LMFDB - Embedded newform 4032.2.v.e.1583.1

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			1583.1
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.e.3599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.96859 - 2.96859i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(-2.96859 - 2.96859i) q^{5} -1.00000 q^{7} +(-0.569860 + 0.569860i) q^{11} +(2.53650 + 2.53650i) q^{13} +1.03528i q^{17} +(5.23568 - 5.23568i) q^{19} -8.71217i q^{23} +12.6251i q^{25} +(6.05233 - 6.05233i) q^{29} -3.00413i q^{31} +(2.96859 + 2.96859i) q^{35} +(-0.149739 + 0.149739i) q^{37} -8.63459 q^{41} +(-1.73121 - 1.73121i) q^{43} +4.10865 q^{47} +1.00000 q^{49} +(6.04092 + 6.04092i) q^{53} +3.38337 q^{55} +(-6.05121 + 6.05121i) q^{59} +(5.81897 + 5.81897i) q^{61} -15.0597i q^{65} +(-0.0256948 + 0.0256948i) q^{67} -14.5257i q^{71} -5.38496i q^{73} +(0.569860 - 0.569860i) q^{77} -3.89876i q^{79} +(-1.40691 - 1.40691i) q^{83} +(3.07333 - 3.07333i) q^{85} -17.0358 q^{89} +(-2.53650 - 2.53650i) q^{91} -31.0852 q^{95} +3.75210 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.96859	−	2.96859i	−1.32760	−	1.32760i	−0.907463	−	0.420133i	$-0.861983\pi$
$5$	−2.96859	−	2.96859i	−1.32760	−	1.32760i	−0.420133	−	0.907463i	$-0.638017\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.569860	+	0.569860i	−0.171819	+	0.171819i	−0.787778	−	0.615959i	$-0.788769\pi$
$11$	−0.569860	+	0.569860i	−0.171819	+	0.171819i	0.615959	+	0.787778i	$0.288769\pi$
$12$		0			0
$13$	2.53650	+	2.53650i	0.703498	+	0.703498i	0.965160	−	0.261661i	$-0.0842704\pi$
$13$	2.53650	+	2.53650i	0.703498	+	0.703498i	−0.261661	+	0.965160i	$0.584270\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.03528i			0.251092i	0.992088	+	0.125546i	$0.0400683\pi$
$17$			1.03528i			0.251092i	−0.992088	+	0.125546i	$0.959932\pi$
$18$		0			0
$19$	5.23568	−	5.23568i	1.20115	−	1.20115i	0.227329	−	0.973818i	$-0.427001\pi$
$19$	5.23568	−	5.23568i	1.20115	−	1.20115i	0.973818	−	0.227329i	$-0.0729993\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.71217i		−	1.81661i	−0.418304	−	0.908307i	$-0.637375\pi$
$23$		−	8.71217i		−	1.81661i	0.418304	−	0.908307i	$-0.362625\pi$
$24$		0			0
$25$			12.6251i			2.52502i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.05233	−	6.05233i	1.12389	−	1.12389i	0.132739	−	0.991151i	$-0.457623\pi$
$29$	6.05233	−	6.05233i	1.12389	−	1.12389i	0.991151	−	0.132739i	$-0.0423772\pi$
$30$		0			0
$31$		−	3.00413i		−	0.539558i	−0.962922	−	0.269779i	$-0.913049\pi$
$31$		−	3.00413i		−	0.539558i	0.962922	−	0.269779i	$-0.0869506\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.96859	+	2.96859i	0.501784	+	0.501784i
$36$		0			0
$37$	−0.149739	+	0.149739i	−0.0246169	+	0.0246169i	−0.719308	−	0.694691i	$-0.755541\pi$
$37$	−0.149739	+	0.149739i	−0.0246169	+	0.0246169i	0.694691	+	0.719308i	$0.255541\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.63459			−1.34850			−0.674248	−	0.738505i	$-0.735532\pi$
$41$	−8.63459			−1.34850			−0.674248	+	0.738505i	$0.735532\pi$
$42$		0			0
$43$	−1.73121	−	1.73121i	−0.264007	−	0.264007i	0.562673	−	0.826680i	$-0.309773\pi$
$43$	−1.73121	−	1.73121i	−0.264007	−	0.264007i	−0.826680	+	0.562673i	$0.809773\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.10865			0.599308			0.299654	−	0.954048i	$-0.403129\pi$
$47$	4.10865			0.599308			0.299654	+	0.954048i	$0.403129\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.04092	+	6.04092i	0.829783	+	0.829783i	0.987487	−	0.157703i	$-0.0504089\pi$
$53$	6.04092	+	6.04092i	0.829783	+	0.829783i	−0.157703	+	0.987487i	$0.550409\pi$
$54$		0			0
$55$	3.38337			0.456213
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.05121	+	6.05121i	−0.787800	+	0.787800i	−0.981133	−	0.193333i	$-0.938070\pi$
$59$	−6.05121	+	6.05121i	−0.787800	+	0.787800i	0.193333	+	0.981133i	$0.438070\pi$
$60$		0			0
$61$	5.81897	+	5.81897i	0.745042	+	0.745042i	0.973544	−	0.228501i	$-0.0733826\pi$
$61$	5.81897	+	5.81897i	0.745042	+	0.745042i	−0.228501	+	0.973544i	$0.573383\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	15.0597i		−	1.86792i
$66$		0			0
$67$	−0.0256948	+	0.0256948i	−0.00313912	+	0.00313912i	−0.708675	−	0.705535i	$-0.750707\pi$
$67$	−0.0256948	+	0.0256948i	−0.00313912	+	0.00313912i	0.705535	+	0.708675i	$0.250707\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	14.5257i		−	1.72388i	−0.507008	−	0.861941i	$-0.669248\pi$
$71$		−	14.5257i		−	1.72388i	0.507008	−	0.861941i	$-0.330752\pi$
$72$		0			0
$73$		−	5.38496i		−	0.630261i	−0.949048	−	0.315131i	$-0.897952\pi$
$73$		−	5.38496i		−	0.630261i	0.949048	−	0.315131i	$-0.102048\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.569860	−	0.569860i	0.0649416	−	0.0649416i
$78$		0			0
$79$		−	3.89876i		−	0.438645i	−0.975652	−	0.219322i	$-0.929615\pi$
$79$		−	3.89876i		−	0.438645i	0.975652	−	0.219322i	$-0.0703846\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.40691	−	1.40691i	−0.154428	−	0.154428i	0.625664	−	0.780092i	$-0.284828\pi$
$83$	−1.40691	−	1.40691i	−0.154428	−	0.154428i	−0.780092	+	0.625664i	$0.784828\pi$
$84$		0			0
$85$	3.07333	−	3.07333i	0.333349	−	0.333349i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−17.0358			−1.80579			−0.902897	−	0.429856i	$-0.858564\pi$
$89$	−17.0358			−1.80579			−0.902897	+	0.429856i	$0.858564\pi$
$90$		0			0
$91$	−2.53650	−	2.53650i	−0.265897	−	0.265897i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−31.0852			−3.18928
$96$		0			0
$97$	3.75210			0.380968			0.190484	−	0.981690i	$-0.438994\pi$
$97$	3.75210			0.380968			0.190484	+	0.981690i	$0.438994\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.23327	−	1.23327i	−0.122715	−	0.122715i	0.643082	−	0.765797i	$-0.277655\pi$
$101$	−1.23327	−	1.23327i	−0.122715	−	0.122715i	−0.765797	+	0.643082i	$0.777655\pi$
$102$		0			0
$103$	−14.3330			−1.41227			−0.706134	−	0.708078i	$-0.749562\pi$
$103$	−14.3330			−1.41227			−0.706134	+	0.708078i	$0.749562\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.33395	−	7.33395i	0.709000	−	0.709000i	−0.257325	−	0.966325i	$-0.582841\pi$
$107$	7.33395	−	7.33395i	0.709000	−	0.709000i	0.966325	+	0.257325i	$0.0828410\pi$
$108$		0			0
$109$	−4.12218	−	4.12218i	−0.394833	−	0.394833i	0.481573	−	0.876406i	$-0.340066\pi$
$109$	−4.12218	−	4.12218i	−0.394833	−	0.394833i	−0.876406	+	0.481573i	$0.840066\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			12.1524i			1.14320i	0.820533	+	0.571599i	$0.193677\pi$
$113$			12.1524i			1.14320i	−0.820533	+	0.571599i	$0.806323\pi$
$114$		0			0
$115$	−25.8629	+	25.8629i	−2.41173	+	2.41173i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	1.03528i		−	0.0949040i
$120$		0			0
$121$			10.3505i			0.940956i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	22.6358	−	22.6358i	2.02461	−	2.02461i
$126$		0			0
$127$		−	1.32089i		−	0.117210i	−0.998281	−	0.0586052i	$-0.981335\pi$
$127$		−	1.32089i		−	0.117210i	0.998281	−	0.0586052i	$-0.0186653\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.67723	+	1.67723i	0.146540	+	0.146540i	0.776571	−	0.630030i	$-0.216957\pi$
$131$	1.67723	+	1.67723i	0.146540	+	0.146540i	−0.630030	+	0.776571i	$0.716957\pi$
$132$		0			0
$133$	−5.23568	+	5.23568i	−0.453991	+	0.453991i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.49554			0.725823			0.362911	−	0.931824i	$-0.381783\pi$
$137$	8.49554			0.725823			0.362911	+	0.931824i	$0.381783\pi$
$138$		0			0
$139$	7.80995	+	7.80995i	0.662432	+	0.662432i	0.955953	−	0.293521i	$-0.0948271\pi$
$139$	7.80995	+	7.80995i	0.662432	+	0.662432i	−0.293521	+	0.955953i	$0.594827\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.89090			−0.241749
$144$		0			0
$145$	−35.9338			−2.98414
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.1113	−	13.1113i	−1.07412	−	1.07412i	−0.997024	−	0.0770980i	$-0.975435\pi$
$149$	−13.1113	−	13.1113i	−1.07412	−	1.07412i	−0.0770980	−	0.997024i	$-0.524565\pi$
$150$		0			0
$151$	−13.4389			−1.09364			−0.546822	−	0.837249i	$-0.684162\pi$
$151$	−13.4389			−1.09364			−0.546822	+	0.837249i	$0.684162\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.91804	+	8.91804i	−0.716314	+	0.716314i
$156$		0			0
$157$	−5.92492	−	5.92492i	−0.472860	−	0.472860i	0.429979	−	0.902839i	$-0.358521\pi$
$157$	−5.92492	−	5.92492i	−0.472860	−	0.472860i	−0.902839	+	0.429979i	$0.858521\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			8.71217i			0.686615i
$162$		0			0
$163$	1.82469	−	1.82469i	0.142921	−	0.142921i	−0.632026	−	0.774947i	$-0.717776\pi$
$163$	1.82469	−	1.82469i	0.142921	−	0.142921i	0.774947	+	0.632026i	$0.217776\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	5.49283i		−	0.425048i	−0.977156	−	0.212524i	$-0.931832\pi$
$167$		−	5.49283i		−	0.425048i	0.977156	−	0.212524i	$-0.0681683\pi$
$168$		0			0
$169$		−	0.132347i		−	0.0101805i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.4180	+	11.4180i	−0.868093	+	0.868093i	−0.992261	−	0.124168i	$-0.960374\pi$
$173$	−11.4180	+	11.4180i	−0.868093	+	0.868093i	0.124168	+	0.992261i	$0.460374\pi$
$174$		0			0
$175$		−	12.6251i		−	0.954368i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.47860	+	5.47860i	0.409490	+	0.409490i	0.881561	−	0.472071i	$-0.156493\pi$
$179$	5.47860	+	5.47860i	0.409490	+	0.409490i	−0.472071	+	0.881561i	$0.656493\pi$
$180$		0			0
$181$	−9.42734	+	9.42734i	−0.700728	+	0.700728i	−0.964567	−	0.263838i	$-0.915011\pi$
$181$	−9.42734	+	9.42734i	−0.700728	+	0.700728i	0.263838	+	0.964567i	$0.415011\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.889027			0.0653625
$186$		0			0
$187$	−0.589965	−	0.589965i	−0.0431425	−	0.0431425i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.1433			−0.733946			−0.366973	−	0.930232i	$-0.619606\pi$
$191$	−10.1433			−0.733946			−0.366973	+	0.930232i	$0.619606\pi$
$192$		0			0
$193$	−12.6529			−0.910774			−0.455387	−	0.890294i	$-0.650499\pi$
$193$	−12.6529			−0.910774			−0.455387	+	0.890294i	$0.650499\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.32030	−	4.32030i	−0.307808	−	0.307808i	0.536250	−	0.844059i	$-0.319840\pi$
$197$	−4.32030	−	4.32030i	−0.307808	−	0.307808i	−0.844059	+	0.536250i	$0.819840\pi$
$198$		0			0
$199$	−7.22338			−0.512052			−0.256026	−	0.966670i	$-0.582413\pi$
$199$	−7.22338			−0.512052			−0.256026	+	0.966670i	$0.582413\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.05233	+	6.05233i	−0.424790	+	0.424790i
$204$		0			0
$205$	25.6326	+	25.6326i	1.79026	+	1.79026i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			5.96721i			0.412760i
$210$		0			0
$211$	−9.51875	+	9.51875i	−0.655298	+	0.655298i	−0.954264	−	0.298966i	$-0.903358\pi$
$211$	−9.51875	+	9.51875i	−0.655298	+	0.655298i	0.298966	+	0.954264i	$0.403358\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			10.2785i			0.700989i
$216$		0			0
$217$			3.00413i			0.203934i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.62599	+	2.62599i	−0.176643	+	0.176643i
$222$		0			0
$223$		−	16.0637i		−	1.07571i	−0.843038	−	0.537853i	$-0.819235\pi$
$223$		−	16.0637i		−	1.07571i	0.843038	−	0.537853i	$-0.180765\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.5695	−	11.5695i	−0.767894	−	0.767894i	0.209842	−	0.977735i	$-0.432705\pi$
$227$	−11.5695	−	11.5695i	−0.767894	−	0.767894i	−0.977735	+	0.209842i	$0.932705\pi$
$228$		0			0
$229$	−4.50533	+	4.50533i	−0.297721	+	0.297721i	−0.840121	−	0.542400i	$-0.817516\pi$
$229$	−4.50533	+	4.50533i	−0.297721	+	0.297721i	0.542400	+	0.840121i	$0.317516\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.2560			−1.06496			−0.532482	−	0.846442i	$-0.678740\pi$
$233$	−16.2560			−1.06496			−0.532482	+	0.846442i	$0.678740\pi$
$234$		0			0
$235$	−12.1969	−	12.1969i	−0.795638	−	0.795638i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.55597			−0.294701			−0.147351	−	0.989084i	$-0.547075\pi$
$239$	−4.55597			−0.294701			−0.147351	+	0.989084i	$0.547075\pi$
$240$		0			0
$241$	−22.0079			−1.41765			−0.708827	−	0.705382i	$-0.750775\pi$
$241$	−22.0079			−1.41765			−0.708827	+	0.705382i	$0.750775\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.96859	−	2.96859i	−0.189657	−	0.189657i
$246$		0			0
$247$	26.5606			1.69001
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.15767	+	3.15767i	−0.199311	+	0.199311i	−0.799704	−	0.600394i	$-0.795010\pi$
$251$	−3.15767	+	3.15767i	−0.199311	+	0.199311i	0.600394	+	0.799704i	$0.295010\pi$
$252$		0			0
$253$	4.96472	+	4.96472i	0.312129	+	0.312129i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	0.930640i		−	0.0580517i	−0.999579	−	0.0290259i	$-0.990759\pi$
$257$		−	0.930640i		−	0.0580517i	0.999579	−	0.0290259i	$-0.00924052\pi$
$258$		0			0
$259$	0.149739	−	0.149739i	0.00930431	−	0.00930431i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			14.8926i			0.918319i	0.888354	+	0.459160i	$0.151849\pi$
$263$			14.8926i			0.918319i	−0.888354	+	0.459160i	$0.848151\pi$
$264$		0			0
$265$		−	35.8660i		−	2.20323i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.20537	−	7.20537i	0.439319	−	0.439319i	−0.452464	−	0.891783i	$-0.649455\pi$
$269$	7.20537	−	7.20537i	0.439319	−	0.439319i	0.891783	+	0.452464i	$0.149455\pi$
$270$		0			0
$271$			25.2262i			1.53238i	0.642611	+	0.766192i	$0.277851\pi$
$271$			25.2262i			1.53238i	−0.642611	+	0.766192i	$0.722149\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.19454	−	7.19454i	−0.433847	−	0.433847i
$276$		0			0
$277$	10.1502	−	10.1502i	0.609866	−	0.609866i	−0.333045	−	0.942911i	$-0.608076\pi$
$277$	10.1502	−	10.1502i	0.609866	−	0.609866i	0.942911	+	0.333045i	$0.108076\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.47979			0.565517			0.282758	−	0.959191i	$-0.408751\pi$
$281$	9.47979			0.565517			0.282758	+	0.959191i	$0.408751\pi$
$282$		0			0
$283$	−2.74974	−	2.74974i	−0.163455	−	0.163455i	0.620640	−	0.784095i	$-0.286873\pi$
$283$	−2.74974	−	2.74974i	−0.163455	−	0.163455i	−0.784095	+	0.620640i	$0.786873\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.63459			0.509683
$288$		0			0
$289$	15.9282			0.936953
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.497276	−	0.497276i	−0.0290512	−	0.0290512i	0.692432	−	0.721483i	$-0.256539\pi$
$293$	−0.497276	−	0.497276i	−0.0290512	−	0.0290512i	−0.721483	+	0.692432i	$0.756539\pi$
$294$		0			0
$295$	35.9272			2.09176
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.0984	−	22.0984i	1.27798	−	1.27798i
$300$		0			0
$301$	1.73121	+	1.73121i	0.0997853	+	0.0997853i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	34.5483i		−	1.97823i
$306$		0			0
$307$	−11.0456	+	11.0456i	−0.630408	+	0.630408i	−0.948170	−	0.317762i	$-0.897069\pi$
$307$	−11.0456	+	11.0456i	−0.630408	+	0.630408i	0.317762	+	0.948170i	$0.397069\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	24.7421i		−	1.40299i	−0.712672	−	0.701497i	$-0.752515\pi$
$311$		−	24.7421i		−	1.40299i	0.712672	−	0.701497i	$-0.247485\pi$
$312$		0			0
$313$		−	20.1653i		−	1.13981i	−0.821710	−	0.569906i	$-0.806980\pi$
$313$		−	20.1653i		−	1.13981i	0.821710	−	0.569906i	$-0.193020\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.70522	−	8.70522i	0.488934	−	0.488934i	−0.419036	−	0.907970i	$-0.637632\pi$
$317$	8.70522	−	8.70522i	0.488934	−	0.488934i	0.907970	+	0.419036i	$0.137632\pi$
$318$		0			0
$319$			6.89796i			0.386212i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.42040	+	5.42040i	0.301599	+	0.301599i
$324$		0			0
$325$	−32.0235	+	32.0235i	−1.77635	+	1.77635i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.10865			−0.226517
$330$		0			0
$331$	2.78793	+	2.78793i	0.153239	+	0.153239i	0.779563	−	0.626324i	$-0.215441\pi$
$331$	2.78793	+	2.78793i	0.153239	+	0.153239i	−0.626324	+	0.779563i	$0.715441\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.152555			0.00833496
$336$		0			0
$337$	−0.465376			−0.0253506			−0.0126753	−	0.999920i	$-0.504035\pi$
$337$	−0.465376			−0.0253506			−0.0126753	+	0.999920i	$0.504035\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.71193	+	1.71193i	0.0927064	+	0.0927064i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.3262	−	21.3262i	1.14485	−	1.14485i	0.157299	−	0.987551i	$-0.449721\pi$
$347$	21.3262	−	21.3262i	1.14485	−	1.14485i	0.987551	−	0.157299i	$-0.0502787\pi$
$348$		0			0
$349$	5.11124	+	5.11124i	0.273598	+	0.273598i	0.830547	−	0.556949i	$-0.188028\pi$
$349$	5.11124	+	5.11124i	0.273598	+	0.273598i	−0.556949	+	0.830547i	$0.688028\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			10.2227i			0.544102i	0.962283	+	0.272051i	$0.0877019\pi$
$353$			10.2227i			0.544102i	−0.962283	+	0.272051i	$0.912298\pi$
$354$		0			0
$355$	−43.1209	+	43.1209i	−2.28862	+	2.28862i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	5.90071i		−	0.311427i	−0.987802	−	0.155714i	$-0.950232\pi$
$359$		−	5.90071i		−	0.311427i	0.987802	−	0.155714i	$-0.0497677\pi$
$360$		0			0
$361$		−	35.8247i		−	1.88551i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.9857	+	15.9857i	−0.836732	+	0.836732i
$366$		0			0
$367$		−	14.1762i		−	0.739994i	−0.929033	−	0.369997i	$-0.879359\pi$
$367$		−	14.1762i		−	0.739994i	0.929033	−	0.369997i	$-0.120641\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.04092	−	6.04092i	−0.313629	−	0.313629i
$372$		0			0
$373$	21.5106	−	21.5106i	1.11378	−	1.11378i	0.121141	−	0.992635i	$-0.461345\pi$
$373$	21.5106	−	21.5106i	1.11378	−	1.11378i	0.992635	−	0.121141i	$-0.0386553\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	30.7035			1.58131
$378$		0			0
$379$	8.15510	+	8.15510i	0.418900	+	0.418900i	0.884824	−	0.465925i	$-0.154278\pi$
$379$	8.15510	+	8.15510i	0.418900	+	0.418900i	−0.465925	+	0.884824i	$0.654278\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.06121			−0.463006			−0.231503	−	0.972834i	$-0.574364\pi$
$383$	−9.06121			−0.463006			−0.231503	+	0.972834i	$0.574364\pi$
$384$		0			0
$385$	−3.38337			−0.172432
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.1084	+	22.1084i	1.12094	+	1.12094i	0.991600	+	0.129339i	$0.0412855\pi$
$389$	22.1084	+	22.1084i	1.12094	+	1.12094i	0.129339	+	0.991600i	$0.458715\pi$
$390$		0			0
$391$	9.01954			0.456138
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.5738	+	11.5738i	−0.582343	+	0.582343i
$396$		0			0
$397$	−25.1321	−	25.1321i	−1.26134	−	1.26134i	−0.950443	−	0.310899i	$-0.899370\pi$
$397$	−25.1321	−	25.1321i	−1.26134	−	1.26134i	−0.310899	−	0.950443i	$-0.600630\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			1.54588i			0.0771975i	0.999255	+	0.0385987i	$0.0122894\pi$
$401$			1.54588i			0.0771975i	−0.999255	+	0.0385987i	$0.987711\pi$
$402$		0			0
$403$	7.61997	−	7.61997i	0.379578	−	0.379578i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.170660i		−	0.00845931i
$408$		0			0
$409$		−	5.72850i		−	0.283256i	−0.989920	−	0.141628i	$-0.954766\pi$
$409$		−	5.72850i		−	0.283256i	0.989920	−	0.141628i	$-0.0452337\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.05121	−	6.05121i	0.297761	−	0.297761i
$414$		0			0
$415$			8.35307i			0.410036i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−26.2299	−	26.2299i	−1.28141	−	1.28141i	−0.939863	−	0.341551i	$-0.889048\pi$
$419$	−26.2299	−	26.2299i	−1.28141	−	1.28141i	−0.341551	−	0.939863i	$-0.610952\pi$
$420$		0			0
$421$	9.38496	−	9.38496i	0.457395	−	0.457395i	−0.440404	−	0.897800i	$-0.645165\pi$
$421$	9.38496	−	9.38496i	0.457395	−	0.457395i	0.897800	+	0.440404i	$0.145165\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.0705			−0.634013
$426$		0			0
$427$	−5.81897	−	5.81897i	−0.281599	−	0.281599i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.9406			−0.767830			−0.383915	−	0.923368i	$-0.625424\pi$
$431$	−15.9406			−0.767830			−0.383915	+	0.923368i	$0.625424\pi$
$432$		0			0
$433$	25.0062			1.20172			0.600862	−	0.799353i	$-0.294824\pi$
$433$	25.0062			1.20172			0.600862	+	0.799353i	$0.294824\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−45.6141	−	45.6141i	−2.18202	−	2.18202i
$438$		0			0
$439$	35.0522			1.67295			0.836474	−	0.548007i	$-0.184613\pi$
$439$	35.0522			1.67295			0.836474	+	0.548007i	$0.184613\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17.1307	−	17.1307i	0.813904	−	0.813904i	−0.171313	−	0.985217i	$-0.554801\pi$
$443$	17.1307	−	17.1307i	0.813904	−	0.813904i	0.985217	+	0.171313i	$0.0548009\pi$
$444$		0			0
$445$	50.5725	+	50.5725i	2.39737	+	2.39737i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			14.7964i			0.698286i	0.937070	+	0.349143i	$0.113527\pi$
$449$			14.7964i			0.698286i	−0.937070	+	0.349143i	$0.886473\pi$
$450$		0			0
$451$	4.92050	−	4.92050i	0.231697	−	0.231697i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			15.0597i			0.706008i
$456$		0			0
$457$			11.9903i			0.560883i	0.959871	+	0.280442i	$0.0904809\pi$
$457$			11.9903i			0.560883i	−0.959871	+	0.280442i	$0.909519\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.9493	+	14.9493i	−0.696260	+	0.696260i	−0.963602	−	0.267342i	$-0.913855\pi$
$461$	−14.9493	+	14.9493i	−0.696260	+	0.696260i	0.267342	+	0.963602i	$0.413855\pi$
$462$		0			0
$463$		−	31.4696i		−	1.46252i	−0.682101	−	0.731258i	$-0.738933\pi$
$463$		−	31.4696i		−	1.46252i	0.682101	−	0.731258i	$-0.261067\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.7699	+	24.7699i	1.14621	+	1.14621i	0.987291	+	0.158922i	$0.0508018\pi$
$467$	24.7699	+	24.7699i	1.14621	+	1.14621i	0.158922	+	0.987291i	$0.449198\pi$
$468$		0			0
$469$	0.0256948	−	0.0256948i	0.00118647	−	0.00118647i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.97310			0.0907230
$474$		0			0
$475$	66.1010	+	66.1010i	3.03292	+	3.03292i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17.7194			−0.809620			−0.404810	−	0.914401i	$-0.632662\pi$
$479$	−17.7194			−0.809620			−0.404810	+	0.914401i	$0.632662\pi$
$480$		0			0
$481$	−0.759624			−0.0346359
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.1385	−	11.1385i	−0.505772	−	0.505772i
$486$		0			0
$487$	33.8639			1.53452			0.767260	−	0.641336i	$-0.221619\pi$
$487$	33.8639			1.53452			0.767260	+	0.641336i	$0.221619\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	25.3486	−	25.3486i	1.14397	−	1.14397i	0.156248	−	0.987718i	$-0.450060\pi$
$491$	25.3486	−	25.3486i	1.14397	−	1.14397i	0.987718	−	0.156248i	$-0.0499400\pi$
$492$		0			0
$493$	6.26586	+	6.26586i	0.282200	+	0.282200i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			14.5257i			0.651566i
$498$		0			0
$499$	7.26690	−	7.26690i	0.325311	−	0.325311i	−0.525489	−	0.850800i	$-0.676118\pi$
$499$	7.26690	−	7.26690i	0.325311	−	0.325311i	0.850800	+	0.525489i	$0.176118\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			12.3031i			0.548567i	0.961649	+	0.274284i	$0.0884407\pi$
$503$			12.3031i			0.548567i	−0.961649	+	0.274284i	$0.911559\pi$
$504$		0			0
$505$			7.32217i			0.325832i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.0082	+	14.0082i	−0.620901	+	0.620901i	−0.945762	−	0.324861i	$-0.894682\pi$
$509$	−14.0082	+	14.0082i	−0.620901	+	0.620901i	0.324861	+	0.945762i	$0.394682\pi$
$510$		0			0
$511$			5.38496i			0.238216i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	42.5487	+	42.5487i	1.87492	+	1.87492i
$516$		0			0
$517$	−2.34135	+	2.34135i	−0.102973	+	0.102973i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−36.8204			−1.61313			−0.806566	−	0.591144i	$-0.798677\pi$
$521$	−36.8204			−1.61313			−0.806566	+	0.591144i	$0.798677\pi$
$522$		0			0
$523$	−2.49783	−	2.49783i	−0.109222	−	0.109222i	0.650384	−	0.759606i	$-0.274608\pi$
$523$	−2.49783	−	2.49783i	−0.109222	−	0.109222i	−0.759606	+	0.650384i	$0.774608\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.11012			0.135479
$528$		0			0
$529$	−52.9020			−2.30009
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−21.9016	−	21.9016i	−0.948664	−	0.948664i
$534$		0			0
$535$	−43.5431			−1.88253
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.569860	+	0.569860i	−0.0245456	+	0.0245456i
$540$		0			0
$541$	14.5049	+	14.5049i	0.623613	+	0.623613i	0.946453	−	0.322841i	$-0.104638\pi$
$541$	14.5049	+	14.5049i	0.623613	+	0.623613i	−0.322841	+	0.946453i	$0.604638\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			24.4742i			1.04836i
$546$		0			0
$547$	−10.3072	+	10.3072i	−0.440704	+	0.440704i	−0.892249	−	0.451545i	$-0.850873\pi$
$547$	−10.3072	+	10.3072i	−0.440704	+	0.440704i	0.451545	+	0.892249i	$0.350873\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	63.3761i		−	2.69991i
$552$		0			0
$553$			3.89876i			0.165792i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.62884	+	8.62884i	−0.365616	+	0.365616i	−0.865875	−	0.500260i	$-0.833238\pi$
$557$	−8.62884	+	8.62884i	−0.365616	+	0.365616i	0.500260	+	0.865875i	$0.333238\pi$
$558$		0			0
$559$		−	8.78243i		−	0.371457i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.3155	−	10.3155i	−0.434748	−	0.434748i	0.455492	−	0.890240i	$-0.349463\pi$
$563$	−10.3155	−	10.3155i	−0.434748	−	0.434748i	−0.890240	+	0.455492i	$0.849463\pi$
$564$		0			0
$565$	36.0755	−	36.0755i	1.51771	−	1.51771i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.23614			0.135666			0.0678330	−	0.997697i	$-0.478391\pi$
$569$	3.23614			0.135666			0.0678330	+	0.997697i	$0.478391\pi$
$570$		0			0
$571$	−27.7822	−	27.7822i	−1.16265	−	1.16265i	−0.983893	−	0.178757i	$-0.942792\pi$
$571$	−27.7822	−	27.7822i	−1.16265	−	1.16265i	−0.178757	−	0.983893i	$-0.557208\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	109.992			4.58699
$576$		0			0
$577$	22.0528			0.918072			0.459036	−	0.888418i	$-0.348195\pi$
$577$	22.0528			0.918072			0.459036	+	0.888418i	$0.348195\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.40691	+	1.40691i	0.0583684	+	0.0583684i
$582$		0			0
$583$	−6.88495			−0.285146
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.9242	+	16.9242i	−0.698538	+	0.698538i	−0.964095	−	0.265557i	$-0.914444\pi$
$587$	−16.9242	+	16.9242i	−0.698538	+	0.698538i	0.265557	+	0.964095i	$0.414444\pi$
$588$		0			0
$589$	−15.7287	−	15.7287i	−0.648088	−	0.648088i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	9.14027i		−	0.375346i	−0.982232	−	0.187673i	$-0.939905\pi$
$593$		−	9.14027i		−	0.375346i	0.982232	−	0.187673i	$-0.0600945\pi$
$594$		0			0
$595$	−3.07333	+	3.07333i	−0.125994	+	0.125994i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			26.4441i			1.08047i	0.841513	+	0.540237i	$0.181666\pi$
$599$			26.4441i			1.08047i	−0.841513	+	0.540237i	$0.818334\pi$
$600$		0			0
$601$		−	28.3857i		−	1.15788i	−0.815371	−	0.578938i	$-0.803467\pi$
$601$		−	28.3857i		−	1.15788i	0.815371	−	0.578938i	$-0.196533\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	30.7265	−	30.7265i	1.24921	−	1.24921i
$606$		0			0
$607$		−	9.53246i		−	0.386911i	−0.981109	−	0.193455i	$-0.938031\pi$
$607$		−	9.53246i		−	0.386911i	0.981109	−	0.193455i	$-0.0619695\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.4216	+	10.4216i	0.421612	+	0.421612i
$612$		0			0
$613$	1.07498	−	1.07498i	0.0434181	−	0.0434181i	−0.685064	−	0.728483i	$-0.740226\pi$
$613$	1.07498	−	1.07498i	0.0434181	−	0.0434181i	0.728483	+	0.685064i	$0.240226\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−35.3828			−1.42446			−0.712229	−	0.701948i	$-0.752314\pi$
$617$	−35.3828			−1.42446			−0.712229	+	0.701948i	$0.752314\pi$
$618$		0			0
$619$	3.97297	+	3.97297i	0.159687	+	0.159687i	0.782428	−	0.622741i	$-0.213981\pi$
$619$	3.97297	+	3.97297i	0.159687	+	0.159687i	−0.622741	+	0.782428i	$0.713981\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17.0358			0.682526
$624$		0			0
$625$	−71.2676			−2.85070
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.155022	−	0.155022i	−0.00618111	−	0.00618111i
$630$		0			0
$631$	25.6289			1.02027			0.510135	−	0.860095i	$-0.329596\pi$
$631$	25.6289			1.02027			0.510135	+	0.860095i	$0.329596\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.92120	+	3.92120i	−0.155608	+	0.155608i
$636$		0			0
$637$	2.53650	+	2.53650i	0.100500	+	0.100500i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			10.5770i			0.417766i	0.977941	+	0.208883i	$0.0669829\pi$
$641$			10.5770i			0.417766i	−0.977941	+	0.208883i	$0.933017\pi$
$642$		0			0
$643$	−8.26445	+	8.26445i	−0.325918	+	0.325918i	−0.851032	−	0.525114i	$-0.824023\pi$
$643$	−8.26445	+	8.26445i	−0.325918	+	0.325918i	0.525114	+	0.851032i	$0.324023\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			23.2954i			0.915835i	0.888995	+	0.457917i	$0.151404\pi$
$647$			23.2954i			0.915835i	−0.888995	+	0.457917i	$0.848596\pi$
$648$		0			0
$649$		−	6.89668i		−	0.270718i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.0365	−	20.0365i	0.784090	−	0.784090i	−0.196429	−	0.980518i	$-0.562934\pi$
$653$	20.0365	−	20.0365i	0.784090	−	0.784090i	0.980518	+	0.196429i	$0.0629344\pi$
$654$		0			0
$655$		−	9.95804i		−	0.389093i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.7397	+	24.7397i	0.963723	+	0.963723i	0.999365	−	0.0356416i	$-0.0113475\pi$
$659$	24.7397	+	24.7397i	0.963723	+	0.963723i	−0.0356416	+	0.999365i	$0.511347\pi$
$660$		0			0
$661$	−6.31446	+	6.31446i	−0.245604	+	0.245604i	−0.819164	−	0.573560i	$-0.805562\pi$
$661$	−6.31446	+	6.31446i	−0.245604	+	0.245604i	0.573560	+	0.819164i	$0.305562\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	31.0852			1.20543
$666$		0			0
$667$	−52.7290	−	52.7290i	−2.04167	−	2.04167i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.63199			−0.256025
$672$		0			0
$673$	2.32930			0.0897877			0.0448939	−	0.998992i	$-0.485705\pi$
$673$	2.32930			0.0897877			0.0448939	+	0.998992i	$0.485705\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.0208	+	17.0208i	0.654163	+	0.654163i	0.953993	−	0.299830i	$-0.0969300\pi$
$677$	17.0208	+	17.0208i	0.654163	+	0.654163i	−0.299830	+	0.953993i	$0.596930\pi$
$678$		0			0
$679$	−3.75210			−0.143993
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.18231	+	5.18231i	−0.198296	+	0.198296i	−0.799269	−	0.600973i	$-0.794780\pi$
$683$	−5.18231	+	5.18231i	−0.198296	+	0.198296i	0.600973	+	0.799269i	$0.294780\pi$
$684$		0			0
$685$	−25.2198	−	25.2198i	−0.963599	−	0.963599i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			30.6455i			1.16750i
$690$		0			0
$691$	23.6472	−	23.6472i	0.899582	−	0.899582i	−0.0958169	−	0.995399i	$-0.530546\pi$
$691$	23.6472	−	23.6472i	0.899582	−	0.899582i	0.995399	+	0.0958169i	$0.0305463\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	46.3692i		−	1.75888i
$696$		0			0
$697$		−	8.93922i		−	0.338597i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.62771	−	4.62771i	0.174786	−	0.174786i	−0.614292	−	0.789078i	$-0.710558\pi$
$701$	4.62771	−	4.62771i	0.174786	−	0.174786i	0.789078	+	0.614292i	$0.210558\pi$
$702$		0			0
$703$			1.56797i			0.0591370i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.23327	+	1.23327i	0.0463820	+	0.0463820i
$708$		0			0
$709$	−27.1441	+	27.1441i	−1.01942	+	1.01942i	−0.0196099	+	0.999808i	$0.506242\pi$
$709$	−27.1441	+	27.1441i	−1.01942	+	1.01942i	−0.999808	+	0.0196099i	$0.993758\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−26.1725			−0.980168
$714$		0			0
$715$	8.58190	+	8.58190i	0.320945	+	0.320945i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−39.3061			−1.46587			−0.732934	−	0.680299i	$-0.761850\pi$
$719$	−39.3061			−1.46587			−0.732934	+	0.680299i	$0.761850\pi$
$720$		0			0
$721$	14.3330			0.533787
$722$		0			0
$723$		0			0
$724$		0			0
$725$	76.4113	+	76.4113i	2.83784	+	2.83784i
$726$		0			0
$727$	−36.1990			−1.34255			−0.671274	−	0.741209i	$-0.734253\pi$
$727$	−36.1990			−1.34255			−0.671274	+	0.741209i	$0.734253\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.79229	−	1.79229i	0.0662902	−	0.0662902i
$732$		0			0
$733$	−3.85554	−	3.85554i	−0.142408	−	0.142408i	0.632309	−	0.774716i	$-0.282107\pi$
$733$	−3.85554	−	3.85554i	−0.142408	−	0.142408i	−0.774716	+	0.632309i	$0.782107\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	0.0292849i		−	0.00107872i
$738$		0			0
$739$	20.1965	−	20.1965i	0.742942	−	0.742942i	−0.230201	−	0.973143i	$-0.573939\pi$
$739$	20.1965	−	20.1965i	0.742942	−	0.742942i	0.973143	+	0.230201i	$0.0739385\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			38.9110i			1.42751i	0.700398	+	0.713753i	$0.253006\pi$
$743$			38.9110i			1.42751i	−0.700398	+	0.713753i	$0.746994\pi$
$744$		0			0
$745$			77.8444i			2.85200i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.33395	+	7.33395i	−0.267977	+	0.267977i
$750$		0			0
$751$		−	1.20106i		−	0.0438274i	−0.999760	−	0.0219137i	$-0.993024\pi$
$751$		−	1.20106i		−	0.0438274i	0.999760	−	0.0219137i	$-0.00697590\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	39.8947	+	39.8947i	1.45192	+	1.45192i
$756$		0			0
$757$	12.9836	−	12.9836i	0.471896	−	0.471896i	−0.430632	−	0.902528i	$-0.641709\pi$
$757$	12.9836	−	12.9836i	0.471896	−	0.471896i	0.902528	+	0.430632i	$0.141709\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.4747			0.597208			0.298604	−	0.954377i	$-0.403479\pi$
$761$	16.4747			0.597208			0.298604	+	0.954377i	$0.403479\pi$
$762$		0			0
$763$	4.12218	+	4.12218i	0.149233	+	0.149233i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−30.6978			−1.10843
$768$		0			0
$769$	33.6226			1.21246			0.606231	−	0.795289i	$-0.292681\pi$
$769$	33.6226			1.21246			0.606231	+	0.795289i	$0.292681\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	20.0820	+	20.0820i	0.722299	+	0.722299i	0.969073	−	0.246774i	$-0.0793707\pi$
$773$	20.0820	+	20.0820i	0.722299	+	0.722299i	−0.246774	+	0.969073i	$0.579371\pi$
$774$		0			0
$775$	37.9274			1.36239
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−45.2079	+	45.2079i	−1.61974	+	1.61974i
$780$		0			0
$781$	8.27761	+	8.27761i	0.296196	+	0.296196i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			35.1774i			1.25553i
$786$		0			0
$787$	1.75329	−	1.75329i	0.0624980	−	0.0624980i	−0.675167	−	0.737665i	$-0.735928\pi$
$787$	1.75329	−	1.75329i	0.0624980	−	0.0624980i	0.737665	+	0.675167i	$0.235928\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	12.1524i		−	0.432089i
$792$		0			0
$793$			29.5196i			1.04827i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16.1165	−	16.1165i	0.570876	−	0.570876i	−0.361497	−	0.932373i	$-0.617734\pi$
$797$	16.1165	−	16.1165i	0.570876	−	0.570876i	0.932373	+	0.361497i	$0.117734\pi$
$798$		0			0
$799$			4.25360i			0.150482i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.06867	+	3.06867i	0.108291	+	0.108291i
$804$		0			0
$805$	25.8629	−	25.8629i	0.911548	−	0.911548i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−30.3586			−1.06735			−0.533676	−	0.845689i	$-0.679190\pi$
$809$	−30.3586			−1.06735			−0.533676	+	0.845689i	$0.679190\pi$
$810$		0			0
$811$	−20.3645	−	20.3645i	−0.715095	−	0.715095i	0.252502	−	0.967596i	$-0.418747\pi$
$811$	−20.3645	−	20.3645i	−0.715095	−	0.715095i	−0.967596	+	0.252502i	$0.918747\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.8336			−0.379483
$816$		0			0
$817$	−18.1281			−0.634223
$818$		0			0
$819$		0			0
$820$		0			0
$821$	35.3970	+	35.3970i	1.23536	+	1.23536i	0.961876	+	0.273487i	$0.0881770\pi$
$821$	35.3970	+	35.3970i	1.23536	+	1.23536i	0.273487	+	0.961876i	$0.411823\pi$
$822$		0			0
$823$	−24.6885			−0.860587			−0.430294	−	0.902689i	$-0.641590\pi$
$823$	−24.6885			−0.860587			−0.430294	+	0.902689i	$0.641590\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−37.8496	+	37.8496i	−1.31616	+	1.31616i	−0.399369	+	0.916790i	$0.630771\pi$
$827$	−37.8496	+	37.8496i	−1.31616	+	1.31616i	−0.916790	+	0.399369i	$0.869229\pi$
$828$		0			0
$829$	−5.82230	−	5.82230i	−0.202217	−	0.202217i	0.598732	−	0.800949i	$-0.295671\pi$
$829$	−5.82230	−	5.82230i	−0.202217	−	0.202217i	−0.800949	+	0.598732i	$0.795671\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1.03528i			0.0358703i
$834$		0			0
$835$	−16.3060	+	16.3060i	−0.564291	+	0.564291i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			24.7552i			0.854645i	0.904099	+	0.427323i	$0.140543\pi$
$839$			24.7552i			0.854645i	−0.904099	+	0.427323i	$0.859457\pi$
$840$		0			0
$841$		−	44.2615i		−	1.52626i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.392885	+	0.392885i	−0.0135156	+	0.0135156i
$846$		0			0
$847$		−	10.3505i		−	0.355648i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.30455	+	1.30455i	0.0447194	+	0.0447194i
$852$		0			0
$853$	9.64611	−	9.64611i	0.330276	−	0.330276i	−0.522415	−	0.852691i	$-0.674969\pi$
$853$	9.64611	−	9.64611i	0.330276	−	0.330276i	0.852691	+	0.522415i	$0.174969\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.92127			0.0997886			0.0498943	−	0.998755i	$-0.484112\pi$
$857$	2.92127			0.0997886			0.0498943	+	0.998755i	$0.484112\pi$
$858$		0			0
$859$	1.63800	+	1.63800i	0.0558880	+	0.0558880i	0.734498	−	0.678610i	$-0.237418\pi$
$859$	1.63800	+	1.63800i	0.0558880	+	0.0558880i	−0.678610	+	0.734498i	$0.737418\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−54.6506			−1.86033			−0.930164	−	0.367145i	$-0.880335\pi$
$863$	−54.6506			−1.86033			−0.930164	+	0.367145i	$0.880335\pi$
$864$		0			0
$865$	67.7907			2.30495
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.22175	+	2.22175i	0.0753676	+	0.0753676i
$870$		0			0
$871$	−0.130350			−0.00441673
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.6358	+	22.6358i	−0.765230	+	0.765230i
$876$		0			0
$877$	−28.8847	−	28.8847i	−0.975368	−	0.975368i	0.0243360	−	0.999704i	$-0.492253\pi$
$877$	−28.8847	−	28.8847i	−0.975368	−	0.975368i	−0.999704	+	0.0243360i	$0.992253\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			22.1474i			0.746166i	0.927798	+	0.373083i	$0.121699\pi$
$881$			22.1474i			0.746166i	−0.927798	+	0.373083i	$0.878301\pi$
$882$		0			0
$883$	34.6797	−	34.6797i	1.16706	−	1.16706i	0.184169	−	0.982895i	$-0.441041\pi$
$883$	34.6797	−	34.6797i	1.16706	−	1.16706i	0.982895	−	0.184169i	$-0.0589594\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			21.6236i			0.726050i	0.931779	+	0.363025i	$0.118256\pi$
$887$			21.6236i			0.726050i	−0.931779	+	0.363025i	$0.881744\pi$
$888$		0			0
$889$			1.32089i			0.0443014i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	21.5116	−	21.5116i	0.719857	−	0.719857i
$894$		0			0
$895$		−	32.5275i		−	1.08727i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−18.1820	−	18.1820i	−0.606403	−	0.606403i
$900$		0			0
$901$	−6.25404	+	6.25404i	−0.208352	+	0.208352i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	55.9719			1.86057
$906$		0			0
$907$	37.3334	+	37.3334i	1.23963	+	1.23963i	0.960150	+	0.279485i	$0.0901637\pi$
$907$	37.3334	+	37.3334i	1.23963	+	1.23963i	0.279485	+	0.960150i	$0.409836\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−22.0091			−0.729194			−0.364597	−	0.931165i	$-0.618793\pi$
$911$	−22.0091			−0.729194			−0.364597	+	0.931165i	$0.618793\pi$
$912$		0			0
$913$	1.60348			0.0530674
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.67723	−	1.67723i	−0.0553871	−	0.0553871i
$918$		0			0
$919$	−35.8652			−1.18308			−0.591542	−	0.806274i	$-0.701481\pi$
$919$	−35.8652			−1.18308			−0.591542	+	0.806274i	$0.701481\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	36.8444	−	36.8444i	1.21275	−	1.21275i
$924$		0			0
$925$	−1.89047	−	1.89047i	−0.0621581	−	0.0621581i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			16.1094i			0.528531i	0.964450	+	0.264265i	$0.0851295\pi$
$929$			16.1094i			0.528531i	−0.964450	+	0.264265i	$0.914870\pi$
$930$		0			0
$931$	5.23568	−	5.23568i	0.171592	−	0.171592i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			3.50273i			0.114552i
$936$		0			0
$937$		−	17.5529i		−	0.573429i	−0.958016	−	0.286714i	$-0.907437\pi$
$937$		−	17.5529i		−	0.573429i	0.958016	−	0.286714i	$-0.0925630\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.41436	−	6.41436i	0.209102	−	0.209102i	−0.594784	−	0.803886i	$-0.702762\pi$
$941$	6.41436	−	6.41436i	0.209102	−	0.209102i	0.803886	+	0.594784i	$0.202762\pi$
$942$		0			0
$943$			75.2260i			2.44970i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.95690	−	7.95690i	−0.258564	−	0.258564i	0.565906	−	0.824470i	$-0.308527\pi$
$947$	−7.95690	−	7.95690i	−0.258564	−	0.258564i	−0.824470	+	0.565906i	$0.808527\pi$
$948$		0			0
$949$	13.6589	−	13.6589i	0.443388	−	0.443388i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.4471			1.47217			0.736087	−	0.676887i	$-0.236671\pi$
$953$	45.4471			1.47217			0.736087	+	0.676887i	$0.236671\pi$
$954$		0			0
$955$	30.1114	+	30.1114i	0.974383	+	0.974383i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.49554			−0.274335
$960$		0			0
$961$	21.9752			0.708878
$962$		0			0
$963$		0			0
$964$		0			0
$965$	37.5612	+	37.5612i	1.20914	+	1.20914i
$966$		0			0
$967$	−62.0308			−1.99478			−0.997389	−	0.0722214i	$-0.976991\pi$
$967$	−62.0308			−1.99478			−0.997389	+	0.0722214i	$0.976991\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.5203	−	11.5203i	0.369704	−	0.369704i	−0.497665	−	0.867369i	$-0.665809\pi$
$971$	11.5203	−	11.5203i	0.369704	−	0.369704i	0.867369	+	0.497665i	$0.165809\pi$
$972$		0			0
$973$	−7.80995	−	7.80995i	−0.250376	−	0.250376i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	30.9755i		−	0.990995i	−0.868609	−	0.495498i	$-0.834986\pi$
$977$		−	30.9755i		−	0.990995i	0.868609	−	0.495498i	$-0.165014\pi$
$978$		0			0
$979$	9.70804	−	9.70804i	0.310270	−	0.310270i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	39.7177i		−	1.26680i	−0.773825	−	0.633399i	$-0.781659\pi$
$983$		−	39.7177i		−	1.26680i	0.773825	−	0.633399i	$-0.218341\pi$
$984$		0			0
$985$			25.6504i			0.817290i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−15.0826	+	15.0826i	−0.479599	+	0.479599i
$990$		0			0
$991$		−	44.2359i		−	1.40520i	−0.711585	−	0.702600i	$-0.752022\pi$
$991$		−	44.2359i		−	1.40520i	0.711585	−	0.702600i	$-0.247978\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	21.4433	+	21.4433i	0.679798	+	0.679798i
$996$		0			0
$997$	−19.7318	+	19.7318i	−0.624913	+	0.624913i	−0.946784	−	0.321871i	$-0.895688\pi$
$997$	−19.7318	+	19.7318i	−0.624913	+	0.624913i	0.321871	+	0.946784i	$0.395688\pi$
$998$		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.1583.1		40
3.2	odd	2	inner	4032.2.v.e.1583.20		40
4.3	odd	2		1008.2.v.e.323.12	yes	40
12.11	even	2		1008.2.v.e.323.9	✓	40
16.5	even	4		1008.2.v.e.827.9	yes	40
16.11	odd	4	inner	4032.2.v.e.3599.20		40
48.5	odd	4		1008.2.v.e.827.12	yes	40
48.11	even	4	inner	4032.2.v.e.3599.1		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.9	✓	40	12.11	even	2
1008.2.v.e.323.12	yes	40	4.3	odd	2
1008.2.v.e.827.9	yes	40	16.5	even	4
1008.2.v.e.827.12	yes	40	48.5	odd	4
4032.2.v.e.1583.1		40	1.1	even	1	trivial
4032.2.v.e.1583.20		40	3.2	odd	2	inner
4032.2.v.e.3599.1		40	48.11	even	4	inner
4032.2.v.e.3599.20		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)

\(f(q)\)	\(=\)	\(q+(-2.96859 - 2.96859i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(-2.96859 - 2.96859i) q^{5} -1.00000 q^{7} +(-0.569860 + 0.569860i) q^{11} +(2.53650 + 2.53650i) q^{13} +1.03528i q^{17} +(5.23568 - 5.23568i) q^{19} -8.71217i q^{23} +12.6251i q^{25} +(6.05233 - 6.05233i) q^{29} -3.00413i q^{31} +(2.96859 + 2.96859i) q^{35} +(-0.149739 + 0.149739i) q^{37} -8.63459 q^{41} +(-1.73121 - 1.73121i) q^{43} +4.10865 q^{47} +1.00000 q^{49} +(6.04092 + 6.04092i) q^{53} +3.38337 q^{55} +(-6.05121 + 6.05121i) q^{59} +(5.81897 + 5.81897i) q^{61} -15.0597i q^{65} +(-0.0256948 + 0.0256948i) q^{67} -14.5257i q^{71} -5.38496i q^{73} +(0.569860 - 0.569860i) q^{77} -3.89876i q^{79} +(-1.40691 - 1.40691i) q^{83} +(3.07333 - 3.07333i) q^{85} -17.0358 q^{89} +(-2.53650 - 2.53650i) q^{91} -31.0852 q^{95} +3.75210 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 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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