LMFDB - Embedded newform 2475.2.f.e.2276.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2475,2,Mod(2276,2475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2475.2276"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,-20,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2276.4
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	2475.2276
Dual form			2475.2.f.e.2276.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +2.44949i q^{7} -1.73205 q^{8} +(1.73205 - 2.82843i) q^{11} +4.89898i q^{13} +4.24264i q^{14} -5.00000 q^{16} +7.34847i q^{19} +(3.00000 - 4.89898i) q^{22} -2.82843i q^{23} +8.48528i q^{26} +2.44949i q^{28} -6.92820 q^{29} -4.00000 q^{31} -5.19615 q^{32} -8.00000 q^{37} +12.7279i q^{38} +6.92820 q^{41} +2.44949i q^{43} +(1.73205 - 2.82843i) q^{44} -4.89898i q^{46} -2.82843i q^{47} +1.00000 q^{49} +4.89898i q^{52} +9.89949i q^{53} -4.24264i q^{56} -12.0000 q^{58} +11.3137i q^{59} +4.89898i q^{61} -6.92820 q^{62} +1.00000 q^{64} +4.00000 q^{67} +2.82843i q^{71} -13.8564 q^{74} +7.34847i q^{76} +(6.92820 + 4.24264i) q^{77} +12.2474i q^{79} +12.0000 q^{82} -13.8564 q^{83} +4.24264i q^{86} +(-3.00000 + 4.89898i) q^{88} +7.07107i q^{89} -12.0000 q^{91} -2.82843i q^{92} -4.89898i q^{94} +10.0000 q^{97} +1.73205 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 20 q^{16} + 12 q^{22} - 16 q^{31} - 32 q^{37} + 4 q^{49} - 48 q^{58} + 4 q^{64} + 16 q^{67} + 48 q^{82} - 12 q^{88} - 48 q^{91} + 40 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 20 * q^16 + 12 * q^22 - 16 * q^31 - 32 * q^37 + 4 * q^49 - 48 * q^58 + 4 * q^64 + 16 * q^67 + 48 * q^82 - 12 * q^88 - 48 * q^91 + 40 * q^97

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 $

$2$	1.73205			1.22474			0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205			1.22474			0.612372	+	0.790569i	$0.290215\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$			2.44949i			0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$			2.44949i			0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$	−1.73205			−0.612372
$9$		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
$13$			4.89898i			1.35873i	0.733799	+	0.679366i	$0.237745\pi$
$13$			4.89898i			1.35873i	−0.733799	+	0.679366i	$0.762255\pi$
$14$			4.24264i			1.13389i
$15$		0			0
$16$	−5.00000			−1.25000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$			7.34847i			1.68585i	0.538028	+	0.842927i	$0.319170\pi$
$19$			7.34847i			1.68585i	−0.538028	+	0.842927i	$0.680830\pi$
$20$		0			0
$21$		0			0
$22$	3.00000	−	4.89898i	0.639602	−	1.04447i
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$		0			0
$26$			8.48528i			1.66410i
$27$		0			0
$28$			2.44949i			0.462910i
$29$	−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	$0.616954\pi$
$32$	−5.19615			−0.918559
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−8.00000			−1.31519			−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000			−1.31519			−0.657596	+	0.753371i	$0.728427\pi$
$38$			12.7279i			2.06474i
$39$		0			0
$40$		0			0
$41$	6.92820			1.08200			0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820			1.08200			0.541002	+	0.841021i	$0.318045\pi$
$42$		0			0
$43$			2.44949i			0.373544i	0.982403	+	0.186772i	$0.0598025\pi$
$43$			2.44949i			0.373544i	−0.982403	+	0.186772i	$0.940197\pi$
$44$	1.73205	−	2.82843i	0.261116	−	0.426401i
$45$		0			0
$46$		−	4.89898i		−	0.722315i
$47$		−	2.82843i		−	0.412568i	−0.978492	−	0.206284i	$-0.933863\pi$
$47$		−	2.82843i		−	0.412568i	0.978492	−	0.206284i	$-0.0661372\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$			4.89898i			0.679366i
$53$			9.89949i			1.35980i	0.733305	+	0.679900i	$0.237977\pi$
$53$			9.89949i			1.35980i	−0.733305	+	0.679900i	$0.762023\pi$
$54$		0			0
$55$		0			0
$56$		−	4.24264i		−	0.566947i
$57$		0			0
$58$	−12.0000			−1.57568
$59$			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	$0.898457\pi$
$62$	−6.92820			−0.879883
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.82843i			0.335673i	0.985815	+	0.167836i	$0.0536780\pi$
$71$			2.82843i			0.335673i	−0.985815	+	0.167836i	$0.946322\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−13.8564			−1.61077
$75$		0			0
$76$			7.34847i			0.842927i
$77$	6.92820	+	4.24264i	0.789542	+	0.483494i
$78$		0			0
$79$			12.2474i			1.37795i	0.724787	+	0.688973i	$0.241938\pi$
$79$			12.2474i			1.37795i	−0.724787	+	0.688973i	$0.758062\pi$
$80$		0			0
$81$		0			0
$82$	12.0000			1.32518
$83$	−13.8564			−1.52094			−0.760469	−	0.649374i	$-0.775031\pi$
$83$	−13.8564			−1.52094			−0.760469	+	0.649374i	$0.775031\pi$
$84$		0			0
$85$		0			0
$86$			4.24264i			0.457496i
$87$		0			0
$88$	−3.00000	+	4.89898i	−0.319801	+	0.522233i
$89$			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
$92$		−	2.82843i		−	0.294884i
$93$		0			0
$94$		−	4.89898i		−	0.505291i
$95$		0			0
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$	1.73205			0.174964
$99$		0			0
$100$		0			0
$101$	13.8564			1.37876			0.689382	−	0.724398i	$-0.257882\pi$
$101$	13.8564			1.37876			0.689382	+	0.724398i	$0.257882\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		−	8.48528i		−	0.832050i
$105$		0			0
$106$			17.1464i			1.66541i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		−	14.6969i		−	1.40771i	−0.710343	−	0.703856i	$-0.751460\pi$
$109$		−	14.6969i		−	1.40771i	0.710343	−	0.703856i	$-0.248540\pi$
$110$		0			0
$111$		0			0
$112$		−	12.2474i		−	1.15728i
$113$			1.41421i			0.133038i	0.997785	+	0.0665190i	$0.0211893\pi$
$113$			1.41421i			0.133038i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$		0			0
$115$		0			0
$116$	−6.92820			−0.643268
$117$		0			0
$118$			19.5959i			1.80395i
$119$		0			0
$120$		0			0
$121$	−5.00000	−	9.79796i	−0.454545	−	0.890724i
$122$			8.48528i			0.768221i
$123$		0			0
$124$	−4.00000			−0.359211
$125$		0			0
$126$		0			0
$127$		−	7.34847i		−	0.652071i	−0.945357	−	0.326036i	$-0.894287\pi$
$127$		−	7.34847i		−	0.652071i	0.945357	−	0.326036i	$-0.105713\pi$
$128$	12.1244			1.07165
$129$		0			0
$130$		0			0
$131$	−3.46410			−0.302660			−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410			−0.302660			−0.151330	+	0.988483i	$0.548356\pi$
$132$		0			0
$133$	−18.0000			−1.56080
$134$	6.92820			0.598506
$135$		0			0
$136$		0			0
$137$		−	15.5563i		−	1.32907i	−0.747258	−	0.664534i	$-0.768630\pi$
$137$		−	15.5563i		−	1.32907i	0.747258	−	0.664534i	$-0.231370\pi$
$138$		0			0
$139$			2.44949i			0.207763i	0.994590	+	0.103882i	$0.0331263\pi$
$139$			2.44949i			0.207763i	−0.994590	+	0.103882i	$0.966874\pi$
$140$		0			0
$141$		0			0
$142$			4.89898i			0.411113i
$143$	13.8564	+	8.48528i	1.15873	+	0.709575i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−8.00000			−0.657596
$149$	−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	$-0.0317782\pi$
$152$		−	12.7279i		−	1.03237i
$153$		0			0
$154$	12.0000	+	7.34847i	0.966988	+	0.592157i
$155$		0			0
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$			21.2132i			1.68763i
$159$		0			0
$160$		0			0
$161$	6.92820			0.546019
$162$		0			0
$163$	−8.00000			−0.626608			−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000			−0.626608			−0.313304	+	0.949653i	$0.601436\pi$
$164$	6.92820			0.541002
$165$		0			0
$166$	−24.0000			−1.86276
$167$	3.46410			0.268060			0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410			0.268060			0.134030	+	0.990977i	$0.457208\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$		0			0
$172$			2.44949i			0.186772i
$173$	6.92820			0.526742			0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820			0.526742			0.263371	+	0.964695i	$0.415166\pi$
$174$		0			0
$175$		0			0
$176$	−8.66025	+	14.1421i	−0.652791	+	1.06600i
$177$		0			0
$178$			12.2474i			0.917985i
$179$		−	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	$0.403908\pi$
$182$	−20.7846			−1.54066
$183$		0			0
$184$			4.89898i			0.361158i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		−	2.82843i		−	0.206284i
$189$		0			0
$190$		0			0
$191$			2.82843i			0.204658i	0.994751	+	0.102329i	$0.0326294\pi$
$191$			2.82843i			0.204658i	−0.994751	+	0.102329i	$0.967371\pi$
$192$		0			0
$193$		−	9.79796i		−	0.705273i	−0.935760	−	0.352636i	$-0.885285\pi$
$193$		−	9.79796i		−	0.705273i	0.935760	−	0.352636i	$-0.114715\pi$
$194$	17.3205			1.24354
$195$		0			0
$196$	1.00000			0.0714286
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$		0			0
$202$	24.0000			1.68863
$203$		−	16.9706i		−	1.19110i
$204$		0			0
$205$		0			0
$206$	6.92820			0.482711
$207$		0			0
$208$		−	24.4949i		−	1.69842i
$209$	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	$-0.138522\pi$
$212$			9.89949i			0.679900i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	9.79796i		−	0.665129i
$218$		−	25.4558i		−	1.72409i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		−	12.7279i		−	0.850420i
$225$		0			0
$226$			2.44949i			0.162938i
$227$	27.7128			1.83936			0.919682	−	0.392664i	$-0.128446\pi$
$227$	27.7128			1.83936			0.919682	+	0.392664i	$0.128446\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$		0			0
$232$	12.0000			0.787839
$233$	20.7846			1.36165			0.680823	−	0.732448i	$-0.261622\pi$
$233$	20.7846			1.36165			0.680823	+	0.732448i	$0.261622\pi$
$234$		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	$0.782587\pi$
$242$	−8.66025	−	16.9706i	−0.556702	−	1.09091i
$243$		0			0
$244$			4.89898i			0.313625i
$245$		0			0
$246$		0			0
$247$	−36.0000			−2.29063
$248$	6.92820			0.439941
$249$		0			0
$250$		0			0
$251$		−	5.65685i		−	0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$		−	5.65685i		−	0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$		0			0
$253$	−8.00000	−	4.89898i	−0.502956	−	0.307996i
$254$		−	12.7279i		−	0.798621i
$255$		0			0
$256$	19.0000			1.18750
$257$			9.89949i			0.617514i	0.951141	+	0.308757i	$0.0999129\pi$
$257$			9.89949i			0.617514i	−0.951141	+	0.308757i	$0.900087\pi$
$258$		0			0
$259$		−	19.5959i		−	1.21763i
$260$		0			0
$261$		0			0
$262$	−6.00000			−0.370681
$263$	−3.46410			−0.213606			−0.106803	−	0.994280i	$-0.534061\pi$
$263$	−3.46410			−0.213606			−0.106803	+	0.994280i	$0.534061\pi$
$264$		0			0
$265$		0			0
$266$	−31.1769			−1.91158
$267$		0			0
$268$	4.00000			0.244339
$269$			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
$277$			24.4949i			1.47176i	0.677114	+	0.735878i	$0.263230\pi$
$277$			24.4949i			1.47176i	−0.677114	+	0.735878i	$0.736770\pi$
$278$			4.24264i			0.254457i
$279$		0			0
$280$		0			0
$281$	13.8564			0.826604			0.413302	−	0.910594i	$-0.364375\pi$
$281$	13.8564			0.826604			0.413302	+	0.910594i	$0.364375\pi$
$282$		0			0
$283$		−	2.44949i		−	0.145607i	−0.997346	−	0.0728035i	$-0.976805\pi$
$283$		−	2.44949i		−	0.145607i	0.997346	−	0.0728035i	$-0.0231946\pi$
$284$			2.82843i			0.167836i
$285$		0			0
$286$	24.0000	+	14.6969i	1.41915	+	0.869048i
$287$			16.9706i			1.00174i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.92820			−0.404750			−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820			−0.404750			−0.202375	+	0.979308i	$0.564866\pi$
$294$		0			0
$295$		0			0
$296$	13.8564			0.805387
$297$		0			0
$298$	−24.0000			−1.39028
$299$	13.8564			0.801337
$300$		0			0
$301$	−6.00000			−0.345834
$302$		−	4.24264i		−	0.244137i
$303$		0			0
$304$		−	36.7423i		−	2.10732i
$305$		0			0
$306$		0			0
$307$		−	7.34847i		−	0.419399i	−0.977766	−	0.209700i	$-0.932751\pi$
$307$		−	7.34847i		−	0.419399i	0.977766	−	0.209700i	$-0.0672486\pi$
$308$	6.92820	+	4.24264i	0.394771	+	0.241747i
$309$		0			0
$310$		0			0
$311$		−	14.1421i		−	0.801927i	−0.916094	−	0.400963i	$-0.868675\pi$
$311$		−	14.1421i		−	0.801927i	0.916094	−	0.400963i	$-0.131325\pi$
$312$		0			0
$313$	16.0000			0.904373			0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000			0.904373			0.452187	+	0.891923i	$0.350644\pi$
$314$	−24.2487			−1.36843
$315$		0			0
$316$			12.2474i			0.688973i
$317$			18.3848i			1.03259i	0.856410	+	0.516296i	$0.172690\pi$
$317$			18.3848i			1.03259i	−0.856410	+	0.516296i	$0.827310\pi$
$318$		0			0
$319$	−12.0000	+	19.5959i	−0.671871	+	1.09716i
$320$		0			0
$321$		0			0
$322$	12.0000			0.668734
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−13.8564			−0.767435
$327$		0			0
$328$	−12.0000			−0.662589
$329$	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	$0.429440\pi$
$332$	−13.8564			−0.760469
$333$		0			0
$334$	6.00000			0.328305
$335$		0			0
$336$		0			0
$337$		−	9.79796i		−	0.533729i	−0.963734	−	0.266864i	$-0.914012\pi$
$337$		−	9.79796i		−	0.533729i	0.963734	−	0.266864i	$-0.0859876\pi$
$338$	−19.0526			−1.03632
$339$		0			0
$340$		0			0
$341$	−6.92820	+	11.3137i	−0.375183	+	0.612672i
$342$		0			0
$343$			19.5959i			1.05808i
$344$		−	4.24264i		−	0.228748i
$345$		0			0
$346$	12.0000			0.645124
$347$	13.8564			0.743851			0.371925	−	0.928263i	$-0.378698\pi$
$347$	13.8564			0.743851			0.371925	+	0.928263i	$0.378698\pi$
$348$		0			0
$349$			4.89898i			0.262236i	0.991367	+	0.131118i	$0.0418567\pi$
$349$			4.89898i			0.262236i	−0.991367	+	0.131118i	$0.958143\pi$
$350$		0			0
$351$		0			0
$352$	−9.00000	+	14.6969i	−0.479702	+	0.783349i
$353$		−	32.5269i		−	1.73123i	−0.500708	−	0.865616i	$-0.666927\pi$
$353$		−	32.5269i		−	1.73123i	0.500708	−	0.865616i	$-0.333073\pi$
$354$		0			0
$355$		0			0
$356$			7.07107i			0.374766i
$357$		0			0
$358$		−	39.1918i		−	2.07135i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−35.0000			−1.84211
$362$	13.8564			0.728277
$363$		0			0
$364$	−12.0000			−0.628971
$365$		0			0
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$			14.1421i			0.737210i
$369$		0			0
$370$		0			0
$371$	−24.2487			−1.25893
$372$		0			0
$373$		−	24.4949i		−	1.26830i	−0.773211	−	0.634149i	$-0.781351\pi$
$373$		−	24.4949i		−	1.26830i	0.773211	−	0.634149i	$-0.218649\pi$
$374$		0			0
$375$		0			0
$376$			4.89898i			0.252646i
$377$		−	33.9411i		−	1.74806i
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$			4.89898i			0.250654i
$383$		−	36.7696i		−	1.87884i	−0.342773	−	0.939418i	$-0.611366\pi$
$383$		−	36.7696i		−	1.87884i	0.342773	−	0.939418i	$-0.388634\pi$
$384$		0			0
$385$		0			0
$386$		−	16.9706i		−	0.863779i
$387$		0			0
$388$	10.0000			0.507673
$389$		−	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
$392$	−1.73205			−0.0874818
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−8.00000			−0.401508			−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000			−0.401508			−0.200754	+	0.979642i	$0.564339\pi$
$398$	13.8564			0.694559
$399$		0			0
$400$		0			0
$401$		−	1.41421i		−	0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$		−	1.41421i		−	0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$		0			0
$403$		−	19.5959i		−	0.976142i
$404$	13.8564			0.689382
$405$		0			0
$406$		−	29.3939i		−	1.45879i
$407$	−13.8564	+	22.6274i	−0.686837	+	1.12160i
$408$		0			0
$409$			39.1918i			1.93791i	0.247234	+	0.968956i	$0.420478\pi$
$409$			39.1918i			1.93791i	−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$	4.00000			0.197066
$413$	−27.7128			−1.36366
$414$		0			0
$415$		0			0
$416$		−	25.4558i		−	1.24808i
$417$		0			0
$418$	36.0000	+	22.0454i	1.76082	+	1.07828i
$419$			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	$0.437546\pi$
$422$		−	21.2132i		−	1.03264i
$423$		0			0
$424$		−	17.1464i		−	0.832704i
$425$		0			0
$426$		0			0
$427$	−12.0000			−0.580721
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.3923			−0.500580			−0.250290	−	0.968171i	$-0.580526\pi$
$431$	−10.3923			−0.500580			−0.250290	+	0.968171i	$0.580526\pi$
$432$		0			0
$433$	16.0000			0.768911			0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000			0.768911			0.384455	+	0.923144i	$0.374389\pi$
$434$		−	16.9706i		−	0.814613i
$435$		0			0
$436$		−	14.6969i		−	0.703856i
$437$	20.7846			0.994263
$438$		0			0
$439$		−	2.44949i		−	0.116908i	−0.998290	−	0.0584539i	$-0.981383\pi$
$439$		−	2.44949i		−	0.116908i	0.998290	−	0.0584539i	$-0.0186171\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			22.6274i			1.07506i	0.843244	+	0.537531i	$0.180643\pi$
$443$			22.6274i			1.07506i	−0.843244	+	0.537531i	$0.819357\pi$
$444$		0			0
$445$		0			0
$446$	27.7128			1.31224
$447$		0			0
$448$			2.44949i			0.115728i
$449$		−	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
$452$			1.41421i			0.0665190i
$453$		0			0
$454$	48.0000			2.25275
$455$		0			0
$456$		0			0
$457$			9.79796i			0.458329i	0.973388	+	0.229165i	$0.0735994\pi$
$457$			9.79796i			0.458329i	−0.973388	+	0.229165i	$0.926401\pi$
$458$	−17.3205			−0.809334
$459$		0			0
$460$		0			0
$461$	−6.92820			−0.322679			−0.161339	−	0.986899i	$-0.551581\pi$
$461$	−6.92820			−0.322679			−0.161339	+	0.986899i	$0.551581\pi$
$462$		0			0
$463$	28.0000			1.30127			0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000			1.30127			0.650635	+	0.759390i	$0.274503\pi$
$464$	34.6410			1.60817
$465$		0			0
$466$	36.0000			1.66767
$467$			22.6274i			1.04707i	0.852004	+	0.523536i	$0.175387\pi$
$467$			22.6274i			1.04707i	−0.852004	+	0.523536i	$0.824613\pi$
$468$		0			0
$469$			9.79796i			0.452428i
$470$		0			0
$471$		0			0
$472$		−	19.5959i		−	0.901975i
$473$	6.92820	+	4.24264i	0.318559	+	0.195077i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	30.0000			1.37217
$479$	−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
$482$			33.9411i			1.54598i
$483$		0			0
$484$	−5.00000	−	9.79796i	−0.227273	−	0.445362i
$485$		0			0
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		−	8.48528i		−	0.384111i
$489$		0			0
$490$		0			0
$491$	27.7128			1.25066			0.625331	−	0.780360i	$-0.284964\pi$
$491$	27.7128			1.25066			0.625331	+	0.780360i	$0.284964\pi$
$492$		0			0
$493$		0			0
$494$	−62.3538			−2.80543
$495$		0			0
$496$	20.0000			0.898027
$497$	−6.92820			−0.310772
$498$		0			0
$499$	32.0000			1.43252			0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000			1.43252			0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$		0			0
$502$		−	9.79796i		−	0.437304i
$503$	31.1769			1.39011			0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769			1.39011			0.695055	+	0.718957i	$0.255380\pi$
$504$		0			0
$505$		0			0
$506$	−13.8564	−	8.48528i	−0.615992	−	0.377217i
$507$		0			0
$508$		−	7.34847i		−	0.326036i
$509$			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
$512$	8.66025			0.382733
$513$		0			0
$514$			17.1464i			0.756297i
$515$		0			0
$516$		0			0
$517$	−8.00000	−	4.89898i	−0.351840	−	0.215457i
$518$		−	33.9411i		−	1.49129i
$519$		0			0
$520$		0			0
$521$			15.5563i			0.681536i	0.940147	+	0.340768i	$0.110687\pi$
$521$			15.5563i			0.681536i	−0.940147	+	0.340768i	$0.889313\pi$
$522$		0			0
$523$		−	22.0454i		−	0.963978i	−0.876177	−	0.481989i	$-0.839914\pi$
$523$		−	22.0454i		−	0.963978i	0.876177	−	0.481989i	$-0.160086\pi$
$524$	−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
$532$	−18.0000			−0.780399
$533$			33.9411i			1.47015i
$534$		0			0
$535$		0			0
$536$	−6.92820			−0.299253
$537$		0			0
$538$			26.9444i			1.16166i
$539$	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	$0.897682\pi$
$542$		−	12.7279i		−	0.546711i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			31.8434i			1.36152i	0.732505	+	0.680762i	$0.238351\pi$
$547$			31.8434i			1.36152i	−0.732505	+	0.680762i	$0.761649\pi$
$548$		−	15.5563i		−	0.664534i
$549$		0			0
$550$		0			0
$551$		−	50.9117i		−	2.16891i
$552$		0			0
$553$	−30.0000			−1.27573
$554$			42.4264i			1.80253i
$555$		0			0
$556$			2.44949i			0.103882i
$557$	−20.7846			−0.880672			−0.440336	−	0.897833i	$-0.645141\pi$
$557$	−20.7846			−0.880672			−0.440336	+	0.897833i	$0.645141\pi$
$558$		0			0
$559$	−12.0000			−0.507546
$560$		0			0
$561$		0			0
$562$	24.0000			1.01238
$563$	−17.3205			−0.729972			−0.364986	−	0.931013i	$-0.618926\pi$
$563$	−17.3205			−0.729972			−0.364986	+	0.931013i	$0.618926\pi$
$564$		0			0
$565$		0			0
$566$		−	4.24264i		−	0.178331i
$567$		0			0
$568$		−	4.89898i		−	0.205557i
$569$	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	$0.983678\pi$
$572$	13.8564	+	8.48528i	0.579365	+	0.354787i
$573$		0			0
$574$			29.3939i			1.22688i
$575$		0			0
$576$		0			0
$577$	10.0000			0.416305			0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000			0.416305			0.208153	+	0.978096i	$0.433255\pi$
$578$	−29.4449			−1.22474
$579$		0			0
$580$		0			0
$581$		−	33.9411i		−	1.40812i
$582$		0			0
$583$	28.0000	+	17.1464i	1.15964	+	0.710132i
$584$		0			0
$585$		0			0
$586$	−12.0000			−0.495715
$587$		−	28.2843i		−	1.16742i	−0.811963	−	0.583708i	$-0.801601\pi$
$587$		−	28.2843i		−	1.16742i	0.811963	−	0.583708i	$-0.198399\pi$
$588$		0			0
$589$		−	29.3939i		−	1.21115i
$590$		0			0
$591$		0			0
$592$	40.0000			1.64399
$593$	−41.5692			−1.70704			−0.853522	−	0.521057i	$-0.825538\pi$
$593$	−41.5692			−1.70704			−0.853522	+	0.521057i	$0.825538\pi$
$594$		0			0
$595$		0			0
$596$	−13.8564			−0.567581
$597$		0			0
$598$	24.0000			0.981433
$599$			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	$0.869126\pi$
$602$	−10.3923			−0.423559
$603$		0			0
$604$		−	2.44949i		−	0.0996683i
$605$		0			0
$606$		0			0
$607$			12.2474i			0.497109i	0.968618	+	0.248554i	$0.0799554\pi$
$607$			12.2474i			0.497109i	−0.968618	+	0.248554i	$0.920045\pi$
$608$		−	38.1838i		−	1.54856i
$609$		0			0
$610$		0			0
$611$	13.8564			0.560570
$612$		0			0
$613$			14.6969i			0.593604i	0.954939	+	0.296802i	$0.0959201\pi$
$613$			14.6969i			0.593604i	−0.954939	+	0.296802i	$0.904080\pi$
$614$		−	12.7279i		−	0.513657i
$615$		0			0
$616$	−12.0000	−	7.34847i	−0.483494	−	0.296078i
$617$			26.8701i			1.08175i	0.841104	+	0.540874i	$0.181906\pi$
$617$			26.8701i			1.08175i	−0.841104	+	0.540874i	$0.818094\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$		0			0
$622$		−	24.4949i		−	0.982156i
$623$	−17.3205			−0.693932
$624$		0			0
$625$		0			0
$626$	27.7128			1.10763
$627$		0			0
$628$	−14.0000			−0.558661
$629$		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	$0.160330\pi$
$632$		−	21.2132i		−	0.843816i
$633$		0			0
$634$			31.8434i			1.26466i
$635$		0			0
$636$		0			0
$637$			4.89898i			0.194105i
$638$	−20.7846	+	33.9411i	−0.822871	+	1.34374i
$639$		0			0
$640$		0			0
$641$			32.5269i			1.28474i	0.766396	+	0.642368i	$0.222048\pi$
$641$			32.5269i			1.28474i	−0.766396	+	0.642368i	$0.777952\pi$
$642$		0			0
$643$	28.0000			1.10421			0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000			1.10421			0.552106	+	0.833774i	$0.313824\pi$
$644$	6.92820			0.273009
$645$		0			0
$646$		0			0
$647$		−	19.7990i		−	0.778379i	−0.921158	−	0.389189i	$-0.872755\pi$
$647$		−	19.7990i		−	0.778379i	0.921158	−	0.389189i	$-0.127245\pi$
$648$		0			0
$649$	32.0000	+	19.5959i	1.25611	+	0.769207i
$650$		0			0
$651$		0			0
$652$	−8.00000			−0.313304
$653$			26.8701i			1.05151i	0.850637	+	0.525753i	$0.176216\pi$
$653$			26.8701i			1.05151i	−0.850637	+	0.525753i	$0.823784\pi$
$654$		0			0
$655$		0			0
$656$	−34.6410			−1.35250
$657$		0			0
$658$	12.0000			0.467809
$659$	−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	$0.640729\pi$
$662$	13.8564			0.538545
$663$		0			0
$664$	24.0000			0.931381
$665$		0			0
$666$		0			0
$667$			19.5959i			0.758757i
$668$	3.46410			0.134030
$669$		0			0
$670$		0			0
$671$	13.8564	+	8.48528i	0.534921	+	0.327571i
$672$		0			0
$673$			39.1918i			1.51073i	0.655302	+	0.755367i	$0.272541\pi$
$673$			39.1918i			1.51073i	−0.655302	+	0.755367i	$0.727459\pi$
$674$		−	16.9706i		−	0.653682i
$675$		0			0
$676$	−11.0000			−0.423077
$677$	27.7128			1.06509			0.532545	−	0.846402i	$-0.321236\pi$
$677$	27.7128			1.06509			0.532545	+	0.846402i	$0.321236\pi$
$678$		0			0
$679$			24.4949i			0.940028i
$680$		0			0
$681$		0			0
$682$	−12.0000	+	19.5959i	−0.459504	+	0.750366i
$683$		−	28.2843i		−	1.08227i	−0.840937	−	0.541134i	$-0.817995\pi$
$683$		−	28.2843i		−	1.08227i	0.840937	−	0.541134i	$-0.182005\pi$
$684$		0			0
$685$		0			0
$686$			33.9411i			1.29588i
$687$		0			0
$688$		−	12.2474i		−	0.466930i
$689$	−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	$0.184354\pi$
$692$	6.92820			0.263371
$693$		0			0
$694$	24.0000			0.911028
$695$		0			0
$696$		0			0
$697$		0			0
$698$			8.48528i			0.321173i
$699$		0			0
$700$		0			0
$701$	−20.7846			−0.785024			−0.392512	−	0.919747i	$-0.628394\pi$
$701$	−20.7846			−0.785024			−0.392512	+	0.919747i	$0.628394\pi$
$702$		0			0
$703$		−	58.7878i		−	2.21722i
$704$	1.73205	−	2.82843i	0.0652791	−	0.106600i
$705$		0			0
$706$		−	56.3383i		−	2.12032i
$707$			33.9411i			1.27649i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$		0			0
$712$		−	12.2474i		−	0.458993i
$713$			11.3137i			0.423702i
$714$		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
$722$	−60.6218			−2.25611
$723$		0			0
$724$	8.00000			0.297318
$725$		0			0
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$	20.7846			0.770329
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$			34.2929i			1.26664i	0.773892	+	0.633318i	$0.218307\pi$
$733$			34.2929i			1.26664i	−0.773892	+	0.633318i	$0.781693\pi$
$734$	−13.8564			−0.511449
$735$		0			0
$736$			14.6969i			0.541736i
$737$	6.92820	−	11.3137i	0.255204	−	0.416746i
$738$		0			0
$739$		−	7.34847i		−	0.270318i	−0.990824	−	0.135159i	$-0.956846\pi$
$739$		−	7.34847i		−	0.270318i	0.990824	−	0.135159i	$-0.0431545\pi$
$740$		0			0
$741$		0			0
$742$	−42.0000			−1.54187
$743$	−27.7128			−1.01668			−0.508342	−	0.861155i	$-0.669742\pi$
$743$	−27.7128			−1.01668			−0.508342	+	0.861155i	$0.669742\pi$
$744$		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	$0.523251\pi$
$752$			14.1421i			0.515711i
$753$		0			0
$754$		−	58.7878i		−	2.14092i
$755$		0			0
$756$		0			0
$757$	−8.00000			−0.290765			−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000			−0.290765			−0.145382	+	0.989376i	$0.546441\pi$
$758$	−27.7128			−1.00657
$759$		0			0
$760$		0			0
$761$	−13.8564			−0.502294			−0.251147	−	0.967949i	$-0.580808\pi$
$761$	−13.8564			−0.502294			−0.251147	+	0.967949i	$0.580808\pi$
$762$		0			0
$763$	36.0000			1.30329
$764$			2.82843i			0.102329i
$765$		0			0
$766$		−	63.6867i		−	2.30110i
$767$	−55.4256			−2.00130
$768$		0			0
$769$		−	19.5959i		−	0.706647i	−0.935501	−	0.353323i	$-0.885052\pi$
$769$		−	19.5959i		−	0.706647i	0.935501	−	0.353323i	$-0.114948\pi$
$770$		0			0
$771$		0			0
$772$		−	9.79796i		−	0.352636i
$773$		−	32.5269i		−	1.16991i	−0.811065	−	0.584956i	$-0.801112\pi$
$773$		−	32.5269i		−	1.16991i	0.811065	−	0.584956i	$-0.198888\pi$
$774$		0			0
$775$		0			0
$776$	−17.3205			−0.621770
$777$		0			0
$778$		−	17.1464i		−	0.614729i
$779$			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
$787$			12.2474i			0.436574i	0.975885	+	0.218287i	$0.0700470\pi$
$787$			12.2474i			0.436574i	−0.975885	+	0.218287i	$0.929953\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.46410			−0.123169
$792$		0			0
$793$	−24.0000			−0.852265
$794$	−13.8564			−0.491745
$795$		0			0
$796$	8.00000			0.283552
$797$			26.8701i			0.951786i	0.879503	+	0.475893i	$0.157875\pi$
$797$			26.8701i			0.951786i	−0.879503	+	0.475893i	$0.842125\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		−	2.44949i		−	0.0864945i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		−	33.9411i		−	1.19553i
$807$		0			0
$808$	−24.0000			−0.844317
$809$	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	$0.873491\pi$
$812$		−	16.9706i		−	0.595550i
$813$		0			0
$814$	−24.0000	+	39.1918i	−0.841200	+	1.37367i
$815$		0			0
$816$		0			0
$817$	−18.0000			−0.629740
$818$			67.8823i			2.37345i
$819$		0			0
$820$		0			0
$821$	13.8564			0.483592			0.241796	−	0.970327i	$-0.422264\pi$
$821$	13.8564			0.483592			0.241796	+	0.970327i	$0.422264\pi$
$822$		0			0
$823$	−8.00000			−0.278862			−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000			−0.278862			−0.139431	+	0.990232i	$0.544527\pi$
$824$	−6.92820			−0.241355
$825$		0			0
$826$	−48.0000			−1.67013
$827$	41.5692			1.44550			0.722752	−	0.691108i	$-0.242877\pi$
$827$	41.5692			1.44550			0.722752	+	0.691108i	$0.242877\pi$
$828$		0			0
$829$	−46.0000			−1.59765			−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000			−1.59765			−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$		0			0
$832$			4.89898i			0.169842i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	20.7846	+	12.7279i	0.718851	+	0.440204i
$837$		0			0
$838$			19.5959i			0.676930i
$839$			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
$842$	13.8564			0.477523
$843$		0			0
$844$		−	12.2474i		−	0.421575i
$845$		0			0
$846$		0			0
$847$	24.0000	−	12.2474i	0.824650	−	0.420827i
$848$		−	49.4975i		−	1.69975i
$849$		0			0
$850$		0			0
$851$			22.6274i			0.775658i
$852$		0			0
$853$		−	34.2929i		−	1.17417i	−0.809527	−	0.587083i	$-0.800276\pi$
$853$		−	34.2929i		−	1.17417i	0.809527	−	0.587083i	$-0.199724\pi$
$854$	−20.7846			−0.711235
$855$		0			0
$856$		0			0
$857$	−34.6410			−1.18331			−0.591657	−	0.806190i	$-0.701526\pi$
$857$	−34.6410			−1.18331			−0.591657	+	0.806190i	$0.701526\pi$
$858$		0			0
$859$	−40.0000			−1.36478			−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000			−1.36478			−0.682391	+	0.730987i	$0.739060\pi$
$860$		0			0
$861$		0			0
$862$	−18.0000			−0.613082
$863$		−	19.7990i		−	0.673965i	−0.941511	−	0.336983i	$-0.890594\pi$
$863$		−	19.7990i		−	0.673965i	0.941511	−	0.336983i	$-0.109406\pi$
$864$		0			0
$865$		0			0
$866$	27.7128			0.941720
$867$		0			0
$868$		−	9.79796i		−	0.332564i
$869$	34.6410	+	21.2132i	1.17512	+	0.719609i
$870$		0			0
$871$			19.5959i			0.663982i
$872$			25.4558i			0.862044i
$873$		0			0
$874$	36.0000			1.21772
$875$		0			0
$876$		0			0
$877$		−	53.8888i		−	1.81969i	−0.414943	−	0.909847i	$-0.636199\pi$
$877$		−	53.8888i		−	1.81969i	0.414943	−	0.909847i	$-0.363801\pi$
$878$		−	4.24264i		−	0.143182i
$879$		0			0
$880$		0			0
$881$		−	9.89949i		−	0.333522i	−0.985997	−	0.166761i	$-0.946669\pi$
$881$		−	9.89949i		−	0.333522i	0.985997	−	0.166761i	$-0.0533309\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$			39.1918i			1.31668i
$887$	55.4256			1.86101			0.930505	−	0.366279i	$-0.119368\pi$
$887$	55.4256			1.86101			0.930505	+	0.366279i	$0.119368\pi$
$888$		0			0
$889$	18.0000			0.603701
$890$		0			0
$891$		0			0
$892$	16.0000			0.535720
$893$	20.7846			0.695530
$894$		0			0
$895$		0			0
$896$			29.6985i			0.992157i
$897$		0			0
$898$		−	17.1464i		−	0.572184i
$899$	27.7128			0.924274
$900$		0			0
$901$		0			0
$902$	20.7846	−	33.9411i	0.692052	−	1.13012i
$903$		0			0
$904$		−	2.44949i		−	0.0814688i
$905$		0			0
$906$		0			0
$907$	52.0000			1.72663			0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000			1.72663			0.863316	+	0.504664i	$0.168384\pi$
$908$	27.7128			0.919682
$909$		0			0
$910$		0			0
$911$			53.7401i			1.78049i	0.455483	+	0.890245i	$0.349467\pi$
$911$			53.7401i			1.78049i	−0.455483	+	0.890245i	$0.650533\pi$
$912$		0			0
$913$	−24.0000	+	39.1918i	−0.794284	+	1.29706i
$914$			16.9706i			0.561336i
$915$		0			0
$916$	−10.0000			−0.330409
$917$		−	8.48528i		−	0.280209i
$918$		0			0
$919$		−	36.7423i		−	1.21202i	−0.795458	−	0.606009i	$-0.792770\pi$
$919$		−	36.7423i		−	1.21202i	0.795458	−	0.606009i	$-0.207230\pi$
$920$		0			0
$921$		0			0
$922$	−12.0000			−0.395199
$923$	−13.8564			−0.456089
$924$		0			0
$925$		0			0
$926$	48.4974			1.59372
$927$		0			0
$928$	36.0000			1.18176
$929$		−	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
$932$	20.7846			0.680823
$933$		0			0
$934$			39.1918i			1.28240i
$935$		0			0
$936$		0			0
$937$			29.3939i			0.960256i	0.877198	+	0.480128i	$0.159410\pi$
$937$			29.3939i			0.960256i	−0.877198	+	0.480128i	$0.840590\pi$
$938$			16.9706i			0.554109i
$939$		0			0
$940$		0			0
$941$	27.7128			0.903412			0.451706	−	0.892167i	$-0.350816\pi$
$941$	27.7128			0.903412			0.451706	+	0.892167i	$0.350816\pi$
$942$		0			0
$943$		−	19.5959i		−	0.638131i
$944$		−	56.5685i		−	1.84115i
$945$		0			0
$946$	12.0000	+	7.34847i	0.390154	+	0.238919i
$947$			22.6274i			0.735292i	0.929966	+	0.367646i	$0.119836\pi$
$947$			22.6274i			0.735292i	−0.929966	+	0.367646i	$0.880164\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20.7846			0.673280			0.336640	−	0.941634i	$-0.390710\pi$
$953$	20.7846			0.673280			0.336640	+	0.941634i	$0.390710\pi$
$954$		0			0
$955$		0			0
$956$	17.3205			0.560185
$957$		0			0
$958$	−66.0000			−2.13236
$959$	38.1051			1.23048
$960$		0			0
$961$	−15.0000			−0.483871
$962$		−	67.8823i		−	2.18861i
$963$		0			0
$964$			19.5959i			0.631142i
$965$		0			0
$966$		0			0
$967$			41.6413i			1.33909i	0.742769	+	0.669547i	$0.233512\pi$
$967$			41.6413i			1.33909i	−0.742769	+	0.669547i	$0.766488\pi$
$968$	8.66025	+	16.9706i	0.278351	+	0.545455i
$969$		0			0
$970$		0			0
$971$		−	56.5685i		−	1.81537i	−0.419651	−	0.907685i	$-0.637848\pi$
$971$		−	56.5685i		−	1.81537i	0.419651	−	0.907685i	$-0.362152\pi$
$972$		0			0
$973$	−6.00000			−0.192351
$974$	−13.8564			−0.443988
$975$		0			0
$976$		−	24.4949i		−	0.784063i
$977$			35.3553i			1.13112i	0.824708	+	0.565559i	$0.191339\pi$
$977$			35.3553i			1.13112i	−0.824708	+	0.565559i	$0.808661\pi$
$978$		0			0
$979$	20.0000	+	12.2474i	0.639203	+	0.391430i
$980$		0			0
$981$		0			0
$982$	48.0000			1.53174
$983$			48.0833i			1.53362i	0.641875	+	0.766809i	$0.278157\pi$
$983$			48.0833i			1.53362i	−0.641875	+	0.766809i	$0.721843\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−36.0000			−1.14531
$989$	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	$0.809342\pi$
$992$	20.7846			0.659912
$993$		0			0
$994$	−12.0000			−0.380617
$995$		0			0
$996$		0			0
$997$			24.4949i			0.775761i	0.921710	+	0.387881i	$0.126793\pi$
$997$			24.4949i			0.775761i	−0.921710	+	0.387881i	$0.873207\pi$
$998$	55.4256			1.75447
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.f.e.2276.4		4
3.2	odd	2	inner	2475.2.f.e.2276.2		4
5.2	odd	4		2475.2.d.a.2474.6		8
5.3	odd	4		2475.2.d.a.2474.4		8
5.4	even	2		99.2.d.a.98.1	✓	4
11.10	odd	2	inner	2475.2.f.e.2276.1		4
15.2	even	4		2475.2.d.a.2474.1		8
15.8	even	4		2475.2.d.a.2474.7		8
15.14	odd	2		99.2.d.a.98.4	yes	4
20.19	odd	2		1584.2.b.e.593.2		4
33.32	even	2	inner	2475.2.f.e.2276.3		4
40.19	odd	2		6336.2.b.t.2177.4		4
40.29	even	2		6336.2.b.s.2177.3		4
45.4	even	6		891.2.g.c.593.3		8
45.14	odd	6		891.2.g.c.593.2		8
45.29	odd	6		891.2.g.c.296.1		8
45.34	even	6		891.2.g.c.296.4		8
55.32	even	4		2475.2.d.a.2474.3		8
55.43	even	4		2475.2.d.a.2474.5		8
55.54	odd	2		99.2.d.a.98.3	yes	4
60.59	even	2		1584.2.b.e.593.4		4
120.29	odd	2		6336.2.b.s.2177.1		4
120.59	even	2		6336.2.b.t.2177.2		4
165.32	odd	4		2475.2.d.a.2474.8		8
165.98	odd	4		2475.2.d.a.2474.2		8
165.164	even	2		99.2.d.a.98.2	yes	4
220.219	even	2		1584.2.b.e.593.1		4
440.109	odd	2		6336.2.b.s.2177.4		4
440.219	even	2		6336.2.b.t.2177.3		4
495.164	even	6		891.2.g.c.296.3		8
495.274	odd	6		891.2.g.c.593.1		8
495.329	even	6		891.2.g.c.593.4		8
495.439	odd	6		891.2.g.c.296.2		8
660.659	odd	2		1584.2.b.e.593.3		4
1320.659	odd	2		6336.2.b.t.2177.1		4
1320.989	even	2		6336.2.b.s.2177.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.d.a.98.1	✓	4	5.4	even	2
99.2.d.a.98.2	yes	4	165.164	even	2
99.2.d.a.98.3	yes	4	55.54	odd	2
99.2.d.a.98.4	yes	4	15.14	odd	2
891.2.g.c.296.1		8	45.29	odd	6
891.2.g.c.296.2		8	495.439	odd	6
891.2.g.c.296.3		8	495.164	even	6
891.2.g.c.296.4		8	45.34	even	6
891.2.g.c.593.1		8	495.274	odd	6
891.2.g.c.593.2		8	45.14	odd	6
891.2.g.c.593.3		8	45.4	even	6
891.2.g.c.593.4		8	495.329	even	6
1584.2.b.e.593.1		4	220.219	even	2
1584.2.b.e.593.2		4	20.19	odd	2
1584.2.b.e.593.3		4	660.659	odd	2
1584.2.b.e.593.4		4	60.59	even	2
2475.2.d.a.2474.1		8	15.2	even	4
2475.2.d.a.2474.2		8	165.98	odd	4
2475.2.d.a.2474.3		8	55.32	even	4
2475.2.d.a.2474.4		8	5.3	odd	4
2475.2.d.a.2474.5		8	55.43	even	4
2475.2.d.a.2474.6		8	5.2	odd	4
2475.2.d.a.2474.7		8	15.8	even	4
2475.2.d.a.2474.8		8	165.32	odd	4
2475.2.f.e.2276.1		4	11.10	odd	2	inner
2475.2.f.e.2276.2		4	3.2	odd	2	inner
2475.2.f.e.2276.3		4	33.32	even	2	inner
2475.2.f.e.2276.4		4	1.1	even	1	trivial
6336.2.b.s.2177.1		4	120.29	odd	2
6336.2.b.s.2177.2		4	1320.989	even	2
6336.2.b.s.2177.3		4	40.29	even	2
6336.2.b.s.2177.4		4	440.109	odd	2
6336.2.b.t.2177.1		4	1320.659	odd	2
6336.2.b.t.2177.2		4	120.59	even	2
6336.2.b.t.2177.3		4	440.219	even	2
6336.2.b.t.2177.4		4	40.19	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +2.44949i q^{7} -1.73205 q^{8} +(1.73205 - 2.82843i) q^{11} +4.89898i q^{13} +4.24264i q^{14} -5.00000 q^{16} +7.34847i q^{19} +(3.00000 - 4.89898i) q^{22} -2.82843i q^{23} +8.48528i q^{26} +2.44949i q^{28} -6.92820 q^{29} -4.00000 q^{31} -5.19615 q^{32} -8.00000 q^{37} +12.7279i q^{38} +6.92820 q^{41} +2.44949i q^{43} +(1.73205 - 2.82843i) q^{44} -4.89898i q^{46} -2.82843i q^{47} +1.00000 q^{49} +4.89898i q^{52} +9.89949i q^{53} -4.24264i q^{56} -12.0000 q^{58} +11.3137i q^{59} +4.89898i q^{61} -6.92820 q^{62} +1.00000 q^{64} +4.00000 q^{67} +2.82843i q^{71} -13.8564 q^{74} +7.34847i q^{76} +(6.92820 + 4.24264i) q^{77} +12.2474i q^{79} +12.0000 q^{82} -13.8564 q^{83} +4.24264i q^{86} +(-3.00000 + 4.89898i) q^{88} +7.07107i q^{89} -12.0000 q^{91} -2.82843i q^{92} -4.89898i q^{94} +10.0000 q^{97} +1.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 20 q^{16} + 12 q^{22} - 16 q^{31} - 32 q^{37} + 4 q^{49} - 48 q^{58} + 4 q^{64} + 16 q^{67} + 48 q^{82} - 12 q^{88} - 48 q^{91} + 40 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 20 * q^16 + 12 * q^22 - 16 * q^31 - 32 * q^37 + 4 * q^49 - 48 * q^58 + 4 * q^64 + 16 * q^67 + 48 * q^82 - 12 * q^88 - 48 * q^91 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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