.1353i	−0.828813	−	0.828813i	−0.987353	+	0.158539i	$0.949322\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.1083			0.618704			0.309352	−	0.950948i	$-0.399888\pi$
$383$	12.1083			0.618704			0.309352	+	0.950948i	$0.399888\pi$
$384$		0			0
$385$	−0.433280			−0.0220820
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.7493	+	12.7493i	0.646414	+	0.646414i	0.952124	−	0.305711i	$-0.0988941\pi$
$389$	12.7493	+	12.7493i	0.646414	+	0.646414i	−0.305711	+	0.952124i	$0.598894\pi$
$390$		0			0
$391$	−3.11704			−0.157636
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.910778	−	0.910778i	0.0458262	−	0.0458262i
$396$		0			0
$397$	16.1299	+	16.1299i	0.809534	+	0.809534i	0.984563	−	0.175029i	$-0.0560019\pi$
$397$	16.1299	+	16.1299i	0.809534	+	0.809534i	−0.175029	+	0.984563i	$0.556002\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.54744i		−	0.0772755i	−0.999253	−	0.0386378i	$-0.987698\pi$
$401$		−	1.54744i		−	0.0772755i	0.999253	−	0.0386378i	$-0.0123018\pi$
$402$		0			0
$403$	21.4760	−	21.4760i	1.06980	−	1.06980i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	30.8170i		−	1.52754i
$408$		0			0
$409$			4.58498i			0.226712i	0.993554	+	0.113356i	$0.0361601\pi$
$409$			4.58498i			0.226712i	−0.993554	+	0.113356i	$0.963840\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.39065	+	3.39065i	−0.166843	+	0.166843i
$414$		0			0
$415$		−	2.01595i		−	0.0989589i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.8750	−	24.8750i	−1.21523	−	1.21523i	−0.969285	−	0.245940i	$-0.920903\pi$
$419$	−24.8750	−	24.8750i	−1.21523	−	1.21523i	−0.245940	−	0.969285i	$-0.579097\pi$
$420$		0			0
$421$	9.42104	−	9.42104i	0.459153	−	0.459153i	−0.439224	−	0.898378i	$-0.644747\pi$
$421$	9.42104	−	9.42104i	0.459153	−	0.459153i	0.898378	+	0.439224i	$0.144747\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.96136			0.192154
$426$		0			0
$427$	−7.51460	−	7.51460i	−0.363657	−	0.363657i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.96634			−0.431893			−0.215947	−	0.976405i	$-0.569284\pi$
$431$	−8.96634			−0.431893			−0.215947	+	0.976405i	$0.569284\pi$
$432$		0			0
$433$	0.679368			0.0326483			0.0163242	−	0.999867i	$-0.494804\pi$
$433$	0.679368			0.0326483			0.0163242	+	0.999867i	$0.494804\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.4018	+	10.4018i	0.497584	+	0.497584i
$438$		0			0
$439$	−35.4959			−1.69413			−0.847065	−	0.531490i	$-0.821632\pi$
$439$	−35.4959			−1.69413			−0.847065	+	0.531490i	$0.821632\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.4369	+	23.4369i	−1.11352	+	1.11352i	−0.120853	+	0.992670i	$0.538563\pi$
$443$	−23.4369	+	23.4369i	−1.11352	+	1.11352i	−0.992670	+	0.120853i	$0.961437\pi$
$444$		0			0
$445$	−0.493876	−	0.493876i	−0.0234120	−	0.0234120i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.45807i		−	0.210389i	−0.994452	−	0.105195i	$-0.966453\pi$
$449$		−	4.45807i		−	0.210389i	0.994452	−	0.105195i	$-0.0335465\pi$
$450$		0			0
$451$	−16.8788	+	16.8788i	−0.794793	+	0.794793i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.690747i		−	0.0323827i
$456$		0			0
$457$			14.9134i			0.697620i	0.937193	+	0.348810i	$0.113414\pi$
$457$			14.9134i			0.697620i	−0.937193	+	0.348810i	$0.886586\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.52217	+	7.52217i	−0.350342	+	0.350342i	−0.860237	−	0.509895i	$-0.829684\pi$
$461$	−7.52217	+	7.52217i	−0.350342	+	0.350342i	0.509895	+	0.860237i	$0.329684\pi$
$462$		0			0
$463$		−	40.0015i		−	1.85903i	−0.368787	−	0.929514i	$-0.620227\pi$
$463$		−	40.0015i		−	1.85903i	0.368787	−	0.929514i	$-0.379773\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.9560	+	18.9560i	0.877177	+	0.877177i	0.993242	−	0.116064i	$-0.0370279\pi$
$467$	18.9560	+	18.9560i	0.877177	+	0.877177i	−0.116064	+	0.993242i	$0.537028\pi$
$468$		0			0
$469$	2.22915	−	2.22915i	0.102933	−	0.102933i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.07732			−0.279435
$474$		0			0
$475$	−13.2193	−	13.2193i	−0.606543	−	0.606543i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.3112			−0.973733			−0.486866	−	0.873476i	$-0.661860\pi$
$479$	−21.3112			−0.973733			−0.486866	+	0.873476i	$0.661860\pi$
$480$		0			0
$481$	49.1294			2.24011
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.387407	+	0.387407i	0.0175912	+	0.0175912i
$486$		0			0
$487$	−13.6167			−0.617031			−0.308515	−	0.951219i	$-0.599832\pi$
$487$	−13.6167			−0.617031			−0.308515	+	0.951219i	$0.599832\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.18965	−	5.18965i	0.234206	−	0.234206i	−0.580240	−	0.814446i	$-0.697041\pi$
$491$	5.18965	−	5.18965i	0.234206	−	0.234206i	0.814446	+	0.580240i	$0.197041\pi$
$492$		0			0
$493$	5.94373	+	5.94373i	0.267692	+	0.267692i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	7.95182i		−	0.356688i
$498$		0			0
$499$	7.15618	−	7.15618i	0.320355	−	0.320355i	−0.528548	−	0.848903i	$-0.677264\pi$
$499$	7.15618	−	7.15618i	0.320355	−	0.320355i	0.848903	+	0.528548i	$0.177264\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			33.5554i			1.49616i	0.663607	+	0.748081i	$0.269025\pi$
$503$			33.5554i			1.49616i	−0.663607	+	0.748081i	$0.730975\pi$
$504$		0			0
$505$		−	1.72650i		−	0.0768282i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.615168	−	0.615168i	0.0272669	−	0.0272669i	−0.693342	−	0.720609i	$-0.743862\pi$
$509$	0.615168	−	0.615168i	0.0272669	−	0.0272669i	0.720609	+	0.693342i	$0.243862\pi$
$510$		0			0
$511$		−	12.9376i		−	0.572324i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.46523	+	1.46523i	0.0645656	+	0.0645656i
$516$		0			0
$517$	15.7199	−	15.7199i	0.691359	−	0.691359i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.91593			0.259182			0.129591	−	0.991568i	$-0.458634\pi$
$521$	5.91593			0.259182			0.129591	+	0.991568i	$0.458634\pi$
$522$		0			0
$523$	−17.7110	−	17.7110i	−0.774449	−	0.774449i	0.204432	−	0.978881i	$-0.434465\pi$
$523$	−17.7110	−	17.7110i	−0.774449	−	0.774449i	−0.978881	+	0.204432i	$0.934465\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.41560			−0.192346
$528$		0			0
$529$	7.61988			0.331299
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−26.9087	−	26.9087i	−1.16555	−	1.16555i
$534$		0			0
$535$	0.980831			0.0424050
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.42480	+	2.42480i	−0.104444	+	0.104444i
$540$		0			0
$541$	−9.13769	−	9.13769i	−0.392860	−	0.392860i	0.482846	−	0.875706i	$-0.339603\pi$
$541$	−9.13769	−	9.13769i	−0.392860	−	0.392860i	−0.875706	+	0.482846i	$0.839603\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.61516i			0.112021i
$546$		0			0
$547$	30.1099	−	30.1099i	1.28741	−	1.28741i	0.351052	−	0.936356i	$-0.385824\pi$
$547$	30.1099	−	30.1099i	1.28741	−	1.28741i	0.936356	−	0.351052i	$-0.114176\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	39.6691i		−	1.68996i
$552$		0			0
$553$		−	10.1941i		−	0.433499i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.3306	−	16.3306i	0.691950	−	0.691950i	−0.270711	−	0.962661i	$-0.587259\pi$
$557$	16.3306	−	16.3306i	0.691950	−	0.691950i	0.962661	+	0.270711i	$0.0872588\pi$
$558$		0			0
$559$		−	9.68863i		−	0.409785i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.288759	+	0.288759i	0.0121697	+	0.0121697i	0.713165	−	0.700996i	$-0.247261\pi$
$563$	0.288759	+	0.288759i	0.0121697	+	0.0121697i	−0.700996	+	0.713165i	$0.747261\pi$
$564$		0			0
$565$	0.739998	−	0.739998i	0.0311319	−	0.0311319i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.8386			0.999365			0.499683	−	0.866209i	$-0.333450\pi$
$569$	23.8386			0.999365			0.499683	+	0.866209i	$0.333450\pi$
$570$		0			0
$571$	−1.86113	−	1.86113i	−0.0778857	−	0.0778857i	0.667091	−	0.744976i	$-0.267539\pi$
$571$	−1.86113	−	1.86113i	−0.0778857	−	0.0778857i	−0.744976	+	0.667091i	$0.767539\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	19.5461			0.815130
$576$		0			0
$577$	−32.0148			−1.33279			−0.666396	−	0.745598i	$-0.732164\pi$
$577$	−32.0148			−1.33279			−0.666396	+	0.745598i	$0.732164\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−11.2820	−	11.2820i	−0.468057	−	0.468057i
$582$		0			0
$583$	3.01009			0.124665
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.73084	+	1.73084i	−0.0714394	+	0.0714394i	−0.741924	−	0.670484i	$-0.766086\pi$
$587$	−1.73084	+	1.73084i	−0.0714394	+	0.0714394i	0.670484	+	0.741924i	$0.266086\pi$
$588$		0			0
$589$	14.7351	+	14.7351i	0.607149	+	0.607149i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			22.8985i			0.940328i	0.882579	+	0.470164i	$0.155805\pi$
$593$			22.8985i			0.940328i	−0.882579	+	0.470164i	$0.844195\pi$
$594$		0			0
$595$	−0.0710109	+	0.0710109i	−0.00291116	+	0.00291116i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	26.0002i		−	1.06234i	−0.847266	−	0.531169i	$-0.821753\pi$
$599$		−	26.0002i		−	1.06234i	0.847266	−	0.531169i	$-0.178247\pi$
$600$		0			0
$601$			21.0502i			0.858654i	0.903149	+	0.429327i	$0.141249\pi$
$601$			21.0502i			0.858654i	−0.903149	+	0.429327i	$0.858751\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.0678415	+	0.0678415i	−0.00275815	+	0.00275815i
$606$		0			0
$607$		−	32.7336i		−	1.32861i	−0.747460	−	0.664307i	$-0.768727\pi$
$607$		−	32.7336i		−	1.32861i	0.747460	−	0.664307i	$-0.231273\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25.0611	+	25.0611i	1.01386	+	1.01386i
$612$		0			0
$613$	7.14939	−	7.14939i	0.288761	−	0.288761i	−0.547829	−	0.836590i	$-0.684546\pi$
$613$	7.14939	−	7.14939i	0.288761	−	0.288761i	0.836590	+	0.547829i	$0.184546\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.8667			0.638769			0.319384	−	0.947625i	$-0.396524\pi$
$617$	15.8667			0.638769			0.319384	+	0.947625i	$0.396524\pi$
$618$		0			0
$619$	20.4351	+	20.4351i	0.821355	+	0.821355i	0.986302	−	0.164947i	$-0.0527453\pi$
$619$	20.4351	+	20.4351i	0.821355	+	0.821355i	−0.164947	+	0.986302i	$0.552745\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.52785			−0.221469
$624$		0			0
$625$	−24.7608			−0.990432
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.05066	−	5.05066i	−0.201383	−	0.201383i
$630$		0			0
$631$	−39.1876			−1.56003			−0.780017	−	0.625758i	$-0.784790\pi$
$631$	−39.1876			−1.56003			−0.780017	+	0.625758i	$0.784790\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.303340	−	0.303340i	0.0120377	−	0.0120377i
$636$		0			0
$637$	−3.86569	−	3.86569i	−0.153164	−	0.153164i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	32.1293i		−	1.26903i	−0.772911	−	0.634515i	$-0.781200\pi$
$641$		−	32.1293i		−	1.26903i	0.772911	−	0.634515i	$-0.218800\pi$
$642$		0			0
$643$	−13.7452	+	13.7452i	−0.542059	+	0.542059i	−0.924132	−	0.382073i	$-0.875210\pi$
$643$	−13.7452	+	13.7452i	−0.542059	+	0.542059i	0.382073	+	0.924132i	$0.375210\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			5.78120i			0.227282i	0.993522	+	0.113641i	$0.0362514\pi$
$647$			5.78120i			0.227282i	−0.993522	+	0.113641i	$0.963749\pi$
$648$		0			0
$649$			16.4433i			0.645457i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.9854	−	13.9854i	0.547293	−	0.547293i	−0.378364	−	0.925657i	$-0.623513\pi$
$653$	13.9854	−	13.9854i	0.547293	−	0.547293i	0.925657	+	0.378364i	$0.123513\pi$
$654$		0			0
$655$		−	1.56075i		−	0.0609834i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−19.6810	−	19.6810i	−0.766661	−	0.766661i	0.210856	−	0.977517i	$-0.432375\pi$
$659$	−19.6810	−	19.6810i	−0.766661	−	0.766661i	−0.977517	+	0.210856i	$0.932375\pi$
$660$		0			0
$661$	12.7655	−	12.7655i	0.496519	−	0.496519i	−0.413833	−	0.910353i	$-0.635810\pi$
$661$	12.7655	−	12.7655i	0.496519	−	0.496519i	0.910353	+	0.413833i	$0.135810\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.473935			0.0183784
$666$		0			0
$667$	29.3275	+	29.3275i	1.13557	+	1.13557i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−36.4429			−1.40686
$672$		0			0
$673$	43.7910			1.68802			0.844010	−	0.536328i	$-0.180189\pi$
$673$	43.7910			1.68802			0.844010	+	0.536328i	$0.180189\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	25.0415	+	25.0415i	0.962422	+	0.962422i	0.999319	−	0.0368968i	$-0.0117473\pi$
$677$	25.0415	+	25.0415i	0.962422	+	0.962422i	−0.0368968	+	0.999319i	$0.511747\pi$
$678$		0			0
$679$	4.33616			0.166407
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.6770	+	12.6770i	−0.485071	+	0.485071i	−0.906747	−	0.421676i	$-0.861442\pi$
$683$	−12.6770	+	12.6770i	−0.485071	+	0.485071i	0.421676	+	0.906747i	$0.361442\pi$
$684$		0			0
$685$	0.433216	+	0.433216i	0.0165523	+	0.0165523i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			4.79877i			0.182818i
$690$		0			0
$691$	−11.5147	+	11.5147i	−0.438041	+	0.438041i	−0.891352	−	0.453312i	$-0.850243\pi$
$691$	−11.5147	+	11.5147i	−0.438041	+	0.438041i	0.453312	+	0.891352i	$0.350243\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	0.392153i		−	0.0148752i
$696$		0			0
$697$			5.53260i			0.209562i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−24.5725	+	24.5725i	−0.928089	+	0.928089i	−0.997582	−	0.0694930i	$-0.977862\pi$
$701$	−24.5725	+	24.5725i	−0.928089	+	0.928089i	0.0694930	+	0.997582i	$0.477862\pi$
$702$		0			0
$703$			33.7087i			1.27135i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.66216	−	9.66216i	−0.363383	−	0.363383i
$708$		0			0
$709$	−11.9488	+	11.9488i	−0.448745	+	0.448745i	−0.894937	−	0.446192i	$-0.852780\pi$
$709$	−11.9488	+	11.9488i	−0.448745	+	0.448745i	0.446192	+	0.894937i	$0.352780\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−21.7874			−0.815946
$714$		0			0
$715$	−1.67492	−	1.67492i	−0.0626386	−	0.0626386i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.8496			0.777559			0.388779	−	0.921331i	$-0.372897\pi$
$719$	20.8496			0.777559			0.388779	+	0.921331i	$0.372897\pi$
$720$		0			0
$721$	16.4000			0.610767
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−37.2715	−	37.2715i	−1.38423	−	1.38423i
$726$		0			0
$727$	49.4099			1.83251			0.916257	−	0.400592i	$-0.131195\pi$
$727$	49.4099			1.83251			0.916257	+	0.400592i	$0.131195\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.996021	+	0.996021i	−0.0368392	+	0.0368392i
$732$		0			0
$733$	−22.8419	−	22.8419i	−0.843686	−	0.843686i	0.145650	−	0.989336i	$-0.453473\pi$
$733$	−22.8419	−	22.8419i	−0.843686	−	0.843686i	−0.989336	+	0.145650i	$0.953473\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	10.8105i		−	0.398210i
$738$		0			0
$739$	3.95695	−	3.95695i	0.145559	−	0.145559i	−0.630572	−	0.776131i	$-0.717180\pi$
$739$	3.95695	−	3.95695i	0.145559	−	0.145559i	0.776131	+	0.630572i	$0.217180\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.72532i		−	0.0632958i	−0.999499	−	0.0316479i	$-0.989924\pi$
$743$		−	1.72532i		−	0.0632958i	0.999499	−	0.0316479i	$-0.0100755\pi$
$744$		0			0
$745$		−	0.507177i		−	0.0185815i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.48912	−	5.48912i	0.200568	−	0.200568i
$750$		0			0
$751$			28.0700i			1.02429i	0.858900	+	0.512144i	$0.171148\pi$
$751$			28.0700i			1.02429i	−0.858900	+	0.512144i	$0.828852\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.69592	−	1.69592i	−0.0617209	−	0.0617209i
$756$		0			0
$757$	−21.3781	+	21.3781i	−0.777001	+	0.777001i	−0.979320	−	0.202319i	$-0.935152\pi$
$757$	−21.3781	+	21.3781i	−0.777001	+	0.777001i	0.202319	+	0.979320i	$0.435152\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.0483			−0.618001			−0.309001	−	0.951062i	$-0.599995\pi$
$761$	−17.0483			−0.618001			−0.309001	+	0.951062i	$0.599995\pi$
$762$		0			0
$763$	14.6355	+	14.6355i	0.529840	+	0.529840i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−26.2144			−0.946547
$768$		0			0
$769$	17.6693			0.637172			0.318586	−	0.947894i	$-0.396792\pi$
$769$	17.6693			0.637172			0.318586	+	0.947894i	$0.396792\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.41723	+	3.41723i	0.122909	+	0.122909i	0.765886	−	0.642977i	$-0.222301\pi$
$773$	3.41723	+	3.41723i	0.122909	+	0.122909i	−0.642977	+	0.765886i	$0.722301\pi$
$774$		0			0
$775$	27.6890			0.994618
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18.4626	−	18.4626i	0.661492	−	0.661492i
$780$		0			0
$781$	−19.2816	−	19.2816i	−0.689949	−	0.689949i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2.20180i			0.0785855i
$786$		0			0
$787$	16.5567	−	16.5567i	0.590183	−	0.590183i	−0.347498	−	0.937681i	$-0.612969\pi$
$787$	16.5567	−	16.5567i	0.590183	−	0.590183i	0.937681	+	0.347498i	$0.112969\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	8.28264i		−	0.294497i
$792$		0			0
$793$		−	58.0982i		−	2.06313i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.4377	+	15.4377i	−0.546833	+	0.546833i	−0.925523	−	0.378691i	$-0.876374\pi$
$797$	−15.4377	+	15.4377i	−0.546833	+	0.546833i	0.378691	+	0.925523i	$0.376374\pi$
$798$		0			0
$799$		−	5.15271i		−	0.182290i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−31.3710	−	31.3710i	−1.10706	−	1.10706i
$804$		0			0
$805$	−0.350382	+	0.350382i	−0.0123493	+	0.0123493i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−49.9730			−1.75696			−0.878478	−	0.477782i	$-0.841441\pi$
$809$	−49.9730			−1.75696			−0.878478	+	0.477782i	$0.841441\pi$
$810$		0			0
$811$	−34.3832	−	34.3832i	−1.20736	−	1.20736i	−0.971880	−	0.235477i	$-0.924335\pi$
$811$	−34.3832	−	34.3832i	−1.20736	−	1.20736i	−0.235477	−	0.971880i	$-0.575665\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.626117			−0.0219319
$816$		0			0
$817$	6.64756			0.232569
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.59874	+	1.59874i	0.0557963	+	0.0557963i	0.734454	−	0.678658i	$-0.237438\pi$
$821$	1.59874	+	1.59874i	0.0557963	+	0.0557963i	−0.678658	+	0.734454i	$0.737438\pi$
$822$		0			0
$823$	37.9226			1.32190			0.660949	−	0.750431i	$-0.270154\pi$
$823$	37.9226			1.32190			0.660949	+	0.750431i	$0.270154\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−34.5300	+	34.5300i	−1.20073	+	1.20073i	−0.226779	+	0.973946i	$0.572820\pi$
$827$	−34.5300	+	34.5300i	−1.20073	+	1.20073i	−0.973946	+	0.226779i	$0.927180\pi$
$828$		0			0
$829$	−16.0890	−	16.0890i	−0.558795	−	0.558795i	0.370169	−	0.928964i	$-0.379300\pi$
$829$	−16.0890	−	16.0890i	−0.558795	−	0.558795i	−0.928964	+	0.370169i	$0.879300\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			0.794810i			0.0275385i
$834$		0			0
$835$	−1.56100	+	1.56100i	−0.0540206	+	0.0540206i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	27.2955i		−	0.942344i	−0.882041	−	0.471172i	$-0.843831\pi$
$839$		−	27.2955i		−	0.942344i	0.882041	−	0.471172i	$-0.156169\pi$
$840$		0			0
$841$		−	82.8462i		−	2.85677i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.50875	−	1.50875i	0.0519025	−	0.0519025i
$846$		0			0
$847$			0.759336i			0.0260911i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.9209	−	24.9209i	−0.854279	−	0.854279i
$852$		0			0
$853$	18.1407	−	18.1407i	0.621125	−	0.621125i	−0.324694	−	0.945819i	$-0.605261\pi$
$853$	18.1407	−	18.1407i	0.621125	−	0.621125i	0.945819	+	0.324694i	$0.105261\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	43.2364			1.47693			0.738464	−	0.674293i	$-0.235551\pi$
$857$	43.2364			1.47693			0.738464	+	0.674293i	$0.235551\pi$
$858$		0			0
$859$	−5.72658	−	5.72658i	−0.195388	−	0.195388i	0.602631	−	0.798020i	$-0.294119\pi$
$859$	−5.72658	−	5.72658i	−0.195388	−	0.195388i	−0.798020	+	0.602631i	$0.794119\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.7413			−0.740082			−0.370041	−	0.929015i	$-0.620656\pi$
$863$	−21.7413			−0.740082			−0.370041	+	0.929015i	$0.620656\pi$
$864$		0			0
$865$	0.260040			0.00884162
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.7188	−	24.7188i	−0.838527	−	0.838527i
$870$		0			0
$871$	17.2344			0.583966
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.892006	−	0.892006i	0.0301553	−	0.0301553i
$876$		0			0
$877$	−23.9822	−	23.9822i	−0.809822	−	0.809822i	0.174785	−	0.984607i	$-0.444077\pi$
$877$	−23.9822	−	23.9822i	−0.809822	−	0.809822i	−0.984607	+	0.174785i	$0.944077\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	35.2047i		−	1.18608i	−0.805175	−	0.593038i	$-0.797929\pi$
$881$		−	35.2047i		−	1.18608i	0.805175	−	0.593038i	$-0.202071\pi$
$882$		0			0
$883$	29.0728	−	29.0728i	0.978377	−	0.978377i	−0.0213942	−	0.999771i	$-0.506810\pi$
$883$	29.0728	−	29.0728i	0.978377	−	0.978377i	0.999771	+	0.0213942i	$0.00681049\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			26.4222i			0.887172i	0.896232	+	0.443586i	$0.146294\pi$
$887$			26.4222i			0.887172i	−0.896232	+	0.443586i	$0.853706\pi$
$888$		0			0
$889$		−	3.39522i		−	0.113872i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−17.1949	+	17.1949i	−0.575405	+	0.575405i
$894$		0			0
$895$			0.397198i			0.0132769i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	41.5453	+	41.5453i	1.38561	+	1.38561i
$900$		0			0
$901$	0.493328	−	0.493328i	0.0164351	−	0.0164351i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.559694			−0.0186049
$906$		0			0
$907$	13.8359	+	13.8359i	0.459413	+	0.459413i	0.898463	−	0.439050i	$-0.144685\pi$
$907$	13.8359	+	13.8359i	0.459413	+	0.459413i	−0.439050	+	0.898463i	$0.644685\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−32.8417			−1.08809			−0.544047	−	0.839055i	$-0.683109\pi$
$911$	−32.8417			−1.08809			−0.544047	+	0.839055i	$0.683109\pi$
$912$		0			0
$913$	−54.7134			−1.81075
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.73455	−	8.73455i	−0.288440	−	0.288440i
$918$		0			0
$919$	−1.13244			−0.0373559			−0.0186779	−	0.999826i	$-0.505946\pi$
$919$	−1.13244			−0.0373559			−0.0186779	+	0.999826i	$0.505946\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	30.7393	−	30.7393i	1.01179	−	1.01179i
$924$		0			0
$925$	31.6713	+	31.6713i	1.04135	+	1.04135i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	47.3464i		−	1.55338i	−0.629880	−	0.776692i	$-0.716896\pi$
$929$		−	47.3464i		−	1.55338i	0.629880	−	0.776692i	$-0.283104\pi$
$930$		0			0
$931$	2.65233	−	2.65233i	0.0869265	−	0.0869265i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			0.344375i			0.0112623i
$936$		0			0
$937$			52.0425i			1.70015i	0.526658	+	0.850077i	$0.323445\pi$
$937$			52.0425i			1.70015i	−0.526658	+	0.850077i	$0.676555\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.8877	−	21.8877i	0.713520	−	0.713520i	−0.253750	−	0.967270i	$-0.581664\pi$
$941$	21.8877	−	21.8877i	0.713520	−	0.713520i	0.967270	+	0.253750i	$0.0816641\pi$
$942$		0			0
$943$			27.2989i			0.888976i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.2385	−	27.2385i	−0.885132	−	0.885132i	0.108919	−	0.994051i	$-0.465261\pi$
$947$	−27.2385	−	27.2385i	−0.885132	−	0.885132i	−0.994051	+	0.108919i	$0.965261\pi$
$948$		0			0
$949$	50.0126	−	50.0126i	1.62348	−	1.62348i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−40.7540			−1.32015			−0.660076	−	0.751199i	$-0.729476\pi$
$953$	−40.7540			−1.32015			−0.660076	+	0.751199i	$0.729476\pi$
$954$		0			0
$955$	−0.322649	−	0.322649i	−0.0104407	−	0.0104407i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.84889			0.156579
$960$		0			0
$961$	0.135984			0.00438657
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.269538	+	0.269538i	0.00867675	+	0.00867675i
$966$		0			0
$967$	−28.9910			−0.932287			−0.466143	−	0.884709i	$-0.654357\pi$
$967$	−28.9910			−0.932287			−0.466143	+	0.884709i	$0.654357\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16.5270	+	16.5270i	−0.530377	+	0.530377i	−0.920685	−	0.390308i	$-0.872369\pi$
$971$	−16.5270	+	16.5270i	−0.530377	+	0.530377i	0.390308	+	0.920685i	$0.372369\pi$
$972$		0			0
$973$	−2.19464	−	2.19464i	−0.0703569	−	0.0703569i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			31.6420i			1.01232i	0.862440	+	0.506159i	$0.168935\pi$
$977$			31.6420i			1.01232i	−0.862440	+	0.506159i	$0.831065\pi$
$978$		0			0
$979$	−13.4040	+	13.4040i	−0.428392	+	0.428392i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			11.2993i			0.360390i	0.983631	+	0.180195i	$0.0576729\pi$
$983$			11.2993i			0.360390i	−0.983631	+	0.180195i	$0.942327\pi$
$984$		0			0
$985$			0.0504089i			0.00160616i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.91457	+	4.91457i	−0.156274	+	0.156274i
$990$		0			0
$991$		−	29.9518i		−	0.951451i	−0.879594	−	0.475725i	$-0.842186\pi$
$991$		−	29.9518i		−	0.951451i	0.879594	−	0.475725i	$-0.157814\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.303780	+	0.303780i	0.00963048	+	0.00963048i
$996$		0			0
$997$	−11.9968	+	11.9968i	−0.379941	+	0.379941i	−0.871081	−	0.491139i	$-0.836581\pi$
$997$	−11.9968	+	11.9968i	−0.379941	+	0.379941i	0.491139	+	0.871081i	$0.336581\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.e.1583.10		40
3.2	odd	2	inner	4032.2.v.e.1583.11		40
4.3	odd	2		1008.2.v.e.323.11	yes	40
12.11	even	2		1008.2.v.e.323.10	✓	40
16.5	even	4		1008.2.v.e.827.10	yes	40
16.11	odd	4	inner	4032.2.v.e.3599.11		40
48.5	odd	4		1008.2.v.e.827.11	yes	40
48.11	even	4	inner	4032.2.v.e.3599.10		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.10	✓	40	12.11	even	2
1008.2.v.e.323.11	yes	40	4.3	odd	2
1008.2.v.e.827.10	yes	40	16.5	even	4
1008.2.v.e.827.11	yes	40	48.5	odd	4
4032.2.v.e.1583.10		40	1.1	even	1	trivial
4032.2.v.e.1583.11		40	3.2	odd	2	inner
4032.2.v.e.3599.10		40	48.11	even	4	inner
4032.2.v.e.3599.11		40	16.11	odd	4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.0893433 - 0.0893433i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(-0.0893433 - 0.0893433i) q^{5} -1.00000 q^{7} +(-2.42480 + 2.42480i) q^{11} +(-3.86569 - 3.86569i) q^{13} +0.794810i q^{17} +(2.65233 - 2.65233i) q^{19} +3.92175i q^{23} -4.98404i q^{25} +(7.47818 - 7.47818i) q^{29} +5.55554i q^{31} +(0.0893433 + 0.0893433i) q^{35} +(-6.35455 + 6.35455i) q^{37} +6.96091 q^{41} +(1.25316 + 1.25316i) q^{43} -6.48295 q^{47} +1.00000 q^{49} +(-0.620687 - 0.620687i) q^{53} +0.433280 q^{55} +(3.39065 - 3.39065i) q^{59} +(7.51460 + 7.51460i) q^{61} +0.690747i q^{65} +(-2.22915 + 2.22915i) q^{67} +7.95182i q^{71} +12.9376i q^{73} +(2.42480 - 2.42480i) q^{77} +10.1941i q^{79} +(11.2820 + 11.2820i) q^{83} +(0.0710109 - 0.0710109i) q^{85} +5.52785 q^{89} +(3.86569 + 3.86569i) q^{91} -0.473935 q^{95} -4.33616 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 40 * q - 40 * q^7	\( 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) \) 40 * q - 40 * q^7 - 24 * q^13 + 32 * q^19 - 8 * q^37 - 32 * q^43 + 40 * q^49 - 48 * q^55 - 24 * q^61 + 64 * q^85 + 24 * q^91 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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