LMFDB - Embedded newform 4032.2.v.e.3599.20

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			3599.20
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.e.1583.20

$q$-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 $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

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

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	0.420133	−	0.907463i	$-0.361983\pi$
$5$	2.96859	−	2.96859i	1.32760	−	1.32760i	0.907463	−	0.420133i	$-0.138017\pi$
$6$		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	$-0.211231\pi$
$11$	0.569860	+	0.569860i	0.171819	+	0.171819i	−0.615959	+	0.787778i	$0.711231\pi$
$12$		0			0
$13$	2.53650	−	2.53650i	0.703498	−	0.703498i	−0.261661	−	0.965160i	$-0.584270\pi$
$13$	2.53650	−	2.53650i	0.703498	−	0.703498i	0.965160	+	0.261661i	$0.0842704\pi$
$14$		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	0.973818	+	0.227329i	$0.0729993\pi$
$19$	5.23568	+	5.23568i	1.20115	+	1.20115i	0.227329	+	0.973818i	$0.427001\pi$
$20$		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	−0.991151	−	0.132739i	$-0.957623\pi$
$29$	−6.05233	−	6.05233i	−1.12389	−	1.12389i	−0.132739	−	0.991151i	$-0.542377\pi$
$30$		0			0
$31$			3.00413i			0.539558i	0.962922	+	0.269779i	$0.0869506\pi$
$31$			3.00413i			0.539558i	−0.962922	+	0.269779i	$0.913049\pi$
$32$		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	0.694691	−	0.719308i	$-0.255541\pi$
$37$	−0.149739	−	0.149739i	−0.0246169	−	0.0246169i	−0.719308	+	0.694691i	$0.755541\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.63459			1.34850			0.674248	−	0.738505i	$-0.264468\pi$
$41$	8.63459			1.34850			0.674248	+	0.738505i	$0.264468\pi$
$42$		0			0
$43$	−1.73121	+	1.73121i	−0.264007	+	0.264007i	−0.826680	−	0.562673i	$-0.809773\pi$
$43$	−1.73121	+	1.73121i	−0.264007	+	0.264007i	0.562673	+	0.826680i	$0.309773\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.10865			−0.599308			−0.299654	−	0.954048i	$-0.596871\pi$
$47$	−4.10865			−0.599308			−0.299654	+	0.954048i	$0.596871\pi$
$48$		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	$-0.949591\pi$
$53$	−6.04092	+	6.04092i	−0.829783	+	0.829783i	0.157703	+	0.987487i	$0.449591\pi$
$54$		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	$-0.0619298\pi$
$59$	6.05121	+	6.05121i	0.787800	+	0.787800i	−0.193333	+	0.981133i	$0.561930\pi$
$60$		0			0
$61$	5.81897	−	5.81897i	0.745042	−	0.745042i	−0.228501	−	0.973544i	$-0.573383\pi$
$61$	5.81897	−	5.81897i	0.745042	−	0.745042i	0.973544	+	0.228501i	$0.0733826\pi$
$62$		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	0.705535	−	0.708675i	$-0.250707\pi$
$67$	−0.0256948	−	0.0256948i	−0.00313912	−	0.00313912i	−0.708675	+	0.705535i	$0.750707\pi$
$68$		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	$0.102048\pi$
$73$			5.38496i			0.630261i	−0.949048	+	0.315131i	$0.897952\pi$
$74$		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	$0.0703846\pi$
$79$			3.89876i			0.438645i	−0.975652	+	0.219322i	$0.929615\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.40691	−	1.40691i	0.154428	−	0.154428i	−0.625664	−	0.780092i	$-0.715172\pi$
$83$	1.40691	−	1.40691i	0.154428	−	0.154428i	0.780092	+	0.625664i	$0.215172\pi$
$84$		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	$-0.141436\pi$
$89$	17.0358			1.80579			0.902897	+	0.429856i	$0.141436\pi$
$90$		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	$-0.722345\pi$
$101$	1.23327	−	1.23327i	0.122715	−	0.122715i	0.765797	+	0.643082i	$0.222345\pi$
$102$		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	$-0.417159\pi$
$107$	−7.33395	−	7.33395i	−0.709000	−	0.709000i	−0.966325	+	0.257325i	$0.917159\pi$
$108$		0			0
$109$	−4.12218	+	4.12218i	−0.394833	+	0.394833i	−0.876406	−	0.481573i	$-0.840066\pi$
$109$	−4.12218	+	4.12218i	−0.394833	+	0.394833i	0.481573	+	0.876406i	$0.340066\pi$
$110$		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	$0.0186653\pi$
$127$			1.32089i			0.117210i	−0.998281	+	0.0586052i	$0.981335\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.67723	+	1.67723i	−0.146540	+	0.146540i	−0.776571	−	0.630030i	$-0.783043\pi$
$131$	−1.67723	+	1.67723i	−0.146540	+	0.146540i	0.630030	+	0.776571i	$0.283043\pi$
$132$		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	$-0.618217\pi$
$137$	−8.49554			−0.725823			−0.362911	+	0.931824i	$0.618217\pi$
$138$		0			0
$139$	7.80995	−	7.80995i	0.662432	−	0.662432i	−0.293521	−	0.955953i	$-0.594827\pi$
$139$	7.80995	−	7.80995i	0.662432	−	0.662432i	0.955953	+	0.293521i	$0.0948271\pi$
$140$		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	0.0770980	−	0.997024i	$-0.475435\pi$
$149$	13.1113	−	13.1113i	1.07412	−	1.07412i	0.997024	−	0.0770980i	$-0.0245654\pi$
$150$		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	−0.902839	−	0.429979i	$-0.858521\pi$
$157$	−5.92492	+	5.92492i	−0.472860	+	0.472860i	0.429979	+	0.902839i	$0.358521\pi$
$158$		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	0.774947	−	0.632026i	$-0.217776\pi$
$163$	1.82469	+	1.82469i	0.142921	+	0.142921i	−0.632026	+	0.774947i	$0.717776\pi$
$164$		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	$-0.0396262\pi$
$173$	11.4180	+	11.4180i	0.868093	+	0.868093i	−0.124168	+	0.992261i	$0.539626\pi$
$174$		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	$-0.843507\pi$
$179$	−5.47860	+	5.47860i	−0.409490	+	0.409490i	0.472071	+	0.881561i	$0.343507\pi$
$180$		0			0
$181$	−9.42734	−	9.42734i	−0.700728	−	0.700728i	0.263838	−	0.964567i	$-0.415011\pi$
$181$	−9.42734	−	9.42734i	−0.700728	−	0.700728i	−0.964567	+	0.263838i	$0.915011\pi$
$182$		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	$-0.380394\pi$
$191$	10.1433			0.733946			0.366973	+	0.930232i	$0.380394\pi$
$192$		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	$-0.680160\pi$
$197$	4.32030	−	4.32030i	0.307808	−	0.307808i	0.844059	+	0.536250i	$0.180160\pi$
$198$		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	0.298966	−	0.954264i	$-0.403358\pi$
$211$	−9.51875	−	9.51875i	−0.655298	−	0.655298i	−0.954264	+	0.298966i	$0.903358\pi$
$212$		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	$0.180765\pi$
$223$			16.0637i			1.07571i	−0.843038	+	0.537853i	$0.819235\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.5695	−	11.5695i	0.767894	−	0.767894i	−0.209842	−	0.977735i	$-0.567295\pi$
$227$	11.5695	−	11.5695i	0.767894	−	0.767894i	0.977735	+	0.209842i	$0.0672949\pi$
$228$		0			0
$229$	−4.50533	−	4.50533i	−0.297721	−	0.297721i	0.542400	−	0.840121i	$-0.317516\pi$
$229$	−4.50533	−	4.50533i	−0.297721	−	0.297721i	−0.840121	+	0.542400i	$0.817516\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.2560			1.06496			0.532482	−	0.846442i	$-0.321260\pi$
$233$	16.2560			1.06496			0.532482	+	0.846442i	$0.321260\pi$
$234$		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	$-0.452925\pi$
$239$	4.55597			0.294701			0.147351	+	0.989084i	$0.452925\pi$
$240$		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	$-0.204990\pi$
$251$	3.15767	+	3.15767i	0.199311	+	0.199311i	−0.600394	+	0.799704i	$0.704990\pi$
$252$		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	$-0.350545\pi$
$269$	−7.20537	−	7.20537i	−0.439319	−	0.439319i	−0.891783	+	0.452464i	$0.850545\pi$
$270$		0			0
$271$		−	25.2262i		−	1.53238i	−0.642611	−	0.766192i	$-0.722149\pi$
$271$		−	25.2262i		−	1.53238i	0.642611	−	0.766192i	$-0.277851\pi$
$272$		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	0.942911	−	0.333045i	$-0.108076\pi$
$277$	10.1502	+	10.1502i	0.609866	+	0.609866i	−0.333045	+	0.942911i	$0.608076\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.47979			−0.565517			−0.282758	−	0.959191i	$-0.591249\pi$
$281$	−9.47979			−0.565517			−0.282758	+	0.959191i	$0.591249\pi$
$282$		0			0
$283$	−2.74974	+	2.74974i	−0.163455	+	0.163455i	−0.784095	−	0.620640i	$-0.786873\pi$
$283$	−2.74974	+	2.74974i	−0.163455	+	0.163455i	0.620640	+	0.784095i	$0.286873\pi$
$284$		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	$-0.743461\pi$
$293$	0.497276	−	0.497276i	0.0290512	−	0.0290512i	0.721483	+	0.692432i	$0.243461\pi$
$294$		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	0.317762	−	0.948170i	$-0.397069\pi$
$307$	−11.0456	−	11.0456i	−0.630408	−	0.630408i	−0.948170	+	0.317762i	$0.897069\pi$
$308$		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	$0.193020\pi$
$313$			20.1653i			1.13981i	−0.821710	+	0.569906i	$0.806980\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.70522	−	8.70522i	−0.488934	−	0.488934i	0.419036	−	0.907970i	$-0.362368\pi$
$317$	−8.70522	−	8.70522i	−0.488934	−	0.488934i	−0.907970	+	0.419036i	$0.862368\pi$
$318$		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	−0.626324	−	0.779563i	$-0.715441\pi$
$331$	2.78793	−	2.78793i	0.153239	−	0.153239i	0.779563	+	0.626324i	$0.215441\pi$
$332$		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	−0.987551	−	0.157299i	$-0.949721\pi$
$347$	−21.3262	−	21.3262i	−1.14485	−	1.14485i	−0.157299	−	0.987551i	$-0.550279\pi$
$348$		0			0
$349$	5.11124	−	5.11124i	0.273598	−	0.273598i	−0.556949	−	0.830547i	$-0.688028\pi$
$349$	5.11124	−	5.11124i	0.273598	−	0.273598i	0.830547	+	0.556949i	$0.188028\pi$
$350$		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	$0.120641\pi$
$367$			14.1762i			0.739994i	−0.929033	+	0.369997i	$0.879359\pi$
$368$		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	0.992635	+	0.121141i	$0.0386553\pi$
$373$	21.5106	+	21.5106i	1.11378	+	1.11378i	0.121141	+	0.992635i	$0.461345\pi$
$374$		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	−0.465925	−	0.884824i	$-0.654278\pi$
$379$	8.15510	−	8.15510i	0.418900	−	0.418900i	0.884824	+	0.465925i	$0.154278\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.06121			0.463006			0.231503	−	0.972834i	$-0.425636\pi$
$383$	9.06121			0.463006			0.231503	+	0.972834i	$0.425636\pi$
$384$		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	−0.129339	+	0.991600i	$0.541285\pi$
$389$	−22.1084	+	22.1084i	−1.12094	+	1.12094i	−0.991600	+	0.129339i	$0.958715\pi$
$390$		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	−0.310899	+	0.950443i	$0.600630\pi$
$397$	−25.1321	+	25.1321i	−1.26134	+	1.26134i	−0.950443	+	0.310899i	$0.899370\pi$
$398$		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	$0.0452337\pi$
$409$			5.72850i			0.283256i	−0.989920	+	0.141628i	$0.954766\pi$
$410$		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	0.341551	−	0.939863i	$-0.389048\pi$
$419$	26.2299	−	26.2299i	1.28141	−	1.28141i	0.939863	−	0.341551i	$-0.110952\pi$
$420$		0			0
$421$	9.38496	+	9.38496i	0.457395	+	0.457395i	0.897800	−	0.440404i	$-0.145165\pi$
$421$	9.38496	+	9.38496i	0.457395	+	0.457395i	−0.440404	+	0.897800i	$0.645165\pi$
$422$		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	$-0.374576\pi$
$431$	15.9406			0.767830			0.383915	+	0.923368i	$0.374576\pi$
$432$		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	$-0.445199\pi$
$443$	−17.1307	−	17.1307i	−0.813904	−	0.813904i	−0.985217	+	0.171313i	$0.945199\pi$
$444$		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	$-0.909519\pi$
$457$		−	11.9903i		−	0.560883i	0.959871	−	0.280442i	$-0.0904809\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.9493	+	14.9493i	0.696260	+	0.696260i	0.963602	−	0.267342i	$-0.0861453\pi$
$461$	14.9493	+	14.9493i	0.696260	+	0.696260i	−0.267342	+	0.963602i	$0.586145\pi$
$462$		0			0
$463$			31.4696i			1.46252i	0.682101	+	0.731258i	$0.261067\pi$
$463$			31.4696i			1.46252i	−0.682101	+	0.731258i	$0.738933\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.7699	+	24.7699i	−1.14621	+	1.14621i	−0.158922	+	0.987291i	$0.550802\pi$
$467$	−24.7699	+	24.7699i	−1.14621	+	1.14621i	−0.987291	+	0.158922i	$0.949198\pi$
$468$		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	$-0.367338\pi$
$479$	17.7194			0.809620			0.404810	+	0.914401i	$0.367338\pi$
$480$		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	−0.987718	−	0.156248i	$-0.950060\pi$
$491$	−25.3486	−	25.3486i	−1.14397	−	1.14397i	−0.156248	−	0.987718i	$-0.549940\pi$
$492$		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	0.850800	−	0.525489i	$-0.176118\pi$
$499$	7.26690	+	7.26690i	0.325311	+	0.325311i	−0.525489	+	0.850800i	$0.676118\pi$
$500$		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	$-0.105318\pi$
$509$	14.0082	+	14.0082i	0.620901	+	0.620901i	−0.324861	+	0.945762i	$0.605318\pi$
$510$		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	$-0.201323\pi$
$521$	36.8204			1.61313			0.806566	+	0.591144i	$0.201323\pi$
$522$		0			0
$523$	−2.49783	+	2.49783i	−0.109222	+	0.109222i	−0.759606	−	0.650384i	$-0.774608\pi$
$523$	−2.49783	+	2.49783i	−0.109222	+	0.109222i	0.650384	+	0.759606i	$0.274608\pi$
$524$		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	−0.322841	−	0.946453i	$-0.604638\pi$
$541$	14.5049	−	14.5049i	0.623613	−	0.623613i	0.946453	+	0.322841i	$0.104638\pi$
$542$		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	0.451545	−	0.892249i	$-0.350873\pi$
$547$	−10.3072	−	10.3072i	−0.440704	−	0.440704i	−0.892249	+	0.451545i	$0.850873\pi$
$548$		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	$-0.166762\pi$
$557$	8.62884	+	8.62884i	0.365616	+	0.365616i	−0.500260	+	0.865875i	$0.666762\pi$
$558$		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	$-0.650537\pi$
$563$	10.3155	−	10.3155i	0.434748	−	0.434748i	0.890240	+	0.455492i	$0.150537\pi$
$564$		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	$-0.521609\pi$
$569$	−3.23614			−0.135666			−0.0678330	+	0.997697i	$0.521609\pi$
$570$		0			0
$571$	−27.7822	+	27.7822i	−1.16265	+	1.16265i	−0.178757	+	0.983893i	$0.557208\pi$
$571$	−27.7822	+	27.7822i	−1.16265	+	1.16265i	−0.983893	+	0.178757i	$0.942792\pi$
$572$		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	$-0.0855560\pi$
$587$	16.9242	+	16.9242i	0.698538	+	0.698538i	−0.265557	+	0.964095i	$0.585556\pi$
$588$		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	$0.196533\pi$
$601$			28.3857i			1.15788i	−0.815371	+	0.578938i	$0.803467\pi$
$602$		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	$0.0619695\pi$
$607$			9.53246i			0.386911i	−0.981109	+	0.193455i	$0.938031\pi$
$608$		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	0.728483	−	0.685064i	$-0.240226\pi$
$613$	1.07498	+	1.07498i	0.0434181	+	0.0434181i	−0.685064	+	0.728483i	$0.740226\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	35.3828			1.42446			0.712229	−	0.701948i	$-0.247686\pi$
$617$	35.3828			1.42446			0.712229	+	0.701948i	$0.247686\pi$
$618$		0			0
$619$	3.97297	−	3.97297i	0.159687	−	0.159687i	−0.622741	−	0.782428i	$-0.713981\pi$
$619$	3.97297	−	3.97297i	0.159687	−	0.159687i	0.782428	+	0.622741i	$0.213981\pi$
$620$		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	0.525114	−	0.851032i	$-0.324023\pi$
$643$	−8.26445	−	8.26445i	−0.325918	−	0.325918i	−0.851032	+	0.525114i	$0.824023\pi$
$644$		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	$-0.437066\pi$
$653$	−20.0365	−	20.0365i	−0.784090	−	0.784090i	−0.980518	+	0.196429i	$0.937066\pi$
$654$		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	$-0.988653\pi$
$659$	−24.7397	+	24.7397i	−0.963723	+	0.963723i	0.0356416	+	0.999365i	$0.488653\pi$
$660$		0			0
$661$	−6.31446	−	6.31446i	−0.245604	−	0.245604i	0.573560	−	0.819164i	$-0.305562\pi$
$661$	−6.31446	−	6.31446i	−0.245604	−	0.245604i	−0.819164	+	0.573560i	$0.805562\pi$
$662$		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	$-0.903070\pi$
$677$	−17.0208	+	17.0208i	−0.654163	+	0.654163i	0.299830	+	0.953993i	$0.403070\pi$
$678$		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	$-0.205220\pi$
$683$	5.18231	+	5.18231i	0.198296	+	0.198296i	−0.600973	+	0.799269i	$0.705220\pi$
$684$		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	0.995399	−	0.0958169i	$-0.0305463\pi$
$691$	23.6472	+	23.6472i	0.899582	+	0.899582i	−0.0958169	+	0.995399i	$0.530546\pi$
$692$		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	$-0.289442\pi$
$701$	−4.62771	−	4.62771i	−0.174786	−	0.174786i	−0.789078	+	0.614292i	$0.789442\pi$
$702$		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	−0.999808	−	0.0196099i	$-0.993758\pi$
$709$	−27.1441	−	27.1441i	−1.01942	−	1.01942i	−0.0196099	−	0.999808i	$-0.506242\pi$
$710$		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	$-0.238150\pi$
$719$	39.3061			1.46587			0.732934	+	0.680299i	$0.238150\pi$
$720$		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	−0.774716	−	0.632309i	$-0.782107\pi$
$733$	−3.85554	+	3.85554i	−0.142408	+	0.142408i	0.632309	+	0.774716i	$0.282107\pi$
$734$		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	0.973143	−	0.230201i	$-0.0739385\pi$
$739$	20.1965	+	20.1965i	0.742942	+	0.742942i	−0.230201	+	0.973143i	$0.573939\pi$
$740$		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	$0.00697590\pi$
$751$			1.20106i			0.0438274i	−0.999760	+	0.0219137i	$0.993024\pi$
$752$		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	0.902528	−	0.430632i	$-0.141709\pi$
$757$	12.9836	+	12.9836i	0.471896	+	0.471896i	−0.430632	+	0.902528i	$0.641709\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16.4747			−0.597208			−0.298604	−	0.954377i	$-0.596521\pi$
$761$	−16.4747			−0.597208			−0.298604	+	0.954377i	$0.596521\pi$
$762$		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	$-0.920629\pi$
$773$	−20.0820	+	20.0820i	−0.722299	+	0.722299i	0.246774	+	0.969073i	$0.420629\pi$
$774$		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	0.737665	−	0.675167i	$-0.235928\pi$
$787$	1.75329	+	1.75329i	0.0624980	+	0.0624980i	−0.675167	+	0.737665i	$0.735928\pi$
$788$		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	$-0.382266\pi$
$797$	−16.1165	−	16.1165i	−0.570876	−	0.570876i	−0.932373	+	0.361497i	$0.882266\pi$
$798$		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	$-0.320810\pi$
$809$	30.3586			1.06735			0.533676	+	0.845689i	$0.320810\pi$
$810$		0			0
$811$	−20.3645	+	20.3645i	−0.715095	+	0.715095i	−0.967596	−	0.252502i	$-0.918747\pi$
$811$	−20.3645	+	20.3645i	−0.715095	+	0.715095i	0.252502	+	0.967596i	$0.418747\pi$
$812$		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	−0.273487	+	0.961876i	$0.588177\pi$
$821$	−35.3970	+	35.3970i	−1.23536	+	1.23536i	−0.961876	+	0.273487i	$0.911823\pi$
$822$		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	0.916790	+	0.399369i	$0.130771\pi$
$827$	37.8496	+	37.8496i	1.31616	+	1.31616i	0.399369	+	0.916790i	$0.369229\pi$
$828$		0			0
$829$	−5.82230	+	5.82230i	−0.202217	+	0.202217i	−0.800949	−	0.598732i	$-0.795671\pi$
$829$	−5.82230	+	5.82230i	−0.202217	+	0.202217i	0.598732	+	0.800949i	$0.295671\pi$
$830$		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	0.852691	−	0.522415i	$-0.174969\pi$
$853$	9.64611	+	9.64611i	0.330276	+	0.330276i	−0.522415	+	0.852691i	$0.674969\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.92127			−0.0997886			−0.0498943	−	0.998755i	$-0.515888\pi$
$857$	−2.92127			−0.0997886			−0.0498943	+	0.998755i	$0.515888\pi$
$858$		0			0
$859$	1.63800	−	1.63800i	0.0558880	−	0.0558880i	−0.678610	−	0.734498i	$-0.737418\pi$
$859$	1.63800	−	1.63800i	0.0558880	−	0.0558880i	0.734498	+	0.678610i	$0.237418\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	54.6506			1.86033			0.930164	−	0.367145i	$-0.119665\pi$
$863$	54.6506			1.86033			0.930164	+	0.367145i	$0.119665\pi$
$864$		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	−0.999704	−	0.0243360i	$-0.992253\pi$
$877$	−28.8847	+	28.8847i	−0.975368	+	0.975368i	0.0243360	+	0.999704i	$0.492253\pi$
$878$		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	0.982895	+	0.184169i	$0.0589594\pi$
$883$	34.6797	+	34.6797i	1.16706	+	1.16706i	0.184169	+	0.982895i	$0.441041\pi$
$884$		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	0.279485	−	0.960150i	$-0.409836\pi$
$907$	37.3334	−	37.3334i	1.23963	−	1.23963i	0.960150	−	0.279485i	$-0.0901637\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22.0091			0.729194			0.364597	−	0.931165i	$-0.381207\pi$
$911$	22.0091			0.729194			0.364597	+	0.931165i	$0.381207\pi$
$912$		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	$0.0925630\pi$
$937$			17.5529i			0.573429i	−0.958016	+	0.286714i	$0.907437\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.41436	−	6.41436i	−0.209102	−	0.209102i	0.594784	−	0.803886i	$-0.297238\pi$
$941$	−6.41436	−	6.41436i	−0.209102	−	0.209102i	−0.803886	+	0.594784i	$0.797238\pi$
$942$		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	$-0.691473\pi$
$947$	7.95690	−	7.95690i	0.258564	−	0.258564i	0.824470	+	0.565906i	$0.191473\pi$
$948$		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	$-0.763329\pi$
$953$	−45.4471			−1.47217			−0.736087	+	0.676887i	$0.763329\pi$
$954$		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	$-0.334191\pi$
$971$	−11.5203	−	11.5203i	−0.369704	−	0.369704i	−0.867369	+	0.497665i	$0.834191\pi$
$972$		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	$0.247978\pi$
$991$			44.2359i			1.40520i	−0.711585	+	0.702600i	$0.752022\pi$
$992$		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	0.321871	−	0.946784i	$-0.395688\pi$
$997$	−19.7318	−	19.7318i	−0.624913	−	0.624913i	−0.946784	+	0.321871i	$0.895688\pi$
$998$		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.3599.20		40
3.2	odd	2	inner	4032.2.v.e.3599.1		40
4.3	odd	2		1008.2.v.e.827.9	yes	40
12.11	even	2		1008.2.v.e.827.12	yes	40
16.3	odd	4	inner	4032.2.v.e.1583.1		40
16.13	even	4		1008.2.v.e.323.12	yes	40
48.29	odd	4		1008.2.v.e.323.9	✓	40
48.35	even	4	inner	4032.2.v.e.1583.20		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.9	✓	40	48.29	odd	4
1008.2.v.e.323.12	yes	40	16.13	even	4
1008.2.v.e.827.9	yes	40	4.3	odd	2
1008.2.v.e.827.12	yes	40	12.11	even	2
4032.2.v.e.1583.1		40	16.3	odd	4	inner
4032.2.v.e.1583.20		40	48.35	even	4	inner
4032.2.v.e.3599.1		40	3.2	odd	2	inner
4032.2.v.e.3599.20		40	1.1	even	1	trivial

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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