LMFDB - Embedded newform 1503.2.c.a.1502.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1502,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1503.1502");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1502.5
Character	$\chi$	$=$	1503.1502
Dual form			1503.2.c.a.1502.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.78744i q^{2} -1.19494 q^{4} +3.10541 q^{5} +3.91211 q^{7} -1.43899i q^{8} +O(q^{10})$	\(q-1.78744i q^{2} -1.19494 q^{4} +3.10541 q^{5} +3.91211 q^{7} -1.43899i q^{8} -5.55073i q^{10} -3.45052i q^{11} -0.615854i q^{13} -6.99266i q^{14} -4.96200 q^{16} -4.80134 q^{17} +3.94082 q^{19} -3.71079 q^{20} -6.16760 q^{22} -5.03817 q^{23} +4.64356 q^{25} -1.10080 q^{26} -4.67475 q^{28} +2.60980i q^{29} -1.21494 q^{31} +5.99130i q^{32} +8.58211i q^{34} +12.1487 q^{35} +3.76909i q^{37} -7.04398i q^{38} -4.46865i q^{40} +10.8643 q^{41} +3.95156i q^{43} +4.12318i q^{44} +9.00543i q^{46} -11.2612i q^{47} +8.30458 q^{49} -8.30009i q^{50} +0.735911i q^{52} -6.37052 q^{53} -10.7153i q^{55} -5.62948i q^{56} +4.66487 q^{58} +9.32292 q^{59} -8.66282 q^{61} +2.17163i q^{62} +0.785091 q^{64} -1.91248i q^{65} +12.5114i q^{67} +5.73733 q^{68} -21.7151i q^{70} -7.34993 q^{71} +3.24263i q^{73} +6.73703 q^{74} -4.70906 q^{76} -13.4988i q^{77} +6.21138i q^{79} -15.4090 q^{80} -19.4194i q^{82} +17.4377 q^{83} -14.9101 q^{85} +7.06319 q^{86} -4.96527 q^{88} +13.0274i q^{89} -2.40929i q^{91} +6.02033 q^{92} -20.1287 q^{94} +12.2379 q^{95} +9.89549 q^{97} -14.8439i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 48 q^{4}+O(q^{10}) $ 56 * q - 48 * q^4	$ 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) $ 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.78744i		−	1.26391i	−0.775005	−	0.631956i	$-0.782252\pi$
$2$		−	1.78744i		−	1.26391i	0.775005	−	0.631956i	$-0.217748\pi$
$3$		0			0
$3$		0			0
$4$	−1.19494			−0.597472
$5$	3.10541			1.38878			0.694390	−	0.719598i	$-0.255674\pi$
$5$	3.10541			1.38878			0.694390	+	0.719598i	$0.255674\pi$
$6$		0			0
$7$	3.91211			1.47864			0.739319	−	0.673356i	$-0.235148\pi$
$7$	3.91211			1.47864			0.739319	+	0.673356i	$0.235148\pi$
$8$		−	1.43899i		−	0.508760i
$9$		0			0
$10$		−	5.55073i		−	1.75530i
$11$		−	3.45052i		−	1.04037i	−0.854053	−	0.520186i	$-0.825863\pi$
$11$		−	3.45052i		−	1.04037i	0.854053	−	0.520186i	$-0.174137\pi$
$12$		0			0
$13$		−	0.615854i		−	0.170807i	−0.996346	−	0.0854036i	$-0.972782\pi$
$13$		−	0.615854i		−	0.170807i	0.996346	−	0.0854036i	$-0.0272180\pi$
$14$		−	6.99266i		−	1.86887i
$15$		0			0
$16$	−4.96200			−1.24050
$17$	−4.80134			−1.16450			−0.582248	−	0.813011i	$-0.697827\pi$
$17$	−4.80134			−1.16450			−0.582248	+	0.813011i	$0.697827\pi$
$18$		0			0
$19$	3.94082			0.904086			0.452043	−	0.891996i	$-0.350695\pi$
$19$	3.94082			0.904086			0.452043	+	0.891996i	$0.350695\pi$
$20$	−3.71079			−0.829758
$21$		0			0
$22$	−6.16760			−1.31494
$23$	−5.03817			−1.05053			−0.525265	−	0.850938i	$-0.676034\pi$
$23$	−5.03817			−1.05053			−0.525265	+	0.850938i	$0.676034\pi$
$24$		0			0
$25$	4.64356			0.928713
$26$	−1.10080			−0.215885
$27$		0			0
$28$	−4.67475			−0.883444
$29$			2.60980i			0.484628i	0.970198	+	0.242314i	$0.0779064\pi$
$29$			2.60980i			0.484628i	−0.970198	+	0.242314i	$0.922094\pi$
$30$		0			0
$31$	−1.21494			−0.218209			−0.109105	−	0.994030i	$-0.534798\pi$
$31$	−1.21494			−0.218209			−0.109105	+	0.994030i	$0.534798\pi$
$32$			5.99130i			1.05912i
$33$		0			0
$34$			8.58211i			1.47182i
$35$	12.1487			2.05350
$36$		0			0
$37$			3.76909i			0.619635i	0.950796	+	0.309818i	$0.100268\pi$
$37$			3.76909i			0.619635i	−0.950796	+	0.309818i	$0.899732\pi$
$38$		−	7.04398i		−	1.14268i
$39$		0			0
$40$		−	4.46865i		−	0.706556i
$41$	10.8643			1.69672			0.848362	−	0.529416i	$-0.177589\pi$
$41$	10.8643			1.69672			0.848362	+	0.529416i	$0.177589\pi$
$42$		0			0
$43$			3.95156i			0.602608i	0.953528	+	0.301304i	$0.0974219\pi$
$43$			3.95156i			0.602608i	−0.953528	+	0.301304i	$0.902578\pi$
$44$			4.12318i			0.621593i
$45$		0			0
$46$			9.00543i			1.32778i
$47$		−	11.2612i		−	1.64261i	−0.570486	−	0.821307i	$-0.693245\pi$
$47$		−	11.2612i		−	1.64261i	0.570486	−	0.821307i	$-0.306755\pi$
$48$		0			0
$49$	8.30458			1.18637
$50$		−	8.30009i		−	1.17381i
$51$		0			0
$52$			0.735911i			0.102052i
$53$	−6.37052			−0.875058			−0.437529	−	0.899204i	$-0.644146\pi$
$53$	−6.37052			−0.875058			−0.437529	+	0.899204i	$0.644146\pi$
$54$		0			0
$55$		−	10.7153i		−	1.44485i
$56$		−	5.62948i		−	0.752271i
$57$		0			0
$58$	4.66487			0.612527
$59$	9.32292			1.21374			0.606870	−	0.794801i	$-0.292425\pi$
$59$	9.32292			1.21374			0.606870	+	0.794801i	$0.292425\pi$
$60$		0			0
$61$	−8.66282			−1.10916			−0.554580	−	0.832130i	$-0.687121\pi$
$61$	−8.66282			−1.10916			−0.554580	+	0.832130i	$0.687121\pi$
$62$			2.17163i			0.275797i
$63$		0			0
$64$	0.785091			0.0981363
$65$		−	1.91248i		−	0.237214i
$66$		0			0
$67$			12.5114i			1.52852i	0.644910	+	0.764258i	$0.276895\pi$
$67$			12.5114i			1.52852i	−0.644910	+	0.764258i	$0.723105\pi$
$68$	5.73733			0.695753
$69$		0			0
$70$		−	21.7151i		−	2.59545i
$71$	−7.34993			−0.872276			−0.436138	−	0.899880i	$-0.643654\pi$
$71$	−7.34993			−0.872276			−0.436138	+	0.899880i	$0.643654\pi$
$72$		0			0
$73$			3.24263i			0.379521i	0.981830	+	0.189760i	$0.0607711\pi$
$73$			3.24263i			0.379521i	−0.981830	+	0.189760i	$0.939229\pi$
$74$	6.73703			0.783164
$75$		0			0
$76$	−4.70906			−0.540166
$77$		−	13.4988i		−	1.53833i
$78$		0			0
$79$			6.21138i			0.698834i	0.936967	+	0.349417i	$0.113620\pi$
$79$			6.21138i			0.698834i	−0.936967	+	0.349417i	$0.886380\pi$
$80$	−15.4090			−1.72278
$81$		0			0
$82$		−	19.4194i		−	2.14451i
$83$	17.4377			1.91404			0.957021	−	0.290019i	$-0.0936618\pi$
$83$	17.4377			1.91404			0.957021	+	0.290019i	$0.0936618\pi$
$84$		0			0
$85$	−14.9101			−1.61723
$86$	7.06319			0.761643
$87$		0			0
$88$	−4.96527			−0.529299
$89$			13.0274i			1.38090i	0.723381	+	0.690450i	$0.242587\pi$
$89$			13.0274i			1.38090i	−0.723381	+	0.690450i	$0.757413\pi$
$90$		0			0
$91$		−	2.40929i		−	0.252562i
$92$	6.02033			0.627663
$93$		0			0
$94$	−20.1287			−2.07612
$95$	12.2379			1.25558
$96$		0			0
$97$	9.89549			1.00473			0.502367	−	0.864654i	$-0.332462\pi$
$97$	9.89549			1.00473			0.502367	+	0.864654i	$0.332462\pi$
$98$		−	14.8439i		−	1.49946i
$99$		0			0
$100$	−5.54880			−0.554880
$101$	11.6588			1.16010			0.580049	−	0.814581i	$-0.303033\pi$
$101$	11.6588			1.16010			0.580049	+	0.814581i	$0.303033\pi$
$102$		0			0
$103$			12.0535i			1.18766i	0.804590	+	0.593831i	$0.202385\pi$
$103$			12.0535i			1.18766i	−0.804590	+	0.593831i	$0.797615\pi$
$104$	−0.886207			−0.0868998
$105$		0			0
$106$			11.3869i			1.10600i
$107$		−	1.04151i		−	0.100686i	−0.998732	−	0.0503432i	$-0.983968\pi$
$107$		−	1.04151i		−	0.100686i	0.998732	−	0.0503432i	$-0.0160315\pi$
$108$		0			0
$109$		−	14.5246i		−	1.39120i	−0.718429	−	0.695600i	$-0.755139\pi$
$109$		−	14.5246i		−	1.39120i	0.718429	−	0.695600i	$-0.244861\pi$
$110$	−19.1529			−1.82616
$111$		0			0
$112$	−19.4119			−1.83425
$113$	−15.5636			−1.46410			−0.732048	−	0.681253i	$-0.761435\pi$
$113$	−15.5636			−1.46410			−0.732048	+	0.681253i	$0.761435\pi$
$114$		0			0
$115$	−15.6456			−1.45896
$116$		−	3.11857i		−	0.289552i
$117$		0			0
$118$		−	16.6642i		−	1.53406i
$119$	−18.7833			−1.72187
$120$		0			0
$121$	−0.906105			−0.0823731
$122$			15.4843i			1.40188i
$123$		0			0
$124$	1.45178			0.130374
$125$	−1.10688			−0.0990026
$126$		0			0
$127$	−6.75410			−0.599329			−0.299665	−	0.954045i	$-0.596875\pi$
$127$	−6.75410			−0.599329			−0.299665	+	0.954045i	$0.596875\pi$
$128$			10.5793i			0.935086i
$129$		0			0
$130$	−3.41844			−0.299817
$131$	−9.38999			−0.820407			−0.410204	−	0.911994i	$-0.634542\pi$
$131$	−9.38999			−0.820407			−0.410204	+	0.911994i	$0.634542\pi$
$132$		0			0
$133$	15.4169			1.33682
$134$	22.3635			1.93191
$135$		0			0
$136$			6.90908i			0.592448i
$137$			5.05799i			0.432134i	0.976379	+	0.216067i	$0.0693229\pi$
$137$			5.05799i			0.432134i	−0.976379	+	0.216067i	$0.930677\pi$
$138$		0			0
$139$		−	8.60527i		−	0.729890i	−0.931029	−	0.364945i	$-0.881088\pi$
$139$		−	8.60527i		−	0.729890i	0.931029	−	0.364945i	$-0.118912\pi$
$140$	−14.5170			−1.22691
$141$		0			0
$142$			13.1376i			1.10248i
$143$	−2.12502			−0.177703
$144$		0			0
$145$			8.10450i			0.673042i
$146$	5.79601			0.479681
$147$		0			0
$148$		−	4.50386i		−	0.370215i
$149$	−11.6058			−0.950782			−0.475391	−	0.879775i	$-0.657693\pi$
$149$	−11.6058			−0.950782			−0.475391	+	0.879775i	$0.657693\pi$
$150$		0			0
$151$		−	1.22746i		−	0.0998897i	−0.998752	−	0.0499448i	$-0.984095\pi$
$151$		−	1.22746i		−	0.0998897i	0.998752	−	0.0499448i	$-0.0159046\pi$
$152$		−	5.67080i		−	0.459963i
$153$		0			0
$154$	−24.1283			−1.94432
$155$	−3.77288			−0.303045
$156$		0			0
$157$	9.67170			0.771885			0.385943	−	0.922523i	$-0.373876\pi$
$157$	9.67170			0.771885			0.385943	+	0.922523i	$0.373876\pi$
$158$	11.1025			0.883265
$159$		0			0
$160$			18.6054i			1.47089i
$161$	−19.7099			−1.55335
$162$		0			0
$163$			2.35267i			0.184276i	0.995746	+	0.0921378i	$0.0293700\pi$
$163$			2.35267i			0.184276i	−0.995746	+	0.0921378i	$0.970630\pi$
$164$	−12.9823			−1.01375
$165$		0			0
$166$		−	31.1689i		−	2.41918i
$167$	−3.73749	−	12.3706i	−0.289216	−	0.957264i
$167$	−3.73749	−	12.3706i	−0.289216	−	0.957264i
$168$		0			0
$169$	12.6207			0.970825
$170$			26.6509i			2.04403i
$171$		0			0
$172$		−	4.72190i		−	0.360041i
$173$		−	0.195437i		−	0.0148588i	−0.999972	−	0.00742942i	$-0.997635\pi$
$173$		−	0.195437i		−	0.0148588i	0.999972	−	0.00742942i	$-0.00236488\pi$
$174$		0			0
$175$	18.1661			1.37323
$176$			17.1215i			1.29058i
$177$		0			0
$178$	23.2857			1.74533
$179$			7.73739i			0.578320i	0.957281	+	0.289160i	$0.0933759\pi$
$179$			7.73739i			0.578320i	−0.957281	+	0.289160i	$0.906624\pi$
$180$		0			0
$181$	0.165868			0.0123289			0.00616444	−	0.999981i	$-0.498038\pi$
$181$	0.165868			0.0123289			0.00616444	+	0.999981i	$0.498038\pi$
$182$	−4.30646			−0.319216
$183$		0			0
$184$			7.24987i			0.534468i
$185$			11.7046i			0.860537i
$186$		0			0
$187$			16.5671i			1.21151i
$188$			13.4565i			0.981416i
$189$		0			0
$190$		−	21.8744i		−	1.58694i
$191$		−	4.80851i		−	0.347931i	−0.984752	−	0.173966i	$-0.944342\pi$
$191$		−	4.80851i		−	0.347931i	0.984752	−	0.173966i	$-0.0556582\pi$
$192$		0			0
$193$		−	4.08762i		−	0.294233i	−0.989119	−	0.147117i	$-0.953001\pi$
$193$		−	4.08762i		−	0.294233i	0.989119	−	0.147117i	$-0.0469993\pi$
$194$		−	17.6876i		−	1.26990i
$195$		0			0
$196$	−9.92350			−0.708822
$197$	−4.80737			−0.342511			−0.171255	−	0.985227i	$-0.554782\pi$
$197$	−4.80737			−0.342511			−0.171255	+	0.985227i	$0.554782\pi$
$198$		0			0
$199$	−8.01787			−0.568372			−0.284186	−	0.958769i	$-0.591723\pi$
$199$	−8.01787			−0.568372			−0.284186	+	0.958769i	$0.591723\pi$
$200$		−	6.68204i		−	0.472492i
$201$		0			0
$202$		−	20.8395i		−	1.46626i
$203$			10.2098i			0.716589i
$204$		0			0
$205$	33.7382			2.35638
$206$	21.5448			1.50110
$207$		0			0
$208$			3.05586i			0.211886i
$209$		−	13.5979i		−	0.940586i
$210$		0			0
$211$	15.3615			1.05753			0.528765	−	0.848768i	$-0.322655\pi$
$211$	15.3615			1.05753			0.528765	+	0.848768i	$0.322655\pi$
$212$	7.61241			0.522822
$213$		0			0
$214$	−1.86164			−0.127259
$215$			12.2712i			0.836891i
$216$		0			0
$217$	−4.75296			−0.322652
$218$	−25.9618			−1.75835
$219$		0			0
$220$			12.8042i			0.863256i
$221$			2.95692i			0.198904i
$222$		0			0
$223$	0.170629			0.0114261			0.00571307	−	0.999984i	$-0.498181\pi$
$223$	0.170629			0.0114261			0.00571307	+	0.999984i	$0.498181\pi$
$224$			23.4386i			1.56606i
$225$		0			0
$226$			27.8189i			1.85049i
$227$	25.2549			1.67622			0.838112	−	0.545497i	$-0.183659\pi$
$227$	25.2549			1.67622			0.838112	+	0.545497i	$0.183659\pi$
$228$		0			0
$229$	−22.1890			−1.46629			−0.733145	−	0.680072i	$-0.761949\pi$
$229$	−22.1890			−1.46629			−0.733145	+	0.680072i	$0.761949\pi$
$230$			27.9655i			1.84399i
$231$		0			0
$232$	3.75548			0.246559
$233$			15.2500i			0.999060i	0.866296	+	0.499530i	$0.166494\pi$
$233$			15.2500i			0.999060i	−0.866296	+	0.499530i	$0.833506\pi$
$234$		0			0
$235$		−	34.9706i		−	2.28123i
$236$	−11.1404			−0.725176
$237$		0			0
$238$			33.5741i			2.17629i
$239$		−	17.1422i		−	1.10883i	−0.832239	−	0.554417i	$-0.812941\pi$
$239$		−	17.1422i		−	1.10883i	0.832239	−	0.554417i	$-0.187059\pi$
$240$		0			0
$241$		−	24.5043i		−	1.57846i	−0.614095	−	0.789232i	$-0.710479\pi$
$241$		−	24.5043i		−	1.57846i	0.614095	−	0.789232i	$-0.289521\pi$
$242$			1.61961i			0.104112i
$243$		0			0
$244$	10.3516			0.662693
$245$	25.7891			1.64761
$246$		0			0
$247$		−	2.42697i		−	0.154424i
$248$			1.74828i			0.111016i
$249$		0			0
$250$			1.97849i			0.125131i
$251$			15.4612i			0.975902i	0.872871	+	0.487951i	$0.162255\pi$
$251$			15.4612i			0.975902i	−0.872871	+	0.487951i	$0.837745\pi$
$252$		0			0
$253$			17.3843i			1.09294i
$254$			12.0726i			0.757499i
$255$		0			0
$256$	20.4800			1.28000
$257$	−21.3976			−1.33475			−0.667374	−	0.744723i	$-0.732582\pi$
$257$	−21.3976			−1.33475			−0.667374	+	0.744723i	$0.732582\pi$
$258$		0			0
$259$			14.7451i			0.916216i
$260$			2.28530i			0.141729i
$261$		0			0
$262$			16.7841i			1.03692i
$263$			32.3404i			1.99419i	0.0761460	+	0.997097i	$0.475738\pi$
$263$			32.3404i			1.99419i	−0.0761460	+	0.997097i	$0.524262\pi$
$264$		0			0
$265$	−19.7831			−1.21526
$266$		−	27.5568i		−	1.68962i
$267$		0			0
$268$		−	14.9505i		−	0.913246i
$269$	−1.83973			−0.112170			−0.0560852	−	0.998426i	$-0.517862\pi$
$269$	−1.83973			−0.112170			−0.0560852	+	0.998426i	$0.517862\pi$
$270$		0			0
$271$			5.54360i			0.336750i	0.985723	+	0.168375i	$0.0538519\pi$
$271$			5.54360i			0.336750i	−0.985723	+	0.168375i	$0.946148\pi$
$272$	23.8242			1.44456
$273$		0			0
$274$	9.04086			0.546179
$275$		−	16.0227i		−	0.966206i
$276$		0			0
$277$		−	10.9295i		−	0.656688i	−0.944558	−	0.328344i	$-0.893510\pi$
$277$		−	10.9295i		−	0.656688i	0.944558	−	0.328344i	$-0.106490\pi$
$278$	−15.3814			−0.922516
$279$		0			0
$280$		−	17.4818i		−	1.04474i
$281$		−	23.2713i		−	1.38825i	−0.719855	−	0.694124i	$-0.755792\pi$
$281$		−	23.2713i		−	1.38825i	0.719855	−	0.694124i	$-0.244208\pi$
$282$		0			0
$283$	17.3309			1.03022			0.515108	−	0.857125i	$-0.327752\pi$
$283$	17.3309			1.03022			0.515108	+	0.857125i	$0.327752\pi$
$284$	8.78275			0.521160
$285$		0			0
$286$			3.79834i			0.224601i
$287$	42.5024			2.50884
$288$		0			0
$289$	6.05284			0.356049
$290$	14.4863			0.850666
$291$		0			0
$292$		−	3.87476i		−	0.226753i
$293$			11.5717i			0.676028i	0.941141	+	0.338014i	$0.109755\pi$
$293$			11.5717i			0.676028i	−0.941141	+	0.338014i	$0.890245\pi$
$294$		0			0
$295$	28.9515			1.68562
$296$	5.42369			0.315245
$297$		0			0
$298$			20.7446i			1.20170i
$299$			3.10278i			0.179438i
$300$		0			0
$301$			15.4589i			0.891039i
$302$	−2.19402			−0.126252
$303$		0			0
$304$	−19.5543			−1.12152
$305$	−26.9016			−1.54038
$306$		0			0
$307$			27.5338i			1.57144i	0.618584	+	0.785719i	$0.287707\pi$
$307$			27.5338i			1.57144i	−0.618584	+	0.785719i	$0.712293\pi$
$308$			16.1303i			0.919110i
$309$		0			0
$310$			6.74379i			0.383022i
$311$		−	32.1551i		−	1.82335i	−0.410911	−	0.911675i	$-0.634789\pi$
$311$		−	32.1551i		−	1.82335i	0.410911	−	0.911675i	$-0.365211\pi$
$312$		0			0
$313$		−	14.3685i		−	0.812155i	−0.913839	−	0.406078i	$-0.866896\pi$
$313$		−	14.3685i		−	0.812155i	0.913839	−	0.406078i	$-0.133104\pi$
$314$		−	17.2876i		−	0.975595i
$315$		0			0
$316$		−	7.42225i		−	0.417534i
$317$		−	27.9642i		−	1.57063i	−0.619099	−	0.785313i	$-0.712502\pi$
$317$		−	27.9642i		−	1.57063i	0.619099	−	0.785313i	$-0.287498\pi$
$318$		0			0
$319$	9.00518			0.504193
$320$	2.43803			0.136290
$321$		0			0
$322$			35.2302i			1.96330i
$323$	−18.9212			−1.05280
$324$		0			0
$325$		−	2.85976i		−	0.158631i
$326$	4.20527			0.232908
$327$		0			0
$328$		−	15.6337i		−	0.863225i
$329$		−	44.0550i		−	2.42883i
$330$		0			0
$331$			27.1876i			1.49437i	0.664619	+	0.747183i	$0.268594\pi$
$331$			27.1876i			1.49437i	−0.664619	+	0.747183i	$0.731406\pi$
$332$	−20.8371			−1.14359
$333$		0			0
$334$	−22.1117	+	6.68054i	−1.20990	+	0.365543i
$335$			38.8532i			2.12277i
$336$		0			0
$337$	−2.29045			−0.124769			−0.0623844	−	0.998052i	$-0.519870\pi$
$337$	−2.29045			−0.124769			−0.0623844	+	0.998052i	$0.519870\pi$
$338$		−	22.5588i		−	1.22704i
$339$		0			0
$340$	17.8168			0.966249
$341$			4.19217i			0.227019i
$342$		0			0
$343$	5.10364			0.275570
$344$	5.68626			0.306583
$345$		0			0
$346$	−0.349333			−0.0187803
$347$	−6.97356			−0.374360			−0.187180	−	0.982326i	$-0.559935\pi$
$347$	−6.97356			−0.374360			−0.187180	+	0.982326i	$0.559935\pi$
$348$		0			0
$349$			9.00896i			0.482238i	0.970496	+	0.241119i	$0.0775145\pi$
$349$			9.00896i			0.482238i	−0.970496	+	0.241119i	$0.922486\pi$
$350$		−	32.4708i		−	1.73564i
$351$		0			0
$352$	20.6731			1.10188
$353$			34.4670i			1.83449i	0.398319	+	0.917247i	$0.369594\pi$
$353$			34.4670i			1.83449i	−0.398319	+	0.917247i	$0.630406\pi$
$354$		0			0
$355$	−22.8245			−1.21140
$356$		−	15.5670i		−	0.825049i
$357$		0			0
$358$	13.8301			0.730945
$359$			10.0307i			0.529400i	0.964331	+	0.264700i	$0.0852729\pi$
$359$			10.0307i			0.529400i	−0.964331	+	0.264700i	$0.914727\pi$
$360$		0			0
$361$	−3.46993			−0.182628
$362$		−	0.296479i		−	0.0155826i
$363$		0			0
$364$			2.87896i			0.150899i
$365$			10.0697i			0.527071i
$366$		0			0
$367$	14.3211			0.747556			0.373778	−	0.927518i	$-0.378062\pi$
$367$	14.3211			0.747556			0.373778	+	0.927518i	$0.378062\pi$
$368$	24.9994			1.30318
$369$		0			0
$370$	20.9212			1.08764
$371$	−24.9221			−1.29389
$372$		0			0
$373$		−	13.3073i		−	0.689025i	−0.938782	−	0.344513i	$-0.888044\pi$
$373$		−	13.3073i		−	0.689025i	0.938782	−	0.344513i	$-0.111956\pi$
$374$	29.6127			1.53124
$375$		0			0
$376$	−16.2048			−0.835696
$377$	1.60726			0.0827779
$378$		0			0
$379$		−	27.7264i		−	1.42421i	−0.702074	−	0.712104i	$-0.747742\pi$
$379$		−	27.7264i		−	1.42421i	0.702074	−	0.712104i	$-0.252258\pi$
$380$	−14.6236			−0.750173
$381$		0			0
$382$	−8.59492			−0.439754
$383$			6.41782i			0.327935i	0.986466	+	0.163968i	$0.0524293\pi$
$383$			6.41782i			0.327935i	−0.986466	+	0.163968i	$0.947571\pi$
$384$		0			0
$385$		−	41.9193i		−	2.13641i
$386$	−7.30638			−0.371885
$387$		0			0
$388$	−11.8246			−0.600301
$389$	−1.15006			−0.0583105			−0.0291552	−	0.999575i	$-0.509282\pi$
$389$	−1.15006			−0.0583105			−0.0291552	+	0.999575i	$0.509282\pi$
$390$		0			0
$391$	24.1899			1.22334
$392$		−	11.9502i		−	0.603576i
$393$		0			0
$394$			8.59289i			0.432903i
$395$			19.2889i			0.970528i
$396$		0			0
$397$	−8.47216			−0.425206			−0.212603	−	0.977139i	$-0.568194\pi$
$397$	−8.47216			−0.425206			−0.212603	+	0.977139i	$0.568194\pi$
$398$			14.3315i			0.718371i
$399$		0			0
$400$	−23.0413			−1.15207
$401$	12.1337			0.605926			0.302963	−	0.953002i	$-0.402024\pi$
$401$	12.1337			0.605926			0.302963	+	0.953002i	$0.402024\pi$
$402$		0			0
$403$			0.748224i			0.0372717i
$404$	−13.9317			−0.693126
$405$		0			0
$406$	18.2495			0.905705
$407$	13.0053			0.644651
$408$		0			0
$409$	18.8338			0.931273			0.465636	−	0.884976i	$-0.345825\pi$
$409$	18.8338			0.931273			0.465636	+	0.884976i	$0.345825\pi$
$410$		−	60.3051i		−	2.97825i
$411$		0			0
$412$		−	14.4032i		−	0.709595i
$413$	36.4722			1.79468
$414$		0			0
$415$	54.1513			2.65818
$416$	3.68976			0.180905
$417$		0			0
$418$	−24.3054			−1.18882
$419$		−	8.09014i		−	0.395229i	−0.980280	−	0.197615i	$-0.936681\pi$
$419$		−	8.09014i		−	0.395229i	0.980280	−	0.197615i	$-0.0633195\pi$
$420$		0			0
$421$	21.9928			1.07187			0.535933	−	0.844261i	$-0.319960\pi$
$421$	21.9928			1.07187			0.535933	+	0.844261i	$0.319960\pi$
$422$		−	27.4578i		−	1.33663i
$423$		0			0
$424$			9.16711i			0.445194i
$425$	−22.2953			−1.08148
$426$		0			0
$427$	−33.8899			−1.64005
$428$			1.24454i			0.0601573i
$429$		0			0
$430$	21.9341			1.05776
$431$			19.3765i			0.933331i	0.884434	+	0.466665i	$0.154545\pi$
$431$			19.3765i			0.933331i	−0.884434	+	0.466665i	$0.845455\pi$
$432$		0			0
$433$	15.9619			0.767082			0.383541	−	0.923524i	$-0.374705\pi$
$433$	15.9619			0.767082			0.383541	+	0.923524i	$0.374705\pi$
$434$			8.49564i			0.407804i
$435$		0			0
$436$			17.3560i			0.831203i
$437$	−19.8545			−0.949771
$438$		0			0
$439$			12.2274i			0.583584i	0.956482	+	0.291792i	$0.0942515\pi$
$439$			12.2274i			0.583584i	−0.956482	+	0.291792i	$0.905749\pi$
$440$	−15.4192			−0.735081
$441$		0			0
$442$	5.28532			0.251397
$443$	−26.4304			−1.25575			−0.627873	−	0.778315i	$-0.716074\pi$
$443$	−26.4304			−1.25575			−0.627873	+	0.778315i	$0.716074\pi$
$444$		0			0
$445$			40.4553i			1.91777i
$446$		−	0.304989i		−	0.0144416i
$447$		0			0
$448$	3.07136			0.145108
$449$		−	26.2065i		−	1.23676i	−0.785879	−	0.618381i	$-0.787789\pi$
$449$		−	26.2065i		−	1.23676i	0.785879	−	0.618381i	$-0.212211\pi$
$450$		0			0
$451$		−	37.4876i		−	1.76522i
$452$	18.5976			0.874756
$453$		0			0
$454$		−	45.1416i		−	2.11860i
$455$		−	7.48182i		−	0.350753i
$456$		0			0
$457$		−	33.8337i		−	1.58267i	−0.611382	−	0.791336i	$-0.709386\pi$
$457$		−	33.8337i		−	1.58267i	0.611382	−	0.791336i	$-0.290614\pi$
$458$			39.6615i			1.85326i
$459$		0			0
$460$	18.6956			0.871686
$461$			41.9537i			1.95398i	0.213285	+	0.976990i	$0.431584\pi$
$461$			41.9537i			1.95398i	−0.213285	+	0.976990i	$0.568416\pi$
$462$		0			0
$463$			31.0888i			1.44482i	0.691466	+	0.722409i	$0.256965\pi$
$463$			31.0888i			1.44482i	−0.691466	+	0.722409i	$0.743035\pi$
$464$		−	12.9498i		−	0.601181i
$465$		0			0
$466$	27.2585			1.26272
$467$			19.8657i			0.919274i	0.888107	+	0.459637i	$0.152020\pi$
$467$			19.8657i			0.919274i	−0.888107	+	0.459637i	$0.847980\pi$
$468$		0			0
$469$			48.9461i			2.26012i
$470$	−62.5079			−2.88328
$471$		0			0
$472$		−	13.4156i		−	0.617502i
$473$	13.6350			0.626936
$474$		0			0
$475$	18.2995			0.839636
$476$	22.4450			1.02877
$477$		0			0
$478$	−30.6406			−1.40147
$479$	18.9641			0.866492			0.433246	−	0.901276i	$-0.357368\pi$
$479$	18.9641			0.866492			0.433246	+	0.901276i	$0.357368\pi$
$480$		0			0
$481$	2.32121			0.105838
$482$	−43.8001			−1.99504
$483$		0			0
$484$	1.08274			0.0492156
$485$	30.7295			1.39536
$486$		0			0
$487$		−	6.09676i		−	0.276270i	−0.990413	−	0.138135i	$-0.955889\pi$
$487$		−	6.09676i		−	0.276270i	0.990413	−	0.138135i	$-0.0441109\pi$
$488$			12.4657i			0.564296i
$489$		0			0
$490$		−	46.0965i		−	2.08243i
$491$			21.5465i			0.972379i	0.873853	+	0.486189i	$0.161614\pi$
$491$			21.5465i			0.972379i	−0.873853	+	0.486189i	$0.838386\pi$
$492$		0			0
$493$		−	12.5305i		−	0.564347i
$494$	−4.33806			−0.195179
$495$		0			0
$496$	6.02852			0.270688
$497$	−28.7537			−1.28978
$498$		0			0
$499$		−	4.40102i		−	0.197017i	−0.995136	−	0.0985083i	$-0.968593\pi$
$499$		−	4.40102i		−	0.197017i	0.995136	−	0.0985083i	$-0.0314071\pi$
$500$	1.32266			0.0591513
$501$		0			0
$502$	27.6360			1.23345
$503$			9.92600i			0.442578i	0.975208	+	0.221289i	$0.0710265\pi$
$503$			9.92600i			0.442578i	−0.975208	+	0.221289i	$0.928974\pi$
$504$		0			0
$505$	36.2055			1.61112
$506$	31.0734			1.38138
$507$		0			0
$508$	8.07077			0.358083
$509$		−	30.5411i		−	1.35371i	−0.736116	−	0.676856i	$-0.763342\pi$
$509$		−	30.5411i		−	1.35371i	0.736116	−	0.676856i	$-0.236658\pi$
$510$		0			0
$511$			12.6855i			0.561174i
$512$		−	15.4483i		−	0.682723i
$513$		0			0
$514$			38.2470i			1.68700i
$515$			37.4309i			1.64940i
$516$		0			0
$517$	−38.8570			−1.70893
$518$	26.3560			1.15802
$519$		0			0
$520$	−2.75204			−0.120685
$521$	−28.6091			−1.25339			−0.626693	−	0.779266i	$-0.715592\pi$
$521$	−28.6091			−1.25339			−0.626693	+	0.779266i	$0.715592\pi$
$522$		0			0
$523$	22.7761			0.995929			0.497965	−	0.867197i	$-0.334081\pi$
$523$	22.7761			0.995929			0.497965	+	0.867197i	$0.334081\pi$
$524$	11.2205			0.490171
$525$		0			0
$526$	57.8065			2.52048
$527$	5.83333			0.254104
$528$		0			0
$529$	2.38315			0.103615
$530$			35.3610i			1.53599i
$531$		0			0
$532$	−18.4223			−0.798710
$533$		−	6.69084i		−	0.289813i
$534$		0			0
$535$		−	3.23431i		−	0.139831i
$536$	18.0038			0.777648
$537$		0			0
$538$			3.28841i			0.141774i
$539$		−	28.6551i		−	1.23426i
$540$		0			0
$541$			12.8773i			0.553639i	0.960922	+	0.276820i	$0.0892804\pi$
$541$			12.8773i			0.553639i	−0.960922	+	0.276820i	$0.910720\pi$
$542$	9.90886			0.425622
$543$		0			0
$544$		−	28.7662i		−	1.23334i
$545$		−	45.1047i		−	1.93207i
$546$		0			0
$547$			25.9971i			1.11156i	0.831331	+	0.555778i	$0.187579\pi$
$547$			25.9971i			1.11156i	−0.831331	+	0.555778i	$0.812421\pi$
$548$		−	6.04402i		−	0.258188i
$549$		0			0
$550$	−28.6397			−1.22120
$551$			10.2848i			0.438146i
$552$		0			0
$553$			24.2996i			1.03332i
$554$	−19.5358			−0.829995
$555$		0			0
$556$			10.2828i			0.436089i
$557$		−	15.6301i		−	0.662269i	−0.943584	−	0.331134i	$-0.892569\pi$
$557$		−	15.6301i		−	0.662269i	0.943584	−	0.331134i	$-0.107431\pi$
$558$		0			0
$559$	2.43359			0.102930
$560$	−60.2818			−2.54737
$561$		0			0
$562$	−41.5960			−1.75462
$563$			23.4044i			0.986380i	0.869922	+	0.493190i	$0.164169\pi$
$563$			23.4044i			0.986380i	−0.869922	+	0.493190i	$0.835831\pi$
$564$		0			0
$565$	−48.3312			−2.03331
$566$		−	30.9780i		−	1.30210i
$567$		0			0
$568$			10.5765i			0.443779i
$569$	−4.04680			−0.169650			−0.0848252	−	0.996396i	$-0.527033\pi$
$569$	−4.04680			−0.169650			−0.0848252	+	0.996396i	$0.527033\pi$
$570$		0			0
$571$			30.7330i			1.28614i	0.765809	+	0.643068i	$0.222339\pi$
$571$			30.7330i			1.28614i	−0.765809	+	0.643068i	$0.777661\pi$
$572$	2.53928			0.106172
$573$		0			0
$574$		−	75.9706i		−	3.17095i
$575$	−23.3951			−0.975641
$576$		0			0
$577$	6.47274			0.269464			0.134732	−	0.990882i	$-0.456983\pi$
$577$	6.47274			0.269464			0.134732	+	0.990882i	$0.456983\pi$
$578$		−	10.8191i		−	0.450015i
$579$		0			0
$580$		−	9.68443i		−	0.402124i
$581$	68.2183			2.83017
$582$		0			0
$583$			21.9816i			0.910385i
$584$	4.66611			0.193085
$585$		0			0
$586$	20.6838			0.854440
$587$	−16.1692			−0.667376			−0.333688	−	0.942684i	$-0.608293\pi$
$587$	−16.1692			−0.667376			−0.333688	+	0.942684i	$0.608293\pi$
$588$		0			0
$589$	−4.78785			−0.197280
$590$		−	51.7490i		−	2.13047i
$591$		0			0
$592$		−	18.7022i		−	0.768657i
$593$	−42.1087			−1.72920			−0.864598	−	0.502465i	$-0.832427\pi$
$593$	−42.1087			−1.72920			−0.864598	+	0.502465i	$0.832427\pi$
$594$		0			0
$595$	−58.3300			−2.39129
$596$	13.8683			0.568066
$597$		0			0
$598$	5.54603			0.226794
$599$		−	15.6648i		−	0.640044i	−0.947410	−	0.320022i	$-0.896310\pi$
$599$		−	15.6648i		−	0.640044i	0.947410	−	0.320022i	$-0.103690\pi$
$600$		0			0
$601$	−8.27893			−0.337705			−0.168852	−	0.985641i	$-0.554006\pi$
$601$	−8.27893			−0.337705			−0.168852	+	0.985641i	$0.554006\pi$
$602$	27.6319			1.12619
$603$		0			0
$604$			1.46675i			0.0596813i
$605$	−2.81382			−0.114398
$606$		0			0
$607$		−	47.9617i		−	1.94671i	−0.229313	−	0.973353i	$-0.573648\pi$
$607$		−	47.9617i		−	1.94671i	0.229313	−	0.973353i	$-0.426352\pi$
$608$			23.6106i			0.957537i
$609$		0			0
$610$			48.0850i			1.94691i
$611$	−6.93525			−0.280570
$612$		0			0
$613$	−45.0655			−1.82018			−0.910089	−	0.414414i	$-0.863987\pi$
$613$	−45.0655			−1.82018			−0.910089	+	0.414414i	$0.863987\pi$
$614$	49.2151			1.98616
$615$		0			0
$616$	−19.4246			−0.782641
$617$			24.2154i			0.974877i	0.873157	+	0.487438i	$0.162069\pi$
$617$			24.2154i			0.974877i	−0.873157	+	0.487438i	$0.837931\pi$
$618$		0			0
$619$			43.0119i			1.72879i	0.502810	+	0.864397i	$0.332300\pi$
$619$			43.0119i			1.72879i	−0.502810	+	0.864397i	$0.667700\pi$
$620$	4.50838			0.181061
$621$		0			0
$622$	−57.4754			−2.30455
$623$			50.9645i			2.04185i
$624$		0			0
$625$	−26.6551			−1.06621
$626$	−25.6828			−1.02649
$627$		0			0
$628$	−11.5571			−0.461180
$629$		−	18.0967i		−	0.721562i
$630$		0			0
$631$	23.4138			0.932090			0.466045	−	0.884761i	$-0.345678\pi$
$631$	23.4138			0.932090			0.466045	+	0.884761i	$0.345678\pi$
$632$	8.93811			0.355539
$633$		0			0
$634$	−49.9844			−1.98513
$635$	−20.9742			−0.832337
$636$		0			0
$637$		−	5.11440i		−	0.202640i
$638$		−	16.0962i		−	0.637256i
$639$		0			0
$640$			32.8530i			1.29863i
$641$	7.42763			0.293374			0.146687	−	0.989183i	$-0.453139\pi$
$641$	7.42763			0.293374			0.146687	+	0.989183i	$0.453139\pi$
$642$		0			0
$643$			4.76716i			0.187999i	0.995572	+	0.0939993i	$0.0299651\pi$
$643$			4.76716i			0.187999i	−0.995572	+	0.0939993i	$0.970035\pi$
$644$	23.5522			0.928086
$645$		0			0
$646$			33.8205i			1.33065i
$647$	10.9750			0.431472			0.215736	−	0.976452i	$-0.430785\pi$
$647$	10.9750			0.431472			0.215736	+	0.976452i	$0.430785\pi$
$648$		0			0
$649$		−	32.1689i		−	1.26274i
$650$	−5.11164			−0.200495
$651$		0			0
$652$		−	2.81131i		−	0.110100i
$653$			39.8868i			1.56089i	0.625224	+	0.780445i	$0.285007\pi$
$653$			39.8868i			1.56089i	−0.625224	+	0.780445i	$0.714993\pi$
$654$		0			0
$655$	−29.1598			−1.13937
$656$	−53.9088			−2.10479
$657$		0			0
$658$	−78.7457			−3.06983
$659$	−18.9865			−0.739610			−0.369805	−	0.929109i	$-0.620576\pi$
$659$	−18.9865			−0.739610			−0.369805	+	0.929109i	$0.620576\pi$
$660$		0			0
$661$		−	16.5068i		−	0.642041i	−0.947072	−	0.321021i	$-0.895974\pi$
$661$		−	16.5068i		−	0.642041i	0.947072	−	0.321021i	$-0.104026\pi$
$662$	48.5962			1.88875
$663$		0			0
$664$		−	25.0927i		−	0.973787i
$665$	47.8758			1.85654
$666$		0			0
$667$		−	13.1486i		−	0.509117i
$668$	4.46609	+	14.7821i	0.172798	+	0.571938i
$669$		0			0
$670$	69.4477			2.68300
$671$			29.8913i			1.15394i
$672$		0			0
$673$		−	2.28352i		−	0.0880230i	−0.999031	−	0.0440115i	$-0.985986\pi$
$673$		−	2.28352i		−	0.0880230i	0.999031	−	0.0440115i	$-0.0140138\pi$
$674$			4.09405i			0.157697i
$675$		0			0
$676$	−15.0811			−0.580041
$677$		−	16.1000i		−	0.618775i	−0.950936	−	0.309388i	$-0.899876\pi$
$677$		−	16.1000i		−	0.618775i	0.950936	−	0.309388i	$-0.100124\pi$
$678$		0			0
$679$	38.7122			1.48564
$680$			21.4555i			0.822781i
$681$		0			0
$682$	7.49325			0.286932
$683$	11.3426			0.434012			0.217006	−	0.976170i	$-0.430371\pi$
$683$	11.3426			0.434012			0.217006	+	0.976170i	$0.430371\pi$
$684$		0			0
$685$			15.7071i			0.600139i
$686$		−	9.12245i		−	0.348297i
$687$		0			0
$688$		−	19.6077i		−	0.747535i
$689$			3.92331i			0.149466i
$690$		0			0
$691$		−	43.0837i		−	1.63898i	−0.573092	−	0.819491i	$-0.694256\pi$
$691$		−	43.0837i		−	1.63898i	0.573092	−	0.819491i	$-0.305744\pi$
$692$			0.233537i			0.00887774i
$693$		0			0
$694$			12.4648i			0.473158i
$695$		−	26.7229i		−	1.01366i
$696$		0			0
$697$	−52.1634			−1.97583
$698$	16.1030			0.609507
$699$		0			0
$700$	−21.7075			−0.820466
$701$			19.3920i			0.732424i	0.930531	+	0.366212i	$0.119345\pi$
$701$			19.3920i			0.732424i	−0.930531	+	0.366212i	$0.880655\pi$
$702$		0			0
$703$			14.8533i			0.560204i
$704$		−	2.70897i		−	0.102098i
$705$		0			0
$706$	61.6077			2.31864
$707$	45.6106			1.71536
$708$		0			0
$709$		−	46.2857i		−	1.73830i	−0.494552	−	0.869148i	$-0.664668\pi$
$709$		−	46.2857i		−	1.73830i	0.494552	−	0.869148i	$-0.335332\pi$
$710$			40.7975i			1.53110i
$711$		0			0
$712$	18.7463			0.702546
$713$	6.12106			0.229236
$714$		0			0
$715$	−6.59905			−0.246790
$716$		−	9.24575i		−	0.345530i
$717$		0			0
$718$	17.9293			0.669115
$719$	−8.66131			−0.323012			−0.161506	−	0.986872i	$-0.551635\pi$
$719$	−8.66131			−0.323012			−0.161506	+	0.986872i	$0.551635\pi$
$720$		0			0
$721$			47.1544i			1.75612i
$722$			6.20229i			0.230826i
$723$		0			0
$724$	−0.198203			−0.00736616
$725$			12.1188i			0.450080i
$726$		0			0
$727$		−	5.08276i		−	0.188509i	−0.995548	−	0.0942546i	$-0.969953\pi$
$727$		−	5.08276i		−	0.188509i	0.995548	−	0.0942546i	$-0.0300468\pi$
$728$	−3.46694			−0.128493
$729$		0			0
$730$	17.9990			0.666172
$731$		−	18.9728i		−	0.701734i
$732$		0			0
$733$	−48.9453			−1.80784			−0.903919	−	0.427705i	$-0.859322\pi$
$733$	−48.9453			−1.80784			−0.903919	+	0.427705i	$0.859322\pi$
$734$		−	25.5981i		−	0.944845i
$735$		0			0
$736$		−	30.1852i		−	1.11264i
$737$	43.1710			1.59023
$738$		0			0
$739$			11.6984i			0.430333i	0.976577	+	0.215166i	$0.0690294\pi$
$739$			11.6984i			0.430333i	−0.976577	+	0.215166i	$0.930971\pi$
$740$		−	13.9863i		−	0.514147i
$741$		0			0
$742$			44.5468i			1.63537i
$743$			20.7942i			0.762866i	0.924397	+	0.381433i	$0.124569\pi$
$743$			20.7942i			0.762866i	−0.924397	+	0.381433i	$0.875431\pi$
$744$		0			0
$745$	−36.0407			−1.32043
$746$	−23.7860			−0.870867
$747$		0			0
$748$		−	19.7968i		−	0.723842i
$749$		−	4.07449i		−	0.148879i
$750$		0			0
$751$			24.0774i			0.878595i	0.898342	+	0.439298i	$0.144773\pi$
$751$			24.0774i			0.878595i	−0.898342	+	0.439298i	$0.855227\pi$
$752$			55.8780i			2.03766i
$753$		0			0
$754$		−	2.87288i		−	0.104624i
$755$		−	3.81178i		−	0.138725i
$756$		0			0
$757$	−21.7557			−0.790726			−0.395363	−	0.918525i	$-0.629381\pi$
$757$	−21.7557			−0.790726			−0.395363	+	0.918525i	$0.629381\pi$
$758$	−49.5593			−1.80007
$759$		0			0
$760$		−	17.6102i		−	0.638787i
$761$		−	17.8942i		−	0.648664i	−0.945943	−	0.324332i	$-0.894861\pi$
$761$		−	17.8942i		−	0.648664i	0.945943	−	0.324332i	$-0.105139\pi$
$762$		0			0
$763$		−	56.8216i		−	2.05708i
$764$			5.74590i			0.207879i
$765$		0			0
$766$	11.4715			0.414481
$767$		−	5.74155i		−	0.207315i
$768$		0			0
$769$		−	17.8587i		−	0.644003i	−0.946739	−	0.322001i	$-0.895644\pi$
$769$		−	17.8587i		−	0.644003i	0.946739	−	0.322001i	$-0.104356\pi$
$770$	−74.9283			−2.70023
$771$		0			0
$772$			4.88448i			0.175796i
$773$	−34.9327			−1.25644			−0.628221	−	0.778035i	$-0.716217\pi$
$773$	−34.9327			−1.25644			−0.628221	+	0.778035i	$0.716217\pi$
$774$		0			0
$775$	−5.64164			−0.202654
$776$		−	14.2395i		−	0.511169i
$777$		0			0
$778$			2.05567i			0.0736993i
$779$	42.8144			1.53399
$780$		0			0
$781$			25.3611i			0.907491i
$782$		−	43.2381i		−	1.54619i
$783$		0			0
$784$	−41.2073			−1.47169
$785$	30.0346			1.07198
$786$		0			0
$787$		−	27.6833i		−	0.986801i	−0.869802	−	0.493401i	$-0.835754\pi$
$787$		−	27.6833i		−	0.986801i	0.869802	−	0.493401i	$-0.164246\pi$
$788$	5.74454			0.204641
$789$		0			0
$790$	34.4777			1.22666
$791$	−60.8863			−2.16487
$792$		0			0
$793$			5.33503i			0.189453i
$794$			15.1435i			0.537422i
$795$		0			0
$796$	9.58090			0.339586
$797$	39.1003			1.38501			0.692503	−	0.721415i	$-0.256508\pi$
$797$	39.1003			1.38501			0.692503	+	0.721415i	$0.256508\pi$
$798$		0			0
$799$			54.0688i			1.91282i
$800$			27.8210i			0.983619i
$801$		0			0
$802$		−	21.6882i		−	0.765837i
$803$	11.1888			0.394843
$804$		0			0
$805$	−61.2071			−2.15727
$806$	1.33741			0.0471081
$807$		0			0
$808$		−	16.7770i		−	0.590211i
$809$			17.6286i			0.619790i	0.950771	+	0.309895i	$0.100294\pi$
$809$			17.6286i			0.619790i	−0.950771	+	0.309895i	$0.899706\pi$
$810$		0			0
$811$		−	38.4996i		−	1.35190i	−0.736945	−	0.675952i	$-0.763732\pi$
$811$		−	38.4996i		−	1.35190i	0.736945	−	0.675952i	$-0.236268\pi$
$812$		−	12.2002i		−	0.428142i
$813$		0			0
$814$		−	23.2463i		−	0.814781i
$815$			7.30601i			0.255919i
$816$		0			0
$817$			15.5724i			0.544810i
$818$		−	33.6644i		−	1.17705i
$819$		0			0
$820$	−40.3153			−1.40787
$821$	−10.3200			−0.360172			−0.180086	−	0.983651i	$-0.557638\pi$
$821$	−10.3200			−0.360172			−0.180086	+	0.983651i	$0.557638\pi$
$822$		0			0
$823$		−	36.4315i		−	1.26992i	−0.772544	−	0.634961i	$-0.781016\pi$
$823$		−	36.4315i		−	1.26992i	0.772544	−	0.634961i	$-0.218984\pi$
$824$	17.3448			0.604235
$825$		0			0
$826$		−	65.1920i		−	2.26832i
$827$	−43.1118			−1.49914			−0.749571	−	0.661923i	$-0.769740\pi$
$827$	−43.1118			−1.49914			−0.749571	+	0.661923i	$0.769740\pi$
$828$		0			0
$829$		−	29.1261i		−	1.01159i	−0.862653	−	0.505796i	$-0.831199\pi$
$829$		−	29.1261i		−	1.01159i	0.862653	−	0.505796i	$-0.168801\pi$
$830$		−	96.7923i		−	3.35971i
$831$		0			0
$832$		−	0.483501i		−	0.0167624i
$833$	−39.8731			−1.38152
$834$		0			0
$835$	−11.6064	−	38.4157i	−0.401657	−	1.32943i
$836$			16.2487i			0.561974i
$837$		0			0
$838$	−14.4606			−0.499535
$839$			22.5459i			0.778369i	0.921160	+	0.389185i	$0.127243\pi$
$839$			22.5459i			0.778369i	−0.921160	+	0.389185i	$0.872757\pi$
$840$		0			0
$841$	22.1889			0.765136
$842$		−	39.3109i		−	1.35474i
$843$		0			0
$844$	−18.3562			−0.631845
$845$	39.1925			1.34826
$846$		0			0
$847$	−3.54478			−0.121800
$848$	31.6105			1.08551
$849$		0			0
$850$			39.8515i			1.36690i
$851$		−	18.9893i		−	0.650946i
$852$		0			0
$853$	−51.7211			−1.77090			−0.885449	−	0.464737i	$-0.846149\pi$
$853$	−51.7211			−1.77090			−0.885449	+	0.464737i	$0.846149\pi$
$854$			60.5762i			2.07287i
$855$		0			0
$856$	−1.49872			−0.0512252
$857$		−	5.72885i		−	0.195694i	−0.995201	−	0.0978469i	$-0.968804\pi$
$857$		−	5.72885i		−	0.195694i	0.995201	−	0.0978469i	$-0.0311956\pi$
$858$		0			0
$859$	−41.1489			−1.40398			−0.701990	−	0.712186i	$-0.747705\pi$
$859$	−41.1489			−1.40398			−0.701990	+	0.712186i	$0.747705\pi$
$860$		−	14.6634i		−	0.500019i
$861$		0			0
$862$	34.6343			1.17965
$863$		−	40.6796i		−	1.38475i	−0.721539	−	0.692374i	$-0.756565\pi$
$863$		−	40.6796i		−	1.38475i	0.721539	−	0.692374i	$-0.243435\pi$
$864$		0			0
$865$		−	0.606913i		−	0.0206357i
$866$		−	28.5310i		−	0.969523i
$867$		0			0
$868$	5.67953			0.192776
$869$	21.4325			0.727047
$870$		0			0
$871$	7.70522			0.261082
$872$	−20.9007			−0.707787
$873$		0			0
$874$			35.4888i			1.20043i
$875$	−4.33024			−0.146389
$876$		0			0
$877$	−12.7367			−0.430089			−0.215044	−	0.976604i	$-0.568990\pi$
$877$	−12.7367			−0.430089			−0.215044	+	0.976604i	$0.568990\pi$
$878$	21.8558			0.737598
$879$		0			0
$880$			53.1692i			1.79233i
$881$	1.97800			0.0666405			0.0333202	−	0.999445i	$-0.489392\pi$
$881$	1.97800			0.0666405			0.0333202	+	0.999445i	$0.489392\pi$
$882$		0			0
$883$	40.0795			1.34878			0.674392	−	0.738374i	$-0.264406\pi$
$883$	40.0795			1.34878			0.674392	+	0.738374i	$0.264406\pi$
$884$		−	3.53336i		−	0.118840i
$885$		0			0
$886$			47.2428i			1.58715i
$887$	−46.1504			−1.54958			−0.774790	−	0.632219i	$-0.782144\pi$
$887$	−46.1504			−1.54958			−0.774790	+	0.632219i	$0.782144\pi$
$888$		0			0
$889$	−26.4228			−0.886191
$890$	72.3115			2.42389
$891$		0			0
$892$	−0.203892			−0.00682680
$893$		−	44.3784i		−	1.48507i
$894$		0			0
$895$			24.0278i			0.803160i
$896$			41.3873i			1.38265i
$897$		0			0
$898$	−46.8426			−1.56316
$899$		−	3.17075i		−	0.105750i
$900$		0			0
$901$	30.5870			1.01900
$902$	−67.0069			−2.23109
$903$		0			0
$904$			22.3958i			0.744873i
$905$	0.515088			0.0171221
$906$		0			0
$907$	13.6786			0.454189			0.227095	−	0.973873i	$-0.427077\pi$
$907$	13.6786			0.454189			0.227095	+	0.973873i	$0.427077\pi$
$908$	−30.1782			−1.00150
$909$		0			0
$910$	−13.3733			−0.443321
$911$			32.2170i			1.06740i	0.845675	+	0.533698i	$0.179198\pi$
$911$			32.2170i			1.06740i	−0.845675	+	0.533698i	$0.820802\pi$
$912$		0			0
$913$		−	60.1693i		−	1.99131i
$914$	−60.4757			−2.00036
$915$		0			0
$916$	26.5146			0.876068
$917$	−36.7346			−1.21309
$918$		0			0
$919$	10.3456			0.341271			0.170636	−	0.985334i	$-0.445418\pi$
$919$	10.3456			0.341271			0.170636	+	0.985334i	$0.445418\pi$
$920$			22.5138i			0.742259i
$921$		0			0
$922$	74.9898			2.46966
$923$			4.52648i			0.148991i
$924$		0			0
$925$			17.5020i			0.575463i
$926$	55.5693			1.82612
$927$		0			0
$928$	−15.6361			−0.513280
$929$			42.9526i			1.40923i	0.709590	+	0.704615i	$0.248880\pi$
$929$			42.9526i			1.40923i	−0.709590	+	0.704615i	$0.751120\pi$
$930$		0			0
$931$	32.7268			1.07258
$932$		−	18.2229i		−	0.596911i
$933$		0			0
$934$	35.5087			1.16188
$935$			51.4477i			1.68252i
$936$		0			0
$937$			14.2255i			0.464728i	0.972629	+	0.232364i	$0.0746460\pi$
$937$			14.2255i			0.464728i	−0.972629	+	0.232364i	$0.925354\pi$
$938$	87.4883			2.85659
$939$		0			0
$940$			41.7879i			1.36297i
$941$	−46.3225			−1.51007			−0.755035	−	0.655684i	$-0.772380\pi$
$941$	−46.3225			−1.51007			−0.755035	+	0.655684i	$0.772380\pi$
$942$		0			0
$943$	−54.7364			−1.78246
$944$	−46.2603			−1.50564
$945$		0			0
$946$		−	24.3717i		−	0.792392i
$947$		−	16.6958i		−	0.542542i	−0.962503	−	0.271271i	$-0.912556\pi$
$947$		−	16.6958i		−	0.542542i	0.962503	−	0.271271i	$-0.0874439\pi$
$948$		0			0
$949$	1.99699			0.0648249
$950$		−	32.7092i		−	1.06123i
$951$		0			0
$952$			27.0290i			0.876016i
$953$	14.1283			0.457661			0.228830	−	0.973466i	$-0.426510\pi$
$953$	14.1283			0.457661			0.228830	+	0.973466i	$0.426510\pi$
$954$		0			0
$955$		−	14.9324i		−	0.483201i
$956$			20.4839i			0.662498i
$957$		0			0
$958$		−	33.8972i		−	1.09517i
$959$			19.7874i			0.638969i
$960$		0			0
$961$	−29.5239			−0.952385
$962$		−	4.14903i		−	0.133770i
$963$		0			0
$964$			29.2813i			0.943088i
$965$		−	12.6937i		−	0.408626i
$966$		0			0
$967$	2.32921			0.0749024			0.0374512	−	0.999298i	$-0.488076\pi$
$967$	2.32921			0.0749024			0.0374512	+	0.999298i	$0.488076\pi$
$968$			1.30388i			0.0419081i
$969$		0			0
$970$		−	54.9272i		−	1.76361i
$971$	−9.20860			−0.295518			−0.147759	−	0.989023i	$-0.547206\pi$
$971$	−9.20860			−0.295518			−0.147759	+	0.989023i	$0.547206\pi$
$972$		0			0
$973$		−	33.6647i		−	1.07924i
$974$	−10.8976			−0.349181
$975$		0			0
$976$	42.9849			1.37591
$977$	12.1347			0.388222			0.194111	−	0.980980i	$-0.437818\pi$
$977$	12.1347			0.388222			0.194111	+	0.980980i	$0.437818\pi$
$978$		0			0
$979$	44.9513			1.43665
$980$	−30.8165			−0.984398
$981$		0			0
$982$	38.5131			1.22900
$983$	−23.7996			−0.759089			−0.379545	−	0.925173i	$-0.623919\pi$
$983$	−23.7996			−0.759089			−0.379545	+	0.925173i	$0.623919\pi$
$984$		0			0
$985$	−14.9288			−0.475673
$986$	−22.3976			−0.713285
$987$		0			0
$988$			2.90009i			0.0922642i
$989$		−	19.9087i		−	0.633058i
$990$		0			0
$991$			28.8051i			0.915023i	0.889204	+	0.457512i	$0.151259\pi$
$991$			28.8051i			0.915023i	−0.889204	+	0.457512i	$0.848741\pi$
$992$		−	7.27905i		−	0.231110i
$993$		0			0
$994$			51.3955i			1.63017i
$995$	−24.8988			−0.789344
$996$		0			0
$997$	−11.1834			−0.354183			−0.177092	−	0.984194i	$-0.556669\pi$
$997$	−11.1834			−0.354183			−0.177092	+	0.984194i	$0.556669\pi$
$998$	−7.86656			−0.249012
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.c.a.1502.5	✓	56
3.2	odd	2	inner	1503.2.c.a.1502.52	yes	56
167.166	odd	2	inner	1503.2.c.a.1502.51	yes	56
501.500	even	2	inner	1503.2.c.a.1502.6	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.2.c.a.1502.5	✓	56	1.1	even	1	trivial
1503.2.c.a.1502.6	yes	56	501.500	even	2	inner
1503.2.c.a.1502.51	yes	56	167.166	odd	2	inner
1503.2.c.a.1502.52	yes	56	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.78744i q^{2} -1.19494 q^{4} +3.10541 q^{5} +3.91211 q^{7} -1.43899i q^{8} +O(q^{10})\)	\(q-1.78744i q^{2} -1.19494 q^{4} +3.10541 q^{5} +3.91211 q^{7} -1.43899i q^{8} -5.55073i q^{10} -3.45052i q^{11} -0.615854i q^{13} -6.99266i q^{14} -4.96200 q^{16} -4.80134 q^{17} +3.94082 q^{19} -3.71079 q^{20} -6.16760 q^{22} -5.03817 q^{23} +4.64356 q^{25} -1.10080 q^{26} -4.67475 q^{28} +2.60980i q^{29} -1.21494 q^{31} +5.99130i q^{32} +8.58211i q^{34} +12.1487 q^{35} +3.76909i q^{37} -7.04398i q^{38} -4.46865i q^{40} +10.8643 q^{41} +3.95156i q^{43} +4.12318i q^{44} +9.00543i q^{46} -11.2612i q^{47} +8.30458 q^{49} -8.30009i q^{50} +0.735911i q^{52} -6.37052 q^{53} -10.7153i q^{55} -5.62948i q^{56} +4.66487 q^{58} +9.32292 q^{59} -8.66282 q^{61} +2.17163i q^{62} +0.785091 q^{64} -1.91248i q^{65} +12.5114i q^{67} +5.73733 q^{68} -21.7151i q^{70} -7.34993 q^{71} +3.24263i q^{73} +6.73703 q^{74} -4.70906 q^{76} -13.4988i q^{77} +6.21138i q^{79} -15.4090 q^{80} -19.4194i q^{82} +17.4377 q^{83} -14.9101 q^{85} +7.06319 q^{86} -4.96527 q^{88} +13.0274i q^{89} -2.40929i q^{91} +6.02033 q^{92} -20.1287 q^{94} +12.2379 q^{95} +9.89549 q^{97} -14.8439i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 48 q^{4}+O(q^{10}) \) 56 * q - 48 * q^4	\( 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) \) 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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