LMFDB - Embedded newform 675.4.b.c.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (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:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.c.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +4.00000 q^{4} +24.0000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} +4.00000 q^{4} +24.0000i q^{8} -10.0000 q^{11} +80.0000i q^{13} -16.0000 q^{16} -7.00000i q^{17} +113.000 q^{19} -20.0000i q^{22} -81.0000i q^{23} -160.000 q^{26} -220.000 q^{29} -189.000 q^{31} +160.000i q^{32} +14.0000 q^{34} +170.000i q^{37} +226.000i q^{38} +130.000 q^{41} -10.0000i q^{43} -40.0000 q^{44} +162.000 q^{46} -160.000i q^{47} +343.000 q^{49} +320.000i q^{52} +631.000i q^{53} -440.000i q^{58} -560.000 q^{59} +229.000 q^{61} -378.000i q^{62} -448.000 q^{64} +750.000i q^{67} -28.0000i q^{68} -890.000 q^{71} +890.000i q^{73} -340.000 q^{74} +452.000 q^{76} +27.0000 q^{79} +260.000i q^{82} +429.000i q^{83} +20.0000 q^{86} -240.000i q^{88} -750.000 q^{89} -324.000i q^{92} +320.000 q^{94} -1480.00i q^{97} +686.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{4}+O(q^{10}) $ 2 * q + 8 * q^4	$ 2 q + 8 q^{4} - 20 q^{11} - 32 q^{16} + 226 q^{19} - 320 q^{26} - 440 q^{29} - 378 q^{31} + 28 q^{34} + 260 q^{41} - 80 q^{44} + 324 q^{46} + 686 q^{49} - 1120 q^{59} + 458 q^{61} - 896 q^{64} - 1780 q^{71} - 680 q^{74} + 904 q^{76} + 54 q^{79} + 40 q^{86} - 1500 q^{89} + 640 q^{94}+O(q^{100}) $ 2 * q + 8 * q^4 - 20 * q^11 - 32 * q^16 + 226 * q^19 - 320 * q^26 - 440 * q^29 - 378 * q^31 + 28 * q^34 + 260 * q^41 - 80 * q^44 + 324 * q^46 + 686 * q^49 - 1120 * q^59 + 458 * q^61 - 896 * q^64 - 1780 * q^71 - 680 * q^74 + 904 * q^76 + 54 * q^79 + 40 * q^86 - 1500 * q^89 + 640 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	2.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	24.0000i	1.06066i
$9$	0	0
$10$	0	0
$11$	−10.0000	−0.274101	−0.137051	−	0.990564i	$-0.543762\pi$
$11$	−10.0000	−0.274101	−0.137051	+	0.990564i	$0.543762\pi$
$12$	0	0
$13$	80.0000i	1.70677i	0.521281	+	0.853385i	$0.325454\pi$
$13$	80.0000i	1.70677i	−0.521281	+	0.853385i	$0.674546\pi$
$14$	0	0
$15$	0	0
$16$	−16.0000	−0.250000
$17$	− 7.00000i	− 0.0998676i	−0.998753	−	0.0499338i	$-0.984099\pi$
$17$	− 7.00000i	− 0.0998676i	0.998753	−	0.0499338i	$-0.0159010\pi$
$18$	0	0
$19$	113.000	1.36442	0.682210	−	0.731156i	$-0.261019\pi$
$19$	113.000	1.36442	0.682210	+	0.731156i	$0.261019\pi$
$20$	0	0
$21$	0	0
$22$	− 20.0000i	− 0.193819i
$23$	− 81.0000i	− 0.734333i	−0.930155	−	0.367167i	$-0.880328\pi$
$23$	− 81.0000i	− 0.734333i	0.930155	−	0.367167i	$-0.119672\pi$
$24$	0	0
$25$	0	0
$26$	−160.000	−1.20687
$27$	0	0
$28$	0	0
$29$	−220.000	−1.40872	−0.704362	−	0.709841i	$-0.748767\pi$
$29$	−220.000	−1.40872	−0.704362	+	0.709841i	$0.748767\pi$
$30$	0	0
$31$	−189.000	−1.09501	−0.547506	−	0.836801i	$-0.684423\pi$
$31$	−189.000	−1.09501	−0.547506	+	0.836801i	$0.684423\pi$
$32$	160.000i	0.883883i
$33$	0	0
$34$	14.0000	0.0706171
$35$	0	0
$36$	0	0
$37$	170.000i	0.755347i	0.925939	+	0.377673i	$0.123276\pi$
$37$	170.000i	0.755347i	−0.925939	+	0.377673i	$0.876724\pi$
$38$	226.000i	0.964791i
$39$	0	0
$40$	0	0
$41$	130.000	0.495185	0.247593	−	0.968864i	$-0.420361\pi$
$41$	130.000	0.495185	0.247593	+	0.968864i	$0.420361\pi$
$42$	0	0
$43$	− 10.0000i	− 0.0354648i	−0.999843	−	0.0177324i	$-0.994355\pi$
$43$	− 10.0000i	− 0.0354648i	0.999843	−	0.0177324i	$-0.00564469\pi$
$44$	−40.0000	−0.137051
$45$	0	0
$46$	162.000	0.519252
$47$	− 160.000i	− 0.496562i	−0.968688	−	0.248281i	$-0.920134\pi$
$47$	− 160.000i	− 0.496562i	0.968688	−	0.248281i	$-0.0798656\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	0	0
$51$	0	0
$52$	320.000i	0.853385i
$53$	631.000i	1.63537i	0.575667	+	0.817684i	$0.304742\pi$
$53$	631.000i	1.63537i	−0.575667	+	0.817684i	$0.695258\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 440.000i	− 0.996118i
$59$	−560.000	−1.23569	−0.617846	−	0.786299i	$-0.711994\pi$
$59$	−560.000	−1.23569	−0.617846	+	0.786299i	$0.711994\pi$
$60$	0	0
$61$	229.000	0.480663	0.240332	−	0.970691i	$-0.422744\pi$
$61$	229.000	0.480663	0.240332	+	0.970691i	$0.422744\pi$
$62$	− 378.000i	− 0.774291i
$63$	0	0
$64$	−448.000	−0.875000
$65$	0	0
$66$	0	0
$67$	750.000i	1.36757i	0.729684	+	0.683784i	$0.239667\pi$
$67$	750.000i	1.36757i	−0.729684	+	0.683784i	$0.760333\pi$
$68$	− 28.0000i	− 0.0499338i
$69$	0	0
$70$	0	0
$71$	−890.000	−1.48766	−0.743828	−	0.668371i	$-0.766992\pi$
$71$	−890.000	−1.48766	−0.743828	+	0.668371i	$0.766992\pi$
$72$	0	0
$73$	890.000i	1.42694i	0.700686	+	0.713470i	$0.252878\pi$
$73$	890.000i	1.42694i	−0.700686	+	0.713470i	$0.747122\pi$
$74$	−340.000	−0.534111
$75$	0	0
$76$	452.000	0.682210
$77$	0	0
$78$	0	0
$79$	27.0000	0.0384524	0.0192262	−	0.999815i	$-0.493880\pi$
$79$	27.0000	0.0384524	0.0192262	+	0.999815i	$0.493880\pi$
$80$	0	0
$81$	0	0
$82$	260.000i	0.350149i
$83$	429.000i	0.567336i	0.958923	+	0.283668i	$0.0915513\pi$
$83$	429.000i	0.567336i	−0.958923	+	0.283668i	$0.908449\pi$
$84$	0	0
$85$	0	0
$86$	20.0000	0.0250774
$87$	0	0
$88$	− 240.000i	− 0.290728i
$89$	−750.000	−0.893257	−0.446628	−	0.894720i	$-0.647375\pi$
$89$	−750.000	−0.893257	−0.446628	+	0.894720i	$0.647375\pi$
$90$	0	0
$91$	0	0
$92$	− 324.000i	− 0.367167i
$93$	0	0
$94$	320.000	0.351122
$95$	0	0
$96$	0	0
$97$	− 1480.00i	− 1.54919i	−0.632459	−	0.774594i	$-0.717954\pi$
$97$	− 1480.00i	− 1.54919i	0.632459	−	0.774594i	$-0.282046\pi$
$98$	686.000i	0.707107i
$99$	0	0
$100$	0	0
$101$	1500.00	1.47778	0.738889	−	0.673827i	$-0.235351\pi$
$101$	1500.00	1.47778	0.738889	+	0.673827i	$0.235351\pi$
$102$	0	0
$103$	460.000i	0.440050i	0.975494	+	0.220025i	$0.0706139\pi$
$103$	460.000i	0.440050i	−0.975494	+	0.220025i	$0.929386\pi$
$104$	−1920.00	−1.81030
$105$	0	0
$106$	−1262.00	−1.15638
$107$	420.000i	0.379467i	0.981836	+	0.189733i	$0.0607623\pi$
$107$	420.000i	0.379467i	−0.981836	+	0.189733i	$0.939238\pi$
$108$	0	0
$109$	607.000	0.533395	0.266698	−	0.963780i	$-0.414068\pi$
$109$	607.000	0.533395	0.266698	+	0.963780i	$0.414068\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2170.00i	1.80652i	0.429097	+	0.903259i	$0.358832\pi$
$113$	2170.00i	1.80652i	−0.429097	+	0.903259i	$0.641168\pi$
$114$	0	0
$115$	0	0
$116$	−880.000	−0.704362
$117$	0	0
$118$	− 1120.00i	− 0.873766i
$119$	0	0
$120$	0	0
$121$	−1231.00	−0.924869
$122$	458.000i	0.339880i
$123$	0	0
$124$	−756.000	−0.547506
$125$	0	0
$126$	0	0
$127$	− 1610.00i	− 1.12492i	−0.826826	−	0.562458i	$-0.809856\pi$
$127$	− 1610.00i	− 1.12492i	0.826826	−	0.562458i	$-0.190144\pi$
$128$	384.000i	0.265165i
$129$	0	0
$130$	0	0
$131$	−2370.00	−1.58067	−0.790335	−	0.612674i	$-0.790094\pi$
$131$	−2370.00	−1.58067	−0.790335	+	0.612674i	$0.790094\pi$
$132$	0	0
$133$	0	0
$134$	−1500.00	−0.967017
$135$	0	0
$136$	168.000	0.105926
$137$	− 1797.00i	− 1.12064i	−0.828275	−	0.560321i	$-0.810678\pi$
$137$	− 1797.00i	− 1.12064i	0.828275	−	0.560321i	$-0.189322\pi$
$138$	0	0
$139$	124.000	0.0756658	0.0378329	−	0.999284i	$-0.487955\pi$
$139$	124.000	0.0756658	0.0378329	+	0.999284i	$0.487955\pi$
$140$	0	0
$141$	0	0
$142$	− 1780.00i	− 1.05193i
$143$	− 800.000i	− 0.467828i
$144$	0	0
$145$	0	0
$146$	−1780.00	−1.00900
$147$	0	0
$148$	680.000i	0.377673i
$149$	−70.0000	−0.0384874	−0.0192437	−	0.999815i	$-0.506126\pi$
$149$	−70.0000	−0.0384874	−0.0192437	+	0.999815i	$0.506126\pi$
$150$	0	0
$151$	2248.00	1.21152	0.605760	−	0.795647i	$-0.292869\pi$
$151$	2248.00	1.21152	0.605760	+	0.795647i	$0.292869\pi$
$152$	2712.00i	1.44719i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1010.00i	0.513419i	0.966489	+	0.256709i	$0.0826384\pi$
$157$	1010.00i	0.513419i	−0.966489	+	0.256709i	$0.917362\pi$
$158$	54.0000i	0.0271899i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 590.000i	− 0.283511i	−0.989902	−	0.141756i	$-0.954725\pi$
$163$	− 590.000i	− 0.283511i	0.989902	−	0.141756i	$-0.0452748\pi$
$164$	520.000	0.247593
$165$	0	0
$166$	−858.000	−0.401167
$167$	− 2403.00i	− 1.11347i	−0.830690	−	0.556736i	$-0.812054\pi$
$167$	− 2403.00i	− 1.11347i	0.830690	−	0.556736i	$-0.187946\pi$
$168$	0	0
$169$	−4203.00	−1.91306
$170$	0	0
$171$	0	0
$172$	− 40.0000i	− 0.0177324i
$173$	801.000i	0.352017i	0.984389	+	0.176008i	$0.0563186\pi$
$173$	801.000i	0.352017i	−0.984389	+	0.176008i	$0.943681\pi$
$174$	0	0
$175$	0	0
$176$	160.000	0.0685253
$177$	0	0
$178$	− 1500.00i	− 0.631628i
$179$	−2360.00	−0.985445	−0.492723	−	0.870186i	$-0.663998\pi$
$179$	−2360.00	−0.985445	−0.492723	+	0.870186i	$0.663998\pi$
$180$	0	0
$181$	1241.00	0.509629	0.254814	−	0.966990i	$-0.417986\pi$
$181$	1241.00	0.509629	0.254814	+	0.966990i	$0.417986\pi$
$182$	0	0
$183$	0	0
$184$	1944.00	0.778878
$185$	0	0
$186$	0	0
$187$	70.0000i	0.0273738i
$188$	− 640.000i	− 0.248281i
$189$	0	0
$190$	0	0
$191$	4990.00	1.89039	0.945193	−	0.326512i	$-0.105873\pi$
$191$	4990.00	1.89039	0.945193	+	0.326512i	$0.105873\pi$
$192$	0	0
$193$	2260.00i	0.842893i	0.906853	+	0.421447i	$0.138477\pi$
$193$	2260.00i	0.842893i	−0.906853	+	0.421447i	$0.861523\pi$
$194$	2960.00	1.09544
$195$	0	0
$196$	1372.00	0.500000
$197$	2247.00i	0.812650i	0.913729	+	0.406325i	$0.133190\pi$
$197$	2247.00i	0.812650i	−0.913729	+	0.406325i	$0.866810\pi$
$198$	0	0
$199$	−4564.00	−1.62580	−0.812898	−	0.582406i	$-0.802111\pi$
$199$	−4564.00	−1.62580	−0.812898	+	0.582406i	$0.802111\pi$
$200$	0	0
$201$	0	0
$202$	3000.00i	1.04495i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−920.000	−0.311162
$207$	0	0
$208$	− 1280.00i	− 0.426692i
$209$	−1130.00	−0.373989
$210$	0	0
$211$	4949.00	1.61471	0.807354	−	0.590068i	$-0.200899\pi$
$211$	4949.00	1.61471	0.807354	+	0.590068i	$0.200899\pi$
$212$	2524.00i	0.817684i
$213$	0	0
$214$	−840.000	−0.268323
$215$	0	0
$216$	0	0
$217$	0	0
$218$	1214.00i	0.377167i
$219$	0	0
$220$	0	0
$221$	560.000	0.170451
$222$	0	0
$223$	− 3890.00i	− 1.16813i	−0.811706	−	0.584067i	$-0.801461\pi$
$223$	− 3890.00i	− 1.16813i	0.811706	−	0.584067i	$-0.198539\pi$
$224$	0	0
$225$	0	0
$226$	−4340.00	−1.27740
$227$	− 2453.00i	− 0.717231i	−0.933485	−	0.358615i	$-0.883249\pi$
$227$	− 2453.00i	− 0.717231i	0.933485	−	0.358615i	$-0.116751\pi$
$228$	0	0
$229$	6213.00	1.79287	0.896434	−	0.443178i	$-0.146149\pi$
$229$	6213.00	1.79287	0.896434	+	0.443178i	$0.146149\pi$
$230$	0	0
$231$	0	0
$232$	− 5280.00i	− 1.49418i
$233$	− 3450.00i	− 0.970030i	−0.874506	−	0.485015i	$-0.838814\pi$
$233$	− 3450.00i	− 0.970030i	0.874506	−	0.485015i	$-0.161186\pi$
$234$	0	0
$235$	0	0
$236$	−2240.00	−0.617846
$237$	0	0
$238$	0	0
$239$	6490.00	1.75650	0.878249	−	0.478203i	$-0.158712\pi$
$239$	6490.00	1.75650	0.878249	+	0.478203i	$0.158712\pi$
$240$	0	0
$241$	−3401.00	−0.909036	−0.454518	−	0.890738i	$-0.650188\pi$
$241$	−3401.00	−0.909036	−0.454518	+	0.890738i	$0.650188\pi$
$242$	− 2462.00i	− 0.653981i
$243$	0	0
$244$	916.000	0.240332
$245$	0	0
$246$	0	0
$247$	9040.00i	2.32875i
$248$	− 4536.00i	− 1.16144i
$249$	0	0
$250$	0	0
$251$	4980.00	1.25233	0.626165	−	0.779691i	$-0.284624\pi$
$251$	4980.00	1.25233	0.626165	+	0.779691i	$0.284624\pi$
$252$	0	0
$253$	810.000i	0.201282i
$254$	3220.00	0.795436
$255$	0	0
$256$	−4352.00	−1.06250
$257$	− 3357.00i	− 0.814801i	−0.913250	−	0.407401i	$-0.866435\pi$
$257$	− 3357.00i	− 0.814801i	0.913250	−	0.407401i	$-0.133565\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 4740.00i	− 1.11770i
$263$	− 4540.00i	− 1.06444i	−0.846605	−	0.532221i	$-0.821357\pi$
$263$	− 4540.00i	− 1.06444i	0.846605	−	0.532221i	$-0.178643\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	3000.00i	0.683784i
$269$	8410.00	1.90620	0.953098	−	0.302662i	$-0.0978752\pi$
$269$	8410.00	1.90620	0.953098	+	0.302662i	$0.0978752\pi$
$270$	0	0
$271$	259.000	0.0580558	0.0290279	−	0.999579i	$-0.490759\pi$
$271$	259.000	0.0580558	0.0290279	+	0.999579i	$0.490759\pi$
$272$	112.000i	0.0249669i
$273$	0	0
$274$	3594.00	0.792414
$275$	0	0
$276$	0	0
$277$	− 4170.00i	− 0.904516i	−0.891887	−	0.452258i	$-0.850619\pi$
$277$	− 4170.00i	− 0.904516i	0.891887	−	0.452258i	$-0.149381\pi$
$278$	248.000i	0.0535038i
$279$	0	0
$280$	0	0
$281$	1740.00	0.369394	0.184697	−	0.982796i	$-0.440870\pi$
$281$	1740.00	0.369394	0.184697	+	0.982796i	$0.440870\pi$
$282$	0	0
$283$	5070.00i	1.06495i	0.846446	+	0.532474i	$0.178738\pi$
$283$	5070.00i	1.06495i	−0.846446	+	0.532474i	$0.821262\pi$
$284$	−3560.00	−0.743828
$285$	0	0
$286$	1600.00	0.330804
$287$	0	0
$288$	0	0
$289$	4864.00	0.990026
$290$	0	0
$291$	0	0
$292$	3560.00i	0.713470i
$293$	− 159.000i	− 0.0317027i	−0.999874	−	0.0158513i	$-0.994954\pi$
$293$	− 159.000i	− 0.0317027i	0.999874	−	0.0158513i	$-0.00504585\pi$
$294$	0	0
$295$	0	0
$296$	−4080.00	−0.801166
$297$	0	0
$298$	− 140.000i	− 0.0272147i
$299$	6480.00	1.25334
$300$	0	0
$301$	0	0
$302$	4496.00i	0.856675i
$303$	0	0
$304$	−1808.00	−0.341105
$305$	0	0
$306$	0	0
$307$	6490.00i	1.20653i	0.797542	+	0.603264i	$0.206133\pi$
$307$	6490.00i	1.20653i	−0.797542	+	0.603264i	$0.793867\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8220.00	1.49876	0.749379	−	0.662142i	$-0.230352\pi$
$311$	8220.00	1.49876	0.749379	+	0.662142i	$0.230352\pi$
$312$	0	0
$313$	4660.00i	0.841530i	0.907170	+	0.420765i	$0.138238\pi$
$313$	4660.00i	0.841530i	−0.907170	+	0.420765i	$0.861762\pi$
$314$	−2020.00	−0.363042
$315$	0	0
$316$	108.000	0.0192262
$317$	6817.00i	1.20783i	0.797050	+	0.603913i	$0.206393\pi$
$317$	6817.00i	1.20783i	−0.797050	+	0.603913i	$0.793607\pi$
$318$	0	0
$319$	2200.00	0.386133
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 791.000i	− 0.136261i
$324$	0	0
$325$	0	0
$326$	1180.00	0.200473
$327$	0	0
$328$	3120.00i	0.525223i
$329$	0	0
$330$	0	0
$331$	192.000	0.0318830	0.0159415	−	0.999873i	$-0.494925\pi$
$331$	192.000	0.0318830	0.0159415	+	0.999873i	$0.494925\pi$
$332$	1716.00i	0.283668i
$333$	0	0
$334$	4806.00	0.787343
$335$	0	0
$336$	0	0
$337$	4840.00i	0.782349i	0.920317	+	0.391174i	$0.127931\pi$
$337$	4840.00i	0.782349i	−0.920317	+	0.391174i	$0.872069\pi$
$338$	− 8406.00i	− 1.35274i
$339$	0	0
$340$	0	0
$341$	1890.00	0.300144
$342$	0	0
$343$	0	0
$344$	240.000	0.0376161
$345$	0	0
$346$	−1602.00	−0.248913
$347$	860.000i	0.133047i	0.997785	+	0.0665234i	$0.0211907\pi$
$347$	860.000i	0.133047i	−0.997785	+	0.0665234i	$0.978809\pi$
$348$	0	0
$349$	−5377.00	−0.824711	−0.412356	−	0.911023i	$-0.635294\pi$
$349$	−5377.00	−0.824711	−0.412356	+	0.911023i	$0.635294\pi$
$350$	0	0
$351$	0	0
$352$	− 1600.00i	− 0.242274i
$353$	− 8010.00i	− 1.20773i	−0.797086	−	0.603866i	$-0.793626\pi$
$353$	− 8010.00i	− 1.20773i	0.797086	−	0.603866i	$-0.206374\pi$
$354$	0	0
$355$	0	0
$356$	−3000.00	−0.446628
$357$	0	0
$358$	− 4720.00i	− 0.696815i
$359$	12930.0	1.90089	0.950445	−	0.310894i	$-0.100628\pi$
$359$	12930.0	1.90089	0.950445	+	0.310894i	$0.100628\pi$
$360$	0	0
$361$	5910.00	0.861642
$362$	2482.00i	0.360362i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6000.00i	− 0.853399i	−0.904393	−	0.426700i	$-0.859676\pi$
$367$	− 6000.00i	− 0.853399i	0.904393	−	0.426700i	$-0.140324\pi$
$368$	1296.00i	0.183583i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	140.000i	0.0194341i	0.999953	+	0.00971706i	$0.00309308\pi$
$373$	140.000i	0.0194341i	−0.999953	+	0.00971706i	$0.996907\pi$
$374$	−140.000	−0.0193562
$375$	0	0
$376$	3840.00	0.526683
$377$	− 17600.0i	− 2.40437i
$378$	0	0
$379$	−6217.00	−0.842601	−0.421301	−	0.906921i	$-0.638426\pi$
$379$	−6217.00	−0.842601	−0.421301	+	0.906921i	$0.638426\pi$
$380$	0	0
$381$	0	0
$382$	9980.00i	1.33670i
$383$	4551.00i	0.607168i	0.952805	+	0.303584i	$0.0981833\pi$
$383$	4551.00i	0.607168i	−0.952805	+	0.303584i	$0.901817\pi$
$384$	0	0
$385$	0	0
$386$	−4520.00	−0.596015
$387$	0	0
$388$	− 5920.00i	− 0.774594i
$389$	−2310.00	−0.301084	−0.150542	−	0.988604i	$-0.548102\pi$
$389$	−2310.00	−0.301084	−0.150542	+	0.988604i	$0.548102\pi$
$390$	0	0
$391$	−567.000	−0.0733361
$392$	8232.00i	1.06066i
$393$	0	0
$394$	−4494.00	−0.574631
$395$	0	0
$396$	0	0
$397$	− 2900.00i	− 0.366617i	−0.983055	−	0.183308i	$-0.941319\pi$
$397$	− 2900.00i	− 0.366617i	0.983055	−	0.183308i	$-0.0586807\pi$
$398$	− 9128.00i	− 1.14961i
$399$	0	0
$400$	0	0
$401$	−2250.00	−0.280199	−0.140099	−	0.990137i	$-0.544742\pi$
$401$	−2250.00	−0.280199	−0.140099	+	0.990137i	$0.544742\pi$
$402$	0	0
$403$	− 15120.0i	− 1.86894i
$404$	6000.00	0.738889
$405$	0	0
$406$	0	0
$407$	− 1700.00i	− 0.207041i
$408$	0	0
$409$	11263.0	1.36166	0.680831	−	0.732441i	$-0.261619\pi$
$409$	11263.0	1.36166	0.680831	+	0.732441i	$0.261619\pi$
$410$	0	0
$411$	0	0
$412$	1840.00i	0.220025i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−12800.0	−1.50859
$417$	0	0
$418$	− 2260.00i	− 0.264450i
$419$	−6910.00	−0.805670	−0.402835	−	0.915273i	$-0.631975\pi$
$419$	−6910.00	−0.805670	−0.402835	+	0.915273i	$0.631975\pi$
$420$	0	0
$421$	−5249.00	−0.607650	−0.303825	−	0.952728i	$-0.598264\pi$
$421$	−5249.00	−0.607650	−0.303825	+	0.952728i	$0.598264\pi$
$422$	9898.00i	1.14177i
$423$	0	0
$424$	−15144.0	−1.73457
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1680.00i	0.189733i
$429$	0	0
$430$	0	0
$431$	11880.0	1.32770	0.663851	−	0.747865i	$-0.268921\pi$
$431$	11880.0	1.32770	0.663851	+	0.747865i	$0.268921\pi$
$432$	0	0
$433$	4280.00i	0.475020i	0.971385	+	0.237510i	$0.0763313\pi$
$433$	4280.00i	0.475020i	−0.971385	+	0.237510i	$0.923669\pi$
$434$	0	0
$435$	0	0
$436$	2428.00	0.266698
$437$	− 9153.00i	− 1.00194i
$438$	0	0
$439$	−6463.00	−0.702647	−0.351324	−	0.936254i	$-0.614268\pi$
$439$	−6463.00	−0.702647	−0.351324	+	0.936254i	$0.614268\pi$
$440$	0	0
$441$	0	0
$442$	1120.00i	0.120527i
$443$	11721.0i	1.25707i	0.777782	+	0.628534i	$0.216345\pi$
$443$	11721.0i	1.25707i	−0.777782	+	0.628534i	$0.783655\pi$
$444$	0	0
$445$	0	0
$446$	7780.00	0.825995
$447$	0	0
$448$	0	0
$449$	−2180.00	−0.229133	−0.114566	−	0.993416i	$-0.536548\pi$
$449$	−2180.00	−0.229133	−0.114566	+	0.993416i	$0.536548\pi$
$450$	0	0
$451$	−1300.00	−0.135731
$452$	8680.00i	0.903259i
$453$	0	0
$454$	4906.00	0.507159
$455$	0	0
$456$	0	0
$457$	− 17840.0i	− 1.82608i	−0.407866	−	0.913042i	$-0.633727\pi$
$457$	− 17840.0i	− 1.82608i	0.407866	−	0.913042i	$-0.366273\pi$
$458$	12426.0i	1.26775i
$459$	0	0
$460$	0	0
$461$	2250.00	0.227317	0.113658	−	0.993520i	$-0.463743\pi$
$461$	2250.00	0.227317	0.113658	+	0.993520i	$0.463743\pi$
$462$	0	0
$463$	− 1230.00i	− 0.123462i	−0.998093	−	0.0617310i	$-0.980338\pi$
$463$	− 1230.00i	− 0.123462i	0.998093	−	0.0617310i	$-0.0196621\pi$
$464$	3520.00	0.352181
$465$	0	0
$466$	6900.00	0.685915
$467$	− 5813.00i	− 0.576003i	−0.957630	−	0.288002i	$-0.907009\pi$
$467$	− 5813.00i	− 0.576003i	0.957630	−	0.288002i	$-0.0929909\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	− 13440.0i	− 1.31065i
$473$	100.000i	0.00972094i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	12980.0i	1.24203i
$479$	−6750.00	−0.643873	−0.321937	−	0.946761i	$-0.604334\pi$
$479$	−6750.00	−0.643873	−0.321937	+	0.946761i	$0.604334\pi$
$480$	0	0
$481$	−13600.0	−1.28920
$482$	− 6802.00i	− 0.642785i
$483$	0	0
$484$	−4924.00	−0.462434
$485$	0	0
$486$	0	0
$487$	− 6610.00i	− 0.615047i	−0.951541	−	0.307523i	$-0.900500\pi$
$487$	− 6610.00i	− 0.615047i	0.951541	−	0.307523i	$-0.0995002\pi$
$488$	5496.00i	0.509820i
$489$	0	0
$490$	0	0
$491$	4990.00	0.458647	0.229323	−	0.973350i	$-0.426349\pi$
$491$	4990.00	0.458647	0.229323	+	0.973350i	$0.426349\pi$
$492$	0	0
$493$	1540.00i	0.140686i
$494$	−18080.0	−1.64668
$495$	0	0
$496$	3024.00	0.273753
$497$	0	0
$498$	0	0
$499$	−1483.00	−0.133042	−0.0665212	−	0.997785i	$-0.521190\pi$
$499$	−1483.00	−0.133042	−0.0665212	+	0.997785i	$0.521190\pi$
$500$	0	0
$501$	0	0
$502$	9960.00i	0.885531i
$503$	− 11641.0i	− 1.03190i	−0.856618	−	0.515951i	$-0.827439\pi$
$503$	− 11641.0i	− 1.03190i	0.856618	−	0.515951i	$-0.172561\pi$
$504$	0	0
$505$	0	0
$506$	−1620.00	−0.142328
$507$	0	0
$508$	− 6440.00i	− 0.562458i
$509$	2620.00	0.228152	0.114076	−	0.993472i	$-0.463609\pi$
$509$	2620.00	0.228152	0.114076	+	0.993472i	$0.463609\pi$
$510$	0	0
$511$	0	0
$512$	− 5632.00i	− 0.486136i
$513$	0	0
$514$	6714.00	0.576151
$515$	0	0
$516$	0	0
$517$	1600.00i	0.136108i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13690.0	1.15119	0.575595	−	0.817735i	$-0.304771\pi$
$521$	13690.0	1.15119	0.575595	+	0.817735i	$0.304771\pi$
$522$	0	0
$523$	10220.0i	0.854473i	0.904140	+	0.427237i	$0.140513\pi$
$523$	10220.0i	0.854473i	−0.904140	+	0.427237i	$0.859487\pi$
$524$	−9480.00	−0.790335
$525$	0	0
$526$	9080.00	0.752675
$527$	1323.00i	0.109356i
$528$	0	0
$529$	5606.00	0.460754
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10400.0i	0.845167i
$534$	0	0
$535$	0	0
$536$	−18000.0	−1.45053
$537$	0	0
$538$	16820.0i	1.34788i
$539$	−3430.00	−0.274101
$540$	0	0
$541$	−2778.00	−0.220768	−0.110384	−	0.993889i	$-0.535208\pi$
$541$	−2778.00	−0.220768	−0.110384	+	0.993889i	$0.535208\pi$
$542$	518.000i	0.0410517i
$543$	0	0
$544$	1120.00	0.0882713
$545$	0	0
$546$	0	0
$547$	− 12830.0i	− 1.00287i	−0.865195	−	0.501436i	$-0.832805\pi$
$547$	− 12830.0i	− 1.00287i	0.865195	−	0.501436i	$-0.167195\pi$
$548$	− 7188.00i	− 0.560321i
$549$	0	0
$550$	0	0
$551$	−24860.0	−1.92209
$552$	0	0
$553$	0	0
$554$	8340.00	0.639590
$555$	0	0
$556$	496.000	0.0378329
$557$	4950.00i	0.376550i	0.982116	+	0.188275i	$0.0602896\pi$
$557$	4950.00i	0.376550i	−0.982116	+	0.188275i	$0.939710\pi$
$558$	0	0
$559$	800.000	0.0605302
$560$	0	0
$561$	0	0
$562$	3480.00i	0.261201i
$563$	6540.00i	0.489570i	0.969577	+	0.244785i	$0.0787174\pi$
$563$	6540.00i	0.489570i	−0.969577	+	0.244785i	$0.921283\pi$
$564$	0	0
$565$	0	0
$566$	−10140.0	−0.753032
$567$	0	0
$568$	− 21360.0i	− 1.57790i
$569$	−15240.0	−1.12284	−0.561418	−	0.827532i	$-0.689744\pi$
$569$	−15240.0	−1.12284	−0.561418	+	0.827532i	$0.689744\pi$
$570$	0	0
$571$	−5281.00	−0.387045	−0.193523	−	0.981096i	$-0.561991\pi$
$571$	−5281.00	−0.387045	−0.193523	+	0.981096i	$0.561991\pi$
$572$	− 3200.00i	− 0.233914i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 10510.0i	− 0.758296i	−0.925336	−	0.379148i	$-0.876217\pi$
$577$	− 10510.0i	− 0.758296i	0.925336	−	0.379148i	$-0.123783\pi$
$578$	9728.00i	0.700054i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 6310.00i	− 0.448256i
$584$	−21360.0	−1.51350
$585$	0	0
$586$	318.000	0.0224172
$587$	− 4107.00i	− 0.288780i	−0.989521	−	0.144390i	$-0.953878\pi$
$587$	− 4107.00i	− 0.288780i	0.989521	−	0.144390i	$-0.0461220\pi$
$588$	0	0
$589$	−21357.0	−1.49406
$590$	0	0
$591$	0	0
$592$	− 2720.00i	− 0.188837i
$593$	26129.0i	1.80943i	0.426022	+	0.904713i	$0.359915\pi$
$593$	26129.0i	1.80943i	−0.426022	+	0.904713i	$0.640085\pi$
$594$	0	0
$595$	0	0
$596$	−280.000	−0.0192437
$597$	0	0
$598$	12960.0i	0.886244i
$599$	−4360.00	−0.297404	−0.148702	−	0.988882i	$-0.547509\pi$
$599$	−4360.00	−0.297404	−0.148702	+	0.988882i	$0.547509\pi$
$600$	0	0
$601$	−16639.0	−1.12932	−0.564658	−	0.825325i	$-0.690992\pi$
$601$	−16639.0	−1.12932	−0.564658	+	0.825325i	$0.690992\pi$
$602$	0	0
$603$	0	0
$604$	8992.00	0.605760
$605$	0	0
$606$	0	0
$607$	490.000i	0.0327652i	0.999866	+	0.0163826i	$0.00521498\pi$
$607$	490.000i	0.0327652i	−0.999866	+	0.0163826i	$0.994785\pi$
$608$	18080.0i	1.20599i
$609$	0	0
$610$	0	0
$611$	12800.0	0.847516
$612$	0	0
$613$	− 18400.0i	− 1.21235i	−0.795332	−	0.606174i	$-0.792704\pi$
$613$	− 18400.0i	− 1.21235i	0.795332	−	0.606174i	$-0.207296\pi$
$614$	−12980.0	−0.853144
$615$	0	0
$616$	0	0
$617$	− 7827.00i	− 0.510702i	−0.966848	−	0.255351i	$-0.917809\pi$
$617$	− 7827.00i	− 0.510702i	0.966848	−	0.255351i	$-0.0821910\pi$
$618$	0	0
$619$	19756.0	1.28281	0.641406	−	0.767202i	$-0.278351\pi$
$619$	19756.0	1.28281	0.641406	+	0.767202i	$0.278351\pi$
$620$	0	0
$621$	0	0
$622$	16440.0i	1.05978i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−9320.00	−0.595051
$627$	0	0
$628$	4040.00i	0.256709i
$629$	1190.00	0.0754347
$630$	0	0
$631$	9829.00	0.620105	0.310053	−	0.950719i	$-0.399653\pi$
$631$	9829.00	0.620105	0.310053	+	0.950719i	$0.399653\pi$
$632$	648.000i	0.0407849i
$633$	0	0
$634$	−13634.0	−0.854062
$635$	0	0
$636$	0	0
$637$	27440.0i	1.70677i
$638$	4400.00i	0.273037i
$639$	0	0
$640$	0	0
$641$	6000.00	0.369713	0.184856	−	0.982766i	$-0.440818\pi$
$641$	6000.00	0.369713	0.184856	+	0.982766i	$0.440818\pi$
$642$	0	0
$643$	− 8280.00i	− 0.507825i	−0.967227	−	0.253912i	$-0.918283\pi$
$643$	− 8280.00i	− 0.507825i	0.967227	−	0.253912i	$-0.0817175\pi$
$644$	0	0
$645$	0	0
$646$	1582.00	0.0963513
$647$	16637.0i	1.01092i	0.862849	+	0.505462i	$0.168678\pi$
$647$	16637.0i	1.01092i	−0.862849	+	0.505462i	$0.831322\pi$
$648$	0	0
$649$	5600.00	0.338705
$650$	0	0
$651$	0	0
$652$	− 2360.00i	− 0.141756i
$653$	19751.0i	1.18364i	0.806070	+	0.591820i	$0.201590\pi$
$653$	19751.0i	1.18364i	−0.806070	+	0.591820i	$0.798410\pi$
$654$	0	0
$655$	0	0
$656$	−2080.00	−0.123796
$657$	0	0
$658$	0	0
$659$	14260.0	0.842930	0.421465	−	0.906845i	$-0.361516\pi$
$659$	14260.0	0.842930	0.421465	+	0.906845i	$0.361516\pi$
$660$	0	0
$661$	22318.0	1.31327	0.656634	−	0.754210i	$-0.271980\pi$
$661$	22318.0	1.31327	0.656634	+	0.754210i	$0.271980\pi$
$662$	384.000i	0.0225447i
$663$	0	0
$664$	−10296.0	−0.601750
$665$	0	0
$666$	0	0
$667$	17820.0i	1.03447i
$668$	− 9612.00i	− 0.556736i
$669$	0	0
$670$	0	0
$671$	−2290.00	−0.131750
$672$	0	0
$673$	− 20040.0i	− 1.14782i	−0.818917	−	0.573912i	$-0.805425\pi$
$673$	− 20040.0i	− 1.14782i	0.818917	−	0.573912i	$-0.194575\pi$
$674$	−9680.00	−0.553204
$675$	0	0
$676$	−16812.0	−0.956532
$677$	2310.00i	0.131138i	0.997848	+	0.0655691i	$0.0208863\pi$
$677$	2310.00i	0.131138i	−0.997848	+	0.0655691i	$0.979114\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	3780.00i	0.212234i
$683$	− 26739.0i	− 1.49801i	−0.662566	−	0.749004i	$-0.730532\pi$
$683$	− 26739.0i	− 1.49801i	0.662566	−	0.749004i	$-0.269468\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	160.000i	0.00886620i
$689$	−50480.0	−2.79120
$690$	0	0
$691$	5101.00	0.280827	0.140413	−	0.990093i	$-0.455157\pi$
$691$	5101.00	0.280827	0.140413	+	0.990093i	$0.455157\pi$
$692$	3204.00i	0.176008i
$693$	0	0
$694$	−1720.00	−0.0940783
$695$	0	0
$696$	0	0
$697$	− 910.000i	− 0.0494530i
$698$	− 10754.0i	− 0.583159i
$699$	0	0
$700$	0	0
$701$	−26030.0	−1.40248	−0.701241	−	0.712925i	$-0.747370\pi$
$701$	−26030.0	−1.40248	−0.701241	+	0.712925i	$0.747370\pi$
$702$	0	0
$703$	19210.0i	1.03061i
$704$	4480.00	0.239839
$705$	0	0
$706$	16020.0	0.853995
$707$	0	0
$708$	0	0
$709$	3854.00	0.204147	0.102073	−	0.994777i	$-0.467452\pi$
$709$	3854.00	0.204147	0.102073	+	0.994777i	$0.467452\pi$
$710$	0	0
$711$	0	0
$712$	− 18000.0i	− 0.947442i
$713$	15309.0i	0.804105i
$714$	0	0
$715$	0	0
$716$	−9440.00	−0.492723
$717$	0	0
$718$	25860.0i	1.34413i
$719$	−870.000	−0.0451259	−0.0225630	−	0.999745i	$-0.507183\pi$
$719$	−870.000	−0.0451259	−0.0225630	+	0.999745i	$0.507183\pi$
$720$	0	0
$721$	0	0
$722$	11820.0i	0.609273i
$723$	0	0
$724$	4964.00	0.254814
$725$	0	0
$726$	0	0
$727$	35780.0i	1.82532i	0.408721	+	0.912659i	$0.365975\pi$
$727$	35780.0i	1.82532i	−0.408721	+	0.912659i	$0.634025\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−70.0000	−0.00354178
$732$	0	0
$733$	3400.00i	0.171326i	0.996324	+	0.0856629i	$0.0273008\pi$
$733$	3400.00i	0.171326i	−0.996324	+	0.0856629i	$0.972699\pi$
$734$	12000.0	0.603444
$735$	0	0
$736$	12960.0	0.649065
$737$	− 7500.00i	− 0.374852i
$738$	0	0
$739$	683.000	0.0339981	0.0169990	−	0.999856i	$-0.494589\pi$
$739$	683.000	0.0339981	0.0169990	+	0.999856i	$0.494589\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 13400.0i	− 0.661640i	−0.943694	−	0.330820i	$-0.892675\pi$
$743$	− 13400.0i	− 0.661640i	0.943694	−	0.330820i	$-0.107325\pi$
$744$	0	0
$745$	0	0
$746$	−280.000	−0.0137420
$747$	0	0
$748$	280.000i	0.0136869i
$749$	0	0
$750$	0	0
$751$	−23219.0	−1.12819	−0.564097	−	0.825709i	$-0.690776\pi$
$751$	−23219.0	−1.12819	−0.564097	+	0.825709i	$0.690776\pi$
$752$	2560.00i	0.124140i
$753$	0	0
$754$	35200.0	1.70014
$755$	0	0
$756$	0	0
$757$	19630.0i	0.942489i	0.882003	+	0.471245i	$0.156195\pi$
$757$	19630.0i	0.942489i	−0.882003	+	0.471245i	$0.843805\pi$
$758$	− 12434.0i	− 0.595809i
$759$	0	0
$760$	0	0
$761$	−2940.00	−0.140046	−0.0700229	−	0.997545i	$-0.522307\pi$
$761$	−2940.00	−0.140046	−0.0700229	+	0.997545i	$0.522307\pi$
$762$	0	0
$763$	0	0
$764$	19960.0	0.945193
$765$	0	0
$766$	−9102.00	−0.429332
$767$	− 44800.0i	− 2.10904i
$768$	0	0
$769$	13987.0	0.655896	0.327948	−	0.944696i	$-0.393643\pi$
$769$	13987.0	0.655896	0.327948	+	0.944696i	$0.393643\pi$
$770$	0	0
$771$	0	0
$772$	9040.00i	0.421447i
$773$	19839.0i	0.923104i	0.887113	+	0.461552i	$0.152707\pi$
$773$	19839.0i	0.923104i	−0.887113	+	0.461552i	$0.847293\pi$
$774$	0	0
$775$	0	0
$776$	35520.0	1.64316
$777$	0	0
$778$	− 4620.00i	− 0.212898i
$779$	14690.0	0.675640
$780$	0	0
$781$	8900.00	0.407768
$782$	− 1134.00i	− 0.0518565i
$783$	0	0
$784$	−5488.00	−0.250000
$785$	0	0
$786$	0	0
$787$	− 38390.0i	− 1.73883i	−0.494086	−	0.869413i	$-0.664497\pi$
$787$	− 38390.0i	− 1.73883i	0.494086	−	0.869413i	$-0.335503\pi$
$788$	8988.00i	0.406325i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18320.0i	0.820381i
$794$	5800.00	0.259237
$795$	0	0
$796$	−18256.0	−0.812898
$797$	28027.0i	1.24563i	0.782369	+	0.622815i	$0.214011\pi$
$797$	28027.0i	1.24563i	−0.782369	+	0.622815i	$0.785989\pi$
$798$	0	0
$799$	−1120.00	−0.0495904
$800$	0	0
$801$	0	0
$802$	− 4500.00i	− 0.198130i
$803$	− 8900.00i	− 0.391126i
$804$	0	0
$805$	0	0
$806$	30240.0	1.32154
$807$	0	0
$808$	36000.0i	1.56742i
$809$	8630.00	0.375049	0.187525	−	0.982260i	$-0.439954\pi$
$809$	8630.00	0.375049	0.187525	+	0.982260i	$0.439954\pi$
$810$	0	0
$811$	−1932.00	−0.0836519	−0.0418260	−	0.999125i	$-0.513317\pi$
$811$	−1932.00	−0.0836519	−0.0418260	+	0.999125i	$0.513317\pi$
$812$	0	0
$813$	0	0
$814$	3400.00	0.146400
$815$	0	0
$816$	0	0
$817$	− 1130.00i	− 0.0483889i
$818$	22526.0i	0.962840i
$819$	0	0
$820$	0	0
$821$	18090.0	0.768996	0.384498	−	0.923126i	$-0.374375\pi$
$821$	18090.0	0.768996	0.384498	+	0.923126i	$0.374375\pi$
$822$	0	0
$823$	− 12890.0i	− 0.545950i	−0.962021	−	0.272975i	$-0.911992\pi$
$823$	− 12890.0i	− 0.545950i	0.962021	−	0.272975i	$-0.0880077\pi$
$824$	−11040.0	−0.466743
$825$	0	0
$826$	0	0
$827$	− 14887.0i	− 0.625963i	−0.949759	−	0.312982i	$-0.898672\pi$
$827$	− 14887.0i	− 0.625963i	0.949759	−	0.312982i	$-0.101328\pi$
$828$	0	0
$829$	−12666.0	−0.530649	−0.265325	−	0.964159i	$-0.585479\pi$
$829$	−12666.0	−0.530649	−0.265325	+	0.964159i	$0.585479\pi$
$830$	0	0
$831$	0	0
$832$	− 35840.0i	− 1.49342i
$833$	− 2401.00i	− 0.0998676i
$834$	0	0
$835$	0	0
$836$	−4520.00	−0.186995
$837$	0	0
$838$	− 13820.0i	− 0.569694i
$839$	43820.0	1.80314	0.901570	−	0.432633i	$-0.142416\pi$
$839$	43820.0	1.80314	0.901570	+	0.432633i	$0.142416\pi$
$840$	0	0
$841$	24011.0	0.984501
$842$	− 10498.0i	− 0.429673i
$843$	0	0
$844$	19796.0	0.807354
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 10096.0i	− 0.408842i
$849$	0	0
$850$	0	0
$851$	13770.0	0.554676
$852$	0	0
$853$	− 19320.0i	− 0.775503i	−0.921764	−	0.387752i	$-0.873252\pi$
$853$	− 19320.0i	− 0.775503i	0.921764	−	0.387752i	$-0.126748\pi$
$854$	0	0
$855$	0	0
$856$	−10080.0	−0.402485
$857$	3653.00i	0.145606i	0.997346	+	0.0728029i	$0.0231944\pi$
$857$	3653.00i	0.145606i	−0.997346	+	0.0728029i	$0.976806\pi$
$858$	0	0
$859$	−24373.0	−0.968098	−0.484049	−	0.875041i	$-0.660834\pi$
$859$	−24373.0	−0.968098	−0.484049	+	0.875041i	$0.660834\pi$
$860$	0	0
$861$	0	0
$862$	23760.0i	0.938827i
$863$	− 17629.0i	− 0.695363i	−0.937613	−	0.347681i	$-0.886969\pi$
$863$	− 17629.0i	− 0.695363i	0.937613	−	0.347681i	$-0.113031\pi$
$864$	0	0
$865$	0	0
$866$	−8560.00	−0.335890
$867$	0	0
$868$	0	0
$869$	−270.000	−0.0105398
$870$	0	0
$871$	−60000.0	−2.33412
$872$	14568.0i	0.565751i
$873$	0	0
$874$	18306.0	0.708478
$875$	0	0
$876$	0	0
$877$	− 21210.0i	− 0.816660i	−0.912834	−	0.408330i	$-0.866111\pi$
$877$	− 21210.0i	− 0.816660i	0.912834	−	0.408330i	$-0.133889\pi$
$878$	− 12926.0i	− 0.496847i
$879$	0	0
$880$	0	0
$881$	−39340.0	−1.50442	−0.752212	−	0.658921i	$-0.771013\pi$
$881$	−39340.0	−1.50442	−0.752212	+	0.658921i	$0.771013\pi$
$882$	0	0
$883$	4240.00i	0.161594i	0.996731	+	0.0807969i	$0.0257465\pi$
$883$	4240.00i	0.161594i	−0.996731	+	0.0807969i	$0.974253\pi$
$884$	2240.00	0.0852255
$885$	0	0
$886$	−23442.0	−0.888882
$887$	15933.0i	0.603132i	0.953445	+	0.301566i	$0.0975093\pi$
$887$	15933.0i	0.603132i	−0.953445	+	0.301566i	$0.902491\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 15560.0i	− 0.584067i
$893$	− 18080.0i	− 0.677519i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 4360.00i	− 0.162021i
$899$	41580.0	1.54257
$900$	0	0
$901$	4417.00	0.163320
$902$	− 2600.00i	− 0.0959762i
$903$	0	0
$904$	−52080.0	−1.91610
$905$	0	0
$906$	0	0
$907$	− 6780.00i	− 0.248210i	−0.992269	−	0.124105i	$-0.960394\pi$
$907$	− 6780.00i	− 0.248210i	0.992269	−	0.124105i	$-0.0396059\pi$
$908$	− 9812.00i	− 0.358615i
$909$	0	0
$910$	0	0
$911$	24740.0	0.899751	0.449875	−	0.893091i	$-0.351468\pi$
$911$	24740.0	0.899751	0.449875	+	0.893091i	$0.351468\pi$
$912$	0	0
$913$	− 4290.00i	− 0.155507i
$914$	35680.0	1.29124
$915$	0	0
$916$	24852.0	0.896434
$917$	0	0
$918$	0	0
$919$	48344.0	1.73528	0.867640	−	0.497194i	$-0.165636\pi$
$919$	48344.0	1.73528	0.867640	+	0.497194i	$0.165636\pi$
$920$	0	0
$921$	0	0
$922$	4500.00i	0.160737i
$923$	− 71200.0i	− 2.53909i
$924$	0	0
$925$	0	0
$926$	2460.00	0.0873009
$927$	0	0
$928$	− 35200.0i	− 1.24515i
$929$	−29650.0	−1.04713	−0.523566	−	0.851985i	$-0.675399\pi$
$929$	−29650.0	−1.04713	−0.523566	+	0.851985i	$0.675399\pi$
$930$	0	0
$931$	38759.0	1.36442
$932$	− 13800.0i	− 0.485015i
$933$	0	0
$934$	11626.0	0.407296
$935$	0	0
$936$	0	0
$937$	10260.0i	0.357716i	0.983875	+	0.178858i	$0.0572402\pi$
$937$	10260.0i	0.357716i	−0.983875	+	0.178858i	$0.942760\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	16270.0	0.563642	0.281821	−	0.959467i	$-0.409062\pi$
$941$	16270.0	0.563642	0.281821	+	0.959467i	$0.409062\pi$
$942$	0	0
$943$	− 10530.0i	− 0.363631i
$944$	8960.00	0.308923
$945$	0	0
$946$	−200.000	−0.00687374
$947$	23103.0i	0.792763i	0.918086	+	0.396382i	$0.129734\pi$
$947$	23103.0i	0.792763i	−0.918086	+	0.396382i	$0.870266\pi$
$948$	0	0
$949$	−71200.0	−2.43546
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 32090.0i	− 1.09076i	−0.838188	−	0.545381i	$-0.816385\pi$
$953$	− 32090.0i	− 1.09076i	0.838188	−	0.545381i	$-0.183615\pi$
$954$	0	0
$955$	0	0
$956$	25960.0	0.878249
$957$	0	0
$958$	− 13500.0i	− 0.455287i
$959$	0	0
$960$	0	0
$961$	5930.00	0.199053
$962$	− 27200.0i	− 0.911604i
$963$	0	0
$964$	−13604.0	−0.454518
$965$	0	0
$966$	0	0
$967$	− 42010.0i	− 1.39705i	−0.715584	−	0.698527i	$-0.753839\pi$
$967$	− 42010.0i	− 1.39705i	0.715584	−	0.698527i	$-0.246161\pi$
$968$	− 29544.0i	− 0.980971i
$969$	0	0
$970$	0	0
$971$	−17490.0	−0.578044	−0.289022	−	0.957322i	$-0.593330\pi$
$971$	−17490.0	−0.578044	−0.289022	+	0.957322i	$0.593330\pi$
$972$	0	0
$973$	0	0
$974$	13220.0	0.434904
$975$	0	0
$976$	−3664.00	−0.120166
$977$	22130.0i	0.724669i	0.932048	+	0.362334i	$0.118020\pi$
$977$	22130.0i	0.724669i	−0.932048	+	0.362334i	$0.881980\pi$
$978$	0	0
$979$	7500.00	0.244843
$980$	0	0
$981$	0	0
$982$	9980.00i	0.324312i
$983$	40959.0i	1.32898i	0.747296	+	0.664491i	$0.231352\pi$
$983$	40959.0i	1.32898i	−0.747296	+	0.664491i	$0.768648\pi$
$984$	0	0
$985$	0	0
$986$	−3080.00	−0.0994799
$987$	0	0
$988$	36160.0i	1.16438i
$989$	−810.000	−0.0260430
$990$	0	0
$991$	61169.0	1.96074	0.980372	−	0.197157i	$-0.0631707\pi$
$991$	61169.0	1.96074	0.980372	+	0.197157i	$0.0631707\pi$
$992$	− 30240.0i	− 0.967864i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26190.0i	0.831941i	0.909378	+	0.415971i	$0.136558\pi$
$997$	26190.0i	0.831941i	−0.909378	+	0.415971i	$0.863442\pi$
$998$	− 2966.00i	− 0.0940752i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.c.649.2		2
3.2	odd	2		675.4.b.d.649.1		2
5.2	odd	4		675.4.a.b.1.1		1
5.3	odd	4		135.4.a.d.1.1	yes	1
5.4	even	2	inner	675.4.b.c.649.1		2
15.2	even	4		675.4.a.i.1.1		1
15.8	even	4		135.4.a.a.1.1	✓	1
15.14	odd	2		675.4.b.d.649.2		2
20.3	even	4		2160.4.a.d.1.1		1
45.13	odd	12		405.4.e.e.136.1		2
45.23	even	12		405.4.e.j.136.1		2
45.38	even	12		405.4.e.j.271.1		2
45.43	odd	12		405.4.e.e.271.1		2
60.23	odd	4		2160.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.a.1.1	✓	1	15.8	even	4
135.4.a.d.1.1	yes	1	5.3	odd	4
405.4.e.e.136.1		2	45.13	odd	12
405.4.e.e.271.1		2	45.43	odd	12
405.4.e.j.136.1		2	45.23	even	12
405.4.e.j.271.1		2	45.38	even	12
675.4.a.b.1.1		1	5.2	odd	4
675.4.a.i.1.1		1	15.2	even	4
675.4.b.c.649.1		2	5.4	even	2	inner
675.4.b.c.649.2		2	1.1	even	1	trivial
675.4.b.d.649.1		2	3.2	odd	2
675.4.b.d.649.2		2	15.14	odd	2
2160.4.a.d.1.1		1	20.3	even	4
2160.4.a.n.1.1		1	60.23	odd	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +4.00000 q^{4} +24.0000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} +4.00000 q^{4} +24.0000i q^{8} -10.0000 q^{11} +80.0000i q^{13} -16.0000 q^{16} -7.00000i q^{17} +113.000 q^{19} -20.0000i q^{22} -81.0000i q^{23} -160.000 q^{26} -220.000 q^{29} -189.000 q^{31} +160.000i q^{32} +14.0000 q^{34} +170.000i q^{37} +226.000i q^{38} +130.000 q^{41} -10.0000i q^{43} -40.0000 q^{44} +162.000 q^{46} -160.000i q^{47} +343.000 q^{49} +320.000i q^{52} +631.000i q^{53} -440.000i q^{58} -560.000 q^{59} +229.000 q^{61} -378.000i q^{62} -448.000 q^{64} +750.000i q^{67} -28.0000i q^{68} -890.000 q^{71} +890.000i q^{73} -340.000 q^{74} +452.000 q^{76} +27.0000 q^{79} +260.000i q^{82} +429.000i q^{83} +20.0000 q^{86} -240.000i q^{88} -750.000 q^{89} -324.000i q^{92} +320.000 q^{94} -1480.00i q^{97} +686.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4}+O(q^{10}) \) 2 * q + 8 * q^4	\( 2 q + 8 q^{4} - 20 q^{11} - 32 q^{16} + 226 q^{19} - 320 q^{26} - 440 q^{29} - 378 q^{31} + 28 q^{34} + 260 q^{41} - 80 q^{44} + 324 q^{46} + 686 q^{49} - 1120 q^{59} + 458 q^{61} - 896 q^{64} - 1780 q^{71} - 680 q^{74} + 904 q^{76} + 54 q^{79} + 40 q^{86} - 1500 q^{89} + 640 q^{94}+O(q^{100}) \) 2 * q + 8 * q^4 - 20 * q^11 - 32 * q^16 + 226 * q^19 - 320 * q^26 - 440 * q^29 - 378 * q^31 + 28 * q^34 + 260 * q^41 - 80 * q^44 + 324 * q^46 + 686 * q^49 - 1120 * q^59 + 458 * q^61 - 896 * q^64 - 1780 * q^71 - 680 * q^74 + 904 * q^76 + 54 * q^79 + 40 * q^86 - 1500 * q^89 + 640 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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