LMFDB - Embedded newform 1521.2.b.k.1351.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.b (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:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.6
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.k.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69202i q^{2} -5.24698 q^{4} -1.04892i q^{5} +0.554958i q^{7} -8.74094i q^{8} +O(q^{10})$	$q+2.69202i q^{2} -5.24698 q^{4} -1.04892i q^{5} +0.554958i q^{7} -8.74094i q^{8} +2.82371 q^{10} -2.91185i q^{11} -1.49396 q^{14} +13.0368 q^{16} -4.85086 q^{17} +0.753020i q^{19} +5.50365i q^{20} +7.83877 q^{22} +5.76271 q^{23} +3.89977 q^{25} -2.91185i q^{28} +1.91185 q^{29} +9.51573i q^{31} +17.6136i q^{32} -13.0586i q^{34} +0.582105 q^{35} +5.75302i q^{37} -2.02715 q^{38} -9.16852 q^{40} +4.91185i q^{41} +11.0978 q^{43} +15.2784i q^{44} +15.5133i q^{46} -0.753020i q^{47} +6.69202 q^{49} +10.4983i q^{50} +7.58211 q^{53} -3.05429 q^{55} +4.85086 q^{56} +5.14675i q^{58} -4.09783i q^{59} -3.42327 q^{61} -25.6165 q^{62} -21.3424 q^{64} +1.87263i q^{67} +25.4523 q^{68} +1.56704i q^{70} -10.5036i q^{71} +10.4765i q^{73} -15.4873 q^{74} -3.95108i q^{76} +1.61596 q^{77} +1.33513 q^{79} -13.6746i q^{80} -13.2228 q^{82} +2.64310i q^{83} +5.08815i q^{85} +29.8756i q^{86} -25.4523 q^{88} -9.92692i q^{89} -30.2368 q^{92} +2.02715 q^{94} +0.789856 q^{95} -17.0737i q^{97} +18.0151i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 22 q^{4}+O(q^{10}) $ 6 * q - 22 * q^4	$ 6 q - 22 q^{4} + 2 q^{10} + 10 q^{14} + 22 q^{16} - 2 q^{17} - 18 q^{22} - 22 q^{25} + 4 q^{29} - 8 q^{35} + 6 q^{40} + 30 q^{43} + 30 q^{49} + 34 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 4 q^{62} + 26 q^{68} - 30 q^{74} + 30 q^{77} + 6 q^{79} + 6 q^{82} - 26 q^{88} - 14 q^{92} - 42 q^{95}+O(q^{100}) $ 6 * q - 22 * q^4 + 2 * q^10 + 10 * q^14 + 22 * q^16 - 2 * q^17 - 18 * q^22 - 22 * q^25 + 4 * q^29 - 8 * q^35 + 6 * q^40 + 30 * q^43 + 30 * q^49 + 34 * q^53 + 6 * q^55 + 2 * q^56 - 26 * q^61 - 4 * q^62 + 26 * q^68 - 30 * q^74 + 30 * q^77 + 6 * q^79 + 6 * q^82 - 26 * q^88 - 14 * q^92 - 42 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.69202i	1.90355i	0.306802	+	0.951773i	$0.400741\pi$
$2$	2.69202i	1.90355i	−0.306802	+	0.951773i	$0.599259\pi$
$3$	0	0
$3$	0	0
$4$	−5.24698	−2.62349
$5$	− 1.04892i	− 0.469090i	−0.972105	−	0.234545i	$-0.924640\pi$
$5$	− 1.04892i	− 0.469090i	0.972105	−	0.234545i	$-0.0753600\pi$
$6$	0	0
$7$	0.554958i	0.209754i	0.994485	+	0.104877i	$0.0334450\pi$
$7$	0.554958i	0.209754i	−0.994485	+	0.104877i	$0.966555\pi$
$8$	− 8.74094i	− 3.09039i
$9$	0	0
$10$	2.82371	0.892935
$11$	− 2.91185i	− 0.877957i	−0.898498	−	0.438979i	$-0.855340\pi$
$11$	− 2.91185i	− 0.877957i	0.898498	−	0.438979i	$-0.144660\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.49396	−0.399277
$15$	0	0
$16$	13.0368	3.25921
$17$	−4.85086	−1.17651	−0.588253	−	0.808677i	$-0.700184\pi$
$17$	−4.85086	−1.17651	−0.588253	+	0.808677i	$0.700184\pi$
$18$	0	0
$19$	0.753020i	0.172755i	0.996262	+	0.0863774i	$0.0275291\pi$
$19$	0.753020i	0.172755i	−0.996262	+	0.0863774i	$0.972471\pi$
$20$	5.50365i	1.23065i
$21$	0	0
$22$	7.83877	1.67123
$23$	5.76271	1.20161	0.600804	−	0.799396i	$-0.294847\pi$
$23$	5.76271	1.20161	0.600804	+	0.799396i	$0.294847\pi$
$24$	0	0
$25$	3.89977	0.779954
$26$	0	0
$27$	0	0
$28$	− 2.91185i	− 0.550289i
$29$	1.91185	0.355022	0.177511	−	0.984119i	$-0.443195\pi$
$29$	1.91185	0.355022	0.177511	+	0.984119i	$0.443195\pi$
$30$	0	0
$31$	9.51573i	1.70908i	0.519389	+	0.854538i	$0.326159\pi$
$31$	9.51573i	1.70908i	−0.519389	+	0.854538i	$0.673841\pi$
$32$	17.6136i	3.11367i
$33$	0	0
$34$	− 13.0586i	− 2.23953i
$35$	0.582105	0.0983937
$36$	0	0
$37$	5.75302i	0.945791i	0.881119	+	0.472895i	$0.156791\pi$
$37$	5.75302i	0.945791i	−0.881119	+	0.472895i	$0.843209\pi$
$38$	−2.02715	−0.328847
$39$	0	0
$40$	−9.16852	−1.44967
$41$	4.91185i	0.767103i	0.923520	+	0.383551i	$0.125299\pi$
$41$	4.91185i	0.767103i	−0.923520	+	0.383551i	$0.874701\pi$
$42$	0	0
$43$	11.0978	1.69240	0.846202	−	0.532862i	$-0.178884\pi$
$43$	11.0978	1.69240	0.846202	+	0.532862i	$0.178884\pi$
$44$	15.2784i	2.30331i
$45$	0	0
$46$	15.5133i	2.28732i
$47$	− 0.753020i	− 0.109839i	−0.998491	−	0.0549197i	$-0.982510\pi$
$47$	− 0.753020i	− 0.109839i	0.998491	−	0.0549197i	$-0.0174903\pi$
$48$	0	0
$49$	6.69202	0.956003
$50$	10.4983i	1.48468i
$51$	0	0
$52$	0	0
$53$	7.58211	1.04148	0.520741	−	0.853715i	$-0.325656\pi$
$53$	7.58211	1.04148	0.520741	+	0.853715i	$0.325656\pi$
$54$	0	0
$55$	−3.05429	−0.411841
$56$	4.85086	0.648223
$57$	0	0
$58$	5.14675i	0.675802i
$59$	− 4.09783i	− 0.533493i	−0.963767	−	0.266746i	$-0.914051\pi$
$59$	− 4.09783i	− 0.533493i	0.963767	−	0.266746i	$-0.0859486\pi$
$60$	0	0
$61$	−3.42327	−0.438305	−0.219153	−	0.975691i	$-0.570329\pi$
$61$	−3.42327	−0.438305	−0.219153	+	0.975691i	$0.570329\pi$
$62$	−25.6165	−3.25330
$63$	0	0
$64$	−21.3424	−2.66780
$65$	0	0
$66$	0	0
$67$	1.87263i	0.228778i	0.993436	+	0.114389i	$0.0364910\pi$
$67$	1.87263i	0.228778i	−0.993436	+	0.114389i	$0.963509\pi$
$68$	25.4523	3.08655
$69$	0	0
$70$	1.56704i	0.187297i
$71$	− 10.5036i	− 1.24655i	−0.782001	−	0.623277i	$-0.785801\pi$
$71$	− 10.5036i	− 1.24655i	0.782001	−	0.623277i	$-0.214199\pi$
$72$	0	0
$73$	10.4765i	1.22618i	0.790012	+	0.613091i	$0.210074\pi$
$73$	10.4765i	1.22618i	−0.790012	+	0.613091i	$0.789926\pi$
$74$	−15.4873	−1.80036
$75$	0	0
$76$	− 3.95108i	− 0.453220i
$77$	1.61596	0.184155
$78$	0	0
$79$	1.33513	0.150213	0.0751067	−	0.997176i	$-0.476070\pi$
$79$	1.33513	0.150213	0.0751067	+	0.997176i	$0.476070\pi$
$80$	− 13.6746i	− 1.52886i
$81$	0	0
$82$	−13.2228	−1.46022
$83$	2.64310i	0.290118i	0.989423	+	0.145059i	$0.0463373\pi$
$83$	2.64310i	0.290118i	−0.989423	+	0.145059i	$0.953663\pi$
$84$	0	0
$85$	5.08815i	0.551887i
$86$	29.8756i	3.22157i
$87$	0	0
$88$	−25.4523	−2.71323
$89$	− 9.92692i	− 1.05225i	−0.850407	−	0.526126i	$-0.823644\pi$
$89$	− 9.92692i	− 1.05225i	0.850407	−	0.526126i	$-0.176356\pi$
$90$	0	0
$91$	0	0
$92$	−30.2368	−3.15241
$93$	0	0
$94$	2.02715	0.209084
$95$	0.789856	0.0810375
$96$	0	0
$97$	− 17.0737i	− 1.73357i	−0.498683	−	0.866784i	$-0.666183\pi$
$97$	− 17.0737i	− 1.73357i	0.498683	−	0.866784i	$-0.333817\pi$
$98$	18.0151i	1.81980i
$99$	0	0
$100$	−20.4620	−2.04620
$101$	7.32304	0.728670	0.364335	−	0.931268i	$-0.381296\pi$
$101$	7.32304	0.728670	0.364335	+	0.931268i	$0.381296\pi$
$102$	0	0
$103$	4.21983	0.415792	0.207896	−	0.978151i	$-0.433338\pi$
$103$	4.21983	0.415792	0.207896	+	0.978151i	$0.433338\pi$
$104$	0	0
$105$	0	0
$106$	20.4112i	1.98251i
$107$	−6.39373	−0.618105	−0.309053	−	0.951045i	$-0.600012\pi$
$107$	−6.39373	−0.618105	−0.309053	+	0.951045i	$0.600012\pi$
$108$	0	0
$109$	3.46011i	0.331418i	0.986175	+	0.165709i	$0.0529913\pi$
$109$	3.46011i	0.331418i	−0.986175	+	0.165709i	$0.947009\pi$
$110$	− 8.22223i	− 0.783958i
$111$	0	0
$112$	7.23490i	0.683634i
$113$	−9.35690	−0.880223	−0.440111	−	0.897943i	$-0.645061\pi$
$113$	−9.35690	−0.880223	−0.440111	+	0.897943i	$0.645061\pi$
$114$	0	0
$115$	− 6.04461i	− 0.563662i
$116$	−10.0315	−0.931398
$117$	0	0
$118$	11.0315	1.01553
$119$	− 2.69202i	− 0.246777i
$120$	0	0
$121$	2.52111	0.229191
$122$	− 9.21552i	− 0.834334i
$123$	0	0
$124$	− 49.9288i	− 4.48374i
$125$	− 9.33513i	− 0.834959i
$126$	0	0
$127$	4.48188	0.397702	0.198851	−	0.980030i	$-0.436279\pi$
$127$	4.48188	0.397702	0.198851	+	0.980030i	$0.436279\pi$
$128$	− 22.2271i	− 1.96462i
$129$	0	0
$130$	0	0
$131$	9.21744	0.805331	0.402666	−	0.915347i	$-0.368084\pi$
$131$	9.21744	0.805331	0.402666	+	0.915347i	$0.368084\pi$
$132$	0	0
$133$	−0.417895	−0.0362361
$134$	−5.04115	−0.435489
$135$	0	0
$136$	42.4010i	3.63586i
$137$	7.46980i	0.638188i	0.947723	+	0.319094i	$0.103379\pi$
$137$	7.46980i	0.638188i	−0.947723	+	0.319094i	$0.896621\pi$
$138$	0	0
$139$	−17.9976	−1.52654	−0.763269	−	0.646081i	$-0.776407\pi$
$139$	−17.9976	−1.52654	−0.763269	+	0.646081i	$0.776407\pi$
$140$	−3.05429	−0.258135
$141$	0	0
$142$	28.2760	2.37287
$143$	0	0
$144$	0	0
$145$	− 2.00538i	− 0.166537i
$146$	−28.2030	−2.33409
$147$	0	0
$148$	− 30.1860i	− 2.48127i
$149$	15.3351i	1.25630i	0.778091	+	0.628151i	$0.216188\pi$
$149$	15.3351i	1.25630i	−0.778091	+	0.628151i	$0.783812\pi$
$150$	0	0
$151$	2.53079i	0.205953i	0.994684	+	0.102977i	$0.0328367\pi$
$151$	2.53079i	0.205953i	−0.994684	+	0.102977i	$0.967163\pi$
$152$	6.58211	0.533879
$153$	0	0
$154$	4.35019i	0.350548i
$155$	9.98121	0.801710
$156$	0	0
$157$	17.2392	1.37584	0.687919	−	0.725787i	$-0.258524\pi$
$157$	17.2392	1.37584	0.687919	+	0.725787i	$0.258524\pi$
$158$	3.59419i	0.285938i
$159$	0	0
$160$	18.4752	1.46059
$161$	3.19806i	0.252043i
$162$	0	0
$163$	15.7071i	1.23027i	0.788420	+	0.615137i	$0.210899\pi$
$163$	15.7071i	1.23027i	−0.788420	+	0.615137i	$0.789101\pi$
$164$	− 25.7724i	− 2.01249i
$165$	0	0
$166$	−7.11529	−0.552254
$167$	− 5.39612i	− 0.417565i	−0.977962	−	0.208782i	$-0.933050\pi$
$167$	− 5.39612i	− 0.417565i	0.977962	−	0.208782i	$-0.0669500\pi$
$168$	0	0
$169$	0	0
$170$	−13.6974	−1.05054
$171$	0	0
$172$	−58.2301	−4.44000
$173$	23.9420	1.82028	0.910138	−	0.414306i	$-0.135976\pi$
$173$	23.9420	1.82028	0.910138	+	0.414306i	$0.135976\pi$
$174$	0	0
$175$	2.16421i	0.163599i
$176$	− 37.9614i	− 2.86145i
$177$	0	0
$178$	26.7235	2.00301
$179$	18.4088	1.37594	0.687969	−	0.725740i	$-0.258502\pi$
$179$	18.4088	1.37594	0.687969	+	0.725740i	$0.258502\pi$
$180$	0	0
$181$	3.63342	0.270070	0.135035	−	0.990841i	$-0.456885\pi$
$181$	3.63342	0.270070	0.135035	+	0.990841i	$0.456885\pi$
$182$	0	0
$183$	0	0
$184$	− 50.3715i	− 3.71344i
$185$	6.03444	0.443661
$186$	0	0
$187$	14.1250i	1.03292i
$188$	3.95108i	0.288162i
$189$	0	0
$190$	2.12631i	0.154259i
$191$	21.1782	1.53240	0.766201	−	0.642601i	$-0.222145\pi$
$191$	21.1782	1.53240	0.766201	+	0.642601i	$0.222145\pi$
$192$	0	0
$193$	− 17.6112i	− 1.26768i	−0.773464	−	0.633840i	$-0.781478\pi$
$193$	− 17.6112i	− 1.26768i	0.773464	−	0.633840i	$-0.218522\pi$
$194$	45.9627	3.29993
$195$	0	0
$196$	−35.1129	−2.50806
$197$	− 4.66248i	− 0.332188i	−0.986110	−	0.166094i	$-0.946884\pi$
$197$	− 4.66248i	− 0.332188i	0.986110	−	0.166094i	$-0.0531155\pi$
$198$	0	0
$199$	−15.0368	−1.06593	−0.532967	−	0.846136i	$-0.678923\pi$
$199$	−15.0368	−1.06593	−0.532967	+	0.846136i	$0.678923\pi$
$200$	− 34.0877i	− 2.41036i
$201$	0	0
$202$	19.7138i	1.38706i
$203$	1.06100i	0.0744675i
$204$	0	0
$205$	5.15213	0.359840
$206$	11.3599i	0.791480i
$207$	0	0
$208$	0	0
$209$	2.19269	0.151671
$210$	0	0
$211$	−0.460107	−0.0316751	−0.0158375	−	0.999875i	$-0.505041\pi$
$211$	−0.460107	−0.0316751	−0.0158375	+	0.999875i	$0.505041\pi$
$212$	−39.7832	−2.73232
$213$	0	0
$214$	− 17.2121i	− 1.17659i
$215$	− 11.6407i	− 0.793890i
$216$	0	0
$217$	−5.28083	−0.358486
$218$	−9.31468	−0.630870
$219$	0	0
$220$	16.0258	1.08046
$221$	0	0
$222$	0	0
$223$	16.3502i	1.09489i	0.836842	+	0.547445i	$0.184399\pi$
$223$	16.3502i	1.09489i	−0.836842	+	0.547445i	$0.815601\pi$
$224$	−9.77479	−0.653106
$225$	0	0
$226$	− 25.1890i	− 1.67555i
$227$	6.56033i	0.435425i	0.976013	+	0.217712i	$0.0698595\pi$
$227$	6.56033i	0.435425i	−0.976013	+	0.217712i	$0.930141\pi$
$228$	0	0
$229$	− 3.95539i	− 0.261380i	−0.991423	−	0.130690i	$-0.958281\pi$
$229$	− 3.95539i	− 0.261380i	0.991423	−	0.130690i	$-0.0417192\pi$
$230$	16.2722	1.07296
$231$	0	0
$232$	− 16.7114i	− 1.09716i
$233$	8.35690	0.547478	0.273739	−	0.961804i	$-0.411739\pi$
$233$	8.35690	0.547478	0.273739	+	0.961804i	$0.411739\pi$
$234$	0	0
$235$	−0.789856	−0.0515245
$236$	21.5013i	1.39961i
$237$	0	0
$238$	7.24698	0.469752
$239$	20.1008i	1.30021i	0.759843	+	0.650107i	$0.225276\pi$
$239$	20.1008i	1.30021i	−0.759843	+	0.650107i	$0.774724\pi$
$240$	0	0
$241$	− 19.0127i	− 1.22471i	−0.790581	−	0.612357i	$-0.790222\pi$
$241$	− 19.0127i	− 1.22471i	0.790581	−	0.612357i	$-0.209778\pi$
$242$	6.78687i	0.436277i
$243$	0	0
$244$	17.9618	1.14989
$245$	− 7.01938i	− 0.448452i
$246$	0	0
$247$	0	0
$248$	83.1764	5.28171
$249$	0	0
$250$	25.1304	1.58938
$251$	−0.763774	−0.0482090	−0.0241045	−	0.999709i	$-0.507673\pi$
$251$	−0.763774	−0.0482090	−0.0241045	+	0.999709i	$0.507673\pi$
$252$	0	0
$253$	− 16.7802i	− 1.05496i
$254$	12.0653i	0.757045i
$255$	0	0
$256$	17.1511	1.07194
$257$	−13.0911	−0.816602	−0.408301	−	0.912847i	$-0.633879\pi$
$257$	−13.0911	−0.816602	−0.408301	+	0.912847i	$0.633879\pi$
$258$	0	0
$259$	−3.19269	−0.198384
$260$	0	0
$261$	0	0
$262$	24.8135i	1.53299i
$263$	−18.3773	−1.13320	−0.566598	−	0.823995i	$-0.691741\pi$
$263$	−18.3773	−1.13320	−0.566598	+	0.823995i	$0.691741\pi$
$264$	0	0
$265$	− 7.95300i	− 0.488549i
$266$	− 1.12498i	− 0.0689771i
$267$	0	0
$268$	− 9.82563i	− 0.600196i
$269$	−23.6625	−1.44273	−0.721363	−	0.692557i	$-0.756484\pi$
$269$	−23.6625	−1.44273	−0.721363	+	0.692557i	$0.756484\pi$
$270$	0	0
$271$	− 19.7530i	− 1.19991i	−0.800034	−	0.599955i	$-0.795185\pi$
$271$	− 19.7530i	− 1.19991i	0.800034	−	0.599955i	$-0.204815\pi$
$272$	−63.2398	−3.83448
$273$	0	0
$274$	−20.1089	−1.21482
$275$	− 11.3556i	− 0.684767i
$276$	0	0
$277$	−1.77777	−0.106816	−0.0534081	−	0.998573i	$-0.517008\pi$
$277$	−1.77777	−0.106816	−0.0534081	+	0.998573i	$0.517008\pi$
$278$	− 48.4499i	− 2.90583i
$279$	0	0
$280$	− 5.08815i	− 0.304075i
$281$	− 1.62133i	− 0.0967207i	−0.998830	−	0.0483603i	$-0.984600\pi$
$281$	− 1.62133i	− 0.0967207i	0.998830	−	0.0483603i	$-0.0153996\pi$
$282$	0	0
$283$	−4.98361	−0.296245	−0.148122	−	0.988969i	$-0.547323\pi$
$283$	−4.98361	−0.296245	−0.148122	+	0.988969i	$0.547323\pi$
$284$	55.1124i	3.27032i
$285$	0	0
$286$	0	0
$287$	−2.72587	−0.160903
$288$	0	0
$289$	6.53079	0.384164
$290$	5.39852	0.317012
$291$	0	0
$292$	− 54.9700i	− 3.21688i
$293$	0.0717525i	0.00419183i	0.999998	+	0.00209591i	$0.000667151\pi$
$293$	0.0717525i	0.00419183i	−0.999998	+	0.00209591i	$0.999333\pi$
$294$	0	0
$295$	−4.29829	−0.250256
$296$	50.2868	2.92286
$297$	0	0
$298$	−41.2825	−2.39143
$299$	0	0
$300$	0	0
$301$	6.15883i	0.354989i
$302$	−6.81295	−0.392041
$303$	0	0
$304$	9.81700i	0.563044i
$305$	3.59073i	0.205605i
$306$	0	0
$307$	− 5.19806i	− 0.296669i	−0.988937	−	0.148335i	$-0.952609\pi$
$307$	− 5.19806i	− 0.296669i	0.988937	−	0.148335i	$-0.0473912\pi$
$308$	−8.47889	−0.483130
$309$	0	0
$310$	26.8696i	1.52609i
$311$	−22.5429	−1.27829	−0.639145	−	0.769087i	$-0.720711\pi$
$311$	−22.5429	−1.27829	−0.639145	+	0.769087i	$0.720711\pi$
$312$	0	0
$313$	22.6612	1.28088	0.640442	−	0.768006i	$-0.278751\pi$
$313$	22.6612	1.28088	0.640442	+	0.768006i	$0.278751\pi$
$314$	46.4083i	2.61897i
$315$	0	0
$316$	−7.00538	−0.394083
$317$	26.3424i	1.47954i	0.672861	+	0.739769i	$0.265065\pi$
$317$	26.3424i	1.47954i	−0.672861	+	0.739769i	$0.734935\pi$
$318$	0	0
$319$	− 5.56704i	− 0.311694i
$320$	22.3864i	1.25144i
$321$	0	0
$322$	−8.60925	−0.479775
$323$	− 3.65279i	− 0.203247i
$324$	0	0
$325$	0	0
$326$	−42.2838	−2.34188
$327$	0	0
$328$	42.9342	2.37065
$329$	0.417895	0.0230393
$330$	0	0
$331$	11.2295i	0.617230i	0.951187	+	0.308615i	$0.0998655\pi$
$331$	11.2295i	0.617230i	−0.951187	+	0.308615i	$0.900134\pi$
$332$	− 13.8683i	− 0.761123i
$333$	0	0
$334$	14.5265	0.794854
$335$	1.96423	0.107317
$336$	0	0
$337$	−2.30798	−0.125724	−0.0628618	−	0.998022i	$-0.520023\pi$
$337$	−2.30798	−0.125724	−0.0628618	+	0.998022i	$0.520023\pi$
$338$	0	0
$339$	0	0
$340$	− 26.6974i	− 1.44787i
$341$	27.7084	1.50049
$342$	0	0
$343$	7.59850i	0.410280i
$344$	− 97.0055i	− 5.23019i
$345$	0	0
$346$	64.4523i	3.46498i
$347$	9.13706	0.490503	0.245252	−	0.969459i	$-0.421129\pi$
$347$	9.13706	0.490503	0.245252	+	0.969459i	$0.421129\pi$
$348$	0	0
$349$	23.9758i	1.28340i	0.766957	+	0.641699i	$0.221770\pi$
$349$	23.9758i	1.28340i	−0.766957	+	0.641699i	$0.778230\pi$
$350$	−5.82610	−0.311418
$351$	0	0
$352$	51.2881	2.73367
$353$	− 27.1239i	− 1.44366i	−0.692070	−	0.721830i	$-0.743301\pi$
$353$	− 27.1239i	− 1.44366i	0.692070	−	0.721830i	$-0.256699\pi$
$354$	0	0
$355$	−11.0175	−0.584746
$356$	52.0863i	2.76057i
$357$	0	0
$358$	49.5569i	2.61916i
$359$	26.0790i	1.37640i	0.725521	+	0.688200i	$0.241599\pi$
$359$	26.0790i	1.37640i	−0.725521	+	0.688200i	$0.758401\pi$
$360$	0	0
$361$	18.4330	0.970156
$362$	9.78123i	0.514090i
$363$	0	0
$364$	0	0
$365$	10.9890	0.575190
$366$	0	0
$367$	9.57434	0.499776	0.249888	−	0.968275i	$-0.419606\pi$
$367$	9.57434	0.499776	0.249888	+	0.968275i	$0.419606\pi$
$368$	75.1275	3.91629
$369$	0	0
$370$	16.2448i	0.844530i
$371$	4.20775i	0.218456i
$372$	0	0
$373$	28.1497	1.45754	0.728769	−	0.684760i	$-0.240093\pi$
$373$	28.1497	1.45754	0.728769	+	0.684760i	$0.240093\pi$
$374$	−38.0248	−1.96621
$375$	0	0
$376$	−6.58211	−0.339446
$377$	0	0
$378$	0	0
$379$	− 16.0465i	− 0.824255i	−0.911126	−	0.412127i	$-0.864786\pi$
$379$	− 16.0465i	− 0.824255i	0.911126	−	0.412127i	$-0.135214\pi$
$380$	−4.14436	−0.212601
$381$	0	0
$382$	57.0122i	2.91700i
$383$	− 24.6165i	− 1.25785i	−0.777467	−	0.628923i	$-0.783496\pi$
$383$	− 24.6165i	− 1.25785i	0.777467	−	0.628923i	$-0.216504\pi$
$384$	0	0
$385$	− 1.69501i	− 0.0863855i
$386$	47.4097	2.41309
$387$	0	0
$388$	89.5852i	4.54800i
$389$	−17.2198	−0.873080	−0.436540	−	0.899685i	$-0.643796\pi$
$389$	−17.2198	−0.873080	−0.436540	+	0.899685i	$0.643796\pi$
$390$	0	0
$391$	−27.9541	−1.41370
$392$	− 58.4946i	− 2.95442i
$393$	0	0
$394$	12.5515	0.632335
$395$	− 1.40044i	− 0.0704636i
$396$	0	0
$397$	− 2.03923i	− 0.102346i	−0.998690	−	0.0511730i	$-0.983704\pi$
$397$	− 2.03923i	− 0.102346i	0.998690	−	0.0511730i	$-0.0162960\pi$
$398$	− 40.4795i	− 2.02905i
$399$	0	0
$400$	50.8407	2.54203
$401$	1.46144i	0.0729806i	0.999334	+	0.0364903i	$0.0116178\pi$
$401$	1.46144i	0.0729806i	−0.999334	+	0.0364903i	$0.988382\pi$
$402$	0	0
$403$	0	0
$404$	−38.4239	−1.91166
$405$	0	0
$406$	−2.85623	−0.141752
$407$	16.7520	0.830364
$408$	0	0
$409$	29.9390i	1.48039i	0.672393	+	0.740194i	$0.265266\pi$
$409$	29.9390i	1.48039i	−0.672393	+	0.740194i	$0.734734\pi$
$410$	13.8696i	0.684973i
$411$	0	0
$412$	−22.1414	−1.09083
$413$	2.27413	0.111902
$414$	0	0
$415$	2.77240	0.136092
$416$	0	0
$417$	0	0
$418$	5.90276i	0.288713i
$419$	−6.64742	−0.324748	−0.162374	−	0.986729i	$-0.551915\pi$
$419$	−6.64742	−0.324748	−0.162374	+	0.986729i	$0.551915\pi$
$420$	0	0
$421$	− 13.5646i	− 0.661100i	−0.943788	−	0.330550i	$-0.892766\pi$
$421$	− 13.5646i	− 0.661100i	0.943788	−	0.330550i	$-0.107234\pi$
$422$	− 1.23862i	− 0.0602950i
$423$	0	0
$424$	− 66.2747i	− 3.21858i
$425$	−18.9172	−0.917620
$426$	0	0
$427$	− 1.89977i	− 0.0919364i
$428$	33.5478	1.62159
$429$	0	0
$430$	31.3370	1.51121
$431$	35.9463i	1.73147i	0.500501	+	0.865736i	$0.333149\pi$
$431$	35.9463i	1.73147i	−0.500501	+	0.865736i	$0.666851\pi$
$432$	0	0
$433$	32.4741	1.56061	0.780303	−	0.625402i	$-0.215065\pi$
$433$	32.4741	1.56061	0.780303	+	0.625402i	$0.215065\pi$
$434$	− 14.2161i	− 0.682395i
$435$	0	0
$436$	− 18.1551i	− 0.869472i
$437$	4.33944i	0.207583i
$438$	0	0
$439$	−12.8321	−0.612441	−0.306221	−	0.951961i	$-0.599065\pi$
$439$	−12.8321	−0.612441	−0.306221	+	0.951961i	$0.599065\pi$
$440$	26.6974i	1.27275i
$441$	0	0
$442$	0	0
$443$	−11.9608	−0.568273	−0.284137	−	0.958784i	$-0.591707\pi$
$443$	−11.9608	−0.568273	−0.284137	+	0.958784i	$0.591707\pi$
$444$	0	0
$445$	−10.4125	−0.493601
$446$	−44.0151	−2.08417
$447$	0	0
$448$	− 11.8442i	− 0.559584i
$449$	12.4789i	0.588915i	0.955665	+	0.294458i	$0.0951390\pi$
$449$	12.4789i	0.588915i	−0.955665	+	0.294458i	$0.904861\pi$
$450$	0	0
$451$	14.3026	0.673483
$452$	49.0954	2.30926
$453$	0	0
$454$	−17.6606	−0.828851
$455$	0	0
$456$	0	0
$457$	− 32.4523i	− 1.51806i	−0.651058	−	0.759028i	$-0.725674\pi$
$457$	− 32.4523i	− 1.51806i	0.651058	−	0.759028i	$-0.274326\pi$
$458$	10.6480	0.497549
$459$	0	0
$460$	31.7159i	1.47876i
$461$	24.4034i	1.13658i	0.822828	+	0.568290i	$0.192395\pi$
$461$	24.4034i	1.13658i	−0.822828	+	0.568290i	$0.807605\pi$
$462$	0	0
$463$	− 33.1836i	− 1.54217i	−0.636731	−	0.771086i	$-0.719714\pi$
$463$	− 33.1836i	− 1.54217i	0.636731	−	0.771086i	$-0.280286\pi$
$464$	24.9245	1.15709
$465$	0	0
$466$	22.4969i	1.04215i
$467$	−38.5206	−1.78252	−0.891261	−	0.453490i	$-0.850179\pi$
$467$	−38.5206	−1.78252	−0.891261	+	0.453490i	$0.850179\pi$
$468$	0	0
$469$	−1.03923	−0.0479871
$470$	− 2.12631i	− 0.0980794i
$471$	0	0
$472$	−35.8189	−1.64870
$473$	− 32.3153i	− 1.48586i
$474$	0	0
$475$	2.93661i	0.134741i
$476$	14.1250i	0.647417i
$477$	0	0
$478$	−54.1118	−2.47502
$479$	8.34481i	0.381284i	0.981660	+	0.190642i	$0.0610570\pi$
$479$	8.34481i	0.381284i	−0.981660	+	0.190642i	$0.938943\pi$
$480$	0	0
$481$	0	0
$482$	51.1825	2.33130
$483$	0	0
$484$	−13.2282	−0.601282
$485$	−17.9089	−0.813200
$486$	0	0
$487$	14.8586i	0.673309i	0.941628	+	0.336654i	$0.109295\pi$
$487$	14.8586i	0.673309i	−0.941628	+	0.336654i	$0.890705\pi$
$488$	29.9226i	1.35453i
$489$	0	0
$490$	18.8963	0.853648
$491$	4.99894	0.225599	0.112799	−	0.993618i	$-0.464018\pi$
$491$	4.99894	0.225599	0.112799	+	0.993618i	$0.464018\pi$
$492$	0	0
$493$	−9.27413	−0.417686
$494$	0	0
$495$	0	0
$496$	124.055i	5.57023i
$497$	5.82908	0.261470
$498$	0	0
$499$	0.385371i	0.0172516i	0.999963	+	0.00862579i	$0.00274571\pi$
$499$	0.385371i	0.0172516i	−0.999963	+	0.00862579i	$0.997254\pi$
$500$	48.9812i	2.19051i
$501$	0	0
$502$	− 2.05610i	− 0.0917681i
$503$	−22.6179	−1.00848	−0.504241	−	0.863563i	$-0.668228\pi$
$503$	−22.6179	−1.00848	−0.504241	+	0.863563i	$0.668228\pi$
$504$	0	0
$505$	− 7.68127i	− 0.341812i
$506$	45.1726	2.00817
$507$	0	0
$508$	−23.5163	−1.04337
$509$	− 11.6039i	− 0.514333i	−0.966367	−	0.257166i	$-0.917211\pi$
$509$	− 11.6039i	− 0.514333i	0.966367	−	0.257166i	$-0.0827888\pi$
$510$	0	0
$511$	−5.81402	−0.257197
$512$	1.71678i	0.0758715i
$513$	0	0
$514$	− 35.2416i	− 1.55444i
$515$	− 4.42626i	− 0.195044i
$516$	0	0
$517$	−2.19269	−0.0964342
$518$	− 8.59478i	− 0.377633i
$519$	0	0
$520$	0	0
$521$	1.62671	0.0712675	0.0356337	−	0.999365i	$-0.488655\pi$
$521$	1.62671	0.0712675	0.0356337	+	0.999365i	$0.488655\pi$
$522$	0	0
$523$	10.0718	0.440407	0.220203	−	0.975454i	$-0.429328\pi$
$523$	10.0718	0.440407	0.220203	+	0.975454i	$0.429328\pi$
$524$	−48.3637	−2.11278
$525$	0	0
$526$	− 49.4722i	− 2.15709i
$527$	− 46.1594i	− 2.01074i
$528$	0	0
$529$	10.2088	0.443862
$530$	21.4097	0.929976
$531$	0	0
$532$	2.19269	0.0950650
$533$	0	0
$534$	0	0
$535$	6.70650i	0.289947i
$536$	16.3685	0.707012
$537$	0	0
$538$	− 63.6999i	− 2.74630i
$539$	− 19.4862i	− 0.839330i
$540$	0	0
$541$	− 20.4674i	− 0.879962i	−0.898007	−	0.439981i	$-0.854985\pi$
$541$	− 20.4674i	− 0.879962i	0.898007	−	0.439981i	$-0.145015\pi$
$542$	53.1756	2.28409
$543$	0	0
$544$	− 85.4408i	− 3.66325i
$545$	3.62937	0.155465
$546$	0	0
$547$	−27.5478	−1.17786	−0.588929	−	0.808185i	$-0.700450\pi$
$547$	−27.5478	−1.17786	−0.588929	+	0.808185i	$0.700450\pi$
$548$	− 39.1939i	− 1.67428i
$549$	0	0
$550$	30.5694	1.30348
$551$	1.43967i	0.0613318i
$552$	0	0
$553$	0.740939i	0.0315079i
$554$	− 4.78581i	− 0.203329i
$555$	0	0
$556$	94.4331	4.00485
$557$	− 37.9855i	− 1.60950i	−0.593615	−	0.804749i	$-0.702300\pi$
$557$	− 37.9855i	− 1.60950i	0.593615	−	0.804749i	$-0.297700\pi$
$558$	0	0
$559$	0	0
$560$	7.58881	0.320686
$561$	0	0
$562$	4.36467	0.184112
$563$	−29.7724	−1.25476	−0.627378	−	0.778714i	$-0.715872\pi$
$563$	−29.7724	−1.25476	−0.627378	+	0.778714i	$0.715872\pi$
$564$	0	0
$565$	9.81461i	0.412904i
$566$	− 13.4160i	− 0.563916i
$567$	0	0
$568$	−91.8117	−3.85234
$569$	−21.9541	−0.920362	−0.460181	−	0.887825i	$-0.652216\pi$
$569$	−21.9541	−0.920362	−0.460181	+	0.887825i	$0.652216\pi$
$570$	0	0
$571$	−2.46575	−0.103188	−0.0515942	−	0.998668i	$-0.516430\pi$
$571$	−2.46575	−0.103188	−0.0515942	+	0.998668i	$0.516430\pi$
$572$	0	0
$573$	0	0
$574$	− 7.33811i	− 0.306287i
$575$	22.4733	0.937199
$576$	0	0
$577$	− 17.4547i	− 0.726650i	−0.931662	−	0.363325i	$-0.881641\pi$
$577$	− 17.4547i	− 0.726650i	0.931662	−	0.363325i	$-0.118359\pi$
$578$	17.5810i	0.731275i
$579$	0	0
$580$	10.5222i	0.436909i
$581$	−1.46681	−0.0608536
$582$	0	0
$583$	− 22.0780i	− 0.914377i
$584$	91.5745	3.78938
$585$	0	0
$586$	−0.193159	−0.00797934
$587$	6.26337i	0.258517i	0.991611	+	0.129259i	$0.0412597\pi$
$587$	6.26337i	0.258517i	−0.991611	+	0.129259i	$0.958740\pi$
$588$	0	0
$589$	−7.16554	−0.295251
$590$	− 11.5711i	− 0.476374i
$591$	0	0
$592$	75.0012i	3.08253i
$593$	− 22.8745i	− 0.939345i	−0.882841	−	0.469672i	$-0.844372\pi$
$593$	− 22.8745i	− 0.939345i	0.882841	−	0.469672i	$-0.155628\pi$
$594$	0	0
$595$	−2.82371	−0.115761
$596$	− 80.4631i	− 3.29590i
$597$	0	0
$598$	0	0
$599$	1.05621	0.0431557	0.0215778	−	0.999767i	$-0.493131\pi$
$599$	1.05621	0.0431557	0.0215778	+	0.999767i	$0.493131\pi$
$600$	0	0
$601$	−33.3236	−1.35930	−0.679650	−	0.733537i	$-0.737868\pi$
$601$	−33.3236	−1.35930	−0.679650	+	0.733537i	$0.737868\pi$
$602$	−16.5797	−0.675739
$603$	0	0
$604$	− 13.2790i	− 0.540316i
$605$	− 2.64443i	− 0.107511i
$606$	0	0
$607$	16.2403	0.659172	0.329586	−	0.944125i	$-0.393091\pi$
$607$	16.2403	0.659172	0.329586	+	0.944125i	$0.393091\pi$
$608$	−13.2634	−0.537901
$609$	0	0
$610$	−9.66632	−0.391378
$611$	0	0
$612$	0	0
$613$	− 11.0479i	− 0.446219i	−0.974793	−	0.223109i	$-0.928379\pi$
$613$	− 11.0479i	− 0.446219i	0.974793	−	0.223109i	$-0.0716207\pi$
$614$	13.9933	0.564723
$615$	0	0
$616$	− 14.1250i	− 0.569112i
$617$	− 4.65950i	− 0.187584i	−0.995592	−	0.0937922i	$-0.970101\pi$
$617$	− 4.65950i	− 0.187584i	0.995592	−	0.0937922i	$-0.0298989\pi$
$618$	0	0
$619$	31.9259i	1.28321i	0.767036	+	0.641604i	$0.221731\pi$
$619$	31.9259i	1.28321i	−0.767036	+	0.641604i	$0.778269\pi$
$620$	−52.3712	−2.10328
$621$	0	0
$622$	− 60.6859i	− 2.43328i
$623$	5.50902	0.220714
$624$	0	0
$625$	9.70709	0.388283
$626$	61.0043i	2.43822i
$627$	0	0
$628$	−90.4538	−3.60950
$629$	− 27.9071i	− 1.11273i
$630$	0	0
$631$	− 39.4413i	− 1.57013i	−0.619411	−	0.785067i	$-0.712628\pi$
$631$	− 39.4413i	− 1.57013i	0.619411	−	0.785067i	$-0.287372\pi$
$632$	− 11.6703i	− 0.464218i
$633$	0	0
$634$	−70.9144	−2.81637
$635$	− 4.70112i	− 0.186558i
$636$	0	0
$637$	0	0
$638$	14.9866	0.593325
$639$	0	0
$640$	−23.3144	−0.921583
$641$	45.4510	1.79521	0.897603	−	0.440804i	$-0.145307\pi$
$641$	45.4510	1.79521	0.897603	+	0.440804i	$0.145307\pi$
$642$	0	0
$643$	29.7469i	1.17310i	0.809912	+	0.586552i	$0.199515\pi$
$643$	29.7469i	1.17310i	−0.809912	+	0.586552i	$0.800485\pi$
$644$	− 16.7802i	− 0.661231i
$645$	0	0
$646$	9.83340	0.386890
$647$	−16.9312	−0.665635	−0.332818	−	0.942991i	$-0.607999\pi$
$647$	−16.9312	−0.665635	−0.332818	+	0.942991i	$0.607999\pi$
$648$	0	0
$649$	−11.9323	−0.468384
$650$	0	0
$651$	0	0
$652$	− 82.4148i	− 3.22761i
$653$	33.1976	1.29912	0.649561	−	0.760309i	$-0.274953\pi$
$653$	33.1976	1.29912	0.649561	+	0.760309i	$0.274953\pi$
$654$	0	0
$655$	− 9.66833i	− 0.377773i
$656$	64.0350i	2.50015i
$657$	0	0
$658$	1.12498i	0.0438564i
$659$	42.6571	1.66168	0.830842	−	0.556508i	$-0.187859\pi$
$659$	42.6571	1.66168	0.830842	+	0.556508i	$0.187859\pi$
$660$	0	0
$661$	38.6902i	1.50488i	0.658664	+	0.752438i	$0.271122\pi$
$661$	38.6902i	1.50488i	−0.658664	+	0.752438i	$0.728878\pi$
$662$	−30.2301	−1.17493
$663$	0	0
$664$	23.1032	0.896578
$665$	0.438337i	0.0169980i
$666$	0	0
$667$	11.0175	0.426598
$668$	28.3134i	1.09548i
$669$	0	0
$670$	5.28775i	0.204283i
$671$	9.96807i	0.384813i
$672$	0	0
$673$	2.59419	0.0999986	0.0499993	−	0.998749i	$-0.484078\pi$
$673$	2.59419	0.0999986	0.0499993	+	0.998749i	$0.484078\pi$
$674$	− 6.21313i	− 0.239321i
$675$	0	0
$676$	0	0
$677$	−1.75302	−0.0673740	−0.0336870	−	0.999432i	$-0.510725\pi$
$677$	−1.75302	−0.0673740	−0.0336870	+	0.999432i	$0.510725\pi$
$678$	0	0
$679$	9.47517	0.363624
$680$	44.4752	1.70555
$681$	0	0
$682$	74.5916i	2.85626i
$683$	− 16.3351i	− 0.625046i	−0.949910	−	0.312523i	$-0.898826\pi$
$683$	− 16.3351i	− 0.625046i	0.949910	−	0.312523i	$-0.101174\pi$
$684$	0	0
$685$	7.83520	0.299368
$686$	−20.4553	−0.780988
$687$	0	0
$688$	144.681	5.51590
$689$	0	0
$690$	0	0
$691$	− 15.9105i	− 0.605265i	−0.953107	−	0.302632i	$-0.902135\pi$
$691$	− 15.9105i	− 0.605265i	0.953107	−	0.302632i	$-0.0978655\pi$
$692$	−125.623	−4.77547
$693$	0	0
$694$	24.5972i	0.933696i
$695$	18.8780i	0.716083i
$696$	0	0
$697$	− 23.8267i	− 0.902500i
$698$	−64.5435	−2.44301
$699$	0	0
$700$	− 11.3556i	− 0.429200i
$701$	−20.8635	−0.788005	−0.394002	−	0.919109i	$-0.628910\pi$
$701$	−20.8635	−0.788005	−0.394002	+	0.919109i	$0.628910\pi$
$702$	0	0
$703$	−4.33214	−0.163390
$704$	62.1460i	2.34222i
$705$	0	0
$706$	73.0182	2.74807
$707$	4.06398i	0.152842i
$708$	0	0
$709$	15.6485i	0.587691i	0.955853	+	0.293846i	$0.0949351\pi$
$709$	15.6485i	0.587691i	−0.955853	+	0.293846i	$0.905065\pi$
$710$	− 29.6592i	− 1.11309i
$711$	0	0
$712$	−86.7706	−3.25187
$713$	54.8364i	2.05364i
$714$	0	0
$715$	0	0
$716$	−96.5906	−3.60976
$717$	0	0
$718$	−70.2054	−2.62004
$719$	−27.1594	−1.01288	−0.506438	−	0.862276i	$-0.669038\pi$
$719$	−27.1594	−1.01288	−0.506438	+	0.862276i	$0.669038\pi$
$720$	0	0
$721$	2.34183i	0.0872143i
$722$	49.6219i	1.84674i
$723$	0	0
$724$	−19.0645	−0.708525
$725$	7.45580	0.276901
$726$	0	0
$727$	−31.7784	−1.17859	−0.589297	−	0.807916i	$-0.700595\pi$
$727$	−31.7784	−1.17859	−0.589297	+	0.807916i	$0.700595\pi$
$728$	0	0
$729$	0	0
$730$	29.5826i	1.09490i
$731$	−53.8340	−1.99112
$732$	0	0
$733$	− 46.8907i	− 1.73195i	−0.500090	−	0.865973i	$-0.666700\pi$
$733$	− 46.8907i	− 1.73195i	0.500090	−	0.865973i	$-0.333300\pi$
$734$	25.7743i	0.951347i
$735$	0	0
$736$	101.502i	3.74141i
$737$	5.45281	0.200857
$738$	0	0
$739$	17.0479i	0.627115i	0.949569	+	0.313558i	$0.101521\pi$
$739$	17.0479i	0.627115i	−0.949569	+	0.313558i	$0.898479\pi$
$740$	−31.6626	−1.16394
$741$	0	0
$742$	−11.3274	−0.415840
$743$	− 11.6324i	− 0.426750i	−0.976970	−	0.213375i	$-0.931554\pi$
$743$	− 11.6324i	− 0.426750i	0.976970	−	0.213375i	$-0.0684455\pi$
$744$	0	0
$745$	16.0853	0.589319
$746$	75.7797i	2.77449i
$747$	0	0
$748$	− 74.1135i	− 2.70986i
$749$	− 3.54825i	− 0.129650i
$750$	0	0
$751$	13.0295	0.475455	0.237727	−	0.971332i	$-0.423598\pi$
$751$	13.0295	0.475455	0.237727	+	0.971332i	$0.423598\pi$
$752$	− 9.81700i	− 0.357989i
$753$	0	0
$754$	0	0
$755$	2.65459	0.0966106
$756$	0	0
$757$	22.7899	0.828311	0.414156	−	0.910206i	$-0.364077\pi$
$757$	22.7899	0.828311	0.414156	+	0.910206i	$0.364077\pi$
$758$	43.1976	1.56901
$759$	0	0
$760$	− 6.90408i	− 0.250437i
$761$	38.3424i	1.38991i	0.719052	+	0.694956i	$0.244576\pi$
$761$	38.3424i	1.38991i	−0.719052	+	0.694956i	$0.755424\pi$
$762$	0	0
$763$	−1.92021	−0.0695164
$764$	−111.122	−4.02024
$765$	0	0
$766$	66.2683	2.39437
$767$	0	0
$768$	0	0
$769$	3.63879i	0.131218i	0.997845	+	0.0656091i	$0.0208990\pi$
$769$	3.63879i	0.131218i	−0.997845	+	0.0656091i	$0.979101\pi$
$770$	4.56299	0.164439
$771$	0	0
$772$	92.4055i	3.32575i
$773$	− 39.3424i	− 1.41505i	−0.706689	−	0.707524i	$-0.749812\pi$
$773$	− 39.3424i	− 1.41505i	0.706689	−	0.707524i	$-0.250188\pi$
$774$	0	0
$775$	37.1092i	1.33300i
$776$	−149.240	−5.35740
$777$	0	0
$778$	− 46.3562i	− 1.66195i
$779$	−3.69873	−0.132521
$780$	0	0
$781$	−30.5851	−1.09442
$782$	− 75.2529i	− 2.69104i
$783$	0	0
$784$	87.2428	3.11581
$785$	− 18.0825i	− 0.645392i
$786$	0	0
$787$	− 25.4252i	− 0.906310i	−0.891432	−	0.453155i	$-0.850298\pi$
$787$	− 25.4252i	− 0.906310i	0.891432	−	0.453155i	$-0.149702\pi$
$788$	24.4639i	0.871492i
$789$	0	0
$790$	3.77000	0.134131
$791$	− 5.19269i	− 0.184631i
$792$	0	0
$793$	0	0
$794$	5.48965	0.194820
$795$	0	0
$796$	78.8980	2.79646
$797$	20.7138	0.733720	0.366860	−	0.930276i	$-0.380433\pi$
$797$	20.7138	0.733720	0.366860	+	0.930276i	$0.380433\pi$
$798$	0	0
$799$	3.65279i	0.129227i
$800$	68.6889i	2.42852i
$801$	0	0
$802$	−3.93422	−0.138922
$803$	30.5060	1.07653
$804$	0	0
$805$	3.35450	0.118231
$806$	0	0
$807$	0	0
$808$	− 64.0103i	− 2.25187i
$809$	−11.0978	−0.390179	−0.195090	−	0.980785i	$-0.562500\pi$
$809$	−11.0978	−0.390179	−0.195090	+	0.980785i	$0.562500\pi$
$810$	0	0
$811$	4.84223i	0.170034i	0.996380	+	0.0850169i	$0.0270944\pi$
$811$	4.84223i	0.170034i	−0.996380	+	0.0850169i	$0.972906\pi$
$812$	− 5.56704i	− 0.195365i
$813$	0	0
$814$	45.0966i	1.58064i
$815$	16.4754	0.577109
$816$	0	0
$817$	8.35690i	0.292371i
$818$	−80.5964	−2.81799
$819$	0	0
$820$	−27.0331	−0.944037
$821$	− 43.7381i	− 1.52647i	−0.646121	−	0.763235i	$-0.723610\pi$
$821$	− 43.7381i	− 1.52647i	0.646121	−	0.763235i	$-0.276390\pi$
$822$	0	0
$823$	−12.0998	−0.421771	−0.210885	−	0.977511i	$-0.567635\pi$
$823$	−12.0998	−0.421771	−0.210885	+	0.977511i	$0.567635\pi$
$824$	− 36.8853i	− 1.28496i
$825$	0	0
$826$	6.12200i	0.213012i
$827$	35.3212i	1.22824i	0.789213	+	0.614120i	$0.210489\pi$
$827$	35.3212i	1.22824i	−0.789213	+	0.614120i	$0.789511\pi$
$828$	0	0
$829$	−53.4946	−1.85794	−0.928971	−	0.370152i	$-0.879306\pi$
$829$	−53.4946	−1.85794	−0.928971	+	0.370152i	$0.879306\pi$
$830$	7.46335i	0.259057i
$831$	0	0
$832$	0	0
$833$	−32.4620	−1.12474
$834$	0	0
$835$	−5.66009	−0.195875
$836$	−11.5050	−0.397908
$837$	0	0
$838$	− 17.8950i	− 0.618172i
$839$	− 23.1521i	− 0.799300i	−0.916668	−	0.399650i	$-0.869132\pi$
$839$	− 23.1521i	− 0.799300i	0.916668	−	0.399650i	$-0.130868\pi$
$840$	0	0
$841$	−25.3448	−0.873959
$842$	36.5163	1.25844
$843$	0	0
$844$	2.41417	0.0830993
$845$	0	0
$846$	0	0
$847$	1.39911i	0.0480739i
$848$	98.8467	3.39441
$849$	0	0
$850$	− 50.9256i	− 1.74673i
$851$	33.1530i	1.13647i
$852$	0	0
$853$	26.7265i	0.915097i	0.889185	+	0.457548i	$0.151272\pi$
$853$	26.7265i	0.915097i	−0.889185	+	0.457548i	$0.848728\pi$
$854$	5.11423	0.175005
$855$	0	0
$856$	55.8872i	1.91019i
$857$	42.6064	1.45541	0.727703	−	0.685892i	$-0.240588\pi$
$857$	42.6064	1.45541	0.727703	+	0.685892i	$0.240588\pi$
$858$	0	0
$859$	33.6079	1.14669	0.573344	−	0.819315i	$-0.305646\pi$
$859$	33.6079	1.14669	0.573344	+	0.819315i	$0.305646\pi$
$860$	61.0786i	2.08276i
$861$	0	0
$862$	−96.7682	−3.29594
$863$	18.7047i	0.636715i	0.947971	+	0.318358i	$0.103131\pi$
$863$	18.7047i	0.636715i	−0.947971	+	0.318358i	$0.896869\pi$
$864$	0	0
$865$	− 25.1132i	− 0.853873i
$866$	87.4210i	2.97069i
$867$	0	0
$868$	27.7084	0.940485
$869$	− 3.88769i	− 0.131881i
$870$	0	0
$871$	0	0
$872$	30.2446	1.02421
$873$	0	0
$874$	−11.6819	−0.395145
$875$	5.18060	0.175136
$876$	0	0
$877$	− 36.8237i	− 1.24345i	−0.783236	−	0.621724i	$-0.786433\pi$
$877$	− 36.8237i	− 1.24345i	0.783236	−	0.621724i	$-0.213567\pi$
$878$	− 34.5442i	− 1.16581i
$879$	0	0
$880$	−39.8183	−1.34228
$881$	−41.1250	−1.38554	−0.692768	−	0.721161i	$-0.743609\pi$
$881$	−41.1250	−1.38554	−0.692768	+	0.721161i	$0.743609\pi$
$882$	0	0
$883$	−30.7482	−1.03476	−0.517380	−	0.855756i	$-0.673093\pi$
$883$	−30.7482	−1.03476	−0.517380	+	0.855756i	$0.673093\pi$
$884$	0	0
$885$	0	0
$886$	− 32.1987i	− 1.08173i
$887$	−7.58940	−0.254827	−0.127414	−	0.991850i	$-0.540668\pi$
$887$	−7.58940	−0.254827	−0.127414	+	0.991850i	$0.540668\pi$
$888$	0	0
$889$	2.48725i	0.0834198i
$890$	− 28.0307i	− 0.939592i
$891$	0	0
$892$	− 85.7891i	− 2.87243i
$893$	0.567040	0.0189753
$894$	0	0
$895$	− 19.3093i	− 0.645439i
$896$	12.3351	0.412088
$897$	0	0
$898$	−33.5934	−1.12103
$899$	18.1927i	0.606760i
$900$	0	0
$901$	−36.7797	−1.22531
$902$	38.5029i	1.28201i
$903$	0	0
$904$	81.7881i	2.72023i
$905$	− 3.81115i	− 0.126687i
$906$	0	0
$907$	−19.3333	−0.641952	−0.320976	−	0.947087i	$-0.604011\pi$
$907$	−19.3333	−0.641952	−0.320976	+	0.947087i	$0.604011\pi$
$908$	− 34.4219i	− 1.14233i
$909$	0	0
$910$	0	0
$911$	−22.7149	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$911$	−22.7149	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$912$	0	0
$913$	7.69633	0.254711
$914$	87.3624	2.88969
$915$	0	0
$916$	20.7539i	0.685727i
$917$	5.11529i	0.168922i
$918$	0	0
$919$	14.2911	0.471420	0.235710	−	0.971823i	$-0.424258\pi$
$919$	14.2911	0.471420	0.235710	+	0.971823i	$0.424258\pi$
$920$	−52.8355	−1.74194
$921$	0	0
$922$	−65.6945	−2.16353
$923$	0	0
$924$	0	0
$925$	22.4355i	0.737674i
$926$	89.3309	2.93560
$927$	0	0
$928$	33.6746i	1.10542i
$929$	32.7211i	1.07354i	0.843727	+	0.536772i	$0.180356\pi$
$929$	32.7211i	1.07354i	−0.843727	+	0.536772i	$0.819644\pi$
$930$	0	0
$931$	5.03923i	0.165154i
$932$	−43.8485	−1.43630
$933$	0	0
$934$	− 103.698i	− 3.39311i
$935$	14.8159	0.484533
$936$	0	0
$937$	−4.01400	−0.131132	−0.0655658	−	0.997848i	$-0.520885\pi$
$937$	−4.01400	−0.131132	−0.0655658	+	0.997848i	$0.520885\pi$
$938$	− 2.79763i	− 0.0913457i
$939$	0	0
$940$	4.14436	0.135174
$941$	− 28.2669i	− 0.921476i	−0.887536	−	0.460738i	$-0.847585\pi$
$941$	− 28.2669i	− 0.921476i	0.887536	−	0.460738i	$-0.152415\pi$
$942$	0	0
$943$	28.3056i	0.921757i
$944$	− 53.4228i	− 1.73876i
$945$	0	0
$946$	86.9934	2.82840
$947$	37.8455i	1.22981i	0.788600	+	0.614906i	$0.210806\pi$
$947$	37.8455i	1.22981i	−0.788600	+	0.614906i	$0.789194\pi$
$948$	0	0
$949$	0	0
$950$	−7.90541	−0.256485
$951$	0	0
$952$	−23.5308	−0.762637
$953$	−40.8256	−1.32247	−0.661236	−	0.750178i	$-0.729968\pi$
$953$	−40.8256	−1.32247	−0.661236	+	0.750178i	$0.729968\pi$
$954$	0	0
$955$	− 22.2142i	− 0.718834i
$956$	− 105.469i	− 3.41110i
$957$	0	0
$958$	−22.4644	−0.725792
$959$	−4.14542	−0.133863
$960$	0	0
$961$	−59.5491	−1.92094
$962$	0	0
$963$	0	0
$964$	99.7591i	3.21302i
$965$	−18.4727	−0.594656
$966$	0	0
$967$	− 10.2798i	− 0.330575i	−0.986245	−	0.165288i	$-0.947145\pi$
$967$	− 10.2798i	− 0.330575i	0.986245	−	0.165288i	$-0.0528552\pi$
$968$	− 22.0368i	− 0.708291i
$969$	0	0
$970$	− 48.2111i	− 1.54796i
$971$	19.4805	0.625161	0.312580	−	0.949891i	$-0.398807\pi$
$971$	19.4805	0.625161	0.312580	+	0.949891i	$0.398807\pi$
$972$	0	0
$973$	− 9.98792i	− 0.320198i
$974$	−39.9997	−1.28167
$975$	0	0
$976$	−44.6286	−1.42853
$977$	47.0538i	1.50539i	0.658372	+	0.752693i	$0.271245\pi$
$977$	47.0538i	1.50539i	−0.658372	+	0.752693i	$0.728755\pi$
$978$	0	0
$979$	−28.9057	−0.923831
$980$	36.8305i	1.17651i
$981$	0	0
$982$	13.4572i	0.429438i
$983$	− 34.4295i	− 1.09813i	−0.835779	−	0.549065i	$-0.814984\pi$
$983$	− 34.4295i	− 1.09813i	0.835779	−	0.549065i	$-0.185016\pi$
$984$	0	0
$985$	−4.89056	−0.155826
$986$	− 24.9661i	− 0.795084i
$987$	0	0
$988$	0	0
$989$	63.9536	2.03361
$990$	0	0
$991$	7.30798	0.232146	0.116073	−	0.993241i	$-0.462969\pi$
$991$	7.30798	0.232146	0.116073	+	0.993241i	$0.462969\pi$
$992$	−167.606	−5.32149
$993$	0	0
$994$	15.6920i	0.497721i
$995$	15.7724i	0.500019i
$996$	0	0
$997$	−35.6915	−1.13036	−0.565181	−	0.824967i	$-0.691194\pi$
$997$	−35.6915	−1.13036	−0.565181	+	0.824967i	$0.691194\pi$
$998$	−1.03743	−0.0328392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.k.1351.6		6
3.2	odd	2		507.2.b.f.337.1		6
13.5	odd	4		1521.2.a.s.1.3		3
13.8	odd	4		1521.2.a.n.1.1		3
13.12	even	2	inner	1521.2.b.k.1351.1		6
39.2	even	12		507.2.e.l.22.3		6
39.5	even	4		507.2.a.i.1.1	✓	3
39.8	even	4		507.2.a.l.1.3	yes	3
39.11	even	12		507.2.e.i.22.1		6
39.17	odd	6		507.2.j.i.361.6		12
39.20	even	12		507.2.e.i.484.1		6
39.23	odd	6		507.2.j.i.316.1		12
39.29	odd	6		507.2.j.i.316.6		12
39.32	even	12		507.2.e.l.484.3		6
39.35	odd	6		507.2.j.i.361.1		12
39.38	odd	2		507.2.b.f.337.6		6
156.47	odd	4		8112.2.a.cp.1.1		3
156.83	odd	4		8112.2.a.cg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.1	✓	3	39.5	even	4
507.2.a.l.1.3	yes	3	39.8	even	4
507.2.b.f.337.1		6	3.2	odd	2
507.2.b.f.337.6		6	39.38	odd	2
507.2.e.i.22.1		6	39.11	even	12
507.2.e.i.484.1		6	39.20	even	12
507.2.e.l.22.3		6	39.2	even	12
507.2.e.l.484.3		6	39.32	even	12
507.2.j.i.316.1		12	39.23	odd	6
507.2.j.i.316.6		12	39.29	odd	6
507.2.j.i.361.1		12	39.35	odd	6
507.2.j.i.361.6		12	39.17	odd	6
1521.2.a.n.1.1		3	13.8	odd	4
1521.2.a.s.1.3		3	13.5	odd	4
1521.2.b.k.1351.1		6	13.12	even	2	inner
1521.2.b.k.1351.6		6	1.1	even	1	trivial
8112.2.a.cg.1.3		3	156.83	odd	4
8112.2.a.cp.1.1		3	156.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.69202i q^{2} -5.24698 q^{4} -1.04892i q^{5} +0.554958i q^{7} -8.74094i q^{8} +O(q^{10})\)	\(q+2.69202i q^{2} -5.24698 q^{4} -1.04892i q^{5} +0.554958i q^{7} -8.74094i q^{8} +2.82371 q^{10} -2.91185i q^{11} -1.49396 q^{14} +13.0368 q^{16} -4.85086 q^{17} +0.753020i q^{19} +5.50365i q^{20} +7.83877 q^{22} +5.76271 q^{23} +3.89977 q^{25} -2.91185i q^{28} +1.91185 q^{29} +9.51573i q^{31} +17.6136i q^{32} -13.0586i q^{34} +0.582105 q^{35} +5.75302i q^{37} -2.02715 q^{38} -9.16852 q^{40} +4.91185i q^{41} +11.0978 q^{43} +15.2784i q^{44} +15.5133i q^{46} -0.753020i q^{47} +6.69202 q^{49} +10.4983i q^{50} +7.58211 q^{53} -3.05429 q^{55} +4.85086 q^{56} +5.14675i q^{58} -4.09783i q^{59} -3.42327 q^{61} -25.6165 q^{62} -21.3424 q^{64} +1.87263i q^{67} +25.4523 q^{68} +1.56704i q^{70} -10.5036i q^{71} +10.4765i q^{73} -15.4873 q^{74} -3.95108i q^{76} +1.61596 q^{77} +1.33513 q^{79} -13.6746i q^{80} -13.2228 q^{82} +2.64310i q^{83} +5.08815i q^{85} +29.8756i q^{86} -25.4523 q^{88} -9.92692i q^{89} -30.2368 q^{92} +2.02715 q^{94} +0.789856 q^{95} -17.0737i q^{97} +18.0151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 22 q^{4}+O(q^{10}) \) 6 * q - 22 * q^4	\( 6 q - 22 q^{4} + 2 q^{10} + 10 q^{14} + 22 q^{16} - 2 q^{17} - 18 q^{22} - 22 q^{25} + 4 q^{29} - 8 q^{35} + 6 q^{40} + 30 q^{43} + 30 q^{49} + 34 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 4 q^{62} + 26 q^{68} - 30 q^{74} + 30 q^{77} + 6 q^{79} + 6 q^{82} - 26 q^{88} - 14 q^{92} - 42 q^{95}+O(q^{100}) \) 6 * q - 22 * q^4 + 2 * q^10 + 10 * q^14 + 22 * q^16 - 2 * q^17 - 18 * q^22 - 22 * q^25 + 4 * q^29 - 8 * q^35 + 6 * q^40 + 30 * q^43 + 30 * q^49 + 34 * q^53 + 6 * q^55 + 2 * q^56 - 26 * q^61 - 4 * q^62 + 26 * q^68 - 30 * q^74 + 30 * q^77 + 6 * q^79 + 6 * q^82 - 26 * q^88 - 14 * q^92 - 42 * q^95

Introduction

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