LMFDB - Embedded newform 169.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +2.00000 q^{3} -5.00000 q^{4} +1.73205 q^{5} +3.46410 q^{6} -13.8564 q^{7} -22.5167 q^{8} -23.0000 q^{9} +O(q^{10})$	\(q+1.73205 q^{2} +2.00000 q^{3} -5.00000 q^{4} +1.73205 q^{5} +3.46410 q^{6} -13.8564 q^{7} -22.5167 q^{8} -23.0000 q^{9} +3.00000 q^{10} -13.8564 q^{11} -10.0000 q^{12} -24.0000 q^{14} +3.46410 q^{15} +1.00000 q^{16} -117.000 q^{17} -39.8372 q^{18} +114.315 q^{19} -8.66025 q^{20} -27.7128 q^{21} -24.0000 q^{22} +78.0000 q^{23} -45.0333 q^{24} -122.000 q^{25} -100.000 q^{27} +69.2820 q^{28} -141.000 q^{29} +6.00000 q^{30} -155.885 q^{31} +181.865 q^{32} -27.7128 q^{33} -202.650 q^{34} -24.0000 q^{35} +115.000 q^{36} +143.760 q^{37} +198.000 q^{38} -39.0000 q^{40} +271.932 q^{41} -48.0000 q^{42} -104.000 q^{43} +69.2820 q^{44} -39.8372 q^{45} +135.100 q^{46} +301.377 q^{47} +2.00000 q^{48} -151.000 q^{49} -211.310 q^{50} -234.000 q^{51} +93.0000 q^{53} -173.205 q^{54} -24.0000 q^{55} +312.000 q^{56} +228.631 q^{57} -244.219 q^{58} -284.056 q^{59} -17.3205 q^{60} +145.000 q^{61} -270.000 q^{62} +318.697 q^{63} +307.000 q^{64} -48.0000 q^{66} -786.351 q^{67} +585.000 q^{68} +156.000 q^{69} -41.5692 q^{70} -1056.55 q^{71} +517.883 q^{72} +458.993 q^{73} +249.000 q^{74} -244.000 q^{75} -571.577 q^{76} +192.000 q^{77} +1276.00 q^{79} +1.73205 q^{80} +421.000 q^{81} +471.000 q^{82} +789.815 q^{83} +138.564 q^{84} -202.650 q^{85} -180.133 q^{86} -282.000 q^{87} +312.000 q^{88} +976.877 q^{89} -69.0000 q^{90} -390.000 q^{92} -311.769 q^{93} +522.000 q^{94} +198.000 q^{95} +363.731 q^{96} -200.918 q^{97} -261.540 q^{98} +318.697 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 10 q^{4} - 46 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 10 * q^4 - 46 * q^9	$ 2 q + 4 q^{3} - 10 q^{4} - 46 q^{9} + 6 q^{10} - 20 q^{12} - 48 q^{14} + 2 q^{16} - 234 q^{17} - 48 q^{22} + 156 q^{23} - 244 q^{25} - 200 q^{27} - 282 q^{29} + 12 q^{30} - 48 q^{35} + 230 q^{36} + 396 q^{38} - 78 q^{40} - 96 q^{42} - 208 q^{43} + 4 q^{48} - 302 q^{49} - 468 q^{51} + 186 q^{53} - 48 q^{55} + 624 q^{56} + 290 q^{61} - 540 q^{62} + 614 q^{64} - 96 q^{66} + 1170 q^{68} + 312 q^{69} + 498 q^{74} - 488 q^{75} + 384 q^{77} + 2552 q^{79} + 842 q^{81} + 942 q^{82} - 564 q^{87} + 624 q^{88} - 138 q^{90} - 780 q^{92} + 1044 q^{94} + 396 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 - 10 * q^4 - 46 * q^9 + 6 * q^10 - 20 * q^12 - 48 * q^14 + 2 * q^16 - 234 * q^17 - 48 * q^22 + 156 * q^23 - 244 * q^25 - 200 * q^27 - 282 * q^29 + 12 * q^30 - 48 * q^35 + 230 * q^36 + 396 * q^38 - 78 * q^40 - 96 * q^42 - 208 * q^43 + 4 * q^48 - 302 * q^49 - 468 * q^51 + 186 * q^53 - 48 * q^55 + 624 * q^56 + 290 * q^61 - 540 * q^62 + 614 * q^64 - 96 * q^66 + 1170 * q^68 + 312 * q^69 + 498 * q^74 - 488 * q^75 + 384 * q^77 + 2552 * q^79 + 842 * q^81 + 942 * q^82 - 564 * q^87 + 624 * q^88 - 138 * q^90 - 780 * q^92 + 1044 * q^94 + 396 * q^95

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 $

$2$	1.73205	0.612372	0.306186	−	0.951972i	$-0.400947\pi$
$2$	1.73205	0.612372	0.306186	+	0.951972i	$0.400947\pi$
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	−5.00000	−0.625000
$5$	1.73205	0.154919	0.0774597	−	0.996995i	$-0.475319\pi$
$5$	1.73205	0.154919	0.0774597	+	0.996995i	$0.475319\pi$
$6$	3.46410	0.235702
$7$	−13.8564	−0.748176	−0.374088	−	0.927393i	$-0.622044\pi$
$7$	−13.8564	−0.748176	−0.374088	+	0.927393i	$0.622044\pi$
$8$	−22.5167	−0.995105
$9$	−23.0000	−0.851852
$10$	3.00000	0.0948683
$11$	−13.8564	−0.379806	−0.189903	−	0.981803i	$-0.560817\pi$
$11$	−13.8564	−0.379806	−0.189903	+	0.981803i	$0.560817\pi$
$12$	−10.0000	−0.240563
$13$	0	0
$13$	0	0
$14$	−24.0000	−0.458162
$15$	3.46410	0.0596285
$16$	1.00000	0.0156250
$17$	−117.000	−1.66922	−0.834608	−	0.550845i	$-0.814306\pi$
$17$	−117.000	−1.66922	−0.834608	+	0.550845i	$0.814306\pi$
$18$	−39.8372	−0.521651
$19$	114.315	1.38030	0.690151	−	0.723665i	$-0.257544\pi$
$19$	114.315	1.38030	0.690151	+	0.723665i	$0.257544\pi$
$20$	−8.66025	−0.0968246
$21$	−27.7128	−0.287973
$22$	−24.0000	−0.232583
$23$	78.0000	0.707136	0.353568	−	0.935409i	$-0.384968\pi$
$23$	78.0000	0.707136	0.353568	+	0.935409i	$0.384968\pi$
$24$	−45.0333	−0.383016
$25$	−122.000	−0.976000
$26$	0	0
$27$	−100.000	−0.712778
$28$	69.2820	0.467610
$29$	−141.000	−0.902864	−0.451432	−	0.892306i	$-0.649087\pi$
$29$	−141.000	−0.902864	−0.451432	+	0.892306i	$0.649087\pi$
$30$	6.00000	0.0365148
$31$	−155.885	−0.903151	−0.451576	−	0.892233i	$-0.649138\pi$
$31$	−155.885	−0.903151	−0.451576	+	0.892233i	$0.649138\pi$
$32$	181.865	1.00467
$33$	−27.7128	−0.146187
$34$	−202.650	−1.02218
$35$	−24.0000	−0.115907
$36$	115.000	0.532407
$37$	143.760	0.638758	0.319379	−	0.947627i	$-0.396526\pi$
$37$	143.760	0.638758	0.319379	+	0.947627i	$0.396526\pi$
$38$	198.000	0.845259
$39$	0	0
$40$	−39.0000	−0.154161
$41$	271.932	1.03582	0.517910	−	0.855435i	$-0.326710\pi$
$41$	271.932	1.03582	0.517910	+	0.855435i	$0.326710\pi$
$42$	−48.0000	−0.176347
$43$	−104.000	−0.368834	−0.184417	−	0.982848i	$-0.559040\pi$
$43$	−104.000	−0.368834	−0.184417	+	0.982848i	$0.559040\pi$
$44$	69.2820	0.237379
$45$	−39.8372	−0.131968
$46$	135.100	0.433030
$47$	301.377	0.935326	0.467663	−	0.883907i	$-0.345096\pi$
$47$	301.377	0.935326	0.467663	+	0.883907i	$0.345096\pi$
$48$	2.00000	0.00601407
$49$	−151.000	−0.440233
$50$	−211.310	−0.597675
$51$	−234.000	−0.642481
$52$	0	0
$53$	93.0000	0.241029	0.120514	−	0.992712i	$-0.461546\pi$
$53$	93.0000	0.241029	0.120514	+	0.992712i	$0.461546\pi$
$54$	−173.205	−0.436486
$55$	−24.0000	−0.0588393
$56$	312.000	0.744513
$57$	228.631	0.531279
$58$	−244.219	−0.552889
$59$	−284.056	−0.626796	−0.313398	−	0.949622i	$-0.601467\pi$
$59$	−284.056	−0.626796	−0.313398	+	0.949622i	$0.601467\pi$
$60$	−17.3205	−0.0372678
$61$	145.000	0.304350	0.152175	−	0.988354i	$-0.451372\pi$
$61$	145.000	0.304350	0.152175	+	0.988354i	$0.451372\pi$
$62$	−270.000	−0.553065
$63$	318.697	0.637335
$64$	307.000	0.599609
$65$	0	0
$66$	−48.0000	−0.0895211
$67$	−786.351	−1.43385	−0.716926	−	0.697149i	$-0.754451\pi$
$67$	−786.351	−1.43385	−0.716926	+	0.697149i	$0.754451\pi$
$68$	585.000	1.04326
$69$	156.000	0.272177
$70$	−41.5692	−0.0709782
$71$	−1056.55	−1.76605	−0.883025	−	0.469326i	$-0.844497\pi$
$71$	−1056.55	−1.76605	−0.883025	+	0.469326i	$0.844497\pi$
$72$	517.883	0.847682
$73$	458.993	0.735906	0.367953	−	0.929844i	$-0.380059\pi$
$73$	458.993	0.735906	0.367953	+	0.929844i	$0.380059\pi$
$74$	249.000	0.391158
$75$	−244.000	−0.375663
$76$	−571.577	−0.862689
$77$	192.000	0.284161
$78$	0	0
$79$	1276.00	1.81723	0.908615	−	0.417634i	$-0.137141\pi$
$79$	1276.00	1.81723	0.908615	+	0.417634i	$0.137141\pi$
$80$	1.73205	0.00242061
$81$	421.000	0.577503
$82$	471.000	0.634308
$83$	789.815	1.04450	0.522250	−	0.852793i	$-0.325093\pi$
$83$	789.815	1.04450	0.522250	+	0.852793i	$0.325093\pi$
$84$	138.564	0.179983
$85$	−202.650	−0.258594
$86$	−180.133	−0.225864
$87$	−282.000	−0.347512
$88$	312.000	0.377947
$89$	976.877	1.16347	0.581734	−	0.813379i	$-0.302374\pi$
$89$	976.877	1.16347	0.581734	+	0.813379i	$0.302374\pi$
$90$	−69.0000	−0.0808138
$91$	0	0
$92$	−390.000	−0.441960
$93$	−311.769	−0.347623
$94$	522.000	0.572768
$95$	198.000	0.213835
$96$	363.731	0.386699
$97$	−200.918	−0.210311	−0.105155	−	0.994456i	$-0.533534\pi$
$97$	−200.918	−0.210311	−0.105155	+	0.994456i	$0.533534\pi$
$98$	−261.540	−0.269587
$99$	318.697	0.323538
$100$	610.000	0.610000
$101$	−429.000	−0.422645	−0.211322	−	0.977416i	$-0.567777\pi$
$101$	−429.000	−0.422645	−0.211322	+	0.977416i	$0.567777\pi$
$102$	−405.300	−0.393438
$103$	−182.000	−0.174107	−0.0870534	−	0.996204i	$-0.527745\pi$
$103$	−182.000	−0.174107	−0.0870534	+	0.996204i	$0.527745\pi$
$104$	0	0
$105$	−48.0000	−0.0446126
$106$	161.081	0.147599
$107$	−1506.00	−1.36066	−0.680330	−	0.732906i	$-0.738163\pi$
$107$	−1506.00	−1.36066	−0.680330	+	0.732906i	$0.738163\pi$
$108$	500.000	0.445486
$109$	−1551.92	−1.36373	−0.681866	−	0.731477i	$-0.738831\pi$
$109$	−1551.92	−1.36373	−0.681866	+	0.731477i	$0.738831\pi$
$110$	−41.5692	−0.0360315
$111$	287.520	0.245858
$112$	−13.8564	−0.0116902
$113$	−687.000	−0.571925	−0.285962	−	0.958241i	$-0.592313\pi$
$113$	−687.000	−0.571925	−0.285962	+	0.958241i	$0.592313\pi$
$114$	396.000	0.325340
$115$	135.100	0.109549
$116$	705.000	0.564290
$117$	0	0
$118$	−492.000	−0.383833
$119$	1621.20	1.24887
$120$	−78.0000	−0.0593366
$121$	−1139.00	−0.855748
$122$	251.147	0.186376
$123$	543.864	0.398687
$124$	779.423	0.564470
$125$	−427.817	−0.306121
$126$	552.000	0.390286
$127$	−286.000	−0.199830	−0.0999149	−	0.994996i	$-0.531857\pi$
$127$	−286.000	−0.199830	−0.0999149	+	0.994996i	$0.531857\pi$
$128$	−923.183	−0.637489
$129$	−208.000	−0.141964
$130$	0	0
$131$	−1974.00	−1.31656	−0.658279	−	0.752774i	$-0.728715\pi$
$131$	−1974.00	−1.31656	−0.658279	+	0.752774i	$0.728715\pi$
$132$	138.564	0.0913671
$133$	−1584.00	−1.03271
$134$	−1362.00	−0.878051
$135$	−173.205	−0.110423
$136$	2634.45	1.66105
$137$	−846.973	−0.528188	−0.264094	−	0.964497i	$-0.585073\pi$
$137$	−846.973	−0.528188	−0.264094	+	0.964497i	$0.585073\pi$
$138$	270.200	0.166674
$139$	236.000	0.144009	0.0720045	−	0.997404i	$-0.477060\pi$
$139$	236.000	0.144009	0.0720045	+	0.997404i	$0.477060\pi$
$140$	120.000	0.0724418
$141$	602.754	0.360007
$142$	−1830.00	−1.08148
$143$	0	0
$144$	−23.0000	−0.0133102
$145$	−244.219	−0.139871
$146$	795.000	0.450648
$147$	−302.000	−0.169446
$148$	−718.801	−0.399224
$149$	46.7654	0.0257125	0.0128563	−	0.999917i	$-0.495908\pi$
$149$	46.7654	0.0257125	0.0128563	+	0.999917i	$0.495908\pi$
$150$	−422.620	−0.230045
$151$	−1770.16	−0.953995	−0.476998	−	0.878905i	$-0.658275\pi$
$151$	−1770.16	−0.953995	−0.476998	+	0.878905i	$0.658275\pi$
$152$	−2574.00	−1.37355
$153$	2691.00	1.42192
$154$	332.554	0.174013
$155$	−270.000	−0.139916
$156$	0	0
$157$	1211.00	0.615594	0.307797	−	0.951452i	$-0.400408\pi$
$157$	1211.00	0.615594	0.307797	+	0.951452i	$0.400408\pi$
$158$	2210.10	1.11282
$159$	186.000	0.0927721
$160$	315.000	0.155643
$161$	−1080.80	−0.529062
$162$	729.193	0.353647
$163$	−1004.59	−0.482733	−0.241367	−	0.970434i	$-0.577596\pi$
$163$	−1004.59	−0.482733	−0.241367	+	0.970434i	$0.577596\pi$
$164$	−1359.66	−0.647388
$165$	−48.0000	−0.0226472
$166$	1368.00	0.639623
$167$	−914.523	−0.423760	−0.211880	−	0.977296i	$-0.567959\pi$
$167$	−914.523	−0.423760	−0.211880	+	0.977296i	$0.567959\pi$
$168$	624.000	0.286563
$169$	0	0
$170$	−351.000	−0.158356
$171$	−2629.25	−1.17581
$172$	520.000	0.230521
$173$	−2574.00	−1.13120	−0.565600	−	0.824680i	$-0.691355\pi$
$173$	−2574.00	−1.13120	−0.565600	+	0.824680i	$0.691355\pi$
$174$	−488.438	−0.212807
$175$	1690.48	0.730219
$176$	−13.8564	−0.00593447
$177$	−568.113	−0.241254
$178$	1692.00	0.712476
$179$	−3744.00	−1.56335	−0.781675	−	0.623686i	$-0.785634\pi$
$179$	−3744.00	−1.56335	−0.781675	+	0.623686i	$0.785634\pi$
$180$	199.186	0.0824802
$181$	637.000	0.261590	0.130795	−	0.991409i	$-0.458247\pi$
$181$	637.000	0.261590	0.130795	+	0.991409i	$0.458247\pi$
$182$	0	0
$183$	290.000	0.117144
$184$	−1756.30	−0.703675
$185$	249.000	0.0989559
$186$	−540.000	−0.212875
$187$	1621.20	0.633978
$188$	−1506.88	−0.584579
$189$	1385.64	0.533283
$190$	342.946	0.130947
$191$	−2598.00	−0.984213	−0.492106	−	0.870535i	$-0.663773\pi$
$191$	−2598.00	−0.984213	−0.492106	+	0.870535i	$0.663773\pi$
$192$	614.000	0.230790
$193$	−1117.17	−0.416662	−0.208331	−	0.978058i	$-0.566803\pi$
$193$	−1117.17	−0.416662	−0.208331	+	0.978058i	$0.566803\pi$
$194$	−348.000	−0.128788
$195$	0	0
$196$	755.000	0.275146
$197$	2050.75	0.741674	0.370837	−	0.928698i	$-0.379071\pi$
$197$	2050.75	0.741674	0.370837	+	0.928698i	$0.379071\pi$
$198$	552.000	0.198126
$199$	2522.00	0.898391	0.449196	−	0.893433i	$-0.351711\pi$
$199$	2522.00	0.898391	0.449196	+	0.893433i	$0.351711\pi$
$200$	2747.03	0.971223
$201$	−1572.70	−0.551890
$202$	−743.050	−0.258816
$203$	1953.75	0.675500
$204$	1170.00	0.401551
$205$	471.000	0.160469
$206$	−315.233	−0.106618
$207$	−1794.00	−0.602375
$208$	0	0
$209$	−1584.00	−0.524247
$210$	−83.1384	−0.0273195
$211$	1042.00	0.339973	0.169986	−	0.985446i	$-0.445628\pi$
$211$	1042.00	0.339973	0.169986	+	0.985446i	$0.445628\pi$
$212$	−465.000	−0.150643
$213$	−2113.10	−0.679753
$214$	−2608.47	−0.833230
$215$	−180.133	−0.0571395
$216$	2251.67	0.709289
$217$	2160.00	0.675716
$218$	−2688.00	−0.835112
$219$	917.987	0.283250
$220$	120.000	0.0367745
$221$	0	0
$222$	498.000	0.150557
$223$	−2407.55	−0.722966	−0.361483	−	0.932379i	$-0.617730\pi$
$223$	−2407.55	−0.722966	−0.361483	+	0.932379i	$0.617730\pi$
$224$	−2520.00	−0.751672
$225$	2806.00	0.831407
$226$	−1189.92	−0.350231
$227$	−2407.55	−0.703942	−0.351971	−	0.936011i	$-0.614488\pi$
$227$	−2407.55	−0.703942	−0.351971	+	0.936011i	$0.614488\pi$
$228$	−1143.15	−0.332049
$229$	2508.01	0.723729	0.361864	−	0.932231i	$-0.382140\pi$
$229$	2508.01	0.723729	0.361864	+	0.932231i	$0.382140\pi$
$230$	234.000	0.0670848
$231$	384.000	0.109374
$232$	3174.85	0.898444
$233$	5850.00	1.64483	0.822417	−	0.568885i	$-0.192625\pi$
$233$	5850.00	1.64483	0.822417	+	0.568885i	$0.192625\pi$
$234$	0	0
$235$	522.000	0.144900
$236$	1420.28	0.391748
$237$	2552.00	0.699452
$238$	2808.00	0.764771
$239$	5383.21	1.45695	0.728475	−	0.685072i	$-0.240229\pi$
$239$	5383.21	1.45695	0.728475	+	0.685072i	$0.240229\pi$
$240$	3.46410	0.000931695 0
$241$	4917.29	1.31432	0.657159	−	0.753752i	$-0.271758\pi$
$241$	4917.29	1.31432	0.657159	+	0.753752i	$0.271758\pi$
$242$	−1972.81	−0.524036
$243$	3542.00	0.935059
$244$	−725.000	−0.190219
$245$	−261.540	−0.0682006
$246$	942.000	0.244145
$247$	0	0
$248$	3510.00	0.898731
$249$	1579.63	0.402028
$250$	−741.000	−0.187460
$251$	3978.00	1.00036	0.500178	−	0.865923i	$-0.333268\pi$
$251$	3978.00	1.00036	0.500178	+	0.865923i	$0.333268\pi$
$252$	−1593.49	−0.398334
$253$	−1080.80	−0.268574
$254$	−495.367	−0.122370
$255$	−405.300	−0.0995328
$256$	−4055.00	−0.989990
$257$	−2067.00	−0.501696	−0.250848	−	0.968026i	$-0.580709\pi$
$257$	−2067.00	−0.501696	−0.250848	+	0.968026i	$0.580709\pi$
$258$	−360.267	−0.0869349
$259$	−1992.00	−0.477903
$260$	0	0
$261$	3243.00	0.769106
$262$	−3419.07	−0.806224
$263$	−2052.00	−0.481109	−0.240555	−	0.970636i	$-0.577329\pi$
$263$	−2052.00	−0.481109	−0.240555	+	0.970636i	$0.577329\pi$
$264$	624.000	0.145472
$265$	161.081	0.0373400
$266$	−2743.57	−0.632402
$267$	1953.75	0.447819
$268$	3931.76	0.896157
$269$	3330.00	0.754772	0.377386	−	0.926056i	$-0.376823\pi$
$269$	3330.00	0.754772	0.377386	+	0.926056i	$0.376823\pi$
$270$	−300.000	−0.0676201
$271$	−2805.92	−0.628958	−0.314479	−	0.949264i	$-0.601830\pi$
$271$	−2805.92	−0.628958	−0.314479	+	0.949264i	$0.601830\pi$
$272$	−117.000	−0.0260815
$273$	0	0
$274$	−1467.00	−0.323448
$275$	1690.48	0.370690
$276$	−780.000	−0.170110
$277$	−377.000	−0.0817752	−0.0408876	−	0.999164i	$-0.513019\pi$
$277$	−377.000	−0.0817752	−0.0408876	+	0.999164i	$0.513019\pi$
$278$	408.764	0.0881872
$279$	3585.35	0.769351
$280$	540.400	0.115340
$281$	−36.3731	−0.00772183	−0.00386092	−	0.999993i	$-0.501229\pi$
$281$	−36.3731	−0.00772183	−0.00386092	+	0.999993i	$0.501229\pi$
$282$	1044.00	0.220458
$283$	7124.00	1.49639	0.748194	−	0.663480i	$-0.230921\pi$
$283$	7124.00	1.49639	0.748194	+	0.663480i	$0.230921\pi$
$284$	5282.75	1.10378
$285$	396.000	0.0823053
$286$	0	0
$287$	−3768.00	−0.774976
$288$	−4182.90	−0.855833
$289$	8776.00	1.78628
$290$	−423.000	−0.0856532
$291$	−401.836	−0.0809486
$292$	−2294.97	−0.459941
$293$	−8322.50	−1.65941	−0.829703	−	0.558205i	$-0.811490\pi$
$293$	−8322.50	−1.65941	−0.829703	+	0.558205i	$0.811490\pi$
$294$	−523.079	−0.103764
$295$	−492.000	−0.0971029
$296$	−3237.00	−0.635631
$297$	1385.64	0.270717
$298$	81.0000	0.0157457
$299$	0	0
$300$	1220.00	0.234789
$301$	1441.07	0.275952
$302$	−3066.00	−0.584200
$303$	−858.000	−0.162676
$304$	114.315	0.0215672
$305$	251.147	0.0471497
$306$	4660.95	0.870747
$307$	−2220.49	−0.412801	−0.206401	−	0.978468i	$-0.566175\pi$
$307$	−2220.49	−0.412801	−0.206401	+	0.978468i	$0.566175\pi$
$308$	−960.000	−0.177601
$309$	−364.000	−0.0670137
$310$	−467.654	−0.0856805
$311$	−4914.00	−0.895972	−0.447986	−	0.894041i	$-0.647859\pi$
$311$	−4914.00	−0.895972	−0.447986	+	0.894041i	$0.647859\pi$
$312$	0	0
$313$	−518.000	−0.0935434	−0.0467717	−	0.998906i	$-0.514893\pi$
$313$	−518.000	−0.0935434	−0.0467717	+	0.998906i	$0.514893\pi$
$314$	2097.51	0.376973
$315$	552.000	0.0987355
$316$	−6380.00	−1.13577
$317$	−3916.17	−0.693861	−0.346930	−	0.937891i	$-0.612776\pi$
$317$	−3916.17	−0.693861	−0.346930	+	0.937891i	$0.612776\pi$
$318$	322.161	0.0568111
$319$	1953.75	0.342913
$320$	531.740	0.0928911
$321$	−3012.00	−0.523718
$322$	−1872.00	−0.323983
$323$	−13374.9	−2.30402
$324$	−2105.00	−0.360940
$325$	0	0
$326$	−1740.00	−0.295613
$327$	−3103.84	−0.524901
$328$	−6123.00	−1.03075
$329$	−4176.00	−0.699788
$330$	−83.1384	−0.0138685
$331$	7454.75	1.23792	0.618958	−	0.785424i	$-0.287555\pi$
$331$	7454.75	1.23792	0.618958	+	0.785424i	$0.287555\pi$
$332$	−3949.08	−0.652812
$333$	−3306.48	−0.544127
$334$	−1584.00	−0.259499
$335$	−1362.00	−0.222131
$336$	−27.7128	−0.00449958
$337$	3575.00	0.577871	0.288936	−	0.957349i	$-0.406699\pi$
$337$	3575.00	0.577871	0.288936	+	0.957349i	$0.406699\pi$
$338$	0	0
$339$	−1374.00	−0.220134
$340$	1013.25	0.161621
$341$	2160.00	0.343022
$342$	−4554.00	−0.720035
$343$	6845.06	1.07755
$344$	2341.73	0.367028
$345$	270.200	0.0421654
$346$	−4458.30	−0.692716
$347$	−6966.00	−1.07768	−0.538839	−	0.842409i	$-0.681137\pi$
$347$	−6966.00	−1.07768	−0.538839	+	0.842409i	$0.681137\pi$
$348$	1410.00	0.217195
$349$	6651.08	1.02013	0.510063	−	0.860137i	$-0.329622\pi$
$349$	6651.08	1.02013	0.510063	+	0.860137i	$0.329622\pi$
$350$	2928.00	0.447166
$351$	0	0
$352$	−2520.00	−0.381581
$353$	5630.90	0.849015	0.424508	−	0.905424i	$-0.360447\pi$
$353$	5630.90	0.849015	0.424508	+	0.905424i	$0.360447\pi$
$354$	−984.000	−0.147737
$355$	−1830.00	−0.273595
$356$	−4884.38	−0.727168
$357$	3242.40	0.480689
$358$	−6484.80	−0.957353
$359$	−7129.12	−1.04808	−0.524040	−	0.851694i	$-0.675576\pi$
$359$	−7129.12	−1.04808	−0.524040	+	0.851694i	$0.675576\pi$
$360$	897.000	0.131322
$361$	6209.00	0.905234
$362$	1103.32	0.160191
$363$	−2278.00	−0.329377
$364$	0	0
$365$	795.000	0.114006
$366$	502.295	0.0717360
$367$	2.00000	0.000284466 0	0.000142233	−	1.00000i	$-0.499955\pi$
$367$	2.00000	0.000284466 0	0.000142233		1.00000i	$0.499955\pi$
$368$	78.0000	0.0110490
$369$	−6254.44	−0.882366
$370$	431.281	0.0605979
$371$	−1288.65	−0.180332
$372$	1558.85	0.217264
$373$	3499.00	0.485714	0.242857	−	0.970062i	$-0.421915\pi$
$373$	3499.00	0.485714	0.242857	+	0.970062i	$0.421915\pi$
$374$	2808.00	0.388231
$375$	−855.633	−0.117826
$376$	−6786.00	−0.930748
$377$	0	0
$378$	2400.00	0.326568
$379$	5518.31	0.747907	0.373953	−	0.927447i	$-0.378002\pi$
$379$	5518.31	0.747907	0.373953	+	0.927447i	$0.378002\pi$
$380$	−990.000	−0.133647
$381$	−572.000	−0.0769146
$382$	−4499.87	−0.602705
$383$	7364.68	0.982552	0.491276	−	0.871004i	$-0.336531\pi$
$383$	7364.68	0.982552	0.491276	+	0.871004i	$0.336531\pi$
$384$	−1846.37	−0.245370
$385$	332.554	0.0440221
$386$	−1935.00	−0.255153
$387$	2392.00	0.314192
$388$	1004.59	0.131444
$389$	1209.00	0.157580	0.0787901	−	0.996891i	$-0.474894\pi$
$389$	1209.00	0.157580	0.0787901	+	0.996891i	$0.474894\pi$
$390$	0	0
$391$	−9126.00	−1.18036
$392$	3400.02	0.438078
$393$	−3948.00	−0.506744
$394$	3552.00	0.454181
$395$	2210.10	0.281524
$396$	−1593.49	−0.202211
$397$	11694.8	1.47845	0.739226	−	0.673457i	$-0.235191\pi$
$397$	11694.8	1.47845	0.739226	+	0.673457i	$0.235191\pi$
$398$	4368.23	0.550150
$399$	−3168.00	−0.397490
$400$	−122.000	−0.0152500
$401$	2980.86	0.371215	0.185607	−	0.982624i	$-0.440575\pi$
$401$	2980.86	0.371215	0.185607	+	0.982624i	$0.440575\pi$
$402$	−2724.00	−0.337962
$403$	0	0
$404$	2145.00	0.264153
$405$	729.193	0.0894664
$406$	3384.00	0.413658
$407$	−1992.00	−0.242604
$408$	5268.90	0.639337
$409$	−43.3013	−0.00523499	−0.00261749	−	0.999997i	$-0.500833\pi$
$409$	−43.3013	−0.00523499	−0.00261749	+	0.999997i	$0.500833\pi$
$410$	815.796	0.0982666
$411$	−1693.95	−0.203300
$412$	910.000	0.108817
$413$	3936.00	0.468954
$414$	−3107.30	−0.368878
$415$	1368.00	0.161813
$416$	0	0
$417$	472.000	0.0554291
$418$	−2743.57	−0.321034
$419$	−9462.00	−1.10322	−0.551610	−	0.834102i	$-0.685986\pi$
$419$	−9462.00	−1.10322	−0.551610	+	0.834102i	$0.685986\pi$
$420$	240.000	0.0278829
$421$	−7068.50	−0.818284	−0.409142	−	0.912471i	$-0.634172\pi$
$421$	−7068.50	−0.818284	−0.409142	+	0.912471i	$0.634172\pi$
$422$	1804.80	0.208190
$423$	−6931.67	−0.796759
$424$	−2094.05	−0.239849
$425$	14274.0	1.62915
$426$	−3660.00	−0.416262
$427$	−2009.18	−0.227707
$428$	7530.00	0.850412
$429$	0	0
$430$	−312.000	−0.0349906
$431$	−9928.12	−1.10956	−0.554780	−	0.831997i	$-0.687198\pi$
$431$	−9928.12	−1.10956	−0.554780	+	0.831997i	$0.687198\pi$
$432$	−100.000	−0.0111372
$433$	6617.00	0.734394	0.367197	−	0.930143i	$-0.380317\pi$
$433$	6617.00	0.734394	0.367197	+	0.930143i	$0.380317\pi$
$434$	3741.23	0.413790
$435$	−488.438	−0.0538364
$436$	7759.59	0.852332
$437$	8916.60	0.976061
$438$	1590.00	0.173455
$439$	−13988.0	−1.52075	−0.760377	−	0.649482i	$-0.774986\pi$
$439$	−13988.0	−1.52075	−0.760377	+	0.649482i	$0.774986\pi$
$440$	540.400	0.0585513
$441$	3473.00	0.375013
$442$	0	0
$443$	2004.00	0.214928	0.107464	−	0.994209i	$-0.465727\pi$
$443$	2004.00	0.214928	0.107464	+	0.994209i	$0.465727\pi$
$444$	−1437.60	−0.153661
$445$	1692.00	0.180244
$446$	−4170.00	−0.442725
$447$	93.5307	0.00989676
$448$	−4253.92	−0.448613
$449$	9082.87	0.954671	0.477336	−	0.878721i	$-0.341603\pi$
$449$	9082.87	0.954671	0.477336	+	0.878721i	$0.341603\pi$
$450$	4860.13	0.509131
$451$	−3768.00	−0.393411
$452$	3435.00	0.357453
$453$	−3540.31	−0.367193
$454$	−4170.00	−0.431074
$455$	0	0
$456$	−5148.00	−0.528678
$457$	2523.60	0.258313	0.129156	−	0.991624i	$-0.458773\pi$
$457$	2523.60	0.258313	0.129156	+	0.991624i	$0.458773\pi$
$458$	4344.00	0.443192
$459$	11700.0	1.18978
$460$	−675.500	−0.0684681
$461$	−19587.8	−1.97894	−0.989472	−	0.144725i	$-0.953770\pi$
$461$	−19587.8	−1.97894	−0.989472	+	0.144725i	$0.953770\pi$
$462$	665.108	0.0669775
$463$	8632.54	0.866497	0.433249	−	0.901274i	$-0.357367\pi$
$463$	8632.54	0.866497	0.433249	+	0.901274i	$0.357367\pi$
$464$	−141.000	−0.0141072
$465$	−540.000	−0.0538535
$466$	10132.5	1.00725
$467$	−5460.00	−0.541025	−0.270512	−	0.962716i	$-0.587193\pi$
$467$	−5460.00	−0.541025	−0.270512	+	0.962716i	$0.587193\pi$
$468$	0	0
$469$	10896.0	1.07277
$470$	904.131	0.0887328
$471$	2422.00	0.236942
$472$	6396.00	0.623728
$473$	1441.07	0.140085
$474$	4420.19	0.428325
$475$	−13946.5	−1.34717
$476$	−8106.00	−0.780542
$477$	−2139.00	−0.205321
$478$	9324.00	0.892196
$479$	2553.04	0.243531	0.121766	−	0.992559i	$-0.461144\pi$
$479$	2553.04	0.243531	0.121766	+	0.992559i	$0.461144\pi$
$480$	630.000	0.0599072
$481$	0	0
$482$	8517.00	0.804852
$483$	−2161.60	−0.203636
$484$	5695.00	0.534842
$485$	−348.000	−0.0325812
$486$	6134.92	0.572605
$487$	−10828.8	−1.00760	−0.503798	−	0.863822i	$-0.668064\pi$
$487$	−10828.8	−1.00760	−0.503798	+	0.863822i	$0.668064\pi$
$488$	−3264.92	−0.302860
$489$	−2009.18	−0.185804
$490$	−453.000	−0.0417642
$491$	−11388.0	−1.04671	−0.523354	−	0.852116i	$-0.675319\pi$
$491$	−11388.0	−1.04671	−0.523354	+	0.852116i	$0.675319\pi$
$492$	−2719.32	−0.249180
$493$	16497.0	1.50707
$494$	0	0
$495$	552.000	0.0501223
$496$	−155.885	−0.0141117
$497$	14640.0	1.32132
$498$	2736.00	0.246191
$499$	17677.3	1.58586	0.792931	−	0.609311i	$-0.208554\pi$
$499$	17677.3	1.58586	0.792931	+	0.609311i	$0.208554\pi$
$500$	2139.08	0.191325
$501$	−1829.05	−0.163105
$502$	6890.10	0.612590
$503$	3876.00	0.343583	0.171792	−	0.985133i	$-0.445044\pi$
$503$	3876.00	0.343583	0.171792	+	0.985133i	$0.445044\pi$
$504$	−7176.00	−0.634215
$505$	−743.050	−0.0654758
$506$	−1872.00	−0.164467
$507$	0	0
$508$	1430.00	0.124894
$509$	−17065.9	−1.48612	−0.743058	−	0.669228i	$-0.766625\pi$
$509$	−17065.9	−1.48612	−0.743058	+	0.669228i	$0.766625\pi$
$510$	−702.000	−0.0609511
$511$	−6360.00	−0.550587
$512$	361.999	0.0312465
$513$	−11431.5	−0.983849
$514$	−3580.15	−0.307225
$515$	−315.233	−0.0269725
$516$	1040.00	0.0887276
$517$	−4176.00	−0.355242
$518$	−3450.25	−0.292655
$519$	−5148.00	−0.435399
$520$	0	0
$521$	2121.00	0.178355	0.0891773	−	0.996016i	$-0.471576\pi$
$521$	2121.00	0.178355	0.0891773	+	0.996016i	$0.471576\pi$
$522$	5617.04	0.470979
$523$	−11464.0	−0.958481	−0.479241	−	0.877684i	$-0.659088\pi$
$523$	−11464.0	−0.958481	−0.479241	+	0.877684i	$0.659088\pi$
$524$	9870.00	0.822849
$525$	3380.96	0.281062
$526$	−3554.17	−0.294618
$527$	18238.5	1.50755
$528$	−27.7128	−0.00228418
$529$	−6083.00	−0.499959
$530$	279.000	0.0228660
$531$	6533.30	0.533938
$532$	7920.00	0.645443
$533$	0	0
$534$	3384.00	0.274232
$535$	−2608.47	−0.210792
$536$	17706.0	1.42683
$537$	−7488.00	−0.601734
$538$	5767.73	0.462202
$539$	2092.32	0.167203
$540$	866.025	0.0690144
$541$	−4764.87	−0.378665	−0.189333	−	0.981913i	$-0.560632\pi$
$541$	−4764.87	−0.378665	−0.189333	+	0.981913i	$0.560632\pi$
$542$	−4860.00	−0.385157
$543$	1274.00	0.100686
$544$	−21278.2	−1.67702
$545$	−2688.00	−0.211268
$546$	0	0
$547$	6554.00	0.512301	0.256151	−	0.966637i	$-0.417546\pi$
$547$	6554.00	0.512301	0.256151	+	0.966637i	$0.417546\pi$
$548$	4234.86	0.330118
$549$	−3335.00	−0.259261
$550$	2928.00	0.227001
$551$	−16118.5	−1.24622
$552$	−3512.60	−0.270844
$553$	−17680.8	−1.35961
$554$	−652.983	−0.0500769
$555$	498.000	0.0380881
$556$	−1180.00	−0.0900057
$557$	18112.1	1.37780	0.688898	−	0.724858i	$-0.258095\pi$
$557$	18112.1	1.37780	0.688898	+	0.724858i	$0.258095\pi$
$558$	6210.00	0.471130
$559$	0	0
$560$	−24.0000	−0.00181104
$561$	3242.40	0.244018
$562$	−63.0000	−0.00472864
$563$	12168.0	0.910870	0.455435	−	0.890269i	$-0.349484\pi$
$563$	12168.0	0.910870	0.455435	+	0.890269i	$0.349484\pi$
$564$	−3013.77	−0.225005
$565$	−1189.92	−0.0886022
$566$	12339.1	0.916347
$567$	−5833.55	−0.432074
$568$	23790.0	1.75741
$569$	7722.00	0.568933	0.284467	−	0.958686i	$-0.408183\pi$
$569$	7722.00	0.568933	0.284467	+	0.958686i	$0.408183\pi$
$570$	685.892	0.0504015
$571$	−11440.0	−0.838440	−0.419220	−	0.907885i	$-0.637696\pi$
$571$	−11440.0	−0.838440	−0.419220	+	0.907885i	$0.637696\pi$
$572$	0	0
$573$	−5196.00	−0.378824
$574$	−6526.37	−0.474574
$575$	−9516.00	−0.690165
$576$	−7061.00	−0.510778
$577$	−15444.7	−1.11433	−0.557167	−	0.830400i	$-0.688112\pi$
$577$	−15444.7	−1.11433	−0.557167	+	0.830400i	$0.688112\pi$
$578$	15200.5	1.09387
$579$	−2234.35	−0.160373
$580$	1221.10	0.0874194
$581$	−10944.0	−0.781469
$582$	−696.000	−0.0495707
$583$	−1288.65	−0.0915442
$584$	−10335.0	−0.732304
$585$	0	0
$586$	−14415.0	−1.01617
$587$	−14071.2	−0.989403	−0.494702	−	0.869063i	$-0.664723\pi$
$587$	−14071.2	−0.989403	−0.494702	+	0.869063i	$0.664723\pi$
$588$	1510.00	0.105904
$589$	−17820.0	−1.24662
$590$	−852.169	−0.0594631
$591$	4101.50	0.285470
$592$	143.760	0.00998059
$593$	26938.6	1.86549	0.932745	−	0.360538i	$-0.117407\pi$
$593$	26938.6	1.86549	0.932745	+	0.360538i	$0.117407\pi$
$594$	2400.00	0.165780
$595$	2808.00	0.193474
$596$	−233.827	−0.0160703
$597$	5044.00	0.345791
$598$	0	0
$599$	−10554.0	−0.719908	−0.359954	−	0.932970i	$-0.617208\pi$
$599$	−10554.0	−0.719908	−0.359954	+	0.932970i	$0.617208\pi$
$600$	5494.07	0.373824
$601$	−14831.0	−1.00660	−0.503302	−	0.864111i	$-0.667882\pi$
$601$	−14831.0	−1.00660	−0.503302	+	0.864111i	$0.667882\pi$
$602$	2496.00	0.168986
$603$	18086.1	1.22143
$604$	8850.78	0.596247
$605$	−1972.81	−0.132572
$606$	−1486.10	−0.0996183
$607$	−7954.00	−0.531866	−0.265933	−	0.963991i	$-0.585680\pi$
$607$	−7954.00	−0.531866	−0.265933	+	0.963991i	$0.585680\pi$
$608$	20790.0	1.38675
$609$	3907.51	0.260000
$610$	435.000	0.0288732
$611$	0	0
$612$	−13455.0	−0.888703
$613$	25220.4	1.66173	0.830866	−	0.556472i	$-0.187845\pi$
$613$	25220.4	1.66173	0.830866	+	0.556472i	$0.187845\pi$
$614$	−3846.00	−0.252788
$615$	942.000	0.0617644
$616$	−4323.20	−0.282771
$617$	17384.6	1.13432	0.567162	−	0.823607i	$-0.308041\pi$
$617$	17384.6	1.13432	0.567162	+	0.823607i	$0.308041\pi$
$618$	−630.466	−0.0410373
$619$	−8209.92	−0.533093	−0.266547	−	0.963822i	$-0.585883\pi$
$619$	−8209.92	−0.533093	−0.266547	+	0.963822i	$0.585883\pi$
$620$	1350.00	0.0874473
$621$	−7800.00	−0.504031
$622$	−8511.30	−0.548669
$623$	−13536.0	−0.870479
$624$	0	0
$625$	14509.0	0.928576
$626$	−897.202	−0.0572834
$627$	−3168.00	−0.201783
$628$	−6055.00	−0.384747
$629$	−16819.9	−1.06622
$630$	956.092	0.0604629
$631$	12865.7	0.811687	0.405843	−	0.913943i	$-0.366978\pi$
$631$	12865.7	0.811687	0.405843	+	0.913943i	$0.366978\pi$
$632$	−28731.3	−1.80834
$633$	2084.00	0.130856
$634$	−6783.00	−0.424901
$635$	−495.367	−0.0309575
$636$	−930.000	−0.0579825
$637$	0	0
$638$	3384.00	0.209990
$639$	24300.7	1.50441
$640$	−1599.00	−0.0987594
$641$	−6201.00	−0.382098	−0.191049	−	0.981581i	$-0.561189\pi$
$641$	−6201.00	−0.382098	−0.191049	+	0.981581i	$0.561189\pi$
$642$	−5216.94	−0.320710
$643$	16821.7	1.03170	0.515849	−	0.856679i	$-0.327476\pi$
$643$	16821.7	1.03170	0.515849	+	0.856679i	$0.327476\pi$
$644$	5404.00	0.330664
$645$	−360.267	−0.0219930
$646$	−23166.0	−1.41092
$647$	13494.0	0.819944	0.409972	−	0.912098i	$-0.365538\pi$
$647$	13494.0	0.819944	0.409972	+	0.912098i	$0.365538\pi$
$648$	−9479.51	−0.574677
$649$	3936.00	0.238061
$650$	0	0
$651$	4320.00	0.260083
$652$	5022.95	0.301708
$653$	−11334.0	−0.679225	−0.339612	−	0.940566i	$-0.610296\pi$
$653$	−11334.0	−0.679225	−0.339612	+	0.940566i	$0.610296\pi$
$654$	−5376.00	−0.321435
$655$	−3419.07	−0.203960
$656$	271.932	0.0161847
$657$	−10556.8	−0.626883
$658$	−7233.04	−0.428531
$659$	13236.0	0.782400	0.391200	−	0.920306i	$-0.372060\pi$
$659$	13236.0	0.782400	0.391200	+	0.920306i	$0.372060\pi$
$660$	240.000	0.0141545
$661$	11852.4	0.697437	0.348718	−	0.937228i	$-0.386617\pi$
$661$	11852.4	0.697437	0.348718	+	0.937228i	$0.386617\pi$
$662$	12912.0	0.758065
$663$	0	0
$664$	−17784.0	−1.03939
$665$	−2743.57	−0.159986
$666$	−5727.00	−0.333208
$667$	−10998.0	−0.638447
$668$	4572.61	0.264850
$669$	−4815.10	−0.278270
$670$	−2359.05	−0.136027
$671$	−2009.18	−0.115594
$672$	−5040.00	−0.289319
$673$	−8021.00	−0.459416	−0.229708	−	0.973260i	$-0.573777\pi$
$673$	−8021.00	−0.459416	−0.229708	+	0.973260i	$0.573777\pi$
$674$	6192.08	0.353873
$675$	12200.0	0.695671
$676$	0	0
$677$	−21630.0	−1.22793	−0.613965	−	0.789333i	$-0.710426\pi$
$677$	−21630.0	−1.22793	−0.613965	+	0.789333i	$0.710426\pi$
$678$	−2379.84	−0.134804
$679$	2784.00	0.157349
$680$	4563.00	0.257328
$681$	−4815.10	−0.270947
$682$	3741.23	0.210057
$683$	−26538.5	−1.48677	−0.743387	−	0.668861i	$-0.766782\pi$
$683$	−26538.5	−1.48677	−0.743387	+	0.668861i	$0.766782\pi$
$684$	13146.3	0.734883
$685$	−1467.00	−0.0818266
$686$	11856.0	0.659860
$687$	5016.02	0.278563
$688$	−104.000	−0.00576303
$689$	0	0
$690$	468.000	0.0258210
$691$	−831.384	−0.0457704	−0.0228852	−	0.999738i	$-0.507285\pi$
$691$	−831.384	−0.0457704	−0.0228852	+	0.999738i	$0.507285\pi$
$692$	12870.0	0.707000
$693$	−4416.00	−0.242063
$694$	−12065.5	−0.659941
$695$	408.764	0.0223098
$696$	6349.70	0.345811
$697$	−31816.0	−1.72901
$698$	11520.0	0.624697
$699$	11700.0	0.633097
$700$	−8452.41	−0.456387
$701$	−30186.0	−1.62640	−0.813202	−	0.581981i	$-0.802278\pi$
$701$	−30186.0	−1.62640	−0.813202	+	0.581981i	$0.802278\pi$
$702$	0	0
$703$	16434.0	0.881679
$704$	−4253.92	−0.227735
$705$	1044.00	0.0557721
$706$	9753.00	0.519914
$707$	5944.40	0.316212
$708$	2840.56	0.150784
$709$	−11880.1	−0.629292	−0.314646	−	0.949209i	$-0.601886\pi$
$709$	−11880.1	−0.629292	−0.314646	+	0.949209i	$0.601886\pi$
$710$	−3169.65	−0.167542
$711$	−29348.0	−1.54801
$712$	−21996.0	−1.15777
$713$	−12159.0	−0.638651
$714$	5616.00	0.294361
$715$	0	0
$716$	18720.0	0.977094
$717$	10766.4	0.560780
$718$	−12348.0	−0.641815
$719$	−18408.0	−0.954802	−0.477401	−	0.878686i	$-0.658421\pi$
$719$	−18408.0	−0.954802	−0.477401	+	0.878686i	$0.658421\pi$
$720$	−39.8372	−0.00206201
$721$	2521.87	0.130262
$722$	10754.3	0.554340
$723$	9834.58	0.505881
$724$	−3185.00	−0.163494
$725$	17202.0	0.881195
$726$	−3945.61	−0.201702
$727$	−21112.0	−1.07703	−0.538515	−	0.842616i	$-0.681014\pi$
$727$	−21112.0	−1.07703	−0.538515	+	0.842616i	$0.681014\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	1376.98	0.0698142
$731$	12168.0	0.615663
$732$	−1450.00	−0.0732152
$733$	23959.5	1.20732	0.603658	−	0.797243i	$-0.293709\pi$
$733$	23959.5	1.20732	0.603658	+	0.797243i	$0.293709\pi$
$734$	3.46410	0.000174199 0
$735$	−523.079	−0.0262504
$736$	14185.5	0.710441
$737$	10896.0	0.544585
$738$	−10833.0	−0.540336
$739$	−3166.19	−0.157605	−0.0788025	−	0.996890i	$-0.525110\pi$
$739$	−3166.19	−0.157605	−0.0788025	+	0.996890i	$0.525110\pi$
$740$	−1245.00	−0.0618474
$741$	0	0
$742$	−2232.00	−0.110430
$743$	−30103.0	−1.48637	−0.743185	−	0.669086i	$-0.766686\pi$
$743$	−30103.0	−1.48637	−0.743185	+	0.669086i	$0.766686\pi$
$744$	7020.00	0.345922
$745$	81.0000	0.00398337
$746$	6060.45	0.297438
$747$	−18165.7	−0.889759
$748$	−8106.00	−0.396236
$749$	20867.7	1.01801
$750$	−1482.00	−0.0721533
$751$	−28496.0	−1.38460	−0.692299	−	0.721610i	$-0.743402\pi$
$751$	−28496.0	−1.38460	−0.692299	+	0.721610i	$0.743402\pi$
$752$	301.377	0.0146145
$753$	7956.00	0.385037
$754$	0	0
$755$	−3066.00	−0.147792
$756$	−6928.20	−0.333302
$757$	17422.0	0.836477	0.418239	−	0.908337i	$-0.362648\pi$
$757$	17422.0	0.836477	0.418239	+	0.908337i	$0.362648\pi$
$758$	9558.00	0.457998
$759$	−2161.60	−0.103374
$760$	−4458.30	−0.212789
$761$	41326.7	1.96858	0.984292	−	0.176547i	$-0.0564926\pi$
$761$	41326.7	1.96858	0.984292	+	0.176547i	$0.0564926\pi$
$762$	−990.733	−0.0471004
$763$	21504.0	1.02031
$764$	12990.0	0.615133
$765$	4660.95	0.220284
$766$	12756.0	0.601688
$767$	0	0
$768$	−8110.00	−0.381047
$769$	−14071.2	−0.659844	−0.329922	−	0.944008i	$-0.607022\pi$
$769$	−14071.2	−0.659844	−0.329922	+	0.944008i	$0.607022\pi$
$770$	576.000	0.0269579
$771$	−4134.00	−0.193103
$772$	5585.86	0.260414
$773$	−200.918	−0.00934866	−0.00467433	−	0.999989i	$-0.501488\pi$
$773$	−200.918	−0.00934866	−0.00467433	+	0.999989i	$0.501488\pi$
$774$	4143.07	0.192402
$775$	19017.9	0.881476
$776$	4524.00	0.209281
$777$	−3984.00	−0.183945
$778$	2094.05	0.0964978
$779$	31086.0	1.42975
$780$	0	0
$781$	14640.0	0.670756
$782$	−15806.7	−0.722821
$783$	14100.0	0.643541
$784$	−151.000	−0.00687864
$785$	2097.51	0.0953675
$786$	−6838.14	−0.310316
$787$	−6903.95	−0.312706	−0.156353	−	0.987701i	$-0.549974\pi$
$787$	−6903.95	−0.312706	−0.156353	+	0.987701i	$0.549974\pi$
$788$	−10253.7	−0.463546
$789$	−4104.00	−0.185179
$790$	3828.00	0.172398
$791$	9519.35	0.427900
$792$	−7176.00	−0.321955
$793$	0	0
$794$	20256.0	0.905363
$795$	322.161	0.0143722
$796$	−12610.0	−0.561494
$797$	31278.0	1.39012	0.695059	−	0.718953i	$-0.255378\pi$
$797$	31278.0	1.39012	0.695059	+	0.718953i	$0.255378\pi$
$798$	−5487.14	−0.243412
$799$	−35261.1	−1.56126
$800$	−22187.6	−0.980561
$801$	−22468.2	−0.991103
$802$	5163.00	0.227322
$803$	−6360.00	−0.279501
$804$	7863.51	0.344931
$805$	−1872.00	−0.0819619
$806$	0	0
$807$	6660.00	0.290512
$808$	9659.65	0.420576
$809$	8049.00	0.349799	0.174900	−	0.984586i	$-0.444040\pi$
$809$	8049.00	0.349799	0.174900	+	0.984586i	$0.444040\pi$
$810$	1263.00	0.0547868
$811$	−14026.1	−0.607305	−0.303653	−	0.952783i	$-0.598206\pi$
$811$	−14026.1	−0.607305	−0.303653	+	0.952783i	$0.598206\pi$
$812$	−9768.77	−0.422188
$813$	−5611.84	−0.242086
$814$	−3450.25	−0.148564
$815$	−1740.00	−0.0747847
$816$	−234.000	−0.0100388
$817$	−11888.8	−0.509102
$818$	−75.0000	−0.00320576
$819$	0	0
$820$	−2355.00	−0.100293
$821$	−8036.72	−0.341636	−0.170818	−	0.985303i	$-0.554641\pi$
$821$	−8036.72	−0.341636	−0.170818	+	0.985303i	$0.554641\pi$
$822$	−2934.00	−0.124495
$823$	40300.0	1.70689	0.853445	−	0.521184i	$-0.174509\pi$
$823$	40300.0	1.70689	0.853445	+	0.521184i	$0.174509\pi$
$824$	4098.03	0.173255
$825$	3380.96	0.142679
$826$	6817.35	0.287174
$827$	39525.4	1.66195	0.830975	−	0.556310i	$-0.187783\pi$
$827$	39525.4	1.66195	0.830975	+	0.556310i	$0.187783\pi$
$828$	8970.00	0.376484
$829$	12311.0	0.515776	0.257888	−	0.966175i	$-0.416973\pi$
$829$	12311.0	0.515776	0.257888	+	0.966175i	$0.416973\pi$
$830$	2369.45	0.0990899
$831$	−754.000	−0.0314753
$832$	0	0
$833$	17667.0	0.734844
$834$	817.528	0.0339433
$835$	−1584.00	−0.0656486
$836$	7920.00	0.327654
$837$	15588.5	0.643747
$838$	−16388.7	−0.675581
$839$	21467.0	0.883343	0.441671	−	0.897177i	$-0.354386\pi$
$839$	21467.0	0.883343	0.441671	+	0.897177i	$0.354386\pi$
$840$	1080.80	0.0443942
$841$	−4508.00	−0.184837
$842$	−12243.0	−0.501095
$843$	−72.7461	−0.00297214
$844$	−5210.00	−0.212483
$845$	0	0
$846$	−12006.0	−0.487913
$847$	15782.4	0.640249
$848$	93.0000	0.00376608
$849$	14248.0	0.575960
$850$	24723.3	0.997649
$851$	11213.3	0.451688
$852$	10565.5	0.424846
$853$	774.227	0.0310774	0.0155387	−	0.999879i	$-0.495054\pi$
$853$	774.227	0.0310774	0.0155387	+	0.999879i	$0.495054\pi$
$854$	−3480.00	−0.139442
$855$	−4554.00	−0.182156
$856$	33910.1	1.35400
$857$	−13923.0	−0.554960	−0.277480	−	0.960731i	$-0.589499\pi$
$857$	−13923.0	−0.554960	−0.277480	+	0.960731i	$0.589499\pi$
$858$	0	0
$859$	−22358.0	−0.888062	−0.444031	−	0.896011i	$-0.646452\pi$
$859$	−22358.0	−0.888062	−0.444031	+	0.896011i	$0.646452\pi$
$860$	900.666	0.0357122
$861$	−7536.00	−0.298288
$862$	−17196.0	−0.679464
$863$	2230.88	0.0879955	0.0439977	−	0.999032i	$-0.485991\pi$
$863$	2230.88	0.0879955	0.0439977	+	0.999032i	$0.485991\pi$
$864$	−18186.5	−0.716109
$865$	−4458.30	−0.175245
$866$	11461.0	0.449723
$867$	17552.0	0.687540
$868$	−10800.0	−0.422322
$869$	−17680.8	−0.690195
$870$	−846.000	−0.0329679
$871$	0	0
$872$	34944.0	1.35706
$873$	4621.11	0.179153
$874$	15444.0	0.597713
$875$	5928.00	0.229032
$876$	−4589.93	−0.177031
$877$	16754.1	0.645093	0.322547	−	0.946554i	$-0.395461\pi$
$877$	16754.1	0.645093	0.322547	+	0.946554i	$0.395461\pi$
$878$	−24227.9	−0.931268
$879$	−16645.0	−0.638706
$880$	−24.0000	−0.000919363 0
$881$	−17355.0	−0.663683	−0.331842	−	0.943335i	$-0.607670\pi$
$881$	−17355.0	−0.663683	−0.331842	+	0.943335i	$0.607670\pi$
$882$	6015.41	0.229648
$883$	46982.0	1.79057	0.895283	−	0.445497i	$-0.146973\pi$
$883$	46982.0	1.79057	0.895283	+	0.445497i	$0.146973\pi$
$884$	0	0
$885$	−984.000	−0.0373749
$886$	3471.03	0.131616
$887$	−8916.00	−0.337508	−0.168754	−	0.985658i	$-0.553974\pi$
$887$	−8916.00	−0.337508	−0.168754	+	0.985658i	$0.553974\pi$
$888$	−6474.00	−0.244655
$889$	3962.93	0.149508
$890$	2930.63	0.110376
$891$	−5833.55	−0.219339
$892$	12037.8	0.451854
$893$	34452.0	1.29103
$894$	162.000	0.00606050
$895$	−6484.80	−0.242193
$896$	12792.0	0.476954
$897$	0	0
$898$	15732.0	0.584614
$899$	21979.7	0.815423
$900$	−14030.0	−0.519630
$901$	−10881.0	−0.402329
$902$	−6526.37	−0.240914
$903$	2882.13	0.106214
$904$	15468.9	0.569126
$905$	1103.32	0.0405254
$906$	−6132.00	−0.224859
$907$	30836.0	1.12888	0.564439	−	0.825475i	$-0.309092\pi$
$907$	30836.0	1.12888	0.564439	+	0.825475i	$0.309092\pi$
$908$	12037.8	0.439964
$909$	9867.00	0.360031
$910$	0	0
$911$	−27480.0	−0.999400	−0.499700	−	0.866199i	$-0.666556\pi$
$911$	−27480.0	−0.999400	−0.499700	+	0.866199i	$0.666556\pi$
$912$	228.631	0.00830123
$913$	−10944.0	−0.396707
$914$	4371.00	0.158184
$915$	502.295	0.0181479
$916$	−12540.0	−0.452331
$917$	27352.5	0.985017
$918$	20265.0	0.728589
$919$	−28442.0	−1.02091	−0.510454	−	0.859905i	$-0.670523\pi$
$919$	−28442.0	−1.02091	−0.510454	+	0.859905i	$0.670523\pi$
$920$	−3042.00	−0.109013
$921$	−4440.98	−0.158887
$922$	−33927.0	−1.21185
$923$	0	0
$924$	−1920.00	−0.0683586
$925$	−17538.7	−0.623427
$926$	14952.0	0.530619
$927$	4186.00	0.148313
$928$	−25643.0	−0.907083
$929$	−6978.43	−0.246453	−0.123227	−	0.992379i	$-0.539324\pi$
$929$	−6978.43	−0.246453	−0.123227	+	0.992379i	$0.539324\pi$
$930$	−935.307	−0.0329784
$931$	−17261.6	−0.607655
$932$	−29250.0	−1.02802
$933$	−9828.00	−0.344860
$934$	−9457.00	−0.331309
$935$	2808.00	0.0982154
$936$	0	0
$937$	−38465.0	−1.34109	−0.670543	−	0.741871i	$-0.733939\pi$
$937$	−38465.0	−1.34109	−0.670543	+	0.741871i	$0.733939\pi$
$938$	18872.4	0.656937
$939$	−1036.00	−0.0360049
$940$	−2610.00	−0.0905626
$941$	−4884.38	−0.169210	−0.0846049	−	0.996415i	$-0.526963\pi$
$941$	−4884.38	−0.169210	−0.0846049	+	0.996415i	$0.526963\pi$
$942$	4195.03	0.145097
$943$	21210.7	0.732466
$944$	−284.056	−0.00979369
$945$	2400.00	0.0826159
$946$	2496.00	0.0857843
$947$	−21765.0	−0.746849	−0.373424	−	0.927661i	$-0.621816\pi$
$947$	−21765.0	−0.746849	−0.373424	+	0.927661i	$0.621816\pi$
$948$	−12760.0	−0.437158
$949$	0	0
$950$	−24156.0	−0.824973
$951$	−7832.33	−0.267067
$952$	−36504.0	−1.24275
$953$	−6474.00	−0.220056	−0.110028	−	0.993928i	$-0.535094\pi$
$953$	−6474.00	−0.220056	−0.110028	+	0.993928i	$0.535094\pi$
$954$	−3704.86	−0.125733
$955$	−4499.87	−0.152474
$956$	−26916.1	−0.910594
$957$	3907.51	0.131987
$958$	4422.00	0.149132
$959$	11736.0	0.395177
$960$	1063.48	0.0357538
$961$	−5491.00	−0.184317
$962$	0	0
$963$	34638.0	1.15908
$964$	−24586.5	−0.821449
$965$	−1935.00	−0.0645491
$966$	−3744.00	−0.124701
$967$	−7541.35	−0.250789	−0.125395	−	0.992107i	$-0.540020\pi$
$967$	−7541.35	−0.250789	−0.125395	+	0.992107i	$0.540020\pi$
$968$	25646.5	0.851559
$969$	−26749.8	−0.886819
$970$	−602.754	−0.0199518
$971$	34998.0	1.15668	0.578342	−	0.815795i	$-0.303700\pi$
$971$	34998.0	1.15668	0.578342	+	0.815795i	$0.303700\pi$
$972$	−17710.0	−0.584412
$973$	−3270.11	−0.107744
$974$	−18756.0	−0.617024
$975$	0	0
$976$	145.000	0.00475547
$977$	−25216.9	−0.825753	−0.412877	−	0.910787i	$-0.635476\pi$
$977$	−25216.9	−0.825753	−0.412877	+	0.910787i	$0.635476\pi$
$978$	−3480.00	−0.113781
$979$	−13536.0	−0.441892
$980$	1307.70	0.0426254
$981$	35694.1	1.16170
$982$	−19724.6	−0.640975
$983$	−56440.6	−1.83131	−0.915654	−	0.401967i	$-0.868327\pi$
$983$	−56440.6	−1.83131	−0.915654	+	0.401967i	$0.868327\pi$
$984$	−12246.0	−0.396736
$985$	3552.00	0.114900
$986$	28573.6	0.922891
$987$	−8352.00	−0.269349
$988$	0	0
$989$	−8112.00	−0.260816
$990$	956.092	0.0306935
$991$	59282.0	1.90026	0.950129	−	0.311859i	$-0.100951\pi$
$991$	59282.0	1.90026	0.950129	+	0.311859i	$0.100951\pi$
$992$	−28350.0	−0.907372
$993$	14909.5	0.476474
$994$	25357.2	0.809137
$995$	4368.23	0.139178
$996$	−7898.15	−0.251268
$997$	−37711.0	−1.19791	−0.598957	−	0.800782i	$-0.704418\pi$
$997$	−37711.0	−1.19791	−0.598957	+	0.800782i	$0.704418\pi$
$998$	30618.0	0.971138
$999$	−14376.0	−0.455292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.i.1.2		2
3.2	odd	2		1521.4.a.o.1.1		2
13.2	odd	12		13.4.e.b.4.1	✓	2
13.3	even	3		169.4.c.h.22.1		4
13.4	even	6		169.4.c.h.146.2		4
13.5	odd	4		169.4.b.d.168.1		2
13.6	odd	12		169.4.e.a.23.1		2
13.7	odd	12		13.4.e.b.10.1	yes	2
13.8	odd	4		169.4.b.d.168.2		2
13.9	even	3		169.4.c.h.146.1		4
13.10	even	6		169.4.c.h.22.2		4
13.11	odd	12		169.4.e.a.147.1		2
13.12	even	2	inner	169.4.a.i.1.1		2
39.2	even	12		117.4.q.a.82.1		2
39.20	even	12		117.4.q.a.10.1		2
39.38	odd	2		1521.4.a.o.1.2		2
52.7	even	12		208.4.w.b.49.1		2
52.15	even	12		208.4.w.b.17.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.e.b.4.1	✓	2	13.2	odd	12
13.4.e.b.10.1	yes	2	13.7	odd	12
117.4.q.a.10.1		2	39.20	even	12
117.4.q.a.82.1		2	39.2	even	12
169.4.a.i.1.1		2	13.12	even	2	inner
169.4.a.i.1.2		2	1.1	even	1	trivial
169.4.b.d.168.1		2	13.5	odd	4
169.4.b.d.168.2		2	13.8	odd	4
169.4.c.h.22.1		4	13.3	even	3
169.4.c.h.22.2		4	13.10	even	6
169.4.c.h.146.1		4	13.9	even	3
169.4.c.h.146.2		4	13.4	even	6
169.4.e.a.23.1		2	13.6	odd	12
169.4.e.a.147.1		2	13.11	odd	12
208.4.w.b.17.1		2	52.15	even	12
208.4.w.b.49.1		2	52.7	even	12
1521.4.a.o.1.1		2	3.2	odd	2
1521.4.a.o.1.2		2	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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +2.00000 q^{3} -5.00000 q^{4} +1.73205 q^{5} +3.46410 q^{6} -13.8564 q^{7} -22.5167 q^{8} -23.0000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{2} +2.00000 q^{3} -5.00000 q^{4} +1.73205 q^{5} +3.46410 q^{6} -13.8564 q^{7} -22.5167 q^{8} -23.0000 q^{9} +3.00000 q^{10} -13.8564 q^{11} -10.0000 q^{12} -24.0000 q^{14} +3.46410 q^{15} +1.00000 q^{16} -117.000 q^{17} -39.8372 q^{18} +114.315 q^{19} -8.66025 q^{20} -27.7128 q^{21} -24.0000 q^{22} +78.0000 q^{23} -45.0333 q^{24} -122.000 q^{25} -100.000 q^{27} +69.2820 q^{28} -141.000 q^{29} +6.00000 q^{30} -155.885 q^{31} +181.865 q^{32} -27.7128 q^{33} -202.650 q^{34} -24.0000 q^{35} +115.000 q^{36} +143.760 q^{37} +198.000 q^{38} -39.0000 q^{40} +271.932 q^{41} -48.0000 q^{42} -104.000 q^{43} +69.2820 q^{44} -39.8372 q^{45} +135.100 q^{46} +301.377 q^{47} +2.00000 q^{48} -151.000 q^{49} -211.310 q^{50} -234.000 q^{51} +93.0000 q^{53} -173.205 q^{54} -24.0000 q^{55} +312.000 q^{56} +228.631 q^{57} -244.219 q^{58} -284.056 q^{59} -17.3205 q^{60} +145.000 q^{61} -270.000 q^{62} +318.697 q^{63} +307.000 q^{64} -48.0000 q^{66} -786.351 q^{67} +585.000 q^{68} +156.000 q^{69} -41.5692 q^{70} -1056.55 q^{71} +517.883 q^{72} +458.993 q^{73} +249.000 q^{74} -244.000 q^{75} -571.577 q^{76} +192.000 q^{77} +1276.00 q^{79} +1.73205 q^{80} +421.000 q^{81} +471.000 q^{82} +789.815 q^{83} +138.564 q^{84} -202.650 q^{85} -180.133 q^{86} -282.000 q^{87} +312.000 q^{88} +976.877 q^{89} -69.0000 q^{90} -390.000 q^{92} -311.769 q^{93} +522.000 q^{94} +198.000 q^{95} +363.731 q^{96} -200.918 q^{97} -261.540 q^{98} +318.697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 10 q^{4} - 46 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 10 * q^4 - 46 * q^9	\( 2 q + 4 q^{3} - 10 q^{4} - 46 q^{9} + 6 q^{10} - 20 q^{12} - 48 q^{14} + 2 q^{16} - 234 q^{17} - 48 q^{22} + 156 q^{23} - 244 q^{25} - 200 q^{27} - 282 q^{29} + 12 q^{30} - 48 q^{35} + 230 q^{36} + 396 q^{38} - 78 q^{40} - 96 q^{42} - 208 q^{43} + 4 q^{48} - 302 q^{49} - 468 q^{51} + 186 q^{53} - 48 q^{55} + 624 q^{56} + 290 q^{61} - 540 q^{62} + 614 q^{64} - 96 q^{66} + 1170 q^{68} + 312 q^{69} + 498 q^{74} - 488 q^{75} + 384 q^{77} + 2552 q^{79} + 842 q^{81} + 942 q^{82} - 564 q^{87} + 624 q^{88} - 138 q^{90} - 780 q^{92} + 1044 q^{94} + 396 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 - 10 * q^4 - 46 * q^9 + 6 * q^10 - 20 * q^12 - 48 * q^14 + 2 * q^16 - 234 * q^17 - 48 * q^22 + 156 * q^23 - 244 * q^25 - 200 * q^27 - 282 * q^29 + 12 * q^30 - 48 * q^35 + 230 * q^36 + 396 * q^38 - 78 * q^40 - 96 * q^42 - 208 * q^43 + 4 * q^48 - 302 * q^49 - 468 * q^51 + 186 * q^53 - 48 * q^55 + 624 * q^56 + 290 * q^61 - 540 * q^62 + 614 * q^64 - 96 * q^66 + 1170 * q^68 + 312 * q^69 + 498 * q^74 - 488 * q^75 + 384 * q^77 + 2552 * q^79 + 842 * q^81 + 942 * q^82 - 564 * q^87 + 624 * q^88 - 138 * q^90 - 780 * q^92 + 1044 * q^94 + 396 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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