LMFDB - Embedded newform 4032.2.v.e.1583.17

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			1583.17
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.e.3599.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.12043 + 2.12043i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(2.12043 + 2.12043i) q^{5} -1.00000 q^{7} +(-4.03827 + 4.03827i) q^{11} +(-4.91665 - 4.91665i) q^{13} -4.76253i q^{17} +(2.21762 - 2.21762i) q^{19} +6.91384i q^{23} +3.99241i q^{25} +(2.61285 - 2.61285i) q^{29} -0.712246i q^{31} +(-2.12043 - 2.12043i) q^{35} +(5.73777 - 5.73777i) q^{37} -3.59400 q^{41} +(-3.36325 - 3.36325i) q^{43} -6.04590 q^{47} +1.00000 q^{49} +(4.53235 + 4.53235i) q^{53} -17.1257 q^{55} +(4.82118 - 4.82118i) q^{59} +(-6.72581 - 6.72581i) q^{61} -20.8508i q^{65} +(3.87274 - 3.87274i) q^{67} -14.7448i q^{71} +4.61241i q^{73} +(4.03827 - 4.03827i) q^{77} -7.43429i q^{79} +(-2.44737 - 2.44737i) q^{83} +(10.0986 - 10.0986i) q^{85} +4.23689 q^{89} +(4.91665 + 4.91665i) q^{91} +9.40458 q^{95} -6.76292 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.12043	+	2.12043i	0.948283	+	0.948283i	0.998727	−	0.0504437i	$-0.0160635\pi$
$5$	2.12043	+	2.12043i	0.948283	+	0.948283i	−0.0504437	+	0.998727i	$0.516064\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.03827	+	4.03827i	−1.21759	+	1.21759i	−0.249110	+	0.968475i	$0.580138\pi$
$11$	−4.03827	+	4.03827i	−1.21759	+	1.21759i	−0.968475	+	0.249110i	$0.919862\pi$
$12$		0			0
$13$	−4.91665	−	4.91665i	−1.36363	−	1.36363i	−0.869237	−	0.494396i	$-0.835389\pi$
$13$	−4.91665	−	4.91665i	−1.36363	−	1.36363i	−0.494396	−	0.869237i	$-0.664611\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.76253i		−	1.15508i	−0.816361	−	0.577542i	$-0.804012\pi$
$17$		−	4.76253i		−	1.15508i	0.816361	−	0.577542i	$-0.195988\pi$
$18$		0			0
$19$	2.21762	−	2.21762i	0.508756	−	0.508756i	−0.405388	−	0.914145i	$-0.632864\pi$
$19$	2.21762	−	2.21762i	0.508756	−	0.508756i	0.914145	+	0.405388i	$0.132864\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.91384i			1.44164i	0.693125	+	0.720818i	$0.256233\pi$
$23$			6.91384i			1.44164i	−0.693125	+	0.720818i	$0.743767\pi$
$24$		0			0
$25$			3.99241i			0.798482i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.61285	−	2.61285i	0.485193	−	0.485193i	−0.421592	−	0.906786i	$-0.638529\pi$
$29$	2.61285	−	2.61285i	0.485193	−	0.485193i	0.906786	+	0.421592i	$0.138529\pi$
$30$		0			0
$31$		−	0.712246i		−	0.127923i	−0.997952	−	0.0639615i	$-0.979627\pi$
$31$		−	0.712246i		−	0.127923i	0.997952	−	0.0639615i	$-0.0203735\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.12043	−	2.12043i	−0.358417	−	0.358417i
$36$		0			0
$37$	5.73777	−	5.73777i	0.943283	−	0.943283i	−0.0551925	−	0.998476i	$-0.517577\pi$
$37$	5.73777	−	5.73777i	0.943283	−	0.943283i	0.998476	+	0.0551925i	$0.0175773\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.59400			−0.561288			−0.280644	−	0.959812i	$-0.590548\pi$
$41$	−3.59400			−0.561288			−0.280644	+	0.959812i	$0.590548\pi$
$42$		0			0
$43$	−3.36325	−	3.36325i	−0.512891	−	0.512891i	0.402520	−	0.915411i	$-0.368134\pi$
$43$	−3.36325	−	3.36325i	−0.512891	−	0.512891i	−0.915411	+	0.402520i	$0.868134\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.04590			−0.881885			−0.440942	−	0.897535i	$-0.645356\pi$
$47$	−6.04590			−0.881885			−0.440942	+	0.897535i	$0.645356\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.53235	+	4.53235i	0.622566	+	0.622566i	0.946187	−	0.323621i	$-0.104900\pi$
$53$	4.53235	+	4.53235i	0.622566	+	0.622566i	−0.323621	+	0.946187i	$0.604900\pi$
$54$		0			0
$55$	−17.1257			−2.30923
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.82118	−	4.82118i	0.627664	−	0.627664i	−0.319815	−	0.947480i	$-0.603621\pi$
$59$	4.82118	−	4.82118i	0.627664	−	0.627664i	0.947480	+	0.319815i	$0.103621\pi$
$60$		0			0
$61$	−6.72581	−	6.72581i	−0.861151	−	0.861151i	0.130321	−	0.991472i	$-0.458399\pi$
$61$	−6.72581	−	6.72581i	−0.861151	−	0.861151i	−0.991472	+	0.130321i	$0.958399\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	20.8508i		−	2.58622i
$66$		0			0
$67$	3.87274	−	3.87274i	0.473130	−	0.473130i	−0.429796	−	0.902926i	$-0.641414\pi$
$67$	3.87274	−	3.87274i	0.473130	−	0.473130i	0.902926	+	0.429796i	$0.141414\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	14.7448i		−	1.74988i	−0.484229	−	0.874941i	$-0.660900\pi$
$71$		−	14.7448i		−	1.74988i	0.484229	−	0.874941i	$-0.339100\pi$
$72$		0			0
$73$			4.61241i			0.539842i	0.962882	+	0.269921i	$0.0869976\pi$
$73$			4.61241i			0.539842i	−0.962882	+	0.269921i	$0.913002\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.03827	−	4.03827i	0.460204	−	0.460204i
$78$		0			0
$79$		−	7.43429i		−	0.836423i	−0.908350	−	0.418211i	$-0.862657\pi$
$79$		−	7.43429i		−	0.836423i	0.908350	−	0.418211i	$-0.137343\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.44737	−	2.44737i	−0.268633	−	0.268633i	0.559916	−	0.828549i	$-0.310833\pi$
$83$	−2.44737	−	2.44737i	−0.268633	−	0.268633i	−0.828549	+	0.559916i	$0.810833\pi$
$84$		0			0
$85$	10.0986	−	10.0986i	1.09535	−	1.09535i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.23689			0.449109			0.224555	−	0.974461i	$-0.427907\pi$
$89$	4.23689			0.449109			0.224555	+	0.974461i	$0.427907\pi$
$90$		0			0
$91$	4.91665	+	4.91665i	0.515405	+	0.515405i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.40458			0.964890
$96$		0			0
$97$	−6.76292			−0.686670			−0.343335	−	0.939213i	$-0.611557\pi$
$97$	−6.76292			−0.686670			−0.343335	+	0.939213i	$0.611557\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.7637	−	12.7637i	−1.27003	−	1.27003i	−0.946070	−	0.323963i	$-0.894985\pi$
$101$	−12.7637	−	12.7637i	−1.27003	−	1.27003i	−0.323963	−	0.946070i	$-0.605015\pi$
$102$		0			0
$103$	−5.09658			−0.502181			−0.251091	−	0.967964i	$-0.580789\pi$
$103$	−5.09658			−0.502181			−0.251091	+	0.967964i	$0.580789\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.45316	−	3.45316i	0.333829	−	0.333829i	−0.520209	−	0.854039i	$-0.674146\pi$
$107$	3.45316	−	3.45316i	0.333829	−	0.333829i	0.854039	+	0.520209i	$0.174146\pi$
$108$		0			0
$109$	12.6452	+	12.6452i	1.21119	+	1.21119i	0.970635	+	0.240558i	$0.0773304\pi$
$109$	12.6452	+	12.6452i	1.21119	+	1.21119i	0.240558	+	0.970635i	$0.422670\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.90657i			0.555644i	0.960633	+	0.277822i	$0.0896125\pi$
$113$			5.90657i			0.555644i	−0.960633	+	0.277822i	$0.910388\pi$
$114$		0			0
$115$	−14.6603	+	14.6603i	−1.36708	+	1.36708i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.76253i			0.436581i
$120$		0			0
$121$		−	21.6153i		−	1.96503i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.13652	−	2.13652i	0.191096	−	0.191096i
$126$		0			0
$127$			3.07038i			0.272452i	0.990678	+	0.136226i	$0.0434974\pi$
$127$			3.07038i			0.272452i	−0.990678	+	0.136226i	$0.956503\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.95401	+	6.95401i	0.607575	+	0.607575i	0.942312	−	0.334737i	$-0.108647\pi$
$131$	6.95401	+	6.95401i	0.607575	+	0.607575i	−0.334737	+	0.942312i	$0.608647\pi$
$132$		0			0
$133$	−2.21762	+	2.21762i	−0.192292	+	0.192292i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.5548			0.987193			0.493596	−	0.869691i	$-0.335682\pi$
$137$	11.5548			0.987193			0.493596	+	0.869691i	$0.335682\pi$
$138$		0			0
$139$	−6.19355	−	6.19355i	−0.525330	−	0.525330i	0.393846	−	0.919176i	$-0.371144\pi$
$139$	−6.19355	−	6.19355i	−0.525330	−	0.525330i	−0.919176	+	0.393846i	$0.871144\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	39.7095			3.32068
$144$		0			0
$145$	11.0807			0.920201
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.38965	+	2.38965i	0.195767	+	0.195767i	0.798183	−	0.602415i	$-0.205795\pi$
$149$	2.38965	+	2.38965i	0.195767	+	0.195767i	−0.602415	+	0.798183i	$0.705795\pi$
$150$		0			0
$151$	−12.2639			−0.998018			−0.499009	−	0.866597i	$-0.666303\pi$
$151$	−12.2639			−0.998018			−0.499009	+	0.866597i	$0.666303\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.51026	−	1.51026i	0.121307	−	0.121307i
$156$		0			0
$157$	−4.81144	−	4.81144i	−0.383995	−	0.383995i	0.488544	−	0.872539i	$-0.337528\pi$
$157$	−4.81144	−	4.81144i	−0.383995	−	0.383995i	−0.872539	+	0.488544i	$0.837528\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.91384i		−	0.544887i
$162$		0			0
$163$	6.02781	−	6.02781i	0.472134	−	0.472134i	−0.430470	−	0.902605i	$-0.641652\pi$
$163$	6.02781	−	6.02781i	0.472134	−	0.472134i	0.902605	+	0.430470i	$0.141652\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4.75117i			0.367656i	0.982958	+	0.183828i	$0.0588490\pi$
$167$			4.75117i			0.367656i	−0.982958	+	0.183828i	$0.941151\pi$
$168$		0			0
$169$			35.3469i			2.71899i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.6615	−	13.6615i	1.03867	−	1.03867i	0.0394449	−	0.999222i	$-0.487441\pi$
$173$	13.6615	−	13.6615i	1.03867	−	1.03867i	0.999222	−	0.0394449i	$-0.0125590\pi$
$174$		0			0
$175$		−	3.99241i		−	0.301798i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.71026	+	9.71026i	0.725779	+	0.725779i	0.969776	−	0.243997i	$-0.0784588\pi$
$179$	9.71026	+	9.71026i	0.725779	+	0.725779i	−0.243997	+	0.969776i	$0.578459\pi$
$180$		0			0
$181$	−3.99969	+	3.99969i	−0.297295	+	0.297295i	−0.839953	−	0.542659i	$-0.817418\pi$
$181$	−3.99969	+	3.99969i	−0.297295	+	0.297295i	0.542659	+	0.839953i	$0.317418\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	24.3330			1.78900
$186$		0			0
$187$	19.2324	+	19.2324i	1.40641	+	1.40641i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.9802			−1.15629			−0.578143	−	0.815936i	$-0.696222\pi$
$191$	−15.9802			−1.15629			−0.578143	+	0.815936i	$0.696222\pi$
$192$		0			0
$193$	−3.55024			−0.255552			−0.127776	−	0.991803i	$-0.540784\pi$
$193$	−3.55024			−0.255552			−0.127776	+	0.991803i	$0.540784\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.4049	−	10.4049i	−0.741320	−	0.741320i	0.231512	−	0.972832i	$-0.425633\pi$
$197$	−10.4049	−	10.4049i	−0.741320	−	0.741320i	−0.972832	+	0.231512i	$0.925633\pi$
$198$		0			0
$199$	−20.0463			−1.42104			−0.710521	−	0.703676i	$-0.751541\pi$
$199$	−20.0463			−1.42104			−0.710521	+	0.703676i	$0.751541\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.61285	+	2.61285i	−0.183386	+	0.183386i
$204$		0			0
$205$	−7.62081	−	7.62081i	−0.532260	−	0.532260i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			17.9107i			1.23891i
$210$		0			0
$211$	14.2434	−	14.2434i	0.980558	−	0.980558i	−0.0192564	−	0.999815i	$-0.506130\pi$
$211$	14.2434	−	14.2434i	0.980558	−	0.980558i	0.999815	+	0.0192564i	$0.00612987\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	14.2630i		−	0.972731i
$216$		0			0
$217$			0.712246i			0.0483504i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−23.4157	+	23.4157i	−1.57511	+	1.57511i
$222$		0			0
$223$		−	17.0811i		−	1.14384i	−0.820311	−	0.571918i	$-0.806199\pi$
$223$		−	17.0811i		−	1.14384i	0.820311	−	0.571918i	$-0.193801\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.8882	−	15.8882i	−1.05453	−	1.05453i	−0.998425	−	0.0561098i	$-0.982130\pi$
$227$	−15.8882	−	15.8882i	−1.05453	−	1.05453i	−0.0561098	−	0.998425i	$-0.517870\pi$
$228$		0			0
$229$	−17.2648	+	17.2648i	−1.14089	+	1.14089i	−0.152605	+	0.988287i	$0.548766\pi$
$229$	−17.2648	+	17.2648i	−1.14089	+	1.14089i	−0.988287	+	0.152605i	$0.951234\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.37677			0.548780			0.274390	−	0.961618i	$-0.411524\pi$
$233$	8.37677			0.548780			0.274390	+	0.961618i	$0.411524\pi$
$234$		0			0
$235$	−12.8199	−	12.8199i	−0.836276	−	0.836276i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.24270			−0.403807			−0.201903	−	0.979405i	$-0.564713\pi$
$239$	−6.24270			−0.403807			−0.201903	+	0.979405i	$0.564713\pi$
$240$		0			0
$241$	−2.39859			−0.154507			−0.0772535	−	0.997011i	$-0.524615\pi$
$241$	−2.39859			−0.154507			−0.0772535	+	0.997011i	$0.524615\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.12043	+	2.12043i	0.135469	+	0.135469i
$246$		0			0
$247$	−21.8065			−1.38751
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.5027	−	10.5027i	0.662927	−	0.662927i	−0.293142	−	0.956069i	$-0.594701\pi$
$251$	10.5027	−	10.5027i	0.662927	−	0.662927i	0.956069	+	0.293142i	$0.0947008\pi$
$252$		0			0
$253$	−27.9200	−	27.9200i	−1.75531	−	1.75531i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1.99198i			0.124256i	0.998068	+	0.0621282i	$0.0197888\pi$
$257$			1.99198i			0.124256i	−0.998068	+	0.0621282i	$0.980211\pi$
$258$		0			0
$259$	−5.73777	+	5.73777i	−0.356528	+	0.356528i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.76661i			0.293922i	0.989142	+	0.146961i	$0.0469492\pi$
$263$			4.76661i			0.293922i	−0.989142	+	0.146961i	$0.953051\pi$
$264$		0			0
$265$			19.2210i			1.18074i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.1764	−	10.1764i	0.620468	−	0.620468i	−0.325183	−	0.945651i	$-0.605426\pi$
$269$	10.1764	−	10.1764i	0.620468	−	0.620468i	0.945651	+	0.325183i	$0.105426\pi$
$270$		0			0
$271$			29.8185i			1.81135i	0.423977	+	0.905673i	$0.360634\pi$
$271$			29.8185i			1.81135i	−0.423977	+	0.905673i	$0.639366\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.1224	−	16.1224i	−0.972220	−	0.972220i
$276$		0			0
$277$	10.3689	−	10.3689i	0.623007	−	0.623007i	−0.323292	−	0.946299i	$-0.604790\pi$
$277$	10.3689	−	10.3689i	0.623007	−	0.623007i	0.946299	+	0.323292i	$0.104790\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.1234			0.603909			0.301954	−	0.953322i	$-0.402361\pi$
$281$	10.1234			0.603909			0.301954	+	0.953322i	$0.402361\pi$
$282$		0			0
$283$	−13.7990	−	13.7990i	−0.820266	−	0.820266i	0.165880	−	0.986146i	$-0.446954\pi$
$283$	−13.7990	−	13.7990i	−0.820266	−	0.820266i	−0.986146	+	0.165880i	$0.946954\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.59400			0.212147
$288$		0			0
$289$	−5.68174			−0.334220
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.85735	−	4.85735i	−0.283770	−	0.283770i	0.550841	−	0.834610i	$-0.314307\pi$
$293$	−4.85735	−	4.85735i	−0.283770	−	0.283770i	−0.834610	+	0.550841i	$0.814307\pi$
$294$		0			0
$295$	20.4459			1.19041
$296$		0			0
$297$		0			0
$298$		0			0
$299$	33.9929	−	33.9929i	1.96586	−	1.96586i
$300$		0			0
$301$	3.36325	+	3.36325i	0.193854	+	0.193854i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	28.5231i		−	1.63323i
$306$		0			0
$307$	21.4100	−	21.4100i	1.22193	−	1.22193i	0.254989	−	0.966944i	$-0.417928\pi$
$307$	21.4100	−	21.4100i	1.22193	−	1.22193i	0.966944	−	0.254989i	$-0.0820719\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3.23033i			0.183175i	0.995797	+	0.0915876i	$0.0291941\pi$
$311$			3.23033i			0.183175i	−0.995797	+	0.0915876i	$0.970806\pi$
$312$		0			0
$313$		−	6.53030i		−	0.369114i	−0.982822	−	0.184557i	$-0.940915\pi$
$313$		−	6.53030i		−	0.369114i	0.982822	−	0.184557i	$-0.0590851\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.7296	+	12.7296i	−0.714965	+	0.714965i	−0.967570	−	0.252605i	$-0.918713\pi$
$317$	−12.7296	+	12.7296i	−0.714965	+	0.714965i	0.252605	+	0.967570i	$0.418713\pi$
$318$		0			0
$319$			21.1028i			1.18153i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.5615	−	10.5615i	−0.587656	−	0.587656i
$324$		0			0
$325$	19.6293	−	19.6293i	1.08884	−	1.08884i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.04590			0.333321
$330$		0			0
$331$	−7.98204	−	7.98204i	−0.438732	−	0.438732i	0.452853	−	0.891585i	$-0.350406\pi$
$331$	−7.98204	−	7.98204i	−0.438732	−	0.438732i	−0.891585	+	0.452853i	$0.850406\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.4237			0.897323
$336$		0			0
$337$	−1.43831			−0.0783498			−0.0391749	−	0.999232i	$-0.512473\pi$
$337$	−1.43831			−0.0783498			−0.0391749	+	0.999232i	$0.512473\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.87624	+	2.87624i	0.155757	+	0.155757i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.3928	−	13.3928i	0.718966	−	0.718966i	−0.249428	−	0.968393i	$-0.580242\pi$
$347$	13.3928	−	13.3928i	0.718966	−	0.718966i	0.968393	+	0.249428i	$0.0802425\pi$
$348$		0			0
$349$	−2.01384	−	2.01384i	−0.107798	−	0.107798i	0.651150	−	0.758949i	$-0.274287\pi$
$349$	−2.01384	−	2.01384i	−0.107798	−	0.107798i	−0.758949	+	0.651150i	$0.774287\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			10.9746i			0.584119i	0.956400	+	0.292060i	$0.0943406\pi$
$353$			10.9746i			0.584119i	−0.956400	+	0.292060i	$0.905659\pi$
$354$		0			0
$355$	31.2652	−	31.2652i	1.65938	−	1.65938i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			13.8112i			0.728926i	0.931218	+	0.364463i	$0.118747\pi$
$359$			13.8112i			0.728926i	−0.931218	+	0.364463i	$0.881253\pi$
$360$		0			0
$361$			9.16435i			0.482334i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.78028	+	9.78028i	−0.511923	+	0.511923i
$366$		0			0
$367$			1.72353i			0.0899673i	0.998988	+	0.0449837i	$0.0143236\pi$
$367$			1.72353i			0.0899673i	−0.998988	+	0.0449837i	$0.985676\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.53235	−	4.53235i	−0.235308	−	0.235308i
$372$		0			0
$373$	5.17122	−	5.17122i	0.267756	−	0.267756i	−0.560440	−	0.828195i	$-0.689368\pi$
$373$	5.17122	−	5.17122i	0.267756	−	0.267756i	0.828195	+	0.560440i	$0.189368\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−25.6929			−1.32325
$378$		0			0
$379$	4.73174	+	4.73174i	0.243053	+	0.243053i	0.818112	−	0.575059i	$-0.195021\pi$
$379$	4.73174	+	4.73174i	0.243053	+	0.243053i	−0.575059	+	0.818112i	$0.695021\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.9260			−0.609388			−0.304694	−	0.952450i	$-0.598554\pi$
$383$	−11.9260			−0.609388			−0.304694	+	0.952450i	$0.598554\pi$
$384$		0			0
$385$	17.1257			0.872807
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−21.5999	−	21.5999i	−1.09516	−	1.09516i	−0.994968	−	0.100189i	$-0.968055\pi$
$389$	−21.5999	−	21.5999i	−1.09516	−	1.09516i	−0.100189	−	0.994968i	$-0.531945\pi$
$390$		0			0
$391$	32.9274			1.66521
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.7639	−	15.7639i	0.793165	−	0.793165i
$396$		0			0
$397$	−24.1706	−	24.1706i	−1.21309	−	1.21309i	−0.970005	−	0.243084i	$-0.921841\pi$
$397$	−24.1706	−	24.1706i	−1.21309	−	1.21309i	−0.243084	−	0.970005i	$-0.578159\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	32.3793i		−	1.61694i	−0.588536	−	0.808471i	$-0.700295\pi$
$401$		−	32.3793i		−	1.61694i	0.588536	−	0.808471i	$-0.299705\pi$
$402$		0			0
$403$	−3.50186	+	3.50186i	−0.174440	+	0.174440i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			46.3413i			2.29706i
$408$		0			0
$409$			15.5253i			0.767676i	0.923400	+	0.383838i	$0.125398\pi$
$409$			15.5253i			0.767676i	−0.923400	+	0.383838i	$0.874602\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.82118	+	4.82118i	−0.237235	+	0.237235i
$414$		0			0
$415$		−	10.3789i		−	0.509481i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.1080	−	24.1080i	−1.17775	−	1.17775i	−0.980315	−	0.197438i	$-0.936738\pi$
$419$	−24.1080	−	24.1080i	−1.17775	−	1.17775i	−0.197438	−	0.980315i	$-0.563262\pi$
$420$		0			0
$421$	−21.4110	+	21.4110i	−1.04351	+	1.04351i	−0.0444969	+	0.999010i	$0.514168\pi$
$421$	−21.4110	+	21.4110i	−1.04351	+	1.04351i	−0.999010	+	0.0444969i	$0.985832\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	19.0140			0.922314
$426$		0			0
$427$	6.72581	+	6.72581i	0.325485	+	0.325485i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.7984			−0.905485			−0.452742	−	0.891641i	$-0.649554\pi$
$431$	−18.7984			−0.905485			−0.452742	+	0.891641i	$0.649554\pi$
$432$		0			0
$433$	26.4453			1.27088			0.635440	−	0.772150i	$-0.280819\pi$
$433$	26.4453			1.27088			0.635440	+	0.772150i	$0.280819\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.3322	+	15.3322i	0.733441	+	0.733441i
$438$		0			0
$439$	−7.11296			−0.339483			−0.169742	−	0.985489i	$-0.554293\pi$
$439$	−7.11296			−0.339483			−0.169742	+	0.985489i	$0.554293\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.0108636	+	0.0108636i	−0.000516147	+	0.000516147i	−0.707365	−	0.706849i	$-0.750116\pi$
$443$	−0.0108636	+	0.0108636i	−0.000516147	+	0.000516147i	0.706849	+	0.707365i	$0.250116\pi$
$444$		0			0
$445$	8.98401	+	8.98401i	0.425883	+	0.425883i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			32.7577i			1.54593i	0.634448	+	0.772966i	$0.281228\pi$
$449$			32.7577i			1.54593i	−0.634448	+	0.772966i	$0.718772\pi$
$450$		0			0
$451$	14.5135	−	14.5135i	0.683416	−	0.683416i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			20.8508i			0.977499i
$456$		0			0
$457$		−	10.0528i		−	0.470252i	−0.971965	−	0.235126i	$-0.924450\pi$
$457$		−	10.0528i		−	0.470252i	0.971965	−	0.235126i	$-0.0755502\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.1351	+	15.1351i	−0.704910	+	0.704910i	−0.965460	−	0.260550i	$-0.916096\pi$
$461$	−15.1351	+	15.1351i	−0.704910	+	0.704910i	0.260550	+	0.965460i	$0.416096\pi$
$462$		0			0
$463$		−	10.6714i		−	0.495940i	−0.968768	−	0.247970i	$-0.920236\pi$
$463$		−	10.6714i		−	0.495940i	0.968768	−	0.247970i	$-0.0797635\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.67348	+	3.67348i	0.169989	+	0.169989i	0.786974	−	0.616986i	$-0.211646\pi$
$467$	3.67348	+	3.67348i	0.169989	+	0.169989i	−0.616986	+	0.786974i	$0.711646\pi$
$468$		0			0
$469$	−3.87274	+	3.87274i	−0.178826	+	0.178826i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	27.1634			1.24898
$474$		0			0
$475$	8.85363	+	8.85363i	0.406233	+	0.406233i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.41155			−0.201569			−0.100784	−	0.994908i	$-0.532135\pi$
$479$	−4.41155			−0.201569			−0.100784	+	0.994908i	$0.532135\pi$
$480$		0			0
$481$	−56.4212			−2.57258
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.3403	−	14.3403i	−0.651158	−	0.651158i
$486$		0			0
$487$	−9.97116			−0.451837			−0.225918	−	0.974146i	$-0.572538\pi$
$487$	−9.97116			−0.451837			−0.225918	+	0.974146i	$0.572538\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.2732	+	30.2732i	−1.36621	+	1.36621i	−0.500440	+	0.865771i	$0.666828\pi$
$491$	−30.2732	+	30.2732i	−1.36621	+	1.36621i	−0.865771	+	0.500440i	$0.833172\pi$
$492$		0			0
$493$	−12.4438	−	12.4438i	−0.560439	−	0.560439i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			14.7448i			0.661394i
$498$		0			0
$499$	−10.4161	+	10.4161i	−0.466289	+	0.466289i	−0.900710	−	0.434421i	$-0.856953\pi$
$499$	−10.4161	+	10.4161i	−0.466289	+	0.466289i	0.434421	+	0.900710i	$0.356953\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			8.37172i			0.373277i	0.982429	+	0.186638i	$0.0597592\pi$
$503$			8.37172i			0.373277i	−0.982429	+	0.186638i	$0.940241\pi$
$504$		0			0
$505$		−	54.1288i		−	2.40870i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.0580	+	17.0580i	−0.756083	+	0.756083i	−0.975607	−	0.219524i	$-0.929550\pi$
$509$	−17.0580	+	17.0580i	−0.756083	+	0.756083i	0.219524	+	0.975607i	$0.429550\pi$
$510$		0			0
$511$		−	4.61241i		−	0.204041i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.8069	−	10.8069i	−0.476210	−	0.476210i
$516$		0			0
$517$	24.4150	−	24.4150i	1.07377	−	1.07377i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.2025			−0.534601			−0.267300	−	0.963613i	$-0.586132\pi$
$521$	−12.2025			−0.534601			−0.267300	+	0.963613i	$0.586132\pi$
$522$		0			0
$523$	8.77429	+	8.77429i	0.383673	+	0.383673i	0.872423	−	0.488751i	$-0.162547\pi$
$523$	8.77429	+	8.77429i	0.383673	+	0.383673i	−0.488751	+	0.872423i	$0.662547\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.39209			−0.147762
$528$		0			0
$529$	−24.8012			−1.07831
$530$		0			0
$531$		0			0
$532$		0			0
$533$	17.6704	+	17.6704i	0.765391	+	0.765391i
$534$		0			0
$535$	14.6443			0.633129
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.03827	+	4.03827i	−0.173941	+	0.173941i
$540$		0			0
$541$	−23.2625	−	23.2625i	−1.00013	−	1.00013i	−1.00000		0.000132738i	$-0.999958\pi$
$541$	−23.2625	−	23.2625i	−1.00013	−	1.00013i	−0.000132738	−	1.00000i	$-0.500042\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			53.6265i			2.29711i
$546$		0			0
$547$	−24.7224	+	24.7224i	−1.05705	+	1.05705i	−0.0587833	+	0.998271i	$0.518722\pi$
$547$	−24.7224	+	24.7224i	−1.05705	+	1.05705i	−0.998271	+	0.0587833i	$0.981278\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	11.5886i		−	0.493690i
$552$		0			0
$553$			7.43429i			0.316138i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.9473	+	15.9473i	−0.675708	+	0.675708i	−0.959026	−	0.283318i	$-0.908565\pi$
$557$	−15.9473	+	15.9473i	−0.675708	+	0.675708i	0.283318	+	0.959026i	$0.408565\pi$
$558$		0			0
$559$			33.0718i			1.39879i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.75478	+	8.75478i	0.368970	+	0.368970i	0.867101	−	0.498132i	$-0.165980\pi$
$563$	8.75478	+	8.75478i	0.368970	+	0.368970i	−0.498132	+	0.867101i	$0.665980\pi$
$564$		0			0
$565$	−12.5244	+	12.5244i	−0.526908	+	0.526908i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	24.3805			1.02209			0.511043	−	0.859555i	$-0.329259\pi$
$569$	24.3805			1.02209			0.511043	+	0.859555i	$0.329259\pi$
$570$		0			0
$571$	9.52629	+	9.52629i	0.398663	+	0.398663i	0.877761	−	0.479098i	$-0.159036\pi$
$571$	9.52629	+	9.52629i	0.398663	+	0.398663i	−0.479098	+	0.877761i	$0.659036\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−27.6029			−1.15112
$576$		0			0
$577$	1.46394			0.0609448			0.0304724	−	0.999536i	$-0.490299\pi$
$577$	1.46394			0.0609448			0.0304724	+	0.999536i	$0.490299\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.44737	+	2.44737i	0.101534	+	0.101534i
$582$		0			0
$583$	−36.6057			−1.51605
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.13883	+	7.13883i	−0.294651	+	0.294651i	−0.838914	−	0.544263i	$-0.816809\pi$
$587$	−7.13883	+	7.13883i	−0.294651	+	0.294651i	0.544263	+	0.838914i	$0.316809\pi$
$588$		0			0
$589$	−1.57949	−	1.57949i	−0.0650816	−	0.0650816i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	14.7385i		−	0.605237i	−0.953112	−	0.302619i	$-0.902139\pi$
$593$		−	14.7385i		−	0.605237i	0.953112	−	0.302619i	$-0.0978609\pi$
$594$		0			0
$595$	−10.0986	+	10.0986i	−0.414002	+	0.414002i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	17.8924i		−	0.731062i	−0.930799	−	0.365531i	$-0.880887\pi$
$599$		−	17.8924i		−	0.731062i	0.930799	−	0.365531i	$-0.119113\pi$
$600$		0			0
$601$			6.69170i			0.272960i	0.990643	+	0.136480i	$0.0435790\pi$
$601$			6.69170i			0.272960i	−0.990643	+	0.136480i	$0.956421\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	45.8336	−	45.8336i	1.86340	−	1.86340i
$606$		0			0
$607$		−	27.0776i		−	1.09904i	−0.835479	−	0.549522i	$-0.814810\pi$
$607$		−	27.0776i		−	1.09904i	0.835479	−	0.549522i	$-0.185190\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29.7255	+	29.7255i	1.20257	+	1.20257i
$612$		0			0
$613$	22.9059	−	22.9059i	0.925159	−	0.925159i	−0.0722294	−	0.997388i	$-0.523011\pi$
$613$	22.9059	−	22.9059i	0.925159	−	0.925159i	0.997388	+	0.0722294i	$0.0230114\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.3945			−0.418466			−0.209233	−	0.977866i	$-0.567097\pi$
$617$	−10.3945			−0.418466			−0.209233	+	0.977866i	$0.567097\pi$
$618$		0			0
$619$	8.53232	+	8.53232i	0.342943	+	0.342943i	0.857473	−	0.514530i	$-0.172033\pi$
$619$	8.53232	+	8.53232i	0.342943	+	0.342943i	−0.514530	+	0.857473i	$0.672033\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.23689			−0.169747
$624$		0			0
$625$	29.0227			1.16091
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27.3263	−	27.3263i	−1.08957	−	1.08957i
$630$		0			0
$631$	3.74199			0.148966			0.0744831	−	0.997222i	$-0.476269\pi$
$631$	3.74199			0.148966			0.0744831	+	0.997222i	$0.476269\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.51051	+	6.51051i	−0.258362	+	0.258362i
$636$		0			0
$637$	−4.91665	−	4.91665i	−0.194805	−	0.194805i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	15.1283i		−	0.597531i	−0.954326	−	0.298766i	$-0.903425\pi$
$641$		−	15.1283i		−	0.597531i	0.954326	−	0.298766i	$-0.0965749\pi$
$642$		0			0
$643$	0.845317	−	0.845317i	0.0333361	−	0.0333361i	−0.690242	−	0.723578i	$-0.742496\pi$
$643$	0.845317	−	0.845317i	0.0333361	−	0.0333361i	0.723578	+	0.690242i	$0.242496\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	23.9117i		−	0.940064i	−0.882649	−	0.470032i	$-0.844242\pi$
$647$		−	23.9117i		−	0.940064i	0.882649	−	0.470032i	$-0.155758\pi$
$648$		0			0
$649$			38.9385i			1.52847i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.8544	−	19.8544i	0.776964	−	0.776964i	−0.202350	−	0.979313i	$-0.564858\pi$
$653$	19.8544	−	19.8544i	0.776964	−	0.776964i	0.979313	+	0.202350i	$0.0648578\pi$
$654$		0			0
$655$			29.4909i			1.15231i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.03903	+	9.03903i	0.352111	+	0.352111i	0.860894	−	0.508784i	$-0.169905\pi$
$659$	9.03903	+	9.03903i	0.352111	+	0.352111i	−0.508784	+	0.860894i	$0.669905\pi$
$660$		0			0
$661$	−31.4210	+	31.4210i	−1.22213	+	1.22213i	−0.255261	+	0.966872i	$0.582161\pi$
$661$	−31.4210	+	31.4210i	−1.22213	+	1.22213i	−0.966872	+	0.255261i	$0.917839\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.40458			−0.364694
$666$		0			0
$667$	18.0648	+	18.0648i	0.699472	+	0.699472i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	54.3213			2.09705
$672$		0			0
$673$	16.8543			0.649684			0.324842	−	0.945768i	$-0.394689\pi$
$673$	16.8543			0.649684			0.324842	+	0.945768i	$0.394689\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.8816	+	20.8816i	0.802545	+	0.802545i	0.983493	−	0.180948i	$-0.0579165\pi$
$677$	20.8816	+	20.8816i	0.802545	+	0.802545i	−0.180948	+	0.983493i	$0.557916\pi$
$678$		0			0
$679$	6.76292			0.259537
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.9351	+	18.9351i	−0.724533	+	0.724533i	−0.969525	−	0.244992i	$-0.921215\pi$
$683$	−18.9351	+	18.9351i	−0.724533	+	0.724533i	0.244992	+	0.969525i	$0.421215\pi$
$684$		0			0
$685$	24.5011	+	24.5011i	0.936138	+	0.936138i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	44.5679i		−	1.69790i
$690$		0			0
$691$	2.24775	−	2.24775i	0.0855083	−	0.0855083i	−0.663059	−	0.748567i	$-0.730742\pi$
$691$	2.24775	−	2.24775i	0.0855083	−	0.0855083i	0.748567	+	0.663059i	$0.230742\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	26.2659i		−	0.996323i
$696$		0			0
$697$			17.1165i			0.648335i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−8.17318	+	8.17318i	−0.308697	+	0.308697i	−0.844404	−	0.535707i	$-0.820045\pi$
$701$	−8.17318	+	8.17318i	−0.308697	+	0.308697i	0.535707	+	0.844404i	$0.320045\pi$
$702$		0			0
$703$		−	25.4483i		−	0.959802i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.7637	+	12.7637i	0.480027	+	0.480027i
$708$		0			0
$709$	−11.4595	+	11.4595i	−0.430372	+	0.430372i	−0.888755	−	0.458383i	$-0.848429\pi$
$709$	−11.4595	+	11.4595i	−0.430372	+	0.430372i	0.458383	+	0.888755i	$0.348429\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.92435			0.184418
$714$		0			0
$715$	84.2011	+	84.2011i	3.14894	+	3.14894i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−23.6156			−0.880714			−0.440357	−	0.897823i	$-0.645148\pi$
$719$	−23.6156			−0.880714			−0.440357	+	0.897823i	$0.645148\pi$
$720$		0			0
$721$	5.09658			0.189807
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.4316	+	10.4316i	0.387418	+	0.387418i
$726$		0			0
$727$	11.2623			0.417695			0.208848	−	0.977948i	$-0.433029\pi$
$727$	11.2623			0.417695			0.208848	+	0.977948i	$0.433029\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16.0176	+	16.0176i	−0.592432	+	0.592432i
$732$		0			0
$733$	8.10898	+	8.10898i	0.299512	+	0.299512i	0.840823	−	0.541311i	$-0.182072\pi$
$733$	8.10898	+	8.10898i	0.299512	+	0.299512i	−0.541311	+	0.840823i	$0.682072\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			31.2783i			1.15215i
$738$		0			0
$739$	35.7952	−	35.7952i	1.31675	−	1.31675i	0.400410	−	0.916336i	$-0.368868\pi$
$739$	35.7952	−	35.7952i	1.31675	−	1.31675i	0.916336	−	0.400410i	$-0.131132\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			24.0620i			0.882748i	0.897323	+	0.441374i	$0.145509\pi$
$743$			24.0620i			0.882748i	−0.897323	+	0.441374i	$0.854491\pi$
$744$		0			0
$745$			10.1341i			0.371286i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.45316	+	3.45316i	−0.126176	+	0.126176i
$750$		0			0
$751$		−	39.8810i		−	1.45528i	−0.685961	−	0.727639i	$-0.740618\pi$
$751$		−	39.8810i		−	1.45528i	0.685961	−	0.727639i	$-0.259382\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−26.0046	−	26.0046i	−0.946404	−	0.946404i
$756$		0			0
$757$	−11.5955	+	11.5955i	−0.421446	+	0.421446i	−0.885701	−	0.464255i	$-0.846322\pi$
$757$	−11.5955	+	11.5955i	−0.421446	+	0.421446i	0.464255	+	0.885701i	$0.346322\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.8050			−0.645430			−0.322715	−	0.946496i	$-0.604596\pi$
$761$	−17.8050			−0.645430			−0.322715	+	0.946496i	$0.604596\pi$
$762$		0			0
$763$	−12.6452	−	12.6452i	−0.457788	−	0.457788i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−47.4081			−1.71181
$768$		0			0
$769$	−6.78949			−0.244835			−0.122418	−	0.992479i	$-0.539065\pi$
$769$	−6.78949			−0.244835			−0.122418	+	0.992479i	$0.539065\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.3304	−	20.3304i	−0.731235	−	0.731235i	0.239629	−	0.970864i	$-0.422974\pi$
$773$	−20.3304	−	20.3304i	−0.731235	−	0.731235i	−0.970864	+	0.239629i	$0.922974\pi$
$774$		0			0
$775$	2.84358			0.102144
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.97011	+	7.97011i	−0.285559	+	0.285559i
$780$		0			0
$781$	59.5434	+	59.5434i	2.13063	+	2.13063i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	20.4046i		−	0.728272i
$786$		0			0
$787$	11.8828	−	11.8828i	0.423577	−	0.423577i	−0.462856	−	0.886433i	$-0.653175\pi$
$787$	11.8828	−	11.8828i	0.423577	−	0.423577i	0.886433	+	0.462856i	$0.153175\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	5.90657i		−	0.210014i
$792$		0			0
$793$			66.1368i			2.34859i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.57631	−	4.57631i	0.162101	−	0.162101i	−0.621396	−	0.783497i	$-0.713434\pi$
$797$	4.57631	−	4.57631i	0.162101	−	0.162101i	0.783497	+	0.621396i	$0.213434\pi$
$798$		0			0
$799$			28.7938i			1.01865i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18.6262	−	18.6262i	−0.657303	−	0.657303i
$804$		0			0
$805$	14.6603	−	14.6603i	0.516707	−	0.516707i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−25.1471			−0.884123			−0.442062	−	0.896985i	$-0.645753\pi$
$809$	−25.1471			−0.884123			−0.442062	+	0.896985i	$0.645753\pi$
$810$		0			0
$811$	−24.5249	−	24.5249i	−0.861187	−	0.861187i	0.130289	−	0.991476i	$-0.458409\pi$
$811$	−24.5249	−	24.5249i	−0.861187	−	0.861187i	−0.991476	+	0.130289i	$0.958409\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	25.5630			0.895434
$816$		0			0
$817$	−14.9168			−0.521873
$818$		0			0
$819$		0			0
$820$		0			0
$821$	24.4111	+	24.4111i	0.851955	+	0.851955i	0.990374	−	0.138419i	$-0.0442021\pi$
$821$	24.4111	+	24.4111i	0.851955	+	0.851955i	−0.138419	+	0.990374i	$0.544202\pi$
$822$		0			0
$823$	−15.7338			−0.548445			−0.274223	−	0.961666i	$-0.588421\pi$
$823$	−15.7338			−0.548445			−0.274223	+	0.961666i	$0.588421\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.0353	−	25.0353i	0.870564	−	0.870564i	−0.121970	−	0.992534i	$-0.538921\pi$
$827$	25.0353	−	25.0353i	0.870564	−	0.870564i	0.992534	+	0.121970i	$0.0389212\pi$
$828$		0			0
$829$	−21.0067	−	21.0067i	−0.729592	−	0.729592i	0.240947	−	0.970538i	$-0.422542\pi$
$829$	−21.0067	−	21.0067i	−0.729592	−	0.729592i	−0.970538	+	0.240947i	$0.922542\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.76253i		−	0.165012i
$834$		0			0
$835$	−10.0745	+	10.0745i	−0.348642	+	0.348642i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	7.34682i		−	0.253640i	−0.991926	−	0.126820i	$-0.959523\pi$
$839$		−	7.34682i		−	0.253640i	0.991926	−	0.126820i	$-0.0404771\pi$
$840$		0			0
$841$			15.3461i			0.529175i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−74.9504	+	74.9504i	−2.57837	+	2.57837i
$846$		0			0
$847$			21.6153i			0.742710i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	39.6700	+	39.6700i	1.35987	+	1.35987i
$852$		0			0
$853$	27.5322	−	27.5322i	0.942685	−	0.942685i	−0.0557594	−	0.998444i	$-0.517758\pi$
$853$	27.5322	−	27.5322i	0.942685	−	0.942685i	0.998444	+	0.0557594i	$0.0177580\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.8983			0.508915			0.254457	−	0.967084i	$-0.418103\pi$
$857$	14.8983			0.508915			0.254457	+	0.967084i	$0.418103\pi$
$858$		0			0
$859$	−13.6121	−	13.6121i	−0.464439	−	0.464439i	0.435668	−	0.900107i	$-0.356512\pi$
$859$	−13.6121	−	13.6121i	−0.464439	−	0.464439i	−0.900107	+	0.435668i	$0.856512\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	43.8205			1.49167			0.745834	−	0.666132i	$-0.232051\pi$
$863$	43.8205			1.49167			0.745834	+	0.666132i	$0.232051\pi$
$864$		0			0
$865$	57.9365			1.96990
$866$		0			0
$867$		0			0
$868$		0			0
$869$	30.0217	+	30.0217i	1.01842	+	1.01842i
$870$		0			0
$871$	−38.0818			−1.29035
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.13652	+	2.13652i	−0.0722275	+	0.0722275i
$876$		0			0
$877$	15.3765	+	15.3765i	0.519229	+	0.519229i	0.917338	−	0.398109i	$-0.130333\pi$
$877$	15.3765	+	15.3765i	0.519229	+	0.519229i	−0.398109	+	0.917338i	$0.630333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			19.9612i			0.672509i	0.941771	+	0.336254i	$0.109160\pi$
$881$			19.9612i			0.672509i	−0.941771	+	0.336254i	$0.890840\pi$
$882$		0			0
$883$	16.8680	−	16.8680i	0.567654	−	0.567654i	−0.363816	−	0.931471i	$-0.618526\pi$
$883$	16.8680	−	16.8680i	0.567654	−	0.567654i	0.931471	+	0.363816i	$0.118526\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	25.0819i		−	0.842167i	−0.907022	−	0.421084i	$-0.861650\pi$
$887$		−	25.0819i		−	0.842167i	0.907022	−	0.421084i	$-0.138350\pi$
$888$		0			0
$889$		−	3.07038i		−	0.102977i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.4075	+	13.4075i	−0.448664	+	0.448664i
$894$		0			0
$895$			41.1797i			1.37649i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.86099	−	1.86099i	−0.0620674	−	0.0620674i
$900$		0			0
$901$	21.5855	−	21.5855i	0.719116	−	0.719116i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.9621			−0.563839
$906$		0			0
$907$	6.00255	+	6.00255i	0.199311	+	0.199311i	0.799705	−	0.600393i	$-0.204989\pi$
$907$	6.00255	+	6.00255i	0.199311	+	0.199311i	−0.600393	+	0.799705i	$0.704989\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14.6496			0.485363			0.242682	−	0.970106i	$-0.421973\pi$
$911$	14.6496			0.485363			0.242682	+	0.970106i	$0.421973\pi$
$912$		0			0
$913$	19.7663			0.654168
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.95401	−	6.95401i	−0.229642	−	0.229642i
$918$		0			0
$919$	−4.17227			−0.137631			−0.0688153	−	0.997629i	$-0.521922\pi$
$919$	−4.17227			−0.137631			−0.0688153	+	0.997629i	$0.521922\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−72.4949	+	72.4949i	−2.38620	+	2.38620i
$924$		0			0
$925$	22.9075	+	22.9075i	0.753195	+	0.753195i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			38.1876i			1.25290i	0.779463	+	0.626448i	$0.215492\pi$
$929$			38.1876i			1.25290i	−0.779463	+	0.626448i	$0.784508\pi$
$930$		0			0
$931$	2.21762	−	2.21762i	0.0726794	−	0.0726794i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			81.5618i			2.66736i
$936$		0			0
$937$			45.0796i			1.47269i	0.676609	+	0.736343i	$0.263449\pi$
$937$			45.0796i			1.47269i	−0.676609	+	0.736343i	$0.736551\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.09966	+	3.09966i	−0.101046	+	0.101046i	−0.755822	−	0.654777i	$-0.772763\pi$
$941$	−3.09966	+	3.09966i	−0.101046	+	0.101046i	0.654777	+	0.755822i	$0.272763\pi$
$942$		0			0
$943$		−	24.8483i		−	0.809173i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.6180	+	16.6180i	0.540012	+	0.540012i	0.923532	−	0.383520i	$-0.125288\pi$
$947$	16.6180	+	16.6180i	0.540012	+	0.540012i	−0.383520	+	0.923532i	$0.625288\pi$
$948$		0			0
$949$	22.6776	−	22.6776i	0.736146	−	0.736146i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−57.9097			−1.87588			−0.937939	−	0.346801i	$-0.887268\pi$
$953$	−57.9097			−1.87588			−0.937939	+	0.346801i	$0.887268\pi$
$954$		0			0
$955$	−33.8848	−	33.8848i	−1.09649	−	1.09649i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.5548			−0.373124
$960$		0			0
$961$	30.4927			0.983636
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.52801	−	7.52801i	−0.242335	−	0.242335i
$966$		0			0
$967$	−37.9630			−1.22081			−0.610404	−	0.792090i	$-0.708993\pi$
$967$	−37.9630			−1.22081			−0.610404	+	0.792090i	$0.708993\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.00112	+	1.00112i	−0.0321273	+	0.0321273i	−0.722988	−	0.690861i	$-0.757232\pi$
$971$	−1.00112	+	1.00112i	−0.0321273	+	0.0321273i	0.690861	+	0.722988i	$0.257232\pi$
$972$		0			0
$973$	6.19355	+	6.19355i	0.198556	+	0.198556i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			24.6595i			0.788928i	0.918911	+	0.394464i	$0.129070\pi$
$977$			24.6595i			0.788928i	−0.918911	+	0.394464i	$0.870930\pi$
$978$		0			0
$979$	−17.1097	+	17.1097i	−0.546829	+	0.546829i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			20.2723i			0.646586i	0.946299	+	0.323293i	$0.104790\pi$
$983$			20.2723i			0.646586i	−0.946299	+	0.323293i	$0.895210\pi$
$984$		0			0
$985$		−	44.1258i		−	1.40596i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	23.2530	−	23.2530i	0.739401	−	0.739401i
$990$		0			0
$991$			50.4804i			1.60356i	0.597618	+	0.801781i	$0.296114\pi$
$991$			50.4804i			1.60356i	−0.597618	+	0.801781i	$0.703886\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−42.5066	−	42.5066i	−1.34755	−	1.34755i
$996$		0			0
$997$	28.1175	−	28.1175i	0.890492	−	0.890492i	−0.104077	−	0.994569i	$-0.533189\pi$
$997$	28.1175	−	28.1175i	0.890492	−	0.890492i	0.994569	+	0.104077i	$0.0331890\pi$
$998$		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.1583.17		40
3.2	odd	2	inner	4032.2.v.e.1583.4		40
4.3	odd	2		1008.2.v.e.323.4	✓	40
12.11	even	2		1008.2.v.e.323.17	yes	40
16.5	even	4		1008.2.v.e.827.17	yes	40
16.11	odd	4	inner	4032.2.v.e.3599.4		40
48.5	odd	4		1008.2.v.e.827.4	yes	40
48.11	even	4	inner	4032.2.v.e.3599.17		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.4	✓	40	4.3	odd	2
1008.2.v.e.323.17	yes	40	12.11	even	2
1008.2.v.e.827.4	yes	40	48.5	odd	4
1008.2.v.e.827.17	yes	40	16.5	even	4
4032.2.v.e.1583.4		40	3.2	odd	2	inner
4032.2.v.e.1583.17		40	1.1	even	1	trivial
4032.2.v.e.3599.4		40	16.11	odd	4	inner
4032.2.v.e.3599.17		40	48.11	even	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)

\(f(q)\)	\(=\)	\(q+(2.12043 + 2.12043i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(2.12043 + 2.12043i) q^{5} -1.00000 q^{7} +(-4.03827 + 4.03827i) q^{11} +(-4.91665 - 4.91665i) q^{13} -4.76253i q^{17} +(2.21762 - 2.21762i) q^{19} +6.91384i q^{23} +3.99241i q^{25} +(2.61285 - 2.61285i) q^{29} -0.712246i q^{31} +(-2.12043 - 2.12043i) q^{35} +(5.73777 - 5.73777i) q^{37} -3.59400 q^{41} +(-3.36325 - 3.36325i) q^{43} -6.04590 q^{47} +1.00000 q^{49} +(4.53235 + 4.53235i) q^{53} -17.1257 q^{55} +(4.82118 - 4.82118i) q^{59} +(-6.72581 - 6.72581i) q^{61} -20.8508i q^{65} +(3.87274 - 3.87274i) q^{67} -14.7448i q^{71} +4.61241i q^{73} +(4.03827 - 4.03827i) q^{77} -7.43429i q^{79} +(-2.44737 - 2.44737i) q^{83} +(10.0986 - 10.0986i) q^{85} +4.23689 q^{89} +(4.91665 + 4.91665i) q^{91} +9.40458 q^{95} -6.76292 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 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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