LMFDB - Embedded newform 1521.4.a.bc.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,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, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 $ x^8 - 52x^6 + 805x^4 - 4210*x^2 + 4992
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.28670$ 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)

$f(q)$	$=$	$q+1.28670 q^{2} -6.34441 q^{4} +12.4484 q^{5} -9.00273 q^{7} -18.4569 q^{8} +O(q^{10})$	$q+1.28670 q^{2} -6.34441 q^{4} +12.4484 q^{5} -9.00273 q^{7} -18.4569 q^{8} +16.0173 q^{10} -51.0774 q^{11} -11.5838 q^{14} +27.0069 q^{16} -6.86199 q^{17} -83.1170 q^{19} -78.9779 q^{20} -65.7211 q^{22} +187.579 q^{23} +29.9631 q^{25} +57.1170 q^{28} -223.614 q^{29} +57.3882 q^{31} +182.405 q^{32} -8.82930 q^{34} -112.070 q^{35} -156.259 q^{37} -106.946 q^{38} -229.759 q^{40} +222.281 q^{41} +347.974 q^{43} +324.056 q^{44} +241.357 q^{46} -45.0185 q^{47} -261.951 q^{49} +38.5534 q^{50} +473.516 q^{53} -635.833 q^{55} +166.163 q^{56} -287.724 q^{58} +615.494 q^{59} +195.325 q^{61} +73.8412 q^{62} +18.6446 q^{64} -355.097 q^{67} +43.5353 q^{68} -144.200 q^{70} +763.420 q^{71} +331.595 q^{73} -201.057 q^{74} +527.329 q^{76} +459.836 q^{77} -207.777 q^{79} +336.193 q^{80} +286.008 q^{82} -251.185 q^{83} -85.4209 q^{85} +447.737 q^{86} +942.731 q^{88} +719.366 q^{89} -1190.08 q^{92} -57.9252 q^{94} -1034.68 q^{95} -1556.16 q^{97} -337.051 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{4} - 22 q^{7}+O(q^{10}) $ 8 * q + 40 * q^4 - 22 * q^7	$ 8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100}) $ 8 * q + 40 * q^4 - 22 * q^7 + 36 * q^10 + 204 * q^16 + 244 * q^19 + 136 * q^22 + 354 * q^25 - 452 * q^28 + 242 * q^31 - 1292 * q^34 + 1018 * q^37 + 1700 * q^40 + 74 * q^43 - 896 * q^46 + 298 * q^49 + 1300 * q^55 + 812 * q^58 + 1148 * q^61 + 3636 * q^64 - 2198 * q^67 + 2200 * q^70 - 2176 * q^73 + 6936 * q^76 + 1862 * q^79 + 5436 * q^82 - 890 * q^85 + 3528 * q^88 - 3104 * q^94 - 4370 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.28670	0.454916	0.227458	−	0.973788i	$-0.426959\pi$
$2$	1.28670	0.454916	0.227458	+	0.973788i	$0.426959\pi$
$3$	0	0
$3$	0	0
$4$	−6.34441	−0.793052
$5$	12.4484	1.11342	0.556710	−	0.830707i	$-0.312063\pi$
$5$	12.4484	1.11342	0.556710	+	0.830707i	$0.312063\pi$
$6$	0	0
$7$	−9.00273	−0.486102	−0.243051	−	0.970014i	$-0.578148\pi$
$7$	−9.00273	−0.486102	−0.243051	+	0.970014i	$0.578148\pi$
$8$	−18.4569	−0.815687
$9$	0	0
$10$	16.0173	0.506512
$11$	−51.0774	−1.40004	−0.700019	−	0.714124i	$-0.746825\pi$
$11$	−51.0774	−1.40004	−0.700019	+	0.714124i	$0.746825\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−11.5838	−0.221135
$15$	0	0
$16$	27.0069	0.421983
$17$	−6.86199	−0.0978987	−0.0489493	−	0.998801i	$-0.515587\pi$
$17$	−6.86199	−0.0978987	−0.0489493	+	0.998801i	$0.515587\pi$
$18$	0	0
$19$	−83.1170	−1.00360	−0.501799	−	0.864984i	$-0.667328\pi$
$19$	−83.1170	−1.00360	−0.501799	+	0.864984i	$0.667328\pi$
$20$	−78.9779	−0.883000
$21$	0	0
$22$	−65.7211	−0.636900
$23$	187.579	1.70056	0.850279	−	0.526332i	$-0.176433\pi$
$23$	187.579	1.70056	0.850279	+	0.526332i	$0.176433\pi$
$24$	0	0
$25$	29.9631	0.239705
$26$	0	0
$27$	0	0
$28$	57.1170	0.385504
$29$	−223.614	−1.43187	−0.715933	−	0.698169i	$-0.753998\pi$
$29$	−223.614	−1.43187	−0.715933	+	0.698169i	$0.753998\pi$
$30$	0	0
$31$	57.3882	0.332491	0.166246	−	0.986084i	$-0.446836\pi$
$31$	57.3882	0.332491	0.166246	+	0.986084i	$0.446836\pi$
$32$	182.405	1.00765
$33$	0	0
$34$	−8.82930	−0.0445357
$35$	−112.070	−0.541236
$36$	0	0
$37$	−156.259	−0.694291	−0.347146	−	0.937811i	$-0.612849\pi$
$37$	−156.259	−0.694291	−0.347146	+	0.937811i	$0.612849\pi$
$38$	−106.946	−0.456552
$39$	0	0
$40$	−229.759	−0.908203
$41$	222.281	0.846695	0.423348	−	0.905967i	$-0.360855\pi$
$41$	222.281	0.846695	0.423348	+	0.905967i	$0.360855\pi$
$42$	0	0
$43$	347.974	1.23408	0.617041	−	0.786931i	$-0.288331\pi$
$43$	347.974	1.23408	0.617041	+	0.786931i	$0.288331\pi$
$44$	324.056	1.11030
$45$	0	0
$46$	241.357	0.773611
$47$	−45.0185	−0.139715	−0.0698577	−	0.997557i	$-0.522255\pi$
$47$	−45.0185	−0.139715	−0.0698577	+	0.997557i	$0.522255\pi$
$48$	0	0
$49$	−261.951	−0.763705
$50$	38.5534	0.109045
$51$	0	0
$52$	0	0
$53$	473.516	1.22722	0.613608	−	0.789611i	$-0.289718\pi$
$53$	473.516	1.22722	0.613608	+	0.789611i	$0.289718\pi$
$54$	0	0
$55$	−635.833	−1.55883
$56$	166.163	0.396507
$57$	0	0
$58$	−287.724	−0.651379
$59$	615.494	1.35814	0.679072	−	0.734071i	$-0.262382\pi$
$59$	615.494	1.35814	0.679072	+	0.734071i	$0.262382\pi$
$60$	0	0
$61$	195.325	0.409981	0.204990	−	0.978764i	$-0.434284\pi$
$61$	195.325	0.409981	0.204990	+	0.978764i	$0.434284\pi$
$62$	73.8412	0.151255
$63$	0	0
$64$	18.6446	0.0364151
$65$	0	0
$66$	0	0
$67$	−355.097	−0.647492	−0.323746	−	0.946144i	$-0.604942\pi$
$67$	−355.097	−0.647492	−0.323746	+	0.946144i	$0.604942\pi$
$68$	43.5353	0.0776387
$69$	0	0
$70$	−144.200	−0.246217
$71$	763.420	1.27608	0.638038	−	0.770005i	$-0.279746\pi$
$71$	763.420	1.27608	0.638038	+	0.770005i	$0.279746\pi$
$72$	0	0
$73$	331.595	0.531647	0.265824	−	0.964022i	$-0.414356\pi$
$73$	331.595	0.531647	0.265824	+	0.964022i	$0.414356\pi$
$74$	−201.057	−0.315844
$75$	0	0
$76$	527.329	0.795905
$77$	459.836	0.680561
$78$	0	0
$79$	−207.777	−0.295908	−0.147954	−	0.988994i	$-0.547269\pi$
$79$	−207.777	−0.295908	−0.147954	+	0.988994i	$0.547269\pi$
$80$	336.193	0.469844
$81$	0	0
$82$	286.008	0.385175
$83$	−251.185	−0.332183	−0.166091	−	0.986110i	$-0.553115\pi$
$83$	−251.185	−0.332183	−0.166091	+	0.986110i	$0.553115\pi$
$84$	0	0
$85$	−85.4209	−0.109002
$86$	447.737	0.561403
$87$	0	0
$88$	942.731	1.14199
$89$	719.366	0.856771	0.428385	−	0.903596i	$-0.359083\pi$
$89$	719.366	0.856771	0.428385	+	0.903596i	$0.359083\pi$
$90$	0	0
$91$	0	0
$92$	−1190.08	−1.34863
$93$	0	0
$94$	−57.9252	−0.0635588
$95$	−1034.68	−1.11743
$96$	0	0
$97$	−1556.16	−1.62891	−0.814454	−	0.580228i	$-0.802964\pi$
$97$	−1556.16	−1.62891	−0.814454	+	0.580228i	$0.802964\pi$
$98$	−337.051	−0.347421
$99$	0	0
$100$	−190.098	−0.190098
$101$	−783.385	−0.771779	−0.385890	−	0.922545i	$-0.626105\pi$
$101$	−783.385	−0.771779	−0.385890	+	0.922545i	$0.626105\pi$
$102$	0	0
$103$	1033.54	0.988720	0.494360	−	0.869257i	$-0.335402\pi$
$103$	1033.54	0.988720	0.494360	+	0.869257i	$0.335402\pi$
$104$	0	0
$105$	0	0
$106$	609.271	0.558279
$107$	−241.358	−0.218065	−0.109033	−	0.994038i	$-0.534775\pi$
$107$	−241.358	−0.218065	−0.109033	+	0.994038i	$0.534775\pi$
$108$	0	0
$109$	1763.93	1.55003	0.775016	−	0.631941i	$-0.217742\pi$
$109$	1763.93	1.55003	0.775016	+	0.631941i	$0.217742\pi$
$110$	−818.124	−0.709137
$111$	0	0
$112$	−243.136	−0.205126
$113$	1867.83	1.55496	0.777482	−	0.628905i	$-0.216497\pi$
$113$	1867.83	1.55496	0.777482	+	0.628905i	$0.216497\pi$
$114$	0	0
$115$	2335.06	1.89344
$116$	1418.70	1.13554
$117$	0	0
$118$	791.954	0.617841
$119$	61.7767	0.0475887
$120$	0	0
$121$	1277.90	0.960108
$122$	251.324	0.186507
$123$	0	0
$124$	−364.095	−0.263683
$125$	−1183.06	−0.846528
$126$	0	0
$127$	238.885	0.166910	0.0834551	−	0.996512i	$-0.473404\pi$
$127$	238.885	0.166910	0.0834551	+	0.996512i	$0.473404\pi$
$128$	−1435.25	−0.991088
$129$	0	0
$130$	0	0
$131$	−1158.72	−0.772807	−0.386403	−	0.922330i	$-0.626283\pi$
$131$	−1158.72	−0.772807	−0.386403	+	0.922330i	$0.626283\pi$
$132$	0	0
$133$	748.280	0.487851
$134$	−456.902	−0.294554
$135$	0	0
$136$	126.651	0.0798547
$137$	−912.520	−0.569064	−0.284532	−	0.958666i	$-0.591838\pi$
$137$	−912.520	−0.569064	−0.284532	+	0.958666i	$0.591838\pi$
$138$	0	0
$139$	2315.04	1.41265	0.706326	−	0.707886i	$-0.250351\pi$
$139$	2315.04	1.41265	0.706326	+	0.707886i	$0.250351\pi$
$140$	711.017	0.429228
$141$	0	0
$142$	982.290	0.580507
$143$	0	0
$144$	0	0
$145$	−2783.64	−1.59427
$146$	426.662	0.241855
$147$	0	0
$148$	991.370	0.550609
$149$	−925.408	−0.508808	−0.254404	−	0.967098i	$-0.581879\pi$
$149$	−925.408	−0.508808	−0.254404	+	0.967098i	$0.581879\pi$
$150$	0	0
$151$	−3523.78	−1.89908	−0.949541	−	0.313644i	$-0.898450\pi$
$151$	−3523.78	−1.89908	−0.949541	+	0.313644i	$0.898450\pi$
$152$	1534.08	0.818622
$153$	0	0
$154$	591.670	0.309598
$155$	714.392	0.370202
$156$	0	0
$157$	1615.83	0.821384	0.410692	−	0.911774i	$-0.365287\pi$
$157$	1615.83	0.821384	0.410692	+	0.911774i	$0.365287\pi$
$158$	−267.346	−0.134613
$159$	0	0
$160$	2270.65	1.12194
$161$	−1688.72	−0.826644
$162$	0	0
$163$	1008.01	0.484375	0.242188	−	0.970229i	$-0.422135\pi$
$163$	1008.01	0.484375	0.242188	+	0.970229i	$0.422135\pi$
$164$	−1410.24	−0.671473
$165$	0	0
$166$	−323.199	−0.151115
$167$	2354.04	1.09078	0.545392	−	0.838181i	$-0.316381\pi$
$167$	2354.04	1.09078	0.545392	+	0.838181i	$0.316381\pi$
$168$	0	0
$169$	0	0
$170$	−109.911	−0.0495869
$171$	0	0
$172$	−2207.69	−0.978691
$173$	3132.32	1.37657	0.688284	−	0.725442i	$-0.258365\pi$
$173$	3132.32	1.37657	0.688284	+	0.725442i	$0.258365\pi$
$174$	0	0
$175$	−269.750	−0.116521
$176$	−1379.44	−0.590792
$177$	0	0
$178$	925.605	0.389759
$179$	−3180.82	−1.32819	−0.664093	−	0.747650i	$-0.731182\pi$
$179$	−3180.82	−1.32819	−0.664093	+	0.747650i	$0.731182\pi$
$180$	0	0
$181$	4230.52	1.73731	0.868653	−	0.495421i	$-0.164986\pi$
$181$	4230.52	1.73731	0.868653	+	0.495421i	$0.164986\pi$
$182$	0	0
$183$	0	0
$184$	−3462.12	−1.38712
$185$	−1945.17	−0.773038
$186$	0	0
$187$	350.493	0.137062
$188$	285.616	0.110802
$189$	0	0
$190$	−1331.31	−0.508335
$191$	−3952.80	−1.49746	−0.748729	−	0.662876i	$-0.769335\pi$
$191$	−3952.80	−1.49746	−0.748729	+	0.662876i	$0.769335\pi$
$192$	0	0
$193$	5042.45	1.88064	0.940319	−	0.340294i	$-0.110527\pi$
$193$	5042.45	1.88064	0.940319	+	0.340294i	$0.110527\pi$
$194$	−2002.31	−0.741016
$195$	0	0
$196$	1661.92	0.605658
$197$	3572.12	1.29189	0.645947	−	0.763382i	$-0.276463\pi$
$197$	3572.12	1.29189	0.645947	+	0.763382i	$0.276463\pi$
$198$	0	0
$199$	5006.84	1.78355	0.891773	−	0.452482i	$-0.149461\pi$
$199$	5006.84	1.78355	0.891773	+	0.452482i	$0.149461\pi$
$200$	−553.026	−0.195524
$201$	0	0
$202$	−1007.98	−0.351095
$203$	2013.14	0.696033
$204$	0	0
$205$	2767.05	0.942728
$206$	1329.86	0.449784
$207$	0	0
$208$	0	0
$209$	4245.40	1.40508
$210$	0	0
$211$	1759.23	0.573983	0.286992	−	0.957933i	$-0.407345\pi$
$211$	1759.23	0.573983	0.286992	+	0.957933i	$0.407345\pi$
$212$	−3004.18	−0.973245
$213$	0	0
$214$	−310.555	−0.0992014
$215$	4331.72	1.37405
$216$	0	0
$217$	−516.651	−0.161625
$218$	2269.64	0.705134
$219$	0	0
$220$	4033.99	1.23623
$221$	0	0
$222$	0	0
$223$	1712.52	0.514255	0.257127	−	0.966377i	$-0.417224\pi$
$223$	1712.52	0.514255	0.257127	+	0.966377i	$0.417224\pi$
$224$	−1642.14	−0.489822
$225$	0	0
$226$	2403.33	0.707378
$227$	2576.65	0.753385	0.376693	−	0.926338i	$-0.377061\pi$
$227$	2576.65	0.753385	0.376693	+	0.926338i	$0.377061\pi$
$228$	0	0
$229$	−1963.10	−0.566486	−0.283243	−	0.959048i	$-0.591410\pi$
$229$	−1963.10	−0.566486	−0.283243	+	0.959048i	$0.591410\pi$
$230$	3004.51	0.861354
$231$	0	0
$232$	4127.23	1.16796
$233$	−1078.11	−0.303131	−0.151565	−	0.988447i	$-0.548431\pi$
$233$	−1078.11	−0.303131	−0.151565	+	0.988447i	$0.548431\pi$
$234$	0	0
$235$	−560.409	−0.155562
$236$	−3904.95	−1.07708
$237$	0	0
$238$	79.4878	0.0216489
$239$	3998.43	1.08216	0.541081	−	0.840970i	$-0.318015\pi$
$239$	3998.43	1.08216	0.541081	+	0.840970i	$0.318015\pi$
$240$	0	0
$241$	−3574.72	−0.955470	−0.477735	−	0.878504i	$-0.658542\pi$
$241$	−3574.72	−0.955470	−0.477735	+	0.878504i	$0.658542\pi$
$242$	1644.27	0.436768
$243$	0	0
$244$	−1239.22	−0.325136
$245$	−3260.87	−0.850325
$246$	0	0
$247$	0	0
$248$	−1059.21	−0.271209
$249$	0	0
$250$	−1522.24	−0.385099
$251$	2150.69	0.540839	0.270419	−	0.962743i	$-0.412838\pi$
$251$	2150.69	0.540839	0.270419	+	0.962743i	$0.412838\pi$
$252$	0	0
$253$	−9581.03	−2.38085
$254$	307.372	0.0759301
$255$	0	0
$256$	−1995.89	−0.487277
$257$	4274.80	1.03757	0.518784	−	0.854906i	$-0.326385\pi$
$257$	4274.80	1.03757	0.518784	+	0.854906i	$0.326385\pi$
$258$	0	0
$259$	1406.75	0.337496
$260$	0	0
$261$	0	0
$262$	−1490.92	−0.351562
$263$	8371.65	1.96281	0.981403	−	0.191960i	$-0.0614845\pi$
$263$	8371.65	1.96281	0.981403	+	0.191960i	$0.0614845\pi$
$264$	0	0
$265$	5894.52	1.36641
$266$	962.810	0.221931
$267$	0	0
$268$	2252.88	0.513495
$269$	−5079.64	−1.15134	−0.575672	−	0.817681i	$-0.695259\pi$
$269$	−5079.64	−1.15134	−0.575672	+	0.817681i	$0.695259\pi$
$270$	0	0
$271$	4931.93	1.10551	0.552756	−	0.833343i	$-0.313576\pi$
$271$	4931.93	1.10551	0.552756	+	0.833343i	$0.313576\pi$
$272$	−185.321	−0.0413115
$273$	0	0
$274$	−1174.14	−0.258876
$275$	−1530.44	−0.335596
$276$	0	0
$277$	5172.52	1.12197	0.560987	−	0.827825i	$-0.310422\pi$
$277$	5172.52	1.12197	0.560987	+	0.827825i	$0.310422\pi$
$278$	2978.75	0.642638
$279$	0	0
$280$	2068.46	0.441479
$281$	−217.515	−0.0461773	−0.0230887	−	0.999733i	$-0.507350\pi$
$281$	−217.515	−0.0461773	−0.0230887	+	0.999733i	$0.507350\pi$
$282$	0	0
$283$	4922.98	1.03407	0.517033	−	0.855966i	$-0.327036\pi$
$283$	4922.98	1.03407	0.517033	+	0.855966i	$0.327036\pi$
$284$	−4843.45	−1.01199
$285$	0	0
$286$	0	0
$287$	−2001.14	−0.411580
$288$	0	0
$289$	−4865.91	−0.990416
$290$	−3581.70	−0.725258
$291$	0	0
$292$	−2103.78	−0.421624
$293$	−2642.25	−0.526832	−0.263416	−	0.964682i	$-0.584849\pi$
$293$	−2642.25	−0.526832	−0.263416	+	0.964682i	$0.584849\pi$
$294$	0	0
$295$	7661.93	1.51219
$296$	2884.05	0.566325
$297$	0	0
$298$	−1190.72	−0.231465
$299$	0	0
$300$	0	0
$301$	−3132.72	−0.599889
$302$	−4534.04	−0.863922
$303$	0	0
$304$	−2244.73	−0.423501
$305$	2431.49	0.456481
$306$	0	0
$307$	−3386.22	−0.629517	−0.314759	−	0.949172i	$-0.601924\pi$
$307$	−3386.22	−0.629517	−0.314759	+	0.949172i	$0.601924\pi$
$308$	−2917.39	−0.539720
$309$	0	0
$310$	919.206	0.168411
$311$	−3.29813	−0.000601349 0	−0.000300675	−	1.00000i	$-0.500096\pi$
$311$	−3.29813	−0.000601349 0	−0.000300675		1.00000i	$0.500096\pi$
$312$	0	0
$313$	2882.62	0.520560	0.260280	−	0.965533i	$-0.416185\pi$
$313$	2882.62	0.520560	0.260280	+	0.965533i	$0.416185\pi$
$314$	2079.08	0.373660
$315$	0	0
$316$	1318.22	0.234671
$317$	−1442.70	−0.255616	−0.127808	−	0.991799i	$-0.540794\pi$
$317$	−1442.70	−0.255616	−0.127808	+	0.991799i	$0.540794\pi$
$318$	0	0
$319$	11421.6	2.00467
$320$	232.095	0.0405454
$321$	0	0
$322$	−2172.87	−0.376054
$323$	570.349	0.0982509
$324$	0	0
$325$	0	0
$326$	1297.00	0.220350
$327$	0	0
$328$	−4102.62	−0.690639
$329$	405.290	0.0679159
$330$	0	0
$331$	−3824.98	−0.635166	−0.317583	−	0.948230i	$-0.602871\pi$
$331$	−3824.98	−0.635166	−0.317583	+	0.948230i	$0.602871\pi$
$332$	1593.62	0.263438
$333$	0	0
$334$	3028.93	0.496215
$335$	−4420.39	−0.720931
$336$	0	0
$337$	2290.22	0.370197	0.185098	−	0.982720i	$-0.440740\pi$
$337$	2290.22	0.370197	0.185098	+	0.982720i	$0.440740\pi$
$338$	0	0
$339$	0	0
$340$	541.946	0.0864445
$341$	−2931.24	−0.465500
$342$	0	0
$343$	5446.21	0.857340
$344$	−6422.52	−1.00663
$345$	0	0
$346$	4030.35	0.626222
$347$	−334.993	−0.0518252	−0.0259126	−	0.999664i	$-0.508249\pi$
$347$	−334.993	−0.0518252	−0.0259126	+	0.999664i	$0.508249\pi$
$348$	0	0
$349$	−8975.27	−1.37661	−0.688303	−	0.725424i	$-0.741644\pi$
$349$	−8975.27	−1.37661	−0.688303	+	0.725424i	$0.741644\pi$
$350$	−347.086	−0.0530072
$351$	0	0
$352$	−9316.77	−1.41075
$353$	−9925.87	−1.49660	−0.748302	−	0.663359i	$-0.769130\pi$
$353$	−9925.87	−1.49660	−0.748302	+	0.663359i	$0.769130\pi$
$354$	0	0
$355$	9503.38	1.42081
$356$	−4563.95	−0.679463
$357$	0	0
$358$	−4092.74	−0.604213
$359$	215.661	0.0317052	0.0158526	−	0.999874i	$-0.494954\pi$
$359$	215.661	0.0317052	0.0158526	+	0.999874i	$0.494954\pi$
$360$	0	0
$361$	49.4437	0.00720858
$362$	5443.40	0.790328
$363$	0	0
$364$	0	0
$365$	4127.83	0.591947
$366$	0	0
$367$	−2861.52	−0.407003	−0.203502	−	0.979075i	$-0.565232\pi$
$367$	−2861.52	−0.407003	−0.203502	+	0.979075i	$0.565232\pi$
$368$	5065.91	0.717606
$369$	0	0
$370$	−2502.85	−0.351667
$371$	−4262.94	−0.596551
$372$	0	0
$373$	−17.0145	−0.00236186	−0.00118093	−	0.999999i	$-0.500376\pi$
$373$	−17.0145	−0.00236186	−0.00118093	+	0.999999i	$0.500376\pi$
$374$	450.978	0.0623516
$375$	0	0
$376$	830.902	0.113964
$377$	0	0
$378$	0	0
$379$	−1901.97	−0.257777	−0.128888	−	0.991659i	$-0.541141\pi$
$379$	−1901.97	−0.257777	−0.128888	+	0.991659i	$0.541141\pi$
$380$	6564.41	0.886177
$381$	0	0
$382$	−5086.05	−0.681217
$383$	−7729.78	−1.03126	−0.515631	−	0.856811i	$-0.672442\pi$
$383$	−7729.78	−1.03126	−0.515631	+	0.856811i	$0.672442\pi$
$384$	0	0
$385$	5724.23	0.757751
$386$	6488.09	0.855532
$387$	0	0
$388$	9872.92	1.29181
$389$	−14445.5	−1.88282	−0.941408	−	0.337271i	$-0.890496\pi$
$389$	−14445.5	−1.88282	−0.941408	+	0.337271i	$0.890496\pi$
$390$	0	0
$391$	−1287.16	−0.166482
$392$	4834.80	0.622945
$393$	0	0
$394$	4596.23	0.587703
$395$	−2586.50	−0.329470
$396$	0	0
$397$	1033.07	0.130600	0.0653000	−	0.997866i	$-0.479200\pi$
$397$	1033.07	0.130600	0.0653000	+	0.997866i	$0.479200\pi$
$398$	6442.29	0.811364
$399$	0	0
$400$	809.210	0.101151
$401$	−4611.73	−0.574311	−0.287155	−	0.957884i	$-0.592710\pi$
$401$	−4611.73	−0.574311	−0.287155	+	0.957884i	$0.592710\pi$
$402$	0	0
$403$	0	0
$404$	4970.12	0.612061
$405$	0	0
$406$	2590.30	0.316636
$407$	7981.29	0.972034
$408$	0	0
$409$	2668.56	0.322621	0.161311	−	0.986904i	$-0.448428\pi$
$409$	2668.56	0.322621	0.161311	+	0.986904i	$0.448428\pi$
$410$	3560.35	0.428862
$411$	0	0
$412$	−6557.23	−0.784106
$413$	−5541.13	−0.660197
$414$	0	0
$415$	−3126.86	−0.369859
$416$	0	0
$417$	0	0
$418$	5462.55	0.639191
$419$	6189.83	0.721701	0.360851	−	0.932624i	$-0.382486\pi$
$419$	6189.83	0.721701	0.360851	+	0.932624i	$0.382486\pi$
$420$	0	0
$421$	8991.13	1.04086	0.520429	−	0.853905i	$-0.325772\pi$
$421$	8991.13	1.04086	0.520429	+	0.853905i	$0.325772\pi$
$422$	2263.60	0.261114
$423$	0	0
$424$	−8739.63	−1.00102
$425$	−205.606	−0.0234668
$426$	0	0
$427$	−1758.46	−0.199292
$428$	1531.28	0.172937
$429$	0	0
$430$	5573.61	0.625078
$431$	2069.01	0.231232	0.115616	−	0.993294i	$-0.463116\pi$
$431$	2069.01	0.231232	0.115616	+	0.993294i	$0.463116\pi$
$432$	0	0
$433$	−14934.7	−1.65754	−0.828769	−	0.559592i	$-0.810958\pi$
$433$	−14934.7	−1.65754	−0.828769	+	0.559592i	$0.810958\pi$
$434$	−664.772	−0.0735256
$435$	0	0
$436$	−11191.1	−1.22926
$437$	−15591.0	−1.70668
$438$	0	0
$439$	−6000.24	−0.652336	−0.326168	−	0.945312i	$-0.605758\pi$
$439$	−6000.24	−0.652336	−0.326168	+	0.945312i	$0.605758\pi$
$440$	11735.5	1.27152
$441$	0	0
$442$	0	0
$443$	−5392.60	−0.578352	−0.289176	−	0.957276i	$-0.593381\pi$
$443$	−5392.60	−0.578352	−0.289176	+	0.957276i	$0.593381\pi$
$444$	0	0
$445$	8954.96	0.953946
$446$	2203.49	0.233943
$447$	0	0
$448$	−167.852	−0.0177015
$449$	−6046.60	−0.635538	−0.317769	−	0.948168i	$-0.602934\pi$
$449$	−6046.60	−0.635538	−0.317769	+	0.948168i	$0.602934\pi$
$450$	0	0
$451$	−11353.6	−1.18541
$452$	−11850.3	−1.23317
$453$	0	0
$454$	3315.37	0.342727
$455$	0	0
$456$	0	0
$457$	1014.57	0.103851	0.0519253	−	0.998651i	$-0.483464\pi$
$457$	1014.57	0.103851	0.0519253	+	0.998651i	$0.483464\pi$
$458$	−2525.91	−0.257704
$459$	0	0
$460$	−14814.6	−1.50159
$461$	−15978.9	−1.61434	−0.807171	−	0.590318i	$-0.799003\pi$
$461$	−15978.9	−1.61434	−0.807171	+	0.590318i	$0.799003\pi$
$462$	0	0
$463$	−128.805	−0.0129289	−0.00646445	−	0.999979i	$-0.502058\pi$
$463$	−128.805	−0.0129289	−0.00646445	+	0.999979i	$0.502058\pi$
$464$	−6039.12	−0.604223
$465$	0	0
$466$	−1387.20	−0.137899
$467$	18832.4	1.86608	0.933039	−	0.359776i	$-0.117147\pi$
$467$	18832.4	1.86608	0.933039	+	0.359776i	$0.117147\pi$
$468$	0	0
$469$	3196.84	0.314747
$470$	−721.076	−0.0707676
$471$	0	0
$472$	−11360.1	−1.10782
$473$	−17773.6	−1.72776
$474$	0	0
$475$	−2490.44	−0.240567
$476$	−391.937	−0.0377403
$477$	0	0
$478$	5144.76	0.492293
$479$	−9231.59	−0.880589	−0.440294	−	0.897853i	$-0.645126\pi$
$479$	−9231.59	−0.880589	−0.440294	+	0.897853i	$0.645126\pi$
$480$	0	0
$481$	0	0
$482$	−4599.58	−0.434658
$483$	0	0
$484$	−8107.55	−0.761415
$485$	−19371.7	−1.81366
$486$	0	0
$487$	7233.01	0.673016	0.336508	−	0.941681i	$-0.390754\pi$
$487$	7233.01	0.673016	0.336508	+	0.941681i	$0.390754\pi$
$488$	−3605.10	−0.334416
$489$	0	0
$490$	−4195.75	−0.386826
$491$	−655.131	−0.0602152	−0.0301076	−	0.999547i	$-0.509585\pi$
$491$	−655.131	−0.0602152	−0.0301076	+	0.999547i	$0.509585\pi$
$492$	0	0
$493$	1534.44	0.140178
$494$	0	0
$495$	0	0
$496$	1549.88	0.140305
$497$	−6872.87	−0.620302
$498$	0	0
$499$	−4560.07	−0.409092	−0.204546	−	0.978857i	$-0.565572\pi$
$499$	−4560.07	−0.409092	−0.204546	+	0.978857i	$0.565572\pi$
$500$	7505.82	0.671341
$501$	0	0
$502$	2767.29	0.246036
$503$	9068.67	0.803881	0.401941	−	0.915666i	$-0.368336\pi$
$503$	9068.67	0.803881	0.401941	+	0.915666i	$0.368336\pi$
$504$	0	0
$505$	−9751.90	−0.859315
$506$	−12327.9	−1.08308
$507$	0	0
$508$	−1515.58	−0.132368
$509$	−15880.9	−1.38292	−0.691461	−	0.722413i	$-0.743033\pi$
$509$	−15880.9	−1.38292	−0.691461	+	0.722413i	$0.743033\pi$
$510$	0	0
$511$	−2985.26	−0.258435
$512$	8913.89	0.769418
$513$	0	0
$514$	5500.37	0.472006
$515$	12866.0	1.10086
$516$	0	0
$517$	2299.43	0.195607
$518$	1810.07	0.153532
$519$	0	0
$520$	0	0
$521$	−10998.0	−0.924821	−0.462410	−	0.886666i	$-0.653015\pi$
$521$	−10998.0	−0.924821	−0.462410	+	0.886666i	$0.653015\pi$
$522$	0	0
$523$	18631.8	1.55776	0.778882	−	0.627171i	$-0.215787\pi$
$523$	18631.8	1.55776	0.778882	+	0.627171i	$0.215787\pi$
$524$	7351.39	0.612875
$525$	0	0
$526$	10771.8	0.892911
$527$	−393.797	−0.0325504
$528$	0	0
$529$	23018.7	1.89190
$530$	7584.46	0.621600
$531$	0	0
$532$	−4747.40	−0.386891
$533$	0	0
$534$	0	0
$535$	−3004.53	−0.242798
$536$	6553.99	0.528151
$537$	0	0
$538$	−6535.96	−0.523764
$539$	13379.8	1.06922
$540$	0	0
$541$	−14270.1	−1.13405	−0.567023	−	0.823702i	$-0.691905\pi$
$541$	−14270.1	−1.13405	−0.567023	+	0.823702i	$0.691905\pi$
$542$	6345.90	0.502915
$543$	0	0
$544$	−1251.66	−0.0986480
$545$	21958.1	1.72584
$546$	0	0
$547$	14379.7	1.12400	0.562002	−	0.827136i	$-0.310031\pi$
$547$	14379.7	1.12400	0.562002	+	0.827136i	$0.310031\pi$
$548$	5789.40	0.451298
$549$	0	0
$550$	−1969.21	−0.152668
$551$	18586.2	1.43702
$552$	0	0
$553$	1870.56	0.143842
$554$	6655.46	0.510404
$555$	0	0
$556$	−14687.5	−1.12031
$557$	−9035.10	−0.687306	−0.343653	−	0.939097i	$-0.611664\pi$
$557$	−9035.10	−0.687306	−0.343653	+	0.939097i	$0.611664\pi$
$558$	0	0
$559$	0	0
$560$	−3026.65	−0.228392
$561$	0	0
$562$	−279.875	−0.0210068
$563$	−7293.47	−0.545973	−0.272987	−	0.962018i	$-0.588012\pi$
$563$	−7293.47	−0.545973	−0.272987	+	0.962018i	$0.588012\pi$
$564$	0	0
$565$	23251.6	1.73133
$566$	6334.37	0.470413
$567$	0	0
$568$	−14090.4	−1.04088
$569$	26415.8	1.94623	0.973117	−	0.230310i	$-0.0739740\pi$
$569$	26415.8	1.94623	0.973117	+	0.230310i	$0.0739740\pi$
$570$	0	0
$571$	−23940.9	−1.75463	−0.877317	−	0.479912i	$-0.840669\pi$
$571$	−23940.9	−1.75463	−0.877317	+	0.479912i	$0.840669\pi$
$572$	0	0
$573$	0	0
$574$	−2574.86	−0.187234
$575$	5620.43	0.407632
$576$	0	0
$577$	10182.8	0.734691	0.367346	−	0.930085i	$-0.380267\pi$
$577$	10182.8	0.734691	0.367346	+	0.930085i	$0.380267\pi$
$578$	−6260.95	−0.450556
$579$	0	0
$580$	17660.6	1.26434
$581$	2261.35	0.161475
$582$	0	0
$583$	−24186.0	−1.71815
$584$	−6120.22	−0.433658
$585$	0	0
$586$	−3399.77	−0.239664
$587$	19107.5	1.34353	0.671763	−	0.740766i	$-0.265537\pi$
$587$	19107.5	1.34353	0.671763	+	0.740766i	$0.265537\pi$
$588$	0	0
$589$	−4769.94	−0.333687
$590$	9858.58	0.687917
$591$	0	0
$592$	−4220.06	−0.292979
$593$	−13597.5	−0.941621	−0.470810	−	0.882234i	$-0.656038\pi$
$593$	−13597.5	−0.941621	−0.470810	+	0.882234i	$0.656038\pi$
$594$	0	0
$595$	769.022	0.0529863
$596$	5871.17	0.403511
$597$	0	0
$598$	0	0
$599$	4150.37	0.283104	0.141552	−	0.989931i	$-0.454791\pi$
$599$	4150.37	0.283104	0.141552	+	0.989931i	$0.454791\pi$
$600$	0	0
$601$	19177.5	1.30161	0.650803	−	0.759247i	$-0.274433\pi$
$601$	19177.5	1.30161	0.650803	+	0.759247i	$0.274433\pi$
$602$	−4030.85	−0.272899
$603$	0	0
$604$	22356.3	1.50607
$605$	15907.9	1.06900
$606$	0	0
$607$	7994.36	0.534565	0.267282	−	0.963618i	$-0.413874\pi$
$607$	7994.36	0.534565	0.267282	+	0.963618i	$0.413874\pi$
$608$	−15161.0	−1.01128
$609$	0	0
$610$	3128.59	0.207660
$611$	0	0
$612$	0	0
$613$	15613.4	1.02874	0.514371	−	0.857568i	$-0.328026\pi$
$613$	15613.4	1.02874	0.514371	+	0.857568i	$0.328026\pi$
$614$	−4357.04	−0.286377
$615$	0	0
$616$	−8487.15	−0.555125
$617$	21636.2	1.41174	0.705868	−	0.708343i	$-0.250557\pi$
$617$	21636.2	1.41174	0.705868	+	0.708343i	$0.250557\pi$
$618$	0	0
$619$	10394.0	0.674911	0.337456	−	0.941341i	$-0.390434\pi$
$619$	10394.0	0.674911	0.337456	+	0.941341i	$0.390434\pi$
$620$	−4532.40	−0.293590
$621$	0	0
$622$	−4.24369	−0.000273563 0
$623$	−6476.25	−0.416478
$624$	0	0
$625$	−18472.6	−1.18225
$626$	3709.06	0.236811
$627$	0	0
$628$	−10251.5	−0.651400
$629$	1072.25	0.0679702
$630$	0	0
$631$	4644.14	0.292996	0.146498	−	0.989211i	$-0.453200\pi$
$631$	4644.14	0.292996	0.146498	+	0.989211i	$0.453200\pi$
$632$	3834.92	0.241369
$633$	0	0
$634$	−1856.32	−0.116284
$635$	2973.74	0.185841
$636$	0	0
$637$	0	0
$638$	14696.2	0.911955
$639$	0	0
$640$	−17866.6	−1.10350
$641$	153.922	0.00948451	0.00474225	−	0.999989i	$-0.498490\pi$
$641$	153.922	0.00948451	0.00474225	+	0.999989i	$0.498490\pi$
$642$	0	0
$643$	−8438.05	−0.517518	−0.258759	−	0.965942i	$-0.583314\pi$
$643$	−8438.05	−0.517518	−0.258759	+	0.965942i	$0.583314\pi$
$644$	10713.9	0.655572
$645$	0	0
$646$	733.865	0.0446959
$647$	665.236	0.0404221	0.0202111	−	0.999796i	$-0.493566\pi$
$647$	665.236	0.0404221	0.0202111	+	0.999796i	$0.493566\pi$
$648$	0	0
$649$	−31437.9	−1.90145
$650$	0	0
$651$	0	0
$652$	−6395.21	−0.384134
$653$	−22414.7	−1.34327	−0.671636	−	0.740882i	$-0.734408\pi$
$653$	−22414.7	−1.34327	−0.671636	+	0.740882i	$0.734408\pi$
$654$	0	0
$655$	−14424.2	−0.860458
$656$	6003.12	0.357291
$657$	0	0
$658$	521.485	0.0308960
$659$	−25585.6	−1.51240	−0.756202	−	0.654338i	$-0.772947\pi$
$659$	−25585.6	−1.51240	−0.756202	+	0.654338i	$0.772947\pi$
$660$	0	0
$661$	−13317.5	−0.783647	−0.391824	−	0.920040i	$-0.628156\pi$
$661$	−13317.5	−0.783647	−0.391824	+	0.920040i	$0.628156\pi$
$662$	−4921.59	−0.288947
$663$	0	0
$664$	4636.10	0.270957
$665$	9314.91	0.543183
$666$	0	0
$667$	−41945.2	−2.43497
$668$	−14935.0	−0.865048
$669$	0	0
$670$	−5687.70	−0.327963
$671$	−9976.71	−0.573989
$672$	0	0
$673$	9778.28	0.560067	0.280034	−	0.959990i	$-0.409654\pi$
$673$	9778.28	0.560067	0.280034	+	0.959990i	$0.409654\pi$
$674$	2946.82	0.168408
$675$	0	0
$676$	0	0
$677$	12392.8	0.703537	0.351769	−	0.936087i	$-0.385580\pi$
$677$	12392.8	0.703537	0.351769	+	0.936087i	$0.385580\pi$
$678$	0	0
$679$	14009.7	0.791815
$680$	1576.61	0.0889119
$681$	0	0
$682$	−3771.62	−0.211763
$683$	−18941.7	−1.06118	−0.530588	−	0.847630i	$-0.678029\pi$
$683$	−18941.7	−1.06118	−0.530588	+	0.847630i	$0.678029\pi$
$684$	0	0
$685$	−11359.4	−0.633608
$686$	7007.62	0.390018
$687$	0	0
$688$	9397.69	0.520761
$689$	0	0
$690$	0	0
$691$	32377.6	1.78249	0.891247	−	0.453519i	$-0.149832\pi$
$691$	32377.6	1.78249	0.891247	+	0.453519i	$0.149832\pi$
$692$	−19872.8	−1.09169
$693$	0	0
$694$	−431.034	−0.0235761
$695$	28818.5	1.57288
$696$	0	0
$697$	−1525.29	−0.0828903
$698$	−11548.4	−0.626240
$699$	0	0
$700$	1711.40	0.0924071
$701$	6072.34	0.327174	0.163587	−	0.986529i	$-0.447694\pi$
$701$	6072.34	0.327174	0.163587	+	0.986529i	$0.447694\pi$
$702$	0	0
$703$	12987.8	0.696789
$704$	−952.316	−0.0509826
$705$	0	0
$706$	−12771.6	−0.680829
$707$	7052.61	0.375163
$708$	0	0
$709$	−3398.52	−0.180020	−0.0900100	−	0.995941i	$-0.528690\pi$
$709$	−3398.52	−0.180020	−0.0900100	+	0.995941i	$0.528690\pi$
$710$	12228.0	0.646348
$711$	0	0
$712$	−13277.3	−0.698857
$713$	10764.8	0.565421
$714$	0	0
$715$	0	0
$716$	20180.4	1.05332
$717$	0	0
$718$	277.491	0.0144232
$719$	28833.6	1.49556	0.747782	−	0.663944i	$-0.231119\pi$
$719$	28833.6	1.49556	0.747782	+	0.663944i	$0.231119\pi$
$720$	0	0
$721$	−9304.72	−0.480619
$722$	63.6190	0.00327930
$723$	0	0
$724$	−26840.2	−1.37777
$725$	−6700.17	−0.343225
$726$	0	0
$727$	−12894.7	−0.657822	−0.328911	−	0.944361i	$-0.606682\pi$
$727$	−12894.7	−0.657822	−0.328911	+	0.944361i	$0.606682\pi$
$728$	0	0
$729$	0	0
$730$	5311.27	0.269286
$731$	−2387.79	−0.120815
$732$	0	0
$733$	−23754.7	−1.19700	−0.598499	−	0.801124i	$-0.704236\pi$
$733$	−23754.7	−1.19700	−0.598499	+	0.801124i	$0.704236\pi$
$734$	−3681.91	−0.185152
$735$	0	0
$736$	34215.2	1.71357
$737$	18137.4	0.906514
$738$	0	0
$739$	−26047.2	−1.29656	−0.648282	−	0.761400i	$-0.724512\pi$
$739$	−26047.2	−1.29656	−0.648282	+	0.761400i	$0.724512\pi$
$740$	12341.0	0.613059
$741$	0	0
$742$	−5485.10	−0.271381
$743$	15919.3	0.786033	0.393016	−	0.919531i	$-0.371432\pi$
$743$	15919.3	0.786033	0.393016	+	0.919531i	$0.371432\pi$
$744$	0	0
$745$	−11519.9	−0.566517
$746$	−21.8924	−0.00107445
$747$	0	0
$748$	−2223.67	−0.108697
$749$	2172.88	0.106002
$750$	0	0
$751$	2237.59	0.108723	0.0543613	−	0.998521i	$-0.482688\pi$
$751$	2237.59	0.108723	0.0543613	+	0.998521i	$0.482688\pi$
$752$	−1215.81	−0.0589575
$753$	0	0
$754$	0	0
$755$	−43865.5	−2.11448
$756$	0	0
$757$	3263.91	0.156709	0.0783545	−	0.996926i	$-0.475033\pi$
$757$	3263.91	0.156709	0.0783545	+	0.996926i	$0.475033\pi$
$758$	−2447.25	−0.117267
$759$	0	0
$760$	19096.9	0.911471
$761$	−1904.91	−0.0907398	−0.0453699	−	0.998970i	$-0.514447\pi$
$761$	−1904.91	−0.0907398	−0.0453699	+	0.998970i	$0.514447\pi$
$762$	0	0
$763$	−15880.2	−0.753474
$764$	25078.2	1.18756
$765$	0	0
$766$	−9945.88	−0.469137
$767$	0	0
$768$	0	0
$769$	3429.87	0.160838	0.0804190	−	0.996761i	$-0.474374\pi$
$769$	3429.87	0.160838	0.0804190	+	0.996761i	$0.474374\pi$
$770$	7365.35	0.344713
$771$	0	0
$772$	−31991.4	−1.49144
$773$	−28243.5	−1.31416	−0.657082	−	0.753819i	$-0.728210\pi$
$773$	−28243.5	−1.31416	−0.657082	+	0.753819i	$0.728210\pi$
$774$	0	0
$775$	1719.53	0.0796997
$776$	28721.9	1.32868
$777$	0	0
$778$	−18587.0	−0.856522
$779$	−18475.4	−0.849741
$780$	0	0
$781$	−38993.5	−1.78655
$782$	−1656.19	−0.0757355
$783$	0	0
$784$	−7074.48	−0.322270
$785$	20114.5	0.914545
$786$	0	0
$787$	6626.72	0.300149	0.150074	−	0.988675i	$-0.452049\pi$
$787$	6626.72	0.300149	0.150074	+	0.988675i	$0.452049\pi$
$788$	−22663.0	−1.02454
$789$	0	0
$790$	−3328.03	−0.149881
$791$	−16815.6	−0.755871
$792$	0	0
$793$	0	0
$794$	1329.25	0.0594120
$795$	0	0
$796$	−31765.5	−1.41444
$797$	−20750.8	−0.922249	−0.461125	−	0.887335i	$-0.652554\pi$
$797$	−20750.8	−0.922249	−0.461125	+	0.887335i	$0.652554\pi$
$798$	0	0
$799$	308.917	0.0136780
$800$	5465.41	0.241539
$801$	0	0
$802$	−5933.89	−0.261263
$803$	−16937.0	−0.744327
$804$	0	0
$805$	−21021.9	−0.920403
$806$	0	0
$807$	0	0
$808$	14458.9	0.629531
$809$	10005.3	0.434818	0.217409	−	0.976081i	$-0.430239\pi$
$809$	10005.3	0.434818	0.217409	+	0.976081i	$0.430239\pi$
$810$	0	0
$811$	32773.5	1.41903	0.709515	−	0.704690i	$-0.248914\pi$
$811$	32773.5	1.41903	0.709515	+	0.704690i	$0.248914\pi$
$812$	−12772.2	−0.551990
$813$	0	0
$814$	10269.5	0.442194
$815$	12548.1	0.539313
$816$	0	0
$817$	−28922.6	−1.23852
$818$	3433.63	0.146765
$819$	0	0
$820$	−17555.3	−0.747632
$821$	−3031.74	−0.128878	−0.0644388	−	0.997922i	$-0.520526\pi$
$821$	−3031.74	−0.128878	−0.0644388	+	0.997922i	$0.520526\pi$
$822$	0	0
$823$	31057.0	1.31541	0.657704	−	0.753277i	$-0.271528\pi$
$823$	31057.0	1.31541	0.657704	+	0.753277i	$0.271528\pi$
$824$	−19076.0	−0.806486
$825$	0	0
$826$	−7129.75	−0.300334
$827$	−8770.13	−0.368764	−0.184382	−	0.982855i	$-0.559028\pi$
$827$	−8770.13	−0.368764	−0.184382	+	0.982855i	$0.559028\pi$
$828$	0	0
$829$	−38698.9	−1.62131	−0.810656	−	0.585523i	$-0.800889\pi$
$829$	−38698.9	−1.62131	−0.810656	+	0.585523i	$0.800889\pi$
$830$	−4023.32	−0.168255
$831$	0	0
$832$	0	0
$833$	1797.50	0.0747657
$834$	0	0
$835$	29304.0	1.21450
$836$	−26934.6	−1.11430
$837$	0	0
$838$	7964.43	0.328313
$839$	−22543.7	−0.927647	−0.463824	−	0.885928i	$-0.653523\pi$
$839$	−22543.7	−0.927647	−0.463824	+	0.885928i	$0.653523\pi$
$840$	0	0
$841$	25614.3	1.05024
$842$	11568.9	0.473502
$843$	0	0
$844$	−11161.3	−0.455198
$845$	0	0
$846$	0	0
$847$	−11504.6	−0.466710
$848$	12788.2	0.517863
$849$	0	0
$850$	−264.553	−0.0106754
$851$	−29310.8	−1.18068
$852$	0	0
$853$	−7751.80	−0.311156	−0.155578	−	0.987824i	$-0.549724\pi$
$853$	−7751.80	−0.311156	−0.155578	+	0.987824i	$0.549724\pi$
$854$	−2262.60	−0.0906613
$855$	0	0
$856$	4454.73	0.177873
$857$	30235.6	1.20517	0.602583	−	0.798056i	$-0.294138\pi$
$857$	30235.6	1.20517	0.602583	+	0.798056i	$0.294138\pi$
$858$	0	0
$859$	3568.09	0.141725	0.0708624	−	0.997486i	$-0.477425\pi$
$859$	3568.09	0.141725	0.0708624	+	0.997486i	$0.477425\pi$
$860$	−27482.2	−1.08969
$861$	0	0
$862$	2662.19	0.105191
$863$	−406.509	−0.0160345	−0.00801723	−	0.999968i	$-0.502552\pi$
$863$	−406.509	−0.0160345	−0.00801723	+	0.999968i	$0.502552\pi$
$864$	0	0
$865$	38992.5	1.53270
$866$	−19216.4	−0.754040
$867$	0	0
$868$	3277.85	0.128177
$869$	10612.7	0.414283
$870$	0	0
$871$	0	0
$872$	−32556.6	−1.26434
$873$	0	0
$874$	−20060.8	−0.776394
$875$	10650.8	0.411499
$876$	0	0
$877$	18961.9	0.730102	0.365051	−	0.930988i	$-0.381052\pi$
$877$	18961.9	0.730102	0.365051	+	0.930988i	$0.381052\pi$
$878$	−7720.48	−0.296758
$879$	0	0
$880$	−17171.9	−0.657800
$881$	−31028.9	−1.18660	−0.593298	−	0.804983i	$-0.702174\pi$
$881$	−31028.9	−1.18660	−0.593298	+	0.804983i	$0.702174\pi$
$882$	0	0
$883$	33896.2	1.29184	0.645922	−	0.763404i	$-0.276473\pi$
$883$	33896.2	1.29184	0.645922	+	0.763404i	$0.276473\pi$
$884$	0	0
$885$	0	0
$886$	−6938.64	−0.263102
$887$	−1773.35	−0.0671288	−0.0335644	−	0.999437i	$-0.510686\pi$
$887$	−1773.35	−0.0671288	−0.0335644	+	0.999437i	$0.510686\pi$
$888$	0	0
$889$	−2150.62	−0.0811354
$890$	11522.3	0.433965
$891$	0	0
$892$	−10864.9	−0.407831
$893$	3741.81	0.140218
$894$	0	0
$895$	−39596.1	−1.47883
$896$	12921.2	0.481770
$897$	0	0
$898$	−7780.14	−0.289116
$899$	−12832.8	−0.476083
$900$	0	0
$901$	−3249.26	−0.120143
$902$	−14608.6	−0.539260
$903$	0	0
$904$	−34474.4	−1.26836
$905$	52663.3	1.93435
$906$	0	0
$907$	14401.6	0.527231	0.263615	−	0.964628i	$-0.415085\pi$
$907$	14401.6	0.527231	0.263615	+	0.964628i	$0.415085\pi$
$908$	−16347.3	−0.597473
$909$	0	0
$910$	0	0
$911$	−19324.5	−0.702799	−0.351400	−	0.936226i	$-0.614294\pi$
$911$	−19324.5	−0.702799	−0.351400	+	0.936226i	$0.614294\pi$
$912$	0	0
$913$	12829.9	0.465069
$914$	1305.45	0.0472433
$915$	0	0
$916$	12454.7	0.449253
$917$	10431.6	0.375663
$918$	0	0
$919$	18649.1	0.669399	0.334700	−	0.942325i	$-0.391365\pi$
$919$	18649.1	0.669399	0.334700	+	0.942325i	$0.391365\pi$
$920$	−43097.9	−1.54445
$921$	0	0
$922$	−20560.0	−0.734390
$923$	0	0
$924$	0	0
$925$	−4681.99	−0.166425
$926$	−165.733	−0.00588156
$927$	0	0
$928$	−40788.3	−1.44283
$929$	27326.4	0.965070	0.482535	−	0.875877i	$-0.339716\pi$
$929$	27326.4	0.965070	0.482535	+	0.875877i	$0.339716\pi$
$930$	0	0
$931$	21772.6	0.766453
$932$	6839.98	0.240398
$933$	0	0
$934$	24231.5	0.848908
$935$	4363.08	0.152608
$936$	0	0
$937$	28699.1	1.00060	0.500298	−	0.865853i	$-0.333224\pi$
$937$	28699.1	1.00060	0.500298	+	0.865853i	$0.333224\pi$
$938$	4113.36	0.143183
$939$	0	0
$940$	3555.47	0.123369
$941$	−55401.1	−1.91926	−0.959630	−	0.281264i	$-0.909246\pi$
$941$	−55401.1	−1.91926	−0.959630	+	0.281264i	$0.909246\pi$
$942$	0	0
$943$	41695.2	1.43985
$944$	16622.6	0.573113
$945$	0	0
$946$	−22869.2	−0.785986
$947$	33258.4	1.14124	0.570619	−	0.821215i	$-0.306703\pi$
$947$	33258.4	1.14124	0.570619	+	0.821215i	$0.306703\pi$
$948$	0	0
$949$	0	0
$950$	−3204.44	−0.109438
$951$	0	0
$952$	−1140.21	−0.0388175
$953$	4411.89	0.149963	0.0749817	−	0.997185i	$-0.476110\pi$
$953$	4411.89	0.149963	0.0749817	+	0.997185i	$0.476110\pi$
$954$	0	0
$955$	−49206.1	−1.66730
$956$	−25367.7	−0.858211
$957$	0	0
$958$	−11878.3	−0.400594
$959$	8215.17	0.276623
$960$	0	0
$961$	−26497.6	−0.889450
$962$	0	0
$963$	0	0
$964$	22679.5	0.757737
$965$	62770.5	2.09394
$966$	0	0
$967$	26014.3	0.865114	0.432557	−	0.901607i	$-0.357611\pi$
$967$	26014.3	0.865114	0.432557	+	0.901607i	$0.357611\pi$
$968$	−23586.1	−0.783148
$969$	0	0
$970$	−24925.5	−0.825063
$971$	−36204.9	−1.19657	−0.598286	−	0.801283i	$-0.704151\pi$
$971$	−36204.9	−1.19657	−0.598286	+	0.801283i	$0.704151\pi$
$972$	0	0
$973$	−20841.6	−0.686693
$974$	9306.69	0.306166
$975$	0	0
$976$	5275.12	0.173005
$977$	−13916.4	−0.455706	−0.227853	−	0.973696i	$-0.573170\pi$
$977$	−13916.4	−0.455706	−0.227853	+	0.973696i	$0.573170\pi$
$978$	0	0
$979$	−36743.3	−1.19951
$980$	20688.3	0.674351
$981$	0	0
$982$	−842.954	−0.0273928
$983$	−13585.3	−0.440796	−0.220398	−	0.975410i	$-0.570736\pi$
$983$	−13585.3	−0.440796	−0.220398	+	0.975410i	$0.570736\pi$
$984$	0	0
$985$	44467.2	1.43842
$986$	1974.36	0.0637691
$987$	0	0
$988$	0	0
$989$	65272.4	2.09863
$990$	0	0
$991$	−1024.49	−0.0328396	−0.0164198	−	0.999865i	$-0.505227\pi$
$991$	−1024.49	−0.0328396	−0.0164198	+	0.999865i	$0.505227\pi$
$992$	10467.9	0.335036
$993$	0	0
$994$	−8843.29	−0.282185
$995$	62327.3	1.98584
$996$	0	0
$997$	18514.9	0.588136	0.294068	−	0.955784i	$-0.404991\pi$
$997$	18514.9	0.588136	0.294068	+	0.955784i	$0.404991\pi$
$998$	−5867.43	−0.186102
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bc.1.5		8
3.2	odd	2	inner	1521.4.a.bc.1.4		8
13.3	even	3		117.4.g.f.100.4	yes	16
13.9	even	3		117.4.g.f.55.4	✓	16
13.12	even	2		1521.4.a.bd.1.4		8
39.29	odd	6		117.4.g.f.100.5	yes	16
39.35	odd	6		117.4.g.f.55.5	yes	16
39.38	odd	2		1521.4.a.bd.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.g.f.55.4	✓	16	13.9	even	3
117.4.g.f.55.5	yes	16	39.35	odd	6
117.4.g.f.100.4	yes	16	13.3	even	3
117.4.g.f.100.5	yes	16	39.29	odd	6
1521.4.a.bc.1.4		8	3.2	odd	2	inner
1521.4.a.bc.1.5		8	1.1	even	1	trivial
1521.4.a.bd.1.4		8	13.12	even	2
1521.4.a.bd.1.5		8	39.38	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28670 q^{2} -6.34441 q^{4} +12.4484 q^{5} -9.00273 q^{7} -18.4569 q^{8} +O(q^{10})\)	\(q+1.28670 q^{2} -6.34441 q^{4} +12.4484 q^{5} -9.00273 q^{7} -18.4569 q^{8} +16.0173 q^{10} -51.0774 q^{11} -11.5838 q^{14} +27.0069 q^{16} -6.86199 q^{17} -83.1170 q^{19} -78.9779 q^{20} -65.7211 q^{22} +187.579 q^{23} +29.9631 q^{25} +57.1170 q^{28} -223.614 q^{29} +57.3882 q^{31} +182.405 q^{32} -8.82930 q^{34} -112.070 q^{35} -156.259 q^{37} -106.946 q^{38} -229.759 q^{40} +222.281 q^{41} +347.974 q^{43} +324.056 q^{44} +241.357 q^{46} -45.0185 q^{47} -261.951 q^{49} +38.5534 q^{50} +473.516 q^{53} -635.833 q^{55} +166.163 q^{56} -287.724 q^{58} +615.494 q^{59} +195.325 q^{61} +73.8412 q^{62} +18.6446 q^{64} -355.097 q^{67} +43.5353 q^{68} -144.200 q^{70} +763.420 q^{71} +331.595 q^{73} -201.057 q^{74} +527.329 q^{76} +459.836 q^{77} -207.777 q^{79} +336.193 q^{80} +286.008 q^{82} -251.185 q^{83} -85.4209 q^{85} +447.737 q^{86} +942.731 q^{88} +719.366 q^{89} -1190.08 q^{92} -57.9252 q^{94} -1034.68 q^{95} -1556.16 q^{97} -337.051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{4} - 22 q^{7}+O(q^{10}) \) 8 * q + 40 * q^4 - 22 * q^7	\( 8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100}) \) 8 * q + 40 * q^4 - 22 * q^7 + 36 * q^10 + 204 * q^16 + 244 * q^19 + 136 * q^22 + 354 * q^25 - 452 * q^28 + 242 * q^31 - 1292 * q^34 + 1018 * q^37 + 1700 * q^40 + 74 * q^43 - 896 * q^46 + 298 * q^49 + 1300 * q^55 + 812 * q^58 + 1148 * q^61 + 3636 * q^64 - 2198 * q^67 + 2200 * q^70 - 2176 * q^73 + 6936 * q^76 + 1862 * q^79 + 5436 * q^82 - 890 * q^85 + 3528 * q^88 - 3104 * q^94 - 4370 * q^97

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