LMFDB - Embedded newform 1521.4.a.bd.1.4

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			1.4
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	$-0.573041\pi$
$2$	−1.28670	−0.454916	−0.227458	+	0.973788i	$0.573041\pi$
$3$	0	0
$3$	0	0
$4$	−6.34441	−0.793052
$5$	−12.4484	−1.11342	−0.556710	−	0.830707i	$-0.687937\pi$
$5$	−12.4484	−1.11342	−0.556710	+	0.830707i	$0.687937\pi$
$6$	0	0
$7$	9.00273	0.486102	0.243051	−	0.970014i	$-0.421852\pi$
$7$	9.00273	0.486102	0.243051	+	0.970014i	$0.421852\pi$
$8$	18.4569	0.815687
$9$	0	0
$10$	16.0173	0.506512
$11$	51.0774	1.40004	0.700019	−	0.714124i	$-0.253175\pi$
$11$	51.0774	1.40004	0.700019	+	0.714124i	$0.253175\pi$
$12$	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	$-0.332672\pi$
$19$	83.1170	1.00360	0.501799	+	0.864984i	$0.332672\pi$
$20$	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	$-0.553164\pi$
$31$	−57.3882	−0.332491	−0.166246	+	0.986084i	$0.553164\pi$
$32$	−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	$-0.387151\pi$
$37$	156.259	0.694291	0.347146	+	0.937811i	$0.387151\pi$
$38$	−106.946	−0.456552
$39$	0	0
$40$	−229.759	−0.908203
$41$	−222.281	−0.846695	−0.423348	−	0.905967i	$-0.639145\pi$
$41$	−222.281	−0.846695	−0.423348	+	0.905967i	$0.639145\pi$
$42$	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	$-0.477745\pi$
$47$	45.0185	0.139715	0.0698577	+	0.997557i	$0.477745\pi$
$48$	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	$-0.737618\pi$
$59$	−615.494	−1.35814	−0.679072	+	0.734071i	$0.737618\pi$
$60$	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	$-0.395058\pi$
$67$	355.097	0.647492	0.323746	+	0.946144i	$0.395058\pi$
$68$	43.5353	0.0776387
$69$	0	0
$70$	144.200	0.246217
$71$	−763.420	−1.27608	−0.638038	−	0.770005i	$-0.720254\pi$
$71$	−763.420	−1.27608	−0.638038	+	0.770005i	$0.720254\pi$
$72$	0	0
$73$	−331.595	−0.531647	−0.265824	−	0.964022i	$-0.585644\pi$
$73$	−331.595	−0.531647	−0.265824	+	0.964022i	$0.585644\pi$
$74$	−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	$-0.446885\pi$
$83$	251.185	0.332183	0.166091	+	0.986110i	$0.446885\pi$
$84$	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	$-0.640917\pi$
$89$	−719.366	−0.856771	−0.428385	+	0.903596i	$0.640917\pi$
$90$	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	$-0.197036\pi$
$97$	1556.16	1.62891	0.814454	+	0.580228i	$0.197036\pi$
$98$	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	$-0.782258\pi$
$109$	−1763.93	−1.55003	−0.775016	+	0.631941i	$0.782258\pi$
$110$	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	$-0.408162\pi$
$137$	912.520	0.569064	0.284532	+	0.958666i	$0.408162\pi$
$138$	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	$-0.418121\pi$
$149$	925.408	0.508808	0.254404	+	0.967098i	$0.418121\pi$
$150$	0	0
$151$	3523.78	1.89908	0.949541	−	0.313644i	$-0.101550\pi$
$151$	3523.78	1.89908	0.949541	+	0.313644i	$0.101550\pi$
$152$	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	$-0.577865\pi$
$163$	−1008.01	−0.484375	−0.242188	+	0.970229i	$0.577865\pi$
$164$	1410.24	0.671473
$165$	0	0
$166$	−323.199	−0.151115
$167$	−2354.04	−1.09078	−0.545392	−	0.838181i	$-0.683619\pi$
$167$	−2354.04	−1.09078	−0.545392	+	0.838181i	$0.683619\pi$
$168$	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	$-0.889473\pi$
$193$	−5042.45	−1.88064	−0.940319	+	0.340294i	$0.889473\pi$
$194$	−2002.31	−0.741016
$195$	0	0
$196$	1661.92	0.605658
$197$	−3572.12	−1.29189	−0.645947	−	0.763382i	$-0.723537\pi$
$197$	−3572.12	−1.29189	−0.645947	+	0.763382i	$0.723537\pi$
$198$	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	$-0.582776\pi$
$223$	−1712.52	−0.514255	−0.257127	+	0.966377i	$0.582776\pi$
$224$	−1642.14	−0.489822
$225$	0	0
$226$	−2403.33	−0.707378
$227$	−2576.65	−0.753385	−0.376693	−	0.926338i	$-0.622939\pi$
$227$	−2576.65	−0.753385	−0.376693	+	0.926338i	$0.622939\pi$
$228$	0	0
$229$	1963.10	0.566486	0.283243	−	0.959048i	$-0.408590\pi$
$229$	1963.10	0.566486	0.283243	+	0.959048i	$0.408590\pi$
$230$	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	$-0.681985\pi$
$239$	−3998.43	−1.08216	−0.541081	+	0.840970i	$0.681985\pi$
$240$	0	0
$241$	3574.72	0.955470	0.477735	−	0.878504i	$-0.341458\pi$
$241$	3574.72	0.955470	0.477735	+	0.878504i	$0.341458\pi$
$242$	−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	$-0.686424\pi$
$271$	−4931.93	−1.10551	−0.552756	+	0.833343i	$0.686424\pi$
$272$	−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	$-0.492650\pi$
$281$	217.515	0.0461773	0.0230887	+	0.999733i	$0.492650\pi$
$282$	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	$-0.415151\pi$
$293$	2642.25	0.526832	0.263416	+	0.964682i	$0.415151\pi$
$294$	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	$-0.398076\pi$
$307$	3386.22	0.629517	0.314759	+	0.949172i	$0.398076\pi$
$308$	−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	$-0.459206\pi$
$317$	1442.70	0.255616	0.127808	+	0.991799i	$0.459206\pi$
$318$	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	$-0.397129\pi$
$331$	3824.98	0.635166	0.317583	+	0.948230i	$0.397129\pi$
$332$	−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	$-0.258356\pi$
$349$	8975.27	1.37661	0.688303	+	0.725424i	$0.258356\pi$
$350$	−347.086	−0.0530072
$351$	0	0
$352$	−9316.77	−1.41075
$353$	9925.87	1.49660	0.748302	−	0.663359i	$-0.230870\pi$
$353$	9925.87	1.49660	0.748302	+	0.663359i	$0.230870\pi$
$354$	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	$-0.505046\pi$
$359$	−215.661	−0.0317052	−0.0158526	+	0.999874i	$0.505046\pi$
$360$	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	$-0.458859\pi$
$379$	1901.97	0.257777	0.128888	+	0.991659i	$0.458859\pi$
$380$	6564.41	0.886177
$381$	0	0
$382$	5086.05	0.681217
$383$	7729.78	1.03126	0.515631	−	0.856811i	$-0.327558\pi$
$383$	7729.78	1.03126	0.515631	+	0.856811i	$0.327558\pi$
$384$	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	$-0.520800\pi$
$397$	−1033.07	−0.130600	−0.0653000	+	0.997866i	$0.520800\pi$
$398$	−6442.29	−0.811364
$399$	0	0
$400$	809.210	0.101151
$401$	4611.73	0.574311	0.287155	−	0.957884i	$-0.407290\pi$
$401$	4611.73	0.574311	0.287155	+	0.957884i	$0.407290\pi$
$402$	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	$-0.551572\pi$
$409$	−2668.56	−0.322621	−0.161311	+	0.986904i	$0.551572\pi$
$410$	−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	$-0.674228\pi$
$421$	−8991.13	−1.04086	−0.520429	+	0.853905i	$0.674228\pi$
$422$	−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	$-0.536884\pi$
$431$	−2069.01	−0.231232	−0.115616	+	0.993294i	$0.536884\pi$
$432$	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	$-0.397066\pi$
$449$	6046.60	0.635538	0.317769	+	0.948168i	$0.397066\pi$
$450$	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	$-0.516536\pi$
$457$	−1014.57	−0.103851	−0.0519253	+	0.998651i	$0.516536\pi$
$458$	−2525.91	−0.257704
$459$	0	0
$460$	14814.6	1.50159
$461$	15978.9	1.61434	0.807171	−	0.590318i	$-0.200997\pi$
$461$	15978.9	1.61434	0.807171	+	0.590318i	$0.200997\pi$
$462$	0	0
$463$	128.805	0.0129289	0.00646445	−	0.999979i	$-0.497942\pi$
$463$	128.805	0.0129289	0.00646445	+	0.999979i	$0.497942\pi$
$464$	−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	$-0.354874\pi$
$479$	9231.59	0.880589	0.440294	+	0.897853i	$0.354874\pi$
$480$	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	$-0.609246\pi$
$487$	−7233.01	−0.673016	−0.336508	+	0.941681i	$0.609246\pi$
$488$	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	$-0.434428\pi$
$499$	4560.07	0.409092	0.204546	+	0.978857i	$0.434428\pi$
$500$	−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	$-0.256967\pi$
$509$	15880.9	1.38292	0.691461	+	0.722413i	$0.256967\pi$
$510$	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	$-0.308095\pi$
$541$	14270.1	1.13405	0.567023	+	0.823702i	$0.308095\pi$
$542$	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	$-0.388336\pi$
$557$	9035.10	0.687306	0.343653	+	0.939097i	$0.388336\pi$
$558$	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	$-0.619733\pi$
$577$	−10182.8	−0.734691	−0.367346	+	0.930085i	$0.619733\pi$
$578$	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	$-0.734463\pi$
$587$	−19107.5	−1.34353	−0.671763	+	0.740766i	$0.734463\pi$
$588$	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	$-0.343962\pi$
$593$	13597.5	0.941621	0.470810	+	0.882234i	$0.343962\pi$
$594$	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	$-0.671974\pi$
$613$	−15613.4	−1.02874	−0.514371	+	0.857568i	$0.671974\pi$
$614$	−4357.04	−0.286377
$615$	0	0
$616$	8487.15	0.555125
$617$	−21636.2	−1.41174	−0.705868	−	0.708343i	$-0.749443\pi$
$617$	−21636.2	−1.41174	−0.705868	+	0.708343i	$0.749443\pi$
$618$	0	0
$619$	−10394.0	−0.674911	−0.337456	−	0.941341i	$-0.609566\pi$
$619$	−10394.0	−0.674911	−0.337456	+	0.941341i	$0.609566\pi$
$620$	−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	$-0.546800\pi$
$631$	−4644.14	−0.292996	−0.146498	+	0.989211i	$0.546800\pi$
$632$	−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	$-0.416686\pi$
$643$	8438.05	0.517518	0.258759	+	0.965942i	$0.416686\pi$
$644$	−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	$-0.371844\pi$
$661$	13317.5	0.783647	0.391824	+	0.920040i	$0.371844\pi$
$662$	−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	$-0.321971\pi$
$683$	18941.7	1.06118	0.530588	+	0.847630i	$0.321971\pi$
$684$	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	$-0.850168\pi$
$691$	−32377.6	−1.78249	−0.891247	+	0.453519i	$0.850168\pi$
$692$	−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	$-0.471310\pi$
$709$	3398.52	0.180020	0.0900100	+	0.995941i	$0.471310\pi$
$710$	−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	$-0.295764\pi$
$733$	23754.7	1.19700	0.598499	+	0.801124i	$0.295764\pi$
$734$	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	$-0.275488\pi$
$739$	26047.2	1.29656	0.648282	+	0.761400i	$0.275488\pi$
$740$	12341.0	0.613059
$741$	0	0
$742$	−5485.10	−0.271381
$743$	−15919.3	−0.786033	−0.393016	−	0.919531i	$-0.628568\pi$
$743$	−15919.3	−0.786033	−0.393016	+	0.919531i	$0.628568\pi$
$744$	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	$-0.485553\pi$
$761$	1904.91	0.0907398	0.0453699	+	0.998970i	$0.485553\pi$
$762$	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	$-0.525626\pi$
$769$	−3429.87	−0.160838	−0.0804190	+	0.996761i	$0.525626\pi$
$770$	7365.35	0.344713
$771$	0	0
$772$	31991.4	1.49144
$773$	28243.5	1.31416	0.657082	−	0.753819i	$-0.271790\pi$
$773$	28243.5	1.31416	0.657082	+	0.753819i	$0.271790\pi$
$774$	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	$-0.547951\pi$
$787$	−6626.72	−0.300149	−0.150074	+	0.988675i	$0.547951\pi$
$788$	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	$-0.751086\pi$
$811$	−32773.5	−1.41903	−0.709515	+	0.704690i	$0.751086\pi$
$812$	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	$-0.479474\pi$
$821$	3031.74	0.128878	0.0644388	+	0.997922i	$0.479474\pi$
$822$	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	$-0.440972\pi$
$827$	8770.13	0.368764	0.184382	+	0.982855i	$0.440972\pi$
$828$	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	$-0.346477\pi$
$839$	22543.7	0.927647	0.463824	+	0.885928i	$0.346477\pi$
$840$	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	$-0.450276\pi$
$853$	7751.80	0.311156	0.155578	+	0.987824i	$0.450276\pi$
$854$	−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	$-0.497448\pi$
$863$	406.509	0.0160345	0.00801723	+	0.999968i	$0.497448\pi$
$864$	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	$-0.618948\pi$
$877$	−18961.9	−0.730102	−0.365051	+	0.930988i	$0.618948\pi$
$878$	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	$-0.660284\pi$
$929$	−27326.4	−0.965070	−0.482535	+	0.875877i	$0.660284\pi$
$930$	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	$-0.0907538\pi$
$941$	55401.1	1.91926	0.959630	+	0.281264i	$0.0907538\pi$
$942$	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	$-0.693297\pi$
$947$	−33258.4	−1.14124	−0.570619	+	0.821215i	$0.693297\pi$
$948$	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	$-0.642389\pi$
$967$	−26014.3	−0.865114	−0.432557	+	0.901607i	$0.642389\pi$
$968$	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	$-0.426830\pi$
$977$	13916.4	0.455706	0.227853	+	0.973696i	$0.426830\pi$
$978$	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	$-0.429264\pi$
$983$	13585.3	0.440796	0.220398	+	0.975410i	$0.429264\pi$
$984$	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.bd.1.4		8
3.2	odd	2	inner	1521.4.a.bd.1.5		8
13.4	even	6		117.4.g.f.55.4	✓	16
13.10	even	6		117.4.g.f.100.4	yes	16
13.12	even	2		1521.4.a.bc.1.5		8
39.17	odd	6		117.4.g.f.55.5	yes	16
39.23	odd	6		117.4.g.f.100.5	yes	16
39.38	odd	2		1521.4.a.bc.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.g.f.55.4	✓	16	13.4	even	6
117.4.g.f.55.5	yes	16	39.17	odd	6
117.4.g.f.100.4	yes	16	13.10	even	6
117.4.g.f.100.5	yes	16	39.23	odd	6
1521.4.a.bc.1.4		8	39.38	odd	2
1521.4.a.bc.1.5		8	13.12	even	2
1521.4.a.bd.1.4		8	1.1	even	1	trivial
1521.4.a.bd.1.5		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
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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