LMFDB - Embedded newform 1521.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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, 0]))

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

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.a (trivial)

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:	yes
Analytic conductor:	$12.1452461474$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69202 q^{2} +5.24698 q^{4} +1.04892 q^{5} +0.554958 q^{7} -8.74094 q^{8} -2.82371 q^{10} -2.91185 q^{11} -1.49396 q^{14} +13.0368 q^{16} +4.85086 q^{17} -0.753020 q^{19} +5.50365 q^{20} +7.83877 q^{22} -5.76271 q^{23} -3.89977 q^{25} +2.91185 q^{28} +1.91185 q^{29} -9.51573 q^{31} -17.6136 q^{32} -13.0586 q^{34} +0.582105 q^{35} +5.75302 q^{37} +2.02715 q^{38} -9.16852 q^{40} -4.91185 q^{41} -11.0978 q^{43} -15.2784 q^{44} +15.5133 q^{46} -0.753020 q^{47} -6.69202 q^{49} +10.4983 q^{50} +7.58211 q^{53} -3.05429 q^{55} -4.85086 q^{56} -5.14675 q^{58} -4.09783 q^{59} -3.42327 q^{61} +25.6165 q^{62} +21.3424 q^{64} -1.87263 q^{67} +25.4523 q^{68} -1.56704 q^{70} +10.5036 q^{71} +10.4765 q^{73} -15.4873 q^{74} -3.95108 q^{76} -1.61596 q^{77} +1.33513 q^{79} +13.6746 q^{80} +13.2228 q^{82} -2.64310 q^{83} +5.08815 q^{85} +29.8756 q^{86} +25.4523 q^{88} -9.92692 q^{89} -30.2368 q^{92} +2.02715 q^{94} -0.789856 q^{95} +17.0737 q^{97} +18.0151 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8} - q^{10} - 5 q^{11} + 5 q^{14} + 11 q^{16} + q^{17} - 7 q^{19} - 15 q^{20} - 9 q^{22} + 11 q^{25} + 5 q^{28} + 2 q^{29} - 16 q^{31} - 22 q^{32}+ \cdots + 29 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 11 * q^4 - 6 * q^5 + 2 * q^7 - 12 * q^8 - q^10 - 5 * q^11 + 5 * q^14 + 11 * q^16 + q^17 - 7 * q^19 - 15 * q^20 - 9 * q^22 + 11 * q^25 + 5 * q^28 + 2 * q^29 - 16 * q^31 - 22 * q^32 - 8 * q^34 - 4 * q^35 + 22 * q^37 + 3 * q^40 - 11 * q^41 - 15 * q^43 - 16 * q^44 - 7 * q^46 - 7 * q^47 - 15 * q^49 + 3 * q^50 + 17 * q^53 + 3 * q^55 - q^56 + 12 * q^58 + 6 * q^59 - 13 * q^61 + 2 * q^62 + 11 * q^67 + 13 * q^68 - 24 * q^70 + 6 * q^73 - 15 * q^74 - 21 * q^76 - 15 * q^77 + 3 * q^79 + 20 * q^80 - 3 * q^82 - 12 * q^83 + 19 * q^85 + 29 * q^86 + 13 * q^88 - q^89 - 7 * q^92 + 21 * q^95 - 5 * q^97 + 29 * q^98

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 $

$2$	−2.69202	−1.90355	−0.951773	−	0.306802i	$-0.900741\pi$
$2$	−2.69202	−1.90355	−0.951773	+	0.306802i	$0.900741\pi$
$3$	0	0
$3$	0	0
$4$	5.24698	2.62349
$5$	1.04892	0.469090	0.234545	−	0.972105i	$-0.424640\pi$
$5$	1.04892	0.469090	0.234545	+	0.972105i	$0.424640\pi$
$6$	0	0
$7$	0.554958	0.209754	0.104877	−	0.994485i	$-0.466555\pi$
$7$	0.554958	0.209754	0.104877	+	0.994485i	$0.466555\pi$
$8$	−8.74094	−3.09039
$9$	0	0
$10$	−2.82371	−0.892935
$11$	−2.91185	−0.877957	−0.438979	−	0.898498i	$-0.644660\pi$
$11$	−2.91185	−0.877957	−0.438979	+	0.898498i	$0.644660\pi$
$12$	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	$-0.299816\pi$
$17$	4.85086	1.17651	0.588253	+	0.808677i	$0.299816\pi$
$18$	0	0
$19$	−0.753020	−0.172755	−0.0863774	−	0.996262i	$-0.527529\pi$
$19$	−0.753020	−0.172755	−0.0863774	+	0.996262i	$0.527529\pi$
$20$	5.50365	1.23065
$21$	0	0
$22$	7.83877	1.67123
$23$	−5.76271	−1.20161	−0.600804	−	0.799396i	$-0.705153\pi$
$23$	−5.76271	−1.20161	−0.600804	+	0.799396i	$0.705153\pi$
$24$	0	0
$25$	−3.89977	−0.779954
$26$	0	0
$27$	0	0
$28$	2.91185	0.550289
$29$	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
$31$	−9.51573	−1.70908	−0.854538	−	0.519389i	$-0.826159\pi$
$31$	−9.51573	−1.70908	−0.854538	+	0.519389i	$0.826159\pi$
$32$	−17.6136	−3.11367
$33$	0	0
$34$	−13.0586	−2.23953
$35$	0.582105	0.0983937
$36$	0	0
$37$	5.75302	0.945791	0.472895	−	0.881119i	$-0.343209\pi$
$37$	5.75302	0.945791	0.472895	+	0.881119i	$0.343209\pi$
$38$	2.02715	0.328847
$39$	0	0
$40$	−9.16852	−1.44967
$41$	−4.91185	−0.767103	−0.383551	−	0.923520i	$-0.625299\pi$
$41$	−4.91185	−0.767103	−0.383551	+	0.923520i	$0.625299\pi$
$42$	0	0
$43$	−11.0978	−1.69240	−0.846202	−	0.532862i	$-0.821116\pi$
$43$	−11.0978	−1.69240	−0.846202	+	0.532862i	$0.821116\pi$
$44$	−15.2784	−2.30331
$45$	0	0
$46$	15.5133	2.28732
$47$	−0.753020	−0.109839	−0.0549197	−	0.998491i	$-0.517490\pi$
$47$	−0.753020	−0.109839	−0.0549197	+	0.998491i	$0.517490\pi$
$48$	0	0
$49$	−6.69202	−0.956003
$50$	10.4983	1.48468
$51$	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
$58$	−5.14675	−0.675802
$59$	−4.09783	−0.533493	−0.266746	−	0.963767i	$-0.585949\pi$
$59$	−4.09783	−0.533493	−0.266746	+	0.963767i	$0.585949\pi$
$60$	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
$67$	−1.87263	−0.228778	−0.114389	−	0.993436i	$-0.536491\pi$
$67$	−1.87263	−0.228778	−0.114389	+	0.993436i	$0.536491\pi$
$68$	25.4523	3.08655
$69$	0	0
$70$	−1.56704	−0.187297
$71$	10.5036	1.24655	0.623277	−	0.782001i	$-0.285801\pi$
$71$	10.5036	1.24655	0.623277	+	0.782001i	$0.285801\pi$
$72$	0	0
$73$	10.4765	1.22618	0.613091	−	0.790012i	$-0.289926\pi$
$73$	10.4765	1.22618	0.613091	+	0.790012i	$0.289926\pi$
$74$	−15.4873	−1.80036
$75$	0	0
$76$	−3.95108	−0.453220
$77$	−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	$0.476070\pi$
$80$	13.6746	1.52886
$81$	0	0
$82$	13.2228	1.46022
$83$	−2.64310	−0.290118	−0.145059	−	0.989423i	$-0.546337\pi$
$83$	−2.64310	−0.290118	−0.145059	+	0.989423i	$0.546337\pi$
$84$	0	0
$85$	5.08815	0.551887
$86$	29.8756	3.22157
$87$	0	0
$88$	25.4523	2.71323
$89$	−9.92692	−1.05225	−0.526126	−	0.850407i	$-0.676356\pi$
$89$	−9.92692	−1.05225	−0.526126	+	0.850407i	$0.676356\pi$
$90$	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
$97$	17.0737	1.73357	0.866784	−	0.498683i	$-0.166183\pi$
$97$	17.0737	1.73357	0.866784	+	0.498683i	$0.166183\pi$
$98$	18.0151	1.81980
$99$	0	0
$100$	−20.4620	−2.04620
$101$	−7.32304	−0.728670	−0.364335	−	0.931268i	$-0.618704\pi$
$101$	−7.32304	−0.728670	−0.364335	+	0.931268i	$0.618704\pi$
$102$	0	0
$103$	−4.21983	−0.415792	−0.207896	−	0.978151i	$-0.566662\pi$
$103$	−4.21983	−0.415792	−0.207896	+	0.978151i	$0.566662\pi$
$104$	0	0
$105$	0	0
$106$	−20.4112	−1.98251
$107$	−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
$109$	−3.46011	−0.331418	−0.165709	−	0.986175i	$-0.552991\pi$
$109$	−3.46011	−0.331418	−0.165709	+	0.986175i	$0.552991\pi$
$110$	8.22223	0.783958
$111$	0	0
$112$	7.23490	0.683634
$113$	−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
$115$	−6.04461	−0.563662
$116$	10.0315	0.931398
$117$	0	0
$118$	11.0315	1.01553
$119$	2.69202	0.246777
$120$	0	0
$121$	−2.52111	−0.229191
$122$	9.21552	0.834334
$123$	0	0
$124$	−49.9288	−4.48374
$125$	−9.33513	−0.834959
$126$	0	0
$127$	−4.48188	−0.397702	−0.198851	−	0.980030i	$-0.563721\pi$
$127$	−4.48188	−0.397702	−0.198851	+	0.980030i	$0.563721\pi$
$128$	−22.2271	−1.96462
$129$	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
$136$	−42.4010	−3.63586
$137$	7.46980	0.638188	0.319094	−	0.947723i	$-0.396621\pi$
$137$	7.46980	0.638188	0.319094	+	0.947723i	$0.396621\pi$
$138$	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
$145$	2.00538	0.166537
$146$	−28.2030	−2.33409
$147$	0	0
$148$	30.1860	2.48127
$149$	−15.3351	−1.25630	−0.628151	−	0.778091i	$-0.716188\pi$
$149$	−15.3351	−1.25630	−0.628151	+	0.778091i	$0.716188\pi$
$150$	0	0
$151$	2.53079	0.205953	0.102977	−	0.994684i	$-0.467163\pi$
$151$	2.53079	0.205953	0.102977	+	0.994684i	$0.467163\pi$
$152$	6.58211	0.533879
$153$	0	0
$154$	4.35019	0.350548
$155$	−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	$0.258524\pi$
$158$	−3.59419	−0.285938
$159$	0	0
$160$	−18.4752	−1.46059
$161$	−3.19806	−0.252043
$162$	0	0
$163$	15.7071	1.23027	0.615137	−	0.788420i	$-0.289101\pi$
$163$	15.7071	1.23027	0.615137	+	0.788420i	$0.289101\pi$
$164$	−25.7724	−2.01249
$165$	0	0
$166$	7.11529	0.552254
$167$	−5.39612	−0.417565	−0.208782	−	0.977962i	$-0.566950\pi$
$167$	−5.39612	−0.417565	−0.208782	+	0.977962i	$0.566950\pi$
$168$	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	$-0.864024\pi$
$173$	−23.9420	−1.82028	−0.910138	+	0.414306i	$0.864024\pi$
$174$	0	0
$175$	−2.16421	−0.163599
$176$	−37.9614	−2.86145
$177$	0	0
$178$	26.7235	2.00301
$179$	−18.4088	−1.37594	−0.687969	−	0.725740i	$-0.741498\pi$
$179$	−18.4088	−1.37594	−0.687969	+	0.725740i	$0.741498\pi$
$180$	0	0
$181$	−3.63342	−0.270070	−0.135035	−	0.990841i	$-0.543115\pi$
$181$	−3.63342	−0.270070	−0.135035	+	0.990841i	$0.543115\pi$
$182$	0	0
$183$	0	0
$184$	50.3715	3.71344
$185$	6.03444	0.443661
$186$	0	0
$187$	−14.1250	−1.03292
$188$	−3.95108	−0.288162
$189$	0	0
$190$	2.12631	0.154259
$191$	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
$193$	−17.6112	−1.26768	−0.633840	−	0.773464i	$-0.718522\pi$
$193$	−17.6112	−1.26768	−0.633840	+	0.773464i	$0.718522\pi$
$194$	−45.9627	−3.29993
$195$	0	0
$196$	−35.1129	−2.50806
$197$	4.66248	0.332188	0.166094	−	0.986110i	$-0.446884\pi$
$197$	4.66248	0.332188	0.166094	+	0.986110i	$0.446884\pi$
$198$	0	0
$199$	15.0368	1.06593	0.532967	−	0.846136i	$-0.321077\pi$
$199$	15.0368	1.06593	0.532967	+	0.846136i	$0.321077\pi$
$200$	34.0877	2.41036
$201$	0	0
$202$	19.7138	1.38706
$203$	1.06100	0.0744675
$204$	0	0
$205$	−5.15213	−0.359840
$206$	11.3599	0.791480
$207$	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
$214$	17.2121	1.17659
$215$	−11.6407	−0.793890
$216$	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
$223$	−16.3502	−1.09489	−0.547445	−	0.836842i	$-0.684399\pi$
$223$	−16.3502	−1.09489	−0.547445	+	0.836842i	$0.684399\pi$
$224$	−9.77479	−0.653106
$225$	0	0
$226$	25.1890	1.67555
$227$	−6.56033	−0.435425	−0.217712	−	0.976013i	$-0.569859\pi$
$227$	−6.56033	−0.435425	−0.217712	+	0.976013i	$0.569859\pi$
$228$	0	0
$229$	−3.95539	−0.261380	−0.130690	−	0.991423i	$-0.541719\pi$
$229$	−3.95539	−0.261380	−0.130690	+	0.991423i	$0.541719\pi$
$230$	16.2722	1.07296
$231$	0	0
$232$	−16.7114	−1.09716
$233$	−8.35690	−0.547478	−0.273739	−	0.961804i	$-0.588261\pi$
$233$	−8.35690	−0.547478	−0.273739	+	0.961804i	$0.588261\pi$
$234$	0	0
$235$	−0.789856	−0.0515245
$236$	−21.5013	−1.39961
$237$	0	0
$238$	−7.24698	−0.469752
$239$	−20.1008	−1.30021	−0.650107	−	0.759843i	$-0.725276\pi$
$239$	−20.1008	−1.30021	−0.650107	+	0.759843i	$0.725276\pi$
$240$	0	0
$241$	−19.0127	−1.22471	−0.612357	−	0.790581i	$-0.709778\pi$
$241$	−19.0127	−1.22471	−0.612357	+	0.790581i	$0.709778\pi$
$242$	6.78687	0.436277
$243$	0	0
$244$	−17.9618	−1.14989
$245$	−7.01938	−0.448452
$246$	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	$-0.492327\pi$
$251$	0.763774	0.0482090	0.0241045	+	0.999709i	$0.492327\pi$
$252$	0	0
$253$	16.7802	1.05496
$254$	12.0653	0.757045
$255$	0	0
$256$	17.1511	1.07194
$257$	13.0911	0.816602	0.408301	−	0.912847i	$-0.366121\pi$
$257$	13.0911	0.816602	0.408301	+	0.912847i	$0.366121\pi$
$258$	0	0
$259$	3.19269	0.198384
$260$	0	0
$261$	0	0
$262$	−24.8135	−1.53299
$263$	−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
$265$	7.95300	0.488549
$266$	1.12498	0.0689771
$267$	0	0
$268$	−9.82563	−0.600196
$269$	−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
$271$	−19.7530	−1.19991	−0.599955	−	0.800034i	$-0.704815\pi$
$271$	−19.7530	−1.19991	−0.599955	+	0.800034i	$0.704815\pi$
$272$	63.2398	3.83448
$273$	0	0
$274$	−20.1089	−1.21482
$275$	11.3556	0.684767
$276$	0	0
$277$	1.77777	0.106816	0.0534081	−	0.998573i	$-0.482992\pi$
$277$	1.77777	0.106816	0.0534081	+	0.998573i	$0.482992\pi$
$278$	48.4499	2.90583
$279$	0	0
$280$	−5.08815	−0.304075
$281$	−1.62133	−0.0967207	−0.0483603	−	0.998830i	$-0.515400\pi$
$281$	−1.62133	−0.0967207	−0.0483603	+	0.998830i	$0.515400\pi$
$282$	0	0
$283$	4.98361	0.296245	0.148122	−	0.988969i	$-0.452677\pi$
$283$	4.98361	0.296245	0.148122	+	0.988969i	$0.452677\pi$
$284$	55.1124	3.27032
$285$	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
$292$	54.9700	3.21688
$293$	0.0717525	0.00419183	0.00209591	−	0.999998i	$-0.499333\pi$
$293$	0.0717525	0.00419183	0.00209591	+	0.999998i	$0.499333\pi$
$294$	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
$301$	−6.15883	−0.354989
$302$	−6.81295	−0.392041
$303$	0	0
$304$	−9.81700	−0.563044
$305$	−3.59073	−0.205605
$306$	0	0
$307$	−5.19806	−0.296669	−0.148335	−	0.988937i	$-0.547391\pi$
$307$	−5.19806	−0.296669	−0.148335	+	0.988937i	$0.547391\pi$
$308$	−8.47889	−0.483130
$309$	0	0
$310$	26.8696	1.52609
$311$	22.5429	1.27829	0.639145	−	0.769087i	$-0.279289\pi$
$311$	22.5429	1.27829	0.639145	+	0.769087i	$0.279289\pi$
$312$	0	0
$313$	22.6612	1.28088	0.640442	−	0.768006i	$-0.278751\pi$
$313$	22.6612	1.28088	0.640442	+	0.768006i	$0.278751\pi$
$314$	−46.4083	−2.61897
$315$	0	0
$316$	7.00538	0.394083
$317$	−26.3424	−1.47954	−0.739769	−	0.672861i	$-0.765065\pi$
$317$	−26.3424	−1.47954	−0.739769	+	0.672861i	$0.765065\pi$
$318$	0	0
$319$	−5.56704	−0.311694
$320$	22.3864	1.25144
$321$	0	0
$322$	8.60925	0.479775
$323$	−3.65279	−0.203247
$324$	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
$331$	−11.2295	−0.617230	−0.308615	−	0.951187i	$-0.599866\pi$
$331$	−11.2295	−0.617230	−0.308615	+	0.951187i	$0.599866\pi$
$332$	−13.8683	−0.761123
$333$	0	0
$334$	14.5265	0.794854
$335$	−1.96423	−0.107317
$336$	0	0
$337$	2.30798	0.125724	0.0628618	−	0.998022i	$-0.479977\pi$
$337$	2.30798	0.125724	0.0628618	+	0.998022i	$0.479977\pi$
$338$	0	0
$339$	0	0
$340$	26.6974	1.44787
$341$	27.7084	1.50049
$342$	0	0
$343$	−7.59850	−0.410280
$344$	97.0055	5.23019
$345$	0	0
$346$	64.4523	3.46498
$347$	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
$349$	23.9758	1.28340	0.641699	−	0.766957i	$-0.278230\pi$
$349$	23.9758	1.28340	0.641699	+	0.766957i	$0.278230\pi$
$350$	5.82610	0.311418
$351$	0	0
$352$	51.2881	2.73367
$353$	27.1239	1.44366	0.721830	−	0.692070i	$-0.243301\pi$
$353$	27.1239	1.44366	0.721830	+	0.692070i	$0.243301\pi$
$354$	0	0
$355$	11.0175	0.584746
$356$	−52.0863	−2.76057
$357$	0	0
$358$	49.5569	2.61916
$359$	26.0790	1.37640	0.688200	−	0.725521i	$-0.258401\pi$
$359$	26.0790	1.37640	0.688200	+	0.725521i	$0.258401\pi$
$360$	0	0
$361$	−18.4330	−0.970156
$362$	9.78123	0.514090
$363$	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
$370$	−16.2448	−0.844530
$371$	4.20775	0.218456
$372$	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
$379$	16.0465	0.824255	0.412127	−	0.911126i	$-0.364786\pi$
$379$	16.0465	0.824255	0.412127	+	0.911126i	$0.364786\pi$
$380$	−4.14436	−0.212601
$381$	0	0
$382$	−57.0122	−2.91700
$383$	24.6165	1.25785	0.628923	−	0.777467i	$-0.283496\pi$
$383$	24.6165	1.25785	0.628923	+	0.777467i	$0.283496\pi$
$384$	0	0
$385$	−1.69501	−0.0863855
$386$	47.4097	2.41309
$387$	0	0
$388$	89.5852	4.54800
$389$	17.2198	0.873080	0.436540	−	0.899685i	$-0.356204\pi$
$389$	17.2198	0.873080	0.436540	+	0.899685i	$0.356204\pi$
$390$	0	0
$391$	−27.9541	−1.41370
$392$	58.4946	2.95442
$393$	0	0
$394$	−12.5515	−0.632335
$395$	1.40044	0.0704636
$396$	0	0
$397$	−2.03923	−0.102346	−0.0511730	−	0.998690i	$-0.516296\pi$
$397$	−2.03923	−0.102346	−0.0511730	+	0.998690i	$0.516296\pi$
$398$	−40.4795	−2.02905
$399$	0	0
$400$	−50.8407	−2.54203
$401$	1.46144	0.0729806	0.0364903	−	0.999334i	$-0.488382\pi$
$401$	1.46144	0.0729806	0.0364903	+	0.999334i	$0.488382\pi$
$402$	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
$409$	−29.9390	−1.48039	−0.740194	−	0.672393i	$-0.765266\pi$
$409$	−29.9390	−1.48039	−0.740194	+	0.672393i	$0.765266\pi$
$410$	13.8696	0.684973
$411$	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
$418$	−5.90276	−0.288713
$419$	−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
$421$	13.5646	0.661100	0.330550	−	0.943788i	$-0.392766\pi$
$421$	13.5646	0.661100	0.330550	+	0.943788i	$0.392766\pi$
$422$	1.23862	0.0602950
$423$	0	0
$424$	−66.2747	−3.21858
$425$	−18.9172	−0.917620
$426$	0	0
$427$	−1.89977	−0.0919364
$428$	−33.5478	−1.62159
$429$	0	0
$430$	31.3370	1.51121
$431$	−35.9463	−1.73147	−0.865736	−	0.500501i	$-0.833149\pi$
$431$	−35.9463	−1.73147	−0.865736	+	0.500501i	$0.833149\pi$
$432$	0	0
$433$	−32.4741	−1.56061	−0.780303	−	0.625402i	$-0.784935\pi$
$433$	−32.4741	−1.56061	−0.780303	+	0.625402i	$0.784935\pi$
$434$	14.2161	0.682395
$435$	0	0
$436$	−18.1551	−0.869472
$437$	4.33944	0.207583
$438$	0	0
$439$	12.8321	0.612441	0.306221	−	0.951961i	$-0.400935\pi$
$439$	12.8321	0.612441	0.306221	+	0.951961i	$0.400935\pi$
$440$	26.6974	1.27275
$441$	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
$448$	11.8442	0.559584
$449$	12.4789	0.588915	0.294458	−	0.955665i	$-0.404861\pi$
$449$	12.4789	0.588915	0.294458	+	0.955665i	$0.404861\pi$
$450$	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
$457$	32.4523	1.51806	0.759028	−	0.651058i	$-0.225674\pi$
$457$	32.4523	1.51806	0.759028	+	0.651058i	$0.225674\pi$
$458$	10.6480	0.497549
$459$	0	0
$460$	−31.7159	−1.47876
$461$	−24.4034	−1.13658	−0.568290	−	0.822828i	$-0.692395\pi$
$461$	−24.4034	−1.13658	−0.568290	+	0.822828i	$0.692395\pi$
$462$	0	0
$463$	−33.1836	−1.54217	−0.771086	−	0.636731i	$-0.780286\pi$
$463$	−33.1836	−1.54217	−0.771086	+	0.636731i	$0.780286\pi$
$464$	24.9245	1.15709
$465$	0	0
$466$	22.4969	1.04215
$467$	38.5206	1.78252	0.891261	−	0.453490i	$-0.149821\pi$
$467$	38.5206	1.78252	0.891261	+	0.453490i	$0.149821\pi$
$468$	0	0
$469$	−1.03923	−0.0479871
$470$	2.12631	0.0980794
$471$	0	0
$472$	35.8189	1.64870
$473$	32.3153	1.48586
$474$	0	0
$475$	2.93661	0.134741
$476$	14.1250	0.647417
$477$	0	0
$478$	54.1118	2.47502
$479$	8.34481	0.381284	0.190642	−	0.981660i	$-0.438943\pi$
$479$	8.34481	0.381284	0.190642	+	0.981660i	$0.438943\pi$
$480$	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
$487$	−14.8586	−0.673309	−0.336654	−	0.941628i	$-0.609295\pi$
$487$	−14.8586	−0.673309	−0.336654	+	0.941628i	$0.609295\pi$
$488$	29.9226	1.35453
$489$	0	0
$490$	18.8963	0.853648
$491$	−4.99894	−0.225599	−0.112799	−	0.993618i	$-0.535982\pi$
$491$	−4.99894	−0.225599	−0.112799	+	0.993618i	$0.535982\pi$
$492$	0	0
$493$	9.27413	0.417686
$494$	0	0
$495$	0	0
$496$	−124.055	−5.57023
$497$	5.82908	0.261470
$498$	0	0
$499$	−0.385371	−0.0172516	−0.00862579	−	0.999963i	$-0.502746\pi$
$499$	−0.385371	−0.0172516	−0.00862579	+	0.999963i	$0.502746\pi$
$500$	−48.9812	−2.19051
$501$	0	0
$502$	−2.05610	−0.0917681
$503$	−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
$505$	−7.68127	−0.341812
$506$	−45.1726	−2.00817
$507$	0	0
$508$	−23.5163	−1.04337
$509$	11.6039	0.514333	0.257166	−	0.966367i	$-0.417211\pi$
$509$	11.6039	0.514333	0.257166	+	0.966367i	$0.417211\pi$
$510$	0	0
$511$	5.81402	0.257197
$512$	−1.71678	−0.0758715
$513$	0	0
$514$	−35.2416	−1.55444
$515$	−4.42626	−0.195044
$516$	0	0
$517$	2.19269	0.0964342
$518$	−8.59478	−0.377633
$519$	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
$526$	49.4722	2.15709
$527$	−46.1594	−2.01074
$528$	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
$535$	−6.70650	−0.289947
$536$	16.3685	0.707012
$537$	0	0
$538$	63.6999	2.74630
$539$	19.4862	0.839330
$540$	0	0
$541$	−20.4674	−0.879962	−0.439981	−	0.898007i	$-0.645015\pi$
$541$	−20.4674	−0.879962	−0.439981	+	0.898007i	$0.645015\pi$
$542$	53.1756	2.28409
$543$	0	0
$544$	−85.4408	−3.66325
$545$	−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	$0.700450\pi$
$548$	39.1939	1.67428
$549$	0	0
$550$	−30.5694	−1.30348
$551$	−1.43967	−0.0613318
$552$	0	0
$553$	0.740939	0.0315079
$554$	−4.78581	−0.203329
$555$	0	0
$556$	−94.4331	−4.00485
$557$	−37.9855	−1.60950	−0.804749	−	0.593615i	$-0.797700\pi$
$557$	−37.9855	−1.60950	−0.804749	+	0.593615i	$0.797700\pi$
$558$	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	$-0.284128\pi$
$563$	29.7724	1.25476	0.627378	+	0.778714i	$0.284128\pi$
$564$	0	0
$565$	−9.81461	−0.412904
$566$	−13.4160	−0.563916
$567$	0	0
$568$	−91.8117	−3.85234
$569$	21.9541	0.920362	0.460181	−	0.887825i	$-0.347784\pi$
$569$	21.9541	0.920362	0.460181	+	0.887825i	$0.347784\pi$
$570$	0	0
$571$	2.46575	0.103188	0.0515942	−	0.998668i	$-0.483570\pi$
$571$	2.46575	0.103188	0.0515942	+	0.998668i	$0.483570\pi$
$572$	0	0
$573$	0	0
$574$	7.33811	0.306287
$575$	22.4733	0.937199
$576$	0	0
$577$	17.4547	0.726650	0.363325	−	0.931662i	$-0.381641\pi$
$577$	17.4547	0.726650	0.363325	+	0.931662i	$0.381641\pi$
$578$	−17.5810	−0.731275
$579$	0	0
$580$	10.5222	0.436909
$581$	−1.46681	−0.0608536
$582$	0	0
$583$	−22.0780	−0.914377
$584$	−91.5745	−3.78938
$585$	0	0
$586$	−0.193159	−0.00797934
$587$	−6.26337	−0.258517	−0.129259	−	0.991611i	$-0.541260\pi$
$587$	−6.26337	−0.258517	−0.129259	+	0.991611i	$0.541260\pi$
$588$	0	0
$589$	7.16554	0.295251
$590$	11.5711	0.476374
$591$	0	0
$592$	75.0012	3.08253
$593$	−22.8745	−0.939345	−0.469672	−	0.882841i	$-0.655628\pi$
$593$	−22.8745	−0.939345	−0.469672	+	0.882841i	$0.655628\pi$
$594$	0	0
$595$	2.82371	0.115761
$596$	−80.4631	−3.29590
$597$	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
$604$	13.2790	0.540316
$605$	−2.64443	−0.107511
$606$	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
$613$	11.0479	0.446219	0.223109	−	0.974793i	$-0.428379\pi$
$613$	11.0479	0.446219	0.223109	+	0.974793i	$0.428379\pi$
$614$	13.9933	0.564723
$615$	0	0
$616$	14.1250	0.569112
$617$	4.65950	0.187584	0.0937922	−	0.995592i	$-0.470101\pi$
$617$	4.65950	0.187584	0.0937922	+	0.995592i	$0.470101\pi$
$618$	0	0
$619$	31.9259	1.28321	0.641604	−	0.767036i	$-0.278269\pi$
$619$	31.9259	1.28321	0.641604	+	0.767036i	$0.278269\pi$
$620$	−52.3712	−2.10328
$621$	0	0
$622$	−60.6859	−2.43328
$623$	−5.50902	−0.220714
$624$	0	0
$625$	9.70709	0.388283
$626$	−61.0043	−2.43822
$627$	0	0
$628$	90.4538	3.60950
$629$	27.9071	1.11273
$630$	0	0
$631$	−39.4413	−1.57013	−0.785067	−	0.619411i	$-0.787372\pi$
$631$	−39.4413	−1.57013	−0.785067	+	0.619411i	$0.787372\pi$
$632$	−11.6703	−0.464218
$633$	0	0
$634$	70.9144	2.81637
$635$	−4.70112	−0.186558
$636$	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	$-0.854693\pi$
$641$	−45.4510	−1.79521	−0.897603	+	0.440804i	$0.854693\pi$
$642$	0	0
$643$	−29.7469	−1.17310	−0.586552	−	0.809912i	$-0.699515\pi$
$643$	−29.7469	−1.17310	−0.586552	+	0.809912i	$0.699515\pi$
$644$	−16.7802	−0.661231
$645$	0	0
$646$	9.83340	0.386890
$647$	16.9312	0.665635	0.332818	−	0.942991i	$-0.392001\pi$
$647$	16.9312	0.665635	0.332818	+	0.942991i	$0.392001\pi$
$648$	0	0
$649$	11.9323	0.468384
$650$	0	0
$651$	0	0
$652$	82.4148	3.22761
$653$	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
$655$	9.66833	0.377773
$656$	−64.0350	−2.50015
$657$	0	0
$658$	1.12498	0.0438564
$659$	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
$661$	38.6902	1.50488	0.752438	−	0.658664i	$-0.228878\pi$
$661$	38.6902	1.50488	0.752438	+	0.658664i	$0.228878\pi$
$662$	30.2301	1.17493
$663$	0	0
$664$	23.1032	0.896578
$665$	−0.438337	−0.0169980
$666$	0	0
$667$	−11.0175	−0.426598
$668$	−28.3134	−1.09548
$669$	0	0
$670$	5.28775	0.204283
$671$	9.96807	0.384813
$672$	0	0
$673$	−2.59419	−0.0999986	−0.0499993	−	0.998749i	$-0.515922\pi$
$673$	−2.59419	−0.0999986	−0.0499993	+	0.998749i	$0.515922\pi$
$674$	−6.21313	−0.239321
$675$	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
$682$	−74.5916	−2.85626
$683$	−16.3351	−0.625046	−0.312523	−	0.949910i	$-0.601174\pi$
$683$	−16.3351	−0.625046	−0.312523	+	0.949910i	$0.601174\pi$
$684$	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
$691$	15.9105	0.605265	0.302632	−	0.953107i	$-0.402135\pi$
$691$	15.9105	0.605265	0.302632	+	0.953107i	$0.402135\pi$
$692$	−125.623	−4.77547
$693$	0	0
$694$	−24.5972	−0.933696
$695$	−18.8780	−0.716083
$696$	0	0
$697$	−23.8267	−0.902500
$698$	−64.5435	−2.44301
$699$	0	0
$700$	−11.3556	−0.429200
$701$	20.8635	0.788005	0.394002	−	0.919109i	$-0.371090\pi$
$701$	20.8635	0.788005	0.394002	+	0.919109i	$0.371090\pi$
$702$	0	0
$703$	−4.33214	−0.163390
$704$	−62.1460	−2.34222
$705$	0	0
$706$	−73.0182	−2.74807
$707$	−4.06398	−0.152842
$708$	0	0
$709$	15.6485	0.587691	0.293846	−	0.955853i	$-0.405065\pi$
$709$	15.6485	0.587691	0.293846	+	0.955853i	$0.405065\pi$
$710$	−29.6592	−1.11309
$711$	0	0
$712$	86.7706	3.25187
$713$	54.8364	2.05364
$714$	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	$-0.330962\pi$
$719$	27.1594	1.01288	0.506438	+	0.862276i	$0.330962\pi$
$720$	0	0
$721$	−2.34183	−0.0872143
$722$	49.6219	1.84674
$723$	0	0
$724$	−19.0645	−0.708525
$725$	−7.45580	−0.276901
$726$	0	0
$727$	31.7784	1.17859	0.589297	−	0.807916i	$-0.299405\pi$
$727$	31.7784	1.17859	0.589297	+	0.807916i	$0.299405\pi$
$728$	0	0
$729$	0	0
$730$	−29.5826	−1.09490
$731$	−53.8340	−1.99112
$732$	0	0
$733$	46.8907	1.73195	0.865973	−	0.500090i	$-0.166700\pi$
$733$	46.8907	1.73195	0.865973	+	0.500090i	$0.166700\pi$
$734$	−25.7743	−0.951347
$735$	0	0
$736$	101.502	3.74141
$737$	5.45281	0.200857
$738$	0	0
$739$	17.0479	0.627115	0.313558	−	0.949569i	$-0.398479\pi$
$739$	17.0479	0.627115	0.313558	+	0.949569i	$0.398479\pi$
$740$	31.6626	1.16394
$741$	0	0
$742$	−11.3274	−0.415840
$743$	11.6324	0.426750	0.213375	−	0.976970i	$-0.431554\pi$
$743$	11.6324	0.426750	0.213375	+	0.976970i	$0.431554\pi$
$744$	0	0
$745$	−16.0853	−0.589319
$746$	−75.7797	−2.77449
$747$	0	0
$748$	−74.1135	−2.70986
$749$	−3.54825	−0.129650
$750$	0	0
$751$	−13.0295	−0.475455	−0.237727	−	0.971332i	$-0.576402\pi$
$751$	−13.0295	−0.475455	−0.237727	+	0.971332i	$0.576402\pi$
$752$	−9.81700	−0.357989
$753$	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
$760$	6.90408	0.250437
$761$	38.3424	1.38991	0.694956	−	0.719052i	$-0.255424\pi$
$761$	38.3424	1.38991	0.694956	+	0.719052i	$0.255424\pi$
$762$	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
$769$	−3.63879	−0.131218	−0.0656091	−	0.997845i	$-0.520899\pi$
$769$	−3.63879	−0.131218	−0.0656091	+	0.997845i	$0.520899\pi$
$770$	4.56299	0.164439
$771$	0	0
$772$	−92.4055	−3.32575
$773$	39.3424	1.41505	0.707524	−	0.706689i	$-0.249812\pi$
$773$	39.3424	1.41505	0.707524	+	0.706689i	$0.249812\pi$
$774$	0	0
$775$	37.1092	1.33300
$776$	−149.240	−5.35740
$777$	0	0
$778$	−46.3562	−1.66195
$779$	3.69873	0.132521
$780$	0	0
$781$	−30.5851	−1.09442
$782$	75.2529	2.69104
$783$	0	0
$784$	−87.2428	−3.11581
$785$	18.0825	0.645392
$786$	0	0
$787$	−25.4252	−0.906310	−0.453155	−	0.891432i	$-0.649702\pi$
$787$	−25.4252	−0.906310	−0.453155	+	0.891432i	$0.649702\pi$
$788$	24.4639	0.871492
$789$	0	0
$790$	−3.77000	−0.134131
$791$	−5.19269	−0.184631
$792$	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	$-0.619567\pi$
$797$	−20.7138	−0.733720	−0.366860	+	0.930276i	$0.619567\pi$
$798$	0	0
$799$	−3.65279	−0.129227
$800$	68.6889	2.42852
$801$	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
$808$	64.0103	2.25187
$809$	−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
$811$	−4.84223	−0.170034	−0.0850169	−	0.996380i	$-0.527094\pi$
$811$	−4.84223	−0.170034	−0.0850169	+	0.996380i	$0.527094\pi$
$812$	5.56704	0.195365
$813$	0	0
$814$	45.0966	1.58064
$815$	16.4754	0.577109
$816$	0	0
$817$	8.35690	0.292371
$818$	80.5964	2.81799
$819$	0	0
$820$	−27.0331	−0.944037
$821$	43.7381	1.52647	0.763235	−	0.646121i	$-0.223610\pi$
$821$	43.7381	1.52647	0.763235	+	0.646121i	$0.223610\pi$
$822$	0	0
$823$	12.0998	0.421771	0.210885	−	0.977511i	$-0.432365\pi$
$823$	12.0998	0.421771	0.210885	+	0.977511i	$0.432365\pi$
$824$	36.8853	1.28496
$825$	0	0
$826$	6.12200	0.213012
$827$	35.3212	1.22824	0.614120	−	0.789213i	$-0.289511\pi$
$827$	35.3212	1.22824	0.614120	+	0.789213i	$0.289511\pi$
$828$	0	0
$829$	53.4946	1.85794	0.928971	−	0.370152i	$-0.120694\pi$
$829$	53.4946	1.85794	0.928971	+	0.370152i	$0.120694\pi$
$830$	7.46335	0.259057
$831$	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
$838$	17.8950	0.618172
$839$	−23.1521	−0.799300	−0.399650	−	0.916668i	$-0.630868\pi$
$839$	−23.1521	−0.799300	−0.399650	+	0.916668i	$0.630868\pi$
$840$	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
$847$	−1.39911	−0.0480739
$848$	98.8467	3.39441
$849$	0	0
$850$	50.9256	1.74673
$851$	−33.1530	−1.13647
$852$	0	0
$853$	26.7265	0.915097	0.457548	−	0.889185i	$-0.348728\pi$
$853$	26.7265	0.915097	0.457548	+	0.889185i	$0.348728\pi$
$854$	5.11423	0.175005
$855$	0	0
$856$	55.8872	1.91019
$857$	−42.6064	−1.45541	−0.727703	−	0.685892i	$-0.759412\pi$
$857$	−42.6064	−1.45541	−0.727703	+	0.685892i	$0.759412\pi$
$858$	0	0
$859$	33.6079	1.14669	0.573344	−	0.819315i	$-0.305646\pi$
$859$	33.6079	1.14669	0.573344	+	0.819315i	$0.305646\pi$
$860$	−61.0786	−2.08276
$861$	0	0
$862$	96.7682	3.29594
$863$	−18.7047	−0.636715	−0.318358	−	0.947971i	$-0.603131\pi$
$863$	−18.7047	−0.636715	−0.318358	+	0.947971i	$0.603131\pi$
$864$	0	0
$865$	−25.1132	−0.853873
$866$	87.4210	2.97069
$867$	0	0
$868$	−27.7084	−0.940485
$869$	−3.88769	−0.131881
$870$	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
$877$	36.8237	1.24345	0.621724	−	0.783236i	$-0.286433\pi$
$877$	36.8237	1.24345	0.621724	+	0.783236i	$0.286433\pi$
$878$	−34.5442	−1.16581
$879$	0	0
$880$	−39.8183	−1.34228
$881$	41.1250	1.38554	0.692768	−	0.721161i	$-0.256391\pi$
$881$	41.1250	1.38554	0.692768	+	0.721161i	$0.256391\pi$
$882$	0	0
$883$	30.7482	1.03476	0.517380	−	0.855756i	$-0.326907\pi$
$883$	30.7482	1.03476	0.517380	+	0.855756i	$0.326907\pi$
$884$	0	0
$885$	0	0
$886$	32.1987	1.08173
$887$	−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
$889$	−2.48725	−0.0834198
$890$	28.0307	0.939592
$891$	0	0
$892$	−85.7891	−2.87243
$893$	0.567040	0.0189753
$894$	0	0
$895$	−19.3093	−0.645439
$896$	−12.3351	−0.412088
$897$	0	0
$898$	−33.5934	−1.12103
$899$	−18.1927	−0.606760
$900$	0	0
$901$	36.7797	1.22531
$902$	−38.5029	−1.28201
$903$	0	0
$904$	81.7881	2.72023
$905$	−3.81115	−0.126687
$906$	0	0
$907$	19.3333	0.641952	0.320976	−	0.947087i	$-0.395989\pi$
$907$	19.3333	0.641952	0.320976	+	0.947087i	$0.395989\pi$
$908$	−34.4219	−1.14233
$909$	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
$916$	−20.7539	−0.685727
$917$	5.11529	0.168922
$918$	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
$925$	−22.4355	−0.737674
$926$	89.3309	2.93560
$927$	0	0
$928$	−33.6746	−1.10542
$929$	−32.7211	−1.07354	−0.536772	−	0.843727i	$-0.680356\pi$
$929$	−32.7211	−1.07354	−0.536772	+	0.843727i	$0.680356\pi$
$930$	0	0
$931$	5.03923	0.165154
$932$	−43.8485	−1.43630
$933$	0	0
$934$	−103.698	−3.39311
$935$	−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	$0.520885\pi$
$938$	2.79763	0.0913457
$939$	0	0
$940$	−4.14436	−0.135174
$941$	28.2669	0.921476	0.460738	−	0.887536i	$-0.347585\pi$
$941$	28.2669	0.921476	0.460738	+	0.887536i	$0.347585\pi$
$942$	0	0
$943$	28.3056	0.921757
$944$	−53.4228	−1.73876
$945$	0	0
$946$	−86.9934	−2.82840
$947$	37.8455	1.22981	0.614906	−	0.788600i	$-0.289194\pi$
$947$	37.8455	1.22981	0.614906	+	0.788600i	$0.289194\pi$
$948$	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	$-0.270032\pi$
$953$	40.8256	1.32247	0.661236	+	0.750178i	$0.270032\pi$
$954$	0	0
$955$	22.2142	0.718834
$956$	−105.469	−3.41110
$957$	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
$964$	−99.7591	−3.21302
$965$	−18.4727	−0.594656
$966$	0	0
$967$	10.2798	0.330575	0.165288	−	0.986245i	$-0.447145\pi$
$967$	10.2798	0.330575	0.165288	+	0.986245i	$0.447145\pi$
$968$	22.0368	0.708291
$969$	0	0
$970$	−48.2111	−1.54796
$971$	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
$973$	−9.98792	−0.320198
$974$	39.9997	1.28167
$975$	0	0
$976$	−44.6286	−1.42853
$977$	−47.0538	−1.50539	−0.752693	−	0.658372i	$-0.771245\pi$
$977$	−47.0538	−1.50539	−0.752693	+	0.658372i	$0.771245\pi$
$978$	0	0
$979$	28.9057	0.923831
$980$	−36.8305	−1.17651
$981$	0	0
$982$	13.4572	0.429438
$983$	−34.4295	−1.09813	−0.549065	−	0.835779i	$-0.685016\pi$
$983$	−34.4295	−1.09813	−0.549065	+	0.835779i	$0.685016\pi$
$984$	0	0
$985$	4.89056	0.155826
$986$	−24.9661	−0.795084
$987$	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
$994$	−15.6920	−0.497721
$995$	15.7724	0.500019
$996$	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.a.n.1.1		3
3.2	odd	2		507.2.a.l.1.3	yes	3
12.11	even	2		8112.2.a.cp.1.1		3
13.5	odd	4		1521.2.b.k.1351.6		6
13.8	odd	4		1521.2.b.k.1351.1		6
13.12	even	2		1521.2.a.s.1.3		3
39.2	even	12		507.2.j.i.316.6		12
39.5	even	4		507.2.b.f.337.1		6
39.8	even	4		507.2.b.f.337.6		6
39.11	even	12		507.2.j.i.316.1		12
39.17	odd	6		507.2.e.l.484.3		6
39.20	even	12		507.2.j.i.361.6		12
39.23	odd	6		507.2.e.l.22.3		6
39.29	odd	6		507.2.e.i.22.1		6
39.32	even	12		507.2.j.i.361.1		12
39.35	odd	6		507.2.e.i.484.1		6
39.38	odd	2		507.2.a.i.1.1	✓	3
156.155	even	2		8112.2.a.cg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.1	✓	3	39.38	odd	2
507.2.a.l.1.3	yes	3	3.2	odd	2
507.2.b.f.337.1		6	39.5	even	4
507.2.b.f.337.6		6	39.8	even	4
507.2.e.i.22.1		6	39.29	odd	6
507.2.e.i.484.1		6	39.35	odd	6
507.2.e.l.22.3		6	39.23	odd	6
507.2.e.l.484.3		6	39.17	odd	6
507.2.j.i.316.1		12	39.11	even	12
507.2.j.i.316.6		12	39.2	even	12
507.2.j.i.361.1		12	39.32	even	12
507.2.j.i.361.6		12	39.20	even	12
1521.2.a.n.1.1		3	1.1	even	1	trivial
1521.2.a.s.1.3		3	13.12	even	2
1521.2.b.k.1351.1		6	13.8	odd	4
1521.2.b.k.1351.6		6	13.5	odd	4
8112.2.a.cg.1.3		3	156.155	even	2
8112.2.a.cp.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69202 q^{2} +5.24698 q^{4} +1.04892 q^{5} +0.554958 q^{7} -8.74094 q^{8} -2.82371 q^{10} -2.91185 q^{11} -1.49396 q^{14} +13.0368 q^{16} +4.85086 q^{17} -0.753020 q^{19} +5.50365 q^{20} +7.83877 q^{22} -5.76271 q^{23} -3.89977 q^{25} +2.91185 q^{28} +1.91185 q^{29} -9.51573 q^{31} -17.6136 q^{32} -13.0586 q^{34} +0.582105 q^{35} +5.75302 q^{37} +2.02715 q^{38} -9.16852 q^{40} -4.91185 q^{41} -11.0978 q^{43} -15.2784 q^{44} +15.5133 q^{46} -0.753020 q^{47} -6.69202 q^{49} +10.4983 q^{50} +7.58211 q^{53} -3.05429 q^{55} -4.85086 q^{56} -5.14675 q^{58} -4.09783 q^{59} -3.42327 q^{61} +25.6165 q^{62} +21.3424 q^{64} -1.87263 q^{67} +25.4523 q^{68} -1.56704 q^{70} +10.5036 q^{71} +10.4765 q^{73} -15.4873 q^{74} -3.95108 q^{76} -1.61596 q^{77} +1.33513 q^{79} +13.6746 q^{80} +13.2228 q^{82} -2.64310 q^{83} +5.08815 q^{85} +29.8756 q^{86} +25.4523 q^{88} -9.92692 q^{89} -30.2368 q^{92} +2.02715 q^{94} -0.789856 q^{95} +17.0737 q^{97} +18.0151 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 11 q^{4} - 6 q^{5} + 2 q^{7} - 12 q^{8} - q^{10} - 5 q^{11} + 5 q^{14} + 11 q^{16} + q^{17} - 7 q^{19} - 15 q^{20} - 9 q^{22} + 11 q^{25} + 5 q^{28} + 2 q^{29} - 16 q^{31} - 22 q^{32}+ \cdots + 29 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 11 * q^4 - 6 * q^5 + 2 * q^7 - 12 * q^8 - q^10 - 5 * q^11 + 5 * q^14 + 11 * q^16 + q^17 - 7 * q^19 - 15 * q^20 - 9 * q^22 + 11 * q^25 + 5 * q^28 + 2 * q^29 - 16 * q^31 - 22 * q^32 - 8 * q^34 - 4 * q^35 + 22 * q^37 + 3 * q^40 - 11 * q^41 - 15 * q^43 - 16 * q^44 - 7 * q^46 - 7 * q^47 - 15 * q^49 + 3 * q^50 + 17 * q^53 + 3 * q^55 - q^56 + 12 * q^58 + 6 * q^59 - 13 * q^61 + 2 * q^62 + 11 * q^67 + 13 * q^68 - 24 * q^70 + 6 * q^73 - 15 * q^74 - 21 * q^76 - 15 * q^77 + 3 * q^79 + 20 * q^80 - 3 * q^82 - 12 * q^83 + 19 * q^85 + 29 * q^86 + 13 * q^88 - q^89 - 7 * q^92 + 21 * q^95 - 5 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more