LMFDB - Embedded newform 2475.2.d.a.2474.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(2474,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2475.2474");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.d (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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2474.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2475.2474
Dual form			2475.2.d.a.2474.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} -1.00000 q^{4} -2.44949 q^{7} -1.73205i q^{8} +(-1.73205 + 2.82843i) q^{11} +4.89898 q^{13} +4.24264i q^{14} -5.00000 q^{16} -7.34847i q^{19} +(4.89898 + 3.00000i) q^{22} +2.82843 q^{23} -8.48528i q^{26} +2.44949 q^{28} -6.92820 q^{29} -4.00000 q^{31} +5.19615i q^{32} -8.00000i q^{37} -12.7279 q^{38} -6.92820 q^{41} +2.44949 q^{43} +(1.73205 - 2.82843i) q^{44} -4.89898i q^{46} -2.82843 q^{47} -1.00000 q^{49} -4.89898 q^{52} -9.89949 q^{53} +4.24264i q^{56} +12.0000i q^{58} +11.3137i q^{59} +4.89898i q^{61} +6.92820i q^{62} -1.00000 q^{64} +4.00000i q^{67} -2.82843i q^{71} -13.8564 q^{74} +7.34847i q^{76} +(4.24264 - 6.92820i) q^{77} -12.2474i q^{79} +12.0000i q^{82} -13.8564i q^{83} -4.24264i q^{86} +(4.89898 + 3.00000i) q^{88} +7.07107i q^{89} -12.0000 q^{91} -2.82843 q^{92} +4.89898i q^{94} +10.0000i q^{97} +1.73205i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 40 q^{16} - 32 q^{31} - 8 q^{49} - 8 q^{64} - 96 q^{91}+O(q^{100}) $ 8 * q - 8 * q^4 - 40 * q^16 - 32 * q^31 - 8 * q^49 - 8 * q^64 - 96 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$		−	1.73205i		−	1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$		−	1.73205i		−	1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.44949			−0.925820			−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949			−0.925820			−0.462910	+	0.886405i	$0.653195\pi$
$8$		−	1.73205i		−	0.612372i
$9$		0			0
$10$		0			0
$11$	−1.73205	+	2.82843i	−0.522233	+	0.852803i
$11$	−1.73205	+	2.82843i	−0.522233	+	0.852803i
$12$		0			0
$13$	4.89898			1.35873			0.679366	−	0.733799i	$-0.262255\pi$
$13$	4.89898			1.35873			0.679366	+	0.733799i	$0.262255\pi$
$14$			4.24264i			1.13389i
$15$		0			0
$16$	−5.00000			−1.25000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		−	7.34847i		−	1.68585i	−0.538028	−	0.842927i	$-0.680830\pi$
$19$		−	7.34847i		−	1.68585i	0.538028	−	0.842927i	$-0.319170\pi$
$20$		0			0
$21$		0			0
$22$	4.89898	+	3.00000i	1.04447	+	0.639602i
$23$	2.82843			0.589768			0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843			0.589768			0.294884	+	0.955533i	$0.404719\pi$
$24$		0			0
$25$		0			0
$26$		−	8.48528i		−	1.66410i
$27$		0			0
$28$	2.44949			0.462910
$29$	−6.92820			−1.28654			−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820			−1.28654			−0.643268	+	0.765641i	$0.722422\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$			5.19615i			0.918559i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	−12.7279			−2.06474
$39$		0			0
$40$		0			0
$41$	−6.92820			−1.08200			−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820			−1.08200			−0.541002	+	0.841021i	$0.681955\pi$
$42$		0			0
$43$	2.44949			0.373544			0.186772	−	0.982403i	$-0.440197\pi$
$43$	2.44949			0.373544			0.186772	+	0.982403i	$0.440197\pi$
$44$	1.73205	−	2.82843i	0.261116	−	0.426401i
$45$		0			0
$46$		−	4.89898i		−	0.722315i
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$	−4.89898			−0.679366
$53$	−9.89949			−1.35980			−0.679900	−	0.733305i	$-0.737977\pi$
$53$	−9.89949			−1.35980			−0.679900	+	0.733305i	$0.737977\pi$
$54$		0			0
$55$		0			0
$56$			4.24264i			0.566947i
$57$		0			0
$58$			12.0000i			1.57568i
$59$			11.3137i			1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$			11.3137i			1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$		0			0
$61$			4.89898i			0.627250i	0.949547	+	0.313625i	$0.101543\pi$
$61$			4.89898i			0.627250i	−0.949547	+	0.313625i	$0.898457\pi$
$62$			6.92820i			0.879883i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.82843i		−	0.335673i	−0.985815	−	0.167836i	$-0.946322\pi$
$71$		−	2.82843i		−	0.335673i	0.985815	−	0.167836i	$-0.0536780\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	−13.8564			−1.61077
$75$		0			0
$76$			7.34847i			0.842927i
$77$	4.24264	−	6.92820i	0.483494	−	0.789542i
$78$		0			0
$79$		−	12.2474i		−	1.37795i	−0.724787	−	0.688973i	$-0.758062\pi$
$79$		−	12.2474i		−	1.37795i	0.724787	−	0.688973i	$-0.241938\pi$
$80$		0			0
$81$		0			0
$82$			12.0000i			1.32518i
$83$		−	13.8564i		−	1.52094i	−0.649374	−	0.760469i	$-0.724969\pi$
$83$		−	13.8564i		−	1.52094i	0.649374	−	0.760469i	$-0.275031\pi$
$84$		0			0
$85$		0			0
$86$		−	4.24264i		−	0.457496i
$87$		0			0
$88$	4.89898	+	3.00000i	0.522233	+	0.319801i
$89$			7.07107i			0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$			7.07107i			0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$	−2.82843			−0.294884
$93$		0			0
$94$			4.89898i			0.505291i
$95$		0			0
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$			1.73205i			0.174964i
$99$		0			0
$100$		0			0
$101$	−13.8564			−1.37876			−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564			−1.37876			−0.689382	+	0.724398i	$0.742118\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		−	8.48528i		−	0.832050i
$105$		0			0
$106$			17.1464i			1.66541i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$			14.6969i			1.40771i	0.710343	+	0.703856i	$0.248540\pi$
$109$			14.6969i			1.40771i	−0.710343	+	0.703856i	$0.751460\pi$
$110$		0			0
$111$		0			0
$112$	12.2474			1.15728
$113$	−1.41421			−0.133038			−0.0665190	−	0.997785i	$-0.521189\pi$
$113$	−1.41421			−0.133038			−0.0665190	+	0.997785i	$0.521189\pi$
$114$		0			0
$115$		0			0
$116$	6.92820			0.643268
$117$		0			0
$118$	19.5959			1.80395
$119$		0			0
$120$		0			0
$121$	−5.00000	−	9.79796i	−0.454545	−	0.890724i
$122$	8.48528			0.768221
$123$		0			0
$124$	4.00000			0.359211
$125$		0			0
$126$		0			0
$127$	7.34847			0.652071			0.326036	−	0.945357i	$-0.394287\pi$
$127$	7.34847			0.652071			0.326036	+	0.945357i	$0.394287\pi$
$128$			12.1244i			1.07165i
$129$		0			0
$130$		0			0
$131$	3.46410			0.302660			0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410			0.302660			0.151330	+	0.988483i	$0.451644\pi$
$132$		0			0
$133$			18.0000i			1.56080i
$134$	6.92820			0.598506
$135$		0			0
$136$		0			0
$137$	−15.5563			−1.32907			−0.664534	−	0.747258i	$-0.731370\pi$
$137$	−15.5563			−1.32907			−0.664534	+	0.747258i	$0.731370\pi$
$138$		0			0
$139$		−	2.44949i		−	0.207763i	−0.994590	−	0.103882i	$-0.966874\pi$
$139$		−	2.44949i		−	0.207763i	0.994590	−	0.103882i	$-0.0331263\pi$
$140$		0			0
$141$		0			0
$142$	−4.89898			−0.411113
$143$	−8.48528	+	13.8564i	−0.709575	+	1.15873i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$			8.00000i			0.657596i
$149$	−13.8564			−1.13516			−0.567581	−	0.823318i	$-0.692120\pi$
$149$	−13.8564			−1.13516			−0.567581	+	0.823318i	$0.692120\pi$
$150$		0			0
$151$		−	2.44949i		−	0.199337i	−0.995021	−	0.0996683i	$-0.968222\pi$
$151$		−	2.44949i		−	0.199337i	0.995021	−	0.0996683i	$-0.0317782\pi$
$152$	−12.7279			−1.03237
$153$		0			0
$154$	−12.0000	−	7.34847i	−0.966988	−	0.592157i
$155$		0			0
$156$		0			0
$157$		−	14.0000i		−	1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$		−	14.0000i		−	1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	−21.2132			−1.68763
$159$		0			0
$160$		0			0
$161$	−6.92820			−0.546019
$162$		0			0
$163$			8.00000i			0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$			8.00000i			0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	6.92820			0.541002
$165$		0			0
$166$	−24.0000			−1.86276
$167$		−	3.46410i		−	0.268060i	−0.990977	−	0.134030i	$-0.957208\pi$
$167$		−	3.46410i		−	0.268060i	0.990977	−	0.134030i	$-0.0427919\pi$
$168$		0			0
$169$	11.0000			0.846154
$170$		0			0
$171$		0			0
$172$	−2.44949			−0.186772
$173$			6.92820i			0.526742i	0.964695	+	0.263371i	$0.0848343\pi$
$173$			6.92820i			0.526742i	−0.964695	+	0.263371i	$0.915166\pi$
$174$		0			0
$175$		0			0
$176$	8.66025	−	14.1421i	0.652791	−	1.06600i
$177$		0			0
$178$	12.2474			0.917985
$179$		−	22.6274i		−	1.69125i	−0.533775	−	0.845626i	$-0.679227\pi$
$179$		−	22.6274i		−	1.69125i	0.533775	−	0.845626i	$-0.320773\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$			20.7846i			1.54066i
$183$		0			0
$184$		−	4.89898i		−	0.361158i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	2.82843			0.206284
$189$		0			0
$190$		0			0
$191$		−	2.82843i		−	0.204658i	−0.994751	−	0.102329i	$-0.967371\pi$
$191$		−	2.82843i		−	0.204658i	0.994751	−	0.102329i	$-0.0326294\pi$
$192$		0			0
$193$	−9.79796			−0.705273			−0.352636	−	0.935760i	$-0.614715\pi$
$193$	−9.79796			−0.705273			−0.352636	+	0.935760i	$0.614715\pi$
$194$	17.3205			1.24354
$195$		0			0
$196$	1.00000			0.0714286
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$		0			0
$202$			24.0000i			1.68863i
$203$	16.9706			1.19110
$204$		0			0
$205$		0			0
$206$	−6.92820			−0.482711
$207$		0			0
$208$	−24.4949			−1.69842
$209$	20.7846	+	12.7279i	1.43770	+	0.880409i
$210$		0			0
$211$		−	12.2474i		−	0.843149i	−0.906794	−	0.421575i	$-0.861478\pi$
$211$		−	12.2474i		−	0.843149i	0.906794	−	0.421575i	$-0.138522\pi$
$212$	9.89949			0.679900
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	9.79796			0.665129
$218$	25.4558			1.72409
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		−	12.7279i		−	0.850420i
$225$		0			0
$226$			2.44949i			0.162938i
$227$		−	27.7128i		−	1.83936i	−0.392664	−	0.919682i	$-0.628446\pi$
$227$		−	27.7128i		−	1.83936i	0.392664	−	0.919682i	$-0.371554\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$			12.0000i			0.787839i
$233$			20.7846i			1.36165i	0.732448	+	0.680823i	$0.238378\pi$
$233$			20.7846i			1.36165i	−0.732448	+	0.680823i	$0.761622\pi$
$234$		0			0
$235$		0			0
$236$		−	11.3137i		−	0.736460i
$237$		0			0
$238$		0			0
$239$	17.3205			1.12037			0.560185	−	0.828367i	$-0.310730\pi$
$239$	17.3205			1.12037			0.560185	+	0.828367i	$0.310730\pi$
$240$		0			0
$241$			19.5959i			1.26228i	0.775667	+	0.631142i	$0.217413\pi$
$241$			19.5959i			1.26228i	−0.775667	+	0.631142i	$0.782587\pi$
$242$	−16.9706	+	8.66025i	−1.09091	+	0.556702i
$243$		0			0
$244$		−	4.89898i		−	0.313625i
$245$		0			0
$246$		0			0
$247$		−	36.0000i		−	2.29063i
$248$			6.92820i			0.439941i
$249$		0			0
$250$		0			0
$251$			5.65685i			0.357057i	0.983935	+	0.178529i	$0.0571337\pi$
$251$			5.65685i			0.357057i	−0.983935	+	0.178529i	$0.942866\pi$
$252$		0			0
$253$	−4.89898	+	8.00000i	−0.307996	+	0.502956i
$254$		−	12.7279i		−	0.798621i
$255$		0			0
$256$	19.0000			1.18750
$257$	9.89949			0.617514			0.308757	−	0.951141i	$-0.400087\pi$
$257$	9.89949			0.617514			0.308757	+	0.951141i	$0.400087\pi$
$258$		0			0
$259$			19.5959i			1.21763i
$260$		0			0
$261$		0			0
$262$		−	6.00000i		−	0.370681i
$263$		−	3.46410i		−	0.213606i	−0.994280	−	0.106803i	$-0.965939\pi$
$263$		−	3.46410i		−	0.213606i	0.994280	−	0.106803i	$-0.0340614\pi$
$264$		0			0
$265$		0			0
$266$	31.1769			1.91158
$267$		0			0
$268$		−	4.00000i		−	0.244339i
$269$			15.5563i			0.948487i	0.880394	+	0.474244i	$0.157278\pi$
$269$			15.5563i			0.948487i	−0.880394	+	0.474244i	$0.842722\pi$
$270$		0			0
$271$		−	7.34847i		−	0.446388i	−0.974774	−	0.223194i	$-0.928352\pi$
$271$		−	7.34847i		−	0.446388i	0.974774	−	0.223194i	$-0.0716483\pi$
$272$		0			0
$273$		0			0
$274$			26.9444i			1.62777i
$275$		0			0
$276$		0			0
$277$	−24.4949			−1.47176			−0.735878	−	0.677114i	$-0.763230\pi$
$277$	−24.4949			−1.47176			−0.735878	+	0.677114i	$0.763230\pi$
$278$	−4.24264			−0.254457
$279$		0			0
$280$		0			0
$281$	−13.8564			−0.826604			−0.413302	−	0.910594i	$-0.635625\pi$
$281$	−13.8564			−0.826604			−0.413302	+	0.910594i	$0.635625\pi$
$282$		0			0
$283$	−2.44949			−0.145607			−0.0728035	−	0.997346i	$-0.523195\pi$
$283$	−2.44949			−0.145607			−0.0728035	+	0.997346i	$0.523195\pi$
$284$			2.82843i			0.167836i
$285$		0			0
$286$	24.0000	+	14.6969i	1.41915	+	0.869048i
$287$	16.9706			1.00174
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.92820i		−	0.404750i	−0.979308	−	0.202375i	$-0.935134\pi$
$293$		−	6.92820i		−	0.404750i	0.979308	−	0.202375i	$-0.0648660\pi$
$294$		0			0
$295$		0			0
$296$	−13.8564			−0.805387
$297$		0			0
$298$			24.0000i			1.39028i
$299$	13.8564			0.801337
$300$		0			0
$301$	−6.00000			−0.345834
$302$	−4.24264			−0.244137
$303$		0			0
$304$			36.7423i			2.10732i
$305$		0			0
$306$		0			0
$307$	7.34847			0.419399			0.209700	−	0.977766i	$-0.432751\pi$
$307$	7.34847			0.419399			0.209700	+	0.977766i	$0.432751\pi$
$308$	−4.24264	+	6.92820i	−0.241747	+	0.394771i
$309$		0			0
$310$		0			0
$311$			14.1421i			0.801927i	0.916094	+	0.400963i	$0.131325\pi$
$311$			14.1421i			0.801927i	−0.916094	+	0.400963i	$0.868675\pi$
$312$		0			0
$313$		−	16.0000i		−	0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$		−	16.0000i		−	0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	−24.2487			−1.36843
$315$		0			0
$316$			12.2474i			0.688973i
$317$	18.3848			1.03259			0.516296	−	0.856410i	$-0.327310\pi$
$317$	18.3848			1.03259			0.516296	+	0.856410i	$0.327310\pi$
$318$		0			0
$319$	12.0000	−	19.5959i	0.671871	−	1.09716i
$320$		0			0
$321$		0			0
$322$			12.0000i			0.668734i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	13.8564			0.767435
$327$		0			0
$328$			12.0000i			0.662589i
$329$	6.92820			0.381964
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$			13.8564i			0.760469i
$333$		0			0
$334$	−6.00000			−0.328305
$335$		0			0
$336$		0			0
$337$	9.79796			0.533729			0.266864	−	0.963734i	$-0.414012\pi$
$337$	9.79796			0.533729			0.266864	+	0.963734i	$0.414012\pi$
$338$		−	19.0526i		−	1.03632i
$339$		0			0
$340$		0			0
$341$	6.92820	−	11.3137i	0.375183	−	0.612672i
$342$		0			0
$343$	19.5959			1.05808
$344$		−	4.24264i		−	0.228748i
$345$		0			0
$346$	12.0000			0.645124
$347$		−	13.8564i		−	0.743851i	−0.928263	−	0.371925i	$-0.878698\pi$
$347$		−	13.8564i		−	0.743851i	0.928263	−	0.371925i	$-0.121302\pi$
$348$		0			0
$349$		−	4.89898i		−	0.262236i	−0.991367	−	0.131118i	$-0.958143\pi$
$349$		−	4.89898i		−	0.262236i	0.991367	−	0.131118i	$-0.0418567\pi$
$350$		0			0
$351$		0			0
$352$	−14.6969	−	9.00000i	−0.783349	−	0.479702i
$353$	32.5269			1.73123			0.865616	−	0.500708i	$-0.166927\pi$
$353$	32.5269			1.73123			0.865616	+	0.500708i	$0.166927\pi$
$354$		0			0
$355$		0			0
$356$		−	7.07107i		−	0.374766i
$357$		0			0
$358$	−39.1918			−2.07135
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−35.0000			−1.84211
$362$		−	13.8564i		−	0.728277i
$363$		0			0
$364$	12.0000			0.628971
$365$		0			0
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	−14.1421			−0.737210
$369$		0			0
$370$		0			0
$371$	24.2487			1.25893
$372$		0			0
$373$	−24.4949			−1.26830			−0.634149	−	0.773211i	$-0.718649\pi$
$373$	−24.4949			−1.26830			−0.634149	+	0.773211i	$0.718649\pi$
$374$		0			0
$375$		0			0
$376$			4.89898i			0.252646i
$377$	−33.9411			−1.74806
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$	−4.89898			−0.250654
$383$	36.7696			1.87884			0.939418	−	0.342773i	$-0.111366\pi$
$383$	36.7696			1.87884			0.939418	+	0.342773i	$0.111366\pi$
$384$		0			0
$385$		0			0
$386$			16.9706i			0.863779i
$387$		0			0
$388$		−	10.0000i		−	0.507673i
$389$		−	9.89949i		−	0.501924i	−0.967997	−	0.250962i	$-0.919253\pi$
$389$		−	9.89949i		−	0.501924i	0.967997	−	0.250962i	$-0.0807470\pi$
$390$		0			0
$391$		0			0
$392$			1.73205i			0.0874818i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$			13.8564i			0.694559i
$399$		0			0
$400$		0			0
$401$			1.41421i			0.0706225i	0.999376	+	0.0353112i	$0.0112422\pi$
$401$			1.41421i			0.0706225i	−0.999376	+	0.0353112i	$0.988758\pi$
$402$		0			0
$403$	−19.5959			−0.976142
$404$	13.8564			0.689382
$405$		0			0
$406$		−	29.3939i		−	1.45879i
$407$	22.6274	+	13.8564i	1.12160	+	0.686837i
$408$		0			0
$409$		−	39.1918i		−	1.93791i	−0.247234	−	0.968956i	$-0.579522\pi$
$409$		−	39.1918i		−	1.93791i	0.247234	−	0.968956i	$-0.420478\pi$
$410$		0			0
$411$		0			0
$412$			4.00000i			0.197066i
$413$		−	27.7128i		−	1.36366i
$414$		0			0
$415$		0			0
$416$			25.4558i			1.24808i
$417$		0			0
$418$	22.0454	−	36.0000i	1.07828	−	1.76082i
$419$			11.3137i			0.552711i	0.961056	+	0.276355i	$0.0891267\pi$
$419$			11.3137i			0.552711i	−0.961056	+	0.276355i	$0.910873\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$	−21.2132			−1.03264
$423$		0			0
$424$			17.1464i			0.832704i
$425$		0			0
$426$		0			0
$427$		−	12.0000i		−	0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	10.3923			0.500580			0.250290	−	0.968171i	$-0.419474\pi$
$431$	10.3923			0.500580			0.250290	+	0.968171i	$0.419474\pi$
$432$		0			0
$433$		−	16.0000i		−	0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$		−	16.0000i		−	0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$		−	16.9706i		−	0.814613i
$435$		0			0
$436$		−	14.6969i		−	0.703856i
$437$		−	20.7846i		−	0.994263i
$438$		0			0
$439$			2.44949i			0.116908i	0.998290	+	0.0584539i	$0.0186171\pi$
$439$			2.44949i			0.116908i	−0.998290	+	0.0584539i	$0.981383\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.6274			−1.07506			−0.537531	−	0.843244i	$-0.680643\pi$
$443$	−22.6274			−1.07506			−0.537531	+	0.843244i	$0.680643\pi$
$444$		0			0
$445$		0			0
$446$	−27.7128			−1.31224
$447$		0			0
$448$	2.44949			0.115728
$449$		−	9.89949i		−	0.467186i	−0.972334	−	0.233593i	$-0.924952\pi$
$449$		−	9.89949i		−	0.467186i	0.972334	−	0.233593i	$-0.0750483\pi$
$450$		0			0
$451$	12.0000	−	19.5959i	0.565058	−	0.922736i
$452$	1.41421			0.0665190
$453$		0			0
$454$	−48.0000			−2.25275
$455$		0			0
$456$		0			0
$457$	−9.79796			−0.458329			−0.229165	−	0.973388i	$-0.573599\pi$
$457$	−9.79796			−0.458329			−0.229165	+	0.973388i	$0.573599\pi$
$458$		−	17.3205i		−	0.809334i
$459$		0			0
$460$		0			0
$461$	6.92820			0.322679			0.161339	−	0.986899i	$-0.448419\pi$
$461$	6.92820			0.322679			0.161339	+	0.986899i	$0.448419\pi$
$462$		0			0
$463$		−	28.0000i		−	1.30127i	−0.759390	−	0.650635i	$-0.774503\pi$
$463$		−	28.0000i		−	1.30127i	0.759390	−	0.650635i	$-0.225497\pi$
$464$	34.6410			1.60817
$465$		0			0
$466$	36.0000			1.66767
$467$	22.6274			1.04707			0.523536	−	0.852004i	$-0.324613\pi$
$467$	22.6274			1.04707			0.523536	+	0.852004i	$0.324613\pi$
$468$		0			0
$469$		−	9.79796i		−	0.452428i
$470$		0			0
$471$		0			0
$472$	19.5959			0.901975
$473$	−4.24264	+	6.92820i	−0.195077	+	0.318559i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		−	30.0000i		−	1.37217i
$479$	−38.1051			−1.74107			−0.870534	−	0.492109i	$-0.836226\pi$
$479$	−38.1051			−1.74107			−0.870534	+	0.492109i	$0.836226\pi$
$480$		0			0
$481$		−	39.1918i		−	1.78699i
$482$	33.9411			1.54598
$483$		0			0
$484$	5.00000	+	9.79796i	0.227273	+	0.445362i
$485$		0			0
$486$		0			0
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	8.48528			0.384111
$489$		0			0
$490$		0			0
$491$	−27.7128			−1.25066			−0.625331	−	0.780360i	$-0.715036\pi$
$491$	−27.7128			−1.25066			−0.625331	+	0.780360i	$0.715036\pi$
$492$		0			0
$493$		0			0
$494$	−62.3538			−2.80543
$495$		0			0
$496$	20.0000			0.898027
$497$			6.92820i			0.310772i
$498$		0			0
$499$	−32.0000			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$	9.79796			0.437304
$503$			31.1769i			1.39011i	0.718957	+	0.695055i	$0.244620\pi$
$503$			31.1769i			1.39011i	−0.718957	+	0.695055i	$0.755380\pi$
$504$		0			0
$505$		0			0
$506$	13.8564	+	8.48528i	0.615992	+	0.377217i
$507$		0			0
$508$	−7.34847			−0.326036
$509$			15.5563i			0.689523i	0.938690	+	0.344762i	$0.112040\pi$
$509$			15.5563i			0.689523i	−0.938690	+	0.344762i	$0.887960\pi$
$510$		0			0
$511$		0			0
$512$		−	8.66025i		−	0.382733i
$513$		0			0
$514$		−	17.1464i		−	0.756297i
$515$		0			0
$516$		0			0
$517$	4.89898	−	8.00000i	0.215457	−	0.351840i
$518$	33.9411			1.49129
$519$		0			0
$520$		0			0
$521$		−	15.5563i		−	0.681536i	−0.940147	−	0.340768i	$-0.889313\pi$
$521$		−	15.5563i		−	0.681536i	0.940147	−	0.340768i	$-0.110687\pi$
$522$		0			0
$523$	−22.0454			−0.963978			−0.481989	−	0.876177i	$-0.660086\pi$
$523$	−22.0454			−0.963978			−0.481989	+	0.876177i	$0.660086\pi$
$524$	−3.46410			−0.151330
$525$		0			0
$526$	−6.00000			−0.261612
$527$		0			0
$528$		0			0
$529$	−15.0000			−0.652174
$530$		0			0
$531$		0			0
$532$		−	18.0000i		−	0.780399i
$533$	−33.9411			−1.47015
$534$		0			0
$535$		0			0
$536$	6.92820			0.299253
$537$		0			0
$538$	26.9444			1.16166
$539$	1.73205	−	2.82843i	0.0746047	−	0.121829i
$540$		0			0
$541$			14.6969i			0.631871i	0.948781	+	0.315935i	$0.102318\pi$
$541$			14.6969i			0.631871i	−0.948781	+	0.315935i	$0.897682\pi$
$542$	−12.7279			−0.546711
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−31.8434			−1.36152			−0.680762	−	0.732505i	$-0.738351\pi$
$547$	−31.8434			−1.36152			−0.680762	+	0.732505i	$0.738351\pi$
$548$	15.5563			0.664534
$549$		0			0
$550$		0			0
$551$			50.9117i			2.16891i
$552$		0			0
$553$			30.0000i			1.27573i
$554$			42.4264i			1.80253i
$555$		0			0
$556$			2.44949i			0.103882i
$557$			20.7846i			0.880672i	0.897833	+	0.440336i	$0.145141\pi$
$557$			20.7846i			0.880672i	−0.897833	+	0.440336i	$0.854859\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$		0			0
$562$			24.0000i			1.01238i
$563$		−	17.3205i		−	0.729972i	−0.931013	−	0.364986i	$-0.881074\pi$
$563$		−	17.3205i		−	0.729972i	0.931013	−	0.364986i	$-0.118926\pi$
$564$		0			0
$565$		0			0
$566$			4.24264i			0.178331i
$567$		0			0
$568$	−4.89898			−0.205557
$569$	13.8564			0.580891			0.290445	−	0.956892i	$-0.406197\pi$
$569$	13.8564			0.580891			0.290445	+	0.956892i	$0.406197\pi$
$570$		0			0
$571$			2.44949i			0.102508i	0.998686	+	0.0512540i	$0.0163218\pi$
$571$			2.44949i			0.102508i	−0.998686	+	0.0512540i	$0.983678\pi$
$572$	8.48528	−	13.8564i	0.354787	−	0.579365i
$573$		0			0
$574$		−	29.3939i		−	1.22688i
$575$		0			0
$576$		0			0
$577$			10.0000i			0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$			10.0000i			0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$		−	29.4449i		−	1.22474i
$579$		0			0
$580$		0			0
$581$			33.9411i			1.40812i
$582$		0			0
$583$	17.1464	−	28.0000i	0.710132	−	1.15964i
$584$		0			0
$585$		0			0
$586$	−12.0000			−0.495715
$587$	−28.2843			−1.16742			−0.583708	−	0.811963i	$-0.698399\pi$
$587$	−28.2843			−1.16742			−0.583708	+	0.811963i	$0.698399\pi$
$588$		0			0
$589$			29.3939i			1.21115i
$590$		0			0
$591$		0			0
$592$			40.0000i			1.64399i
$593$		−	41.5692i		−	1.70704i	−0.521057	−	0.853522i	$-0.674462\pi$
$593$		−	41.5692i		−	1.70704i	0.521057	−	0.853522i	$-0.325538\pi$
$594$		0			0
$595$		0			0
$596$	13.8564			0.567581
$597$		0			0
$598$		−	24.0000i		−	0.981433i
$599$			2.82843i			0.115566i	0.998329	+	0.0577832i	$0.0184032\pi$
$599$			2.82843i			0.115566i	−0.998329	+	0.0577832i	$0.981597\pi$
$600$		0			0
$601$			19.5959i			0.799334i	0.916660	+	0.399667i	$0.130874\pi$
$601$			19.5959i			0.799334i	−0.916660	+	0.399667i	$0.869126\pi$
$602$			10.3923i			0.423559i
$603$		0			0
$604$			2.44949i			0.0996683i
$605$		0			0
$606$		0			0
$607$	−12.2474			−0.497109			−0.248554	−	0.968618i	$-0.579955\pi$
$607$	−12.2474			−0.497109			−0.248554	+	0.968618i	$0.579955\pi$
$608$	38.1838			1.54856
$609$		0			0
$610$		0			0
$611$	−13.8564			−0.560570
$612$		0			0
$613$	14.6969			0.593604			0.296802	−	0.954939i	$-0.404080\pi$
$613$	14.6969			0.593604			0.296802	+	0.954939i	$0.404080\pi$
$614$		−	12.7279i		−	0.513657i
$615$		0			0
$616$	−12.0000	−	7.34847i	−0.483494	−	0.296078i
$617$	26.8701			1.08175			0.540874	−	0.841104i	$-0.318094\pi$
$617$	26.8701			1.08175			0.540874	+	0.841104i	$0.318094\pi$
$618$		0			0
$619$	28.0000			1.12542			0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000			1.12542			0.562708	+	0.826656i	$0.309760\pi$
$620$		0			0
$621$		0			0
$622$	24.4949			0.982156
$623$		−	17.3205i		−	0.693932i
$624$		0			0
$625$		0			0
$626$	−27.7128			−1.10763
$627$		0			0
$628$			14.0000i			0.558661i
$629$		0			0
$630$		0			0
$631$	44.0000			1.75161			0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000			1.75161			0.875806	+	0.482663i	$0.160330\pi$
$632$	−21.2132			−0.843816
$633$		0			0
$634$		−	31.8434i		−	1.26466i
$635$		0			0
$636$		0			0
$637$	−4.89898			−0.194105
$638$	−33.9411	−	20.7846i	−1.34374	−	0.822871i
$639$		0			0
$640$		0			0
$641$		−	32.5269i		−	1.28474i	−0.766396	−	0.642368i	$-0.777952\pi$
$641$		−	32.5269i		−	1.28474i	0.766396	−	0.642368i	$-0.222048\pi$
$642$		0			0
$643$		−	28.0000i		−	1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$		−	28.0000i		−	1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	6.92820			0.273009
$645$		0			0
$646$		0			0
$647$	−19.7990			−0.778379			−0.389189	−	0.921158i	$-0.627245\pi$
$647$	−19.7990			−0.778379			−0.389189	+	0.921158i	$0.627245\pi$
$648$		0			0
$649$	−32.0000	−	19.5959i	−1.25611	−	0.769207i
$650$		0			0
$651$		0			0
$652$		−	8.00000i		−	0.313304i
$653$	−26.8701			−1.05151			−0.525753	−	0.850637i	$-0.676216\pi$
$653$	−26.8701			−1.05151			−0.525753	+	0.850637i	$0.676216\pi$
$654$		0			0
$655$		0			0
$656$	34.6410			1.35250
$657$		0			0
$658$		−	12.0000i		−	0.467809i
$659$	−38.1051			−1.48436			−0.742182	−	0.670198i	$-0.766209\pi$
$659$	−38.1051			−1.48436			−0.742182	+	0.670198i	$0.766209\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$		−	13.8564i		−	0.538545i
$663$		0			0
$664$	−24.0000			−0.931381
$665$		0			0
$666$		0			0
$667$	−19.5959			−0.758757
$668$			3.46410i			0.134030i
$669$		0			0
$670$		0			0
$671$	−13.8564	−	8.48528i	−0.534921	−	0.327571i
$672$		0			0
$673$	39.1918			1.51073			0.755367	−	0.655302i	$-0.227459\pi$
$673$	39.1918			1.51073			0.755367	+	0.655302i	$0.227459\pi$
$674$		−	16.9706i		−	0.653682i
$675$		0			0
$676$	−11.0000			−0.423077
$677$		−	27.7128i		−	1.06509i	−0.846402	−	0.532545i	$-0.821236\pi$
$677$		−	27.7128i		−	1.06509i	0.846402	−	0.532545i	$-0.178764\pi$
$678$		0			0
$679$		−	24.4949i		−	0.940028i
$680$		0			0
$681$		0			0
$682$	−19.5959	−	12.0000i	−0.750366	−	0.459504i
$683$	28.2843			1.08227			0.541134	−	0.840937i	$-0.317995\pi$
$683$	28.2843			1.08227			0.541134	+	0.840937i	$0.317995\pi$
$684$		0			0
$685$		0			0
$686$		−	33.9411i		−	1.29588i
$687$		0			0
$688$	−12.2474			−0.466930
$689$	−48.4974			−1.84760
$690$		0			0
$691$	44.0000			1.67384			0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000			1.67384			0.836919	+	0.547326i	$0.184354\pi$
$692$		−	6.92820i		−	0.263371i
$693$		0			0
$694$	−24.0000			−0.911028
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−8.48528			−0.321173
$699$		0			0
$700$		0			0
$701$	20.7846			0.785024			0.392512	−	0.919747i	$-0.371606\pi$
$701$	20.7846			0.785024			0.392512	+	0.919747i	$0.371606\pi$
$702$		0			0
$703$	−58.7878			−2.21722
$704$	1.73205	−	2.82843i	0.0652791	−	0.106600i
$705$		0			0
$706$		−	56.3383i		−	2.12032i
$707$	33.9411			1.27649
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$		0			0
$712$	12.2474			0.458993
$713$	−11.3137			−0.423702
$714$		0			0
$715$		0			0
$716$			22.6274i			0.845626i
$717$		0			0
$718$		0			0
$719$			2.82843i			0.105483i	0.998608	+	0.0527413i	$0.0167959\pi$
$719$			2.82843i			0.105483i	−0.998608	+	0.0527413i	$0.983204\pi$
$720$		0			0
$721$			9.79796i			0.364895i
$722$			60.6218i			2.25611i
$723$		0			0
$724$	−8.00000			−0.297318
$725$		0			0
$726$		0			0
$727$		−	8.00000i		−	0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$		−	8.00000i		−	0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$			20.7846i			0.770329i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	34.2929			1.26664			0.633318	−	0.773892i	$-0.281693\pi$
$733$	34.2929			1.26664			0.633318	+	0.773892i	$0.281693\pi$
$734$	−13.8564			−0.511449
$735$		0			0
$736$			14.6969i			0.541736i
$737$	−11.3137	−	6.92820i	−0.416746	−	0.255204i
$738$		0			0
$739$			7.34847i			0.270318i	0.990824	+	0.135159i	$0.0431545\pi$
$739$			7.34847i			0.270318i	−0.990824	+	0.135159i	$0.956846\pi$
$740$		0			0
$741$		0			0
$742$		−	42.0000i		−	1.54187i
$743$		−	27.7128i		−	1.01668i	−0.861155	−	0.508342i	$-0.830258\pi$
$743$		−	27.7128i		−	1.01668i	0.861155	−	0.508342i	$-0.169742\pi$
$744$		0			0
$745$		0			0
$746$			42.4264i			1.55334i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$	14.1421			0.515711
$753$		0			0
$754$			58.7878i			2.14092i
$755$		0			0
$756$		0			0
$757$		−	8.00000i		−	0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$		−	8.00000i		−	0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$		−	27.7128i		−	1.00657i
$759$		0			0
$760$		0			0
$761$	13.8564			0.502294			0.251147	−	0.967949i	$-0.419192\pi$
$761$	13.8564			0.502294			0.251147	+	0.967949i	$0.419192\pi$
$762$		0			0
$763$		−	36.0000i		−	1.30329i
$764$			2.82843i			0.102329i
$765$		0			0
$766$		−	63.6867i		−	2.30110i
$767$			55.4256i			2.00130i
$768$		0			0
$769$			19.5959i			0.706647i	0.935501	+	0.353323i	$0.114948\pi$
$769$			19.5959i			0.706647i	−0.935501	+	0.353323i	$0.885052\pi$
$770$		0			0
$771$		0			0
$772$	9.79796			0.352636
$773$	32.5269			1.16991			0.584956	−	0.811065i	$-0.301112\pi$
$773$	32.5269			1.16991			0.584956	+	0.811065i	$0.301112\pi$
$774$		0			0
$775$		0			0
$776$	17.3205			0.621770
$777$		0			0
$778$	−17.1464			−0.614729
$779$			50.9117i			1.82410i
$780$		0			0
$781$	8.00000	+	4.89898i	0.286263	+	0.175299i
$782$		0			0
$783$		0			0
$784$	5.00000			0.178571
$785$		0			0
$786$		0			0
$787$	−12.2474			−0.436574			−0.218287	−	0.975885i	$-0.570047\pi$
$787$	−12.2474			−0.436574			−0.218287	+	0.975885i	$0.570047\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.46410			0.123169
$792$		0			0
$793$			24.0000i			0.852265i
$794$	−13.8564			−0.491745
$795$		0			0
$796$	8.00000			0.283552
$797$	26.8701			0.951786			0.475893	−	0.879503i	$-0.342125\pi$
$797$	26.8701			0.951786			0.475893	+	0.879503i	$0.342125\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	2.44949			0.0864945
$803$		0			0
$804$		0			0
$805$		0			0
$806$			33.9411i			1.19553i
$807$		0			0
$808$			24.0000i			0.844317i
$809$	20.7846			0.730748			0.365374	−	0.930861i	$-0.380941\pi$
$809$	20.7846			0.730748			0.365374	+	0.930861i	$0.380941\pi$
$810$		0			0
$811$			22.0454i			0.774119i	0.922055	+	0.387059i	$0.126509\pi$
$811$			22.0454i			0.774119i	−0.922055	+	0.387059i	$0.873491\pi$
$812$	−16.9706			−0.595550
$813$		0			0
$814$	24.0000	−	39.1918i	0.841200	−	1.37367i
$815$		0			0
$816$		0			0
$817$		−	18.0000i		−	0.629740i
$818$	−67.8823			−2.37345
$819$		0			0
$820$		0			0
$821$	−13.8564			−0.483592			−0.241796	−	0.970327i	$-0.577736\pi$
$821$	−13.8564			−0.483592			−0.241796	+	0.970327i	$0.577736\pi$
$822$		0			0
$823$			8.00000i			0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$			8.00000i			0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	−6.92820			−0.241355
$825$		0			0
$826$	−48.0000			−1.67013
$827$		−	41.5692i		−	1.44550i	−0.691108	−	0.722752i	$-0.742877\pi$
$827$		−	41.5692i		−	1.44550i	0.691108	−	0.722752i	$-0.257123\pi$
$828$		0			0
$829$	46.0000			1.59765			0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000			1.59765			0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$		0			0
$832$	−4.89898			−0.169842
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−20.7846	−	12.7279i	−0.718851	−	0.440204i
$837$		0			0
$838$	19.5959			0.676930
$839$			19.7990i			0.683537i	0.939784	+	0.341769i	$0.111026\pi$
$839$			19.7990i			0.683537i	−0.939784	+	0.341769i	$0.888974\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		−	13.8564i		−	0.477523i
$843$		0			0
$844$			12.2474i			0.421575i
$845$		0			0
$846$		0			0
$847$	12.2474	+	24.0000i	0.420827	+	0.824650i
$848$	49.4975			1.69975
$849$		0			0
$850$		0			0
$851$		−	22.6274i		−	0.775658i
$852$		0			0
$853$	−34.2929			−1.17417			−0.587083	−	0.809527i	$-0.699724\pi$
$853$	−34.2929			−1.17417			−0.587083	+	0.809527i	$0.699724\pi$
$854$	−20.7846			−0.711235
$855$		0			0
$856$		0			0
$857$			34.6410i			1.18331i	0.806190	+	0.591657i	$0.201526\pi$
$857$			34.6410i			1.18331i	−0.806190	+	0.591657i	$0.798474\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$		0			0
$861$		0			0
$862$		−	18.0000i		−	0.613082i
$863$	19.7990			0.673965			0.336983	−	0.941511i	$-0.390594\pi$
$863$	19.7990			0.673965			0.336983	+	0.941511i	$0.390594\pi$
$864$		0			0
$865$		0			0
$866$	−27.7128			−0.941720
$867$		0			0
$868$	−9.79796			−0.332564
$869$	34.6410	+	21.2132i	1.17512	+	0.719609i
$870$		0			0
$871$			19.5959i			0.663982i
$872$	25.4558			0.862044
$873$		0			0
$874$	−36.0000			−1.21772
$875$		0			0
$876$		0			0
$877$	53.8888			1.81969			0.909847	−	0.414943i	$-0.136199\pi$
$877$	53.8888			1.81969			0.909847	+	0.414943i	$0.136199\pi$
$878$	4.24264			0.143182
$879$		0			0
$880$		0			0
$881$			9.89949i			0.333522i	0.985997	+	0.166761i	$0.0533309\pi$
$881$			9.89949i			0.333522i	−0.985997	+	0.166761i	$0.946669\pi$
$882$		0			0
$883$		−	28.0000i		−	0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$		−	28.0000i		−	0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$		0			0
$885$		0			0
$886$			39.1918i			1.31668i
$887$		−	55.4256i		−	1.86101i	−0.366279	−	0.930505i	$-0.619368\pi$
$887$		−	55.4256i		−	1.86101i	0.366279	−	0.930505i	$-0.380632\pi$
$888$		0			0
$889$	−18.0000			−0.603701
$890$		0			0
$891$		0			0
$892$			16.0000i			0.535720i
$893$			20.7846i			0.695530i
$894$		0			0
$895$		0			0
$896$		−	29.6985i		−	0.992157i
$897$		0			0
$898$	−17.1464			−0.572184
$899$	27.7128			0.924274
$900$		0			0
$901$		0			0
$902$	−33.9411	−	20.7846i	−1.13012	−	0.692052i
$903$		0			0
$904$			2.44949i			0.0814688i
$905$		0			0
$906$		0			0
$907$			52.0000i			1.72663i	0.504664	+	0.863316i	$0.331616\pi$
$907$			52.0000i			1.72663i	−0.504664	+	0.863316i	$0.668384\pi$
$908$			27.7128i			0.919682i
$909$		0			0
$910$		0			0
$911$		−	53.7401i		−	1.78049i	−0.455483	−	0.890245i	$-0.650533\pi$
$911$		−	53.7401i		−	1.78049i	0.455483	−	0.890245i	$-0.349467\pi$
$912$		0			0
$913$	39.1918	+	24.0000i	1.29706	+	0.794284i
$914$			16.9706i			0.561336i
$915$		0			0
$916$	−10.0000			−0.330409
$917$	−8.48528			−0.280209
$918$		0			0
$919$			36.7423i			1.21202i	0.795458	+	0.606009i	$0.207230\pi$
$919$			36.7423i			1.21202i	−0.795458	+	0.606009i	$0.792770\pi$
$920$		0			0
$921$		0			0
$922$		−	12.0000i		−	0.395199i
$923$		−	13.8564i		−	0.456089i
$924$		0			0
$925$		0			0
$926$	−48.4974			−1.59372
$927$		0			0
$928$		−	36.0000i		−	1.18176i
$929$		−	9.89949i		−	0.324792i	−0.986726	−	0.162396i	$-0.948078\pi$
$929$		−	9.89949i		−	0.324792i	0.986726	−	0.162396i	$-0.0519222\pi$
$930$		0			0
$931$			7.34847i			0.240836i
$932$		−	20.7846i		−	0.680823i
$933$		0			0
$934$		−	39.1918i		−	1.28240i
$935$		0			0
$936$		0			0
$937$	−29.3939			−0.960256			−0.480128	−	0.877198i	$-0.659410\pi$
$937$	−29.3939			−0.960256			−0.480128	+	0.877198i	$0.659410\pi$
$938$	−16.9706			−0.554109
$939$		0			0
$940$		0			0
$941$	−27.7128			−0.903412			−0.451706	−	0.892167i	$-0.649184\pi$
$941$	−27.7128			−0.903412			−0.451706	+	0.892167i	$0.649184\pi$
$942$		0			0
$943$	−19.5959			−0.638131
$944$		−	56.5685i		−	1.84115i
$945$		0			0
$946$	12.0000	+	7.34847i	0.390154	+	0.238919i
$947$	22.6274			0.735292			0.367646	−	0.929966i	$-0.380164\pi$
$947$	22.6274			0.735292			0.367646	+	0.929966i	$0.380164\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			20.7846i			0.673280i	0.941634	+	0.336640i	$0.109290\pi$
$953$			20.7846i			0.673280i	−0.941634	+	0.336640i	$0.890710\pi$
$954$		0			0
$955$		0			0
$956$	−17.3205			−0.560185
$957$		0			0
$958$			66.0000i			2.13236i
$959$	38.1051			1.23048
$960$		0			0
$961$	−15.0000			−0.483871
$962$	−67.8823			−2.18861
$963$		0			0
$964$		−	19.5959i		−	0.631142i
$965$		0			0
$966$		0			0
$967$	−41.6413			−1.33909			−0.669547	−	0.742769i	$-0.733512\pi$
$967$	−41.6413			−1.33909			−0.669547	+	0.742769i	$0.733512\pi$
$968$	−16.9706	+	8.66025i	−0.545455	+	0.278351i
$969$		0			0
$970$		0			0
$971$			56.5685i			1.81537i	0.419651	+	0.907685i	$0.362152\pi$
$971$			56.5685i			1.81537i	−0.419651	+	0.907685i	$0.637848\pi$
$972$		0			0
$973$			6.00000i			0.192351i
$974$	−13.8564			−0.443988
$975$		0			0
$976$		−	24.4949i		−	0.784063i
$977$	35.3553			1.13112			0.565559	−	0.824708i	$-0.308661\pi$
$977$	35.3553			1.13112			0.565559	+	0.824708i	$0.308661\pi$
$978$		0			0
$979$	−20.0000	−	12.2474i	−0.639203	−	0.391430i
$980$		0			0
$981$		0			0
$982$			48.0000i			1.53174i
$983$	−48.0833			−1.53362			−0.766809	−	0.641875i	$-0.778157\pi$
$983$	−48.0833			−1.53362			−0.766809	+	0.641875i	$0.778157\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$			36.0000i			1.14531i
$989$	6.92820			0.220304
$990$		0			0
$991$	−52.0000			−1.65183			−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000			−1.65183			−0.825917	+	0.563791i	$0.809342\pi$
$992$		−	20.7846i		−	0.659912i
$993$		0			0
$994$	12.0000			0.380617
$995$		0			0
$996$		0			0
$997$	−24.4949			−0.775761			−0.387881	−	0.921710i	$-0.626793\pi$
$997$	−24.4949			−0.775761			−0.387881	+	0.921710i	$0.626793\pi$
$998$			55.4256i			1.75447i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.d.a.2474.1		8
3.2	odd	2	inner	2475.2.d.a.2474.6		8
5.2	odd	4		99.2.d.a.98.4	yes	4
5.3	odd	4		2475.2.f.e.2276.2		4
5.4	even	2	inner	2475.2.d.a.2474.7		8
11.10	odd	2	inner	2475.2.d.a.2474.8		8
15.2	even	4		99.2.d.a.98.1	✓	4
15.8	even	4		2475.2.f.e.2276.4		4
15.14	odd	2	inner	2475.2.d.a.2474.4		8
20.7	even	4		1584.2.b.e.593.4		4
33.32	even	2	inner	2475.2.d.a.2474.3		8
40.27	even	4		6336.2.b.t.2177.2		4
40.37	odd	4		6336.2.b.s.2177.1		4
45.2	even	12		891.2.g.c.296.4		8
45.7	odd	12		891.2.g.c.296.1		8
45.22	odd	12		891.2.g.c.593.2		8
45.32	even	12		891.2.g.c.593.3		8
55.32	even	4		99.2.d.a.98.2	yes	4
55.43	even	4		2475.2.f.e.2276.3		4
55.54	odd	2	inner	2475.2.d.a.2474.2		8
60.47	odd	4		1584.2.b.e.593.2		4
120.77	even	4		6336.2.b.s.2177.3		4
120.107	odd	4		6336.2.b.t.2177.4		4
165.32	odd	4		99.2.d.a.98.3	yes	4
165.98	odd	4		2475.2.f.e.2276.1		4
165.164	even	2	inner	2475.2.d.a.2474.5		8
220.87	odd	4		1584.2.b.e.593.3		4
440.197	even	4		6336.2.b.s.2177.2		4
440.307	odd	4		6336.2.b.t.2177.1		4
495.32	odd	12		891.2.g.c.593.1		8
495.142	even	12		891.2.g.c.296.3		8
495.362	odd	12		891.2.g.c.296.2		8
495.472	even	12		891.2.g.c.593.4		8
660.527	even	4		1584.2.b.e.593.1		4
1320.197	odd	4		6336.2.b.s.2177.4		4
1320.1187	even	4		6336.2.b.t.2177.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.d.a.98.1	✓	4	15.2	even	4
99.2.d.a.98.2	yes	4	55.32	even	4
99.2.d.a.98.3	yes	4	165.32	odd	4
99.2.d.a.98.4	yes	4	5.2	odd	4
891.2.g.c.296.1		8	45.7	odd	12
891.2.g.c.296.2		8	495.362	odd	12
891.2.g.c.296.3		8	495.142	even	12
891.2.g.c.296.4		8	45.2	even	12
891.2.g.c.593.1		8	495.32	odd	12
891.2.g.c.593.2		8	45.22	odd	12
891.2.g.c.593.3		8	45.32	even	12
891.2.g.c.593.4		8	495.472	even	12
1584.2.b.e.593.1		4	660.527	even	4
1584.2.b.e.593.2		4	60.47	odd	4
1584.2.b.e.593.3		4	220.87	odd	4
1584.2.b.e.593.4		4	20.7	even	4
2475.2.d.a.2474.1		8	1.1	even	1	trivial
2475.2.d.a.2474.2		8	55.54	odd	2	inner
2475.2.d.a.2474.3		8	33.32	even	2	inner
2475.2.d.a.2474.4		8	15.14	odd	2	inner
2475.2.d.a.2474.5		8	165.164	even	2	inner
2475.2.d.a.2474.6		8	3.2	odd	2	inner
2475.2.d.a.2474.7		8	5.4	even	2	inner
2475.2.d.a.2474.8		8	11.10	odd	2	inner
2475.2.f.e.2276.1		4	165.98	odd	4
2475.2.f.e.2276.2		4	5.3	odd	4
2475.2.f.e.2276.3		4	55.43	even	4
2475.2.f.e.2276.4		4	15.8	even	4
6336.2.b.s.2177.1		4	40.37	odd	4
6336.2.b.s.2177.2		4	440.197	even	4
6336.2.b.s.2177.3		4	120.77	even	4
6336.2.b.s.2177.4		4	1320.197	odd	4
6336.2.b.t.2177.1		4	440.307	odd	4
6336.2.b.t.2177.2		4	40.27	even	4
6336.2.b.t.2177.3		4	1320.1187	even	4
6336.2.b.t.2177.4		4	120.107	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -1.00000 q^{4} -2.44949 q^{7} -1.73205i q^{8} +(-1.73205 + 2.82843i) q^{11} +4.89898 q^{13} +4.24264i q^{14} -5.00000 q^{16} -7.34847i q^{19} +(4.89898 + 3.00000i) q^{22} +2.82843 q^{23} -8.48528i q^{26} +2.44949 q^{28} -6.92820 q^{29} -4.00000 q^{31} +5.19615i q^{32} -8.00000i q^{37} -12.7279 q^{38} -6.92820 q^{41} +2.44949 q^{43} +(1.73205 - 2.82843i) q^{44} -4.89898i q^{46} -2.82843 q^{47} -1.00000 q^{49} -4.89898 q^{52} -9.89949 q^{53} +4.24264i q^{56} +12.0000i q^{58} +11.3137i q^{59} +4.89898i q^{61} +6.92820i q^{62} -1.00000 q^{64} +4.00000i q^{67} -2.82843i q^{71} -13.8564 q^{74} +7.34847i q^{76} +(4.24264 - 6.92820i) q^{77} -12.2474i q^{79} +12.0000i q^{82} -13.8564i q^{83} -4.24264i q^{86} +(4.89898 + 3.00000i) q^{88} +7.07107i q^{89} -12.0000 q^{91} -2.82843 q^{92} +4.89898i q^{94} +10.0000i q^{97} +1.73205i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 40 q^{16} - 32 q^{31} - 8 q^{49} - 8 q^{64} - 96 q^{91}+O(q^{100}) \) 8 * q - 8 * q^4 - 40 * q^16 - 32 * q^31 - 8 * q^49 - 8 * q^64 - 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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