LMFDB - Embedded newform 513.2.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(1,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("513.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 13 $ x^4 - 8*x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.39417$ of defining polynomial
Character	$\chi$	$=$	513.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39417 q^{2} +3.73205 q^{4} +1.75265 q^{5} +4.73205 q^{7} -4.14682 q^{8} -4.19615 q^{10} +4.14682 q^{11} +3.46410 q^{13} -11.3293 q^{14} +2.46410 q^{16} -1.00000 q^{19} +6.54099 q^{20} -9.92820 q^{22} -6.54099 q^{23} -1.92820 q^{25} -8.29365 q^{26} +17.6603 q^{28} +5.42986 q^{29} -6.66025 q^{31} +2.39417 q^{32} +8.29365 q^{35} -6.19615 q^{37} +2.39417 q^{38} -7.26795 q^{40} -10.6878 q^{41} +0.732051 q^{43} +15.4762 q^{44} +15.6603 q^{46} +7.18251 q^{47} +15.3923 q^{49} +4.61645 q^{50} +12.9282 q^{52} +7.18251 q^{53} +7.26795 q^{55} -19.6230 q^{56} -13.0000 q^{58} +3.03569 q^{59} +3.19615 q^{61} +15.9458 q^{62} -10.6603 q^{64} +6.07137 q^{65} -2.26795 q^{67} -19.8564 q^{70} -6.07137 q^{71} -8.12436 q^{73} +14.8346 q^{74} -3.73205 q^{76} +19.6230 q^{77} -9.00000 q^{79} +4.31872 q^{80} +25.5885 q^{82} +11.9709 q^{83} -1.75265 q^{86} -17.1962 q^{88} -15.9458 q^{89} +16.3923 q^{91} -24.4113 q^{92} -17.1962 q^{94} -1.75265 q^{95} +9.66025 q^{97} -36.8518 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} + 12 q^{7} + 4 q^{10} - 4 q^{16} - 4 q^{19} - 12 q^{22} + 20 q^{25} + 36 q^{28} + 8 q^{31} - 4 q^{37} - 36 q^{40} - 4 q^{43} + 28 q^{46} + 20 q^{49} + 24 q^{52} + 36 q^{55} - 52 q^{58} - 8 q^{61}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q + 8 * q^4 + 12 * q^7 + 4 * q^10 - 4 * q^16 - 4 * q^19 - 12 * q^22 + 20 * q^25 + 36 * q^28 + 8 * q^31 - 4 * q^37 - 36 * q^40 - 4 * q^43 + 28 * q^46 + 20 * q^49 + 24 * q^52 + 36 * q^55 - 52 * q^58 - 8 * q^61 - 8 * q^64 - 16 * q^67 - 24 * q^70 + 16 * q^73 - 8 * q^76 - 36 * q^79 + 40 * q^82 - 48 * q^88 + 24 * q^91 - 48 * q^94 + 4 * q^97

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 $

$2$	−2.39417	−1.69293	−0.846467	−	0.532441i	$-0.821275\pi$
$2$	−2.39417	−1.69293	−0.846467	+	0.532441i	$0.821275\pi$
$3$	0	0
$3$	0	0
$4$	3.73205	1.86603
$5$	1.75265	0.783811	0.391905	−	0.920006i	$-0.371816\pi$
$5$	1.75265	0.783811	0.391905	+	0.920006i	$0.371816\pi$
$6$	0	0
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	−4.14682	−1.46612
$9$	0	0
$10$	−4.19615	−1.32694
$11$	4.14682	1.25031	0.625157	−	0.780499i	$-0.285035\pi$
$11$	4.14682	1.25031	0.625157	+	0.780499i	$0.285035\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	−11.3293	−3.02789
$15$	0	0
$16$	2.46410	0.616025
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	6.54099	1.46261
$21$	0	0
$22$	−9.92820	−2.11670
$23$	−6.54099	−1.36389	−0.681946	−	0.731403i	$-0.738866\pi$
$23$	−6.54099	−1.36389	−0.681946	+	0.731403i	$0.738866\pi$
$24$	0	0
$25$	−1.92820	−0.385641
$26$	−8.29365	−1.62652
$27$	0	0
$28$	17.6603	3.33747
$29$	5.42986	1.00830	0.504150	−	0.863616i	$-0.331806\pi$
$29$	5.42986	1.00830	0.504150	+	0.863616i	$0.331806\pi$
$30$	0	0
$31$	−6.66025	−1.19622	−0.598108	−	0.801415i	$-0.704081\pi$
$31$	−6.66025	−1.19622	−0.598108	+	0.801415i	$0.704081\pi$
$32$	2.39417	0.423233
$33$	0	0
$34$	0	0
$35$	8.29365	1.40188
$36$	0	0
$37$	−6.19615	−1.01864	−0.509321	−	0.860577i	$-0.670103\pi$
$37$	−6.19615	−1.01864	−0.509321	+	0.860577i	$0.670103\pi$
$38$	2.39417	0.388386
$39$	0	0
$40$	−7.26795	−1.14916
$41$	−10.6878	−1.66916	−0.834578	−	0.550889i	$-0.814289\pi$
$41$	−10.6878	−1.66916	−0.834578	+	0.550889i	$0.814289\pi$
$42$	0	0
$43$	0.732051	0.111637	0.0558184	−	0.998441i	$-0.482223\pi$
$43$	0.732051	0.111637	0.0558184	+	0.998441i	$0.482223\pi$
$44$	15.4762	2.33312
$45$	0	0
$46$	15.6603	2.30898
$47$	7.18251	1.04768	0.523838	−	0.851818i	$-0.324500\pi$
$47$	7.18251	1.04768	0.523838	+	0.851818i	$0.324500\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	4.61645	0.652864
$51$	0	0
$52$	12.9282	1.79282
$53$	7.18251	0.986594	0.493297	−	0.869861i	$-0.335792\pi$
$53$	7.18251	0.986594	0.493297	+	0.869861i	$0.335792\pi$
$54$	0	0
$55$	7.26795	0.980010
$56$	−19.6230	−2.62223
$57$	0	0
$58$	−13.0000	−1.70698
$59$	3.03569	0.395213	0.197606	−	0.980281i	$-0.436683\pi$
$59$	3.03569	0.395213	0.197606	+	0.980281i	$0.436683\pi$
$60$	0	0
$61$	3.19615	0.409225	0.204613	−	0.978843i	$-0.434407\pi$
$61$	3.19615	0.409225	0.204613	+	0.978843i	$0.434407\pi$
$62$	15.9458	2.02512
$63$	0	0
$64$	−10.6603	−1.33253
$65$	6.07137	0.753061
$66$	0	0
$67$	−2.26795	−0.277074	−0.138537	−	0.990357i	$-0.544240\pi$
$67$	−2.26795	−0.277074	−0.138537	+	0.990357i	$0.544240\pi$
$68$	0	0
$69$	0	0
$70$	−19.8564	−2.37329
$71$	−6.07137	−0.720539	−0.360270	−	0.932848i	$-0.617315\pi$
$71$	−6.07137	−0.720539	−0.360270	+	0.932848i	$0.617315\pi$
$72$	0	0
$73$	−8.12436	−0.950884	−0.475442	−	0.879747i	$-0.657712\pi$
$73$	−8.12436	−0.950884	−0.475442	+	0.879747i	$0.657712\pi$
$74$	14.8346	1.72449
$75$	0	0
$76$	−3.73205	−0.428096
$77$	19.6230	2.23625
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	4.31872	0.482847
$81$	0	0
$82$	25.5885	2.82577
$83$	11.9709	1.31397	0.656986	−	0.753903i	$-0.271831\pi$
$83$	11.9709	1.31397	0.656986	+	0.753903i	$0.271831\pi$
$84$	0	0
$85$	0	0
$86$	−1.75265	−0.188994
$87$	0	0
$88$	−17.1962	−1.83312
$89$	−15.9458	−1.69025	−0.845125	−	0.534569i	$-0.820474\pi$
$89$	−15.9458	−1.69025	−0.845125	+	0.534569i	$0.820474\pi$
$90$	0	0
$91$	16.3923	1.71838
$92$	−24.4113	−2.54506
$93$	0	0
$94$	−17.1962	−1.77365
$95$	−1.75265	−0.179819
$96$	0	0
$97$	9.66025	0.980850	0.490425	−	0.871483i	$-0.336842\pi$
$97$	9.66025	0.980850	0.490425	+	0.871483i	$0.336842\pi$
$98$	−36.8518	−3.72259
$99$	0	0
$100$	−7.19615	−0.719615
$101$	−10.0463	−0.999645	−0.499822	−	0.866128i	$-0.666601\pi$
$101$	−10.0463	−0.999645	−0.499822	+	0.866128i	$0.666601\pi$
$102$	0	0
$103$	11.4641	1.12959	0.564796	−	0.825231i	$-0.308955\pi$
$103$	11.4641	1.12959	0.564796	+	0.825231i	$0.308955\pi$
$104$	−14.3650	−1.40861
$105$	0	0
$106$	−17.1962	−1.67024
$107$	−12.6124	−1.21928	−0.609642	−	0.792677i	$-0.708687\pi$
$107$	−12.6124	−1.21928	−0.609642	+	0.792677i	$0.708687\pi$
$108$	0	0
$109$	−5.26795	−0.504578	−0.252289	−	0.967652i	$-0.581183\pi$
$109$	−5.26795	−0.504578	−0.252289	+	0.967652i	$0.581183\pi$
$110$	−17.4007	−1.65909
$111$	0	0
$112$	11.6603	1.10179
$113$	18.5118	1.74145	0.870724	−	0.491772i	$-0.163651\pi$
$113$	18.5118	1.74145	0.870724	+	0.491772i	$0.163651\pi$
$114$	0	0
$115$	−11.4641	−1.06903
$116$	20.2645	1.88151
$117$	0	0
$118$	−7.26795	−0.669069
$119$	0	0
$120$	0	0
$121$	6.19615	0.563287
$122$	−7.65213	−0.692792
$123$	0	0
$124$	−24.8564	−2.23217
$125$	−12.1427	−1.08608
$126$	0	0
$127$	−2.92820	−0.259836	−0.129918	−	0.991525i	$-0.541471\pi$
$127$	−2.92820	−0.259836	−0.129918	+	0.991525i	$0.541471\pi$
$128$	20.7341	1.83265
$129$	0	0
$130$	−14.5359	−1.27488
$131$	−16.1177	−1.40821	−0.704104	−	0.710097i	$-0.748651\pi$
$131$	−16.1177	−1.40821	−0.704104	+	0.710097i	$0.748651\pi$
$132$	0	0
$133$	−4.73205	−0.410321
$134$	5.42986	0.469068
$135$	0	0
$136$	0	0
$137$	−6.07137	−0.518712	−0.259356	−	0.965782i	$-0.583510\pi$
$137$	−6.07137	−0.518712	−0.259356	+	0.965782i	$0.583510\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	30.9523	2.61595
$141$	0	0
$142$	14.5359	1.21983
$143$	14.3650	1.20126
$144$	0	0
$145$	9.51666	0.790316
$146$	19.4511	1.60978
$147$	0	0
$148$	−23.1244	−1.90081
$149$	−14.3650	−1.17683	−0.588414	−	0.808560i	$-0.700248\pi$
$149$	−14.3650	−1.17683	−0.588414	+	0.808560i	$0.700248\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	4.14682	0.336352
$153$	0	0
$154$	−46.9808	−3.78582
$155$	−11.6731	−0.937608
$156$	0	0
$157$	−11.5885	−0.924860	−0.462430	−	0.886656i	$-0.653022\pi$
$157$	−11.5885	−0.924860	−0.462430	+	0.886656i	$0.653022\pi$
$158$	21.5475	1.71423
$159$	0	0
$160$	4.19615	0.331735
$161$	−30.9523	−2.43938
$162$	0	0
$163$	21.2679	1.66583	0.832917	−	0.553398i	$-0.186669\pi$
$163$	21.2679	1.66583	0.832917	+	0.553398i	$0.186669\pi$
$164$	−39.8875	−3.11469
$165$	0	0
$166$	−28.6603	−2.22447
$167$	−2.22228	−0.171965	−0.0859825	−	0.996297i	$-0.527403\pi$
$167$	−2.22228	−0.171965	−0.0859825	+	0.996297i	$0.527403\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	2.73205	0.208317
$173$	−5.89948	−0.448529	−0.224265	−	0.974528i	$-0.571998\pi$
$173$	−5.89948	−0.448529	−0.224265	+	0.974528i	$0.571998\pi$
$174$	0	0
$175$	−9.12436	−0.689736
$176$	10.2182	0.770226
$177$	0	0
$178$	38.1769	2.86148
$179$	14.8346	1.10879	0.554397	−	0.832253i	$-0.312949\pi$
$179$	14.8346	1.10879	0.554397	+	0.832253i	$0.312949\pi$
$180$	0	0
$181$	−4.33975	−0.322571	−0.161285	−	0.986908i	$-0.551564\pi$
$181$	−4.33975	−0.322571	−0.161285	+	0.986908i	$0.551564\pi$
$182$	−39.2460	−2.90910
$183$	0	0
$184$	27.1244	1.99963
$185$	−10.8597	−0.798422
$186$	0	0
$187$	0	0
$188$	26.8055	1.95499
$189$	0	0
$190$	4.19615	0.304421
$191$	12.4405	0.900161	0.450081	−	0.892988i	$-0.351395\pi$
$191$	12.4405	0.900161	0.450081	+	0.892988i	$0.351395\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−23.1283	−1.66051
$195$	0	0
$196$	57.4449	4.10320
$197$	2.56606	0.182824	0.0914122	−	0.995813i	$-0.470862\pi$
$197$	2.56606	0.182824	0.0914122	+	0.995813i	$0.470862\pi$
$198$	0	0
$199$	12.5359	0.888646	0.444323	−	0.895867i	$-0.353444\pi$
$199$	12.5359	0.888646	0.444323	+	0.895867i	$0.353444\pi$
$200$	7.99592	0.565397
$201$	0	0
$202$	24.0526	1.69233
$203$	25.6944	1.80339
$204$	0	0
$205$	−18.7321	−1.30830
$206$	−27.4470	−1.91232
$207$	0	0
$208$	8.53590	0.591858
$209$	−4.14682	−0.286842
$210$	0	0
$211$	14.1244	0.972361	0.486180	−	0.873858i	$-0.338390\pi$
$211$	14.1244	0.972361	0.486180	+	0.873858i	$0.338390\pi$
$212$	26.8055	1.84101
$213$	0	0
$214$	30.1962	2.06417
$215$	1.28303	0.0875021
$216$	0	0
$217$	−31.5167	−2.13949
$218$	12.6124	0.854217
$219$	0	0
$220$	27.1244	1.82872
$221$	0	0
$222$	0	0
$223$	22.4641	1.50431	0.752154	−	0.658988i	$-0.229015\pi$
$223$	22.4641	1.50431	0.752154	+	0.658988i	$0.229015\pi$
$224$	11.3293	0.756973
$225$	0	0
$226$	−44.3205	−2.94816
$227$	8.29365	0.550469	0.275234	−	0.961377i	$-0.411245\pi$
$227$	8.29365	0.550469	0.275234	+	0.961377i	$0.411245\pi$
$228$	0	0
$229$	−14.3205	−0.946326	−0.473163	−	0.880975i	$-0.656888\pi$
$229$	−14.3205	−0.946326	−0.473163	+	0.880975i	$0.656888\pi$
$230$	27.4470	1.80980
$231$	0	0
$232$	−22.5167	−1.47829
$233$	10.0463	0.658155	0.329078	−	0.944303i	$-0.393262\pi$
$233$	10.0463	0.658155	0.329078	+	0.944303i	$0.393262\pi$
$234$	0	0
$235$	12.5885	0.821180
$236$	11.3293	0.737477
$237$	0	0
$238$	0	0
$239$	−15.4762	−1.00107	−0.500535	−	0.865716i	$-0.666863\pi$
$239$	−15.4762	−1.00107	−0.500535	+	0.865716i	$0.666863\pi$
$240$	0	0
$241$	8.19615	0.527961	0.263980	−	0.964528i	$-0.414965\pi$
$241$	8.19615	0.527961	0.263980	+	0.964528i	$0.414965\pi$
$242$	−14.8346	−0.953607
$243$	0	0
$244$	11.9282	0.763625
$245$	26.9774	1.72352
$246$	0	0
$247$	−3.46410	−0.220416
$248$	27.6189	1.75380
$249$	0	0
$250$	29.0718	1.83866
$251$	−3.03569	−0.191611	−0.0958054	−	0.995400i	$-0.530543\pi$
$251$	−3.03569	−0.191611	−0.0958054	+	0.995400i	$0.530543\pi$
$252$	0	0
$253$	−27.1244	−1.70529
$254$	7.01062	0.439885
$255$	0	0
$256$	−28.3205	−1.77003
$257$	0.469622	0.0292942	0.0146471	−	0.999893i	$-0.495338\pi$
$257$	0.469622	0.0292942	0.0146471	+	0.999893i	$0.495338\pi$
$258$	0	0
$259$	−29.3205	−1.82189
$260$	22.6587	1.40523
$261$	0	0
$262$	38.5885	2.38400
$263$	−14.6627	−0.904144	−0.452072	−	0.891981i	$-0.649315\pi$
$263$	−14.6627	−0.904144	−0.452072	+	0.891981i	$0.649315\pi$
$264$	0	0
$265$	12.5885	0.773303
$266$	11.3293	0.694646
$267$	0	0
$268$	−8.46410	−0.517027
$269$	13.5516	0.826256	0.413128	−	0.910673i	$-0.364436\pi$
$269$	13.5516	0.826256	0.413128	+	0.910673i	$0.364436\pi$
$270$	0	0
$271$	0.339746	0.0206381	0.0103190	−	0.999947i	$-0.496715\pi$
$271$	0.339746	0.0206381	0.0103190	+	0.999947i	$0.496715\pi$
$272$	0	0
$273$	0	0
$274$	14.5359	0.878146
$275$	−7.99592	−0.482172
$276$	0	0
$277$	−21.3923	−1.28534	−0.642670	−	0.766144i	$-0.722173\pi$
$277$	−21.3923	−1.28534	−0.642670	+	0.766144i	$0.722173\pi$
$278$	−23.9417	−1.43593
$279$	0	0
$280$	−34.3923	−2.05533
$281$	25.3506	1.51229	0.756144	−	0.654405i	$-0.227081\pi$
$281$	25.3506	1.51229	0.756144	+	0.654405i	$0.227081\pi$
$282$	0	0
$283$	28.9282	1.71960	0.859802	−	0.510628i	$-0.170587\pi$
$283$	28.9282	1.71960	0.859802	+	0.510628i	$0.170587\pi$
$284$	−22.6587	−1.34454
$285$	0	0
$286$	−34.3923	−2.03366
$287$	−50.5753	−2.98537
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−22.7845	−1.33795
$291$	0	0
$292$	−30.3205	−1.77437
$293$	−13.7235	−0.801736	−0.400868	−	0.916136i	$-0.631291\pi$
$293$	−13.7235	−0.801736	−0.400868	+	0.916136i	$0.631291\pi$
$294$	0	0
$295$	5.32051	0.309772
$296$	25.6944	1.49345
$297$	0	0
$298$	34.3923	1.99229
$299$	−22.6587	−1.31038
$300$	0	0
$301$	3.46410	0.199667
$302$	−16.7592	−0.964383
$303$	0	0
$304$	−2.46410	−0.141326
$305$	5.60175	0.320755
$306$	0	0
$307$	−3.00000	−0.171219	−0.0856095	−	0.996329i	$-0.527284\pi$
$307$	−3.00000	−0.171219	−0.0856095	+	0.996329i	$0.527284\pi$
$308$	73.2340	4.17289
$309$	0	0
$310$	27.9474	1.58731
$311$	8.93516	0.506667	0.253333	−	0.967379i	$-0.418473\pi$
$311$	8.93516	0.506667	0.253333	+	0.967379i	$0.418473\pi$
$312$	0	0
$313$	−21.1962	−1.19808	−0.599039	−	0.800720i	$-0.704450\pi$
$313$	−21.1962	−1.19808	−0.599039	+	0.800720i	$0.704450\pi$
$314$	27.7447	1.56573
$315$	0	0
$316$	−33.5885	−1.88950
$317$	−11.3293	−0.636319	−0.318159	−	0.948037i	$-0.603065\pi$
$317$	−11.3293	−0.636319	−0.318159	+	0.948037i	$0.603065\pi$
$318$	0	0
$319$	22.5167	1.26069
$320$	−18.6837	−1.04445
$321$	0	0
$322$	74.1051	4.12972
$323$	0	0
$324$	0	0
$325$	−6.67949	−0.370512
$326$	−50.9191	−2.82015
$327$	0	0
$328$	44.3205	2.44719
$329$	33.9880	1.87382
$330$	0	0
$331$	−22.3923	−1.23079	−0.615396	−	0.788218i	$-0.711004\pi$
$331$	−22.3923	−1.23079	−0.615396	+	0.788218i	$0.711004\pi$
$332$	44.6758	2.45190
$333$	0	0
$334$	5.32051	0.291125
$335$	−3.97493	−0.217174
$336$	0	0
$337$	−12.1962	−0.664367	−0.332183	−	0.943215i	$-0.607785\pi$
$337$	−12.1962	−0.664367	−0.332183	+	0.943215i	$0.607785\pi$
$338$	2.39417	0.130226
$339$	0	0
$340$	0	0
$341$	−27.6189	−1.49565
$342$	0	0
$343$	39.7128	2.14429
$344$	−3.03569	−0.163673
$345$	0	0
$346$	14.1244	0.759330
$347$	−36.2103	−1.94387	−0.971935	−	0.235250i	$-0.924409\pi$
$347$	−36.2103	−1.94387	−0.971935	+	0.235250i	$0.924409\pi$
$348$	0	0
$349$	−36.1769	−1.93651	−0.968253	−	0.249973i	$-0.919578\pi$
$349$	−36.1769	−1.93651	−0.968253	+	0.249973i	$0.919578\pi$
$350$	21.8453	1.16768
$351$	0	0
$352$	9.92820	0.529175
$353$	−13.0820	−0.696284	−0.348142	−	0.937442i	$-0.613187\pi$
$353$	−13.0820	−0.696284	−0.348142	+	0.937442i	$0.613187\pi$
$354$	0	0
$355$	−10.6410	−0.564766
$356$	−59.5105	−3.15405
$357$	0	0
$358$	−35.5167	−1.87711
$359$	−2.69190	−0.142073	−0.0710365	−	0.997474i	$-0.522631\pi$
$359$	−2.69190	−0.142073	−0.0710365	+	0.997474i	$0.522631\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	10.3901	0.546091
$363$	0	0
$364$	61.1769	3.20654
$365$	−14.2392	−0.745313
$366$	0	0
$367$	10.5359	0.549969	0.274985	−	0.961449i	$-0.411327\pi$
$367$	10.5359	0.549969	0.274985	+	0.961449i	$0.411327\pi$
$368$	−16.1177	−0.840192
$369$	0	0
$370$	26.0000	1.35168
$371$	33.9880	1.76457
$372$	0	0
$373$	−15.6603	−0.810857	−0.405429	−	0.914127i	$-0.632878\pi$
$373$	−15.6603	−0.810857	−0.405429	+	0.914127i	$0.632878\pi$
$374$	0	0
$375$	0	0
$376$	−29.7846	−1.53602
$377$	18.8096	0.968742
$378$	0	0
$379$	−16.8564	−0.865855	−0.432928	−	0.901429i	$-0.642519\pi$
$379$	−16.8564	−0.865855	−0.432928	+	0.901429i	$0.642519\pi$
$380$	−6.54099	−0.335546
$381$	0	0
$382$	−29.7846	−1.52391
$383$	12.6124	0.644462	0.322231	−	0.946661i	$-0.395567\pi$
$383$	12.6124	0.644462	0.322231	+	0.946661i	$0.395567\pi$
$384$	0	0
$385$	34.3923	1.75279
$386$	−23.9417	−1.21860
$387$	0	0
$388$	36.0526	1.83029
$389$	9.10706	0.461746	0.230873	−	0.972984i	$-0.425842\pi$
$389$	9.10706	0.461746	0.230873	+	0.972984i	$0.425842\pi$
$390$	0	0
$391$	0	0
$392$	−63.8292	−3.22386
$393$	0	0
$394$	−6.14359	−0.309510
$395$	−15.7739	−0.793670
$396$	0	0
$397$	24.8564	1.24751	0.623754	−	0.781621i	$-0.285607\pi$
$397$	24.8564	1.24751	0.623754	+	0.781621i	$0.285607\pi$
$398$	−30.0131	−1.50442
$399$	0	0
$400$	−4.75129	−0.237564
$401$	−1.75265	−0.0875234	−0.0437617	−	0.999042i	$-0.513934\pi$
$401$	−1.75265	−0.0875234	−0.0437617	+	0.999042i	$0.513934\pi$
$402$	0	0
$403$	−23.0718	−1.14929
$404$	−37.4933	−1.86536
$405$	0	0
$406$	−61.5167	−3.05302
$407$	−25.6944	−1.27362
$408$	0	0
$409$	6.87564	0.339979	0.169989	−	0.985446i	$-0.445627\pi$
$409$	6.87564	0.339979	0.169989	+	0.985446i	$0.445627\pi$
$410$	44.8477	2.21487
$411$	0	0
$412$	42.7846	2.10785
$413$	14.3650	0.706856
$414$	0	0
$415$	20.9808	1.02991
$416$	8.29365	0.406630
$417$	0	0
$418$	9.92820	0.485604
$419$	2.39417	0.116963	0.0584814	−	0.998288i	$-0.481374\pi$
$419$	2.39417	0.116963	0.0584814	+	0.998288i	$0.481374\pi$
$420$	0	0
$421$	5.46410	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$422$	−33.8161	−1.64614
$423$	0	0
$424$	−29.7846	−1.44647
$425$	0	0
$426$	0	0
$427$	15.1244	0.731919
$428$	−47.0700	−2.27521
$429$	0	0
$430$	−3.07180	−0.148135
$431$	−13.5516	−0.652758	−0.326379	−	0.945239i	$-0.605829\pi$
$431$	−13.5516	−0.652758	−0.326379	+	0.945239i	$0.605829\pi$
$432$	0	0
$433$	−33.5167	−1.61071	−0.805354	−	0.592794i	$-0.798025\pi$
$433$	−33.5167	−1.61071	−0.805354	+	0.592794i	$0.798025\pi$
$434$	75.4562	3.62202
$435$	0	0
$436$	−19.6603	−0.941555
$437$	6.54099	0.312898
$438$	0	0
$439$	27.7846	1.32609	0.663044	−	0.748581i	$-0.269264\pi$
$439$	27.7846	1.32609	0.663044	+	0.748581i	$0.269264\pi$
$440$	−30.1389	−1.43682
$441$	0	0
$442$	0	0
$443$	29.8412	1.41780	0.708899	−	0.705310i	$-0.249192\pi$
$443$	29.8412	1.41780	0.708899	+	0.705310i	$0.249192\pi$
$444$	0	0
$445$	−27.9474	−1.32484
$446$	−53.7829	−2.54669
$447$	0	0
$448$	−50.4449	−2.38330
$449$	34.6295	1.63427	0.817134	−	0.576448i	$-0.195562\pi$
$449$	34.6295	1.63427	0.817134	+	0.576448i	$0.195562\pi$
$450$	0	0
$451$	−44.3205	−2.08697
$452$	69.0871	3.24959
$453$	0	0
$454$	−19.8564	−0.931907
$455$	28.7300	1.34689
$456$	0	0
$457$	15.5885	0.729197	0.364599	−	0.931165i	$-0.381206\pi$
$457$	15.5885	0.729197	0.364599	+	0.931165i	$0.381206\pi$
$458$	34.2857	1.60207
$459$	0	0
$460$	−42.7846	−1.99484
$461$	−16.5873	−0.772547	−0.386274	−	0.922384i	$-0.626238\pi$
$461$	−16.5873	−0.772547	−0.386274	+	0.922384i	$0.626238\pi$
$462$	0	0
$463$	18.3923	0.854763	0.427381	−	0.904071i	$-0.359436\pi$
$463$	18.3923	0.854763	0.427381	+	0.904071i	$0.359436\pi$
$464$	13.3797	0.621138
$465$	0	0
$466$	−24.0526	−1.11421
$467$	−6.24327	−0.288904	−0.144452	−	0.989512i	$-0.546142\pi$
$467$	−6.24327	−0.288904	−0.144452	+	0.989512i	$0.546142\pi$
$468$	0	0
$469$	−10.7321	−0.495560
$470$	−30.1389	−1.39020
$471$	0	0
$472$	−12.5885	−0.579431
$473$	3.03569	0.139581
$474$	0	0
$475$	1.92820	0.0884720
$476$	0	0
$477$	0	0
$478$	37.0526	1.69474
$479$	14.8346	0.677812	0.338906	−	0.940820i	$-0.389943\pi$
$479$	14.8346	0.677812	0.338906	+	0.940820i	$0.389943\pi$
$480$	0	0
$481$	−21.4641	−0.978679
$482$	−19.6230	−0.893802
$483$	0	0
$484$	23.1244	1.05111
$485$	16.9311	0.768801
$486$	0	0
$487$	11.3397	0.513853	0.256926	−	0.966431i	$-0.417290\pi$
$487$	11.3397	0.513853	0.256926	+	0.966431i	$0.417290\pi$
$488$	−13.2539	−0.599975
$489$	0	0
$490$	−64.5885	−2.91781
$491$	−27.2751	−1.23091	−0.615454	−	0.788173i	$-0.711027\pi$
$491$	−27.2751	−1.23091	−0.615454	+	0.788173i	$0.711027\pi$
$492$	0	0
$493$	0	0
$494$	8.29365	0.373149
$495$	0	0
$496$	−16.4115	−0.736900
$497$	−28.7300	−1.28872
$498$	0	0
$499$	14.3923	0.644288	0.322144	−	0.946691i	$-0.395596\pi$
$499$	14.3923	0.644288	0.322144	+	0.946691i	$0.395596\pi$
$500$	−45.3173	−2.02665
$501$	0	0
$502$	7.26795	0.324384
$503$	6.36910	0.283984	0.141992	−	0.989868i	$-0.454649\pi$
$503$	6.36910	0.283984	0.141992	+	0.989868i	$0.454649\pi$
$504$	0	0
$505$	−17.6077	−0.783532
$506$	64.9403	2.88695
$507$	0	0
$508$	−10.9282	−0.484861
$509$	2.39417	0.106120	0.0530599	−	0.998591i	$-0.483103\pi$
$509$	2.39417	0.106120	0.0530599	+	0.998591i	$0.483103\pi$
$510$	0	0
$511$	−38.4449	−1.70070
$512$	26.3359	1.16389
$513$	0	0
$514$	−1.12436	−0.0495932
$515$	20.0926	0.885386
$516$	0	0
$517$	29.7846	1.30993
$518$	70.1983	3.08434
$519$	0	0
$520$	−25.1769	−1.10408
$521$	4.31872	0.189206	0.0946032	−	0.995515i	$-0.469842\pi$
$521$	4.31872	0.189206	0.0946032	+	0.995515i	$0.469842\pi$
$522$	0	0
$523$	−26.5167	−1.15949	−0.579746	−	0.814797i	$-0.696848\pi$
$523$	−26.5167	−1.15949	−0.579746	+	0.814797i	$0.696848\pi$
$524$	−60.1520	−2.62775
$525$	0	0
$526$	35.1051	1.53066
$527$	0	0
$528$	0	0
$529$	19.7846	0.860200
$530$	−30.1389	−1.30915
$531$	0	0
$532$	−17.6603	−0.765669
$533$	−37.0237	−1.60367
$534$	0	0
$535$	−22.1051	−0.955688
$536$	9.40479	0.406225
$537$	0	0
$538$	−32.4449	−1.39880
$539$	63.8292	2.74932
$540$	0	0
$541$	7.19615	0.309387	0.154693	−	0.987963i	$-0.450561\pi$
$541$	7.19615	0.309387	0.154693	+	0.987963i	$0.450561\pi$
$542$	−0.813410	−0.0349389
$543$	0	0
$544$	0	0
$545$	−9.23289	−0.395494
$546$	0	0
$547$	−33.8564	−1.44760	−0.723798	−	0.690012i	$-0.757605\pi$
$547$	−33.8564	−1.44760	−0.723798	+	0.690012i	$0.757605\pi$
$548$	−22.6587	−0.967930
$549$	0	0
$550$	19.1436	0.816286
$551$	−5.42986	−0.231320
$552$	0	0
$553$	−42.5885	−1.81105
$554$	51.2168	2.17599
$555$	0	0
$556$	37.3205	1.58274
$557$	44.5039	1.88569	0.942846	−	0.333229i	$-0.108138\pi$
$557$	44.5039	1.88569	0.942846	+	0.333229i	$0.108138\pi$
$558$	0	0
$559$	2.53590	0.107257
$560$	20.4364	0.863595
$561$	0	0
$562$	−60.6936	−2.56020
$563$	41.3424	1.74237	0.871187	−	0.490951i	$-0.163350\pi$
$563$	41.3424	1.74237	0.871187	+	0.490951i	$0.163350\pi$
$564$	0	0
$565$	32.4449	1.36497
$566$	−69.2590	−2.91117
$567$	0	0
$568$	25.1769	1.05640
$569$	21.8453	0.915801	0.457900	−	0.889003i	$-0.348602\pi$
$569$	21.8453	0.915801	0.457900	+	0.889003i	$0.348602\pi$
$570$	0	0
$571$	6.05256	0.253292	0.126646	−	0.991948i	$-0.459579\pi$
$571$	6.05256	0.253292	0.126646	+	0.991948i	$0.459579\pi$
$572$	53.6110	2.24159
$573$	0	0
$574$	121.086	5.05403
$575$	12.6124	0.525972
$576$	0	0
$577$	−39.4641	−1.64291	−0.821456	−	0.570272i	$-0.806838\pi$
$577$	−39.4641	−1.64291	−0.821456	+	0.570272i	$0.806838\pi$
$578$	40.7009	1.69293
$579$	0	0
$580$	35.5167	1.47475
$581$	56.6467	2.35010
$582$	0	0
$583$	29.7846	1.23355
$584$	33.6903	1.39411
$585$	0	0
$586$	32.8564	1.35729
$587$	10.3901	0.428845	0.214422	−	0.976741i	$-0.431213\pi$
$587$	10.3901	0.428845	0.214422	+	0.976741i	$0.431213\pi$
$588$	0	0
$589$	6.66025	0.274431
$590$	−12.7382	−0.524423
$591$	0	0
$592$	−15.2679	−0.627509
$593$	−3.03569	−0.124661	−0.0623303	−	0.998056i	$-0.519853\pi$
$593$	−3.03569	−0.124661	−0.0623303	+	0.998056i	$0.519853\pi$
$594$	0	0
$595$	0	0
$596$	−53.6110	−2.19599
$597$	0	0
$598$	54.2487	2.21839
$599$	7.48024	0.305634	0.152817	−	0.988254i	$-0.451165\pi$
$599$	7.48024	0.305634	0.152817	+	0.988254i	$0.451165\pi$
$600$	0	0
$601$	20.9808	0.855823	0.427912	−	0.903821i	$-0.359249\pi$
$601$	20.9808	0.855823	0.427912	+	0.903821i	$0.359249\pi$
$602$	−8.29365	−0.338024
$603$	0	0
$604$	26.1244	1.06298
$605$	10.8597	0.441510
$606$	0	0
$607$	−29.4449	−1.19513	−0.597565	−	0.801820i	$-0.703865\pi$
$607$	−29.4449	−1.19513	−0.597565	+	0.801820i	$0.703865\pi$
$608$	−2.39417	−0.0970964
$609$	0	0
$610$	−13.4115	−0.543017
$611$	24.8809	1.00658
$612$	0	0
$613$	30.9282	1.24918	0.624589	−	0.780953i	$-0.285266\pi$
$613$	30.9282	1.24918	0.624589	+	0.780953i	$0.285266\pi$
$614$	7.18251	0.289863
$615$	0	0
$616$	−81.3731	−3.27861
$617$	−38.7763	−1.56108	−0.780538	−	0.625108i	$-0.785055\pi$
$617$	−38.7763	−1.56108	−0.780538	+	0.625108i	$0.785055\pi$
$618$	0	0
$619$	−7.46410	−0.300008	−0.150004	−	0.988685i	$-0.547929\pi$
$619$	−7.46410	−0.300008	−0.150004	+	0.988685i	$0.547929\pi$
$620$	−43.5647	−1.74960
$621$	0	0
$622$	−21.3923	−0.857753
$623$	−75.4562	−3.02309
$624$	0	0
$625$	−11.6410	−0.465641
$626$	50.7472	2.02827
$627$	0	0
$628$	−43.2487	−1.72581
$629$	0	0
$630$	0	0
$631$	−7.07180	−0.281524	−0.140762	−	0.990043i	$-0.544955\pi$
$631$	−7.07180	−0.281524	−0.140762	+	0.990043i	$0.544955\pi$
$632$	37.3214	1.48457
$633$	0	0
$634$	27.1244	1.07725
$635$	−5.13213	−0.203662
$636$	0	0
$637$	53.3205	2.11264
$638$	−53.9087	−2.13427
$639$	0	0
$640$	36.3397	1.43645
$641$	3.20758	0.126692	0.0633459	−	0.997992i	$-0.479823\pi$
$641$	3.20758	0.126692	0.0633459	+	0.997992i	$0.479823\pi$
$642$	0	0
$643$	−23.5167	−0.927407	−0.463703	−	0.885991i	$-0.653480\pi$
$643$	−23.5167	−0.927407	−0.463703	+	0.885991i	$0.653480\pi$
$644$	−115.516	−4.55195
$645$	0	0
$646$	0	0
$647$	3.33341	0.131050	0.0655250	−	0.997851i	$-0.479128\pi$
$647$	3.33341	0.131050	0.0655250	+	0.997851i	$0.479128\pi$
$648$	0	0
$649$	12.5885	0.494140
$650$	15.9918	0.627252
$651$	0	0
$652$	79.3731	3.10849
$653$	−25.8202	−1.01042	−0.505211	−	0.862996i	$-0.668585\pi$
$653$	−25.8202	−1.01042	−0.505211	+	0.862996i	$0.668585\pi$
$654$	0	0
$655$	−28.2487	−1.10377
$656$	−26.3359	−1.02824
$657$	0	0
$658$	−81.3731	−3.17225
$659$	−35.2710	−1.37396	−0.686982	−	0.726674i	$-0.741065\pi$
$659$	−35.2710	−1.37396	−0.686982	+	0.726674i	$0.741065\pi$
$660$	0	0
$661$	18.2487	0.709793	0.354896	−	0.934906i	$-0.384516\pi$
$661$	18.2487	0.709793	0.354896	+	0.934906i	$0.384516\pi$
$662$	53.6110	2.08365
$663$	0	0
$664$	−49.6410	−1.92645
$665$	−8.29365	−0.321614
$666$	0	0
$667$	−35.5167	−1.37521
$668$	−8.29365	−0.320891
$669$	0	0
$670$	9.51666	0.367661
$671$	13.2539	0.511660
$672$	0	0
$673$	−17.8564	−0.688314	−0.344157	−	0.938912i	$-0.611835\pi$
$673$	−17.8564	−0.688314	−0.344157	+	0.938912i	$0.611835\pi$
$674$	29.1997	1.12473
$675$	0	0
$676$	−3.73205	−0.143540
$677$	−7.82403	−0.300702	−0.150351	−	0.988633i	$-0.548040\pi$
$677$	−7.82403	−0.300702	−0.150351	+	0.988633i	$0.548040\pi$
$678$	0	0
$679$	45.7128	1.75430
$680$	0	0
$681$	0	0
$682$	66.1244	2.53203
$683$	13.5516	0.518538	0.259269	−	0.965805i	$-0.416518\pi$
$683$	13.5516	0.518538	0.259269	+	0.965805i	$0.416518\pi$
$684$	0	0
$685$	−10.6410	−0.406572
$686$	−95.0792	−3.63014
$687$	0	0
$688$	1.80385	0.0687710
$689$	24.8809	0.947889
$690$	0	0
$691$	6.05256	0.230250	0.115125	−	0.993351i	$-0.463273\pi$
$691$	6.05256	0.230250	0.115125	+	0.993351i	$0.463273\pi$
$692$	−22.0172	−0.836967
$693$	0	0
$694$	86.6936	3.29084
$695$	17.5265	0.664820
$696$	0	0
$697$	0	0
$698$	86.6137	3.27838
$699$	0	0
$700$	−34.0526	−1.28707
$701$	7.48024	0.282525	0.141262	−	0.989972i	$-0.454884\pi$
$701$	7.48024	0.282525	0.141262	+	0.989972i	$0.454884\pi$
$702$	0	0
$703$	6.19615	0.233692
$704$	−44.2062	−1.66608
$705$	0	0
$706$	31.3205	1.17876
$707$	−47.5396	−1.78791
$708$	0	0
$709$	−25.7128	−0.965665	−0.482832	−	0.875713i	$-0.660392\pi$
$709$	−25.7128	−0.965665	−0.482832	+	0.875713i	$0.660392\pi$
$710$	25.4764	0.956112
$711$	0	0
$712$	66.1244	2.47811
$713$	43.5647	1.63151
$714$	0	0
$715$	25.1769	0.941563
$716$	55.3636	2.06904
$717$	0	0
$718$	6.44486	0.240520
$719$	−10.6878	−0.398588	−0.199294	−	0.979940i	$-0.563865\pi$
$719$	−10.6878	−0.398588	−0.199294	+	0.979940i	$0.563865\pi$
$720$	0	0
$721$	54.2487	2.02033
$722$	−2.39417	−0.0891018
$723$	0	0
$724$	−16.1962	−0.601925
$725$	−10.4699	−0.388841
$726$	0	0
$727$	−10.2487	−0.380104	−0.190052	−	0.981774i	$-0.560866\pi$
$727$	−10.2487	−0.380104	−0.190052	+	0.981774i	$0.560866\pi$
$728$	−67.9760	−2.51936
$729$	0	0
$730$	34.0910	1.26177
$731$	0	0
$732$	0	0
$733$	0.660254	0.0243870	0.0121935	−	0.999926i	$-0.496119\pi$
$733$	0.660254	0.0243870	0.0121935	+	0.999926i	$0.496119\pi$
$734$	−25.2247	−0.931062
$735$	0	0
$736$	−15.6603	−0.577245
$737$	−9.40479	−0.346430
$738$	0	0
$739$	39.3731	1.44836	0.724181	−	0.689610i	$-0.242218\pi$
$739$	39.3731	1.44836	0.724181	+	0.689610i	$0.242218\pi$
$740$	−40.5290	−1.48988
$741$	0	0
$742$	−81.3731	−2.98730
$743$	20.4364	0.749739	0.374869	−	0.927078i	$-0.377688\pi$
$743$	20.4364	0.749739	0.374869	+	0.927078i	$0.377688\pi$
$744$	0	0
$745$	−25.1769	−0.922411
$746$	37.4933	1.37273
$747$	0	0
$748$	0	0
$749$	−59.6824	−2.18075
$750$	0	0
$751$	−35.4641	−1.29410	−0.647052	−	0.762446i	$-0.723998\pi$
$751$	−35.4641	−1.29410	−0.647052	+	0.762446i	$0.723998\pi$
$752$	17.6984	0.645396
$753$	0	0
$754$	−45.0333	−1.64002
$755$	12.2686	0.446499
$756$	0	0
$757$	−0.0717968	−0.00260950	−0.00130475	−	0.999999i	$-0.500415\pi$
$757$	−0.0717968	−0.00260950	−0.00130475	+	0.999999i	$0.500415\pi$
$758$	40.3571	1.46584
$759$	0	0
$760$	7.26795	0.263636
$761$	−45.6611	−1.65521	−0.827607	−	0.561308i	$-0.810298\pi$
$761$	−45.6611	−1.65521	−0.827607	+	0.561308i	$0.810298\pi$
$762$	0	0
$763$	−24.9282	−0.902462
$764$	46.4285	1.67972
$765$	0	0
$766$	−30.1962	−1.09103
$767$	10.5159	0.379708
$768$	0	0
$769$	−26.7846	−0.965878	−0.482939	−	0.875654i	$-0.660431\pi$
$769$	−26.7846	−0.965878	−0.482939	+	0.875654i	$0.660431\pi$
$770$	−82.3410	−2.96736
$771$	0	0
$772$	37.3205	1.34319
$773$	−11.0316	−0.396779	−0.198390	−	0.980123i	$-0.563571\pi$
$773$	−11.0316	−0.396779	−0.198390	+	0.980123i	$0.563571\pi$
$774$	0	0
$775$	12.8423	0.461310
$776$	−40.0594	−1.43805
$777$	0	0
$778$	−21.8038	−0.781706
$779$	10.6878	0.382931
$780$	0	0
$781$	−25.1769	−0.900901
$782$	0	0
$783$	0	0
$784$	37.9282	1.35458
$785$	−20.3106	−0.724915
$786$	0	0
$787$	15.9808	0.569653	0.284826	−	0.958579i	$-0.408064\pi$
$787$	15.9808	0.569653	0.284826	+	0.958579i	$0.408064\pi$
$788$	9.57668	0.341155
$789$	0	0
$790$	37.7654	1.34363
$791$	87.5990	3.11466
$792$	0	0
$793$	11.0718	0.393171
$794$	−59.5105	−2.11195
$795$	0	0
$796$	46.7846	1.65824
$797$	33.6903	1.19337	0.596685	−	0.802475i	$-0.296484\pi$
$797$	33.6903	1.19337	0.596685	+	0.802475i	$0.296484\pi$
$798$	0	0
$799$	0	0
$800$	−4.61645	−0.163216
$801$	0	0
$802$	4.19615	0.148171
$803$	−33.6903	−1.18890
$804$	0	0
$805$	−54.2487	−1.91202
$806$	55.2378	1.94567
$807$	0	0
$808$	41.6603	1.46560
$809$	−12.2686	−0.431340	−0.215670	−	0.976466i	$-0.569194\pi$
$809$	−12.2686	−0.431340	−0.215670	+	0.976466i	$0.569194\pi$
$810$	0	0
$811$	22.3205	0.783779	0.391890	−	0.920012i	$-0.371822\pi$
$811$	22.3205	0.783779	0.391890	+	0.920012i	$0.371822\pi$
$812$	95.8926	3.36517
$813$	0	0
$814$	61.5167	2.15616
$815$	37.2754	1.30570
$816$	0	0
$817$	−0.732051	−0.0256112
$818$	−16.4615	−0.575562
$819$	0	0
$820$	−69.9090	−2.44133
$821$	−30.1389	−1.05186	−0.525928	−	0.850529i	$-0.676282\pi$
$821$	−30.1389	−1.05186	−0.525928	+	0.850529i	$0.676282\pi$
$822$	0	0
$823$	33.6603	1.17332	0.586661	−	0.809832i	$-0.300442\pi$
$823$	33.6603	1.17332	0.586661	+	0.809832i	$0.300442\pi$
$824$	−47.5396	−1.65612
$825$	0	0
$826$	−34.3923	−1.19666
$827$	−49.2923	−1.71406	−0.857030	−	0.515266i	$-0.827693\pi$
$827$	−49.2923	−1.71406	−0.857030	+	0.515266i	$0.827693\pi$
$828$	0	0
$829$	12.1436	0.421764	0.210882	−	0.977511i	$-0.432366\pi$
$829$	12.1436	0.421764	0.210882	+	0.977511i	$0.432366\pi$
$830$	−50.2315	−1.74356
$831$	0	0
$832$	−36.9282	−1.28026
$833$	0	0
$834$	0	0
$835$	−3.89488	−0.134788
$836$	−15.4762	−0.535254
$837$	0	0
$838$	−5.73205	−0.198010
$839$	−2.22228	−0.0767215	−0.0383608	−	0.999264i	$-0.512214\pi$
$839$	−2.22228	−0.0767215	−0.0383608	+	0.999264i	$0.512214\pi$
$840$	0	0
$841$	0.483340	0.0166669
$842$	−13.0820	−0.450835
$843$	0	0
$844$	52.7128	1.81445
$845$	−1.75265	−0.0602931
$846$	0	0
$847$	29.3205	1.00746
$848$	17.6984	0.607767
$849$	0	0
$850$	0	0
$851$	40.5290	1.38932
$852$	0	0
$853$	22.7128	0.777672	0.388836	−	0.921307i	$-0.372877\pi$
$853$	22.7128	0.777672	0.388836	+	0.921307i	$0.372877\pi$
$854$	−36.2103	−1.23909
$855$	0	0
$856$	52.3013	1.78762
$857$	−5.77364	−0.197224	−0.0986120	−	0.995126i	$-0.531440\pi$
$857$	−5.77364	−0.197224	−0.0986120	+	0.995126i	$0.531440\pi$
$858$	0	0
$859$	−8.39230	−0.286342	−0.143171	−	0.989698i	$-0.545730\pi$
$859$	−8.39230	−0.286342	−0.143171	+	0.989698i	$0.545730\pi$
$860$	4.78834	0.163281
$861$	0	0
$862$	32.4449	1.10508
$863$	−15.3043	−0.520963	−0.260482	−	0.965479i	$-0.583881\pi$
$863$	−15.3043	−0.520963	−0.260482	+	0.965479i	$0.583881\pi$
$864$	0	0
$865$	−10.3397	−0.351562
$866$	80.2446	2.72682
$867$	0	0
$868$	−117.622	−3.99234
$869$	−37.3214	−1.26604
$870$	0	0
$871$	−7.85641	−0.266204
$872$	21.8453	0.739774
$873$	0	0
$874$	−15.6603	−0.529716
$875$	−57.4601	−1.94251
$876$	0	0
$877$	12.7321	0.429931	0.214965	−	0.976622i	$-0.431036\pi$
$877$	12.7321	0.429931	0.214965	+	0.976622i	$0.431036\pi$
$878$	−66.5211	−2.24498
$879$	0	0
$880$	17.9090	0.603711
$881$	−36.6799	−1.23578	−0.617889	−	0.786266i	$-0.712012\pi$
$881$	−36.6799	−1.23578	−0.617889	+	0.786266i	$0.712012\pi$
$882$	0	0
$883$	−1.51666	−0.0510397	−0.0255198	−	0.999674i	$-0.508124\pi$
$883$	−1.51666	−0.0510397	−0.0255198	+	0.999674i	$0.508124\pi$
$884$	0	0
$885$	0	0
$886$	−71.4449	−2.40024
$887$	−19.9668	−0.670419	−0.335209	−	0.942144i	$-0.608807\pi$
$887$	−19.9668	−0.670419	−0.335209	+	0.942144i	$0.608807\pi$
$888$	0	0
$889$	−13.8564	−0.464729
$890$	66.9109	2.24286
$891$	0	0
$892$	83.8372	2.80708
$893$	−7.18251	−0.240354
$894$	0	0
$895$	26.0000	0.869084
$896$	98.1149	3.27779
$897$	0	0
$898$	−82.9090	−2.76671
$899$	−36.1642	−1.20614
$900$	0	0
$901$	0	0
$902$	106.111	3.53310
$903$	0	0
$904$	−76.7654	−2.55318
$905$	−7.60607	−0.252834
$906$	0	0
$907$	−8.24871	−0.273894	−0.136947	−	0.990578i	$-0.543729\pi$
$907$	−8.24871	−0.273894	−0.136947	+	0.990578i	$0.543729\pi$
$908$	30.9523	1.02719
$909$	0	0
$910$	−68.7846	−2.28019
$911$	−28.8559	−0.956038	−0.478019	−	0.878350i	$-0.658645\pi$
$911$	−28.8559	−0.956038	−0.478019	+	0.878350i	$0.658645\pi$
$912$	0	0
$913$	49.6410	1.64288
$914$	−37.3214	−1.23448
$915$	0	0
$916$	−53.4449	−1.76587
$917$	−76.2697	−2.51865
$918$	0	0
$919$	53.1244	1.75241	0.876205	−	0.481938i	$-0.160067\pi$
$919$	53.1244	1.75241	0.876205	+	0.481938i	$0.160067\pi$
$920$	47.5396	1.56733
$921$	0	0
$922$	39.7128	1.30787
$923$	−21.0319	−0.692272
$924$	0	0
$925$	11.9474	0.392829
$926$	−44.0343	−1.44706
$927$	0	0
$928$	13.0000	0.426746
$929$	28.2604	0.927194	0.463597	−	0.886046i	$-0.346559\pi$
$929$	28.2604	0.927194	0.463597	+	0.886046i	$0.346559\pi$
$930$	0	0
$931$	−15.3923	−0.504462
$932$	37.4933	1.22813
$933$	0	0
$934$	14.9474	0.489095
$935$	0	0
$936$	0	0
$937$	−12.0718	−0.394368	−0.197184	−	0.980366i	$-0.563180\pi$
$937$	−12.0718	−0.394368	−0.197184	+	0.980366i	$0.563180\pi$
$938$	25.6944	0.838950
$939$	0	0
$940$	46.9808	1.53234
$941$	23.1283	0.753961	0.376980	−	0.926221i	$-0.376962\pi$
$941$	23.1283	0.753961	0.376980	+	0.926221i	$0.376962\pi$
$942$	0	0
$943$	69.9090	2.27655
$944$	7.48024	0.243461
$945$	0	0
$946$	−7.26795	−0.236301
$947$	33.2207	1.07953	0.539763	−	0.841817i	$-0.318514\pi$
$947$	33.2207	1.07953	0.539763	+	0.841817i	$0.318514\pi$
$948$	0	0
$949$	−28.1436	−0.913580
$950$	−4.61645	−0.149777
$951$	0	0
$952$	0	0
$953$	18.9815	0.614870	0.307435	−	0.951569i	$-0.400529\pi$
$953$	18.9815	0.614870	0.307435	+	0.951569i	$0.400529\pi$
$954$	0	0
$955$	21.8038	0.705556
$956$	−57.7578	−1.86802
$957$	0	0
$958$	−35.5167	−1.14749
$959$	−28.7300	−0.927741
$960$	0	0
$961$	13.3590	0.430935
$962$	51.3887	1.65684
$963$	0	0
$964$	30.5885	0.985188
$965$	17.5265	0.564199
$966$	0	0
$967$	40.5885	1.30524	0.652618	−	0.757687i	$-0.273671\pi$
$967$	40.5885	1.30524	0.652618	+	0.757687i	$0.273671\pi$
$968$	−25.6944	−0.825848
$969$	0	0
$970$	−40.5359	−1.30153
$971$	−20.4364	−0.655835	−0.327918	−	0.944706i	$-0.606347\pi$
$971$	−20.4364	−0.655835	−0.327918	+	0.944706i	$0.606347\pi$
$972$	0	0
$973$	47.3205	1.51703
$974$	−27.1493	−0.869919
$975$	0	0
$976$	7.87564	0.252093
$977$	−17.6984	−0.566223	−0.283112	−	0.959087i	$-0.591367\pi$
$977$	−17.6984	−0.566223	−0.283112	+	0.959087i	$0.591367\pi$
$978$	0	0
$979$	−66.1244	−2.11334
$980$	100.681	3.21614
$981$	0	0
$982$	65.3013	2.08385
$983$	51.5145	1.64306	0.821529	−	0.570166i	$-0.193121\pi$
$983$	51.5145	1.64306	0.821529	+	0.570166i	$0.193121\pi$
$984$	0	0
$985$	4.49742	0.143300
$986$	0	0
$987$	0	0
$988$	−12.9282	−0.411301
$989$	−4.78834	−0.152260
$990$	0	0
$991$	−10.5359	−0.334684	−0.167342	−	0.985899i	$-0.553518\pi$
$991$	−10.5359	−0.334684	−0.167342	+	0.985899i	$0.553518\pi$
$992$	−15.9458	−0.506279
$993$	0	0
$994$	68.7846	2.18172
$995$	21.9711	0.696531
$996$	0	0
$997$	19.0718	0.604010	0.302005	−	0.953306i	$-0.402344\pi$
$997$	19.0718	0.604010	0.302005	+	0.953306i	$0.402344\pi$
$998$	−34.4576	−1.09074
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.i.1.1	✓	4
3.2	odd	2	inner	513.2.a.i.1.4	yes	4
4.3	odd	2		8208.2.a.bt.1.3		4
12.11	even	2		8208.2.a.bt.1.2		4
19.18	odd	2		9747.2.a.bf.1.4		4
57.56	even	2		9747.2.a.bf.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.i.1.1	✓	4	1.1	even	1	trivial
513.2.a.i.1.4	yes	4	3.2	odd	2	inner
8208.2.a.bt.1.2		4	12.11	even	2
8208.2.a.bt.1.3		4	4.3	odd	2
9747.2.a.bf.1.1		4	57.56	even	2
9747.2.a.bf.1.4		4	19.18	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39417 q^{2} +3.73205 q^{4} +1.75265 q^{5} +4.73205 q^{7} -4.14682 q^{8} -4.19615 q^{10} +4.14682 q^{11} +3.46410 q^{13} -11.3293 q^{14} +2.46410 q^{16} -1.00000 q^{19} +6.54099 q^{20} -9.92820 q^{22} -6.54099 q^{23} -1.92820 q^{25} -8.29365 q^{26} +17.6603 q^{28} +5.42986 q^{29} -6.66025 q^{31} +2.39417 q^{32} +8.29365 q^{35} -6.19615 q^{37} +2.39417 q^{38} -7.26795 q^{40} -10.6878 q^{41} +0.732051 q^{43} +15.4762 q^{44} +15.6603 q^{46} +7.18251 q^{47} +15.3923 q^{49} +4.61645 q^{50} +12.9282 q^{52} +7.18251 q^{53} +7.26795 q^{55} -19.6230 q^{56} -13.0000 q^{58} +3.03569 q^{59} +3.19615 q^{61} +15.9458 q^{62} -10.6603 q^{64} +6.07137 q^{65} -2.26795 q^{67} -19.8564 q^{70} -6.07137 q^{71} -8.12436 q^{73} +14.8346 q^{74} -3.73205 q^{76} +19.6230 q^{77} -9.00000 q^{79} +4.31872 q^{80} +25.5885 q^{82} +11.9709 q^{83} -1.75265 q^{86} -17.1962 q^{88} -15.9458 q^{89} +16.3923 q^{91} -24.4113 q^{92} -17.1962 q^{94} -1.75265 q^{95} +9.66025 q^{97} -36.8518 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 12 q^{7} + 4 q^{10} - 4 q^{16} - 4 q^{19} - 12 q^{22} + 20 q^{25} + 36 q^{28} + 8 q^{31} - 4 q^{37} - 36 q^{40} - 4 q^{43} + 28 q^{46} + 20 q^{49} + 24 q^{52} + 36 q^{55} - 52 q^{58} - 8 q^{61}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q + 8 * q^4 + 12 * q^7 + 4 * q^10 - 4 * q^16 - 4 * q^19 - 12 * q^22 + 20 * q^25 + 36 * q^28 + 8 * q^31 - 4 * q^37 - 36 * q^40 - 4 * q^43 + 28 * q^46 + 20 * q^49 + 24 * q^52 + 36 * q^55 - 52 * q^58 - 8 * q^61 - 8 * q^64 - 16 * q^67 - 24 * q^70 + 16 * q^73 - 8 * q^76 - 36 * q^79 + 40 * q^82 - 48 * q^88 + 24 * q^91 - 48 * q^94 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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