LMFDB - Embedded newform 1503.2.c.a.1502.6

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.78744i q^{2} -1.19494 q^{4} +3.10541 q^{5} +3.91211 q^{7} +1.43899i q^{8} +O(q^{10})$	\(q+1.78744i q^{2} -1.19494 q^{4} +3.10541 q^{5} +3.91211 q^{7} +1.43899i q^{8} +5.55073i q^{10} +3.45052i q^{11} +0.615854i q^{13} +6.99266i q^{14} -4.96200 q^{16} -4.80134 q^{17} +3.94082 q^{19} -3.71079 q^{20} -6.16760 q^{22} -5.03817 q^{23} +4.64356 q^{25} -1.10080 q^{26} -4.67475 q^{28} -2.60980i q^{29} -1.21494 q^{31} -5.99130i q^{32} -8.58211i q^{34} +12.1487 q^{35} -3.76909i q^{37} +7.04398i q^{38} +4.46865i q^{40} +10.8643 q^{41} -3.95156i q^{43} -4.12318i q^{44} -9.00543i q^{46} +11.2612i q^{47} +8.30458 q^{49} +8.30009i q^{50} -0.735911i q^{52} -6.37052 q^{53} +10.7153i q^{55} +5.62948i q^{56} +4.66487 q^{58} +9.32292 q^{59} -8.66282 q^{61} -2.17163i q^{62} +0.785091 q^{64} +1.91248i q^{65} -12.5114i q^{67} +5.73733 q^{68} +21.7151i q^{70} -7.34993 q^{71} -3.24263i q^{73} +6.73703 q^{74} -4.70906 q^{76} +13.4988i q^{77} -6.21138i q^{79} -15.4090 q^{80} +19.4194i q^{82} +17.4377 q^{83} -14.9101 q^{85} +7.06319 q^{86} -4.96527 q^{88} -13.0274i q^{89} +2.40929i q^{91} +6.02033 q^{92} -20.1287 q^{94} +12.2379 q^{95} +9.89549 q^{97} +14.8439i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.217748\pi$
$2$			1.78744i			1.26391i	−0.775005	+	0.631956i	$0.782252\pi$
$3$		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	$0.174137\pi$
$11$			3.45052i			1.04037i	−0.854053	+	0.520186i	$0.825863\pi$
$12$		0			0
$13$			0.615854i			0.170807i	0.996346	+	0.0854036i	$0.0272180\pi$
$13$			0.615854i			0.170807i	−0.996346	+	0.0854036i	$0.972782\pi$
$14$			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	$-0.922094\pi$
$29$		−	2.60980i		−	0.484628i	0.970198	−	0.242314i	$-0.0779064\pi$
$30$		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	$-0.899732\pi$
$37$		−	3.76909i		−	0.619635i	0.950796	−	0.309818i	$-0.100268\pi$
$38$			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	$-0.902578\pi$
$43$		−	3.95156i		−	0.602608i	0.953528	−	0.301304i	$-0.0974219\pi$
$44$		−	4.12318i		−	0.621593i
$45$		0			0
$46$		−	9.00543i		−	1.32778i
$47$			11.2612i			1.64261i	0.570486	+	0.821307i	$0.306755\pi$
$47$			11.2612i			1.64261i	−0.570486	+	0.821307i	$0.693245\pi$
$48$		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	$-0.723105\pi$
$67$		−	12.5114i		−	1.52852i	0.644910	−	0.764258i	$-0.276895\pi$
$68$	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	$-0.939229\pi$
$73$		−	3.24263i		−	0.379521i	0.981830	−	0.189760i	$-0.0607711\pi$
$74$	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	$-0.886380\pi$
$79$		−	6.21138i		−	0.698834i	0.936967	−	0.349417i	$-0.113620\pi$
$80$	−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	$-0.757413\pi$
$89$		−	13.0274i		−	1.38090i	0.723381	−	0.690450i	$-0.242587\pi$
$90$		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	$-0.797615\pi$
$103$		−	12.0535i		−	1.18766i	0.804590	−	0.593831i	$-0.202385\pi$
$104$	−0.886207			−0.0868998
$105$		0			0
$106$		−	11.3869i		−	1.10600i
$107$			1.04151i			0.100686i	0.998732	+	0.0503432i	$0.0160315\pi$
$107$			1.04151i			0.100686i	−0.998732	+	0.0503432i	$0.983968\pi$
$108$		0			0
$109$			14.5246i			1.39120i	0.718429	+	0.695600i	$0.244861\pi$
$109$			14.5246i			1.39120i	−0.718429	+	0.695600i	$0.755139\pi$
$110$	−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	$-0.930677\pi$
$137$		−	5.05799i		−	0.432134i	0.976379	−	0.216067i	$-0.0693229\pi$
$138$		0			0
$139$			8.60527i			0.729890i	0.931029	+	0.364945i	$0.118912\pi$
$139$			8.60527i			0.729890i	−0.931029	+	0.364945i	$0.881088\pi$
$140$	−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	$0.0159046\pi$
$151$			1.22746i			0.0998897i	−0.998752	+	0.0499448i	$0.984095\pi$
$152$			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	$-0.970630\pi$
$163$		−	2.35267i		−	0.184276i	0.995746	−	0.0921378i	$-0.0293700\pi$
$164$	−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	$0.00236488\pi$
$173$			0.195437i			0.0148588i	−0.999972	+	0.00742942i	$0.997635\pi$
$174$		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	$-0.906624\pi$
$179$		−	7.73739i		−	0.578320i	0.957281	−	0.289160i	$-0.0933759\pi$
$180$		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	$0.0556582\pi$
$191$			4.80851i			0.347931i	−0.984752	+	0.173966i	$0.944342\pi$
$192$		0			0
$193$			4.08762i			0.294233i	0.989119	+	0.147117i	$0.0469993\pi$
$193$			4.08762i			0.294233i	−0.989119	+	0.147117i	$0.953001\pi$
$194$			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	$-0.833506\pi$
$233$		−	15.2500i		−	0.999060i	0.866296	−	0.499530i	$-0.166494\pi$
$234$		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	$0.187059\pi$
$239$			17.1422i			1.10883i	−0.832239	+	0.554417i	$0.812941\pi$
$240$		0			0
$241$			24.5043i			1.57846i	0.614095	+	0.789232i	$0.289521\pi$
$241$			24.5043i			1.57846i	−0.614095	+	0.789232i	$0.710479\pi$
$242$		−	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	$-0.837745\pi$
$251$		−	15.4612i		−	0.975902i	0.872871	−	0.487951i	$-0.162255\pi$
$252$		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	$-0.524262\pi$
$263$		−	32.3404i		−	1.99419i	0.0761460	−	0.997097i	$-0.475738\pi$
$264$		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	$-0.946148\pi$
$271$		−	5.54360i		−	0.336750i	0.985723	−	0.168375i	$-0.0538519\pi$
$272$	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	$0.106490\pi$
$277$			10.9295i			0.656688i	−0.944558	+	0.328344i	$0.893510\pi$
$278$	−15.3814			−0.922516
$279$		0			0
$280$			17.4818i			1.04474i
$281$			23.2713i			1.38825i	0.719855	+	0.694124i	$0.244208\pi$
$281$			23.2713i			1.38825i	−0.719855	+	0.694124i	$0.755792\pi$
$282$		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	$-0.890245\pi$
$293$		−	11.5717i		−	0.676028i	0.941141	−	0.338014i	$-0.109755\pi$
$294$		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	$-0.712293\pi$
$307$		−	27.5338i		−	1.57144i	0.618584	−	0.785719i	$-0.287707\pi$
$308$		−	16.1303i		−	0.919110i
$309$		0			0
$310$		−	6.74379i		−	0.383022i
$311$			32.1551i			1.82335i	0.410911	+	0.911675i	$0.365211\pi$
$311$			32.1551i			1.82335i	−0.410911	+	0.911675i	$0.634789\pi$
$312$		0			0
$313$			14.3685i			0.812155i	0.913839	+	0.406078i	$0.133104\pi$
$313$			14.3685i			0.812155i	−0.913839	+	0.406078i	$0.866896\pi$
$314$			17.2876i			0.975595i
$315$		0			0
$316$			7.42225i			0.417534i
$317$			27.9642i			1.57063i	0.619099	+	0.785313i	$0.287498\pi$
$317$			27.9642i			1.57063i	−0.619099	+	0.785313i	$0.712502\pi$
$318$		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	$-0.731406\pi$
$331$		−	27.1876i		−	1.49437i	0.664619	−	0.747183i	$-0.268594\pi$
$332$	−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	$-0.922486\pi$
$349$		−	9.00896i		−	0.482238i	0.970496	−	0.241119i	$-0.0775145\pi$
$350$			32.4708i			1.73564i
$351$		0			0
$352$	20.6731			1.10188
$353$		−	34.4670i		−	1.83449i	−0.398319	−	0.917247i	$-0.630406\pi$
$353$		−	34.4670i		−	1.83449i	0.398319	−	0.917247i	$-0.369594\pi$
$354$		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	$-0.914727\pi$
$359$		−	10.0307i		−	0.529400i	0.964331	−	0.264700i	$-0.0852729\pi$
$360$		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	$0.111956\pi$
$373$			13.3073i			0.689025i	−0.938782	+	0.344513i	$0.888044\pi$
$374$	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	$0.252258\pi$
$379$			27.7264i			1.42421i	−0.702074	+	0.712104i	$0.747742\pi$
$380$	−14.6236			−0.750173
$381$		0			0
$382$	−8.59492			−0.439754
$383$		−	6.41782i		−	0.327935i	−0.986466	−	0.163968i	$-0.947571\pi$
$383$		−	6.41782i		−	0.327935i	0.986466	−	0.163968i	$-0.0524293\pi$
$384$		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	$0.0633195\pi$
$419$			8.09014i			0.395229i	−0.980280	+	0.197615i	$0.936681\pi$
$420$		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	$-0.845455\pi$
$431$		−	19.3765i		−	0.933331i	0.884434	−	0.466665i	$-0.154545\pi$
$432$		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	$-0.905749\pi$
$439$		−	12.2274i		−	0.583584i	0.956482	−	0.291792i	$-0.0942515\pi$
$440$	−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	$0.212211\pi$
$449$			26.2065i			1.23676i	−0.785879	+	0.618381i	$0.787789\pi$
$450$		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	$0.290614\pi$
$457$			33.8337i			1.58267i	−0.611382	+	0.791336i	$0.709386\pi$
$458$		−	39.6615i		−	1.85326i
$459$		0			0
$460$	18.6956			0.871686
$461$		−	41.9537i		−	1.95398i	−0.213285	−	0.976990i	$-0.568416\pi$
$461$		−	41.9537i		−	1.95398i	0.213285	−	0.976990i	$-0.431584\pi$
$462$		0			0
$463$		−	31.0888i		−	1.44482i	−0.691466	−	0.722409i	$-0.743035\pi$
$463$		−	31.0888i		−	1.44482i	0.691466	−	0.722409i	$-0.256965\pi$
$464$			12.9498i			0.601181i
$465$		0			0
$466$	27.2585			1.26272
$467$		−	19.8657i		−	0.919274i	−0.888107	−	0.459637i	$-0.847980\pi$
$467$		−	19.8657i		−	0.919274i	0.888107	−	0.459637i	$-0.152020\pi$
$468$		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	$0.0441109\pi$
$487$			6.09676i			0.276270i	−0.990413	+	0.138135i	$0.955889\pi$
$488$		−	12.4657i		−	0.564296i
$489$		0			0
$490$			46.0965i			2.08243i
$491$		−	21.5465i		−	0.972379i	−0.873853	−	0.486189i	$-0.838386\pi$
$491$		−	21.5465i		−	0.972379i	0.873853	−	0.486189i	$-0.161614\pi$
$492$		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	$0.0314071\pi$
$499$			4.40102i			0.197017i	−0.995136	+	0.0985083i	$0.968593\pi$
$500$	1.32266			0.0591513
$501$		0			0
$502$	27.6360			1.23345
$503$		−	9.92600i		−	0.442578i	−0.975208	−	0.221289i	$-0.928974\pi$
$503$		−	9.92600i		−	0.442578i	0.975208	−	0.221289i	$-0.0710265\pi$
$504$		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	$0.236658\pi$
$509$			30.5411i			1.35371i	−0.736116	+	0.676856i	$0.763342\pi$
$510$		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	$-0.910720\pi$
$541$		−	12.8773i		−	0.553639i	0.960922	−	0.276820i	$-0.0892804\pi$
$542$	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	$-0.812421\pi$
$547$		−	25.9971i		−	1.11156i	0.831331	−	0.555778i	$-0.187579\pi$
$548$			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	$0.107431\pi$
$557$			15.6301i			0.662269i	−0.943584	+	0.331134i	$0.892569\pi$
$558$		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	$-0.835831\pi$
$563$		−	23.4044i		−	0.986380i	0.869922	−	0.493190i	$-0.164169\pi$
$564$		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	$-0.777661\pi$
$571$		−	30.7330i		−	1.28614i	0.765809	−	0.643068i	$-0.222339\pi$
$572$	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	$0.103690\pi$
$599$			15.6648i			0.640044i	−0.947410	+	0.320022i	$0.896310\pi$
$600$		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	$0.426352\pi$
$607$			47.9617i			1.94671i	−0.229313	+	0.973353i	$0.573648\pi$
$608$		−	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	$-0.837931\pi$
$617$		−	24.2154i		−	0.974877i	0.873157	−	0.487438i	$-0.162069\pi$
$618$		0			0
$619$		−	43.0119i		−	1.72879i	−0.502810	−	0.864397i	$-0.667700\pi$
$619$		−	43.0119i		−	1.72879i	0.502810	−	0.864397i	$-0.332300\pi$
$620$	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	$-0.970035\pi$
$643$		−	4.76716i		−	0.187999i	0.995572	−	0.0939993i	$-0.0299651\pi$
$644$	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	$-0.714993\pi$
$653$		−	39.8868i		−	1.56089i	0.625224	−	0.780445i	$-0.285007\pi$
$654$		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	$0.104026\pi$
$661$			16.5068i			0.642041i	−0.947072	+	0.321021i	$0.895974\pi$
$662$	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	$0.0140138\pi$
$673$			2.28352i			0.0880230i	−0.999031	+	0.0440115i	$0.985986\pi$
$674$		−	4.09405i		−	0.157697i
$675$		0			0
$676$	−15.0811			−0.580041
$677$			16.1000i			0.618775i	0.950936	+	0.309388i	$0.100124\pi$
$677$			16.1000i			0.618775i	−0.950936	+	0.309388i	$0.899876\pi$
$678$		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	$0.305744\pi$
$691$			43.0837i			1.63898i	−0.573092	+	0.819491i	$0.694256\pi$
$692$		−	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	$-0.880655\pi$
$701$		−	19.3920i		−	0.732424i	0.930531	−	0.366212i	$-0.119345\pi$
$702$		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	$0.335332\pi$
$709$			46.2857i			1.73830i	−0.494552	+	0.869148i	$0.664668\pi$
$710$		−	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	$0.0300468\pi$
$727$			5.08276i			0.188509i	−0.995548	+	0.0942546i	$0.969953\pi$
$728$	−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	$-0.930971\pi$
$739$		−	11.6984i		−	0.430333i	0.976577	−	0.215166i	$-0.0690294\pi$
$740$			13.9863i			0.514147i
$741$		0			0
$742$		−	44.5468i		−	1.63537i
$743$		−	20.7942i		−	0.762866i	−0.924397	−	0.381433i	$-0.875431\pi$
$743$		−	20.7942i		−	0.762866i	0.924397	−	0.381433i	$-0.124569\pi$
$744$		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	$-0.855227\pi$
$751$		−	24.0774i		−	0.878595i	0.898342	−	0.439298i	$-0.144773\pi$
$752$		−	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	$0.105139\pi$
$761$			17.8942i			0.648664i	−0.945943	+	0.324332i	$0.894861\pi$
$762$		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	$0.104356\pi$
$769$			17.8587i			0.644003i	−0.946739	+	0.322001i	$0.895644\pi$
$770$	−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	$0.164246\pi$
$787$			27.6833i			0.986801i	−0.869802	+	0.493401i	$0.835754\pi$
$788$	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	$-0.899706\pi$
$809$		−	17.6286i		−	0.619790i	0.950771	−	0.309895i	$-0.100294\pi$
$810$		0			0
$811$			38.4996i			1.35190i	0.736945	+	0.675952i	$0.236268\pi$
$811$			38.4996i			1.35190i	−0.736945	+	0.675952i	$0.763732\pi$
$812$			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	$0.218984\pi$
$823$			36.4315i			1.26992i	−0.772544	+	0.634961i	$0.781016\pi$
$824$	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	$0.168801\pi$
$829$			29.1261i			1.01159i	−0.862653	+	0.505796i	$0.831199\pi$
$830$			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	$-0.872757\pi$
$839$		−	22.5459i		−	0.778369i	0.921160	−	0.389185i	$-0.127243\pi$
$840$		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	$0.0311956\pi$
$857$			5.72885i			0.195694i	−0.995201	+	0.0978469i	$0.968804\pi$
$858$		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	$0.243435\pi$
$863$			40.6796i			1.38475i	−0.721539	+	0.692374i	$0.756565\pi$
$864$		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	$-0.820802\pi$
$911$		−	32.2170i		−	1.06740i	0.845675	−	0.533698i	$-0.179198\pi$
$912$		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	$-0.751120\pi$
$929$		−	42.9526i		−	1.40923i	0.709590	−	0.704615i	$-0.248880\pi$
$930$		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	$-0.925354\pi$
$937$		−	14.2255i		−	0.464728i	0.972629	−	0.232364i	$-0.0746460\pi$
$938$	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	$0.0874439\pi$
$947$			16.6958i			0.542542i	−0.962503	+	0.271271i	$0.912556\pi$
$948$		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	$-0.848741\pi$
$991$		−	28.8051i		−	0.915023i	0.889204	−	0.457512i	$-0.151259\pi$
$992$			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.1502.6	yes	56
3.2	odd	2	inner	1503.2.c.a.1502.51	yes	56
167.166	odd	2	inner	1503.2.c.a.1502.52	yes	56
501.500	even	2	inner	1503.2.c.a.1502.5	✓	56

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