LMFDB - Embedded newform 1521.4.a.bh.1.2

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.83438$ 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)

$f(q)$	$=$	$q-3.83438 q^{2} +6.70249 q^{4} +11.3710 q^{5} -31.0623 q^{7} +4.97517 q^{8} +O(q^{10})$	$q-3.83438 q^{2} +6.70249 q^{4} +11.3710 q^{5} -31.0623 q^{7} +4.97517 q^{8} -43.6008 q^{10} -20.9478 q^{11} +119.105 q^{14} -72.6966 q^{16} -114.389 q^{17} +45.1768 q^{19} +76.2140 q^{20} +80.3219 q^{22} -73.9590 q^{23} +4.29980 q^{25} -208.194 q^{28} +27.2113 q^{29} -179.587 q^{31} +238.945 q^{32} +438.610 q^{34} -353.209 q^{35} -354.736 q^{37} -173.225 q^{38} +56.5727 q^{40} +81.4187 q^{41} -256.018 q^{43} -140.402 q^{44} +283.587 q^{46} +463.501 q^{47} +621.865 q^{49} -16.4871 q^{50} -76.6055 q^{53} -238.198 q^{55} -154.540 q^{56} -104.338 q^{58} +54.4676 q^{59} -494.496 q^{61} +688.606 q^{62} -334.634 q^{64} +611.991 q^{67} -766.688 q^{68} +1354.34 q^{70} +16.3056 q^{71} -321.825 q^{73} +1360.19 q^{74} +302.797 q^{76} +650.686 q^{77} +385.324 q^{79} -826.633 q^{80} -312.190 q^{82} +663.760 q^{83} -1300.71 q^{85} +981.671 q^{86} -104.219 q^{88} +545.723 q^{89} -495.709 q^{92} -1777.24 q^{94} +513.706 q^{95} +689.189 q^{97} -2384.47 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8}+O(q^{10}) $ 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8	$ 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8} - 147 q^{10} + 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} - 161 q^{19} + 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} - 344 q^{28} - 280 q^{29} - 706 q^{31} + 680 q^{32} - 216 q^{34} - 20 q^{35} - 298 q^{37} + 739 q^{38} + 13 q^{40} + 1201 q^{41} - 533 q^{43} + 355 q^{44} + 840 q^{46} + 956 q^{47} + 403 q^{49} - 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} + 2877 q^{58} + 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} + 931 q^{67} + 1536 q^{68} + 4854 q^{70} + 2046 q^{71} + 45 q^{73} + 1990 q^{74} + 3608 q^{76} + 718 q^{77} + 412 q^{79} - 787 q^{80} + 2757 q^{82} + 3709 q^{83} + 2106 q^{85} + 125 q^{86} - 636 q^{88} + 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} + 1087 q^{97} - 282 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8 - 147 * q^10 + 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 - 161 * q^19 + 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 - 344 * q^28 - 280 * q^29 - 706 * q^31 + 680 * q^32 - 216 * q^34 - 20 * q^35 - 298 * q^37 + 739 * q^38 + 13 * q^40 + 1201 * q^41 - 533 * q^43 + 355 * q^44 + 840 * q^46 + 956 * q^47 + 403 * q^49 - 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 + 2877 * q^58 + 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 + 931 * q^67 + 1536 * q^68 + 4854 * q^70 + 2046 * q^71 + 45 * q^73 + 1990 * q^74 + 3608 * q^76 + 718 * q^77 + 412 * q^79 - 787 * q^80 + 2757 * q^82 + 3709 * q^83 + 2106 * q^85 + 125 * q^86 - 636 * q^88 + 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 + 1087 * q^97 - 282 * q^98

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 $

$2$	−3.83438	−1.35566	−0.677829	−	0.735219i	$-0.737079\pi$
$2$	−3.83438	−1.35566	−0.677829	+	0.735219i	$0.737079\pi$
$3$	0	0
$3$	0	0
$4$	6.70249	0.837811
$5$	11.3710	1.01705	0.508527	−	0.861046i	$-0.330190\pi$
$5$	11.3710	1.01705	0.508527	+	0.861046i	$0.330190\pi$
$6$	0	0
$7$	−31.0623	−1.67721	−0.838603	−	0.544744i	$-0.816627\pi$
$7$	−31.0623	−1.67721	−0.838603	+	0.544744i	$0.816627\pi$
$8$	4.97517	0.219873
$9$	0	0
$10$	−43.6008	−1.37878
$11$	−20.9478	−0.574182	−0.287091	−	0.957903i	$-0.592688\pi$
$11$	−20.9478	−0.574182	−0.287091	+	0.957903i	$0.592688\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	119.105	2.27372
$15$	0	0
$16$	−72.6966	−1.13588
$17$	−114.389	−1.63196	−0.815980	−	0.578080i	$-0.803802\pi$
$17$	−114.389	−1.63196	−0.815980	+	0.578080i	$0.803802\pi$
$18$	0	0
$19$	45.1768	0.545488	0.272744	−	0.962087i	$-0.412069\pi$
$19$	45.1768	0.545488	0.272744	+	0.962087i	$0.412069\pi$
$20$	76.2140	0.852098
$21$	0	0
$22$	80.3219	0.778395
$23$	−73.9590	−0.670501	−0.335250	−	0.942129i	$-0.608821\pi$
$23$	−73.9590	−0.670501	−0.335250	+	0.942129i	$0.608821\pi$
$24$	0	0
$25$	4.29980	0.0343984
$26$	0	0
$27$	0	0
$28$	−208.194	−1.40518
$29$	27.2113	0.174242	0.0871208	−	0.996198i	$-0.472233\pi$
$29$	27.2113	0.174242	0.0871208	+	0.996198i	$0.472233\pi$
$30$	0	0
$31$	−179.587	−1.04048	−0.520239	−	0.854021i	$-0.674157\pi$
$31$	−179.587	−1.04048	−0.520239	+	0.854021i	$0.674157\pi$
$32$	238.945	1.32000
$33$	0	0
$34$	438.610	2.21238
$35$	−353.209	−1.70581
$36$	0	0
$37$	−354.736	−1.57617	−0.788085	−	0.615567i	$-0.788927\pi$
$37$	−354.736	−1.57617	−0.788085	+	0.615567i	$0.788927\pi$
$38$	−173.225	−0.739496
$39$	0	0
$40$	56.5727	0.223623
$41$	81.4187	0.310133	0.155067	−	0.987904i	$-0.450441\pi$
$41$	81.4187	0.310133	0.155067	+	0.987904i	$0.450441\pi$
$42$	0	0
$43$	−256.018	−0.907962	−0.453981	−	0.891011i	$-0.649997\pi$
$43$	−256.018	−0.907962	−0.453981	+	0.891011i	$0.649997\pi$
$44$	−140.402	−0.481056
$45$	0	0
$46$	283.587	0.908970
$47$	463.501	1.43848	0.719240	−	0.694762i	$-0.244490\pi$
$47$	463.501	1.43848	0.719240	+	0.694762i	$0.244490\pi$
$48$	0	0
$49$	621.865	1.81302
$50$	−16.4871	−0.0466325
$51$	0	0
$52$	0	0
$53$	−76.6055	−0.198539	−0.0992695	−	0.995061i	$-0.531651\pi$
$53$	−76.6055	−0.198539	−0.0992695	+	0.995061i	$0.531651\pi$
$54$	0	0
$55$	−238.198	−0.583974
$56$	−154.540	−0.368773
$57$	0	0
$58$	−104.338	−0.236212
$59$	54.4676	0.120188	0.0600939	−	0.998193i	$-0.480860\pi$
$59$	54.4676	0.120188	0.0600939	+	0.998193i	$0.480860\pi$
$60$	0	0
$61$	−494.496	−1.03793	−0.518965	−	0.854795i	$-0.673683\pi$
$61$	−494.496	−1.03793	−0.518965	+	0.854795i	$0.673683\pi$
$62$	688.606	1.41053
$63$	0	0
$64$	−334.634	−0.653582
$65$	0	0
$66$	0	0
$67$	611.991	1.11592	0.557960	−	0.829868i	$-0.311584\pi$
$67$	611.991	1.11592	0.557960	+	0.829868i	$0.311584\pi$
$68$	−766.688	−1.36727
$69$	0	0
$70$	1354.34	2.31249
$71$	16.3056	0.0272551	0.0136276	−	0.999907i	$-0.495662\pi$
$71$	16.3056	0.0272551	0.0136276	+	0.999907i	$0.495662\pi$
$72$	0	0
$73$	−321.825	−0.515983	−0.257992	−	0.966147i	$-0.583061\pi$
$73$	−321.825	−0.515983	−0.257992	+	0.966147i	$0.583061\pi$
$74$	1360.19	2.13675
$75$	0	0
$76$	302.797	0.457016
$77$	650.686	0.963021
$78$	0	0
$79$	385.324	0.548764	0.274382	−	0.961621i	$-0.411527\pi$
$79$	385.324	0.548764	0.274382	+	0.961621i	$0.411527\pi$
$80$	−826.633	−1.15526
$81$	0	0
$82$	−312.190	−0.420435
$83$	663.760	0.877797	0.438899	−	0.898537i	$-0.355369\pi$
$83$	663.760	0.877797	0.438899	+	0.898537i	$0.355369\pi$
$84$	0	0
$85$	−1300.71	−1.65979
$86$	981.671	1.23089
$87$	0	0
$88$	−104.219	−0.126247
$89$	545.723	0.649961	0.324980	−	0.945721i	$-0.394642\pi$
$89$	545.723	0.649961	0.324980	+	0.945721i	$0.394642\pi$
$90$	0	0
$91$	0	0
$92$	−495.709	−0.561753
$93$	0	0
$94$	−1777.24	−1.95009
$95$	513.706	0.554791
$96$	0	0
$97$	689.189	0.721408	0.360704	−	0.932680i	$-0.382536\pi$
$97$	689.189	0.721408	0.360704	+	0.932680i	$0.382536\pi$
$98$	−2384.47	−2.45783
$99$	0	0
$100$	28.8194	0.0288194
$101$	−471.398	−0.464414	−0.232207	−	0.972666i	$-0.574595\pi$
$101$	−471.398	−0.464414	−0.232207	+	0.972666i	$0.574595\pi$
$102$	0	0
$103$	−335.521	−0.320970	−0.160485	−	0.987038i	$-0.551306\pi$
$103$	−335.521	−0.320970	−0.160485	+	0.987038i	$0.551306\pi$
$104$	0	0
$105$	0	0
$106$	293.735	0.269151
$107$	−1629.26	−1.47203	−0.736014	−	0.676967i	$-0.763294\pi$
$107$	−1629.26	−1.47203	−0.736014	+	0.676967i	$0.763294\pi$
$108$	0	0
$109$	1518.95	1.33476	0.667381	−	0.744717i	$-0.267415\pi$
$109$	1518.95	1.33476	0.667381	+	0.744717i	$0.267415\pi$
$110$	913.341	0.791669
$111$	0	0
$112$	2258.12	1.90511
$113$	−1195.41	−0.995173	−0.497586	−	0.867414i	$-0.665780\pi$
$113$	−1195.41	−0.995173	−0.497586	+	0.867414i	$0.665780\pi$
$114$	0	0
$115$	−840.988	−0.681935
$116$	182.383	0.145982
$117$	0	0
$118$	−208.850	−0.162934
$119$	3553.17	2.73713
$120$	0	0
$121$	−892.190	−0.670315
$122$	1896.09	1.40708
$123$	0	0
$124$	−1203.68	−0.871723
$125$	−1372.48	−0.982069
$126$	0	0
$127$	526.425	0.367816	0.183908	−	0.982943i	$-0.441125\pi$
$127$	526.425	0.367816	0.183908	+	0.982943i	$0.441125\pi$
$128$	−628.446	−0.433963
$129$	0	0
$130$	0	0
$131$	834.024	0.556252	0.278126	−	0.960545i	$-0.410287\pi$
$131$	834.024	0.556252	0.278126	+	0.960545i	$0.410287\pi$
$132$	0	0
$133$	−1403.30	−0.914896
$134$	−2346.61	−1.51281
$135$	0	0
$136$	−569.103	−0.358825
$137$	−466.762	−0.291082	−0.145541	−	0.989352i	$-0.546492\pi$
$137$	−466.762	−0.291082	−0.145541	+	0.989352i	$0.546492\pi$
$138$	0	0
$139$	−713.080	−0.435127	−0.217563	−	0.976046i	$-0.569811\pi$
$139$	−713.080	−0.435127	−0.217563	+	0.976046i	$0.569811\pi$
$140$	−2367.38	−1.42914
$141$	0	0
$142$	−62.5218	−0.0369487
$143$	0	0
$144$	0	0
$145$	309.420	0.177213
$146$	1234.00	0.699497
$147$	0	0
$148$	−2377.61	−1.32053
$149$	662.814	0.364428	0.182214	−	0.983259i	$-0.441674\pi$
$149$	662.814	0.364428	0.182214	+	0.983259i	$0.441674\pi$
$150$	0	0
$151$	−190.862	−0.102862	−0.0514310	−	0.998677i	$-0.516378\pi$
$151$	−190.862	−0.102862	−0.0514310	+	0.998677i	$0.516378\pi$
$152$	224.762	0.119938
$153$	0	0
$154$	−2494.98	−1.30553
$155$	−2042.09	−1.05822
$156$	0	0
$157$	−2833.96	−1.44060	−0.720302	−	0.693661i	$-0.755997\pi$
$157$	−2833.96	−1.44060	−0.720302	+	0.693661i	$0.755997\pi$
$158$	−1477.48	−0.743937
$159$	0	0
$160$	2717.05	1.34251
$161$	2297.34	1.12457
$162$	0	0
$163$	−3565.55	−1.71334	−0.856672	−	0.515861i	$-0.827472\pi$
$163$	−3565.55	−1.71334	−0.856672	+	0.515861i	$0.827472\pi$
$164$	545.708	0.259833
$165$	0	0
$166$	−2545.11	−1.18999
$167$	−120.630	−0.0558959	−0.0279480	−	0.999609i	$-0.508897\pi$
$167$	−120.630	−0.0558959	−0.0279480	+	0.999609i	$0.508897\pi$
$168$	0	0
$169$	0	0
$170$	4987.44	2.25011
$171$	0	0
$172$	−1715.96	−0.760700
$173$	2080.28	0.914223	0.457112	−	0.889409i	$-0.348884\pi$
$173$	2080.28	0.914223	0.457112	+	0.889409i	$0.348884\pi$
$174$	0	0
$175$	−133.562	−0.0576932
$176$	1522.83	0.652204
$177$	0	0
$178$	−2092.51	−0.881125
$179$	553.140	0.230970	0.115485	−	0.993309i	$-0.463158\pi$
$179$	553.140	0.230970	0.115485	+	0.993309i	$0.463158\pi$
$180$	0	0
$181$	−3305.19	−1.35731	−0.678655	−	0.734457i	$-0.737437\pi$
$181$	−3305.19	−1.35731	−0.678655	+	0.734457i	$0.737437\pi$
$182$	0	0
$183$	0	0
$184$	−367.958	−0.147425
$185$	−4033.71	−1.60305
$186$	0	0
$187$	2396.19	0.937042
$188$	3106.61	1.20517
$189$	0	0
$190$	−1969.75	−0.752107
$191$	3318.14	1.25703	0.628514	−	0.777798i	$-0.283664\pi$
$191$	3318.14	1.25703	0.628514	+	0.777798i	$0.283664\pi$
$192$	0	0
$193$	3902.18	1.45536	0.727681	−	0.685916i	$-0.240598\pi$
$193$	3902.18	1.45536	0.727681	+	0.685916i	$0.240598\pi$
$194$	−2642.61	−0.977983
$195$	0	0
$196$	4168.04	1.51897
$197$	−3632.99	−1.31391	−0.656953	−	0.753932i	$-0.728155\pi$
$197$	−3632.99	−1.31391	−0.656953	+	0.753932i	$0.728155\pi$
$198$	0	0
$199$	2957.23	1.05343	0.526715	−	0.850042i	$-0.323424\pi$
$199$	2957.23	1.05343	0.526715	+	0.850042i	$0.323424\pi$
$200$	21.3922	0.00756330
$201$	0	0
$202$	1807.52	0.629587
$203$	−845.245	−0.292239
$204$	0	0
$205$	925.813	0.315422
$206$	1286.52	0.435125
$207$	0	0
$208$	0	0
$209$	−946.355	−0.313209
$210$	0	0
$211$	−3194.05	−1.04212	−0.521061	−	0.853520i	$-0.674464\pi$
$211$	−3194.05	−1.04212	−0.521061	+	0.853520i	$0.674464\pi$
$212$	−513.447	−0.166338
$213$	0	0
$214$	6247.22	1.99557
$215$	−2911.18	−0.923446
$216$	0	0
$217$	5578.39	1.74509
$218$	−5824.23	−1.80948
$219$	0	0
$220$	−1596.52	−0.489259
$221$	0	0
$222$	0	0
$223$	−75.1092	−0.0225546	−0.0112773	−	0.999936i	$-0.503590\pi$
$223$	−75.1092	−0.0225546	−0.0112773	+	0.999936i	$0.503590\pi$
$224$	−7422.18	−2.21391
$225$	0	0
$226$	4583.65	1.34911
$227$	4322.79	1.26394	0.631968	−	0.774995i	$-0.282248\pi$
$227$	4322.79	1.26394	0.631968	+	0.774995i	$0.282248\pi$
$228$	0	0
$229$	677.923	0.195626	0.0978132	−	0.995205i	$-0.468815\pi$
$229$	677.923	0.195626	0.0978132	+	0.995205i	$0.468815\pi$
$230$	3224.67	0.924472
$231$	0	0
$232$	135.381	0.0383111
$233$	−92.9061	−0.0261222	−0.0130611	−	0.999915i	$-0.504158\pi$
$233$	−92.9061	−0.0261222	−0.0130611	+	0.999915i	$0.504158\pi$
$234$	0	0
$235$	5270.47	1.46301
$236$	365.068	0.100695
$237$	0	0
$238$	−13624.2	−3.71062
$239$	−2082.56	−0.563638	−0.281819	−	0.959468i	$-0.590938\pi$
$239$	−2082.56	−0.563638	−0.281819	+	0.959468i	$0.590938\pi$
$240$	0	0
$241$	6541.25	1.74838	0.874189	−	0.485586i	$-0.161394\pi$
$241$	6541.25	1.74838	0.874189	+	0.485586i	$0.161394\pi$
$242$	3421.00	0.908719
$243$	0	0
$244$	−3314.35	−0.869589
$245$	7071.23	1.84394
$246$	0	0
$247$	0	0
$248$	−893.476	−0.228773
$249$	0	0
$250$	5262.62	1.33135
$251$	2409.29	0.605868	0.302934	−	0.953012i	$-0.402034\pi$
$251$	2409.29	0.605868	0.302934	+	0.953012i	$0.402034\pi$
$252$	0	0
$253$	1549.28	0.384989
$254$	−2018.51	−0.498633
$255$	0	0
$256$	5086.77	1.24189
$257$	263.587	0.0639770	0.0319885	−	0.999488i	$-0.489816\pi$
$257$	263.587	0.0639770	0.0319885	+	0.999488i	$0.489816\pi$
$258$	0	0
$259$	11018.9	2.64356
$260$	0	0
$261$	0	0
$262$	−3197.97	−0.754088
$263$	7324.62	1.71732	0.858660	−	0.512545i	$-0.171297\pi$
$263$	7324.62	1.71732	0.858660	+	0.512545i	$0.171297\pi$
$264$	0	0
$265$	−871.081	−0.201925
$266$	5380.77	1.24029
$267$	0	0
$268$	4101.86	0.934930
$269$	4683.37	1.06152	0.530762	−	0.847521i	$-0.321906\pi$
$269$	4683.37	1.06152	0.530762	+	0.847521i	$0.321906\pi$
$270$	0	0
$271$	1629.37	0.365231	0.182615	−	0.983184i	$-0.441544\pi$
$271$	1629.37	0.365231	0.182615	+	0.983184i	$0.441544\pi$
$272$	8315.66	1.85372
$273$	0	0
$274$	1789.75	0.394608
$275$	−90.0714	−0.0197510
$276$	0	0
$277$	6200.66	1.34499	0.672494	−	0.740103i	$-0.265223\pi$
$277$	6200.66	1.34499	0.672494	+	0.740103i	$0.265223\pi$
$278$	2734.22	0.589883
$279$	0	0
$280$	−1757.28	−0.375062
$281$	2951.26	0.626538	0.313269	−	0.949664i	$-0.398576\pi$
$281$	2951.26	0.626538	0.313269	+	0.949664i	$0.398576\pi$
$282$	0	0
$283$	5249.17	1.10258	0.551291	−	0.834313i	$-0.314135\pi$
$283$	5249.17	1.10258	0.551291	+	0.834313i	$0.314135\pi$
$284$	109.288	0.0228346
$285$	0	0
$286$	0	0
$287$	−2529.05	−0.520157
$288$	0	0
$289$	8171.77	1.66330
$290$	−1186.43	−0.240241
$291$	0	0
$292$	−2157.03	−0.432296
$293$	−2912.33	−0.580684	−0.290342	−	0.956923i	$-0.593769\pi$
$293$	−2912.33	−0.580684	−0.290342	+	0.956923i	$0.593769\pi$
$294$	0	0
$295$	619.352	0.122237
$296$	−1764.87	−0.346558
$297$	0	0
$298$	−2541.48	−0.494040
$299$	0	0
$300$	0	0
$301$	7952.50	1.52284
$302$	731.840	0.139446
$303$	0	0
$304$	−3284.20	−0.619611
$305$	−5622.92	−1.05563
$306$	0	0
$307$	−5000.10	−0.929546	−0.464773	−	0.885430i	$-0.653864\pi$
$307$	−5000.10	−0.929546	−0.464773	+	0.885430i	$0.653864\pi$
$308$	4361.22	0.806829
$309$	0	0
$310$	7830.14	1.43459
$311$	7840.94	1.42964	0.714822	−	0.699307i	$-0.246508\pi$
$311$	7840.94	1.42964	0.714822	+	0.699307i	$0.246508\pi$
$312$	0	0
$313$	−7518.43	−1.35772	−0.678860	−	0.734267i	$-0.737526\pi$
$313$	−7518.43	−1.35772	−0.678860	+	0.734267i	$0.737526\pi$
$314$	10866.5	1.95297
$315$	0	0
$316$	2582.63	0.459761
$317$	485.238	0.0859738	0.0429869	−	0.999076i	$-0.486313\pi$
$317$	485.238	0.0859738	0.0429869	+	0.999076i	$0.486313\pi$
$318$	0	0
$319$	−570.017	−0.100046
$320$	−3805.13	−0.664728
$321$	0	0
$322$	−8808.86	−1.52453
$323$	−5167.72	−0.890215
$324$	0	0
$325$	0	0
$326$	13671.7	2.32271
$327$	0	0
$328$	405.072	0.0681901
$329$	−14397.4	−2.41263
$330$	0	0
$331$	−8222.88	−1.36547	−0.682735	−	0.730666i	$-0.739210\pi$
$331$	−8222.88	−1.36547	−0.682735	+	0.730666i	$0.739210\pi$
$332$	4448.84	0.735428
$333$	0	0
$334$	462.541	0.0757758
$335$	6958.96	1.13495
$336$	0	0
$337$	7744.78	1.25188	0.625942	−	0.779870i	$-0.284715\pi$
$337$	7744.78	1.25188	0.625942	+	0.779870i	$0.284715\pi$
$338$	0	0
$339$	0	0
$340$	−8718.02	−1.39059
$341$	3761.96	0.597423
$342$	0	0
$343$	−8662.19	−1.36360
$344$	−1273.73	−0.199637
$345$	0	0
$346$	−7976.58	−1.23937
$347$	3532.61	0.546514	0.273257	−	0.961941i	$-0.411899\pi$
$347$	3532.61	0.546514	0.273257	+	0.961941i	$0.411899\pi$
$348$	0	0
$349$	−3366.19	−0.516298	−0.258149	−	0.966105i	$-0.583112\pi$
$349$	−3366.19	−0.516298	−0.258149	+	0.966105i	$0.583112\pi$
$350$	512.127	0.0782123
$351$	0	0
$352$	−5005.37	−0.757919
$353$	−9967.98	−1.50295	−0.751476	−	0.659761i	$-0.770658\pi$
$353$	−9967.98	−1.50295	−0.751476	+	0.659761i	$0.770658\pi$
$354$	0	0
$355$	185.411	0.0277199
$356$	3657.70	0.544544
$357$	0	0
$358$	−2120.95	−0.313117
$359$	2742.72	0.403219	0.201609	−	0.979466i	$-0.435383\pi$
$359$	2742.72	0.403219	0.201609	+	0.979466i	$0.435383\pi$
$360$	0	0
$361$	−4818.05	−0.702443
$362$	12673.4	1.84005
$363$	0	0
$364$	0	0
$365$	−3659.47	−0.524783
$366$	0	0
$367$	−1641.86	−0.233527	−0.116763	−	0.993160i	$-0.537252\pi$
$367$	−1641.86	−0.233527	−0.116763	+	0.993160i	$0.537252\pi$
$368$	5376.57	0.761611
$369$	0	0
$370$	15466.8	2.17319
$371$	2379.54	0.332991
$372$	0	0
$373$	11207.5	1.55577	0.777887	−	0.628404i	$-0.216291\pi$
$373$	11207.5	1.55577	0.777887	+	0.628404i	$0.216291\pi$
$374$	−9187.91	−1.27031
$375$	0	0
$376$	2306.00	0.316284
$377$	0	0
$378$	0	0
$379$	−3844.47	−0.521048	−0.260524	−	0.965467i	$-0.583895\pi$
$379$	−3844.47	−0.521048	−0.260524	+	0.965467i	$0.583895\pi$
$380$	3443.11	0.464810
$381$	0	0
$382$	−12723.0	−1.70410
$383$	12154.3	1.62156	0.810778	−	0.585354i	$-0.199045\pi$
$383$	12154.3	1.62156	0.810778	+	0.585354i	$0.199045\pi$
$384$	0	0
$385$	7398.96	0.979444
$386$	−14962.4	−1.97297
$387$	0	0
$388$	4619.28	0.604403
$389$	5269.41	0.686812	0.343406	−	0.939187i	$-0.388419\pi$
$389$	5269.41	0.686812	0.343406	+	0.939187i	$0.388419\pi$
$390$	0	0
$391$	8460.07	1.09423
$392$	3093.88	0.398634
$393$	0	0
$394$	13930.3	1.78121
$395$	4381.53	0.558123
$396$	0	0
$397$	−655.483	−0.0828659	−0.0414329	−	0.999141i	$-0.513192\pi$
$397$	−655.483	−0.0828659	−0.0414329	+	0.999141i	$0.513192\pi$
$398$	−11339.2	−1.42809
$399$	0	0
$400$	−312.581	−0.0390726
$401$	7699.97	0.958898	0.479449	−	0.877570i	$-0.340837\pi$
$401$	7699.97	0.958898	0.479449	+	0.877570i	$0.340837\pi$
$402$	0	0
$403$	0	0
$404$	−3159.53	−0.389091
$405$	0	0
$406$	3240.99	0.396177
$407$	7430.94	0.905008
$408$	0	0
$409$	555.201	0.0671220	0.0335610	−	0.999437i	$-0.489315\pi$
$409$	555.201	0.0671220	0.0335610	+	0.999437i	$0.489315\pi$
$410$	−3549.92	−0.427605
$411$	0	0
$412$	−2248.83	−0.268912
$413$	−1691.89	−0.201580
$414$	0	0
$415$	7547.62	0.892767
$416$	0	0
$417$	0	0
$418$	3628.69	0.424605
$419$	−3316.99	−0.386744	−0.193372	−	0.981126i	$-0.561942\pi$
$419$	−3316.99	−0.386744	−0.193372	+	0.981126i	$0.561942\pi$
$420$	0	0
$421$	13270.3	1.53623	0.768117	−	0.640310i	$-0.221194\pi$
$421$	13270.3	1.53623	0.768117	+	0.640310i	$0.221194\pi$
$422$	12247.2	1.41276
$423$	0	0
$424$	−381.125	−0.0436535
$425$	−491.849	−0.0561369
$426$	0	0
$427$	15360.2	1.74082
$428$	−10920.1	−1.23328
$429$	0	0
$430$	11162.6	1.25188
$431$	13692.2	1.53023	0.765117	−	0.643892i	$-0.222681\pi$
$431$	13692.2	1.53023	0.765117	+	0.643892i	$0.222681\pi$
$432$	0	0
$433$	−15189.1	−1.68578	−0.842888	−	0.538089i	$-0.819146\pi$
$433$	−15189.1	−1.68578	−0.842888	+	0.538089i	$0.819146\pi$
$434$	−21389.7	−2.36575
$435$	0	0
$436$	10180.7	1.11828
$437$	−3341.23	−0.365750
$438$	0	0
$439$	−2802.04	−0.304634	−0.152317	−	0.988332i	$-0.548673\pi$
$439$	−2802.04	−0.304634	−0.152317	+	0.988332i	$0.548673\pi$
$440$	−1185.07	−0.128400
$441$	0	0
$442$	0	0
$443$	7539.78	0.808636	0.404318	−	0.914618i	$-0.367509\pi$
$443$	7539.78	0.808636	0.404318	+	0.914618i	$0.367509\pi$
$444$	0	0
$445$	6205.42	0.661045
$446$	287.997	0.0305764
$447$	0	0
$448$	10394.5	1.09619
$449$	5558.82	0.584269	0.292135	−	0.956377i	$-0.405634\pi$
$449$	5558.82	0.584269	0.292135	+	0.956377i	$0.405634\pi$
$450$	0	0
$451$	−1705.54	−0.178073
$452$	−8012.20	−0.833766
$453$	0	0
$454$	−16575.2	−1.71347
$455$	0	0
$456$	0	0
$457$	4327.71	0.442980	0.221490	−	0.975163i	$-0.428908\pi$
$457$	4327.71	0.442980	0.221490	+	0.975163i	$0.428908\pi$
$458$	−2599.42	−0.265203
$459$	0	0
$460$	−5636.71	−0.571333
$461$	−11292.7	−1.14090	−0.570450	−	0.821333i	$-0.693231\pi$
$461$	−11292.7	−1.14090	−0.570450	+	0.821333i	$0.693231\pi$
$462$	0	0
$463$	−8582.54	−0.861478	−0.430739	−	0.902477i	$-0.641747\pi$
$463$	−8582.54	−0.861478	−0.430739	+	0.902477i	$0.641747\pi$
$464$	−1978.17	−0.197918
$465$	0	0
$466$	356.238	0.0354128
$467$	−17543.0	−1.73831	−0.869157	−	0.494536i	$-0.835338\pi$
$467$	−17543.0	−1.73831	−0.869157	+	0.494536i	$0.835338\pi$
$468$	0	0
$469$	−19009.8	−1.87163
$470$	−20209.0	−1.98334
$471$	0	0
$472$	270.986	0.0264261
$473$	5363.01	0.521335
$474$	0	0
$475$	194.252	0.0187639
$476$	23815.1	2.29320
$477$	0	0
$478$	7985.32	0.764101
$479$	6299.15	0.600867	0.300434	−	0.953803i	$-0.402869\pi$
$479$	6299.15	0.600867	0.300434	+	0.953803i	$0.402869\pi$
$480$	0	0
$481$	0	0
$482$	−25081.7	−2.37020
$483$	0	0
$484$	−5979.89	−0.561597
$485$	7836.78	0.733711
$486$	0	0
$487$	−3571.73	−0.332342	−0.166171	−	0.986097i	$-0.553140\pi$
$487$	−3571.73	−0.332342	−0.166171	+	0.986097i	$0.553140\pi$
$488$	−2460.20	−0.228213
$489$	0	0
$490$	−27113.8	−2.49975
$491$	13385.2	1.23028	0.615139	−	0.788419i	$-0.289100\pi$
$491$	13385.2	1.23028	0.615139	+	0.788419i	$0.289100\pi$
$492$	0	0
$493$	−3112.66	−0.284356
$494$	0	0
$495$	0	0
$496$	13055.4	1.18186
$497$	−506.488	−0.0457125
$498$	0	0
$499$	19227.7	1.72495	0.862476	−	0.506098i	$-0.168913\pi$
$499$	19227.7	1.72495	0.862476	+	0.506098i	$0.168913\pi$
$500$	−9199.05	−0.822788
$501$	0	0
$502$	−9238.13	−0.821350
$503$	18651.2	1.65332	0.826658	−	0.562705i	$-0.190239\pi$
$503$	18651.2	1.65332	0.826658	+	0.562705i	$0.190239\pi$
$504$	0	0
$505$	−5360.26	−0.472334
$506$	−5940.53	−0.521914
$507$	0	0
$508$	3528.35	0.308160
$509$	−15886.1	−1.38338	−0.691688	−	0.722196i	$-0.743133\pi$
$509$	−15886.1	−1.38338	−0.691688	+	0.722196i	$0.743133\pi$
$510$	0	0
$511$	9996.62	0.865410
$512$	−14477.1	−1.24961
$513$	0	0
$514$	−1010.69	−0.0867310
$515$	−3815.21	−0.326443
$516$	0	0
$517$	−9709.33	−0.825949
$518$	−42250.7	−3.58377
$519$	0	0
$520$	0	0
$521$	−1824.67	−0.153436	−0.0767179	−	0.997053i	$-0.524444\pi$
$521$	−1824.67	−0.153436	−0.0767179	+	0.997053i	$0.524444\pi$
$522$	0	0
$523$	16206.3	1.35498	0.677488	−	0.735534i	$-0.263069\pi$
$523$	16206.3	1.35498	0.677488	+	0.735534i	$0.263069\pi$
$524$	5590.04	0.466034
$525$	0	0
$526$	−28085.4	−2.32810
$527$	20542.7	1.69802
$528$	0	0
$529$	−6697.07	−0.550429
$530$	3340.06	0.273741
$531$	0	0
$532$	−9405.57	−0.766510
$533$	0	0
$534$	0	0
$535$	−18526.4	−1.49713
$536$	3044.76	0.245361
$537$	0	0
$538$	−17957.8	−1.43906
$539$	−13026.7	−1.04100
$540$	0	0
$541$	−2429.83	−0.193099	−0.0965496	−	0.995328i	$-0.530781\pi$
$541$	−2429.83	−0.193099	−0.0965496	+	0.995328i	$0.530781\pi$
$542$	−6247.65	−0.495128
$543$	0	0
$544$	−27332.6	−2.15418
$545$	17272.0	1.35752
$546$	0	0
$547$	−2409.16	−0.188315	−0.0941573	−	0.995557i	$-0.530016\pi$
$547$	−2409.16	−0.188315	−0.0941573	+	0.995557i	$0.530016\pi$
$548$	−3128.47	−0.243871
$549$	0	0
$550$	345.368	0.0267756
$551$	1229.32	0.0950468
$552$	0	0
$553$	−11969.1	−0.920390
$554$	−23775.7	−1.82334
$555$	0	0
$556$	−4779.41	−0.364554
$557$	−8668.70	−0.659434	−0.329717	−	0.944080i	$-0.606953\pi$
$557$	−8668.70	−0.659434	−0.329717	+	0.944080i	$0.606953\pi$
$558$	0	0
$559$	0	0
$560$	25677.1	1.93760
$561$	0	0
$562$	−11316.2	−0.849371
$563$	7818.35	0.585265	0.292632	−	0.956225i	$-0.405469\pi$
$563$	7818.35	0.585265	0.292632	+	0.956225i	$0.405469\pi$
$564$	0	0
$565$	−13593.0	−1.01214
$566$	−20127.3	−1.49473
$567$	0	0
$568$	81.1229	0.00599268
$569$	−9117.24	−0.671730	−0.335865	−	0.941910i	$-0.609029\pi$
$569$	−9117.24	−0.671730	−0.335865	+	0.941910i	$0.609029\pi$
$570$	0	0
$571$	−11842.4	−0.867932	−0.433966	−	0.900929i	$-0.642886\pi$
$571$	−11842.4	−0.867932	−0.433966	+	0.900929i	$0.642886\pi$
$572$	0	0
$573$	0	0
$574$	9697.35	0.705156
$575$	−318.009	−0.0230642
$576$	0	0
$577$	10958.1	0.790623	0.395312	−	0.918547i	$-0.370637\pi$
$577$	10958.1	0.790623	0.395312	+	0.918547i	$0.370637\pi$
$578$	−31333.7	−2.25486
$579$	0	0
$580$	2073.88	0.148471
$581$	−20617.9	−1.47225
$582$	0	0
$583$	1604.72	0.113997
$584$	−1601.13	−0.113451
$585$	0	0
$586$	11167.0	0.787209
$587$	23057.8	1.62129	0.810644	−	0.585539i	$-0.199117\pi$
$587$	23057.8	1.62129	0.810644	+	0.585539i	$0.199117\pi$
$588$	0	0
$589$	−8113.18	−0.567568
$590$	−2374.83	−0.165712
$591$	0	0
$592$	25788.1	1.79035
$593$	−9904.56	−0.685888	−0.342944	−	0.939356i	$-0.611424\pi$
$593$	−9904.56	−0.685888	−0.342944	+	0.939356i	$0.611424\pi$
$594$	0	0
$595$	40403.2	2.78381
$596$	4442.50	0.305322
$597$	0	0
$598$	0	0
$599$	−21334.5	−1.45527	−0.727633	−	0.685967i	$-0.759380\pi$
$599$	−21334.5	−1.45527	−0.727633	+	0.685967i	$0.759380\pi$
$600$	0	0
$601$	−19484.4	−1.32244	−0.661219	−	0.750193i	$-0.729960\pi$
$601$	−19484.4	−1.32244	−0.661219	+	0.750193i	$0.729960\pi$
$602$	−30492.9	−2.06445
$603$	0	0
$604$	−1279.25	−0.0861789
$605$	−10145.1	−0.681747
$606$	0	0
$607$	19855.7	1.32770	0.663852	−	0.747864i	$-0.268921\pi$
$607$	19855.7	1.32770	0.663852	+	0.747864i	$0.268921\pi$
$608$	10794.8	0.720043
$609$	0	0
$610$	21560.4	1.43108
$611$	0	0
$612$	0	0
$613$	−5153.57	−0.339561	−0.169781	−	0.985482i	$-0.554306\pi$
$613$	−5153.57	−0.339561	−0.169781	+	0.985482i	$0.554306\pi$
$614$	19172.3	1.26015
$615$	0	0
$616$	3237.27	0.211743
$617$	−924.218	−0.0603040	−0.0301520	−	0.999545i	$-0.509599\pi$
$617$	−924.218	−0.0603040	−0.0301520	+	0.999545i	$0.509599\pi$
$618$	0	0
$619$	15690.6	1.01883	0.509417	−	0.860520i	$-0.329861\pi$
$619$	15690.6	1.01883	0.509417	+	0.860520i	$0.329861\pi$
$620$	−13687.1	−0.886589
$621$	0	0
$622$	−30065.2	−1.93811
$623$	−16951.4	−1.09012
$624$	0	0
$625$	−16144.0	−1.03322
$626$	28828.5	1.84061
$627$	0	0
$628$	−18994.6	−1.20695
$629$	40577.8	2.57225
$630$	0	0
$631$	−21103.6	−1.33141	−0.665705	−	0.746215i	$-0.731869\pi$
$631$	−21103.6	−1.33141	−0.665705	+	0.746215i	$0.731869\pi$
$632$	1917.05	0.120659
$633$	0	0
$634$	−1860.59	−0.116551
$635$	5985.98	0.374089
$636$	0	0
$637$	0	0
$638$	2185.66	0.135629
$639$	0	0
$640$	−7146.06	−0.441364
$641$	6695.38	0.412561	0.206280	−	0.978493i	$-0.433864\pi$
$641$	6695.38	0.412561	0.206280	+	0.978493i	$0.433864\pi$
$642$	0	0
$643$	9647.69	0.591707	0.295854	−	0.955233i	$-0.404396\pi$
$643$	9647.69	0.591707	0.295854	+	0.955233i	$0.404396\pi$
$644$	15397.9	0.942175
$645$	0	0
$646$	19815.0	1.20683
$647$	23747.5	1.44299	0.721493	−	0.692422i	$-0.243456\pi$
$647$	23747.5	1.44299	0.721493	+	0.692422i	$0.243456\pi$
$648$	0	0
$649$	−1140.98	−0.0690096
$650$	0	0
$651$	0	0
$652$	−23898.0	−1.43546
$653$	−19833.5	−1.18858	−0.594292	−	0.804249i	$-0.702568\pi$
$653$	−19833.5	−1.18858	−0.594292	+	0.804249i	$0.702568\pi$
$654$	0	0
$655$	9483.70	0.565738
$656$	−5918.86	−0.352275
$657$	0	0
$658$	55205.1	3.27070
$659$	−10807.4	−0.638841	−0.319421	−	0.947613i	$-0.603488\pi$
$659$	−10807.4	−0.638841	−0.319421	+	0.947613i	$0.603488\pi$
$660$	0	0
$661$	−13431.5	−0.790356	−0.395178	−	0.918605i	$-0.629317\pi$
$661$	−13431.5	−0.790356	−0.395178	+	0.918605i	$0.629317\pi$
$662$	31529.7	1.85111
$663$	0	0
$664$	3302.32	0.193004
$665$	−15956.9	−0.930498
$666$	0	0
$667$	−2012.52	−0.116829
$668$	−808.520	−0.0468302
$669$	0	0
$670$	−26683.3	−1.53861
$671$	10358.6	0.595961
$672$	0	0
$673$	21581.4	1.23611	0.618054	−	0.786136i	$-0.287921\pi$
$673$	21581.4	1.23611	0.618054	+	0.786136i	$0.287921\pi$
$674$	−29696.4	−1.69713
$675$	0	0
$676$	0	0
$677$	−17482.4	−0.992474	−0.496237	−	0.868187i	$-0.665285\pi$
$677$	−17482.4	−0.992474	−0.496237	+	0.868187i	$0.665285\pi$
$678$	0	0
$679$	−21407.8	−1.20995
$680$	−6471.27	−0.364944
$681$	0	0
$682$	−14424.8	−0.809902
$683$	−2842.23	−0.159231	−0.0796155	−	0.996826i	$-0.525369\pi$
$683$	−2842.23	−0.159231	−0.0796155	+	0.996826i	$0.525369\pi$
$684$	0	0
$685$	−5307.56	−0.296046
$686$	33214.1	1.84857
$687$	0	0
$688$	18611.6	1.03134
$689$	0	0
$690$	0	0
$691$	22875.5	1.25937	0.629686	−	0.776850i	$-0.283184\pi$
$691$	22875.5	1.25937	0.629686	+	0.776850i	$0.283184\pi$
$692$	13943.0	0.765946
$693$	0	0
$694$	−13545.4	−0.740886
$695$	−8108.43	−0.442547
$696$	0	0
$697$	−9313.38	−0.506125
$698$	12907.2	0.699923
$699$	0	0
$700$	−895.195	−0.0483360
$701$	−11980.3	−0.645492	−0.322746	−	0.946486i	$-0.604606\pi$
$701$	−11980.3	−0.645492	−0.322746	+	0.946486i	$0.604606\pi$
$702$	0	0
$703$	−16025.9	−0.859782
$704$	7009.85	0.375275
$705$	0	0
$706$	38221.0	2.03749
$707$	14642.7	0.778918
$708$	0	0
$709$	4204.77	0.222727	0.111363	−	0.993780i	$-0.464478\pi$
$709$	4204.77	0.222727	0.111363	+	0.993780i	$0.464478\pi$
$710$	−710.936	−0.0375788
$711$	0	0
$712$	2715.06	0.142909
$713$	13282.1	0.697641
$714$	0	0
$715$	0	0
$716$	3707.41	0.193509
$717$	0	0
$718$	−10516.7	−0.546627
$719$	11537.0	0.598411	0.299205	−	0.954189i	$-0.403278\pi$
$719$	11537.0	0.598411	0.299205	+	0.954189i	$0.403278\pi$
$720$	0	0
$721$	10422.0	0.538332
$722$	18474.3	0.952272
$723$	0	0
$724$	−22153.0	−1.13717
$725$	117.003	0.00599364
$726$	0	0
$727$	−33899.7	−1.72940	−0.864698	−	0.502293i	$-0.832490\pi$
$727$	−33899.7	−1.72940	−0.864698	+	0.502293i	$0.832490\pi$
$728$	0	0
$729$	0	0
$730$	14031.8	0.711426
$731$	29285.6	1.48176
$732$	0	0
$733$	−378.221	−0.0190585	−0.00952927	−	0.999955i	$-0.503033\pi$
$733$	−378.221	−0.0190585	−0.00952927	+	0.999955i	$0.503033\pi$
$734$	6295.51	0.316582
$735$	0	0
$736$	−17672.1	−0.885059
$737$	−12819.9	−0.640741
$738$	0	0
$739$	22663.6	1.12814	0.564069	−	0.825728i	$-0.309235\pi$
$739$	22663.6	1.12814	0.564069	+	0.825728i	$0.309235\pi$
$740$	−27035.9	−1.34305
$741$	0	0
$742$	−9124.06	−0.451422
$743$	−16407.3	−0.810130	−0.405065	−	0.914288i	$-0.632751\pi$
$743$	−16407.3	−0.810130	−0.405065	+	0.914288i	$0.632751\pi$
$744$	0	0
$745$	7536.86	0.370643
$746$	−42974.0	−2.10910
$747$	0	0
$748$	16060.4	0.785064
$749$	50608.7	2.46889
$750$	0	0
$751$	21957.9	1.06692	0.533458	−	0.845827i	$-0.320892\pi$
$751$	21957.9	1.06692	0.533458	+	0.845827i	$0.320892\pi$
$752$	−33694.9	−1.63395
$753$	0	0
$754$	0	0
$755$	−2170.30	−0.104616
$756$	0	0
$757$	20823.5	0.999792	0.499896	−	0.866085i	$-0.333371\pi$
$757$	20823.5	0.999792	0.499896	+	0.866085i	$0.333371\pi$
$758$	14741.2	0.706364
$759$	0	0
$760$	2555.77	0.121984
$761$	28456.1	1.35550	0.677749	−	0.735294i	$-0.262956\pi$
$761$	28456.1	1.35550	0.677749	+	0.735294i	$0.262956\pi$
$762$	0	0
$763$	−47182.0	−2.23867
$764$	22239.8	1.05315
$765$	0	0
$766$	−46604.2	−2.19828
$767$	0	0
$768$	0	0
$769$	−13450.0	−0.630713	−0.315356	−	0.948973i	$-0.602124\pi$
$769$	−13450.0	−0.630713	−0.315356	+	0.948973i	$0.602124\pi$
$770$	−28370.4	−1.32779
$771$	0	0
$772$	26154.3	1.21932
$773$	−26577.8	−1.23666	−0.618329	−	0.785919i	$-0.712190\pi$
$773$	−26577.8	−1.23666	−0.618329	+	0.785919i	$0.712190\pi$
$774$	0	0
$775$	−772.190	−0.0357908
$776$	3428.83	0.158618
$777$	0	0
$778$	−20204.9	−0.931083
$779$	3678.24	0.169174
$780$	0	0
$781$	−341.566	−0.0156494
$782$	−32439.1	−1.48340
$783$	0	0
$784$	−45207.5	−2.05938
$785$	−32225.0	−1.46517
$786$	0	0
$787$	8281.40	0.375095	0.187548	−	0.982256i	$-0.439946\pi$
$787$	8281.40	0.375095	0.187548	+	0.982256i	$0.439946\pi$
$788$	−24350.0	−1.10080
$789$	0	0
$790$	−16800.4	−0.756624
$791$	37132.1	1.66911
$792$	0	0
$793$	0	0
$794$	2513.37	0.112338
$795$	0	0
$796$	19820.8	0.882575
$797$	−20668.8	−0.918605	−0.459302	−	0.888280i	$-0.651901\pi$
$797$	−20668.8	−0.918605	−0.459302	+	0.888280i	$0.651901\pi$
$798$	0	0
$799$	−53019.3	−2.34754
$800$	1027.42	0.0454059
$801$	0	0
$802$	−29524.6	−1.29994
$803$	6741.52	0.296268
$804$	0	0
$805$	26123.0	1.14375
$806$	0	0
$807$	0	0
$808$	−2345.28	−0.102112
$809$	−7238.85	−0.314591	−0.157296	−	0.987552i	$-0.550278\pi$
$809$	−7238.85	−0.314591	−0.157296	+	0.987552i	$0.550278\pi$
$810$	0	0
$811$	−13101.3	−0.567260	−0.283630	−	0.958934i	$-0.591539\pi$
$811$	−13101.3	−0.567260	−0.283630	+	0.958934i	$0.591539\pi$
$812$	−5665.24	−0.244841
$813$	0	0
$814$	−28493.1	−1.22688
$815$	−40543.9	−1.74256
$816$	0	0
$817$	−11566.1	−0.495283
$818$	−2128.85	−0.0909946
$819$	0	0
$820$	6205.25	0.264264
$821$	12092.9	0.514062	0.257031	−	0.966403i	$-0.417256\pi$
$821$	12092.9	0.514062	0.257031	+	0.966403i	$0.417256\pi$
$822$	0	0
$823$	27041.2	1.14532	0.572659	−	0.819794i	$-0.305912\pi$
$823$	27041.2	1.14532	0.572659	+	0.819794i	$0.305912\pi$
$824$	−1669.27	−0.0705727
$825$	0	0
$826$	6487.35	0.273273
$827$	−25572.3	−1.07526	−0.537628	−	0.843182i	$-0.680680\pi$
$827$	−25572.3	−1.07526	−0.537628	+	0.843182i	$0.680680\pi$
$828$	0	0
$829$	−17505.8	−0.733415	−0.366707	−	0.930336i	$-0.619515\pi$
$829$	−17505.8	−0.733415	−0.366707	+	0.930336i	$0.619515\pi$
$830$	−28940.5	−1.21029
$831$	0	0
$832$	0	0
$833$	−71134.3	−2.95877
$834$	0	0
$835$	−1371.68	−0.0568491
$836$	−6342.93	−0.262410
$837$	0	0
$838$	12718.6	0.524293
$839$	−28293.4	−1.16424	−0.582119	−	0.813103i	$-0.697776\pi$
$839$	−28293.4	−1.16424	−0.582119	+	0.813103i	$0.697776\pi$
$840$	0	0
$841$	−23648.5	−0.969640
$842$	−50883.3	−2.08261
$843$	0	0
$844$	−21408.1	−0.873100
$845$	0	0
$846$	0	0
$847$	27713.4	1.12426
$848$	5568.95	0.225517
$849$	0	0
$850$	1885.94	0.0761025
$851$	26235.9	1.05682
$852$	0	0
$853$	15866.0	0.636861	0.318430	−	0.947946i	$-0.396844\pi$
$853$	15866.0	0.636861	0.318430	+	0.947946i	$0.396844\pi$
$854$	−58896.8	−2.35996
$855$	0	0
$856$	−8105.86	−0.323660
$857$	−23623.3	−0.941607	−0.470804	−	0.882238i	$-0.656036\pi$
$857$	−23623.3	−0.941607	−0.470804	+	0.882238i	$0.656036\pi$
$858$	0	0
$859$	33403.1	1.32677	0.663387	−	0.748277i	$-0.269118\pi$
$859$	33403.1	1.32677	0.663387	+	0.748277i	$0.269118\pi$
$860$	−19512.2	−0.773673
$861$	0	0
$862$	−52501.2	−2.07447
$863$	−45490.5	−1.79434	−0.897169	−	0.441687i	$-0.854380\pi$
$863$	−45490.5	−1.79434	−0.897169	+	0.441687i	$0.854380\pi$
$864$	0	0
$865$	23654.9	0.929814
$866$	58240.8	2.28534
$867$	0	0
$868$	37389.0	1.46206
$869$	−8071.70	−0.315090
$870$	0	0
$871$	0	0
$872$	7557.03	0.293479
$873$	0	0
$874$	12811.6	0.495833
$875$	42632.4	1.64713
$876$	0	0
$877$	19570.1	0.753518	0.376759	−	0.926311i	$-0.377039\pi$
$877$	19570.1	0.753518	0.376759	+	0.926311i	$0.377039\pi$
$878$	10744.1	0.412979
$879$	0	0
$880$	17316.1	0.663326
$881$	3016.95	0.115373	0.0576864	−	0.998335i	$-0.481628\pi$
$881$	3016.95	0.115373	0.0576864	+	0.998335i	$0.481628\pi$
$882$	0	0
$883$	17163.8	0.654141	0.327071	−	0.945000i	$-0.393939\pi$
$883$	17163.8	0.654141	0.327071	+	0.945000i	$0.393939\pi$
$884$	0	0
$885$	0	0
$886$	−28910.4	−1.09623
$887$	−28464.7	−1.07751	−0.538754	−	0.842463i	$-0.681105\pi$
$887$	−28464.7	−1.07751	−0.538754	+	0.842463i	$0.681105\pi$
$888$	0	0
$889$	−16352.0	−0.616903
$890$	−23793.9	−0.896151
$891$	0	0
$892$	−503.418	−0.0188965
$893$	20939.5	0.784674
$894$	0	0
$895$	6289.76	0.234909
$896$	19520.9	0.727845
$897$	0	0
$898$	−21314.6	−0.792070
$899$	−4886.80	−0.181295
$900$	0	0
$901$	8762.80	0.324008
$902$	6539.70	0.241406
$903$	0	0
$904$	−5947.36	−0.218812
$905$	−37583.4	−1.38046
$906$	0	0
$907$	−26610.1	−0.974171	−0.487085	−	0.873354i	$-0.661940\pi$
$907$	−26610.1	−0.974171	−0.487085	+	0.873354i	$0.661940\pi$
$908$	28973.4	1.05894
$909$	0	0
$910$	0	0
$911$	52648.2	1.91472	0.957362	−	0.288890i	$-0.0932863\pi$
$911$	52648.2	1.91472	0.957362	+	0.288890i	$0.0932863\pi$
$912$	0	0
$913$	−13904.3	−0.504015
$914$	−16594.1	−0.600529
$915$	0	0
$916$	4543.77	0.163898
$917$	−25906.7	−0.932949
$918$	0	0
$919$	46923.5	1.68429	0.842146	−	0.539250i	$-0.181292\pi$
$919$	46923.5	1.68429	0.842146	+	0.539250i	$0.181292\pi$
$920$	−4184.06	−0.149939
$921$	0	0
$922$	43300.6	1.54667
$923$	0	0
$924$	0	0
$925$	−1525.30	−0.0542178
$926$	32908.7	1.16787
$927$	0	0
$928$	6502.00	0.229999
$929$	41163.7	1.45376	0.726878	−	0.686767i	$-0.240971\pi$
$929$	41163.7	1.45376	0.726878	+	0.686767i	$0.240971\pi$
$930$	0	0
$931$	28093.9	0.988980
$932$	−622.702	−0.0218855
$933$	0	0
$934$	67266.5	2.35656
$935$	27247.1	0.953022
$936$	0	0
$937$	−23380.3	−0.815156	−0.407578	−	0.913170i	$-0.633627\pi$
$937$	−23380.3	−0.815156	−0.407578	+	0.913170i	$0.633627\pi$
$938$	72891.0	2.53729
$939$	0	0
$940$	35325.3	1.22573
$941$	−34506.2	−1.19540	−0.597700	−	0.801720i	$-0.703918\pi$
$941$	−34506.2	−1.19540	−0.597700	+	0.801720i	$0.703918\pi$
$942$	0	0
$943$	−6021.65	−0.207945
$944$	−3959.61	−0.136519
$945$	0	0
$946$	−20563.8	−0.706753
$947$	−483.439	−0.0165889	−0.00829443	−	0.999966i	$-0.502640\pi$
$947$	−483.439	−0.0165889	−0.00829443	+	0.999966i	$0.502640\pi$
$948$	0	0
$949$	0	0
$950$	−744.835	−0.0254375
$951$	0	0
$952$	17677.6	0.601823
$953$	11962.7	0.406619	0.203310	−	0.979114i	$-0.434830\pi$
$953$	11962.7	0.406619	0.203310	+	0.979114i	$0.434830\pi$
$954$	0	0
$955$	37730.6	1.27846
$956$	−13958.3	−0.472222
$957$	0	0
$958$	−24153.3	−0.814571
$959$	14498.7	0.488204
$960$	0	0
$961$	2460.54	0.0825934
$962$	0	0
$963$	0	0
$964$	43842.6	1.46481
$965$	44371.7	1.48018
$966$	0	0
$967$	−7695.43	−0.255913	−0.127957	−	0.991780i	$-0.540842\pi$
$967$	−7695.43	−0.255913	−0.127957	+	0.991780i	$0.540842\pi$
$968$	−4438.79	−0.147385
$969$	0	0
$970$	−30049.2	−0.994661
$971$	44177.9	1.46008	0.730039	−	0.683406i	$-0.239502\pi$
$971$	44177.9	1.46008	0.730039	+	0.683406i	$0.239502\pi$
$972$	0	0
$973$	22149.9	0.729797
$974$	13695.4	0.450542
$975$	0	0
$976$	35948.2	1.17897
$977$	3412.61	0.111749	0.0558746	−	0.998438i	$-0.482205\pi$
$977$	3412.61	0.111749	0.0558746	+	0.998438i	$0.482205\pi$
$978$	0	0
$979$	−11431.7	−0.373196
$980$	47394.8	1.54487
$981$	0	0
$982$	−51324.0	−1.66784
$983$	6313.78	0.204861	0.102431	−	0.994740i	$-0.467338\pi$
$983$	6313.78	0.204861	0.102431	+	0.994740i	$0.467338\pi$
$984$	0	0
$985$	−41310.7	−1.33631
$986$	11935.1	0.385489
$987$	0	0
$988$	0	0
$989$	18934.8	0.608789
$990$	0	0
$991$	12177.5	0.390344	0.195172	−	0.980769i	$-0.437474\pi$
$991$	12177.5	0.390344	0.195172	+	0.980769i	$0.437474\pi$
$992$	−42911.5	−1.37343
$993$	0	0
$994$	1942.07	0.0619705
$995$	33626.7	1.07139
$996$	0	0
$997$	17506.2	0.556096	0.278048	−	0.960567i	$-0.410313\pi$
$997$	17506.2	0.556096	0.278048	+	0.960567i	$0.410313\pi$
$998$	−73726.4	−2.33845
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bh.1.2		9
3.2	odd	2		169.4.a.k.1.8	✓	9
13.12	even	2		1521.4.a.bg.1.8		9
39.2	even	12		169.4.e.h.147.15		36
39.5	even	4		169.4.b.g.168.4		18
39.8	even	4		169.4.b.g.168.15		18
39.11	even	12		169.4.e.h.147.4		36
39.17	odd	6		169.4.c.k.146.8		18
39.20	even	12		169.4.e.h.23.15		36
39.23	odd	6		169.4.c.k.22.8		18
39.29	odd	6		169.4.c.l.22.2		18
39.32	even	12		169.4.e.h.23.4		36
39.35	odd	6		169.4.c.l.146.2		18
39.38	odd	2		169.4.a.l.1.2	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.8	✓	9	3.2	odd	2
169.4.a.l.1.2	yes	9	39.38	odd	2
169.4.b.g.168.4		18	39.5	even	4
169.4.b.g.168.15		18	39.8	even	4
169.4.c.k.22.8		18	39.23	odd	6
169.4.c.k.146.8		18	39.17	odd	6
169.4.c.l.22.2		18	39.29	odd	6
169.4.c.l.146.2		18	39.35	odd	6
169.4.e.h.23.4		36	39.32	even	12
169.4.e.h.23.15		36	39.20	even	12
169.4.e.h.147.4		36	39.11	even	12
169.4.e.h.147.15		36	39.2	even	12
1521.4.a.bg.1.8		9	13.12	even	2
1521.4.a.bh.1.2		9	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-3.83438 q^{2} +6.70249 q^{4} +11.3710 q^{5} -31.0623 q^{7} +4.97517 q^{8} +O(q^{10})\)	\(q-3.83438 q^{2} +6.70249 q^{4} +11.3710 q^{5} -31.0623 q^{7} +4.97517 q^{8} -43.6008 q^{10} -20.9478 q^{11} +119.105 q^{14} -72.6966 q^{16} -114.389 q^{17} +45.1768 q^{19} +76.2140 q^{20} +80.3219 q^{22} -73.9590 q^{23} +4.29980 q^{25} -208.194 q^{28} +27.2113 q^{29} -179.587 q^{31} +238.945 q^{32} +438.610 q^{34} -353.209 q^{35} -354.736 q^{37} -173.225 q^{38} +56.5727 q^{40} +81.4187 q^{41} -256.018 q^{43} -140.402 q^{44} +283.587 q^{46} +463.501 q^{47} +621.865 q^{49} -16.4871 q^{50} -76.6055 q^{53} -238.198 q^{55} -154.540 q^{56} -104.338 q^{58} +54.4676 q^{59} -494.496 q^{61} +688.606 q^{62} -334.634 q^{64} +611.991 q^{67} -766.688 q^{68} +1354.34 q^{70} +16.3056 q^{71} -321.825 q^{73} +1360.19 q^{74} +302.797 q^{76} +650.686 q^{77} +385.324 q^{79} -826.633 q^{80} -312.190 q^{82} +663.760 q^{83} -1300.71 q^{85} +981.671 q^{86} -104.219 q^{88} +545.723 q^{89} -495.709 q^{92} -1777.24 q^{94} +513.706 q^{95} +689.189 q^{97} -2384.47 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8}+O(q^{10}) \) 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8	\( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8} - 147 q^{10} + 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} - 161 q^{19} + 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} - 344 q^{28} - 280 q^{29} - 706 q^{31} + 680 q^{32} - 216 q^{34} - 20 q^{35} - 298 q^{37} + 739 q^{38} + 13 q^{40} + 1201 q^{41} - 533 q^{43} + 355 q^{44} + 840 q^{46} + 956 q^{47} + 403 q^{49} - 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} + 2877 q^{58} + 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} + 931 q^{67} + 1536 q^{68} + 4854 q^{70} + 2046 q^{71} + 45 q^{73} + 1990 q^{74} + 3608 q^{76} + 718 q^{77} + 412 q^{79} - 787 q^{80} + 2757 q^{82} + 3709 q^{83} + 2106 q^{85} + 125 q^{86} - 636 q^{88} + 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} + 1087 q^{97} - 282 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 37 * q^4 + 30 * q^5 - 38 * q^7 + 60 * q^8 - 147 * q^10 + 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 - 161 * q^19 + 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 - 344 * q^28 - 280 * q^29 - 706 * q^31 + 680 * q^32 - 216 * q^34 - 20 * q^35 - 298 * q^37 + 739 * q^38 + 13 * q^40 + 1201 * q^41 - 533 * q^43 + 355 * q^44 + 840 * q^46 + 956 * q^47 + 403 * q^49 - 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 + 2877 * q^58 + 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 + 931 * q^67 + 1536 * q^68 + 4854 * q^70 + 2046 * q^71 + 45 * q^73 + 1990 * q^74 + 3608 * q^76 + 718 * q^77 + 412 * q^79 - 787 * q^80 + 2757 * q^82 + 3709 * q^83 + 2106 * q^85 + 125 * q^86 - 636 * q^88 + 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 + 1087 * q^97 - 282 * q^98

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