LMFDB - Embedded newform 528.3.i.a.353.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,3,Mod(353,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("528.353");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	528.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$14.3869579582$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			353.1
Root			$0.500000 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	528.353
Dual form			528.3.i.a.353.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -6.63325i q^{5} +8.00000 q^{7} +9.00000 q^{9} -3.31662i q^{11} +4.00000 q^{13} +19.8997i q^{15} +13.2665i q^{17} +6.00000 q^{19} -24.0000 q^{21} -6.63325i q^{23} -19.0000 q^{25} -27.0000 q^{27} -39.7995i q^{29} +26.0000 q^{31} +9.94987i q^{33} -53.0660i q^{35} +30.0000 q^{37} -12.0000 q^{39} +13.2665i q^{41} -42.0000 q^{43} -59.6992i q^{45} -86.2322i q^{47} +15.0000 q^{49} -39.7995i q^{51} -59.6992i q^{53} -22.0000 q^{55} -18.0000 q^{57} -66.3325i q^{59} +12.0000 q^{61} +72.0000 q^{63} -26.5330i q^{65} -2.00000 q^{67} +19.8997i q^{69} +59.6992i q^{71} -74.0000 q^{73} +57.0000 q^{75} -26.5330i q^{77} +40.0000 q^{79} +81.0000 q^{81} +39.7995i q^{83} +88.0000 q^{85} +119.398i q^{87} -119.398i q^{89} +32.0000 q^{91} -78.0000 q^{93} -39.7995i q^{95} +62.0000 q^{97} -29.8496i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 16 q^{7} + 18 q^{9} + 8 q^{13} + 12 q^{19} - 48 q^{21} - 38 q^{25} - 54 q^{27} + 52 q^{31} + 60 q^{37} - 24 q^{39} - 84 q^{43} + 30 q^{49} - 44 q^{55} - 36 q^{57} + 24 q^{61} + 144 q^{63}+ \cdots + 124 q^{97}+O(q^{100}) $ 2 * q - 6 * q^3 + 16 * q^7 + 18 * q^9 + 8 * q^13 + 12 * q^19 - 48 * q^21 - 38 * q^25 - 54 * q^27 + 52 * q^31 + 60 * q^37 - 24 * q^39 - 84 * q^43 + 30 * q^49 - 44 * q^55 - 36 * q^57 + 24 * q^61 + 144 * q^63 - 4 * q^67 - 148 * q^73 + 114 * q^75 + 80 * q^79 + 162 * q^81 + 176 * q^85 + 64 * q^91 - 156 * q^93 + 124 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$.

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−1.00000
$3$	−3.00000	−1.00000
$4$	0	0
$5$	− 6.63325i	− 1.32665i	−0.748331	−	0.663325i	$-0.769145\pi$
$5$	− 6.63325i	− 1.32665i	0.748331	−	0.663325i	$-0.230855\pi$
$6$	0	0
$7$	8.00000	1.14286	0.571429	−	0.820652i	$-0.306389\pi$
$7$	8.00000	1.14286	0.571429	+	0.820652i	$0.306389\pi$
$8$	0	0
$9$	9.00000	1.00000
$10$	0	0
$11$	− 3.31662i	− 0.301511i
$11$	− 3.31662i	− 0.301511i
$12$	0	0
$13$	4.00000	0.307692	0.153846	−	0.988095i	$-0.450834\pi$
$13$	4.00000	0.307692	0.153846	+	0.988095i	$0.450834\pi$
$14$	0	0
$15$	19.8997i	1.32665i
$16$	0	0
$17$	13.2665i	0.780382i	0.920734	+	0.390191i	$0.127591\pi$
$17$	13.2665i	0.780382i	−0.920734	+	0.390191i	$0.872409\pi$
$18$	0	0
$19$	6.00000	0.315789	0.157895	−	0.987456i	$-0.449529\pi$
$19$	6.00000	0.315789	0.157895	+	0.987456i	$0.449529\pi$
$20$	0	0
$21$	−24.0000	−1.14286
$22$	0	0
$23$	− 6.63325i	− 0.288402i	−0.989548	−	0.144201i	$-0.953939\pi$
$23$	− 6.63325i	− 0.288402i	0.989548	−	0.144201i	$-0.0460612\pi$
$24$	0	0
$25$	−19.0000	−0.760000
$26$	0	0
$27$	−27.0000	−1.00000
$28$	0	0
$29$	− 39.7995i	− 1.37240i	−0.727415	−	0.686198i	$-0.759278\pi$
$29$	− 39.7995i	− 1.37240i	0.727415	−	0.686198i	$-0.240722\pi$
$30$	0	0
$31$	26.0000	0.838710	0.419355	−	0.907822i	$-0.362256\pi$
$31$	26.0000	0.838710	0.419355	+	0.907822i	$0.362256\pi$
$32$	0	0
$33$	9.94987i	0.301511i
$34$	0	0
$35$	− 53.0660i	− 1.51617i
$36$	0	0
$37$	30.0000	0.810811	0.405405	−	0.914137i	$-0.367130\pi$
$37$	30.0000	0.810811	0.405405	+	0.914137i	$0.367130\pi$
$38$	0	0
$39$	−12.0000	−0.307692
$40$	0	0
$41$	13.2665i	0.323573i	0.986826	+	0.161787i	$0.0517256\pi$
$41$	13.2665i	0.323573i	−0.986826	+	0.161787i	$0.948274\pi$
$42$	0	0
$43$	−42.0000	−0.976744	−0.488372	−	0.872635i	$-0.662409\pi$
$43$	−42.0000	−0.976744	−0.488372	+	0.872635i	$0.662409\pi$
$44$	0	0
$45$	− 59.6992i	− 1.32665i
$46$	0	0
$47$	− 86.2322i	− 1.83473i	−0.398049	−	0.917364i	$-0.630312\pi$
$47$	− 86.2322i	− 1.83473i	0.398049	−	0.917364i	$-0.369688\pi$
$48$	0	0
$49$	15.0000	0.306122
$50$	0	0
$51$	− 39.7995i	− 0.780382i
$52$	0	0
$53$	− 59.6992i	− 1.12640i	−0.826320	−	0.563200i	$-0.809570\pi$
$53$	− 59.6992i	− 1.12640i	0.826320	−	0.563200i	$-0.190430\pi$
$54$	0	0
$55$	−22.0000	−0.400000
$56$	0	0
$57$	−18.0000	−0.315789
$58$	0	0
$59$	− 66.3325i	− 1.12428i	−0.827042	−	0.562140i	$-0.809978\pi$
$59$	− 66.3325i	− 1.12428i	0.827042	−	0.562140i	$-0.190022\pi$
$60$	0	0
$61$	12.0000	0.196721	0.0983607	−	0.995151i	$-0.468640\pi$
$61$	12.0000	0.196721	0.0983607	+	0.995151i	$0.468640\pi$
$62$	0	0
$63$	72.0000	1.14286
$64$	0	0
$65$	− 26.5330i	− 0.408200i
$66$	0	0
$67$	−2.00000	−0.0298507	−0.0149254	−	0.999889i	$-0.504751\pi$
$67$	−2.00000	−0.0298507	−0.0149254	+	0.999889i	$0.504751\pi$
$68$	0	0
$69$	19.8997i	0.288402i
$70$	0	0
$71$	59.6992i	0.840834i	0.907331	+	0.420417i	$0.138116\pi$
$71$	59.6992i	0.840834i	−0.907331	+	0.420417i	$0.861884\pi$
$72$	0	0
$73$	−74.0000	−1.01370	−0.506849	−	0.862035i	$-0.669190\pi$
$73$	−74.0000	−1.01370	−0.506849	+	0.862035i	$0.669190\pi$
$74$	0	0
$75$	57.0000	0.760000
$76$	0	0
$77$	− 26.5330i	− 0.344584i
$78$	0	0
$79$	40.0000	0.506329	0.253165	−	0.967423i	$-0.418529\pi$
$79$	40.0000	0.506329	0.253165	+	0.967423i	$0.418529\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	39.7995i	0.479512i	0.970833	+	0.239756i	$0.0770674\pi$
$83$	39.7995i	0.479512i	−0.970833	+	0.239756i	$0.922933\pi$
$84$	0	0
$85$	88.0000	1.03529
$86$	0	0
$87$	119.398i	1.37240i
$88$	0	0
$89$	− 119.398i	− 1.34156i	−0.741658	−	0.670778i	$-0.765960\pi$
$89$	− 119.398i	− 1.34156i	0.741658	−	0.670778i	$-0.234040\pi$
$90$	0	0
$91$	32.0000	0.351648
$92$	0	0
$93$	−78.0000	−0.838710
$94$	0	0
$95$	− 39.7995i	− 0.418942i
$96$	0	0
$97$	62.0000	0.639175	0.319588	−	0.947557i	$-0.396456\pi$
$97$	62.0000	0.639175	0.319588	+	0.947557i	$0.396456\pi$
$98$	0	0
$99$	− 29.8496i	− 0.301511i
$100$	0	0
$101$	106.132i	1.05081i	0.850852	+	0.525406i	$0.176087\pi$
$101$	106.132i	1.05081i	−0.850852	+	0.525406i	$0.823913\pi$
$102$	0	0
$103$	−74.0000	−0.718447	−0.359223	−	0.933252i	$-0.616958\pi$
$103$	−74.0000	−0.718447	−0.359223	+	0.933252i	$0.616958\pi$
$104$	0	0
$105$	159.198i	1.51617i
$106$	0	0
$107$	39.7995i	0.371958i	0.982554	+	0.185979i	$0.0595456\pi$
$107$	39.7995i	0.371958i	−0.982554	+	0.185979i	$0.940454\pi$
$108$	0	0
$109$	−200.000	−1.83486	−0.917431	−	0.397894i	$-0.869741\pi$
$109$	−200.000	−1.83486	−0.917431	+	0.397894i	$0.869741\pi$
$110$	0	0
$111$	−90.0000	−0.810811
$112$	0	0
$113$	− 39.7995i	− 0.352208i	−0.984372	−	0.176104i	$-0.943651\pi$
$113$	− 39.7995i	− 0.352208i	0.984372	−	0.176104i	$-0.0563495\pi$
$114$	0	0
$115$	−44.0000	−0.382609
$116$	0	0
$117$	36.0000	0.307692
$118$	0	0
$119$	106.132i	0.891865i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	− 39.7995i	− 0.323573i
$124$	0	0
$125$	− 39.7995i	− 0.318396i
$126$	0	0
$127$	−188.000	−1.48031	−0.740157	−	0.672434i	$-0.765249\pi$
$127$	−188.000	−1.48031	−0.740157	+	0.672434i	$0.765249\pi$
$128$	0	0
$129$	126.000	0.976744
$130$	0	0
$131$	− 39.7995i	− 0.303813i	−0.988395	−	0.151906i	$-0.951459\pi$
$131$	− 39.7995i	− 0.303813i	0.988395	−	0.151906i	$-0.0485413\pi$
$132$	0	0
$133$	48.0000	0.360902
$134$	0	0
$135$	179.098i	1.32665i
$136$	0	0
$137$	106.132i	0.774686i	0.921936	+	0.387343i	$0.126607\pi$
$137$	106.132i	0.774686i	−0.921936	+	0.387343i	$0.873393\pi$
$138$	0	0
$139$	74.0000	0.532374	0.266187	−	0.963921i	$-0.414236\pi$
$139$	74.0000	0.532374	0.266187	+	0.963921i	$0.414236\pi$
$140$	0	0
$141$	258.697i	1.83473i
$142$	0	0
$143$	− 13.2665i	− 0.0927727i
$144$	0	0
$145$	−264.000	−1.82069
$146$	0	0
$147$	−45.0000	−0.306122
$148$	0	0
$149$	− 92.8655i	− 0.623258i	−0.950204	−	0.311629i	$-0.899125\pi$
$149$	− 92.8655i	− 0.623258i	0.950204	−	0.311629i	$-0.100875\pi$
$150$	0	0
$151$	160.000	1.05960	0.529801	−	0.848122i	$-0.322266\pi$
$151$	160.000	1.05960	0.529801	+	0.848122i	$0.322266\pi$
$152$	0	0
$153$	119.398i	0.780382i
$154$	0	0
$155$	− 172.464i	− 1.11267i
$156$	0	0
$157$	182.000	1.15924	0.579618	−	0.814888i	$-0.303202\pi$
$157$	182.000	1.15924	0.579618	+	0.814888i	$0.303202\pi$
$158$	0	0
$159$	179.098i	1.12640i
$160$	0	0
$161$	− 53.0660i	− 0.329602i
$162$	0	0
$163$	290.000	1.77914	0.889571	−	0.456798i	$-0.151004\pi$
$163$	290.000	1.77914	0.889571	+	0.456798i	$0.151004\pi$
$164$	0	0
$165$	66.0000	0.400000
$166$	0	0
$167$	238.797i	1.42992i	0.699164	+	0.714961i	$0.253556\pi$
$167$	238.797i	1.42992i	−0.699164	+	0.714961i	$0.746444\pi$
$168$	0	0
$169$	−153.000	−0.905325
$170$	0	0
$171$	54.0000	0.315789
$172$	0	0
$173$	198.997i	1.15027i	0.818057	+	0.575137i	$0.195051\pi$
$173$	198.997i	1.15027i	−0.818057	+	0.575137i	$0.804949\pi$
$174$	0	0
$175$	−152.000	−0.868571
$176$	0	0
$177$	198.997i	1.12428i
$178$	0	0
$179$	− 198.997i	− 1.11172i	−0.831277	−	0.555859i	$-0.812389\pi$
$179$	− 198.997i	− 1.11172i	0.831277	−	0.555859i	$-0.187611\pi$
$180$	0	0
$181$	10.0000	0.0552486	0.0276243	−	0.999618i	$-0.491206\pi$
$181$	10.0000	0.0552486	0.0276243	+	0.999618i	$0.491206\pi$
$182$	0	0
$183$	−36.0000	−0.196721
$184$	0	0
$185$	− 198.997i	− 1.07566i
$186$	0	0
$187$	44.0000	0.235294
$188$	0	0
$189$	−216.000	−1.14286
$190$	0	0
$191$	112.765i	0.590394i	0.955436	+	0.295197i	$0.0953853\pi$
$191$	112.765i	0.590394i	−0.955436	+	0.295197i	$0.904615\pi$
$192$	0	0
$193$	298.000	1.54404	0.772021	−	0.635597i	$-0.219246\pi$
$193$	298.000	1.54404	0.772021	+	0.635597i	$0.219246\pi$
$194$	0	0
$195$	79.5990i	0.408200i
$196$	0	0
$197$	132.665i	0.673426i	0.941607	+	0.336713i	$0.109315\pi$
$197$	132.665i	0.673426i	−0.941607	+	0.336713i	$0.890685\pi$
$198$	0	0
$199$	42.0000	0.211055	0.105528	−	0.994416i	$-0.466347\pi$
$199$	42.0000	0.211055	0.105528	+	0.994416i	$0.466347\pi$
$200$	0	0
$201$	6.00000	0.0298507
$202$	0	0
$203$	− 318.396i	− 1.56845i
$204$	0	0
$205$	88.0000	0.429268
$206$	0	0
$207$	− 59.6992i	− 0.288402i
$208$	0	0
$209$	− 19.8997i	− 0.0952141i
$210$	0	0
$211$	−246.000	−1.16588	−0.582938	−	0.812516i	$-0.698097\pi$
$211$	−246.000	−1.16588	−0.582938	+	0.812516i	$0.698097\pi$
$212$	0	0
$213$	− 179.098i	− 0.840834i
$214$	0	0
$215$	278.596i	1.29580i
$216$	0	0
$217$	208.000	0.958525
$218$	0	0
$219$	222.000	1.01370
$220$	0	0
$221$	53.0660i	0.240118i
$222$	0	0
$223$	302.000	1.35426	0.677130	−	0.735863i	$-0.263223\pi$
$223$	302.000	1.35426	0.677130	+	0.735863i	$0.263223\pi$
$224$	0	0
$225$	−171.000	−0.760000
$226$	0	0
$227$	198.997i	0.876641i	0.898819	+	0.438320i	$0.144426\pi$
$227$	198.997i	0.876641i	−0.898819	+	0.438320i	$0.855574\pi$
$228$	0	0
$229$	150.000	0.655022	0.327511	−	0.944847i	$-0.393790\pi$
$229$	150.000	0.655022	0.327511	+	0.944847i	$0.393790\pi$
$230$	0	0
$231$	79.5990i	0.344584i
$232$	0	0
$233$	− 437.794i	− 1.87895i	−0.342623	−	0.939473i	$-0.611315\pi$
$233$	− 437.794i	− 1.87895i	0.342623	−	0.939473i	$-0.388685\pi$
$234$	0	0
$235$	−572.000	−2.43404
$236$	0	0
$237$	−120.000	−0.506329
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	130.000	0.539419	0.269710	−	0.962942i	$-0.413072\pi$
$241$	130.000	0.539419	0.269710	+	0.962942i	$0.413072\pi$
$242$	0	0
$243$	−243.000	−1.00000
$244$	0	0
$245$	− 99.4987i	− 0.406117i
$246$	0	0
$247$	24.0000	0.0971660
$248$	0	0
$249$	− 119.398i	− 0.479512i
$250$	0	0
$251$	− 291.863i	− 1.16280i	−0.813618	−	0.581400i	$-0.802505\pi$
$251$	− 291.863i	− 1.16280i	0.813618	−	0.581400i	$-0.197495\pi$
$252$	0	0
$253$	−22.0000	−0.0869565
$254$	0	0
$255$	−264.000	−1.03529
$256$	0	0
$257$	490.860i	1.90996i	0.296665	+	0.954981i	$0.404125\pi$
$257$	490.860i	1.90996i	−0.296665	+	0.954981i	$0.595875\pi$
$258$	0	0
$259$	240.000	0.926641
$260$	0	0
$261$	− 358.195i	− 1.37240i
$262$	0	0
$263$	− 225.530i	− 0.857530i	−0.903416	−	0.428765i	$-0.858949\pi$
$263$	− 225.530i	− 0.857530i	0.903416	−	0.428765i	$-0.141051\pi$
$264$	0	0
$265$	−396.000	−1.49434
$266$	0	0
$267$	358.195i	1.34156i
$268$	0	0
$269$	72.9657i	0.271248i	0.990760	+	0.135624i	$0.0433039\pi$
$269$	72.9657i	0.271248i	−0.990760	+	0.135624i	$0.956696\pi$
$270$	0	0
$271$	448.000	1.65314	0.826568	−	0.562836i	$-0.190290\pi$
$271$	448.000	1.65314	0.826568	+	0.562836i	$0.190290\pi$
$272$	0	0
$273$	−96.0000	−0.351648
$274$	0	0
$275$	63.0159i	0.229149i
$276$	0	0
$277$	−260.000	−0.938628	−0.469314	−	0.883031i	$-0.655499\pi$
$277$	−260.000	−0.938628	−0.469314	+	0.883031i	$0.655499\pi$
$278$	0	0
$279$	234.000	0.838710
$280$	0	0
$281$	− 198.997i	− 0.708176i	−0.935212	−	0.354088i	$-0.884791\pi$
$281$	− 198.997i	− 0.708176i	0.935212	−	0.354088i	$-0.115209\pi$
$282$	0	0
$283$	50.0000	0.176678	0.0883392	−	0.996090i	$-0.471844\pi$
$283$	50.0000	0.176678	0.0883392	+	0.996090i	$0.471844\pi$
$284$	0	0
$285$	119.398i	0.418942i
$286$	0	0
$287$	106.132i	0.369798i
$288$	0	0
$289$	113.000	0.391003
$290$	0	0
$291$	−186.000	−0.639175
$292$	0	0
$293$	477.594i	1.63001i	0.579451	+	0.815007i	$0.303267\pi$
$293$	477.594i	1.63001i	−0.579451	+	0.815007i	$0.696733\pi$
$294$	0	0
$295$	−440.000	−1.49153
$296$	0	0
$297$	89.5489i	0.301511i
$298$	0	0
$299$	− 26.5330i	− 0.0887391i
$300$	0	0
$301$	−336.000	−1.11628
$302$	0	0
$303$	− 318.396i	− 1.05081i
$304$	0	0
$305$	− 79.5990i	− 0.260980i
$306$	0	0
$307$	−86.0000	−0.280130	−0.140065	−	0.990142i	$-0.544731\pi$
$307$	−86.0000	−0.280130	−0.140065	+	0.990142i	$0.544731\pi$
$308$	0	0
$309$	222.000	0.718447
$310$	0	0
$311$	− 19.8997i	− 0.0639863i	−0.999488	−	0.0319932i	$-0.989815\pi$
$311$	− 19.8997i	− 0.0639863i	0.999488	−	0.0319932i	$-0.0101855\pi$
$312$	0	0
$313$	98.0000	0.313099	0.156550	−	0.987670i	$-0.449963\pi$
$313$	98.0000	0.313099	0.156550	+	0.987670i	$0.449963\pi$
$314$	0	0
$315$	− 477.594i	− 1.51617i
$316$	0	0
$317$	311.763i	0.983479i	0.870742	+	0.491739i	$0.163639\pi$
$317$	311.763i	0.983479i	−0.870742	+	0.491739i	$0.836361\pi$
$318$	0	0
$319$	−132.000	−0.413793
$320$	0	0
$321$	− 119.398i	− 0.371958i
$322$	0	0
$323$	79.5990i	0.246437i
$324$	0	0
$325$	−76.0000	−0.233846
$326$	0	0
$327$	600.000	1.83486
$328$	0	0
$329$	− 689.858i	− 2.09683i
$330$	0	0
$331$	218.000	0.658610	0.329305	−	0.944224i	$-0.393186\pi$
$331$	218.000	0.658610	0.329305	+	0.944224i	$0.393186\pi$
$332$	0	0
$333$	270.000	0.810811
$334$	0	0
$335$	13.2665i	0.0396015i
$336$	0	0
$337$	278.000	0.824926	0.412463	−	0.910974i	$-0.364669\pi$
$337$	278.000	0.824926	0.412463	+	0.910974i	$0.364669\pi$
$338$	0	0
$339$	119.398i	0.352208i
$340$	0	0
$341$	− 86.2322i	− 0.252880i
$342$	0	0
$343$	−272.000	−0.793003
$344$	0	0
$345$	132.000	0.382609
$346$	0	0
$347$	− 331.662i	− 0.955800i	−0.878414	−	0.477900i	$-0.841398\pi$
$347$	− 331.662i	− 0.955800i	0.878414	−	0.477900i	$-0.158602\pi$
$348$	0	0
$349$	324.000	0.928367	0.464183	−	0.885739i	$-0.346348\pi$
$349$	324.000	0.928367	0.464183	+	0.885739i	$0.346348\pi$
$350$	0	0
$351$	−108.000	−0.307692
$352$	0	0
$353$	397.995i	1.12746i	0.825958	+	0.563732i	$0.190635\pi$
$353$	397.995i	1.12746i	−0.825958	+	0.563732i	$0.809365\pi$
$354$	0	0
$355$	396.000	1.11549
$356$	0	0
$357$	− 318.396i	− 0.891865i
$358$	0	0
$359$	305.129i	0.849943i	0.905207	+	0.424971i	$0.139716\pi$
$359$	305.129i	0.849943i	−0.905207	+	0.424971i	$0.860284\pi$
$360$	0	0
$361$	−325.000	−0.900277
$362$	0	0
$363$	33.0000	0.0909091
$364$	0	0
$365$	490.860i	1.34482i
$366$	0	0
$367$	278.000	0.757493	0.378747	−	0.925500i	$-0.376355\pi$
$367$	278.000	0.757493	0.378747	+	0.925500i	$0.376355\pi$
$368$	0	0
$369$	119.398i	0.323573i
$370$	0	0
$371$	− 477.594i	− 1.28732i
$372$	0	0
$373$	−68.0000	−0.182306	−0.0911528	−	0.995837i	$-0.529055\pi$
$373$	−68.0000	−0.182306	−0.0911528	+	0.995837i	$0.529055\pi$
$374$	0	0
$375$	119.398i	0.318396i
$376$	0	0
$377$	− 159.198i	− 0.422276i
$378$	0	0
$379$	−670.000	−1.76781	−0.883905	−	0.467666i	$-0.845095\pi$
$379$	−670.000	−1.76781	−0.883905	+	0.467666i	$0.845095\pi$
$380$	0	0
$381$	564.000	1.48031
$382$	0	0
$383$	33.1662i	0.0865959i	0.999062	+	0.0432980i	$0.0137865\pi$
$383$	33.1662i	0.0865959i	−0.999062	+	0.0432980i	$0.986214\pi$
$384$	0	0
$385$	−176.000	−0.457143
$386$	0	0
$387$	−378.000	−0.976744
$388$	0	0
$389$	− 417.895i	− 1.07428i	−0.843493	−	0.537140i	$-0.819505\pi$
$389$	− 417.895i	− 1.07428i	0.843493	−	0.537140i	$-0.180495\pi$
$390$	0	0
$391$	88.0000	0.225064
$392$	0	0
$393$	119.398i	0.303813i
$394$	0	0
$395$	− 265.330i	− 0.671721i
$396$	0	0
$397$	−86.0000	−0.216625	−0.108312	−	0.994117i	$-0.534545\pi$
$397$	−86.0000	−0.216625	−0.108312	+	0.994117i	$0.534545\pi$
$398$	0	0
$399$	−144.000	−0.360902
$400$	0	0
$401$	− 252.063i	− 0.628587i	−0.949326	−	0.314294i	$-0.898232\pi$
$401$	− 252.063i	− 0.628587i	0.949326	−	0.314294i	$-0.101768\pi$
$402$	0	0
$403$	104.000	0.258065
$404$	0	0
$405$	− 537.293i	− 1.32665i
$406$	0	0
$407$	− 99.4987i	− 0.244469i
$408$	0	0
$409$	510.000	1.24694	0.623472	−	0.781846i	$-0.285722\pi$
$409$	510.000	1.24694	0.623472	+	0.781846i	$0.285722\pi$
$410$	0	0
$411$	− 318.396i	− 0.774686i
$412$	0	0
$413$	− 530.660i	− 1.28489i
$414$	0	0
$415$	264.000	0.636145
$416$	0	0
$417$	−222.000	−0.532374
$418$	0	0
$419$	530.660i	1.26649i	0.773951	+	0.633246i	$0.218278\pi$
$419$	530.660i	1.26649i	−0.773951	+	0.633246i	$0.781722\pi$
$420$	0	0
$421$	−170.000	−0.403800	−0.201900	−	0.979406i	$-0.564712\pi$
$421$	−170.000	−0.403800	−0.201900	+	0.979406i	$0.564712\pi$
$422$	0	0
$423$	− 776.090i	− 1.83473i
$424$	0	0
$425$	− 252.063i	− 0.593091i
$426$	0	0
$427$	96.0000	0.224824
$428$	0	0
$429$	39.7995i	0.0927727i
$430$	0	0
$431$	− 278.596i	− 0.646396i	−0.946331	−	0.323198i	$-0.895242\pi$
$431$	− 278.596i	− 0.646396i	0.946331	−	0.323198i	$-0.104758\pi$
$432$	0	0
$433$	−542.000	−1.25173	−0.625866	−	0.779931i	$-0.715254\pi$
$433$	−542.000	−1.25173	−0.625866	+	0.779931i	$0.715254\pi$
$434$	0	0
$435$	792.000	1.82069
$436$	0	0
$437$	− 39.7995i	− 0.0910744i
$438$	0	0
$439$	−328.000	−0.747153	−0.373576	−	0.927599i	$-0.621869\pi$
$439$	−328.000	−0.747153	−0.373576	+	0.927599i	$0.621869\pi$
$440$	0	0
$441$	135.000	0.306122
$442$	0	0
$443$	− 132.665i	− 0.299470i	−0.988726	−	0.149735i	$-0.952158\pi$
$443$	− 132.665i	− 0.299470i	0.988726	−	0.149735i	$-0.0478420\pi$
$444$	0	0
$445$	−792.000	−1.77978
$446$	0	0
$447$	278.596i	0.623258i
$448$	0	0
$449$	− 451.061i	− 1.00459i	−0.864696	−	0.502295i	$-0.832489\pi$
$449$	− 451.061i	− 1.00459i	0.864696	−	0.502295i	$-0.167511\pi$
$450$	0	0
$451$	44.0000	0.0975610
$452$	0	0
$453$	−480.000	−1.05960
$454$	0	0
$455$	− 212.264i	− 0.466514i
$456$	0	0
$457$	342.000	0.748359	0.374179	−	0.927356i	$-0.377924\pi$
$457$	342.000	0.748359	0.374179	+	0.927356i	$0.377924\pi$
$458$	0	0
$459$	− 358.195i	− 0.780382i
$460$	0	0
$461$	− 79.5990i	− 0.172666i	−0.996266	−	0.0863330i	$-0.972485\pi$
$461$	− 79.5990i	− 0.172666i	0.996266	−	0.0863330i	$-0.0275149\pi$
$462$	0	0
$463$	86.0000	0.185745	0.0928726	−	0.995678i	$-0.470395\pi$
$463$	86.0000	0.185745	0.0928726	+	0.995678i	$0.470395\pi$
$464$	0	0
$465$	517.393i	1.11267i
$466$	0	0
$467$	596.992i	1.27836i	0.769059	+	0.639178i	$0.220725\pi$
$467$	596.992i	1.27836i	−0.769059	+	0.639178i	$0.779275\pi$
$468$	0	0
$469$	−16.0000	−0.0341151
$470$	0	0
$471$	−546.000	−1.15924
$472$	0	0
$473$	139.298i	0.294499i
$474$	0	0
$475$	−114.000	−0.240000
$476$	0	0
$477$	− 537.293i	− 1.12640i
$478$	0	0
$479$	− 225.530i	− 0.470836i	−0.971894	−	0.235418i	$-0.924354\pi$
$479$	− 225.530i	− 0.470836i	0.971894	−	0.235418i	$-0.0756459\pi$
$480$	0	0
$481$	120.000	0.249480
$482$	0	0
$483$	159.198i	0.329602i
$484$	0	0
$485$	− 411.261i	− 0.847962i
$486$	0	0
$487$	−446.000	−0.915811	−0.457906	−	0.889001i	$-0.651400\pi$
$487$	−446.000	−0.915811	−0.457906	+	0.889001i	$0.651400\pi$
$488$	0	0
$489$	−870.000	−1.77914
$490$	0	0
$491$	703.124i	1.43203i	0.698087	+	0.716013i	$0.254035\pi$
$491$	703.124i	1.43203i	−0.698087	+	0.716013i	$0.745965\pi$
$492$	0	0
$493$	528.000	1.07099
$494$	0	0
$495$	−198.000	−0.400000
$496$	0	0
$497$	477.594i	0.960954i
$498$	0	0
$499$	58.0000	0.116232	0.0581162	−	0.998310i	$-0.481491\pi$
$499$	58.0000	0.116232	0.0581162	+	0.998310i	$0.481491\pi$
$500$	0	0
$501$	− 716.391i	− 1.42992i
$502$	0	0
$503$	504.127i	1.00224i	0.865378	+	0.501120i	$0.167079\pi$
$503$	504.127i	1.00224i	−0.865378	+	0.501120i	$0.832921\pi$
$504$	0	0
$505$	704.000	1.39406
$506$	0	0
$507$	459.000	0.905325
$508$	0	0
$509$	736.291i	1.44654i	0.690563	+	0.723272i	$0.257363\pi$
$509$	736.291i	1.44654i	−0.690563	+	0.723272i	$0.742637\pi$
$510$	0	0
$511$	−592.000	−1.15851
$512$	0	0
$513$	−162.000	−0.315789
$514$	0	0
$515$	490.860i	0.953127i
$516$	0	0
$517$	−286.000	−0.553191
$518$	0	0
$519$	− 596.992i	− 1.15027i
$520$	0	0
$521$	278.596i	0.534734i	0.963595	+	0.267367i	$0.0861536\pi$
$521$	278.596i	0.534734i	−0.963595	+	0.267367i	$0.913846\pi$
$522$	0	0
$523$	142.000	0.271511	0.135755	−	0.990742i	$-0.456654\pi$
$523$	142.000	0.271511	0.135755	+	0.990742i	$0.456654\pi$
$524$	0	0
$525$	456.000	0.868571
$526$	0	0
$527$	344.929i	0.654514i
$528$	0	0
$529$	485.000	0.916824
$530$	0	0
$531$	− 596.992i	− 1.12428i
$532$	0	0
$533$	53.0660i	0.0995610i
$534$	0	0
$535$	264.000	0.493458
$536$	0	0
$537$	596.992i	1.11172i
$538$	0	0
$539$	− 49.7494i	− 0.0922994i
$540$	0	0
$541$	400.000	0.739372	0.369686	−	0.929157i	$-0.379465\pi$
$541$	400.000	0.739372	0.369686	+	0.929157i	$0.379465\pi$
$542$	0	0
$543$	−30.0000	−0.0552486
$544$	0	0
$545$	1326.65i	2.43422i
$546$	0	0
$547$	−170.000	−0.310786	−0.155393	−	0.987853i	$-0.549664\pi$
$547$	−170.000	−0.310786	−0.155393	+	0.987853i	$0.549664\pi$
$548$	0	0
$549$	108.000	0.196721
$550$	0	0
$551$	− 238.797i	− 0.433388i
$552$	0	0
$553$	320.000	0.578662
$554$	0	0
$555$	596.992i	1.07566i
$556$	0	0
$557$	849.056i	1.52434i	0.647378	+	0.762169i	$0.275865\pi$
$557$	849.056i	1.52434i	−0.647378	+	0.762169i	$0.724135\pi$
$558$	0	0
$559$	−168.000	−0.300537
$560$	0	0
$561$	−132.000	−0.235294
$562$	0	0
$563$	703.124i	1.24889i	0.781069	+	0.624444i	$0.214675\pi$
$563$	703.124i	1.24889i	−0.781069	+	0.624444i	$0.785325\pi$
$564$	0	0
$565$	−264.000	−0.467257
$566$	0	0
$567$	648.000	1.14286
$568$	0	0
$569$	− 119.398i	− 0.209839i	−0.994481	−	0.104920i	$-0.966541\pi$
$569$	− 119.398i	− 0.209839i	0.994481	−	0.104920i	$-0.0334585\pi$
$570$	0	0
$571$	706.000	1.23643	0.618214	−	0.786010i	$-0.287857\pi$
$571$	706.000	1.23643	0.618214	+	0.786010i	$0.287857\pi$
$572$	0	0
$573$	− 338.296i	− 0.590394i
$574$	0	0
$575$	126.032i	0.219186i
$576$	0	0
$577$	−738.000	−1.27903	−0.639515	−	0.768779i	$-0.720865\pi$
$577$	−738.000	−1.27903	−0.639515	+	0.768779i	$0.720865\pi$
$578$	0	0
$579$	−894.000	−1.54404
$580$	0	0
$581$	318.396i	0.548014i
$582$	0	0
$583$	−198.000	−0.339623
$584$	0	0
$585$	− 238.797i	− 0.408200i
$586$	0	0
$587$	941.921i	1.60464i	0.596897	+	0.802318i	$0.296400\pi$
$587$	941.921i	1.60464i	−0.596897	+	0.802318i	$0.703600\pi$
$588$	0	0
$589$	156.000	0.264856
$590$	0	0
$591$	− 397.995i	− 0.673426i
$592$	0	0
$593$	543.926i	0.917245i	0.888631	+	0.458623i	$0.151657\pi$
$593$	543.926i	0.917245i	−0.888631	+	0.458623i	$0.848343\pi$
$594$	0	0
$595$	704.000	1.18319
$596$	0	0
$597$	−126.000	−0.211055
$598$	0	0
$599$	46.4327i	0.0775171i	0.999249	+	0.0387586i	$0.0123403\pi$
$599$	46.4327i	0.0775171i	−0.999249	+	0.0387586i	$0.987660\pi$
$600$	0	0
$601$	542.000	0.901830	0.450915	−	0.892567i	$-0.351098\pi$
$601$	542.000	0.901830	0.450915	+	0.892567i	$0.351098\pi$
$602$	0	0
$603$	−18.0000	−0.0298507
$604$	0	0
$605$	72.9657i	0.120605i
$606$	0	0
$607$	700.000	1.15321	0.576606	−	0.817022i	$-0.304377\pi$
$607$	700.000	1.15321	0.576606	+	0.817022i	$0.304377\pi$
$608$	0	0
$609$	955.188i	1.56845i
$610$	0	0
$611$	− 344.929i	− 0.564532i
$612$	0	0
$613$	764.000	1.24633	0.623165	−	0.782091i	$-0.285847\pi$
$613$	764.000	1.24633	0.623165	+	0.782091i	$0.285847\pi$
$614$	0	0
$615$	−264.000	−0.429268
$616$	0	0
$617$	39.7995i	0.0645049i	0.999480	+	0.0322524i	$0.0102680\pi$
$617$	39.7995i	0.0645049i	−0.999480	+	0.0322524i	$0.989732\pi$
$618$	0	0
$619$	742.000	1.19871	0.599354	−	0.800484i	$-0.295424\pi$
$619$	742.000	1.19871	0.599354	+	0.800484i	$0.295424\pi$
$620$	0	0
$621$	179.098i	0.288402i
$622$	0	0
$623$	− 955.188i	− 1.53321i
$624$	0	0
$625$	−739.000	−1.18240
$626$	0	0
$627$	59.6992i	0.0952141i
$628$	0	0
$629$	397.995i	0.632742i
$630$	0	0
$631$	410.000	0.649762	0.324881	−	0.945755i	$-0.394676\pi$
$631$	410.000	0.649762	0.324881	+	0.945755i	$0.394676\pi$
$632$	0	0
$633$	738.000	1.16588
$634$	0	0
$635$	1247.05i	1.96386i
$636$	0	0
$637$	60.0000	0.0941915
$638$	0	0
$639$	537.293i	0.840834i
$640$	0	0
$641$	− 756.190i	− 1.17970i	−0.807511	−	0.589852i	$-0.799186\pi$
$641$	− 756.190i	− 1.17970i	0.807511	−	0.589852i	$-0.200814\pi$
$642$	0	0
$643$	−890.000	−1.38414	−0.692068	−	0.721832i	$-0.743300\pi$
$643$	−890.000	−1.38414	−0.692068	+	0.721832i	$0.743300\pi$
$644$	0	0
$645$	− 835.789i	− 1.29580i
$646$	0	0
$647$	− 484.227i	− 0.748419i	−0.927344	−	0.374210i	$-0.877914\pi$
$647$	− 484.227i	− 0.748419i	0.927344	−	0.374210i	$-0.122086\pi$
$648$	0	0
$649$	−220.000	−0.338983
$650$	0	0
$651$	−624.000	−0.958525
$652$	0	0
$653$	391.362i	0.599329i	0.954045	+	0.299664i	$0.0968747\pi$
$653$	391.362i	0.599329i	−0.954045	+	0.299664i	$0.903125\pi$
$654$	0	0
$655$	−264.000	−0.403053
$656$	0	0
$657$	−666.000	−1.01370
$658$	0	0
$659$	384.728i	0.583806i	0.956448	+	0.291903i	$0.0942885\pi$
$659$	384.728i	0.583806i	−0.956448	+	0.291903i	$0.905711\pi$
$660$	0	0
$661$	−746.000	−1.12859	−0.564297	−	0.825572i	$-0.690853\pi$
$661$	−746.000	−1.12859	−0.564297	+	0.825572i	$0.690853\pi$
$662$	0	0
$663$	− 159.198i	− 0.240118i
$664$	0	0
$665$	− 318.396i	− 0.478791i
$666$	0	0
$667$	−264.000	−0.395802
$668$	0	0
$669$	−906.000	−1.35426
$670$	0	0
$671$	− 39.7995i	− 0.0593137i
$672$	0	0
$673$	−634.000	−0.942051	−0.471025	−	0.882120i	$-0.656116\pi$
$673$	−634.000	−0.942051	−0.471025	+	0.882120i	$0.656116\pi$
$674$	0	0
$675$	513.000	0.760000
$676$	0	0
$677$	795.990i	1.17576i	0.808948	+	0.587880i	$0.200037\pi$
$677$	795.990i	1.17576i	−0.808948	+	0.587880i	$0.799963\pi$
$678$	0	0
$679$	496.000	0.730486
$680$	0	0
$681$	− 596.992i	− 0.876641i
$682$	0	0
$683$	451.061i	0.660411i	0.943909	+	0.330206i	$0.107118\pi$
$683$	451.061i	0.660411i	−0.943909	+	0.330206i	$0.892882\pi$
$684$	0	0
$685$	704.000	1.02774
$686$	0	0
$687$	−450.000	−0.655022
$688$	0	0
$689$	− 238.797i	− 0.346585i
$690$	0	0
$691$	−458.000	−0.662808	−0.331404	−	0.943489i	$-0.607522\pi$
$691$	−458.000	−0.662808	−0.331404	+	0.943489i	$0.607522\pi$
$692$	0	0
$693$	− 238.797i	− 0.344584i
$694$	0	0
$695$	− 490.860i	− 0.706274i
$696$	0	0
$697$	−176.000	−0.252511
$698$	0	0
$699$	1313.38i	1.87895i
$700$	0	0
$701$	504.127i	0.719154i	0.933115	+	0.359577i	$0.117079\pi$
$701$	504.127i	0.719154i	−0.933115	+	0.359577i	$0.882921\pi$
$702$	0	0
$703$	180.000	0.256046
$704$	0	0
$705$	1716.00	2.43404
$706$	0	0
$707$	849.056i	1.20093i
$708$	0	0
$709$	−562.000	−0.792666	−0.396333	−	0.918107i	$-0.629717\pi$
$709$	−562.000	−0.792666	−0.396333	+	0.918107i	$0.629717\pi$
$710$	0	0
$711$	360.000	0.506329
$712$	0	0
$713$	− 172.464i	− 0.241886i
$714$	0	0
$715$	−88.0000	−0.123077
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 338.296i	− 0.470509i	−0.971934	−	0.235254i	$-0.924408\pi$
$719$	− 338.296i	− 0.470509i	0.971934	−	0.235254i	$-0.0755923\pi$
$720$	0	0
$721$	−592.000	−0.821082
$722$	0	0
$723$	−390.000	−0.539419
$724$	0	0
$725$	756.190i	1.04302i
$726$	0	0
$727$	42.0000	0.0577717	0.0288858	−	0.999583i	$-0.490804\pi$
$727$	42.0000	0.0577717	0.0288858	+	0.999583i	$0.490804\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	0	0
$731$	− 557.193i	− 0.762234i
$732$	0	0
$733$	−624.000	−0.851296	−0.425648	−	0.904889i	$-0.639954\pi$
$733$	−624.000	−0.851296	−0.425648	+	0.904889i	$0.639954\pi$
$734$	0	0
$735$	298.496i	0.406117i
$736$	0	0
$737$	6.63325i	0.00900034i
$738$	0	0
$739$	−686.000	−0.928281	−0.464141	−	0.885761i	$-0.653637\pi$
$739$	−686.000	−0.928281	−0.464141	+	0.885761i	$0.653637\pi$
$740$	0	0
$741$	−72.0000	−0.0971660
$742$	0	0
$743$	862.322i	1.16060i	0.814404	+	0.580298i	$0.197064\pi$
$743$	862.322i	1.16060i	−0.814404	+	0.580298i	$0.802936\pi$
$744$	0	0
$745$	−616.000	−0.826846
$746$	0	0
$747$	358.195i	0.479512i
$748$	0	0
$749$	318.396i	0.425095i
$750$	0	0
$751$	−94.0000	−0.125166	−0.0625832	−	0.998040i	$-0.519934\pi$
$751$	−94.0000	−0.125166	−0.0625832	+	0.998040i	$0.519934\pi$
$752$	0	0
$753$	875.589i	1.16280i
$754$	0	0
$755$	− 1061.32i	− 1.40572i
$756$	0	0
$757$	1118.00	1.47688	0.738441	−	0.674318i	$-0.235562\pi$
$757$	1118.00	1.47688	0.738441	+	0.674318i	$0.235562\pi$
$758$	0	0
$759$	66.0000	0.0869565
$760$	0	0
$761$	− 1154.19i	− 1.51667i	−0.651865	−	0.758335i	$-0.726013\pi$
$761$	− 1154.19i	− 1.51667i	0.651865	−	0.758335i	$-0.273987\pi$
$762$	0	0
$763$	−1600.00	−2.09699
$764$	0	0
$765$	792.000	1.03529
$766$	0	0
$767$	− 265.330i	− 0.345932i
$768$	0	0
$769$	1274.00	1.65670	0.828349	−	0.560213i	$-0.189281\pi$
$769$	1274.00	1.65670	0.828349	+	0.560213i	$0.189281\pi$
$770$	0	0
$771$	− 1472.58i	− 1.90996i
$772$	0	0
$773$	− 935.288i	− 1.20995i	−0.796246	−	0.604973i	$-0.793184\pi$
$773$	− 935.288i	− 1.20995i	0.796246	−	0.604973i	$-0.206816\pi$
$774$	0	0
$775$	−494.000	−0.637419
$776$	0	0
$777$	−720.000	−0.926641
$778$	0	0
$779$	79.5990i	0.102181i
$780$	0	0
$781$	198.000	0.253521
$782$	0	0
$783$	1074.59i	1.37240i
$784$	0	0
$785$	− 1207.25i	− 1.53790i
$786$	0	0
$787$	−298.000	−0.378653	−0.189327	−	0.981914i	$-0.560630\pi$
$787$	−298.000	−0.378653	−0.189327	+	0.981914i	$0.560630\pi$
$788$	0	0
$789$	676.591i	0.857530i
$790$	0	0
$791$	− 318.396i	− 0.402523i
$792$	0	0
$793$	48.0000	0.0605296
$794$	0	0
$795$	1188.00	1.49434
$796$	0	0
$797$	524.027i	0.657499i	0.944417	+	0.328750i	$0.106627\pi$
$797$	524.027i	0.657499i	−0.944417	+	0.328750i	$0.893373\pi$
$798$	0	0
$799$	1144.00	1.43179
$800$	0	0
$801$	− 1074.59i	− 1.34156i
$802$	0	0
$803$	245.430i	0.305642i
$804$	0	0
$805$	−352.000	−0.437267
$806$	0	0
$807$	− 218.897i	− 0.271248i
$808$	0	0
$809$	− 915.388i	− 1.13151i	−0.824575	−	0.565753i	$-0.808586\pi$
$809$	− 915.388i	− 1.13151i	0.824575	−	0.565753i	$-0.191414\pi$
$810$	0	0
$811$	182.000	0.224414	0.112207	−	0.993685i	$-0.464208\pi$
$811$	182.000	0.224414	0.112207	+	0.993685i	$0.464208\pi$
$812$	0	0
$813$	−1344.00	−1.65314
$814$	0	0
$815$	− 1923.64i	− 2.36030i
$816$	0	0
$817$	−252.000	−0.308446
$818$	0	0
$819$	288.000	0.351648
$820$	0	0
$821$	− 822.523i	− 1.00185i	−0.865489	−	0.500927i	$-0.832992\pi$
$821$	− 822.523i	− 1.00185i	0.865489	−	0.500927i	$-0.167008\pi$
$822$	0	0
$823$	246.000	0.298906	0.149453	−	0.988769i	$-0.452249\pi$
$823$	246.000	0.298906	0.149453	+	0.988769i	$0.452249\pi$
$824$	0	0
$825$	− 189.048i	− 0.229149i
$826$	0	0
$827$	− 543.926i	− 0.657710i	−0.944380	−	0.328855i	$-0.893337\pi$
$827$	− 543.926i	− 0.657710i	0.944380	−	0.328855i	$-0.106663\pi$
$828$	0	0
$829$	250.000	0.301568	0.150784	−	0.988567i	$-0.451820\pi$
$829$	250.000	0.301568	0.150784	+	0.988567i	$0.451820\pi$
$830$	0	0
$831$	780.000	0.938628
$832$	0	0
$833$	198.997i	0.238893i
$834$	0	0
$835$	1584.00	1.89701
$836$	0	0
$837$	−702.000	−0.838710
$838$	0	0
$839$	− 470.961i	− 0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$839$	− 470.961i	− 0.561336i	0.959805	−	0.280668i	$-0.0905559\pi$
$840$	0	0
$841$	−743.000	−0.883472
$842$	0	0
$843$	596.992i	0.708176i
$844$	0	0
$845$	1014.89i	1.20105i
$846$	0	0
$847$	−88.0000	−0.103896
$848$	0	0
$849$	−150.000	−0.176678
$850$	0	0
$851$	− 198.997i	− 0.233840i
$852$	0	0
$853$	56.0000	0.0656506	0.0328253	−	0.999461i	$-0.489549\pi$
$853$	56.0000	0.0656506	0.0328253	+	0.999461i	$0.489549\pi$
$854$	0	0
$855$	− 358.195i	− 0.418942i
$856$	0	0
$857$	− 252.063i	− 0.294123i	−0.989127	−	0.147062i	$-0.953018\pi$
$857$	− 252.063i	− 0.294123i	0.989127	−	0.147062i	$-0.0469815\pi$
$858$	0	0
$859$	−1278.00	−1.48778	−0.743888	−	0.668304i	$-0.767021\pi$
$859$	−1278.00	−1.48778	−0.743888	+	0.668304i	$0.767021\pi$
$860$	0	0
$861$	− 318.396i	− 0.369798i
$862$	0	0
$863$	1134.29i	1.31435i	0.753737	+	0.657176i	$0.228249\pi$
$863$	1134.29i	1.31435i	−0.753737	+	0.657176i	$0.771751\pi$
$864$	0	0
$865$	1320.00	1.52601
$866$	0	0
$867$	−339.000	−0.391003
$868$	0	0
$869$	− 132.665i	− 0.152664i
$870$	0	0
$871$	−8.00000	−0.00918485
$872$	0	0
$873$	558.000	0.639175
$874$	0	0
$875$	− 318.396i	− 0.363881i
$876$	0	0
$877$	456.000	0.519954	0.259977	−	0.965615i	$-0.416285\pi$
$877$	456.000	0.519954	0.259977	+	0.965615i	$0.416285\pi$
$878$	0	0
$879$	− 1432.78i	− 1.63001i
$880$	0	0
$881$	− 610.259i	− 0.692689i	−0.938107	−	0.346344i	$-0.887423\pi$
$881$	− 610.259i	− 0.692689i	0.938107	−	0.346344i	$-0.112577\pi$
$882$	0	0
$883$	1094.00	1.23896	0.619479	−	0.785013i	$-0.287344\pi$
$883$	1094.00	1.23896	0.619479	+	0.785013i	$0.287344\pi$
$884$	0	0
$885$	1320.00	1.49153
$886$	0	0
$887$	− 145.931i	− 0.164523i	−0.996611	−	0.0822613i	$-0.973786\pi$
$887$	− 145.931i	− 0.164523i	0.996611	−	0.0822613i	$-0.0262142\pi$
$888$	0	0
$889$	−1504.00	−1.69179
$890$	0	0
$891$	− 268.647i	− 0.301511i
$892$	0	0
$893$	− 517.393i	− 0.579388i
$894$	0	0
$895$	−1320.00	−1.47486
$896$	0	0
$897$	79.5990i	0.0887391i
$898$	0	0
$899$	− 1034.79i	− 1.15104i
$900$	0	0
$901$	792.000	0.879023
$902$	0	0
$903$	1008.00	1.11628
$904$	0	0
$905$	− 66.3325i	− 0.0732956i
$906$	0	0
$907$	−186.000	−0.205072	−0.102536	−	0.994729i	$-0.532696\pi$
$907$	−186.000	−0.205072	−0.102536	+	0.994729i	$0.532696\pi$
$908$	0	0
$909$	955.188i	1.05081i
$910$	0	0
$911$	− 855.689i	− 0.939286i	−0.882857	−	0.469643i	$-0.844383\pi$
$911$	− 855.689i	− 0.939286i	0.882857	−	0.469643i	$-0.155617\pi$
$912$	0	0
$913$	132.000	0.144578
$914$	0	0
$915$	238.797i	0.260980i
$916$	0	0
$917$	− 318.396i	− 0.347215i
$918$	0	0
$919$	428.000	0.465724	0.232862	−	0.972510i	$-0.425191\pi$
$919$	428.000	0.465724	0.232862	+	0.972510i	$0.425191\pi$
$920$	0	0
$921$	258.000	0.280130
$922$	0	0
$923$	238.797i	0.258718i
$924$	0	0
$925$	−570.000	−0.616216
$926$	0	0
$927$	−666.000	−0.718447
$928$	0	0
$929$	636.792i	0.685460i	0.939434	+	0.342730i	$0.111352\pi$
$929$	636.792i	0.685460i	−0.939434	+	0.342730i	$0.888648\pi$
$930$	0	0
$931$	90.0000	0.0966702
$932$	0	0
$933$	59.6992i	0.0639863i
$934$	0	0
$935$	− 291.863i	− 0.312153i
$936$	0	0
$937$	290.000	0.309498	0.154749	−	0.987954i	$-0.450543\pi$
$937$	290.000	0.309498	0.154749	+	0.987954i	$0.450543\pi$
$938$	0	0
$939$	−294.000	−0.313099
$940$	0	0
$941$	875.589i	0.930488i	0.885183	+	0.465244i	$0.154033\pi$
$941$	875.589i	0.930488i	−0.885183	+	0.465244i	$0.845967\pi$
$942$	0	0
$943$	88.0000	0.0933192
$944$	0	0
$945$	1432.78i	1.51617i
$946$	0	0
$947$	79.5990i	0.0840538i	0.999116	+	0.0420269i	$0.0133815\pi$
$947$	79.5990i	0.0840538i	−0.999116	+	0.0420269i	$0.986618\pi$
$948$	0	0
$949$	−296.000	−0.311907
$950$	0	0
$951$	− 935.288i	− 0.983479i
$952$	0	0
$953$	− 782.723i	− 0.821326i	−0.911787	−	0.410663i	$-0.865297\pi$
$953$	− 782.723i	− 0.821326i	0.911787	−	0.410663i	$-0.134703\pi$
$954$	0	0
$955$	748.000	0.783246
$956$	0	0
$957$	396.000	0.413793
$958$	0	0
$959$	849.056i	0.885356i
$960$	0	0
$961$	−285.000	−0.296566
$962$	0	0
$963$	358.195i	0.371958i
$964$	0	0
$965$	− 1976.71i	− 2.04840i
$966$	0	0
$967$	−460.000	−0.475698	−0.237849	−	0.971302i	$-0.576442\pi$
$967$	−460.000	−0.475698	−0.237849	+	0.971302i	$0.576442\pi$
$968$	0	0
$969$	− 238.797i	− 0.246437i
$970$	0	0
$971$	1167.45i	1.20232i	0.799129	+	0.601160i	$0.205294\pi$
$971$	1167.45i	1.20232i	−0.799129	+	0.601160i	$0.794706\pi$
$972$	0	0
$973$	592.000	0.608428
$974$	0	0
$975$	228.000	0.233846
$976$	0	0
$977$	1313.38i	1.34430i	0.740414	+	0.672151i	$0.234630\pi$
$977$	1313.38i	1.34430i	−0.740414	+	0.672151i	$0.765370\pi$
$978$	0	0
$979$	−396.000	−0.404494
$980$	0	0
$981$	−1800.00	−1.83486
$982$	0	0
$983$	− 417.895i	− 0.425122i	−0.977148	−	0.212561i	$-0.931820\pi$
$983$	− 417.895i	− 0.425122i	0.977148	−	0.212561i	$-0.0681804\pi$
$984$	0	0
$985$	880.000	0.893401
$986$	0	0
$987$	2069.57i	2.09683i
$988$	0	0
$989$	278.596i	0.281695i
$990$	0	0
$991$	−838.000	−0.845610	−0.422805	−	0.906221i	$-0.638955\pi$
$991$	−838.000	−0.845610	−0.422805	+	0.906221i	$0.638955\pi$
$992$	0	0
$993$	−654.000	−0.658610
$994$	0	0
$995$	− 278.596i	− 0.279996i
$996$	0	0
$997$	−52.0000	−0.0521565	−0.0260782	−	0.999660i	$-0.508302\pi$
$997$	−52.0000	−0.0521565	−0.0260782	+	0.999660i	$0.508302\pi$
$998$	0	0
$999$	−810.000	−0.810811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.3.i.a.353.1		2
3.2	odd	2	inner	528.3.i.a.353.2		2
4.3	odd	2		33.3.b.a.23.2	yes	2
12.11	even	2		33.3.b.a.23.1	✓	2
44.3	odd	10		363.3.h.e.251.2		8
44.7	even	10		363.3.h.d.269.2		8
44.15	odd	10		363.3.h.e.269.1		8
44.19	even	10		363.3.h.d.251.1		8
44.27	odd	10		363.3.h.e.245.1		8
44.31	odd	10		363.3.h.e.323.2		8
44.35	even	10		363.3.h.d.323.1		8
44.39	even	10		363.3.h.d.245.2		8
44.43	even	2		363.3.b.d.122.1		2
132.35	odd	10		363.3.h.d.323.2		8
132.47	even	10		363.3.h.e.251.1		8
132.59	even	10		363.3.h.e.269.2		8
132.71	even	10		363.3.h.e.245.2		8
132.83	odd	10		363.3.h.d.245.1		8
132.95	odd	10		363.3.h.d.269.1		8
132.107	odd	10		363.3.h.d.251.2		8
132.119	even	10		363.3.h.e.323.1		8
132.131	odd	2		363.3.b.d.122.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.b.a.23.1	✓	2	12.11	even	2
33.3.b.a.23.2	yes	2	4.3	odd	2
363.3.b.d.122.1		2	44.43	even	2
363.3.b.d.122.2		2	132.131	odd	2
363.3.h.d.245.1		8	132.83	odd	10
363.3.h.d.245.2		8	44.39	even	10
363.3.h.d.251.1		8	44.19	even	10
363.3.h.d.251.2		8	132.107	odd	10
363.3.h.d.269.1		8	132.95	odd	10
363.3.h.d.269.2		8	44.7	even	10
363.3.h.d.323.1		8	44.35	even	10
363.3.h.d.323.2		8	132.35	odd	10
363.3.h.e.245.1		8	44.27	odd	10
363.3.h.e.245.2		8	132.71	even	10
363.3.h.e.251.1		8	132.47	even	10
363.3.h.e.251.2		8	44.3	odd	10
363.3.h.e.269.1		8	44.15	odd	10
363.3.h.e.269.2		8	132.59	even	10
363.3.h.e.323.1		8	132.119	even	10
363.3.h.e.323.2		8	44.31	odd	10
528.3.i.a.353.1		2	1.1	even	1	trivial
528.3.i.a.353.2		2	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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	528.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -6.63325i q^{5} +8.00000 q^{7} +9.00000 q^{9} -3.31662i q^{11} +4.00000 q^{13} +19.8997i q^{15} +13.2665i q^{17} +6.00000 q^{19} -24.0000 q^{21} -6.63325i q^{23} -19.0000 q^{25} -27.0000 q^{27} -39.7995i q^{29} +26.0000 q^{31} +9.94987i q^{33} -53.0660i q^{35} +30.0000 q^{37} -12.0000 q^{39} +13.2665i q^{41} -42.0000 q^{43} -59.6992i q^{45} -86.2322i q^{47} +15.0000 q^{49} -39.7995i q^{51} -59.6992i q^{53} -22.0000 q^{55} -18.0000 q^{57} -66.3325i q^{59} +12.0000 q^{61} +72.0000 q^{63} -26.5330i q^{65} -2.00000 q^{67} +19.8997i q^{69} +59.6992i q^{71} -74.0000 q^{73} +57.0000 q^{75} -26.5330i q^{77} +40.0000 q^{79} +81.0000 q^{81} +39.7995i q^{83} +88.0000 q^{85} +119.398i q^{87} -119.398i q^{89} +32.0000 q^{91} -78.0000 q^{93} -39.7995i q^{95} +62.0000 q^{97} -29.8496i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 16 q^{7} + 18 q^{9} + 8 q^{13} + 12 q^{19} - 48 q^{21} - 38 q^{25} - 54 q^{27} + 52 q^{31} + 60 q^{37} - 24 q^{39} - 84 q^{43} + 30 q^{49} - 44 q^{55} - 36 q^{57} + 24 q^{61} + 144 q^{63}+ \cdots + 124 q^{97}+O(q^{100}) \) 2 * q - 6 * q^3 + 16 * q^7 + 18 * q^9 + 8 * q^13 + 12 * q^19 - 48 * q^21 - 38 * q^25 - 54 * q^27 + 52 * q^31 + 60 * q^37 - 24 * q^39 - 84 * q^43 + 30 * q^49 - 44 * q^55 - 36 * q^57 + 24 * q^61 + 144 * q^63 - 4 * q^67 - 148 * q^73 + 114 * q^75 + 80 * q^79 + 162 * q^81 + 176 * q^85 + 64 * q^91 - 156 * q^93 + 124 * q^97

Introduction

L-functions

Modular forms

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Database

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