LMFDB - Embedded newform 1900.2.c.f.1749.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.c (of order $2$, degree $1$, 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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.6594624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 18x^{2} + 1 $ x^6 + 9x^4 + 18x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.4
Root			$2.46050i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.f.1749.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59358i q^{3} +4.51459i q^{7} +0.460505 q^{9} +O(q^{10})$	$q+1.59358i q^{3} +4.51459i q^{7} +0.460505 q^{9} +0.593579 q^{11} +4.05408i q^{13} -5.32743i q^{17} -1.00000 q^{19} -7.19436 q^{21} +7.46050i q^{23} +5.51459i q^{27} +2.32743 q^{29} +6.97509 q^{31} +0.945916i q^{33} +3.40642i q^{37} -6.46050 q^{39} -7.38151 q^{41} +6.18716i q^{43} -12.7089i q^{47} -13.3815 q^{49} +8.48968 q^{51} -8.84202i q^{53} -1.59358i q^{57} -4.10817 q^{59} +9.89610 q^{61} +2.07899i q^{63} -12.0613i q^{67} -11.8889 q^{69} -7.10817 q^{71} +5.27335i q^{73} +2.67977i q^{77} +3.96790 q^{79} -7.40642 q^{81} +7.53950i q^{83} +3.70895i q^{87} -9.19436 q^{89} -18.3025 q^{91} +11.1154i q^{93} -6.42840i q^{97} +0.273346 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{9}+O(q^{10}) $ 6 * q - 10 * q^9	$ 6 q - 10 q^{9} - 2 q^{11} - 6 q^{19} - 16 q^{21} - 6 q^{29} - 2 q^{31} - 26 q^{39} - 6 q^{41} - 42 q^{49} - 24 q^{51} + 12 q^{59} - 10 q^{61} - 18 q^{69} - 6 q^{71} - 4 q^{79} - 50 q^{81} - 28 q^{89} - 46 q^{91}+O(q^{100}) $ 6 * q - 10 * q^9 - 2 * q^11 - 6 * q^19 - 16 * q^21 - 6 * q^29 - 2 * q^31 - 26 * q^39 - 6 * q^41 - 42 * q^49 - 24 * q^51 + 12 * q^59 - 10 * q^61 - 18 * q^69 - 6 * q^71 - 4 * q^79 - 50 * q^81 - 28 * q^89 - 46 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	1.59358i	0.920053i	0.887905	+	0.460027i	$0.152160\pi$
$3$	1.59358i	0.920053i	−0.887905	+	0.460027i	$0.847840\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.51459i	1.70635i	0.521621	+	0.853177i	$0.325327\pi$
$7$	4.51459i	1.70635i	−0.521621	+	0.853177i	$0.674673\pi$
$8$	0	0
$9$	0.460505	0.153502
$10$	0	0
$11$	0.593579	0.178971	0.0894855	−	0.995988i	$-0.471478\pi$
$11$	0.593579	0.178971	0.0894855	+	0.995988i	$0.471478\pi$
$12$	0	0
$13$	4.05408i	1.12440i	0.827001	+	0.562200i	$0.190045\pi$
$13$	4.05408i	1.12440i	−0.827001	+	0.562200i	$0.809955\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.32743i	− 1.29209i	−0.763299	−	0.646046i	$-0.776421\pi$
$17$	− 5.32743i	− 1.29209i	0.763299	−	0.646046i	$-0.223579\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−7.19436	−1.56994
$22$	0	0
$23$	7.46050i	1.55562i	0.628498	+	0.777811i	$0.283670\pi$
$23$	7.46050i	1.55562i	−0.628498	+	0.777811i	$0.716330\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.51459i	1.06128i
$28$	0	0
$29$	2.32743	0.432193	0.216096	−	0.976372i	$-0.430667\pi$
$29$	2.32743	0.432193	0.216096	+	0.976372i	$0.430667\pi$
$30$	0	0
$31$	6.97509	1.25276	0.626382	−	0.779516i	$-0.284535\pi$
$31$	6.97509	1.25276	0.626382	+	0.779516i	$0.284535\pi$
$32$	0	0
$33$	0.945916i	0.164663i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.40642i	0.560012i	0.959998	+	0.280006i	$0.0903365\pi$
$37$	3.40642i	0.560012i	−0.959998	+	0.280006i	$0.909664\pi$
$38$	0	0
$39$	−6.46050	−1.03451
$40$	0	0
$41$	−7.38151	−1.15280	−0.576399	−	0.817168i	$-0.695543\pi$
$41$	−7.38151	−1.15280	−0.576399	+	0.817168i	$0.695543\pi$
$42$	0	0
$43$	6.18716i	0.943533i	0.881724	+	0.471766i	$0.156383\pi$
$43$	6.18716i	0.943533i	−0.881724	+	0.471766i	$0.843617\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.7089i	− 1.85379i	−0.375321	−	0.926895i	$-0.622467\pi$
$47$	− 12.7089i	− 1.85379i	0.375321	−	0.926895i	$-0.377533\pi$
$48$	0	0
$49$	−13.3815	−1.91164
$50$	0	0
$51$	8.48968	1.18879
$52$	0	0
$53$	− 8.84202i	− 1.21454i	−0.794494	−	0.607272i	$-0.792264\pi$
$53$	− 8.84202i	− 1.21454i	0.794494	−	0.607272i	$-0.207736\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.59358i	− 0.211075i
$58$	0	0
$59$	−4.10817	−0.534838	−0.267419	−	0.963580i	$-0.586171\pi$
$59$	−4.10817	−0.534838	−0.267419	+	0.963580i	$0.586171\pi$
$60$	0	0
$61$	9.89610	1.26707	0.633533	−	0.773716i	$-0.281604\pi$
$61$	9.89610	1.26707	0.633533	+	0.773716i	$0.281604\pi$
$62$	0	0
$63$	2.07899i	0.261928i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0613i	− 1.47352i	−0.676154	−	0.736760i	$-0.736355\pi$
$67$	− 12.0613i	− 1.47352i	0.676154	−	0.736760i	$-0.263645\pi$
$68$	0	0
$69$	−11.8889	−1.43126
$70$	0	0
$71$	−7.10817	−0.843584	−0.421792	−	0.906693i	$-0.638599\pi$
$71$	−7.10817	−0.843584	−0.421792	+	0.906693i	$0.638599\pi$
$72$	0	0
$73$	5.27335i	0.617198i	0.951192	+	0.308599i	$0.0998602\pi$
$73$	5.27335i	0.617198i	−0.951192	+	0.308599i	$0.900140\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.67977i	0.305388i
$78$	0	0
$79$	3.96790	0.446423	0.223212	−	0.974770i	$-0.428346\pi$
$79$	3.96790	0.446423	0.223212	+	0.974770i	$0.428346\pi$
$80$	0	0
$81$	−7.40642	−0.822936
$82$	0	0
$83$	7.53950i	0.827567i	0.910375	+	0.413784i	$0.135793\pi$
$83$	7.53950i	0.827567i	−0.910375	+	0.413784i	$0.864207\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.70895i	0.397641i
$88$	0	0
$89$	−9.19436	−0.974600	−0.487300	−	0.873235i	$-0.662018\pi$
$89$	−9.19436	−0.974600	−0.487300	+	0.873235i	$0.662018\pi$
$90$	0	0
$91$	−18.3025	−1.91863
$92$	0	0
$93$	11.1154i	1.15261i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.42840i	− 0.652705i	−0.945248	−	0.326353i	$-0.894180\pi$
$97$	− 6.42840i	− 0.652705i	0.945248	−	0.326353i	$-0.105820\pi$
$98$	0	0
$99$	0.273346	0.0274723
$100$	0	0
$101$	−11.8099	−1.17513	−0.587565	−	0.809177i	$-0.699913\pi$
$101$	−11.8099	−1.17513	−0.587565	+	0.809177i	$0.699913\pi$
$102$	0	0
$103$	18.4648i	1.81939i	0.415279	+	0.909694i	$0.363684\pi$
$103$	18.4648i	1.81939i	−0.415279	+	0.909694i	$0.636316\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.431327i	− 0.0416979i	−0.999783	−	0.0208490i	$-0.993363\pi$
$107$	− 0.431327i	− 0.0416979i	0.999783	−	0.0208490i	$-0.00663691\pi$
$108$	0	0
$109$	7.11537	0.681528	0.340764	−	0.940149i	$-0.389314\pi$
$109$	7.11537	0.681528	0.340764	+	0.940149i	$0.389314\pi$
$110$	0	0
$111$	−5.42840	−0.515241
$112$	0	0
$113$	− 3.16225i	− 0.297480i	−0.988876	−	0.148740i	$-0.952478\pi$
$113$	− 3.16225i	− 0.297480i	0.988876	−	0.148740i	$-0.0475217\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.86693i	0.172597i
$118$	0	0
$119$	24.0512	2.20477
$120$	0	0
$121$	−10.6477	−0.967969
$122$	0	0
$123$	− 11.7630i	− 1.06064i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.4107i	1.36748i	0.729727	+	0.683739i	$0.239647\pi$
$127$	15.4107i	1.36748i	−0.729727	+	0.683739i	$0.760353\pi$
$128$	0	0
$129$	−9.85973	−0.868101
$130$	0	0
$131$	11.0072	0.961703	0.480852	−	0.876802i	$-0.340328\pi$
$131$	11.0072	0.961703	0.480852	+	0.876802i	$0.340328\pi$
$132$	0	0
$133$	− 4.51459i	− 0.391465i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.72665i	− 0.745568i	−0.927918	−	0.372784i	$-0.878403\pi$
$137$	− 8.72665i	− 0.745568i	0.927918	−	0.372784i	$-0.121597\pi$
$138$	0	0
$139$	7.64047	0.648056	0.324028	−	0.946048i	$-0.394963\pi$
$139$	7.64047	0.648056	0.324028	+	0.946048i	$0.394963\pi$
$140$	0	0
$141$	20.2527	1.70559
$142$	0	0
$143$	2.40642i	0.201235i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 21.3245i	− 1.75882i
$148$	0	0
$149$	6.67684	0.546988	0.273494	−	0.961874i	$-0.411821\pi$
$149$	6.67684	0.546988	0.273494	+	0.961874i	$0.411821\pi$
$150$	0	0
$151$	−19.1623	−1.55940	−0.779701	−	0.626152i	$-0.784629\pi$
$151$	−19.1623	−1.55940	−0.779701	+	0.626152i	$0.784629\pi$
$152$	0	0
$153$	− 2.45331i	− 0.198338i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.7558i	1.33726i	0.743595	+	0.668630i	$0.233119\pi$
$157$	16.7558i	1.33726i	−0.743595	+	0.668630i	$0.766881\pi$
$158$	0	0
$159$	14.0905	1.11745
$160$	0	0
$161$	−33.6811	−2.65444
$162$	0	0
$163$	2.32023i	0.181735i	0.995863	+	0.0908673i	$0.0289639\pi$
$163$	2.32023i	0.181735i	−0.995863	+	0.0908673i	$0.971036\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.56867i	− 0.663064i	−0.943444	−	0.331532i	$-0.892435\pi$
$167$	− 8.56867i	− 0.663064i	0.943444	−	0.331532i	$-0.107565\pi$
$168$	0	0
$169$	−3.43560	−0.264277
$170$	0	0
$171$	−0.460505	−0.0352157
$172$	0	0
$173$	− 9.04689i	− 0.687822i	−0.939002	−	0.343911i	$-0.888248\pi$
$173$	− 9.04689i	− 0.687822i	0.939002	−	0.343911i	$-0.111752\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.54669i	− 0.492080i
$178$	0	0
$179$	−13.4969	−1.00880	−0.504402	−	0.863469i	$-0.668287\pi$
$179$	−13.4969	−1.00880	−0.504402	+	0.863469i	$0.668287\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	15.7702i	1.16577i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.16225i	− 0.231247i
$188$	0	0
$189$	−24.8961	−1.81093
$190$	0	0
$191$	−12.4356	−0.899808	−0.449904	−	0.893077i	$-0.648542\pi$
$191$	−12.4356	−0.899808	−0.449904	+	0.893077i	$0.648542\pi$
$192$	0	0
$193$	1.64766i	0.118601i	0.998240	+	0.0593007i	$0.0188871\pi$
$193$	1.64766i	0.118601i	−0.998240	+	0.0593007i	$0.981113\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.3566i	− 1.52160i	−0.648989	−	0.760798i	$-0.724808\pi$
$197$	− 21.3566i	− 1.52160i	0.648989	−	0.760798i	$-0.275192\pi$
$198$	0	0
$199$	12.2632	0.869317	0.434658	−	0.900595i	$-0.356869\pi$
$199$	12.2632	0.869317	0.434658	+	0.900595i	$0.356869\pi$
$200$	0	0
$201$	19.2206	1.35572
$202$	0	0
$203$	10.5074i	0.737474i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.43560i	0.238791i
$208$	0	0
$209$	−0.593579	−0.0410587
$210$	0	0
$211$	19.3743	1.33378	0.666892	−	0.745155i	$-0.267624\pi$
$211$	19.3743	1.33378	0.666892	+	0.745155i	$0.267624\pi$
$212$	0	0
$213$	− 11.3274i	− 0.776143i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	31.4897i	2.13766i
$218$	0	0
$219$	−8.40350	−0.567856
$220$	0	0
$221$	21.5979	1.45283
$222$	0	0
$223$	− 20.5117i	− 1.37356i	−0.726864	−	0.686781i	$-0.759023\pi$
$223$	− 20.5117i	− 1.37356i	0.726864	−	0.686781i	$-0.240977\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.91381i	0.259769i	0.991529	+	0.129884i	$0.0414606\pi$
$227$	3.91381i	0.259769i	−0.991529	+	0.129884i	$0.958539\pi$
$228$	0	0
$229$	−3.58638	−0.236995	−0.118497	−	0.992954i	$-0.537808\pi$
$229$	−3.58638	−0.236995	−0.118497	+	0.992954i	$0.537808\pi$
$230$	0	0
$231$	−4.27042	−0.280973
$232$	0	0
$233$	12.5294i	0.820827i	0.911900	+	0.410413i	$0.134616\pi$
$233$	12.5294i	0.820827i	−0.911900	+	0.410413i	$0.865384\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.32316i	0.410733i
$238$	0	0
$239$	23.5218	1.52150	0.760749	−	0.649046i	$-0.224832\pi$
$239$	23.5218	1.52150	0.760749	+	0.649046i	$0.224832\pi$
$240$	0	0
$241$	9.22353	0.594140	0.297070	−	0.954856i	$-0.403991\pi$
$241$	9.22353	0.594140	0.297070	+	0.954856i	$0.403991\pi$
$242$	0	0
$243$	4.74105i	0.304138i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.05408i	− 0.257955i
$248$	0	0
$249$	−12.0148	−0.761406
$250$	0	0
$251$	18.7237	1.18183	0.590916	−	0.806733i	$-0.298767\pi$
$251$	18.7237	1.18183	0.590916	+	0.806733i	$0.298767\pi$
$252$	0	0
$253$	4.42840i	0.278411i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.1196i	1.69168i	0.533439	+	0.845838i	$0.320899\pi$
$257$	27.1196i	1.69168i	−0.533439	+	0.845838i	$0.679101\pi$
$258$	0	0
$259$	−15.3786	−0.955579
$260$	0	0
$261$	1.07179	0.0663423
$262$	0	0
$263$	− 18.5510i	− 1.14390i	−0.820288	−	0.571951i	$-0.806187\pi$
$263$	− 18.5510i	− 1.14390i	0.820288	−	0.571951i	$-0.193813\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 14.6519i	− 0.896684i
$268$	0	0
$269$	26.1331	1.59336	0.796681	−	0.604400i	$-0.206587\pi$
$269$	26.1331	1.59336	0.796681	+	0.604400i	$0.206587\pi$
$270$	0	0
$271$	1.95311	0.118643	0.0593216	−	0.998239i	$-0.481106\pi$
$271$	1.95311	0.118643	0.0593216	+	0.998239i	$0.481106\pi$
$272$	0	0
$273$	− 29.1665i	− 1.76524i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.46770i	− 0.148270i	−0.997248	−	0.0741349i	$-0.976380\pi$
$277$	− 2.46770i	− 0.148270i	0.997248	−	0.0741349i	$-0.0236195\pi$
$278$	0	0
$279$	3.21206	0.192301
$280$	0	0
$281$	24.8712	1.48369	0.741846	−	0.670571i	$-0.233951\pi$
$281$	24.8712	1.48369	0.741846	+	0.670571i	$0.233951\pi$
$282$	0	0
$283$	− 9.55389i	− 0.567920i	−0.958836	−	0.283960i	$-0.908352\pi$
$283$	− 9.55389i	− 0.567920i	0.958836	−	0.283960i	$-0.0916483\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 33.3245i	− 1.96708i
$288$	0	0
$289$	−11.3815	−0.669501
$290$	0	0
$291$	10.2442	0.600524
$292$	0	0
$293$	11.2527i	0.657390i	0.944436	+	0.328695i	$0.106609\pi$
$293$	11.2527i	0.657390i	−0.944436	+	0.328695i	$0.893391\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.27335i	0.189939i
$298$	0	0
$299$	−30.2455	−1.74914
$300$	0	0
$301$	−27.9325	−1.61000
$302$	0	0
$303$	− 18.8200i	− 1.08118i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.2877i	− 0.701298i	−0.936507	−	0.350649i	$-0.885961\pi$
$307$	− 12.2877i	− 0.701298i	0.936507	−	0.350649i	$-0.114039\pi$
$308$	0	0
$309$	−29.4251	−1.67393
$310$	0	0
$311$	17.1331	0.971528	0.485764	−	0.874090i	$-0.338542\pi$
$311$	17.1331	0.971528	0.485764	+	0.874090i	$0.338542\pi$
$312$	0	0
$313$	− 4.98229i	− 0.281616i	−0.990037	−	0.140808i	$-0.955030\pi$
$313$	− 4.98229i	− 0.281616i	0.990037	−	0.140808i	$-0.0449700\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.3963i	1.25790i	0.777445	+	0.628951i	$0.216515\pi$
$317$	22.3963i	1.25790i	−0.777445	+	0.628951i	$0.783485\pi$
$318$	0	0
$319$	1.38151	0.0773500
$320$	0	0
$321$	0.687353	0.0383643
$322$	0	0
$323$	5.32743i	0.296426i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.3389i	0.627043i
$328$	0	0
$329$	57.3757	3.16322
$330$	0	0
$331$	4.06887	0.223645	0.111823	−	0.993728i	$-0.464331\pi$
$331$	4.06887	0.223645	0.111823	+	0.993728i	$0.464331\pi$
$332$	0	0
$333$	1.56867i	0.0859628i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.52179i	0.518685i	0.965785	+	0.259342i	$0.0835058\pi$
$337$	9.52179i	0.518685i	−0.965785	+	0.259342i	$0.916494\pi$
$338$	0	0
$339$	5.03930	0.273697
$340$	0	0
$341$	4.14027	0.224208
$342$	0	0
$343$	− 28.8099i	− 1.55559i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.1052i	0.757209i	0.925559	+	0.378605i	$0.123596\pi$
$347$	14.1052i	0.757209i	−0.925559	+	0.378605i	$0.876404\pi$
$348$	0	0
$349$	−15.2704	−0.817407	−0.408703	−	0.912667i	$-0.634019\pi$
$349$	−15.2704	−0.817407	−0.408703	+	0.912667i	$0.634019\pi$
$350$	0	0
$351$	−22.3566	−1.19331
$352$	0	0
$353$	31.6591i	1.68505i	0.538661	+	0.842523i	$0.318930\pi$
$353$	31.6591i	1.68505i	−0.538661	+	0.842523i	$0.681070\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	38.3274i	2.02850i
$358$	0	0
$359$	26.5366	1.40055	0.700273	−	0.713875i	$-0.253061\pi$
$359$	26.5366	1.40055	0.700273	+	0.713875i	$0.253061\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 16.9679i	− 0.890584i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 31.2379i	− 1.63061i	−0.579034	−	0.815303i	$-0.696570\pi$
$367$	− 31.2379i	− 1.63061i	0.579034	−	0.815303i	$-0.303430\pi$
$368$	0	0
$369$	−3.39922	−0.176957
$370$	0	0
$371$	39.9181	2.07244
$372$	0	0
$373$	− 29.8961i	− 1.54796i	−0.633209	−	0.773981i	$-0.718263\pi$
$373$	− 29.8961i	− 1.54796i	0.633209	−	0.773981i	$-0.281737\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.43560i	0.485958i
$378$	0	0
$379$	36.6093	1.88049	0.940247	−	0.340492i	$-0.110594\pi$
$379$	36.6093	1.88049	0.940247	+	0.340492i	$0.110594\pi$
$380$	0	0
$381$	−24.5582	−1.25815
$382$	0	0
$383$	22.9751i	1.17397i	0.809597	+	0.586986i	$0.199686\pi$
$383$	22.9751i	1.17397i	−0.809597	+	0.586986i	$0.800314\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.84922i	0.144834i
$388$	0	0
$389$	27.4183	1.39016	0.695081	−	0.718931i	$-0.255369\pi$
$389$	27.4183	1.39016	0.695081	+	0.718931i	$0.255369\pi$
$390$	0	0
$391$	39.7453	2.01001
$392$	0	0
$393$	17.5408i	0.884818i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.05408i	0.454411i	0.973847	+	0.227206i	$0.0729590\pi$
$397$	9.05408i	0.454411i	−0.973847	+	0.227206i	$0.927041\pi$
$398$	0	0
$399$	7.19436	0.360168
$400$	0	0
$401$	7.21926	0.360513	0.180256	−	0.983620i	$-0.442307\pi$
$401$	7.21926	0.360513	0.180256	+	0.983620i	$0.442307\pi$
$402$	0	0
$403$	28.2776i	1.40861i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.02198i	0.100226i
$408$	0	0
$409$	−1.08619	−0.0537085	−0.0268543	−	0.999639i	$-0.508549\pi$
$409$	−1.08619	−0.0537085	−0.0268543	+	0.999639i	$0.508549\pi$
$410$	0	0
$411$	13.9066	0.685963
$412$	0	0
$413$	− 18.5467i	− 0.912623i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.1757i	0.596246i
$418$	0	0
$419$	−10.9899	−0.536891	−0.268445	−	0.963295i	$-0.586510\pi$
$419$	−10.9899	−0.536891	−0.268445	+	0.963295i	$0.586510\pi$
$420$	0	0
$421$	5.30545	0.258572	0.129286	−	0.991607i	$-0.458732\pi$
$421$	5.30545	0.258572	0.129286	+	0.991607i	$0.458732\pi$
$422$	0	0
$423$	− 5.85253i	− 0.284560i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	44.6768i	2.16206i
$428$	0	0
$429$	−3.83482	−0.185147
$430$	0	0
$431$	11.0905	0.534209	0.267104	−	0.963668i	$-0.413933\pi$
$431$	11.0905	0.534209	0.267104	+	0.963668i	$0.413933\pi$
$432$	0	0
$433$	− 12.1652i	− 0.584621i	−0.956323	−	0.292311i	$-0.905576\pi$
$433$	− 12.1652i	− 0.584621i	0.956323	−	0.292311i	$-0.0944241\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.46050i	− 0.356884i
$438$	0	0
$439$	−27.5586	−1.31530	−0.657649	−	0.753325i	$-0.728449\pi$
$439$	−27.5586	−1.31530	−0.657649	+	0.753325i	$0.728449\pi$
$440$	0	0
$441$	−6.16225	−0.293441
$442$	0	0
$443$	14.1800i	0.673710i	0.941556	+	0.336855i	$0.109363\pi$
$443$	14.1800i	0.673710i	−0.941556	+	0.336855i	$0.890637\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.6401i	0.503258i
$448$	0	0
$449$	14.9895	0.707398	0.353699	−	0.935359i	$-0.384924\pi$
$449$	14.9895	0.707398	0.353699	+	0.935359i	$0.384924\pi$
$450$	0	0
$451$	−4.38151	−0.206317
$452$	0	0
$453$	− 30.5366i	− 1.43473i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.1331i	0.988564i	0.869302	+	0.494282i	$0.164569\pi$
$457$	21.1331i	0.988564i	−0.869302	+	0.494282i	$0.835431\pi$
$458$	0	0
$459$	29.3786	1.37128
$460$	0	0
$461$	40.5801	1.89001	0.945003	−	0.327062i	$-0.106059\pi$
$461$	40.5801	1.89001	0.945003	+	0.327062i	$0.106059\pi$
$462$	0	0
$463$	12.7381i	0.591991i	0.955189	+	0.295995i	$0.0956513\pi$
$463$	12.7381i	0.591991i	−0.955189	+	0.295995i	$0.904349\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.95739i	0.414498i	0.978288	+	0.207249i	$0.0664511\pi$
$467$	8.95739i	0.414498i	−0.978288	+	0.207249i	$0.933549\pi$
$468$	0	0
$469$	54.4517	2.51435
$470$	0	0
$471$	−26.7017	−1.23035
$472$	0	0
$473$	3.67257i	0.168865i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.07179i	− 0.186435i
$478$	0	0
$479$	−6.32451	−0.288974	−0.144487	−	0.989507i	$-0.546153\pi$
$479$	−6.32451	−0.288974	−0.144487	+	0.989507i	$0.546153\pi$
$480$	0	0
$481$	−13.8099	−0.629678
$482$	0	0
$483$	− 53.6735i	− 2.44223i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 14.4461i	− 0.654616i	−0.944918	−	0.327308i	$-0.893859\pi$
$487$	− 14.4461i	− 0.654616i	0.944918	−	0.327308i	$-0.106141\pi$
$488$	0	0
$489$	−3.69748	−0.167206
$490$	0	0
$491$	6.28813	0.283779	0.141890	−	0.989882i	$-0.454682\pi$
$491$	6.28813	0.283779	0.141890	+	0.989882i	$0.454682\pi$
$492$	0	0
$493$	− 12.3992i	− 0.558433i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 32.0905i	− 1.43945i
$498$	0	0
$499$	31.9181	1.42885	0.714425	−	0.699712i	$-0.246688\pi$
$499$	31.9181	1.42885	0.714425	+	0.699712i	$0.246688\pi$
$500$	0	0
$501$	13.6549	0.610054
$502$	0	0
$503$	15.7601i	0.702708i	0.936243	+	0.351354i	$0.114279\pi$
$503$	15.7601i	0.702708i	−0.936243	+	0.351354i	$0.885721\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5.47490i	− 0.243149i
$508$	0	0
$509$	−26.1885	−1.16079	−0.580393	−	0.814337i	$-0.697101\pi$
$509$	−26.1885	−1.16079	−0.580393	+	0.814337i	$0.697101\pi$
$510$	0	0
$511$	−23.8070	−1.05316
$512$	0	0
$513$	− 5.51459i	− 0.243475i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7.54377i	− 0.331775i
$518$	0	0
$519$	14.4169	0.632833
$520$	0	0
$521$	−42.3245	−1.85427	−0.927135	−	0.374727i	$-0.877736\pi$
$521$	−42.3245	−1.85427	−0.927135	+	0.374727i	$0.877736\pi$
$522$	0	0
$523$	− 19.3714i	− 0.847052i	−0.905884	−	0.423526i	$-0.860792\pi$
$523$	− 19.3714i	− 0.847052i	0.905884	−	0.423526i	$-0.139208\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 37.1593i	− 1.61869i
$528$	0	0
$529$	−32.6591	−1.41996
$530$	0	0
$531$	−1.89183	−0.0820985
$532$	0	0
$533$	− 29.9253i	− 1.29621i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 21.5083i	− 0.928154i
$538$	0	0
$539$	−7.94299	−0.342129
$540$	0	0
$541$	−5.87451	−0.252565	−0.126282	−	0.991994i	$-0.540305\pi$
$541$	−5.87451	−0.252565	−0.126282	+	0.991994i	$0.540305\pi$
$542$	0	0
$543$	− 11.1551i	− 0.478709i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.2556i	0.866069i	0.901377	+	0.433034i	$0.142557\pi$
$547$	20.2556i	0.866069i	−0.901377	+	0.433034i	$0.857443\pi$
$548$	0	0
$549$	4.55720	0.194497
$550$	0	0
$551$	−2.32743	−0.0991519
$552$	0	0
$553$	17.9134i	0.761756i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.76010i	0.159321i	0.996822	+	0.0796604i	$0.0253836\pi$
$557$	3.76010i	0.159321i	−0.996822	+	0.0796604i	$0.974616\pi$
$558$	0	0
$559$	−25.0833	−1.06091
$560$	0	0
$561$	5.03930	0.212759
$562$	0	0
$563$	11.2484i	0.474065i	0.971502	+	0.237033i	$0.0761748\pi$
$563$	11.2484i	0.474065i	−0.971502	+	0.237033i	$0.923825\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 33.4369i	− 1.40422i
$568$	0	0
$569$	−12.2949	−0.515431	−0.257715	−	0.966221i	$-0.582970\pi$
$569$	−12.2949	−0.515431	−0.257715	+	0.966221i	$0.582970\pi$
$570$	0	0
$571$	13.8784	0.580793	0.290396	−	0.956906i	$-0.406213\pi$
$571$	13.8784	0.580793	0.290396	+	0.956906i	$0.406213\pi$
$572$	0	0
$573$	− 19.8171i	− 0.827872i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 31.1052i	− 1.29493i	−0.762096	−	0.647464i	$-0.775830\pi$
$577$	− 31.1052i	− 1.29493i	0.762096	−	0.647464i	$-0.224170\pi$
$578$	0	0
$579$	−2.62568	−0.109120
$580$	0	0
$581$	−34.0377	−1.41212
$582$	0	0
$583$	− 5.24844i	− 0.217368i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 37.7807i	− 1.55938i	−0.626167	−	0.779689i	$-0.715377\pi$
$587$	− 37.7807i	− 1.55938i	0.626167	−	0.779689i	$-0.284623\pi$
$588$	0	0
$589$	−6.97509	−0.287404
$590$	0	0
$591$	34.0335	1.39995
$592$	0	0
$593$	− 40.5949i	− 1.66703i	−0.552494	−	0.833517i	$-0.686324\pi$
$593$	− 40.5949i	− 1.66703i	0.552494	−	0.833517i	$-0.313676\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.5424i	0.799818i
$598$	0	0
$599$	−18.2776	−0.746803	−0.373402	−	0.927670i	$-0.621809\pi$
$599$	−18.2776	−0.746803	−0.373402	+	0.927670i	$0.621809\pi$
$600$	0	0
$601$	−13.4313	−0.547875	−0.273938	−	0.961747i	$-0.588326\pi$
$601$	−13.4313	−0.547875	−0.273938	+	0.961747i	$0.588326\pi$
$602$	0	0
$603$	− 5.55428i	− 0.226188i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	47.2235i	1.91674i	0.285522	+	0.958372i	$0.407833\pi$
$607$	47.2235i	1.91674i	−0.285522	+	0.958372i	$0.592167\pi$
$608$	0	0
$609$	−16.7444	−0.678516
$610$	0	0
$611$	51.5231	2.08440
$612$	0	0
$613$	10.2268i	0.413059i	0.978440	+	0.206529i	$0.0662169\pi$
$613$	10.2268i	0.413059i	−0.978440	+	0.206529i	$0.933783\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.3829i	1.30368i	0.758354	+	0.651842i	$0.226004\pi$
$617$	32.3829i	1.30368i	−0.758354	+	0.651842i	$0.773996\pi$
$618$	0	0
$619$	−19.8784	−0.798980	−0.399490	−	0.916738i	$-0.630813\pi$
$619$	−19.8784	−0.798980	−0.399490	+	0.916738i	$0.630813\pi$
$620$	0	0
$621$	−41.1416	−1.65096
$622$	0	0
$623$	− 41.5087i	− 1.66301i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 0.945916i	− 0.0377762i
$628$	0	0
$629$	18.1475	0.723587
$630$	0	0
$631$	14.5103	0.577647	0.288823	−	0.957382i	$-0.406736\pi$
$631$	14.5103	0.577647	0.288823	+	0.957382i	$0.406736\pi$
$632$	0	0
$633$	30.8745i	1.22715i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 54.2498i	− 2.14945i
$638$	0	0
$639$	−3.27335	−0.129492
$640$	0	0
$641$	3.15798	0.124733	0.0623664	−	0.998053i	$-0.480135\pi$
$641$	3.15798	0.124733	0.0623664	+	0.998053i	$0.480135\pi$
$642$	0	0
$643$	− 22.0000i	− 0.867595i	−0.901010	−	0.433798i	$-0.857173\pi$
$643$	− 22.0000i	− 0.867595i	0.901010	−	0.433798i	$-0.142827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.36673i	0.0537317i	0.999639	+	0.0268659i	$0.00855270\pi$
$647$	1.36673i	0.0537317i	−0.999639	+	0.0268659i	$0.991447\pi$
$648$	0	0
$649$	−2.43852	−0.0957204
$650$	0	0
$651$	−50.1813	−1.96676
$652$	0	0
$653$	− 17.0220i	− 0.666122i	−0.942905	−	0.333061i	$-0.891919\pi$
$653$	− 17.0220i	− 0.666122i	0.942905	−	0.333061i	$-0.108081\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.42840i	0.0947410i
$658$	0	0
$659$	−14.6883	−0.572175	−0.286088	−	0.958203i	$-0.592355\pi$
$659$	−14.6883	−0.572175	−0.286088	+	0.958203i	$0.592355\pi$
$660$	0	0
$661$	−1.86693	−0.0726150	−0.0363075	−	0.999341i	$-0.511560\pi$
$661$	−1.86693	−0.0726150	−0.0363075	+	0.999341i	$0.511560\pi$
$662$	0	0
$663$	34.4179i	1.33668i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17.3638i	0.672329i
$668$	0	0
$669$	32.6870	1.26375
$670$	0	0
$671$	5.87412	0.226768
$672$	0	0
$673$	− 15.3465i	− 0.591564i	−0.955256	−	0.295782i	$-0.904420\pi$
$673$	− 15.3465i	− 0.591564i	0.955256	−	0.295782i	$-0.0955801\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.3422i	− 0.666515i	−0.942836	−	0.333258i	$-0.891852\pi$
$677$	− 17.3422i	− 0.666515i	0.942836	−	0.333258i	$-0.108148\pi$
$678$	0	0
$679$	29.0216	1.11375
$680$	0	0
$681$	−6.23697	−0.239001
$682$	0	0
$683$	40.9473i	1.56680i	0.621516	+	0.783402i	$0.286517\pi$
$683$	40.9473i	1.56680i	−0.621516	+	0.783402i	$0.713483\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.71518i	− 0.218048i
$688$	0	0
$689$	35.8463	1.36563
$690$	0	0
$691$	5.65525	0.215136	0.107568	−	0.994198i	$-0.465694\pi$
$691$	5.65525	0.215136	0.107568	+	0.994198i	$0.465694\pi$
$692$	0	0
$693$	1.23405i	0.0468775i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	39.3245i	1.48952i
$698$	0	0
$699$	−19.9665	−0.755204
$700$	0	0
$701$	35.3786	1.33623	0.668115	−	0.744058i	$-0.267101\pi$
$701$	35.3786	1.33623	0.668115	+	0.744058i	$0.267101\pi$
$702$	0	0
$703$	− 3.40642i	− 0.128476i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 53.3169i	− 2.00519i
$708$	0	0
$709$	41.2852	1.55050	0.775249	−	0.631655i	$-0.217624\pi$
$709$	41.2852	1.55050	0.775249	+	0.631655i	$0.217624\pi$
$710$	0	0
$711$	1.82724	0.0685267
$712$	0	0
$713$	52.0377i	1.94883i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	37.4838i	1.39986i
$718$	0	0
$719$	−17.0689	−0.636561	−0.318281	−	0.947997i	$-0.603105\pi$
$719$	−17.0689	−0.636561	−0.318281	+	0.947997i	$0.603105\pi$
$720$	0	0
$721$	−83.3609	−3.10452
$722$	0	0
$723$	14.6984i	0.546641i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27.8741i	1.03379i	0.856048	+	0.516897i	$0.172913\pi$
$727$	27.8741i	1.03379i	−0.856048	+	0.516897i	$0.827087\pi$
$728$	0	0
$729$	−29.7745	−1.10276
$730$	0	0
$731$	32.9617	1.21913
$732$	0	0
$733$	− 34.7922i	− 1.28508i	−0.766252	−	0.642540i	$-0.777881\pi$
$733$	− 34.7922i	− 1.28508i	0.766252	−	0.642540i	$-0.222119\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.15933i	− 0.263717i
$738$	0	0
$739$	49.0128	1.80297	0.901483	−	0.432815i	$-0.142480\pi$
$739$	49.0128	1.80297	0.901483	+	0.432815i	$0.142480\pi$
$740$	0	0
$741$	6.46050	0.237333
$742$	0	0
$743$	− 20.8142i	− 0.763599i	−0.924245	−	0.381799i	$-0.875305\pi$
$743$	− 20.8142i	− 0.763599i	0.924245	−	0.381799i	$-0.124695\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.47197i	0.127033i
$748$	0	0
$749$	1.94726	0.0711514
$750$	0	0
$751$	−14.4457	−0.527132	−0.263566	−	0.964641i	$-0.584899\pi$
$751$	−14.4457	−0.527132	−0.263566	+	0.964641i	$0.584899\pi$
$752$	0	0
$753$	29.8377i	1.08735i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 8.11537i	− 0.294958i	−0.989065	−	0.147479i	$-0.952884\pi$
$757$	− 8.11537i	− 0.294958i	0.989065	−	0.147479i	$-0.0471159\pi$
$758$	0	0
$759$	−7.05701	−0.256153
$760$	0	0
$761$	30.2967	1.09825	0.549127	−	0.835739i	$-0.314960\pi$
$761$	30.2967	1.09825	0.549127	+	0.835739i	$0.314960\pi$
$762$	0	0
$763$	32.1230i	1.16293i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 16.6549i	− 0.601372i
$768$	0	0
$769$	−40.4471	−1.45856	−0.729279	−	0.684216i	$-0.760144\pi$
$769$	−40.4471	−1.45856	−0.729279	+	0.684216i	$0.760144\pi$
$770$	0	0
$771$	−43.2173	−1.55643
$772$	0	0
$773$	− 2.16518i	− 0.0778760i	−0.999242	−	0.0389380i	$-0.987603\pi$
$773$	− 2.16518i	− 0.0778760i	0.999242	−	0.0389380i	$-0.0123975\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 24.5070i	− 0.879184i
$778$	0	0
$779$	7.38151	0.264470
$780$	0	0
$781$	−4.21926	−0.150977
$782$	0	0
$783$	12.8348i	0.458679i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.4284i	0.942071i	0.882114	+	0.471035i	$0.156120\pi$
$787$	26.4284i	0.942071i	−0.882114	+	0.471035i	$0.843880\pi$
$788$	0	0
$789$	29.5624	1.05245
$790$	0	0
$791$	14.2763	0.507606
$792$	0	0
$793$	40.1196i	1.42469i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.5041i	1.39931i	0.714483	+	0.699653i	$0.246662\pi$
$797$	39.5041i	1.39931i	−0.714483	+	0.699653i	$0.753338\pi$
$798$	0	0
$799$	−67.7060	−2.39527
$800$	0	0
$801$	−4.23405	−0.149603
$802$	0	0
$803$	3.13015i	0.110461i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	41.6451i	1.46598i
$808$	0	0
$809$	27.4078	0.963606	0.481803	−	0.876280i	$-0.339982\pi$
$809$	27.4078	0.963606	0.481803	+	0.876280i	$0.339982\pi$
$810$	0	0
$811$	−50.7467	−1.78196	−0.890978	−	0.454046i	$-0.849980\pi$
$811$	−50.7467	−1.78196	−0.890978	+	0.454046i	$0.849980\pi$
$812$	0	0
$813$	3.11244i	0.109158i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.18716i	− 0.216461i
$818$	0	0
$819$	−8.42840	−0.294512
$820$	0	0
$821$	12.7410	0.444666	0.222333	−	0.974971i	$-0.428633\pi$
$821$	12.7410	0.444666	0.222333	+	0.974971i	$0.428633\pi$
$822$	0	0
$823$	17.1623i	0.598239i	0.954216	+	0.299119i	$0.0966928\pi$
$823$	17.1623i	0.598239i	−0.954216	+	0.299119i	$0.903307\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 53.6735i	− 1.86641i	−0.359343	−	0.933206i	$-0.616999\pi$
$827$	− 53.6735i	− 1.86641i	0.359343	−	0.933206i	$-0.383001\pi$
$828$	0	0
$829$	−7.45331	−0.258864	−0.129432	−	0.991588i	$-0.541315\pi$
$829$	−7.45331	−0.258864	−0.129432	+	0.991588i	$0.541315\pi$
$830$	0	0
$831$	3.93248	0.136416
$832$	0	0
$833$	71.2891i	2.47002i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	38.4648i	1.32954i
$838$	0	0
$839$	−52.8257	−1.82374	−0.911872	−	0.410474i	$-0.865363\pi$
$839$	−52.8257	−1.82374	−0.911872	+	0.410474i	$0.865363\pi$
$840$	0	0
$841$	−23.5831	−0.813209
$842$	0	0
$843$	39.6342i	1.36508i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 48.0698i	− 1.65170i
$848$	0	0
$849$	15.2249	0.522517
$850$	0	0
$851$	−25.4136	−0.871168
$852$	0	0
$853$	− 16.6663i	− 0.570644i	−0.958432	−	0.285322i	$-0.907899\pi$
$853$	− 16.6663i	− 0.570644i	0.958432	−	0.285322i	$-0.0921006\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 16.6654i	− 0.569278i	−0.958635	−	0.284639i	$-0.908126\pi$
$857$	− 16.6654i	− 0.569278i	0.958635	−	0.284639i	$-0.0918738\pi$
$858$	0	0
$859$	30.0010	1.02362	0.511810	−	0.859099i	$-0.328975\pi$
$859$	30.0010	1.02362	0.511810	+	0.859099i	$0.328975\pi$
$860$	0	0
$861$	53.1052	1.80982
$862$	0	0
$863$	23.0498i	0.784625i	0.919832	+	0.392312i	$0.128325\pi$
$863$	23.0498i	0.784625i	−0.919832	+	0.392312i	$0.871675\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 18.1373i	− 0.615977i
$868$	0	0
$869$	2.35526	0.0798968
$870$	0	0
$871$	48.8975	1.65683
$872$	0	0
$873$	− 2.96031i	− 0.100191i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.9938i	0.843979i	0.906601	+	0.421990i	$0.138668\pi$
$877$	24.9938i	0.843979i	−0.906601	+	0.421990i	$0.861332\pi$
$878$	0	0
$879$	−17.9321	−0.604834
$880$	0	0
$881$	14.2590	0.480396	0.240198	−	0.970724i	$-0.422788\pi$
$881$	14.2590	0.480396	0.240198	+	0.970724i	$0.422788\pi$
$882$	0	0
$883$	47.6021i	1.60194i	0.598705	+	0.800970i	$0.295682\pi$
$883$	47.6021i	1.60194i	−0.598705	+	0.800970i	$0.704318\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 33.6959i	− 1.13140i	−0.824612	−	0.565699i	$-0.808607\pi$
$887$	− 33.6959i	− 1.13140i	0.824612	−	0.565699i	$-0.191393\pi$
$888$	0	0
$889$	−69.5729	−2.33340
$890$	0	0
$891$	−4.39630	−0.147282
$892$	0	0
$893$	12.7089i	0.425289i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 48.1986i	− 1.60931i
$898$	0	0
$899$	16.2340	0.541436
$900$	0	0
$901$	−47.1052	−1.56930
$902$	0	0
$903$	− 44.5126i	− 1.48129i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.05701i	0.267529i	0.991013	+	0.133764i	$0.0427065\pi$
$907$	8.05701i	0.267529i	−0.991013	+	0.133764i	$0.957293\pi$
$908$	0	0
$909$	−5.43852	−0.180384
$910$	0	0
$911$	−21.6942	−0.718760	−0.359380	−	0.933191i	$-0.617012\pi$
$911$	−21.6942	−0.718760	−0.359380	+	0.933191i	$0.617012\pi$
$912$	0	0
$913$	4.47529i	0.148110i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	49.6930i	1.64101i
$918$	0	0
$919$	42.7525	1.41028	0.705138	−	0.709070i	$-0.250885\pi$
$919$	42.7525	1.41028	0.705138	+	0.709070i	$0.250885\pi$
$920$	0	0
$921$	19.5815	0.645232
$922$	0	0
$923$	− 28.8171i	− 0.948527i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.50312i	0.279279i
$928$	0	0
$929$	−32.8290	−1.07708	−0.538542	−	0.842599i	$-0.681025\pi$
$929$	−32.8290	−1.07708	−0.538542	+	0.842599i	$0.681025\pi$
$930$	0	0
$931$	13.3815	0.438561
$932$	0	0
$933$	27.3029i	0.893857i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.6313i	0.673995i	0.941505	+	0.336998i	$0.109411\pi$
$937$	20.6313i	0.673995i	−0.941505	+	0.336998i	$0.890589\pi$
$938$	0	0
$939$	7.93968	0.259102
$940$	0	0
$941$	−2.25271	−0.0734363	−0.0367182	−	0.999326i	$-0.511690\pi$
$941$	−2.25271	−0.0734363	−0.0367182	+	0.999326i	$0.511690\pi$
$942$	0	0
$943$	− 55.0698i	− 1.79332i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.1387i	1.66178i	0.556434	+	0.830892i	$0.312169\pi$
$947$	51.1387i	1.66178i	−0.556434	+	0.830892i	$0.687831\pi$
$948$	0	0
$949$	−21.3786	−0.693978
$950$	0	0
$951$	−35.6903	−1.15734
$952$	0	0
$953$	− 4.14454i	− 0.134255i	−0.997744	−	0.0671275i	$-0.978617\pi$
$953$	− 4.14454i	− 0.134255i	0.997744	−	0.0671275i	$-0.0213834\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.20155i	0.0711661i
$958$	0	0
$959$	39.3973	1.27220
$960$	0	0
$961$	17.6519	0.569417
$962$	0	0
$963$	− 0.198628i	− 0.00640070i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 29.3202i	− 0.942875i	−0.881899	−	0.471438i	$-0.843735\pi$
$967$	− 29.3202i	− 0.942875i	0.881899	−	0.471438i	$-0.156265\pi$
$968$	0	0
$969$	−8.48968	−0.272728
$970$	0	0
$971$	−10.0144	−0.321377	−0.160689	−	0.987005i	$-0.551371\pi$
$971$	−10.0144	−0.321377	−0.160689	+	0.987005i	$0.551371\pi$
$972$	0	0
$973$	34.4936i	1.10581i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 32.2206i	− 1.03083i	−0.856941	−	0.515414i	$-0.827638\pi$
$977$	− 32.2206i	− 1.03083i	0.856941	−	0.515414i	$-0.172362\pi$
$978$	0	0
$979$	−5.45758	−0.174425
$980$	0	0
$981$	3.27666	0.104616
$982$	0	0
$983$	12.7837i	0.407736i	0.978998	+	0.203868i	$0.0653513\pi$
$983$	12.7837i	0.407736i	−0.978998	+	0.203868i	$0.934649\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	91.4327i	2.91033i
$988$	0	0
$989$	−46.1593	−1.46778
$990$	0	0
$991$	−28.4893	−0.904992	−0.452496	−	0.891766i	$-0.649466\pi$
$991$	−28.4893	−0.904992	−0.452496	+	0.891766i	$0.649466\pi$
$992$	0	0
$993$	6.48406i	0.205766i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 42.6122i	− 1.34954i	−0.738027	−	0.674772i	$-0.764242\pi$
$997$	− 42.6122i	− 1.34954i	0.738027	−	0.674772i	$-0.235758\pi$
$998$	0	0
$999$	−18.7850	−0.594331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.f.1749.4		6
5.2	odd	4		1900.2.a.i.1.2	yes	3
5.3	odd	4		1900.2.a.g.1.2	✓	3
5.4	even	2	inner	1900.2.c.f.1749.3		6
20.3	even	4		7600.2.a.ca.1.2		3
20.7	even	4		7600.2.a.bl.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.g.1.2	✓	3	5.3	odd	4
1900.2.a.i.1.2	yes	3	5.2	odd	4
1900.2.c.f.1749.3		6	5.4	even	2	inner
1900.2.c.f.1749.4		6	1.1	even	1	trivial
7600.2.a.bl.1.2		3	20.7	even	4
7600.2.a.ca.1.2		3	20.3	even	4

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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.59358i q^{3} +4.51459i q^{7} +0.460505 q^{9} +O(q^{10})\)	\(q+1.59358i q^{3} +4.51459i q^{7} +0.460505 q^{9} +0.593579 q^{11} +4.05408i q^{13} -5.32743i q^{17} -1.00000 q^{19} -7.19436 q^{21} +7.46050i q^{23} +5.51459i q^{27} +2.32743 q^{29} +6.97509 q^{31} +0.945916i q^{33} +3.40642i q^{37} -6.46050 q^{39} -7.38151 q^{41} +6.18716i q^{43} -12.7089i q^{47} -13.3815 q^{49} +8.48968 q^{51} -8.84202i q^{53} -1.59358i q^{57} -4.10817 q^{59} +9.89610 q^{61} +2.07899i q^{63} -12.0613i q^{67} -11.8889 q^{69} -7.10817 q^{71} +5.27335i q^{73} +2.67977i q^{77} +3.96790 q^{79} -7.40642 q^{81} +7.53950i q^{83} +3.70895i q^{87} -9.19436 q^{89} -18.3025 q^{91} +11.1154i q^{93} -6.42840i q^{97} +0.273346 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{9}+O(q^{10}) \) 6 * q - 10 * q^9	\( 6 q - 10 q^{9} - 2 q^{11} - 6 q^{19} - 16 q^{21} - 6 q^{29} - 2 q^{31} - 26 q^{39} - 6 q^{41} - 42 q^{49} - 24 q^{51} + 12 q^{59} - 10 q^{61} - 18 q^{69} - 6 q^{71} - 4 q^{79} - 50 q^{81} - 28 q^{89} - 46 q^{91}+O(q^{100}) \) 6 * q - 10 * q^9 - 2 * q^11 - 6 * q^19 - 16 * q^21 - 6 * q^29 - 2 * q^31 - 26 * q^39 - 6 * q^41 - 42 * q^49 - 24 * q^51 + 12 * q^59 - 10 * q^61 - 18 * q^69 - 6 * q^71 - 4 * q^79 - 50 * q^81 - 28 * q^89 - 46 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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