LMFDB - Embedded newform 3087.2.c.c.3086.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3087,2,Mod(3086,3087)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3087, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3087.3086");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3087 = 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3087.c (of order $2$, degree $1$, 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:	$24.6498191040$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3086.8
Character	$\chi$	$=$	3087.3086
Dual form			3087.2.c.c.3086.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.06406i q^{2} +0.867767 q^{4} -3.23675 q^{5} +3.05149i q^{8} +O(q^{10})$	$q+1.06406i q^{2} +0.867767 q^{4} -3.23675 q^{5} +3.05149i q^{8} -3.44411i q^{10} +0.699431i q^{11} -2.71931i q^{13} -1.51144 q^{16} +6.50168 q^{17} +0.581246i q^{19} -2.80875 q^{20} -0.744240 q^{22} +0.742427i q^{23} +5.47655 q^{25} +2.89352 q^{26} +6.65157i q^{29} -8.87219i q^{31} +4.49470i q^{32} +6.91820i q^{34} +1.66798 q^{37} -0.618483 q^{38} -9.87691i q^{40} +3.71552 q^{41} -5.59897 q^{43} +0.606944i q^{44} -0.789990 q^{46} +2.32997 q^{47} +5.82740i q^{50} -2.35973i q^{52} +6.07261i q^{53} -2.26388i q^{55} -7.07769 q^{58} +0.355771 q^{59} +11.7351i q^{61} +9.44058 q^{62} -7.80554 q^{64} +8.80172i q^{65} +15.4957 q^{67} +5.64194 q^{68} +8.84942i q^{71} +14.2339i q^{73} +1.77484i q^{74} +0.504386i q^{76} -0.775773 q^{79} +4.89217 q^{80} +3.95356i q^{82} -16.5058 q^{83} -21.0443 q^{85} -5.95766i q^{86} -2.13431 q^{88} -3.83377 q^{89} +0.644254i q^{92} +2.47924i q^{94} -1.88135i q^{95} -10.4610i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) $ 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3087\mathbb{Z}\right)^\times$.

$n$	$344$	$2404$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.06406i	0.752407i	0.926537	+	0.376203i	$0.122771\pi$
$2$	1.06406i	0.752407i	−0.926537	+	0.376203i	$0.877229\pi$
$3$	0	0
$3$	0	0
$4$	0.867767	0.433884
$5$	−3.23675	−1.44752	−0.723759	−	0.690053i	$-0.757587\pi$
$5$	−3.23675	−1.44752	−0.723759	+	0.690053i	$0.757587\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.05149i	1.07886i
$9$	0	0
$10$	− 3.44411i	− 1.08912i
$11$	0.699431i	0.210886i	0.994425	+	0.105443i	$0.0336261\pi$
$11$	0.699431i	0.210886i	−0.994425	+	0.105443i	$0.966374\pi$
$12$	0	0
$13$	− 2.71931i	− 0.754200i	−0.926173	−	0.377100i	$-0.876921\pi$
$13$	− 2.71931i	− 0.754200i	0.926173	−	0.377100i	$-0.123079\pi$
$14$	0	0
$15$	0	0
$16$	−1.51144	−0.377861
$17$	6.50168	1.57689	0.788444	−	0.615107i	$-0.210887\pi$
$17$	6.50168	1.57689	0.788444	+	0.615107i	$0.210887\pi$
$18$	0	0
$19$	0.581246i	0.133347i	0.997775	+	0.0666735i	$0.0212386\pi$
$19$	0.581246i	0.133347i	−0.997775	+	0.0666735i	$0.978761\pi$
$20$	−2.80875	−0.628055
$21$	0	0
$22$	−0.744240	−0.158672
$23$	0.742427i	0.154807i	0.997000	+	0.0774033i	$0.0246629\pi$
$23$	0.742427i	0.154807i	−0.997000	+	0.0774033i	$0.975337\pi$
$24$	0	0
$25$	5.47655	1.09531
$26$	2.89352	0.567465
$27$	0	0
$28$	0	0
$29$	6.65157i	1.23516i	0.786506	+	0.617582i	$0.211888\pi$
$29$	6.65157i	1.23516i	−0.786506	+	0.617582i	$0.788112\pi$
$30$	0	0
$31$	− 8.87219i	− 1.59349i	−0.604314	−	0.796746i	$-0.706553\pi$
$31$	− 8.87219i	− 1.59349i	0.604314	−	0.796746i	$-0.293447\pi$
$32$	4.49470i	0.794559i
$33$	0	0
$34$	6.91820i	1.18646i
$35$	0	0
$36$	0	0
$37$	1.66798	0.274214	0.137107	−	0.990556i	$-0.456220\pi$
$37$	1.66798	0.274214	0.137107	+	0.990556i	$0.456220\pi$
$38$	−0.618483	−0.100331
$39$	0	0
$40$	− 9.87691i	− 1.56168i
$41$	3.71552	0.580267	0.290134	−	0.956986i	$-0.406300\pi$
$41$	3.71552	0.580267	0.290134	+	0.956986i	$0.406300\pi$
$42$	0	0
$43$	−5.59897	−0.853835	−0.426917	−	0.904291i	$-0.640400\pi$
$43$	−5.59897	−0.853835	−0.426917	+	0.904291i	$0.640400\pi$
$44$	0.606944i	0.0915002i
$45$	0	0
$46$	−0.789990	−0.116478
$47$	2.32997	0.339861	0.169931	−	0.985456i	$-0.445646\pi$
$47$	2.32997	0.339861	0.169931	+	0.985456i	$0.445646\pi$
$48$	0	0
$49$	0	0
$50$	5.82740i	0.824119i
$51$	0	0
$52$	− 2.35973i	− 0.327235i
$53$	6.07261i	0.834138i	0.908875	+	0.417069i	$0.136943\pi$
$53$	6.07261i	0.834138i	−0.908875	+	0.417069i	$0.863057\pi$
$54$	0	0
$55$	− 2.26388i	− 0.305262i
$56$	0	0
$57$	0	0
$58$	−7.07769	−0.929347
$59$	0.355771	0.0463175	0.0231587	−	0.999732i	$-0.492628\pi$
$59$	0.355771	0.0463175	0.0231587	+	0.999732i	$0.492628\pi$
$60$	0	0
$61$	11.7351i	1.50252i	0.660007	+	0.751260i	$0.270553\pi$
$61$	11.7351i	1.50252i	−0.660007	+	0.751260i	$0.729447\pi$
$62$	9.44058	1.19895
$63$	0	0
$64$	−7.80554	−0.975693
$65$	8.80172i	1.09172i
$66$	0	0
$67$	15.4957	1.89310	0.946552	−	0.322550i	$-0.104540\pi$
$67$	15.4957	1.89310	0.946552	+	0.322550i	$0.104540\pi$
$68$	5.64194	0.684186
$69$	0	0
$70$	0	0
$71$	8.84942i	1.05023i	0.851030	+	0.525117i	$0.175978\pi$
$71$	8.84942i	1.05023i	−0.851030	+	0.525117i	$0.824022\pi$
$72$	0	0
$73$	14.2339i	1.66596i	0.553305	+	0.832979i	$0.313366\pi$
$73$	14.2339i	1.66596i	−0.553305	+	0.832979i	$0.686634\pi$
$74$	1.77484i	0.206321i
$75$	0	0
$76$	0.504386i	0.0578571i
$77$	0	0
$78$	0	0
$79$	−0.775773	−0.0872813	−0.0436406	−	0.999047i	$-0.513896\pi$
$79$	−0.775773	−0.0872813	−0.0436406	+	0.999047i	$0.513896\pi$
$80$	4.89217	0.546961
$81$	0	0
$82$	3.95356i	0.436597i
$83$	−16.5058	−1.81175	−0.905873	−	0.423549i	$-0.860784\pi$
$83$	−16.5058	−1.81175	−0.905873	+	0.423549i	$0.860784\pi$
$84$	0	0
$85$	−21.0443	−2.28257
$86$	− 5.95766i	− 0.642431i
$87$	0	0
$88$	−2.13431	−0.227518
$89$	−3.83377	−0.406379	−0.203189	−	0.979139i	$-0.565131\pi$
$89$	−3.83377	−0.406379	−0.203189	+	0.979139i	$0.565131\pi$
$90$	0	0
$91$	0	0
$92$	0.644254i	0.0671681i
$93$	0	0
$94$	2.47924i	0.255714i
$95$	− 1.88135i	− 0.193022i
$96$	0	0
$97$	− 10.4610i	− 1.06215i	−0.847325	−	0.531075i	$-0.821788\pi$
$97$	− 10.4610i	− 1.06215i	0.847325	−	0.531075i	$-0.178212\pi$
$98$	0	0
$99$	0	0
$100$	4.75237	0.475237
$101$	−15.7643	−1.56861	−0.784304	−	0.620377i	$-0.786980\pi$
$101$	−15.7643	−1.56861	−0.784304	+	0.620377i	$0.786980\pi$
$102$	0	0
$103$	13.5477i	1.33490i	0.744655	+	0.667449i	$0.232614\pi$
$103$	13.5477i	1.33490i	−0.744655	+	0.667449i	$0.767386\pi$
$104$	8.29794	0.813680
$105$	0	0
$106$	−6.46165	−0.627611
$107$	− 13.3360i	− 1.28924i	−0.764504	−	0.644619i	$-0.777016\pi$
$107$	− 13.3360i	− 1.28924i	0.764504	−	0.644619i	$-0.222984\pi$
$108$	0	0
$109$	9.28866	0.889692	0.444846	−	0.895607i	$-0.353258\pi$
$109$	9.28866	0.889692	0.444846	+	0.895607i	$0.353258\pi$
$110$	2.40892	0.229681
$111$	0	0
$112$	0	0
$113$	5.86335i	0.551578i	0.961218	+	0.275789i	$0.0889391\pi$
$113$	5.86335i	0.551578i	−0.961218	+	0.275789i	$0.911061\pi$
$114$	0	0
$115$	− 2.40305i	− 0.224086i
$116$	5.77201i	0.535918i
$117$	0	0
$118$	0.378564i	0.0348496i
$119$	0	0
$120$	0	0
$121$	10.5108	0.955527
$122$	−12.4868	−1.13051
$123$	0	0
$124$	− 7.69900i	− 0.691390i
$125$	−1.54247	−0.137963
$126$	0	0
$127$	−10.1902	−0.904237	−0.452119	−	0.891958i	$-0.649332\pi$
$127$	−10.1902	−0.904237	−0.452119	+	0.891958i	$0.649332\pi$
$128$	0.683809i	0.0604407i
$129$	0	0
$130$	−9.36559	−0.821417
$131$	−12.3920	−1.08270	−0.541348	−	0.840799i	$-0.682086\pi$
$131$	−12.3920	−1.08270	−0.541348	+	0.840799i	$0.682086\pi$
$132$	0	0
$133$	0	0
$134$	16.4885i	1.42439i
$135$	0	0
$136$	19.8398i	1.70125i
$137$	1.53066i	0.130773i	0.997860	+	0.0653864i	$0.0208280\pi$
$137$	1.53066i	0.130773i	−0.997860	+	0.0653864i	$0.979172\pi$
$138$	0	0
$139$	20.4423i	1.73389i	0.498403	+	0.866945i	$0.333920\pi$
$139$	20.4423i	1.73389i	−0.498403	+	0.866945i	$0.666080\pi$
$140$	0	0
$141$	0	0
$142$	−9.41635	−0.790203
$143$	1.90197	0.159051
$144$	0	0
$145$	− 21.5295i	− 1.78792i
$146$	−15.1458	−1.25348
$147$	0	0
$148$	1.44742	0.118977
$149$	16.0703i	1.31653i	0.752786	+	0.658266i	$0.228710\pi$
$149$	16.0703i	1.31653i	−0.752786	+	0.658266i	$0.771290\pi$
$150$	0	0
$151$	11.6504	0.948098	0.474049	−	0.880499i	$-0.342792\pi$
$151$	11.6504	0.948098	0.474049	+	0.880499i	$0.342792\pi$
$152$	−1.77367	−0.143863
$153$	0	0
$154$	0	0
$155$	28.7171i	2.30661i
$156$	0	0
$157$	15.2146i	1.21426i	0.794603	+	0.607129i	$0.207679\pi$
$157$	15.2146i	1.21426i	−0.794603	+	0.607129i	$0.792321\pi$
$158$	− 0.825472i	− 0.0656711i
$159$	0	0
$160$	− 14.5482i	− 1.15014i
$161$	0	0
$162$	0	0
$163$	17.7335	1.38900	0.694499	−	0.719494i	$-0.255626\pi$
$163$	17.7335	1.38900	0.694499	+	0.719494i	$0.255626\pi$
$164$	3.22421	0.251769
$165$	0	0
$166$	− 17.5632i	− 1.36317i
$167$	12.0363	0.931393	0.465697	−	0.884944i	$-0.345804\pi$
$167$	12.0363	0.931393	0.465697	+	0.884944i	$0.345804\pi$
$168$	0	0
$169$	5.60537	0.431182
$170$	− 22.3925i	− 1.71742i
$171$	0	0
$172$	−4.85860	−0.370465
$173$	−10.9940	−0.835859	−0.417930	−	0.908479i	$-0.637244\pi$
$173$	−10.9940	−0.835859	−0.417930	+	0.908479i	$0.637244\pi$
$174$	0	0
$175$	0	0
$176$	− 1.05715i	− 0.0796858i
$177$	0	0
$178$	− 4.07938i	− 0.305762i
$179$	− 11.2624i	− 0.841792i	−0.907109	−	0.420896i	$-0.861716\pi$
$179$	− 11.2624i	− 0.841792i	0.907109	−	0.420896i	$-0.138284\pi$
$180$	0	0
$181$	1.65897i	0.123310i	0.998098	+	0.0616550i	$0.0196379\pi$
$181$	1.65897i	0.123310i	−0.998098	+	0.0616550i	$0.980362\pi$
$182$	0	0
$183$	0	0
$184$	−2.26551	−0.167015
$185$	−5.39884	−0.396930
$186$	0	0
$187$	4.54748i	0.332544i
$188$	2.02187	0.147460
$189$	0	0
$190$	2.00188	0.145231
$191$	8.47446i	0.613190i	0.951840	+	0.306595i	$0.0991897\pi$
$191$	8.47446i	0.613190i	−0.951840	+	0.306595i	$0.900810\pi$
$192$	0	0
$193$	−23.6179	−1.70005	−0.850025	−	0.526742i	$-0.823413\pi$
$193$	−23.6179	−1.70005	−0.850025	+	0.526742i	$0.823413\pi$
$194$	11.1311	0.799169
$195$	0	0
$196$	0	0
$197$	20.1961i	1.43891i	0.694537	+	0.719457i	$0.255609\pi$
$197$	20.1961i	1.43891i	−0.694537	+	0.719457i	$0.744391\pi$
$198$	0	0
$199$	1.55682i	0.110360i	0.998476	+	0.0551799i	$0.0175732\pi$
$199$	1.55682i	0.110360i	−0.998476	+	0.0551799i	$0.982427\pi$
$200$	16.7116i	1.18169i
$201$	0	0
$202$	− 16.7742i	− 1.18023i
$203$	0	0
$204$	0	0
$205$	−12.0262	−0.839948
$206$	−14.4157	−1.00439
$207$	0	0
$208$	4.11008i	0.284983i
$209$	−0.406542	−0.0281211
$210$	0	0
$211$	6.92376	0.476651	0.238326	−	0.971185i	$-0.423401\pi$
$211$	6.92376	0.476651	0.238326	+	0.971185i	$0.423401\pi$
$212$	5.26962i	0.361919i
$213$	0	0
$214$	14.1903	0.970032
$215$	18.1225	1.23594
$216$	0	0
$217$	0	0
$218$	9.88373i	0.669411i
$219$	0	0
$220$	− 1.96453i	− 0.132448i
$221$	− 17.6801i	− 1.18929i
$222$	0	0
$223$	24.2781i	1.62578i	0.582417	+	0.812890i	$0.302107\pi$
$223$	24.2781i	1.62578i	−0.582417	+	0.812890i	$0.697893\pi$
$224$	0	0
$225$	0	0
$226$	−6.23898	−0.415011
$227$	5.74207	0.381115	0.190557	−	0.981676i	$-0.438971\pi$
$227$	5.74207	0.381115	0.190557	+	0.981676i	$0.438971\pi$
$228$	0	0
$229$	20.2897i	1.34078i	0.742007	+	0.670392i	$0.233874\pi$
$229$	20.2897i	1.34078i	−0.742007	+	0.670392i	$0.766126\pi$
$230$	2.55700	0.168604
$231$	0	0
$232$	−20.2972	−1.33258
$233$	− 12.3784i	− 0.810936i	−0.914109	−	0.405468i	$-0.867109\pi$
$233$	− 12.3784i	− 0.810936i	0.914109	−	0.405468i	$-0.132891\pi$
$234$	0	0
$235$	−7.54154	−0.491956
$236$	0.308727	0.0200964
$237$	0	0
$238$	0	0
$239$	− 18.2647i	− 1.18144i	−0.806875	−	0.590722i	$-0.798843\pi$
$239$	− 18.2647i	− 1.18144i	0.806875	−	0.590722i	$-0.201157\pi$
$240$	0	0
$241$	− 23.2883i	− 1.50013i	−0.661364	−	0.750065i	$-0.730022\pi$
$241$	− 23.2883i	− 1.50013i	0.661364	−	0.750065i	$-0.269978\pi$
$242$	11.1842i	0.718945i
$243$	0	0
$244$	10.1833i	0.651919i
$245$	0	0
$246$	0	0
$247$	1.58059	0.100570
$248$	27.0734	1.71916
$249$	0	0
$250$	− 1.64129i	− 0.103804i
$251$	12.7841	0.806923	0.403462	−	0.914997i	$-0.367807\pi$
$251$	12.7841	0.806923	0.403462	+	0.914997i	$0.367807\pi$
$252$	0	0
$253$	−0.519276	−0.0326466
$254$	− 10.8431i	− 0.680355i
$255$	0	0
$256$	−16.3387	−1.02117
$257$	−6.55775	−0.409061	−0.204530	−	0.978860i	$-0.565567\pi$
$257$	−6.55775	−0.409061	−0.204530	+	0.978860i	$0.565567\pi$
$258$	0	0
$259$	0	0
$260$	7.63784i	0.473679i
$261$	0	0
$262$	− 13.1859i	− 0.814628i
$263$	− 29.7935i	− 1.83715i	−0.395249	−	0.918574i	$-0.629342\pi$
$263$	− 29.7935i	− 1.83715i	0.395249	−	0.918574i	$-0.370658\pi$
$264$	0	0
$265$	− 19.6555i	− 1.20743i
$266$	0	0
$267$	0	0
$268$	13.4467	0.821387
$269$	−1.43750	−0.0876460	−0.0438230	−	0.999039i	$-0.513954\pi$
$269$	−1.43750	−0.0876460	−0.0438230	+	0.999039i	$0.513954\pi$
$270$	0	0
$271$	15.4973i	0.941396i	0.882294	+	0.470698i	$0.155998\pi$
$271$	15.4973i	0.941396i	−0.882294	+	0.470698i	$0.844002\pi$
$272$	−9.82692	−0.595845
$273$	0	0
$274$	−1.62872	−0.0983944
$275$	3.83047i	0.230986i
$276$	0	0
$277$	1.71075	0.102789	0.0513945	−	0.998678i	$-0.483633\pi$
$277$	1.71075	0.102789	0.0513945	+	0.998678i	$0.483633\pi$
$278$	−21.7519	−1.30459
$279$	0	0
$280$	0	0
$281$	4.30417i	0.256765i	0.991725	+	0.128383i	$0.0409785\pi$
$281$	4.30417i	0.256765i	−0.991725	+	0.128383i	$0.959021\pi$
$282$	0	0
$283$	− 8.76848i	− 0.521232i	−0.965443	−	0.260616i	$-0.916074\pi$
$283$	− 8.76848i	− 0.521232i	0.965443	−	0.260616i	$-0.0839257\pi$
$284$	7.67924i	0.455679i
$285$	0	0
$286$	2.02382i	0.119671i
$287$	0	0
$288$	0	0
$289$	25.2718	1.48658
$290$	22.9087	1.34525
$291$	0	0
$292$	12.3518i	0.722832i
$293$	31.5105	1.84086	0.920432	−	0.390903i	$-0.127837\pi$
$293$	31.5105	1.84086	0.920432	+	0.390903i	$0.127837\pi$
$294$	0	0
$295$	−1.15154	−0.0670454
$296$	5.08983i	0.295840i
$297$	0	0
$298$	−17.0998	−0.990568
$299$	2.01889	0.116755
$300$	0	0
$301$	0	0
$302$	12.3968i	0.713355i
$303$	0	0
$304$	− 0.878521i	− 0.0503867i
$305$	− 37.9834i	− 2.17492i
$306$	0	0
$307$	− 2.13806i	− 0.122026i	−0.998137	−	0.0610128i	$-0.980567\pi$
$307$	− 2.13806i	− 0.122026i	0.998137	−	0.0610128i	$-0.0194330\pi$
$308$	0	0
$309$	0	0
$310$	−30.5568	−1.73551
$311$	11.4164	0.647367	0.323683	−	0.946165i	$-0.395079\pi$
$311$	11.4164	0.647367	0.323683	+	0.946165i	$0.395079\pi$
$312$	0	0
$313$	− 9.39656i	− 0.531125i	−0.964094	−	0.265563i	$-0.914442\pi$
$313$	− 9.39656i	− 0.531125i	0.964094	−	0.265563i	$-0.0855577\pi$
$314$	−16.1893	−0.913617
$315$	0	0
$316$	−0.673191	−0.0378699
$317$	6.03447i	0.338930i	0.985536	+	0.169465i	$0.0542039\pi$
$317$	6.03447i	0.338930i	−0.985536	+	0.169465i	$0.945796\pi$
$318$	0	0
$319$	−4.65231	−0.260480
$320$	25.2646	1.41233
$321$	0	0
$322$	0	0
$323$	3.77907i	0.210273i
$324$	0	0
$325$	− 14.8924i	− 0.826083i
$326$	18.8696i	1.04509i
$327$	0	0
$328$	11.3379i	0.626030i
$329$	0	0
$330$	0	0
$331$	−4.51343	−0.248081	−0.124040	−	0.992277i	$-0.539585\pi$
$331$	−4.51343	−0.248081	−0.124040	+	0.992277i	$0.539585\pi$
$332$	−14.3232	−0.786087
$333$	0	0
$334$	12.8073i	0.700787i
$335$	−50.1558	−2.74030
$336$	0	0
$337$	−0.262350	−0.0142911	−0.00714556	−	0.999974i	$-0.502275\pi$
$337$	−0.262350	−0.0142911	−0.00714556	+	0.999974i	$0.502275\pi$
$338$	5.96447i	0.324424i
$339$	0	0
$340$	−18.2616	−0.990372
$341$	6.20549	0.336046
$342$	0	0
$343$	0	0
$344$	− 17.0852i	− 0.921172i
$345$	0	0
$346$	− 11.6983i	− 0.628907i
$347$	5.73283i	0.307755i	0.988090	+	0.153877i	$0.0491761\pi$
$347$	5.73283i	0.307755i	−0.988090	+	0.153877i	$0.950824\pi$
$348$	0	0
$349$	− 11.7082i	− 0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$349$	− 11.7082i	− 0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$350$	0	0
$351$	0	0
$352$	−3.14374	−0.167562
$353$	9.25250	0.492461	0.246230	−	0.969211i	$-0.420808\pi$
$353$	9.25250	0.492461	0.246230	+	0.969211i	$0.420808\pi$
$354$	0	0
$355$	− 28.6434i	− 1.52023i
$356$	−3.32682	−0.176321
$357$	0	0
$358$	11.9839	0.633370
$359$	19.8213i	1.04613i	0.852294	+	0.523064i	$0.175211\pi$
$359$	19.8213i	1.04613i	−0.852294	+	0.523064i	$0.824789\pi$
$360$	0	0
$361$	18.6622	0.982219
$362$	−1.76525	−0.0927793
$363$	0	0
$364$	0	0
$365$	− 46.0717i	− 2.41150i
$366$	0	0
$367$	1.86165i	0.0971773i	0.998819	+	0.0485887i	$0.0154723\pi$
$367$	1.86165i	0.0971773i	−0.998819	+	0.0485887i	$0.984528\pi$
$368$	− 1.12214i	− 0.0584954i
$369$	0	0
$370$	− 5.74471i	− 0.298653i
$371$	0	0
$372$	0	0
$373$	−25.5179	−1.32127	−0.660634	−	0.750708i	$-0.729712\pi$
$373$	−25.5179	−1.32127	−0.660634	+	0.750708i	$0.729712\pi$
$374$	−4.83881	−0.250209
$375$	0	0
$376$	7.10988i	0.366664i
$377$	18.0877	0.931562
$378$	0	0
$379$	−25.7502	−1.32270	−0.661350	−	0.750078i	$-0.730016\pi$
$379$	−25.7502	−1.32270	−0.661350	+	0.750078i	$0.730016\pi$
$380$	− 1.63257i	− 0.0837492i
$381$	0	0
$382$	−9.01736	−0.461369
$383$	12.7085	0.649374	0.324687	−	0.945822i	$-0.394741\pi$
$383$	12.7085	0.649374	0.324687	+	0.945822i	$0.394741\pi$
$384$	0	0
$385$	0	0
$386$	− 25.1309i	− 1.27913i
$387$	0	0
$388$	− 9.07768i	− 0.460849i
$389$	− 27.8265i	− 1.41086i	−0.708781	−	0.705429i	$-0.750754\pi$
$389$	− 27.8265i	− 1.41086i	0.708781	−	0.705429i	$-0.249246\pi$
$390$	0	0
$391$	4.82702i	0.244113i
$392$	0	0
$393$	0	0
$394$	−21.4900	−1.08265
$395$	2.51098	0.126341
$396$	0	0
$397$	− 13.8127i	− 0.693241i	−0.938005	−	0.346620i	$-0.887329\pi$
$397$	− 13.8127i	− 0.693241i	0.938005	−	0.346620i	$-0.112671\pi$
$398$	−1.65655	−0.0830354
$399$	0	0
$400$	−8.27750	−0.413875
$401$	− 2.27378i	− 0.113547i	−0.998387	−	0.0567736i	$-0.981919\pi$
$401$	− 2.27378i	− 0.113547i	0.998387	−	0.0567736i	$-0.0180813\pi$
$402$	0	0
$403$	−24.1262	−1.20181
$404$	−13.6798	−0.680593
$405$	0	0
$406$	0	0
$407$	1.16664i	0.0578281i
$408$	0	0
$409$	− 17.8055i	− 0.880423i	−0.897894	−	0.440212i	$-0.854903\pi$
$409$	− 17.8055i	− 0.880423i	0.897894	−	0.440212i	$-0.145097\pi$
$410$	− 12.7967i	− 0.631983i
$411$	0	0
$412$	11.7563i	0.579191i
$413$	0	0
$414$	0	0
$415$	53.4251	2.62254
$416$	12.2225	0.599256
$417$	0	0
$418$	− 0.432586i	− 0.0211585i
$419$	−4.85817	−0.237337	−0.118668	−	0.992934i	$-0.537863\pi$
$419$	−4.85817	−0.237337	−0.118668	+	0.992934i	$0.537863\pi$
$420$	0	0
$421$	−7.40414	−0.360856	−0.180428	−	0.983588i	$-0.557748\pi$
$421$	−7.40414	−0.360856	−0.180428	+	0.983588i	$0.557748\pi$
$422$	7.36732i	0.358636i
$423$	0	0
$424$	−18.5305	−0.899921
$425$	35.6067	1.72718
$426$	0	0
$427$	0	0
$428$	− 11.5725i	− 0.559380i
$429$	0	0
$430$	19.2835i	0.929931i
$431$	31.0308i	1.49470i	0.664429	+	0.747351i	$0.268675\pi$
$431$	31.0308i	1.49470i	−0.664429	+	0.747351i	$0.731325\pi$
$432$	0	0
$433$	− 27.0645i	− 1.30064i	−0.759662	−	0.650318i	$-0.774636\pi$
$433$	− 27.0645i	− 1.30064i	0.759662	−	0.650318i	$-0.225364\pi$
$434$	0	0
$435$	0	0
$436$	8.06040	0.386023
$437$	−0.431533	−0.0206430
$438$	0	0
$439$	− 26.9454i	− 1.28604i	−0.765851	−	0.643018i	$-0.777682\pi$
$439$	− 26.9454i	− 1.28604i	0.765851	−	0.643018i	$-0.222318\pi$
$440$	6.90822	0.329336
$441$	0	0
$442$	18.8127	0.894829
$443$	0.652931i	0.0310217i	0.999880	+	0.0155108i	$0.00493745\pi$
$443$	0.652931i	0.0310217i	−0.999880	+	0.0155108i	$0.995063\pi$
$444$	0	0
$445$	12.4090	0.588241
$446$	−25.8334	−1.22325
$447$	0	0
$448$	0	0
$449$	23.3074i	1.09994i	0.835183	+	0.549972i	$0.185362\pi$
$449$	23.3074i	1.09994i	−0.835183	+	0.549972i	$0.814638\pi$
$450$	0	0
$451$	2.59875i	0.122371i
$452$	5.08803i	0.239321i
$453$	0	0
$454$	6.10993i	0.286753i
$455$	0	0
$456$	0	0
$457$	−36.4707	−1.70603	−0.853015	−	0.521887i	$-0.825228\pi$
$457$	−36.4707	−1.70603	−0.853015	+	0.521887i	$0.825228\pi$
$458$	−21.5896	−1.00881
$459$	0	0
$460$	− 2.08529i	− 0.0972271i
$461$	26.3678	1.22807	0.614035	−	0.789279i	$-0.289545\pi$
$461$	26.3678	1.22807	0.614035	+	0.789279i	$0.289545\pi$
$462$	0	0
$463$	17.5599	0.816076	0.408038	−	0.912965i	$-0.366213\pi$
$463$	17.5599	0.816076	0.408038	+	0.912965i	$0.366213\pi$
$464$	− 10.0535i	− 0.466721i
$465$	0	0
$466$	13.1714	0.610154
$467$	−25.0005	−1.15689	−0.578444	−	0.815722i	$-0.696340\pi$
$467$	−25.0005	−1.15689	−0.578444	+	0.815722i	$0.696340\pi$
$468$	0	0
$469$	0	0
$470$	− 8.02468i	− 0.370151i
$471$	0	0
$472$	1.08563i	0.0499703i
$473$	− 3.91609i	− 0.180062i
$474$	0	0
$475$	3.18322i	0.146056i
$476$	0	0
$477$	0	0
$478$	19.4348	0.888927
$479$	−1.73499	−0.0792738	−0.0396369	−	0.999214i	$-0.512620\pi$
$479$	−1.73499	−0.0792738	−0.0396369	+	0.999214i	$0.512620\pi$
$480$	0	0
$481$	− 4.53575i	− 0.206813i
$482$	24.7802	1.12871
$483$	0	0
$484$	9.12093	0.414588
$485$	33.8595i	1.53748i
$486$	0	0
$487$	−12.8737	−0.583361	−0.291681	−	0.956516i	$-0.594214\pi$
$487$	−12.8737	−0.583361	−0.291681	+	0.956516i	$0.594214\pi$
$488$	−35.8094	−1.62101
$489$	0	0
$490$	0	0
$491$	− 38.7852i	− 1.75035i	−0.483804	−	0.875176i	$-0.660745\pi$
$491$	− 38.7852i	− 1.75035i	0.483804	−	0.875176i	$-0.339255\pi$
$492$	0	0
$493$	43.2463i	1.94772i
$494$	1.68185i	0.0756698i
$495$	0	0
$496$	13.4098i	0.602119i
$497$	0	0
$498$	0	0
$499$	22.5167	1.00799	0.503993	−	0.863708i	$-0.331864\pi$
$499$	22.5167	1.00799	0.503993	+	0.863708i	$0.331864\pi$
$500$	−1.33851	−0.0598598
$501$	0	0
$502$	13.6031i	0.607135i
$503$	−0.431101	−0.0192218	−0.00961091	−	0.999954i	$-0.503059\pi$
$503$	−0.431101	−0.0192218	−0.00961091	+	0.999954i	$0.503059\pi$
$504$	0	0
$505$	51.0251	2.27059
$506$	− 0.552543i	− 0.0245636i
$507$	0	0
$508$	−8.84276	−0.392334
$509$	−33.6247	−1.49039	−0.745194	−	0.666847i	$-0.767643\pi$
$509$	−33.6247	−1.49039	−0.745194	+	0.666847i	$0.767643\pi$
$510$	0	0
$511$	0	0
$512$	− 16.0178i	− 0.707894i
$513$	0	0
$514$	− 6.97786i	− 0.307780i
$515$	− 43.8506i	− 1.93229i
$516$	0	0
$517$	1.62966i	0.0716722i
$518$	0	0
$519$	0	0
$520$	−26.8583	−1.17782
$521$	20.0638	0.879012	0.439506	−	0.898240i	$-0.355154\pi$
$521$	20.0638	0.879012	0.439506	+	0.898240i	$0.355154\pi$
$522$	0	0
$523$	− 0.242504i	− 0.0106040i	−0.999986	−	0.00530198i	$-0.998312\pi$
$523$	− 0.242504i	− 0.0106040i	0.999986	−	0.00530198i	$-0.00168768\pi$
$524$	−10.7534	−0.469764
$525$	0	0
$526$	31.7022	1.38228
$527$	− 57.6841i	− 2.51276i
$528$	0	0
$529$	22.4488	0.976035
$530$	20.9147	0.908479
$531$	0	0
$532$	0	0
$533$	− 10.1037i	− 0.437638i
$534$	0	0
$535$	43.1653i	1.86620i
$536$	47.2850i	2.04240i
$537$	0	0
$538$	− 1.52959i	− 0.0659455i
$539$	0	0
$540$	0	0
$541$	4.51687	0.194195	0.0970977	−	0.995275i	$-0.469044\pi$
$541$	4.51687	0.194195	0.0970977	+	0.995275i	$0.469044\pi$
$542$	−16.4902	−0.708313
$543$	0	0
$544$	29.2231i	1.25293i
$545$	−30.0651	−1.28785
$546$	0	0
$547$	28.5555	1.22094	0.610472	−	0.792038i	$-0.290980\pi$
$547$	28.5555	1.22094	0.610472	+	0.792038i	$0.290980\pi$
$548$	1.32825i	0.0567402i
$549$	0	0
$550$	−4.07587	−0.173796
$551$	−3.86620	−0.164706
$552$	0	0
$553$	0	0
$554$	1.82035i	0.0773392i
$555$	0	0
$556$	17.7391i	0.752307i
$557$	− 28.7904i	− 1.21989i	−0.792445	−	0.609944i	$-0.791192\pi$
$557$	− 28.7904i	− 1.21989i	0.792445	−	0.609944i	$-0.208808\pi$
$558$	0	0
$559$	15.2253i	0.643962i
$560$	0	0
$561$	0	0
$562$	−4.57991	−0.193192
$563$	−29.9049	−1.26034	−0.630171	−	0.776456i	$-0.717015\pi$
$563$	−29.9049	−1.26034	−0.630171	+	0.776456i	$0.717015\pi$
$564$	0	0
$565$	− 18.9782i	− 0.798419i
$566$	9.33023	0.392179
$567$	0	0
$568$	−27.0039	−1.13306
$569$	25.6498i	1.07529i	0.843170	+	0.537647i	$0.180687\pi$
$569$	25.6498i	1.07529i	−0.843170	+	0.537647i	$0.819313\pi$
$570$	0	0
$571$	30.5885	1.28009	0.640045	−	0.768338i	$-0.278916\pi$
$571$	30.5885	1.28009	0.640045	+	0.768338i	$0.278916\pi$
$572$	1.65047	0.0690095
$573$	0	0
$574$	0	0
$575$	4.06594i	0.169561i
$576$	0	0
$577$	− 33.9163i	− 1.41195i	−0.708236	−	0.705976i	$-0.750509\pi$
$577$	− 33.9163i	− 1.41195i	0.708236	−	0.705976i	$-0.249491\pi$
$578$	26.8908i	1.11851i
$579$	0	0
$580$	− 18.6826i	− 0.775751i
$581$	0	0
$582$	0	0
$583$	−4.24738	−0.175908
$584$	−43.4347	−1.79734
$585$	0	0
$586$	33.5292i	1.38508i
$587$	−13.6861	−0.564885	−0.282443	−	0.959284i	$-0.591145\pi$
$587$	−13.6861	−0.564885	−0.282443	+	0.959284i	$0.591145\pi$
$588$	0	0
$589$	5.15693	0.212487
$590$	− 1.22532i	− 0.0504455i
$591$	0	0
$592$	−2.52106	−0.103615
$593$	45.1939	1.85589	0.927945	−	0.372716i	$-0.121574\pi$
$593$	45.1939	1.85589	0.927945	+	0.372716i	$0.121574\pi$
$594$	0	0
$595$	0	0
$596$	13.9453i	0.571222i
$597$	0	0
$598$	2.14822i	0.0878474i
$599$	− 35.8228i	− 1.46368i	−0.681476	−	0.731841i	$-0.738661\pi$
$599$	− 35.8228i	− 1.46368i	0.681476	−	0.731841i	$-0.261339\pi$
$600$	0	0
$601$	2.05781i	0.0839400i	0.999119	+	0.0419700i	$0.0133634\pi$
$601$	2.05781i	0.0839400i	−0.999119	+	0.0419700i	$0.986637\pi$
$602$	0	0
$603$	0	0
$604$	10.1099	0.411364
$605$	−34.0208	−1.38314
$606$	0	0
$607$	22.8987i	0.929428i	0.885461	+	0.464714i	$0.153843\pi$
$607$	22.8987i	0.929428i	−0.885461	+	0.464714i	$0.846157\pi$
$608$	−2.61253	−0.105952
$609$	0	0
$610$	40.4168	1.63643
$611$	− 6.33591i	− 0.256324i
$612$	0	0
$613$	−8.20196	−0.331274	−0.165637	−	0.986187i	$-0.552968\pi$
$613$	−8.20196	−0.331274	−0.165637	+	0.986187i	$0.552968\pi$
$614$	2.27503	0.0918129
$615$	0	0
$616$	0	0
$617$	− 30.5639i	− 1.23046i	−0.788349	−	0.615228i	$-0.789064\pi$
$617$	− 30.5639i	− 1.23046i	0.788349	−	0.615228i	$-0.210936\pi$
$618$	0	0
$619$	47.4160i	1.90581i	0.303270	+	0.952905i	$0.401922\pi$
$619$	47.4160i	1.90581i	−0.303270	+	0.952905i	$0.598078\pi$
$620$	24.9197i	1.00080i
$621$	0	0
$622$	12.1478i	0.487083i
$623$	0	0
$624$	0	0
$625$	−22.3902	−0.895606
$626$	9.99854	0.399622
$627$	0	0
$628$	13.2028i	0.526847i
$629$	10.8447	0.432405
$630$	0	0
$631$	34.1614	1.35994	0.679972	−	0.733238i	$-0.261992\pi$
$631$	34.1614	1.35994	0.679972	+	0.733238i	$0.261992\pi$
$632$	− 2.36726i	− 0.0941647i
$633$	0	0
$634$	−6.42107	−0.255013
$635$	32.9833	1.30890
$636$	0	0
$637$	0	0
$638$	− 4.95036i	− 0.195987i
$639$	0	0
$640$	− 2.21332i	− 0.0874891i
$641$	33.9597i	1.34133i	0.741761	+	0.670664i	$0.233991\pi$
$641$	33.9597i	1.34133i	−0.741761	+	0.670664i	$0.766009\pi$
$642$	0	0
$643$	27.2852i	1.07602i	0.842937	+	0.538012i	$0.180824\pi$
$643$	27.2852i	1.07602i	−0.842937	+	0.538012i	$0.819176\pi$
$644$	0	0
$645$	0	0
$646$	−4.02118	−0.158211
$647$	15.1250	0.594625	0.297313	−	0.954780i	$-0.403910\pi$
$647$	15.1250	0.594625	0.297313	+	0.954780i	$0.403910\pi$
$648$	0	0
$649$	0.248838i	0.00976773i
$650$	15.8465	0.621551
$651$	0	0
$652$	15.3886	0.602663
$653$	34.6919i	1.35760i	0.734325	+	0.678799i	$0.237499\pi$
$653$	34.6919i	1.35760i	−0.734325	+	0.678799i	$0.762501\pi$
$654$	0	0
$655$	40.1099	1.56722
$656$	−5.61581	−0.219260
$657$	0	0
$658$	0	0
$659$	− 40.0377i	− 1.55965i	−0.625998	−	0.779824i	$-0.715308\pi$
$659$	− 40.0377i	− 1.55965i	0.625998	−	0.779824i	$-0.284692\pi$
$660$	0	0
$661$	24.0584i	0.935763i	0.883791	+	0.467881i	$0.154983\pi$
$661$	24.0584i	0.935763i	−0.883791	+	0.467881i	$0.845017\pi$
$662$	− 4.80258i	− 0.186658i
$663$	0	0
$664$	− 50.3673i	− 1.95463i
$665$	0	0
$666$	0	0
$667$	−4.93830	−0.191212
$668$	10.4447	0.404116
$669$	0	0
$670$	− 53.3690i	− 2.06182i
$671$	−8.20786	−0.316861
$672$	0	0
$673$	−15.1204	−0.582847	−0.291423	−	0.956594i	$-0.594129\pi$
$673$	−15.1204	−0.582847	−0.291423	+	0.956594i	$0.594129\pi$
$674$	− 0.279157i	− 0.0107527i
$675$	0	0
$676$	4.86416	0.187083
$677$	7.65304	0.294130	0.147065	−	0.989127i	$-0.453017\pi$
$677$	7.65304	0.294130	0.147065	+	0.989127i	$0.453017\pi$
$678$	0	0
$679$	0	0
$680$	− 64.2164i	− 2.46259i
$681$	0	0
$682$	6.60304i	0.252843i
$683$	19.1173i	0.731502i	0.930713	+	0.365751i	$0.119188\pi$
$683$	19.1173i	0.731502i	−0.930713	+	0.365751i	$0.880812\pi$
$684$	0	0
$685$	− 4.95435i	− 0.189296i
$686$	0	0
$687$	0	0
$688$	8.46253	0.322631
$689$	16.5133	0.629107
$690$	0	0
$691$	− 21.1450i	− 0.804392i	−0.915554	−	0.402196i	$-0.868247\pi$
$691$	− 21.1450i	− 0.804392i	0.915554	−	0.402196i	$-0.131753\pi$
$692$	−9.54025	−0.362666
$693$	0	0
$694$	−6.10010	−0.231557
$695$	− 66.1665i	− 2.50984i
$696$	0	0
$697$	24.1571	0.915016
$698$	12.4583	0.471553
$699$	0	0
$700$	0	0
$701$	− 5.99602i	− 0.226466i	−0.993568	−	0.113233i	$-0.963879\pi$
$701$	− 5.99602i	− 0.226466i	0.993568	−	0.113233i	$-0.0361207\pi$
$702$	0	0
$703$	0.969507i	0.0365657i
$704$	− 5.45944i	− 0.205760i
$705$	0	0
$706$	9.84525i	0.370531i
$707$	0	0
$708$	0	0
$709$	29.2002	1.09664	0.548318	−	0.836270i	$-0.315268\pi$
$709$	29.2002	1.09664	0.548318	+	0.836270i	$0.315268\pi$
$710$	30.4784	1.14383
$711$	0	0
$712$	− 11.6987i	− 0.438428i
$713$	6.58695	0.246683
$714$	0	0
$715$	−6.15620	−0.230229
$716$	− 9.77315i	− 0.365240i
$717$	0	0
$718$	−21.0911	−0.787113
$719$	7.31249	0.272710	0.136355	−	0.990660i	$-0.456461\pi$
$719$	7.31249	0.272710	0.136355	+	0.990660i	$0.456461\pi$
$720$	0	0
$721$	0	0
$722$	19.8577i	0.739028i
$723$	0	0
$724$	1.43960i	0.0535022i
$725$	36.4276i	1.35289i
$726$	0	0
$727$	18.8932i	0.700710i	0.936617	+	0.350355i	$0.113939\pi$
$727$	18.8932i	0.700710i	−0.936617	+	0.350355i	$0.886061\pi$
$728$	0	0
$729$	0	0
$730$	49.0233	1.81443
$731$	−36.4027	−1.34640
$732$	0	0
$733$	35.2662i	1.30259i	0.758826	+	0.651294i	$0.225773\pi$
$733$	35.2662i	1.30259i	−0.758826	+	0.651294i	$0.774227\pi$
$734$	−1.98091	−0.0731169
$735$	0	0
$736$	−3.33699	−0.123003
$737$	10.8382i	0.399230i
$738$	0	0
$739$	39.4234	1.45021	0.725106	−	0.688637i	$-0.241791\pi$
$739$	39.4234	1.45021	0.725106	+	0.688637i	$0.241791\pi$
$740$	−4.68494	−0.172222
$741$	0	0
$742$	0	0
$743$	− 16.3303i	− 0.599101i	−0.954080	−	0.299551i	$-0.903163\pi$
$743$	− 16.3303i	− 0.599101i	0.954080	−	0.299551i	$-0.0968368\pi$
$744$	0	0
$745$	− 52.0156i	− 1.90570i
$746$	− 27.1527i	− 0.994131i
$747$	0	0
$748$	3.94615i	0.144286i
$749$	0	0
$750$	0	0
$751$	−10.3426	−0.377406	−0.188703	−	0.982034i	$-0.560428\pi$
$751$	−10.3426	−0.377406	−0.188703	+	0.982034i	$0.560428\pi$
$752$	−3.52162	−0.128420
$753$	0	0
$754$	19.2464i	0.700914i
$755$	−37.7095	−1.37239
$756$	0	0
$757$	22.1500	0.805054	0.402527	−	0.915408i	$-0.368132\pi$
$757$	22.1500	0.805054	0.402527	+	0.915408i	$0.368132\pi$
$758$	− 27.3999i	− 0.995209i
$759$	0	0
$760$	5.74091	0.208245
$761$	12.3923	0.449221	0.224610	−	0.974449i	$-0.427889\pi$
$761$	12.3923	0.449221	0.224610	+	0.974449i	$0.427889\pi$
$762$	0	0
$763$	0	0
$764$	7.35386i	0.266053i
$765$	0	0
$766$	13.5227i	0.488593i
$767$	− 0.967452i	− 0.0349327i
$768$	0	0
$769$	− 54.4250i	− 1.96261i	−0.192448	−	0.981307i	$-0.561643\pi$
$769$	− 54.4250i	− 1.96261i	0.192448	−	0.981307i	$-0.438357\pi$
$770$	0	0
$771$	0	0
$772$	−20.4948	−0.737624
$773$	35.6261	1.28138	0.640690	−	0.767800i	$-0.278648\pi$
$773$	35.6261	1.28138	0.640690	+	0.767800i	$0.278648\pi$
$774$	0	0
$775$	− 48.5890i	− 1.74537i
$776$	31.9215	1.14591
$777$	0	0
$778$	29.6092	1.06154
$779$	2.15963i	0.0773769i
$780$	0	0
$781$	−6.18956	−0.221480
$782$	−5.13626	−0.183672
$783$	0	0
$784$	0	0
$785$	− 49.2459i	− 1.75766i
$786$	0	0
$787$	− 29.8669i	− 1.06464i	−0.846543	−	0.532320i	$-0.821320\pi$
$787$	− 29.8669i	− 1.06464i	0.846543	−	0.532320i	$-0.178680\pi$
$788$	17.5255i	0.624322i
$789$	0	0
$790$	2.67185i	0.0950601i
$791$	0	0
$792$	0	0
$793$	31.9112	1.13320
$794$	14.6976	0.521599
$795$	0	0
$796$	1.35095i	0.0478833i
$797$	−39.0290	−1.38248	−0.691239	−	0.722626i	$-0.742935\pi$
$797$	−39.0290	−1.38248	−0.691239	+	0.722626i	$0.742935\pi$
$798$	0	0
$799$	15.1487	0.535923
$800$	24.6155i	0.870288i
$801$	0	0
$802$	2.41945	0.0854337
$803$	−9.95567	−0.351328
$804$	0	0
$805$	0	0
$806$	− 25.6718i	− 0.904252i
$807$	0	0
$808$	− 48.1046i	− 1.69231i
$809$	− 15.5843i	− 0.547914i	−0.961742	−	0.273957i	$-0.911667\pi$
$809$	− 15.5843i	− 0.547914i	0.961742	−	0.273957i	$-0.0883325\pi$
$810$	0	0
$811$	− 19.2285i	− 0.675206i	−0.941289	−	0.337603i	$-0.890384\pi$
$811$	− 19.2285i	− 0.675206i	0.941289	−	0.337603i	$-0.109616\pi$
$812$	0	0
$813$	0	0
$814$	−1.24138	−0.0435103
$815$	−57.3990	−2.01060
$816$	0	0
$817$	− 3.25438i	− 0.113856i
$818$	18.9462	0.662437
$819$	0	0
$820$	−10.4360	−0.364440
$821$	− 5.35524i	− 0.186899i	−0.995624	−	0.0934496i	$-0.970211\pi$
$821$	− 5.35524i	− 0.186899i	0.995624	−	0.0934496i	$-0.0297894\pi$
$822$	0	0
$823$	0.0188996	0.000658797 0	0.000329399	−	1.00000i	$-0.499895\pi$
$823$	0.0188996	0.000658797 0	0.000329399		1.00000i	$0.499895\pi$
$824$	−41.3408	−1.44017
$825$	0	0
$826$	0	0
$827$	− 13.3217i	− 0.463240i	−0.972806	−	0.231620i	$-0.925597\pi$
$827$	− 13.3217i	− 0.463240i	0.972806	−	0.231620i	$-0.0744026\pi$
$828$	0	0
$829$	− 8.78111i	− 0.304981i	−0.988305	−	0.152490i	$-0.951271\pi$
$829$	− 8.78111i	− 0.304981i	0.988305	−	0.152490i	$-0.0487293\pi$
$830$	56.8478i	1.97322i
$831$	0	0
$832$	21.2257i	0.735868i
$833$	0	0
$834$	0	0
$835$	−38.9583	−1.34821
$836$	−0.352784	−0.0122013
$837$	0	0
$838$	− 5.16940i	− 0.178574i
$839$	22.2873	0.769441	0.384721	−	0.923033i	$-0.374298\pi$
$839$	22.2873	0.769441	0.384721	+	0.923033i	$0.374298\pi$
$840$	0	0
$841$	−15.2433	−0.525633
$842$	− 7.87848i	− 0.271510i
$843$	0	0
$844$	6.00821	0.206811
$845$	−18.1432	−0.624144
$846$	0	0
$847$	0	0
$848$	− 9.17842i	− 0.315188i
$849$	0	0
$850$	37.8879i	1.29954i
$851$	1.23835i	0.0424502i
$852$	0	0
$853$	− 2.41899i	− 0.0828248i	−0.999142	−	0.0414124i	$-0.986814\pi$
$853$	− 2.41899i	− 0.0828248i	0.999142	−	0.0414124i	$-0.0131857\pi$
$854$	0	0
$855$	0	0
$856$	40.6946	1.39091
$857$	27.5213	0.940109	0.470054	−	0.882637i	$-0.344234\pi$
$857$	27.5213	0.940109	0.470054	+	0.882637i	$0.344234\pi$
$858$	0	0
$859$	25.5133i	0.870502i	0.900309	+	0.435251i	$0.143340\pi$
$859$	25.5133i	0.870502i	−0.900309	+	0.435251i	$0.856660\pi$
$860$	15.7261	0.536255
$861$	0	0
$862$	−33.0188	−1.12462
$863$	27.2351i	0.927093i	0.886073	+	0.463547i	$0.153423\pi$
$863$	27.2351i	0.927093i	−0.886073	+	0.463547i	$0.846577\pi$
$864$	0	0
$865$	35.5849	1.20992
$866$	28.7984	0.978608
$867$	0	0
$868$	0	0
$869$	− 0.542600i	− 0.0184064i
$870$	0	0
$871$	− 42.1377i	− 1.42778i
$872$	28.3442i	0.959857i
$873$	0	0
$874$	− 0.459178i	− 0.0155319i
$875$	0	0
$876$	0	0
$877$	−47.5486	−1.60560	−0.802800	−	0.596248i	$-0.796658\pi$
$877$	−47.5486	−1.60560	−0.802800	+	0.596248i	$0.796658\pi$
$878$	28.6717	0.967622
$879$	0	0
$880$	3.42174i	0.115347i
$881$	−28.6217	−0.964290	−0.482145	−	0.876092i	$-0.660142\pi$
$881$	−28.6217	−0.964290	−0.482145	+	0.876092i	$0.660142\pi$
$882$	0	0
$883$	6.03859	0.203215	0.101607	−	0.994825i	$-0.467601\pi$
$883$	6.03859	0.203215	0.101607	+	0.994825i	$0.467601\pi$
$884$	− 15.3422i	− 0.516013i
$885$	0	0
$886$	−0.694761	−0.0233409
$887$	−28.9776	−0.972974	−0.486487	−	0.873688i	$-0.661722\pi$
$887$	−28.9776	−0.972974	−0.486487	+	0.873688i	$0.661722\pi$
$888$	0	0
$889$	0	0
$890$	13.2039i	0.442597i
$891$	0	0
$892$	21.0677i	0.705400i
$893$	1.35429i	0.0453195i
$894$	0	0
$895$	36.4536i	1.21851i
$896$	0	0
$897$	0	0
$898$	−24.8006	−0.827606
$899$	59.0140	1.96823
$900$	0	0
$901$	39.4822i	1.31534i
$902$	−2.76524	−0.0920724
$903$	0	0
$904$	−17.8920	−0.595078
$905$	− 5.36966i	− 0.178494i
$906$	0	0
$907$	32.2651	1.07134	0.535672	−	0.844426i	$-0.320058\pi$
$907$	32.2651	1.07134	0.535672	+	0.844426i	$0.320058\pi$
$908$	4.98278	0.165359
$909$	0	0
$910$	0	0
$911$	3.28593i	0.108868i	0.998517	+	0.0544338i	$0.0173354\pi$
$911$	3.28593i	0.108868i	−0.998517	+	0.0544338i	$0.982665\pi$
$912$	0	0
$913$	− 11.5447i	− 0.382073i
$914$	− 38.8072i	− 1.28363i
$915$	0	0
$916$	17.6068i	0.581744i
$917$	0	0
$918$	0	0
$919$	50.1217	1.65336	0.826681	−	0.562670i	$-0.190226\pi$
$919$	50.1217	1.65336	0.826681	+	0.562670i	$0.190226\pi$
$920$	7.33288	0.241758
$921$	0	0
$922$	28.0570i	0.924008i
$923$	24.0643	0.792086
$924$	0	0
$925$	9.13478	0.300350
$926$	18.6848i	0.614021i
$927$	0	0
$928$	−29.8968	−0.981411
$929$	−23.9594	−0.786081	−0.393041	−	0.919521i	$-0.628577\pi$
$929$	−23.9594	−0.786081	−0.393041	+	0.919521i	$0.628577\pi$
$930$	0	0
$931$	0	0
$932$	− 10.7416i	− 0.351852i
$933$	0	0
$934$	− 26.6022i	− 0.870450i
$935$	− 14.7190i	− 0.481364i
$936$	0	0
$937$	31.8400i	1.04017i	0.854115	+	0.520084i	$0.174099\pi$
$937$	31.8400i	1.04017i	−0.854115	+	0.520084i	$0.825901\pi$
$938$	0	0
$939$	0	0
$940$	−6.54430	−0.213452
$941$	11.9469	0.389457	0.194729	−	0.980857i	$-0.437617\pi$
$941$	11.9469	0.389457	0.194729	+	0.980857i	$0.437617\pi$
$942$	0	0
$943$	2.75850i	0.0898293i
$944$	−0.537729	−0.0175016
$945$	0	0
$946$	4.16698	0.135480
$947$	− 29.7866i	− 0.967935i	−0.875086	−	0.483968i	$-0.839195\pi$
$947$	− 29.7866i	− 0.967935i	0.875086	−	0.483968i	$-0.160805\pi$
$948$	0	0
$949$	38.7065	1.25647
$950$	−3.38715	−0.109894
$951$	0	0
$952$	0	0
$953$	− 39.8959i	− 1.29235i	−0.763188	−	0.646177i	$-0.776367\pi$
$953$	− 39.8959i	− 1.29235i	0.763188	−	0.646177i	$-0.223633\pi$
$954$	0	0
$955$	− 27.4297i	− 0.887604i
$956$	− 15.8495i	− 0.512609i
$957$	0	0
$958$	− 1.84614i	− 0.0596462i
$959$	0	0
$960$	0	0
$961$	−47.7158	−1.53922
$962$	4.82633	0.155607
$963$	0	0
$964$	− 20.2088i	− 0.650882i
$965$	76.4451	2.46085
$966$	0	0
$967$	−45.8294	−1.47378	−0.736888	−	0.676015i	$-0.763705\pi$
$967$	−45.8294	−1.47378	−0.736888	+	0.676015i	$0.763705\pi$
$968$	32.0736i	1.03088i
$969$	0	0
$970$	−36.0287	−1.15681
$971$	−35.9104	−1.15242	−0.576209	−	0.817302i	$-0.695469\pi$
$971$	−35.9104	−1.15242	−0.576209	+	0.817302i	$0.695469\pi$
$972$	0	0
$973$	0	0
$974$	− 13.6984i	− 0.438925i
$975$	0	0
$976$	− 17.7369i	− 0.567744i
$977$	10.7437i	0.343722i	0.985121	+	0.171861i	$0.0549780\pi$
$977$	10.7437i	0.343722i	−0.985121	+	0.171861i	$0.945022\pi$
$978$	0	0
$979$	− 2.68146i	− 0.0856998i
$980$	0	0
$981$	0	0
$982$	41.2700	1.31698
$983$	−32.9206	−1.05000	−0.525002	−	0.851101i	$-0.675936\pi$
$983$	−32.9206	−1.05000	−0.525002	+	0.851101i	$0.675936\pi$
$984$	0	0
$985$	− 65.3698i	− 2.08286i
$986$	−46.0169	−1.46548
$987$	0	0
$988$	1.37158	0.0436358
$989$	− 4.15682i	− 0.132179i
$990$	0	0
$991$	16.8152	0.534151	0.267076	−	0.963676i	$-0.413943\pi$
$991$	16.8152	0.534151	0.267076	+	0.963676i	$0.413943\pi$
$992$	39.8779	1.26612
$993$	0	0
$994$	0	0
$995$	− 5.03902i	− 0.159748i
$996$	0	0
$997$	40.9743i	1.29767i	0.760930	+	0.648834i	$0.224743\pi$
$997$	40.9743i	1.29767i	−0.760930	+	0.648834i	$0.775257\pi$
$998$	23.9592i	0.758416i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.c.c.3086.8	yes	24
3.2	odd	2	inner	3087.2.c.c.3086.17	yes	24
7.6	odd	2	inner	3087.2.c.c.3086.18	yes	24
21.20	even	2	inner	3087.2.c.c.3086.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3087.2.c.c.3086.7	✓	24	21.20	even	2	inner
3087.2.c.c.3086.8	yes	24	1.1	even	1	trivial
3087.2.c.c.3086.17	yes	24	3.2	odd	2	inner
3087.2.c.c.3086.18	yes	24	7.6	odd	2	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:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.06406i q^{2} +0.867767 q^{4} -3.23675 q^{5} +3.05149i q^{8} +O(q^{10})\)	\(q+1.06406i q^{2} +0.867767 q^{4} -3.23675 q^{5} +3.05149i q^{8} -3.44411i q^{10} +0.699431i q^{11} -2.71931i q^{13} -1.51144 q^{16} +6.50168 q^{17} +0.581246i q^{19} -2.80875 q^{20} -0.744240 q^{22} +0.742427i q^{23} +5.47655 q^{25} +2.89352 q^{26} +6.65157i q^{29} -8.87219i q^{31} +4.49470i q^{32} +6.91820i q^{34} +1.66798 q^{37} -0.618483 q^{38} -9.87691i q^{40} +3.71552 q^{41} -5.59897 q^{43} +0.606944i q^{44} -0.789990 q^{46} +2.32997 q^{47} +5.82740i q^{50} -2.35973i q^{52} +6.07261i q^{53} -2.26388i q^{55} -7.07769 q^{58} +0.355771 q^{59} +11.7351i q^{61} +9.44058 q^{62} -7.80554 q^{64} +8.80172i q^{65} +15.4957 q^{67} +5.64194 q^{68} +8.84942i q^{71} +14.2339i q^{73} +1.77484i q^{74} +0.504386i q^{76} -0.775773 q^{79} +4.89217 q^{80} +3.95356i q^{82} -16.5058 q^{83} -21.0443 q^{85} -5.95766i q^{86} -2.13431 q^{88} -3.83377 q^{89} +0.644254i q^{92} +2.47924i q^{94} -1.88135i q^{95} -10.4610i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) \) 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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