LMFDB - Embedded newform 4032.2.v.e.1583.19

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			1583.19
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.e.3599.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.62814 + 2.62814i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(2.62814 + 2.62814i) q^{5} -1.00000 q^{7} +(-0.583140 + 0.583140i) q^{11} +(1.76590 + 1.76590i) q^{13} +3.63734i q^{17} +(-0.963353 + 0.963353i) q^{19} -3.77401i q^{23} +8.81423i q^{25} +(4.76058 - 4.76058i) q^{29} +4.89207i q^{31} +(-2.62814 - 2.62814i) q^{35} +(-4.66911 + 4.66911i) q^{37} +3.83668 q^{41} +(3.45278 + 3.45278i) q^{43} -11.6693 q^{47} +1.00000 q^{49} +(5.48624 + 5.48624i) q^{53} -3.06515 q^{55} +(-4.40688 + 4.40688i) q^{59} +(-7.99959 - 7.99959i) q^{61} +9.28208i q^{65} +(11.3524 - 11.3524i) q^{67} +5.85120i q^{71} +0.564936i q^{73} +(0.583140 - 0.583140i) q^{77} +16.4692i q^{79} +(6.92234 + 6.92234i) q^{83} +(-9.55945 + 9.55945i) q^{85} -10.7726 q^{89} +(-1.76590 - 1.76590i) q^{91} -5.06365 q^{95} +8.64969 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.62814	+	2.62814i	1.17534	+	1.17534i	0.980918	+	0.194421i	$0.0622829\pi$
$5$	2.62814	+	2.62814i	1.17534	+	1.17534i	0.194421	+	0.980918i	$0.437717\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.583140	+	0.583140i	−0.175823	+	0.175823i	−0.789532	−	0.613709i	$-0.789677\pi$
$11$	−0.583140	+	0.583140i	−0.175823	+	0.175823i	0.613709	+	0.789532i	$0.289677\pi$
$12$		0			0
$13$	1.76590	+	1.76590i	0.489774	+	0.489774i	0.908235	−	0.418461i	$-0.137430\pi$
$13$	1.76590	+	1.76590i	0.489774	+	0.489774i	−0.418461	+	0.908235i	$0.637430\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.63734i			0.882186i	0.897462	+	0.441093i	$0.145409\pi$
$17$			3.63734i			0.882186i	−0.897462	+	0.441093i	$0.854591\pi$
$18$		0			0
$19$	−0.963353	+	0.963353i	−0.221008	+	0.221008i	−0.808923	−	0.587915i	$-0.799949\pi$
$19$	−0.963353	+	0.963353i	−0.221008	+	0.221008i	0.587915	+	0.808923i	$0.299949\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.77401i		−	0.786935i	−0.919338	−	0.393468i	$-0.871275\pi$
$23$		−	3.77401i		−	0.786935i	0.919338	−	0.393468i	$-0.128725\pi$
$24$		0			0
$25$			8.81423i			1.76285i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.76058	−	4.76058i	0.884018	−	0.884018i	−0.109922	−	0.993940i	$-0.535060\pi$
$29$	4.76058	−	4.76058i	0.884018	−	0.884018i	0.993940	+	0.109922i	$0.0350602\pi$
$30$		0			0
$31$			4.89207i			0.878641i	0.898330	+	0.439320i	$0.144781\pi$
$31$			4.89207i			0.878641i	−0.898330	+	0.439320i	$0.855219\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.62814	−	2.62814i	−0.444237	−	0.444237i
$36$		0			0
$37$	−4.66911	+	4.66911i	−0.767598	+	0.767598i	−0.977683	−	0.210085i	$-0.932626\pi$
$37$	−4.66911	+	4.66911i	−0.767598	+	0.767598i	0.210085	+	0.977683i	$0.432626\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.83668			0.599188			0.299594	−	0.954067i	$-0.403149\pi$
$41$	3.83668			0.599188			0.299594	+	0.954067i	$0.403149\pi$
$42$		0			0
$43$	3.45278	+	3.45278i	0.526544	+	0.526544i	0.919540	−	0.392996i	$-0.128561\pi$
$43$	3.45278	+	3.45278i	0.526544	+	0.526544i	−0.392996	+	0.919540i	$0.628561\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.6693			−1.70214			−0.851070	−	0.525052i	$-0.824046\pi$
$47$	−11.6693			−1.70214			−0.851070	+	0.525052i	$0.824046\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.48624	+	5.48624i	0.753593	+	0.753593i	0.975148	−	0.221555i	$-0.0711133\pi$
$53$	5.48624	+	5.48624i	0.753593	+	0.753593i	−0.221555	+	0.975148i	$0.571113\pi$
$54$		0			0
$55$	−3.06515			−0.413304
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.40688	+	4.40688i	−0.573727	+	0.573727i	−0.933168	−	0.359441i	$-0.882967\pi$
$59$	−4.40688	+	4.40688i	−0.573727	+	0.573727i	0.359441	+	0.933168i	$0.382967\pi$
$60$		0			0
$61$	−7.99959	−	7.99959i	−1.02424	−	1.02424i	−0.999699	−	0.0245439i	$-0.992187\pi$
$61$	−7.99959	−	7.99959i	−1.02424	−	1.02424i	−0.0245439	−	0.999699i	$-0.507813\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			9.28208i			1.15130i
$66$		0			0
$67$	11.3524	−	11.3524i	1.38692	−	1.38692i	0.555213	−	0.831708i	$-0.312637\pi$
$67$	11.3524	−	11.3524i	1.38692	−	1.38692i	0.831708	−	0.555213i	$-0.187363\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.85120i			0.694409i	0.937789	+	0.347205i	$0.112869\pi$
$71$			5.85120i			0.694409i	−0.937789	+	0.347205i	$0.887131\pi$
$72$		0			0
$73$			0.564936i			0.0661207i	0.999453	+	0.0330604i	$0.0105254\pi$
$73$			0.564936i			0.0661207i	−0.999453	+	0.0330604i	$0.989475\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.583140	−	0.583140i	0.0664550	−	0.0664550i
$78$		0			0
$79$			16.4692i			1.85293i	0.376382	+	0.926465i	$0.377168\pi$
$79$			16.4692i			1.85293i	−0.376382	+	0.926465i	$0.622832\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.92234	+	6.92234i	0.759826	+	0.759826i	0.976290	−	0.216465i	$-0.0694526\pi$
$83$	6.92234	+	6.92234i	0.759826	+	0.759826i	−0.216465	+	0.976290i	$0.569453\pi$
$84$		0			0
$85$	−9.55945	+	9.55945i	−1.03687	+	1.03687i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.7726			−1.14189			−0.570946	−	0.820988i	$-0.693423\pi$
$89$	−10.7726			−1.14189			−0.570946	+	0.820988i	$0.693423\pi$
$90$		0			0
$91$	−1.76590	−	1.76590i	−0.185117	−	0.185117i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.06365			−0.519520
$96$		0			0
$97$	8.64969			0.878243			0.439122	−	0.898428i	$-0.355290\pi$
$97$	8.64969			0.878243			0.439122	+	0.898428i	$0.355290\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.9849	−	10.9849i	−1.09303	−	1.09303i	−0.995203	−	0.0978307i	$-0.968810\pi$
$101$	−10.9849	−	10.9849i	−1.09303	−	1.09303i	−0.0978307	−	0.995203i	$-0.531190\pi$
$102$		0			0
$103$	2.31765			0.228365			0.114183	−	0.993460i	$-0.463575\pi$
$103$	2.31765			0.228365			0.114183	+	0.993460i	$0.463575\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.38720	+	4.38720i	−0.424127	+	0.424127i	−0.886622	−	0.462495i	$-0.846954\pi$
$107$	−4.38720	+	4.38720i	−0.424127	+	0.424127i	0.462495	+	0.886622i	$0.346954\pi$
$108$		0			0
$109$	1.97102	+	1.97102i	0.188790	+	0.188790i	0.795173	−	0.606383i	$-0.207380\pi$
$109$	1.97102	+	1.97102i	0.188790	+	0.188790i	−0.606383	+	0.795173i	$0.707380\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	7.79490i		−	0.733283i	−0.930362	−	0.366641i	$-0.880508\pi$
$113$		−	7.79490i		−	0.733283i	0.930362	−	0.366641i	$-0.119492\pi$
$114$		0			0
$115$	9.91862	−	9.91862i	0.924916	−	0.924916i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	3.63734i		−	0.333435i
$120$		0			0
$121$			10.3199i			0.938172i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.0243	+	10.0243i	−0.896603	+	0.896603i
$126$		0			0
$127$			5.11789i			0.454139i	0.973879	+	0.227070i	$0.0729145\pi$
$127$			5.11789i			0.454139i	−0.973879	+	0.227070i	$0.927086\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.29337	−	6.29337i	−0.549854	−	0.549854i	0.376544	−	0.926399i	$-0.377112\pi$
$131$	−6.29337	−	6.29337i	−0.549854	−	0.549854i	−0.926399	+	0.376544i	$0.877112\pi$
$132$		0			0
$133$	0.963353	−	0.963353i	0.0835333	−	0.0835333i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.2421			−1.21678			−0.608392	−	0.793636i	$-0.708185\pi$
$137$	−14.2421			−1.21678			−0.608392	+	0.793636i	$0.708185\pi$
$138$		0			0
$139$	−1.23142	−	1.23142i	−0.104448	−	0.104448i	0.652952	−	0.757400i	$-0.273530\pi$
$139$	−1.23142	−	1.23142i	−0.104448	−	0.104448i	−0.757400	+	0.652952i	$0.773530\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.05954			−0.172227
$144$		0			0
$145$	25.0229			2.07804
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.2649	+	14.2649i	1.16863	+	1.16863i	0.982531	+	0.186098i	$0.0595841\pi$
$149$	14.2649	+	14.2649i	1.16863	+	1.16863i	0.186098	+	0.982531i	$0.440416\pi$
$150$		0			0
$151$	15.2951			1.24470			0.622349	−	0.782740i	$-0.286178\pi$
$151$	15.2951			1.24470			0.622349	+	0.782740i	$0.286178\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.8570	+	12.8570i	−1.03270	+	1.03270i
$156$		0			0
$157$	−4.79120	−	4.79120i	−0.382379	−	0.382379i	0.489580	−	0.871959i	$-0.337150\pi$
$157$	−4.79120	−	4.79120i	−0.382379	−	0.382379i	−0.871959	+	0.489580i	$0.837150\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.77401i			0.297434i
$162$		0			0
$163$	−3.28056	+	3.28056i	−0.256953	+	0.256953i	−0.823814	−	0.566861i	$-0.808158\pi$
$163$	−3.28056	+	3.28056i	−0.256953	+	0.256953i	0.566861	+	0.823814i	$0.308158\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	21.8772i		−	1.69291i	−0.532461	−	0.846455i	$-0.678733\pi$
$167$		−	21.8772i		−	1.69291i	0.532461	−	0.846455i	$-0.321267\pi$
$168$		0			0
$169$		−	6.76317i		−	0.520244i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.4598	+	12.4598i	−0.947304	+	0.947304i	−0.998679	−	0.0513757i	$-0.983639\pi$
$173$	−12.4598	+	12.4598i	−0.947304	+	0.947304i	0.0513757	+	0.998679i	$0.483639\pi$
$174$		0			0
$175$		−	8.81423i		−	0.666293i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.24841	−	9.24841i	−0.691259	−	0.691259i	0.271250	−	0.962509i	$-0.412563\pi$
$179$	−9.24841	−	9.24841i	−0.691259	−	0.691259i	−0.962509	+	0.271250i	$0.912563\pi$
$180$		0			0
$181$	−16.6449	+	16.6449i	−1.23721	+	1.23721i	−0.276072	+	0.961137i	$0.589033\pi$
$181$	−16.6449	+	16.6449i	−1.23721	+	1.23721i	−0.961137	+	0.276072i	$0.910967\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−24.5422			−1.80438
$186$		0			0
$187$	−2.12108	−	2.12108i	−0.155109	−	0.155109i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.93740			−0.140185			−0.0700927	−	0.997540i	$-0.522330\pi$
$191$	−1.93740			−0.140185			−0.0700927	+	0.997540i	$0.522330\pi$
$192$		0			0
$193$	−10.0684			−0.724742			−0.362371	−	0.932034i	$-0.618033\pi$
$193$	−10.0684			−0.724742			−0.362371	+	0.932034i	$0.618033\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.45632	−	1.45632i	−0.103758	−	0.103758i	0.653322	−	0.757080i	$-0.273375\pi$
$197$	−1.45632	−	1.45632i	−0.103758	−	0.103758i	−0.757080	+	0.653322i	$0.773375\pi$
$198$		0			0
$199$	16.3258			1.15730			0.578652	−	0.815574i	$-0.303579\pi$
$199$	16.3258			1.15730			0.578652	+	0.815574i	$0.303579\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.76058	+	4.76058i	−0.334127	+	0.334127i
$204$		0			0
$205$	10.0833	+	10.0833i	0.704250	+	0.704250i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1.12354i		−	0.0777168i
$210$		0			0
$211$	6.81896	−	6.81896i	0.469437	−	0.469437i	−0.432295	−	0.901732i	$-0.642296\pi$
$211$	6.81896	−	6.81896i	0.469437	−	0.469437i	0.901732	+	0.432295i	$0.142296\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			18.1488i			1.23774i
$216$		0			0
$217$		−	4.89207i		−	0.332095i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.42320	+	6.42320i	−0.432071	+	0.432071i
$222$		0			0
$223$		−	2.90066i		−	0.194243i	−0.995273	−	0.0971213i	$-0.969037\pi$
$223$		−	2.90066i		−	0.194243i	0.995273	−	0.0971213i	$-0.0309635\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.48182	−	9.48182i	−0.629331	−	0.629331i	0.318569	−	0.947900i	$-0.396798\pi$
$227$	−9.48182	−	9.48182i	−0.629331	−	0.629331i	−0.947900	+	0.318569i	$0.896798\pi$
$228$		0			0
$229$	14.9933	−	14.9933i	0.990786	−	0.990786i	−0.00917178	−	0.999958i	$-0.502920\pi$
$229$	14.9933	−	14.9933i	0.990786	−	0.990786i	0.999958	+	0.00917178i	$0.00291951\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.9359			0.781945			0.390973	−	0.920402i	$-0.372139\pi$
$233$	11.9359			0.781945			0.390973	+	0.920402i	$0.372139\pi$
$234$		0			0
$235$	−30.6685	−	30.6685i	−2.00059	−	2.00059i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.4892			0.937227			0.468614	−	0.883403i	$-0.344754\pi$
$239$	14.4892			0.937227			0.468614	+	0.883403i	$0.344754\pi$
$240$		0			0
$241$	12.5295			0.807094			0.403547	−	0.914959i	$-0.367777\pi$
$241$	12.5295			0.807094			0.403547	+	0.914959i	$0.367777\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.62814	+	2.62814i	0.167906	+	0.167906i
$246$		0			0
$247$	−3.40238			−0.216488
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.1384	+	12.1384i	−0.766166	+	0.766166i	−0.977429	−	0.211263i	$-0.932242\pi$
$251$	−12.1384	+	12.1384i	−0.766166	+	0.766166i	0.211263	+	0.977429i	$0.432242\pi$
$252$		0			0
$253$	2.20078	+	2.20078i	0.138362	+	0.138362i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	12.8533i		−	0.801764i	−0.916130	−	0.400882i	$-0.868704\pi$
$257$		−	12.8533i		−	0.801764i	0.916130	−	0.400882i	$-0.131296\pi$
$258$		0			0
$259$	4.66911	−	4.66911i	0.290125	−	0.290125i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			25.0594i			1.54523i	0.634877	+	0.772613i	$0.281051\pi$
$263$			25.0594i			1.54523i	−0.634877	+	0.772613i	$0.718949\pi$
$264$		0			0
$265$			28.8372i			1.77145i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.1085	+	21.1085i	−1.28701	+	1.28701i	−0.350409	+	0.936597i	$0.613957\pi$
$269$	−21.1085	+	21.1085i	−1.28701	+	1.28701i	−0.936597	+	0.350409i	$0.886043\pi$
$270$		0			0
$271$			27.0517i			1.64327i	0.570013	+	0.821636i	$0.306938\pi$
$271$			27.0517i			1.64327i	−0.570013	+	0.821636i	$0.693062\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.13993	−	5.13993i	−0.309949	−	0.309949i
$276$		0			0
$277$	9.25373	−	9.25373i	0.556003	−	0.556003i	−0.372164	−	0.928167i	$-0.621384\pi$
$277$	9.25373	−	9.25373i	0.556003	−	0.556003i	0.928167	+	0.372164i	$0.121384\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.2367			0.968600			0.484300	−	0.874902i	$-0.339074\pi$
$281$	16.2367			0.968600			0.484300	+	0.874902i	$0.339074\pi$
$282$		0			0
$283$	9.02844	+	9.02844i	0.536685	+	0.536685i	0.922554	−	0.385869i	$-0.126098\pi$
$283$	9.02844	+	9.02844i	0.536685	+	0.536685i	−0.385869	+	0.922554i	$0.626098\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.83668			−0.226472
$288$		0			0
$289$	3.76973			0.221749
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−20.6256	−	20.6256i	−1.20496	−	1.20496i	−0.972639	−	0.232324i	$-0.925367\pi$
$293$	−20.6256	−	20.6256i	−1.20496	−	1.20496i	−0.232324	−	0.972639i	$-0.574633\pi$
$294$		0			0
$295$	−23.1638			−1.34865
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.66454	−	6.66454i	0.385420	−	0.385420i
$300$		0			0
$301$	−3.45278	−	3.45278i	−0.199015	−	0.199015i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	42.0481i		−	2.40767i
$306$		0			0
$307$	−9.53094	+	9.53094i	−0.543959	+	0.543959i	−0.924687	−	0.380728i	$-0.875673\pi$
$307$	−9.53094	+	9.53094i	−0.543959	+	0.543959i	0.380728	+	0.924687i	$0.375673\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4.39638i		−	0.249296i	−0.992201	−	0.124648i	$-0.960220\pi$
$311$		−	4.39638i		−	0.249296i	0.992201	−	0.124648i	$-0.0397802\pi$
$312$		0			0
$313$		−	27.8579i		−	1.57462i	−0.616558	−	0.787310i	$-0.711473\pi$
$313$		−	27.8579i		−	1.57462i	0.616558	−	0.787310i	$-0.288527\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.5283	+	11.5283i	−0.647491	+	0.647491i	−0.952386	−	0.304895i	$-0.901379\pi$
$317$	−11.5283	+	11.5283i	−0.647491	+	0.647491i	0.304895	+	0.952386i	$0.401379\pi$
$318$		0			0
$319$			5.55217i			0.310862i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.50405	−	3.50405i	−0.194970	−	0.194970i
$324$		0			0
$325$	−15.5651	+	15.5651i	−0.863395	+	0.863395i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.6693			0.643349
$330$		0			0
$331$	21.4888	+	21.4888i	1.18113	+	1.18113i	0.979451	+	0.201682i	$0.0646408\pi$
$331$	21.4888	+	21.4888i	1.18113	+	1.18113i	0.201682	+	0.979451i	$0.435359\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	59.6716			3.26021
$336$		0			0
$337$	6.64762			0.362119			0.181059	−	0.983472i	$-0.442047\pi$
$337$	6.64762			0.362119			0.181059	+	0.983472i	$0.442047\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.85276	−	2.85276i	−0.154486	−	0.154486i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19.4459	−	19.4459i	1.04391	−	1.04391i	0.0449218	−	0.998991i	$-0.485696\pi$
$347$	19.4459	−	19.4459i	1.04391	−	1.04391i	0.998991	−	0.0449218i	$-0.0143039\pi$
$348$		0			0
$349$	−6.38328	−	6.38328i	−0.341689	−	0.341689i	0.515313	−	0.857002i	$-0.327676\pi$
$349$	−6.38328	−	6.38328i	−0.341689	−	0.341689i	−0.857002	+	0.515313i	$0.827676\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			7.05143i			0.375310i	0.982235	+	0.187655i	$0.0600886\pi$
$353$			7.05143i			0.375310i	−0.982235	+	0.187655i	$0.939911\pi$
$354$		0			0
$355$	−15.3778	+	15.3778i	−0.816167	+	0.816167i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			28.7598i			1.51788i	0.651159	+	0.758941i	$0.274283\pi$
$359$			28.7598i			1.51788i	−0.651159	+	0.758941i	$0.725717\pi$
$360$		0			0
$361$			17.1439i			0.902311i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.48473	+	1.48473i	−0.0777143	+	0.0777143i
$366$		0			0
$367$		−	9.89445i		−	0.516486i	−0.966080	−	0.258243i	$-0.916856\pi$
$367$		−	9.89445i		−	0.516486i	0.966080	−	0.258243i	$-0.0831435\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.48624	−	5.48624i	−0.284831	−	0.284831i
$372$		0			0
$373$	−24.6444	+	24.6444i	−1.27604	+	1.27604i	−0.333175	+	0.942865i	$0.608120\pi$
$373$	−24.6444	+	24.6444i	−1.27604	+	1.27604i	−0.942865	+	0.333175i	$0.891880\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	16.8135			0.865937
$378$		0			0
$379$	−3.92064	−	3.92064i	−0.201390	−	0.201390i	0.599206	−	0.800595i	$-0.295483\pi$
$379$	−3.92064	−	3.92064i	−0.201390	−	0.201390i	−0.800595	+	0.599206i	$0.795483\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.5474			0.947730			0.473865	−	0.880598i	$-0.342858\pi$
$383$	18.5474			0.947730			0.473865	+	0.880598i	$0.342858\pi$
$384$		0			0
$385$	3.06515			0.156214
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.7437	+	19.7437i	1.00104	+	1.00104i	0.999999	+	0.00104509i	$0.000332661\pi$
$389$	19.7437	+	19.7437i	1.00104	+	1.00104i	0.00104509	+	0.999999i	$0.499667\pi$
$390$		0			0
$391$	13.7274			0.694223
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−43.2833	+	43.2833i	−2.17782	+	2.17782i
$396$		0			0
$397$	13.4592	+	13.4592i	0.675497	+	0.675497i	0.958978	−	0.283481i	$-0.0914893\pi$
$397$	13.4592	+	13.4592i	0.675497	+	0.675497i	−0.283481	+	0.958978i	$0.591489\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	29.4638i		−	1.47135i	−0.677334	−	0.735676i	$-0.736865\pi$
$401$		−	29.4638i		−	1.47135i	0.677334	−	0.735676i	$-0.263135\pi$
$402$		0			0
$403$	−8.63891	+	8.63891i	−0.430335	+	0.430335i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.44550i		−	0.269923i
$408$		0			0
$409$		−	15.9026i		−	0.786333i	−0.919467	−	0.393167i	$-0.871380\pi$
$409$		−	15.9026i		−	0.786333i	0.919467	−	0.393167i	$-0.128620\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.40688	−	4.40688i	0.216848	−	0.216848i
$414$		0			0
$415$			36.3858i			1.78611i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.45606	+	6.45606i	0.315399	+	0.315399i	0.846997	−	0.531598i	$-0.178408\pi$
$419$	6.45606	+	6.45606i	0.315399	+	0.315399i	−0.531598	+	0.846997i	$0.678408\pi$
$420$		0			0
$421$	11.8827	−	11.8827i	0.579126	−	0.579126i	−0.355536	−	0.934663i	$-0.615702\pi$
$421$	11.8827	−	11.8827i	0.579126	−	0.579126i	0.934663	+	0.355536i	$0.115702\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−32.0604			−1.55516
$426$		0			0
$427$	7.99959	+	7.99959i	0.387127	+	0.387127i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.49791			−0.264825			−0.132412	−	0.991195i	$-0.542272\pi$
$431$	−5.49791			−0.264825			−0.132412	+	0.991195i	$0.542272\pi$
$432$		0			0
$433$	20.0080			0.961522			0.480761	−	0.876852i	$-0.340360\pi$
$433$	20.0080			0.961522			0.480761	+	0.876852i	$0.340360\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.63570	+	3.63570i	0.173919	+	0.173919i
$438$		0			0
$439$	0.00616772			0.000294369			0.000147185	−	1.00000i	$-0.499953\pi$
$439$	0.00616772			0.000294369			0.000147185		1.00000i	$0.499953\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.02537	−	4.02537i	0.191251	−	0.191251i	−0.604985	−	0.796237i	$-0.706821\pi$
$443$	4.02537	−	4.02537i	0.191251	−	0.191251i	0.796237	+	0.604985i	$0.206821\pi$
$444$		0			0
$445$	−28.3118	−	28.3118i	−1.34211	−	1.34211i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			19.6378i			0.926765i	0.886158	+	0.463382i	$0.153364\pi$
$449$			19.6378i			0.926765i	−0.886158	+	0.463382i	$0.846636\pi$
$450$		0			0
$451$	−2.23732	+	2.23732i	−0.105351	+	0.105351i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	9.28208i		−	0.435151i
$456$		0			0
$457$			18.0292i			0.843372i	0.906742	+	0.421686i	$0.138562\pi$
$457$			18.0292i			0.843372i	−0.906742	+	0.421686i	$0.861438\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.06055	+	2.06055i	−0.0959694	+	0.0959694i	−0.753461	−	0.657492i	$-0.771617\pi$
$461$	−2.06055	+	2.06055i	−0.0959694	+	0.0959694i	0.657492	+	0.753461i	$0.271617\pi$
$462$		0			0
$463$			38.8765i			1.80674i	0.428857	+	0.903372i	$0.358916\pi$
$463$			38.8765i			1.80674i	−0.428857	+	0.903372i	$0.641084\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.88850	+	6.88850i	0.318762	+	0.318762i	0.848291	−	0.529530i	$-0.177632\pi$
$467$	6.88850	+	6.88850i	0.318762	+	0.318762i	−0.529530	+	0.848291i	$0.677632\pi$
$468$		0			0
$469$	−11.3524	+	11.3524i	−0.524207	+	0.524207i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.02691			−0.185157
$474$		0			0
$475$	−8.49121	−	8.49121i	−0.389604	−	0.389604i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.25769			−0.148848			−0.0744239	−	0.997227i	$-0.523712\pi$
$479$	−3.25769			−0.148848			−0.0744239	+	0.997227i	$0.523712\pi$
$480$		0			0
$481$	−16.4904			−0.751898
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22.7326	+	22.7326i	1.03223	+	1.03223i
$486$		0			0
$487$	27.6993			1.25517			0.627587	−	0.778547i	$-0.284043\pi$
$487$	27.6993			1.25517			0.627587	+	0.778547i	$0.284043\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20.9676	−	20.9676i	0.946256	−	0.946256i	−0.0523720	−	0.998628i	$-0.516678\pi$
$491$	20.9676	−	20.9676i	0.946256	−	0.946256i	0.998628	+	0.0523720i	$0.0166782\pi$
$492$		0			0
$493$	17.3159	+	17.3159i	0.779868	+	0.779868i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	5.85120i		−	0.262462i
$498$		0			0
$499$	−2.46939	+	2.46939i	−0.110545	+	0.110545i	−0.760216	−	0.649671i	$-0.774907\pi$
$499$	−2.46939	+	2.46939i	−0.110545	+	0.110545i	0.649671	+	0.760216i	$0.274907\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			19.9403i			0.889093i	0.895756	+	0.444547i	$0.146635\pi$
$503$			19.9403i			0.889093i	−0.895756	+	0.444547i	$0.853365\pi$
$504$		0			0
$505$		−	57.7394i		−	2.56937i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.45882	−	7.45882i	0.330606	−	0.330606i	−0.522210	−	0.852817i	$-0.674892\pi$
$509$	7.45882	−	7.45882i	0.330606	−	0.330606i	0.852817	+	0.522210i	$0.174892\pi$
$510$		0			0
$511$		−	0.564936i		−	0.0249913i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.09112	+	6.09112i	0.268407	+	0.268407i
$516$		0			0
$517$	6.80483	−	6.80483i	0.299276	−	0.299276i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	23.2354			1.01796			0.508981	−	0.860778i	$-0.330022\pi$
$521$	23.2354			1.01796			0.508981	+	0.860778i	$0.330022\pi$
$522$		0			0
$523$	−19.1847	−	19.1847i	−0.838888	−	0.838888i	0.149824	−	0.988713i	$-0.452129\pi$
$523$	−19.1847	−	19.1847i	−0.838888	−	0.838888i	−0.988713	+	0.149824i	$0.952129\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17.7941			−0.775124
$528$		0			0
$529$	8.75685			0.380733
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.77520	+	6.77520i	0.293467	+	0.293467i
$534$		0			0
$535$	−23.0604			−0.996986
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.583140	+	0.583140i	−0.0251176	+	0.0251176i
$540$		0			0
$541$	−9.33342	−	9.33342i	−0.401275	−	0.401275i	0.477407	−	0.878682i	$-0.341577\pi$
$541$	−9.33342	−	9.33342i	−0.401275	−	0.401275i	−0.878682	+	0.477407i	$0.841577\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			10.3602i			0.443784i
$546$		0			0
$547$	−12.2760	+	12.2760i	−0.524883	+	0.524883i	−0.919042	−	0.394159i	$-0.871036\pi$
$547$	−12.2760	+	12.2760i	−0.524883	+	0.524883i	0.394159	+	0.919042i	$0.371036\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			9.17224i			0.390751i
$552$		0			0
$553$		−	16.4692i		−	0.700342i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.3475	−	14.3475i	0.607923	−	0.607923i	−0.334480	−	0.942403i	$-0.608561\pi$
$557$	14.3475	−	14.3475i	0.607923	−	0.607923i	0.942403	+	0.334480i	$0.108561\pi$
$558$		0			0
$559$			12.1946i			0.515775i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.8013	−	14.8013i	−0.623802	−	0.623802i	0.322700	−	0.946501i	$-0.395410\pi$
$563$	−14.8013	−	14.8013i	−0.623802	−	0.623802i	−0.946501	+	0.322700i	$0.895410\pi$
$564$		0			0
$565$	20.4861	−	20.4861i	0.861856	−	0.861856i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.30895			−0.348329			−0.174165	−	0.984717i	$-0.555723\pi$
$569$	−8.30895			−0.348329			−0.174165	+	0.984717i	$0.555723\pi$
$570$		0			0
$571$	−9.56981	−	9.56981i	−0.400484	−	0.400484i	0.477920	−	0.878404i	$-0.341391\pi$
$571$	−9.56981	−	9.56981i	−0.400484	−	0.400484i	−0.878404	+	0.477920i	$0.841391\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	33.2650			1.38725
$576$		0			0
$577$	36.5616			1.52208			0.761039	−	0.648706i	$-0.224690\pi$
$577$	36.5616			1.52208			0.761039	+	0.648706i	$0.224690\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.92234	−	6.92234i	−0.287187	−	0.287187i
$582$		0			0
$583$	−6.39849			−0.264998
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.86143	+	2.86143i	−0.118104	+	0.118104i	−0.763689	−	0.645585i	$-0.776614\pi$
$587$	−2.86143	+	2.86143i	−0.118104	+	0.118104i	0.645585	+	0.763689i	$0.276614\pi$
$588$		0			0
$589$	−4.71279	−	4.71279i	−0.194187	−	0.194187i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	40.5600i		−	1.66560i	−0.553575	−	0.832799i	$-0.686737\pi$
$593$		−	40.5600i		−	1.66560i	0.553575	−	0.832799i	$-0.313263\pi$
$594$		0			0
$595$	9.55945	−	9.55945i	0.391899	−	0.391899i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	41.0811i		−	1.67853i	−0.543725	−	0.839263i	$-0.682987\pi$
$599$		−	41.0811i		−	1.67853i	0.543725	−	0.839263i	$-0.317013\pi$
$600$		0			0
$601$		−	26.2778i		−	1.07190i	−0.844251	−	0.535948i	$-0.819954\pi$
$601$		−	26.2778i		−	1.07190i	0.844251	−	0.535948i	$-0.180046\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−27.1221	+	27.1221i	−1.10267	+	1.10267i
$606$		0			0
$607$			47.8974i			1.94410i	0.234784	+	0.972048i	$0.424562\pi$
$607$			47.8974i			1.94410i	−0.234784	+	0.972048i	$0.575438\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−20.6068	−	20.6068i	−0.833663	−	0.833663i
$612$		0			0
$613$	0.694342	−	0.694342i	0.0280442	−	0.0280442i	−0.692946	−	0.720990i	$-0.743687\pi$
$613$	0.694342	−	0.694342i	0.0280442	−	0.0280442i	0.720990	+	0.692946i	$0.243687\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.60105			0.185231			0.0926156	−	0.995702i	$-0.470477\pi$
$617$	4.60105			0.185231			0.0926156	+	0.995702i	$0.470477\pi$
$618$		0			0
$619$	6.50800	+	6.50800i	0.261579	+	0.261579i	0.825695	−	0.564117i	$-0.190783\pi$
$619$	6.50800	+	6.50800i	0.261579	+	0.261579i	−0.564117	+	0.825695i	$0.690783\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.7726			0.431594
$624$		0			0
$625$	−8.61949			−0.344779
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.9832	−	16.9832i	−0.677164	−	0.677164i
$630$		0			0
$631$	−30.8637			−1.22867			−0.614333	−	0.789047i	$-0.710575\pi$
$631$	−30.8637			−1.22867			−0.614333	+	0.789047i	$0.710575\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.4505	+	13.4505i	−0.533768	+	0.533768i
$636$		0			0
$637$	1.76590	+	1.76590i	0.0699676	+	0.0699676i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.57360i			0.0621536i	0.999517	+	0.0310768i	$0.00989365\pi$
$641$			1.57360i			0.0621536i	−0.999517	+	0.0310768i	$0.990106\pi$
$642$		0			0
$643$	−0.203779	+	0.203779i	−0.00803626	+	0.00803626i	−0.711113	−	0.703077i	$-0.751809\pi$
$643$	−0.203779	+	0.203779i	−0.00803626	+	0.00803626i	0.703077	+	0.711113i	$0.251809\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	7.73020i		−	0.303906i	−0.988388	−	0.151953i	$-0.951444\pi$
$647$		−	7.73020i		−	0.303906i	0.988388	−	0.151953i	$-0.0485562\pi$
$648$		0			0
$649$		−	5.13966i		−	0.201749i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.7507	−	25.7507i	1.00770	−	1.00770i	0.00773247	−	0.999970i	$-0.497539\pi$
$653$	25.7507	−	25.7507i	1.00770	−	1.00770i	0.999970	−	0.00773247i	$-0.00246135\pi$
$654$		0			0
$655$		−	33.0797i		−	1.29253i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14.1443	−	14.1443i	−0.550984	−	0.550984i	0.375741	−	0.926725i	$-0.377388\pi$
$659$	−14.1443	−	14.1443i	−0.550984	−	0.550984i	−0.926725	+	0.375741i	$0.877388\pi$
$660$		0			0
$661$	−8.31619	+	8.31619i	−0.323462	+	0.323462i	−0.850094	−	0.526632i	$-0.823455\pi$
$661$	−8.31619	+	8.31619i	−0.323462	+	0.323462i	0.526632	+	0.850094i	$0.323455\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.06365			0.196360
$666$		0			0
$667$	−17.9665	−	17.9665i	−0.695665	−	0.695665i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9.32976			0.360172
$672$		0			0
$673$	34.9355			1.34667			0.673333	−	0.739339i	$-0.264862\pi$
$673$	34.9355			1.34667			0.673333	+	0.739339i	$0.264862\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.44273	−	5.44273i	−0.209181	−	0.209181i	0.594738	−	0.803919i	$-0.297256\pi$
$677$	−5.44273	−	5.44273i	−0.209181	−	0.209181i	−0.803919	+	0.594738i	$0.797256\pi$
$678$		0			0
$679$	−8.64969			−0.331945
$680$		0			0
$681$		0			0
$682$		0			0
$683$	33.7753	−	33.7753i	1.29238	−	1.29238i	0.359063	−	0.933313i	$-0.383096\pi$
$683$	33.7753	−	33.7753i	1.29238	−	1.29238i	0.933313	−	0.359063i	$-0.116904\pi$
$684$		0			0
$685$	−37.4302	−	37.4302i	−1.43014	−	1.43014i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			19.3763i			0.738179i
$690$		0			0
$691$	31.8468	−	31.8468i	1.21151	−	1.21151i	0.240979	−	0.970530i	$-0.422531\pi$
$691$	31.8468	−	31.8468i	1.21151	−	1.21151i	0.970530	−	0.240979i	$-0.0774686\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	6.47269i		−	0.245523i
$696$		0			0
$697$			13.9553i			0.528595i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.0196	−	17.0196i	0.642822	−	0.642822i	−0.308426	−	0.951248i	$-0.599802\pi$
$701$	17.0196	−	17.0196i	0.642822	−	0.642822i	0.951248	+	0.308426i	$0.0998023\pi$
$702$		0			0
$703$		−	8.99601i		−	0.339291i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10.9849	+	10.9849i	0.413128	+	0.413128i
$708$		0			0
$709$	4.45471	−	4.45471i	0.167300	−	0.167300i	−0.618491	−	0.785792i	$-0.712256\pi$
$709$	4.45471	−	4.45471i	0.167300	−	0.167300i	0.785792	+	0.618491i	$0.212256\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.4627			0.691434
$714$		0			0
$715$	−5.41275	−	5.41275i	−0.202425	−	0.202425i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.3426			−0.646771			−0.323385	−	0.946267i	$-0.604821\pi$
$719$	−17.3426			−0.646771			−0.323385	+	0.946267i	$0.604821\pi$
$720$		0			0
$721$	−2.31765			−0.0863139
$722$		0			0
$723$		0			0
$724$		0			0
$725$	41.9609	+	41.9609i	1.55839	+	1.55839i
$726$		0			0
$727$	−0.938724			−0.0348153			−0.0174077	−	0.999848i	$-0.505541\pi$
$727$	−0.938724			−0.0348153			−0.0174077	+	0.999848i	$0.505541\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.5590	+	12.5590i	−0.464510	+	0.464510i
$732$		0			0
$733$	−17.3212	−	17.3212i	−0.639775	−	0.639775i	0.310725	−	0.950500i	$-0.399428\pi$
$733$	−17.3212	−	17.3212i	−0.639775	−	0.639775i	−0.950500	+	0.310725i	$0.899428\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			13.2401i			0.487706i
$738$		0			0
$739$	30.7806	−	30.7806i	1.13228	−	1.13228i	0.142486	−	0.989797i	$-0.454490\pi$
$739$	30.7806	−	30.7806i	1.13228	−	1.13228i	0.989797	−	0.142486i	$-0.0455096\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	41.3295i		−	1.51623i	−0.652120	−	0.758116i	$-0.726120\pi$
$743$		−	41.3295i		−	1.51623i	0.652120	−	0.758116i	$-0.273880\pi$
$744$		0			0
$745$			74.9805i			2.74707i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.38720	−	4.38720i	0.160305	−	0.160305i
$750$		0			0
$751$		−	50.1560i		−	1.83022i	−0.403204	−	0.915110i	$-0.632104\pi$
$751$		−	50.1560i		−	1.83022i	0.403204	−	0.915110i	$-0.367896\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	40.1977	+	40.1977i	1.46294	+	1.46294i
$756$		0			0
$757$	−2.81463	+	2.81463i	−0.102299	+	0.102299i	−0.756404	−	0.654105i	$-0.773046\pi$
$757$	−2.81463	+	2.81463i	−0.102299	+	0.102299i	0.654105	+	0.756404i	$0.273046\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21.6910			−0.786297			−0.393149	−	0.919475i	$-0.628614\pi$
$761$	−21.6910			−0.786297			−0.393149	+	0.919475i	$0.628614\pi$
$762$		0			0
$763$	−1.97102	−	1.97102i	−0.0713558	−	0.0713558i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.5642			−0.561992
$768$		0			0
$769$	−4.53040			−0.163371			−0.0816853	−	0.996658i	$-0.526030\pi$
$769$	−4.53040			−0.163371			−0.0816853	+	0.996658i	$0.526030\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−23.6600	−	23.6600i	−0.850990	−	0.850990i	0.139265	−	0.990255i	$-0.455526\pi$
$773$	−23.6600	−	23.6600i	−0.850990	−	0.850990i	−0.990255	+	0.139265i	$0.955526\pi$
$774$		0			0
$775$	−43.1198			−1.54891
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.69607	+	3.69607i	−0.132426	+	0.132426i
$780$		0			0
$781$	−3.41207	−	3.41207i	−0.122093	−	0.122093i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	25.1839i		−	0.898850i
$786$		0			0
$787$	1.29221	−	1.29221i	0.0460623	−	0.0460623i	−0.683700	−	0.729763i	$-0.739631\pi$
$787$	1.29221	−	1.29221i	0.0460623	−	0.0460623i	0.729763	+	0.683700i	$0.239631\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7.79490i			0.277155i
$792$		0			0
$793$		−	28.2530i		−	1.00329i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.95058	−	6.95058i	0.246202	−	0.246202i	−0.573208	−	0.819410i	$-0.694301\pi$
$797$	6.95058	−	6.95058i	0.246202	−	0.246202i	0.819410	+	0.573208i	$0.194301\pi$
$798$		0			0
$799$		−	42.4452i		−	1.50160i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.329437	−	0.329437i	−0.0116256	−	0.0116256i
$804$		0			0
$805$	−9.91862	+	9.91862i	−0.349586	+	0.349586i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	20.9481			0.736495			0.368248	−	0.929728i	$-0.379958\pi$
$809$	20.9481			0.736495			0.368248	+	0.929728i	$0.379958\pi$
$810$		0			0
$811$	−15.6400	−	15.6400i	−0.549194	−	0.549194i	0.377014	−	0.926208i	$-0.376951\pi$
$811$	−15.6400	−	15.6400i	−0.549194	−	0.549194i	−0.926208	+	0.377014i	$0.876951\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−17.2435			−0.604015
$816$		0			0
$817$	−6.65249			−0.232741
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.17232	+	7.17232i	0.250316	+	0.250316i	0.821100	−	0.570784i	$-0.193361\pi$
$821$	7.17232	+	7.17232i	0.250316	+	0.250316i	−0.570784	+	0.821100i	$0.693361\pi$
$822$		0			0
$823$	3.57692			0.124684			0.0623418	−	0.998055i	$-0.480143\pi$
$823$	3.57692			0.124684			0.0623418	+	0.998055i	$0.480143\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.0753	−	13.0753i	0.454672	−	0.454672i	−0.442230	−	0.896902i	$-0.645812\pi$
$827$	13.0753	−	13.0753i	0.454672	−	0.454672i	0.896902	+	0.442230i	$0.145812\pi$
$828$		0			0
$829$	2.86517	+	2.86517i	0.0995115	+	0.0995115i	0.755110	−	0.655598i	$-0.227584\pi$
$829$	2.86517	+	2.86517i	0.0995115	+	0.0995115i	−0.655598	+	0.755110i	$0.727584\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			3.63734i			0.126027i
$834$		0			0
$835$	57.4963	−	57.4963i	1.98974	−	1.98974i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	53.9862i		−	1.86381i	−0.362703	−	0.931905i	$-0.618146\pi$
$839$		−	53.9862i		−	1.86381i	0.362703	−	0.931905i	$-0.381854\pi$
$840$		0			0
$841$		−	16.3263i		−	0.562975i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17.7746	−	17.7746i	0.611463	−	0.611463i
$846$		0			0
$847$		−	10.3199i		−	0.354596i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	17.6213	+	17.6213i	0.604050	+	0.604050i
$852$		0			0
$853$	25.3458	−	25.3458i	0.867823	−	0.867823i	−0.124408	−	0.992231i	$-0.539703\pi$
$853$	25.3458	−	25.3458i	0.867823	−	0.867823i	0.992231	+	0.124408i	$0.0397033\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.6403			0.773379			0.386689	−	0.922210i	$-0.373619\pi$
$857$	22.6403			0.773379			0.386689	+	0.922210i	$0.373619\pi$
$858$		0			0
$859$	26.6166	+	26.6166i	0.908146	+	0.908146i	0.996123	−	0.0879765i	$-0.0280400\pi$
$859$	26.6166	+	26.6166i	0.908146	+	0.908146i	−0.0879765	+	0.996123i	$0.528040\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.5204			−0.834683			−0.417342	−	0.908750i	$-0.637038\pi$
$863$	−24.5204			−0.834683			−0.417342	+	0.908750i	$0.637038\pi$
$864$		0			0
$865$	−65.4924			−2.22681
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.60385	−	9.60385i	−0.325788	−	0.325788i
$870$		0			0
$871$	40.0946			1.35855
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.0243	−	10.0243i	0.338884	−	0.338884i
$876$		0			0
$877$	28.9688	+	28.9688i	0.978206	+	0.978206i	0.999768	−	0.0215611i	$-0.00686364\pi$
$877$	28.9688	+	28.9688i	0.978206	+	0.978206i	−0.0215611	+	0.999768i	$0.506864\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	30.1989i		−	1.01743i	−0.860936	−	0.508713i	$-0.830122\pi$
$881$		−	30.1989i		−	1.01743i	0.860936	−	0.508713i	$-0.169878\pi$
$882$		0			0
$883$	3.28384	−	3.28384i	0.110510	−	0.110510i	−0.649690	−	0.760200i	$-0.725101\pi$
$883$	3.28384	−	3.28384i	0.110510	−	0.110510i	0.760200	+	0.649690i	$0.225101\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			48.4792i			1.62777i	0.581024	+	0.813886i	$0.302652\pi$
$887$			48.4792i			1.62777i	−0.581024	+	0.813886i	$0.697348\pi$
$888$		0			0
$889$		−	5.11789i		−	0.171648i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.2416	−	11.2416i	0.376187	−	0.376187i
$894$		0			0
$895$		−	48.6122i		−	1.62493i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23.2891	+	23.2891i	0.776734	+	0.776734i
$900$		0			0
$901$	−19.9553	+	19.9553i	−0.664808	+	0.664808i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−87.4904			−2.90828
$906$		0			0
$907$	−33.8236	−	33.8236i	−1.12310	−	1.12310i	−0.991273	−	0.131822i	$-0.957917\pi$
$907$	−33.8236	−	33.8236i	−1.12310	−	1.12310i	−0.131822	−	0.991273i	$-0.542083\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.4793			0.512852			0.256426	−	0.966564i	$-0.417455\pi$
$911$	15.4793			0.512852			0.256426	+	0.966564i	$0.417455\pi$
$912$		0			0
$913$	−8.07339			−0.267190
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.29337	+	6.29337i	0.207825	+	0.207825i
$918$		0			0
$919$	49.5898			1.63582			0.817908	−	0.575349i	$-0.195134\pi$
$919$	49.5898			1.63582			0.817908	+	0.575349i	$0.195134\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.3326	+	10.3326i	−0.340103	+	0.340103i
$924$		0			0
$925$	−41.1546	−	41.1546i	−1.35316	−	1.35316i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			26.2852i			0.862391i	0.902259	+	0.431195i	$0.141908\pi$
$929$			26.2852i			0.862391i	−0.902259	+	0.431195i	$0.858092\pi$
$930$		0			0
$931$	−0.963353	+	0.963353i	−0.0315726	+	0.0315726i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	11.1490i		−	0.364611i
$936$		0			0
$937$			2.51468i			0.0821510i	0.999156	+	0.0410755i	$0.0130784\pi$
$937$			2.51468i			0.0821510i	−0.999156	+	0.0410755i	$0.986922\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.4678	−	15.4678i	0.504237	−	0.504237i	−0.408514	−	0.912752i	$-0.633953\pi$
$941$	15.4678	−	15.4678i	0.504237	−	0.504237i	0.912752	+	0.408514i	$0.133953\pi$
$942$		0			0
$943$		−	14.4797i		−	0.471523i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.9424	−	17.9424i	−0.583049	−	0.583049i	0.352691	−	0.935740i	$-0.385267\pi$
$947$	−17.9424	−	17.9424i	−0.583049	−	0.583049i	−0.935740	+	0.352691i	$0.885267\pi$
$948$		0			0
$949$	−0.997622	+	0.997622i	−0.0323842	+	0.0323842i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.59267			−0.181164			−0.0905821	−	0.995889i	$-0.528873\pi$
$953$	−5.59267			−0.181164			−0.0905821	+	0.995889i	$0.528873\pi$
$954$		0			0
$955$	−5.09176	−	5.09176i	−0.164766	−	0.164766i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.2421			0.459901
$960$		0			0
$961$	7.06770			0.227990
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−26.4612	−	26.4612i	−0.851818	−	0.851818i
$966$		0			0
$967$	−31.6701			−1.01844			−0.509220	−	0.860636i	$-0.670066\pi$
$967$	−31.6701			−1.01844			−0.509220	+	0.860636i	$0.670066\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.7252	−	19.7252i	0.633012	−	0.633012i	−0.315810	−	0.948822i	$-0.602276\pi$
$971$	19.7252	−	19.7252i	0.633012	−	0.633012i	0.948822	+	0.315810i	$0.102276\pi$
$972$		0			0
$973$	1.23142	+	1.23142i	0.0394775	+	0.0394775i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			15.2206i			0.486951i	0.969907	+	0.243475i	$0.0782875\pi$
$977$			15.2206i			0.486951i	−0.969907	+	0.243475i	$0.921713\pi$
$978$		0			0
$979$	6.28192	−	6.28192i	0.200771	−	0.200771i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	39.8828i		−	1.27207i	−0.771662	−	0.636033i	$-0.780574\pi$
$983$		−	39.8828i		−	1.27207i	0.771662	−	0.636033i	$-0.219426\pi$
$984$		0			0
$985$		−	7.65482i		−	0.243903i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.0308	−	13.0308i	0.414356	−	0.414356i
$990$		0			0
$991$			48.0864i			1.52752i	0.645503	+	0.763758i	$0.276648\pi$
$991$			48.0864i			1.52752i	−0.645503	+	0.763758i	$0.723352\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	42.9065	+	42.9065i	1.36023	+	1.36023i
$996$		0			0
$997$	−16.3703	+	16.3703i	−0.518452	+	0.518452i	−0.917103	−	0.398651i	$-0.869479\pi$
$997$	−16.3703	+	16.3703i	−0.518452	+	0.518452i	0.398651	+	0.917103i	$0.369479\pi$
$998$		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.1583.19		40
3.2	odd	2	inner	4032.2.v.e.1583.2		40
4.3	odd	2		1008.2.v.e.323.14	yes	40
12.11	even	2		1008.2.v.e.323.7	✓	40
16.5	even	4		1008.2.v.e.827.7	yes	40
16.11	odd	4	inner	4032.2.v.e.3599.2		40
48.5	odd	4		1008.2.v.e.827.14	yes	40
48.11	even	4	inner	4032.2.v.e.3599.19		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.7	✓	40	12.11	even	2
1008.2.v.e.323.14	yes	40	4.3	odd	2
1008.2.v.e.827.7	yes	40	16.5	even	4
1008.2.v.e.827.14	yes	40	48.5	odd	4
4032.2.v.e.1583.2		40	3.2	odd	2	inner
4032.2.v.e.1583.19		40	1.1	even	1	trivial
4032.2.v.e.3599.2		40	16.11	odd	4	inner
4032.2.v.e.3599.19		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)

\(f(q)\)	\(=\)	\(q+(2.62814 + 2.62814i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(2.62814 + 2.62814i) q^{5} -1.00000 q^{7} +(-0.583140 + 0.583140i) q^{11} +(1.76590 + 1.76590i) q^{13} +3.63734i q^{17} +(-0.963353 + 0.963353i) q^{19} -3.77401i q^{23} +8.81423i q^{25} +(4.76058 - 4.76058i) q^{29} +4.89207i q^{31} +(-2.62814 - 2.62814i) q^{35} +(-4.66911 + 4.66911i) q^{37} +3.83668 q^{41} +(3.45278 + 3.45278i) q^{43} -11.6693 q^{47} +1.00000 q^{49} +(5.48624 + 5.48624i) q^{53} -3.06515 q^{55} +(-4.40688 + 4.40688i) q^{59} +(-7.99959 - 7.99959i) q^{61} +9.28208i q^{65} +(11.3524 - 11.3524i) q^{67} +5.85120i q^{71} +0.564936i q^{73} +(0.583140 - 0.583140i) q^{77} +16.4692i q^{79} +(6.92234 + 6.92234i) q^{83} +(-9.55945 + 9.55945i) q^{85} -10.7726 q^{89} +(-1.76590 - 1.76590i) q^{91} -5.06365 q^{95} +8.64969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 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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