LMFDB - Embedded newform 169.4.b.d.168.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(168,169)]

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([1]))

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

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.d.168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	1.73205i	0.612372i	0.951972	+	0.306186i	$0.0990530\pi$
$2$	1.73205i	0.612372i	−0.951972	+	0.306186i	$0.900947\pi$
$3$	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
$5$	1.73205i	0.154919i	0.996995	+	0.0774597i	$0.0246809\pi$
$5$	1.73205i	0.154919i	−0.996995	+	0.0774597i	$0.975319\pi$
$6$	3.46410i	0.235702i
$7$	13.8564i	0.748176i	0.927393	+	0.374088i	$0.122044\pi$
$7$	13.8564i	0.748176i	−0.927393	+	0.374088i	$0.877956\pi$
$8$	22.5167i	0.995105i
$9$	−23.0000	−0.851852
$10$	−3.00000	−0.0948683
$11$	13.8564i	0.379806i	0.981803	+	0.189903i	$0.0608173\pi$
$11$	13.8564i	0.379806i	−0.981803	+	0.189903i	$0.939183\pi$
$12$	10.0000	0.240563
$13$	0	0
$13$	0	0
$14$	−24.0000	−0.458162
$15$	3.46410i	0.0596285i
$16$	1.00000	0.0156250
$17$	117.000	1.66922	0.834608	−	0.550845i	$-0.185694\pi$
$17$	117.000	1.66922	0.834608	+	0.550845i	$0.185694\pi$
$18$	− 39.8372i	− 0.521651i
$19$	114.315i	1.38030i	0.723665	+	0.690151i	$0.242456\pi$
$19$	114.315i	1.38030i	−0.723665	+	0.690151i	$0.757544\pi$
$20$	8.66025i	0.0968246i
$21$	27.7128i	0.287973i
$22$	−24.0000	−0.232583
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	45.0333i	0.383016i
$25$	122.000	0.976000
$26$	0	0
$27$	−100.000	−0.712778
$28$	69.2820i	0.467610i
$29$	−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
$31$	− 155.885i	− 0.903151i	−0.892233	−	0.451576i	$-0.850862\pi$
$31$	− 155.885i	− 0.903151i	0.892233	−	0.451576i	$-0.149138\pi$
$32$	181.865i	1.00467i
$33$	27.7128i	0.146187i
$34$	202.650i	1.02218i
$35$	−24.0000	−0.115907
$36$	−115.000	−0.532407
$37$	− 143.760i	− 0.638758i	−0.947627	−	0.319379i	$-0.896526\pi$
$37$	− 143.760i	− 0.638758i	0.947627	−	0.319379i	$-0.103474\pi$
$38$	−198.000	−0.845259
$39$	0	0
$40$	−39.0000	−0.154161
$41$	271.932i	1.03582i	0.855435	+	0.517910i	$0.173290\pi$
$41$	271.932i	1.03582i	−0.855435	+	0.517910i	$0.826710\pi$
$42$	−48.0000	−0.176347
$43$	104.000	0.368834	0.184417	−	0.982848i	$-0.440960\pi$
$43$	104.000	0.368834	0.184417	+	0.982848i	$0.440960\pi$
$44$	69.2820i	0.237379i
$45$	− 39.8372i	− 0.131968i
$46$	− 135.100i	− 0.433030i
$47$	− 301.377i	− 0.935326i	−0.883907	−	0.467663i	$-0.845096\pi$
$47$	− 301.377i	− 0.935326i	0.883907	−	0.467663i	$-0.154904\pi$
$48$	2.00000	0.00601407
$49$	151.000	0.440233
$50$	211.310i	0.597675i
$51$	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	$0.461546\pi$
$54$	− 173.205i	− 0.436486i
$55$	−24.0000	−0.0588393
$56$	−312.000	−0.744513
$57$	228.631i	0.531279i
$58$	− 244.219i	− 0.552889i
$59$	284.056i	0.626796i	0.949622	+	0.313398i	$0.101467\pi$
$59$	284.056i	0.626796i	−0.949622	+	0.313398i	$0.898533\pi$
$60$	17.3205i	0.0372678i
$61$	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
$63$	− 318.697i	− 0.637335i
$64$	−307.000	−0.599609
$65$	0	0
$66$	−48.0000	−0.0895211
$67$	− 786.351i	− 1.43385i	−0.697149	−	0.716926i	$-0.745549\pi$
$67$	− 786.351i	− 1.43385i	0.697149	−	0.716926i	$-0.254451\pi$
$68$	585.000	1.04326
$69$	−156.000	−0.272177
$70$	− 41.5692i	− 0.0709782i
$71$	− 1056.55i	− 1.76605i	−0.469326	−	0.883025i	$-0.655503\pi$
$71$	− 1056.55i	− 1.76605i	0.469326	−	0.883025i	$-0.344497\pi$
$72$	− 517.883i	− 0.847682i
$73$	− 458.993i	− 0.735906i	−0.929844	−	0.367953i	$-0.880059\pi$
$73$	− 458.993i	− 0.735906i	0.929844	−	0.367953i	$-0.119941\pi$
$74$	249.000	0.391158
$75$	244.000	0.375663
$76$	571.577i	0.862689i
$77$	−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	$0.137141\pi$
$80$	1.73205i	0.00242061i
$81$	421.000	0.577503
$82$	−471.000	−0.634308
$83$	789.815i	1.04450i	0.852793	+	0.522250i	$0.174907\pi$
$83$	789.815i	1.04450i	−0.852793	+	0.522250i	$0.825093\pi$
$84$	138.564i	0.179983i
$85$	202.650i	0.258594i
$86$	180.133i	0.225864i
$87$	−282.000	−0.347512
$88$	−312.000	−0.377947
$89$	− 976.877i	− 1.16347i	−0.813379	−	0.581734i	$-0.802374\pi$
$89$	− 976.877i	− 1.16347i	0.813379	−	0.581734i	$-0.197626\pi$
$90$	69.0000	0.0808138
$91$	0	0
$92$	−390.000	−0.441960
$93$	− 311.769i	− 0.347623i
$94$	522.000	0.572768
$95$	−198.000	−0.213835
$96$	363.731i	0.386699i
$97$	− 200.918i	− 0.210311i	−0.994456	−	0.105155i	$-0.966466\pi$
$97$	− 200.918i	− 0.210311i	0.994456	−	0.105155i	$-0.0335340\pi$
$98$	261.540i	0.269587i
$99$	− 318.697i	− 0.323538i
$100$	610.000	0.610000
$101$	429.000	0.422645	0.211322	−	0.977416i	$-0.432223\pi$
$101$	429.000	0.422645	0.211322	+	0.977416i	$0.432223\pi$
$102$	405.300i	0.393438i
$103$	182.000	0.174107	0.0870534	−	0.996204i	$-0.472255\pi$
$103$	182.000	0.174107	0.0870534	+	0.996204i	$0.472255\pi$
$104$	0	0
$105$	−48.0000	−0.0446126
$106$	161.081i	0.147599i
$107$	−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
$109$	− 1551.92i	− 1.36373i	−0.731477	−	0.681866i	$-0.761169\pi$
$109$	− 1551.92i	− 1.36373i	0.731477	−	0.681866i	$-0.238831\pi$
$110$	− 41.5692i	− 0.0360315i
$111$	− 287.520i	− 0.245858i
$112$	13.8564i	0.0116902i
$113$	−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
$115$	− 135.100i	− 0.109549i
$116$	−705.000	−0.564290
$117$	0	0
$118$	−492.000	−0.383833
$119$	1621.20i	1.24887i
$120$	−78.0000	−0.0593366
$121$	1139.00	0.855748
$122$	251.147i	0.186376i
$123$	543.864i	0.398687i
$124$	− 779.423i	− 0.564470i
$125$	427.817i	0.306121i
$126$	552.000	0.390286
$127$	286.000	0.199830	0.0999149	−	0.994996i	$-0.468143\pi$
$127$	286.000	0.199830	0.0999149	+	0.994996i	$0.468143\pi$
$128$	923.183i	0.637489i
$129$	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	$0.728715\pi$
$132$	138.564i	0.0913671i
$133$	−1584.00	−1.03271
$134$	1362.00	0.878051
$135$	− 173.205i	− 0.110423i
$136$	2634.45i	1.66105i
$137$	846.973i	0.528188i	0.964497	+	0.264094i	$0.0850729\pi$
$137$	846.973i	0.528188i	−0.964497	+	0.264094i	$0.914927\pi$
$138$	− 270.200i	− 0.166674i
$139$	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
$141$	− 602.754i	− 0.360007i
$142$	1830.00	1.08148
$143$	0	0
$144$	−23.0000	−0.0133102
$145$	− 244.219i	− 0.139871i
$146$	795.000	0.450648
$147$	302.000	0.169446
$148$	− 718.801i	− 0.399224i
$149$	46.7654i	0.0257125i	0.999917	+	0.0128563i	$0.00409239\pi$
$149$	46.7654i	0.0257125i	−0.999917	+	0.0128563i	$0.995908\pi$
$150$	422.620i	0.230045i
$151$	1770.16i	0.953995i	0.878905	+	0.476998i	$0.158275\pi$
$151$	1770.16i	0.953995i	−0.878905	+	0.476998i	$0.841725\pi$
$152$	−2574.00	−1.37355
$153$	−2691.00	−1.42192
$154$	− 332.554i	− 0.174013i
$155$	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	$0.400408\pi$
$158$	2210.10i	1.11282i
$159$	186.000	0.0927721
$160$	−315.000	−0.155643
$161$	− 1080.80i	− 0.529062i
$162$	729.193i	0.353647i
$163$	1004.59i	0.482733i	0.970434	+	0.241367i	$0.0775956\pi$
$163$	1004.59i	0.482733i	−0.970434	+	0.241367i	$0.922404\pi$
$164$	1359.66i	0.647388i
$165$	−48.0000	−0.0226472
$166$	−1368.00	−0.639623
$167$	914.523i	0.423760i	0.977296	+	0.211880i	$0.0679586\pi$
$167$	914.523i	0.423760i	−0.977296	+	0.211880i	$0.932041\pi$
$168$	−624.000	−0.286563
$169$	0	0
$170$	−351.000	−0.158356
$171$	− 2629.25i	− 1.17581i
$172$	520.000	0.230521
$173$	2574.00	1.13120	0.565600	−	0.824680i	$-0.308645\pi$
$173$	2574.00	1.13120	0.565600	+	0.824680i	$0.308645\pi$
$174$	− 488.438i	− 0.212807i
$175$	1690.48i	0.730219i
$176$	13.8564i	0.00593447i
$177$	568.113i	0.241254i
$178$	1692.00	0.712476
$179$	3744.00	1.56335	0.781675	−	0.623686i	$-0.214366\pi$
$179$	3744.00	1.56335	0.781675	+	0.623686i	$0.214366\pi$
$180$	− 199.186i	− 0.0824802i
$181$	−637.000	−0.261590	−0.130795	−	0.991409i	$-0.541753\pi$
$181$	−637.000	−0.261590	−0.130795	+	0.991409i	$0.541753\pi$
$182$	0	0
$183$	290.000	0.117144
$184$	− 1756.30i	− 0.703675i
$185$	249.000	0.0989559
$186$	540.000	0.212875
$187$	1621.20i	0.633978i
$188$	− 1506.88i	− 0.584579i
$189$	− 1385.64i	− 0.533283i
$190$	− 342.946i	− 0.130947i
$191$	−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
$193$	1117.17i	0.416662i	0.978058	+	0.208331i	$0.0668032\pi$
$193$	1117.17i	0.416662i	−0.978058	+	0.208331i	$0.933197\pi$
$194$	348.000	0.128788
$195$	0	0
$196$	755.000	0.275146
$197$	2050.75i	0.741674i	0.928698	+	0.370837i	$0.120929\pi$
$197$	2050.75i	0.741674i	−0.928698	+	0.370837i	$0.879071\pi$
$198$	552.000	0.198126
$199$	−2522.00	−0.898391	−0.449196	−	0.893433i	$-0.648289\pi$
$199$	−2522.00	−0.898391	−0.449196	+	0.893433i	$0.648289\pi$
$200$	2747.03i	0.971223i
$201$	− 1572.70i	− 0.551890i
$202$	743.050i	0.258816i
$203$	− 1953.75i	− 0.675500i
$204$	1170.00	0.401551
$205$	−471.000	−0.160469
$206$	315.233i	0.106618i
$207$	1794.00	0.602375
$208$	0	0
$209$	−1584.00	−0.524247
$210$	− 83.1384i	− 0.0273195i
$211$	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
$213$	− 2113.10i	− 0.679753i
$214$	− 2608.47i	− 0.833230i
$215$	180.133i	0.0571395i
$216$	− 2251.67i	− 0.709289i
$217$	2160.00	0.675716
$218$	2688.00	0.835112
$219$	− 917.987i	− 0.283250i
$220$	−120.000	−0.0367745
$221$	0	0
$222$	498.000	0.150557
$223$	− 2407.55i	− 0.722966i	−0.932379	−	0.361483i	$-0.882270\pi$
$223$	− 2407.55i	− 0.722966i	0.932379	−	0.361483i	$-0.117730\pi$
$224$	−2520.00	−0.751672
$225$	−2806.00	−0.831407
$226$	− 1189.92i	− 0.350231i
$227$	− 2407.55i	− 0.703942i	−0.936011	−	0.351971i	$-0.885512\pi$
$227$	− 2407.55i	− 0.703942i	0.936011	−	0.351971i	$-0.114488\pi$
$228$	1143.15i	0.332049i
$229$	− 2508.01i	− 0.723729i	−0.932231	−	0.361864i	$-0.882140\pi$
$229$	− 2508.01i	− 0.723729i	0.932231	−	0.361864i	$-0.117860\pi$
$230$	234.000	0.0670848
$231$	−384.000	−0.109374
$232$	− 3174.85i	− 0.898444i
$233$	−5850.00	−1.64483	−0.822417	−	0.568885i	$-0.807375\pi$
$233$	−5850.00	−1.64483	−0.822417	+	0.568885i	$0.807375\pi$
$234$	0	0
$235$	522.000	0.144900
$236$	1420.28i	0.391748i
$237$	2552.00	0.699452
$238$	−2808.00	−0.764771
$239$	5383.21i	1.45695i	0.685072	+	0.728475i	$0.259771\pi$
$239$	5383.21i	1.45695i	−0.685072	+	0.728475i	$0.740229\pi$
$240$	3.46410i	0 0.000931695i
$241$	− 4917.29i	− 1.31432i	−0.753752	−	0.657159i	$-0.771758\pi$
$241$	− 4917.29i	− 1.31432i	0.753752	−	0.657159i	$-0.228242\pi$
$242$	1972.81i	0.524036i
$243$	3542.00	0.935059
$244$	725.000	0.190219
$245$	261.540i	0.0682006i
$246$	−942.000	−0.244145
$247$	0	0
$248$	3510.00	0.898731
$249$	1579.63i	0.402028i
$250$	−741.000	−0.187460
$251$	−3978.00	−1.00036	−0.500178	−	0.865923i	$-0.666732\pi$
$251$	−3978.00	−1.00036	−0.500178	+	0.865923i	$0.666732\pi$
$252$	− 1593.49i	− 0.398334i
$253$	− 1080.80i	− 0.268574i
$254$	495.367i	0.122370i
$255$	405.300i	0.0995328i
$256$	−4055.00	−0.989990
$257$	2067.00	0.501696	0.250848	−	0.968026i	$-0.419291\pi$
$257$	2067.00	0.501696	0.250848	+	0.968026i	$0.419291\pi$
$258$	360.267i	0.0869349i
$259$	1992.00	0.477903
$260$	0	0
$261$	3243.00	0.769106
$262$	− 3419.07i	− 0.806224i
$263$	−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
$265$	161.081i	0.0373400i
$266$	− 2743.57i	− 0.632402i
$267$	− 1953.75i	− 0.447819i
$268$	− 3931.76i	− 0.896157i
$269$	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
$271$	2805.92i	0.628958i	0.949264	+	0.314479i	$0.101830\pi$
$271$	2805.92i	0.628958i	−0.949264	+	0.314479i	$0.898170\pi$
$272$	117.000	0.0260815
$273$	0	0
$274$	−1467.00	−0.323448
$275$	1690.48i	0.370690i
$276$	−780.000	−0.170110
$277$	377.000	0.0817752	0.0408876	−	0.999164i	$-0.486981\pi$
$277$	377.000	0.0817752	0.0408876	+	0.999164i	$0.486981\pi$
$278$	408.764i	0.0881872i
$279$	3585.35i	0.769351i
$280$	− 540.400i	− 0.115340i
$281$	36.3731i	0.00772183i	0.999993	+	0.00386092i	$0.00122897\pi$
$281$	36.3731i	0.00772183i	−0.999993	+	0.00386092i	$0.998771\pi$
$282$	1044.00	0.220458
$283$	−7124.00	−1.49639	−0.748194	−	0.663480i	$-0.769079\pi$
$283$	−7124.00	−1.49639	−0.748194	+	0.663480i	$0.769079\pi$
$284$	− 5282.75i	− 1.10378i
$285$	−396.000	−0.0823053
$286$	0	0
$287$	−3768.00	−0.774976
$288$	− 4182.90i	− 0.855833i
$289$	8776.00	1.78628
$290$	423.000	0.0856532
$291$	− 401.836i	− 0.0809486i
$292$	− 2294.97i	− 0.459941i
$293$	8322.50i	1.65941i	0.558205	+	0.829703i	$0.311490\pi$
$293$	8322.50i	1.65941i	−0.558205	+	0.829703i	$0.688510\pi$
$294$	523.079i	0.103764i
$295$	−492.000	−0.0971029
$296$	3237.00	0.635631
$297$	− 1385.64i	− 0.270717i
$298$	−81.0000	−0.0157457
$299$	0	0
$300$	1220.00	0.234789
$301$	1441.07i	0.275952i
$302$	−3066.00	−0.584200
$303$	858.000	0.162676
$304$	114.315i	0.0215672i
$305$	251.147i	0.0471497i
$306$	− 4660.95i	− 0.870747i
$307$	2220.49i	0.412801i	0.978468	+	0.206401i	$0.0661750\pi$
$307$	2220.49i	0.412801i	−0.978468	+	0.206401i	$0.933825\pi$
$308$	−960.000	−0.177601
$309$	364.000	0.0670137
$310$	467.654i	0.0856805i
$311$	4914.00	0.895972	0.447986	−	0.894041i	$-0.352141\pi$
$311$	4914.00	0.895972	0.447986	+	0.894041i	$0.352141\pi$
$312$	0	0
$313$	−518.000	−0.0935434	−0.0467717	−	0.998906i	$-0.514893\pi$
$313$	−518.000	−0.0935434	−0.0467717	+	0.998906i	$0.514893\pi$
$314$	2097.51i	0.376973i
$315$	552.000	0.0987355
$316$	6380.00	1.13577
$317$	− 3916.17i	− 0.693861i	−0.937891	−	0.346930i	$-0.887224\pi$
$317$	− 3916.17i	− 0.693861i	0.937891	−	0.346930i	$-0.112776\pi$
$318$	322.161i	0.0568111i
$319$	− 1953.75i	− 0.342913i
$320$	− 531.740i	− 0.0928911i
$321$	−3012.00	−0.523718
$322$	1872.00	0.323983
$323$	13374.9i	2.30402i
$324$	2105.00	0.360940
$325$	0	0
$326$	−1740.00	−0.295613
$327$	− 3103.84i	− 0.524901i
$328$	−6123.00	−1.03075
$329$	4176.00	0.699788
$330$	− 83.1384i	− 0.0138685i
$331$	7454.75i	1.23792i	0.785424	+	0.618958i	$0.212445\pi$
$331$	7454.75i	1.23792i	−0.785424	+	0.618958i	$0.787555\pi$
$332$	3949.08i	0.652812i
$333$	3306.48i	0.544127i
$334$	−1584.00	−0.259499
$335$	1362.00	0.222131
$336$	27.7128i	0.00449958i
$337$	−3575.00	−0.577871	−0.288936	−	0.957349i	$-0.593301\pi$
$337$	−3575.00	−0.577871	−0.288936	+	0.957349i	$0.593301\pi$
$338$	0	0
$339$	−1374.00	−0.220134
$340$	1013.25i	0.161621i
$341$	2160.00	0.343022
$342$	4554.00	0.720035
$343$	6845.06i	1.07755i
$344$	2341.73i	0.367028i
$345$	− 270.200i	− 0.0421654i
$346$	4458.30i	0.692716i
$347$	−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
$349$	− 6651.08i	− 1.02013i	−0.860137	−	0.510063i	$-0.829622\pi$
$349$	− 6651.08i	− 1.02013i	0.860137	−	0.510063i	$-0.170378\pi$
$350$	−2928.00	−0.447166
$351$	0	0
$352$	−2520.00	−0.381581
$353$	5630.90i	0.849015i	0.905424	+	0.424508i	$0.139553\pi$
$353$	5630.90i	0.849015i	−0.905424	+	0.424508i	$0.860447\pi$
$354$	−984.000	−0.147737
$355$	1830.00	0.273595
$356$	− 4884.38i	− 0.727168i
$357$	3242.40i	0.480689i
$358$	6484.80i	0.957353i
$359$	7129.12i	1.04808i	0.851694	+	0.524040i	$0.175576\pi$
$359$	7129.12i	1.04808i	−0.851694	+	0.524040i	$0.824424\pi$
$360$	897.000	0.131322
$361$	−6209.00	−0.905234
$362$	− 1103.32i	− 0.160191i
$363$	2278.00	0.329377
$364$	0	0
$365$	795.000	0.114006
$366$	502.295i	0.0717360i
$367$	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
$369$	− 6254.44i	− 0.882366i
$370$	431.281i	0.0605979i
$371$	1288.65i	0.180332i
$372$	− 1558.85i	− 0.217264i
$373$	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
$375$	855.633i	0.117826i
$376$	6786.00	0.930748
$377$	0	0
$378$	2400.00	0.326568
$379$	5518.31i	0.747907i	0.927447	+	0.373953i	$0.121998\pi$
$379$	5518.31i	0.747907i	−0.927447	+	0.373953i	$0.878002\pi$
$380$	−990.000	−0.133647
$381$	572.000	0.0769146
$382$	− 4499.87i	− 0.602705i
$383$	7364.68i	0.982552i	0.871004	+	0.491276i	$0.163469\pi$
$383$	7364.68i	0.982552i	−0.871004	+	0.491276i	$0.836531\pi$
$384$	1846.37i	0.245370i
$385$	− 332.554i	− 0.0440221i
$386$	−1935.00	−0.255153
$387$	−2392.00	−0.314192
$388$	− 1004.59i	− 0.131444i
$389$	−1209.00	−0.157580	−0.0787901	−	0.996891i	$-0.525106\pi$
$389$	−1209.00	−0.157580	−0.0787901	+	0.996891i	$0.525106\pi$
$390$	0	0
$391$	−9126.00	−1.18036
$392$	3400.02i	0.438078i
$393$	−3948.00	−0.506744
$394$	−3552.00	−0.454181
$395$	2210.10i	0.281524i
$396$	− 1593.49i	− 0.202211i
$397$	− 11694.8i	− 1.47845i	−0.673457	−	0.739226i	$-0.735191\pi$
$397$	− 11694.8i	− 1.47845i	0.673457	−	0.739226i	$-0.264809\pi$
$398$	− 4368.23i	− 0.550150i
$399$	−3168.00	−0.397490
$400$	122.000	0.0152500
$401$	− 2980.86i	− 0.371215i	−0.982624	−	0.185607i	$-0.940575\pi$
$401$	− 2980.86i	− 0.371215i	0.982624	−	0.185607i	$-0.0594252\pi$
$402$	2724.00	0.337962
$403$	0	0
$404$	2145.00	0.264153
$405$	729.193i	0.0894664i
$406$	3384.00	0.413658
$407$	1992.00	0.242604
$408$	5268.90i	0.639337i
$409$	− 43.3013i	− 0.00523499i	−0.999997	−	0.00261749i	$-0.999167\pi$
$409$	− 43.3013i	− 0.00523499i	0.999997	−	0.00261749i	$-0.000833175\pi$
$410$	− 815.796i	− 0.0982666i
$411$	1693.95i	0.203300i
$412$	910.000	0.108817
$413$	−3936.00	−0.468954
$414$	3107.30i	0.368878i
$415$	−1368.00	−0.161813
$416$	0	0
$417$	472.000	0.0554291
$418$	− 2743.57i	− 0.321034i
$419$	−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
$421$	− 7068.50i	− 0.818284i	−0.912471	−	0.409142i	$-0.865828\pi$
$421$	− 7068.50i	− 0.818284i	0.912471	−	0.409142i	$-0.134172\pi$
$422$	1804.80i	0.208190i
$423$	6931.67i	0.796759i
$424$	2094.05i	0.239849i
$425$	14274.0	1.62915
$426$	3660.00	0.416262
$427$	2009.18i	0.227707i
$428$	−7530.00	−0.850412
$429$	0	0
$430$	−312.000	−0.0349906
$431$	− 9928.12i	− 1.10956i	−0.831997	−	0.554780i	$-0.812802\pi$
$431$	− 9928.12i	− 1.10956i	0.831997	−	0.554780i	$-0.187198\pi$
$432$	−100.000	−0.0111372
$433$	−6617.00	−0.734394	−0.367197	−	0.930143i	$-0.619683\pi$
$433$	−6617.00	−0.734394	−0.367197	+	0.930143i	$0.619683\pi$
$434$	3741.23i	0.413790i
$435$	− 488.438i	− 0.0538364i
$436$	− 7759.59i	− 0.852332i
$437$	− 8916.60i	− 0.976061i
$438$	1590.00	0.173455
$439$	13988.0	1.52075	0.760377	−	0.649482i	$-0.225014\pi$
$439$	13988.0	1.52075	0.760377	+	0.649482i	$0.225014\pi$
$440$	− 540.400i	− 0.0585513i
$441$	−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	$0.465727\pi$
$444$	− 1437.60i	− 0.153661i
$445$	1692.00	0.180244
$446$	4170.00	0.442725
$447$	93.5307i	0.00989676i
$448$	− 4253.92i	− 0.448613i
$449$	− 9082.87i	− 0.954671i	−0.878721	−	0.477336i	$-0.841603\pi$
$449$	− 9082.87i	− 0.954671i	0.878721	−	0.477336i	$-0.158397\pi$
$450$	− 4860.13i	− 0.509131i
$451$	−3768.00	−0.393411
$452$	−3435.00	−0.357453
$453$	3540.31i	0.367193i
$454$	4170.00	0.431074
$455$	0	0
$456$	−5148.00	−0.528678
$457$	2523.60i	0.258313i	0.991624	+	0.129156i	$0.0412269\pi$
$457$	2523.60i	0.258313i	−0.991624	+	0.129156i	$0.958773\pi$
$458$	4344.00	0.443192
$459$	−11700.0	−1.18978
$460$	− 675.500i	− 0.0684681i
$461$	− 19587.8i	− 1.97894i	−0.144725	−	0.989472i	$-0.546230\pi$
$461$	− 19587.8i	− 1.97894i	0.144725	−	0.989472i	$-0.453770\pi$
$462$	− 665.108i	− 0.0669775i
$463$	− 8632.54i	− 0.866497i	−0.901274	−	0.433249i	$-0.857367\pi$
$463$	− 8632.54i	− 0.866497i	0.901274	−	0.433249i	$-0.142633\pi$
$464$	−141.000	−0.0141072
$465$	540.000	0.0538535
$466$	− 10132.5i	− 1.00725i
$467$	5460.00	0.541025	0.270512	−	0.962716i	$-0.412807\pi$
$467$	5460.00	0.541025	0.270512	+	0.962716i	$0.412807\pi$
$468$	0	0
$469$	10896.0	1.07277
$470$	904.131i	0.0887328i
$471$	2422.00	0.236942
$472$	−6396.00	−0.623728
$473$	1441.07i	0.140085i
$474$	4420.19i	0.428325i
$475$	13946.5i	1.34717i
$476$	8106.00i	0.780542i
$477$	−2139.00	−0.205321
$478$	−9324.00	−0.892196
$479$	− 2553.04i	− 0.243531i	−0.992559	−	0.121766i	$-0.961144\pi$
$479$	− 2553.04i	− 0.243531i	0.992559	−	0.121766i	$-0.0388556\pi$
$480$	−630.000	−0.0599072
$481$	0	0
$482$	8517.00	0.804852
$483$	− 2161.60i	− 0.203636i
$484$	5695.00	0.534842
$485$	348.000	0.0325812
$486$	6134.92i	0.572605i
$487$	− 10828.8i	− 1.00760i	−0.863822	−	0.503798i	$-0.831936\pi$
$487$	− 10828.8i	− 1.00760i	0.863822	−	0.503798i	$-0.168064\pi$
$488$	3264.92i	0.302860i
$489$	2009.18i	0.185804i
$490$	−453.000	−0.0417642
$491$	11388.0	1.04671	0.523354	−	0.852116i	$-0.324681\pi$
$491$	11388.0	1.04671	0.523354	+	0.852116i	$0.324681\pi$
$492$	2719.32i	0.249180i
$493$	−16497.0	−1.50707
$494$	0	0
$495$	552.000	0.0501223
$496$	− 155.885i	− 0.0141117i
$497$	14640.0	1.32132
$498$	−2736.00	−0.246191
$499$	17677.3i	1.58586i	0.609311	+	0.792931i	$0.291446\pi$
$499$	17677.3i	1.58586i	−0.609311	+	0.792931i	$0.708554\pi$
$500$	2139.08i	0.191325i
$501$	1829.05i	0.163105i
$502$	− 6890.10i	− 0.612590i
$503$	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
$505$	743.050i	0.0654758i
$506$	1872.00	0.164467
$507$	0	0
$508$	1430.00	0.124894
$509$	− 17065.9i	− 1.48612i	−0.669228	−	0.743058i	$-0.733375\pi$
$509$	− 17065.9i	− 1.48612i	0.669228	−	0.743058i	$-0.266625\pi$
$510$	−702.000	−0.0609511
$511$	6360.00	0.550587
$512$	361.999i	0.0312465i
$513$	− 11431.5i	− 0.983849i
$514$	3580.15i	0.307225i
$515$	315.233i	0.0269725i
$516$	1040.00	0.0887276
$517$	4176.00	0.355242
$518$	3450.25i	0.292655i
$519$	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	$0.471576\pi$
$522$	5617.04i	0.470979i
$523$	−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
$525$	3380.96i	0.281062i
$526$	− 3554.17i	− 0.294618i
$527$	− 18238.5i	− 1.50755i
$528$	27.7128i	0.00228418i
$529$	−6083.00	−0.499959
$530$	−279.000	−0.0228660
$531$	− 6533.30i	− 0.533938i
$532$	−7920.00	−0.645443
$533$	0	0
$534$	3384.00	0.274232
$535$	− 2608.47i	− 0.210792i
$536$	17706.0	1.42683
$537$	7488.00	0.601734
$538$	5767.73i	0.462202i
$539$	2092.32i	0.167203i
$540$	− 866.025i	− 0.0690144i
$541$	4764.87i	0.378665i	0.981913	+	0.189333i	$0.0606324\pi$
$541$	4764.87i	0.378665i	−0.981913	+	0.189333i	$0.939368\pi$
$542$	−4860.00	−0.385157
$543$	−1274.00	−0.100686
$544$	21278.2i	1.67702i
$545$	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	$0.417546\pi$
$548$	4234.86i	0.330118i
$549$	−3335.00	−0.259261
$550$	−2928.00	−0.227001
$551$	− 16118.5i	− 1.24622i
$552$	− 3512.60i	− 0.270844i
$553$	17680.8i	1.35961i
$554$	652.983i	0.0500769i
$555$	498.000	0.0380881
$556$	1180.00	0.0900057
$557$	− 18112.1i	− 1.37780i	−0.724858	−	0.688898i	$-0.758095\pi$
$557$	− 18112.1i	− 1.37780i	0.724858	−	0.688898i	$-0.241905\pi$
$558$	−6210.00	−0.471130
$559$	0	0
$560$	−24.0000	−0.00181104
$561$	3242.40i	0.244018i
$562$	−63.0000	−0.00472864
$563$	−12168.0	−0.910870	−0.455435	−	0.890269i	$-0.650516\pi$
$563$	−12168.0	−0.910870	−0.455435	+	0.890269i	$0.650516\pi$
$564$	− 3013.77i	− 0.225005i
$565$	− 1189.92i	− 0.0886022i
$566$	− 12339.1i	− 0.916347i
$567$	5833.55i	0.432074i
$568$	23790.0	1.75741
$569$	−7722.00	−0.568933	−0.284467	−	0.958686i	$-0.591817\pi$
$569$	−7722.00	−0.568933	−0.284467	+	0.958686i	$0.591817\pi$
$570$	− 685.892i	− 0.0504015i
$571$	11440.0	0.838440	0.419220	−	0.907885i	$-0.362304\pi$
$571$	11440.0	0.838440	0.419220	+	0.907885i	$0.362304\pi$
$572$	0	0
$573$	−5196.00	−0.378824
$574$	− 6526.37i	− 0.474574i
$575$	−9516.00	−0.690165
$576$	7061.00	0.510778
$577$	− 15444.7i	− 1.11433i	−0.830400	−	0.557167i	$-0.811888\pi$
$577$	− 15444.7i	− 1.11433i	0.830400	−	0.557167i	$-0.188112\pi$
$578$	15200.5i	1.09387i
$579$	2234.35i	0.160373i
$580$	− 1221.10i	− 0.0874194i
$581$	−10944.0	−0.781469
$582$	696.000	0.0495707
$583$	1288.65i	0.0915442i
$584$	10335.0	0.732304
$585$	0	0
$586$	−14415.0	−1.01617
$587$	− 14071.2i	− 0.989403i	−0.869063	−	0.494702i	$-0.835277\pi$
$587$	− 14071.2i	− 0.989403i	0.869063	−	0.494702i	$-0.164723\pi$
$588$	1510.00	0.105904
$589$	17820.0	1.24662
$590$	− 852.169i	− 0.0594631i
$591$	4101.50i	0.285470i
$592$	− 143.760i	− 0.00998059i
$593$	− 26938.6i	− 1.86549i	−0.360538	−	0.932745i	$-0.617407\pi$
$593$	− 26938.6i	− 1.86549i	0.360538	−	0.932745i	$-0.382593\pi$
$594$	2400.00	0.165780
$595$	−2808.00	−0.193474
$596$	233.827i	0.0160703i
$597$	−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	$0.617208\pi$
$600$	5494.07i	0.373824i
$601$	−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
$603$	18086.1i	1.22143i
$604$	8850.78i	0.596247i
$605$	1972.81i	0.132572i
$606$	1486.10i	0.0996183i
$607$	−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
$609$	− 3907.51i	− 0.260000i
$610$	−435.000	−0.0288732
$611$	0	0
$612$	−13455.0	−0.888703
$613$	25220.4i	1.66173i	0.556472	+	0.830866i	$0.312155\pi$
$613$	25220.4i	1.66173i	−0.556472	+	0.830866i	$0.687845\pi$
$614$	−3846.00	−0.252788
$615$	−942.000	−0.0617644
$616$	− 4323.20i	− 0.282771i
$617$	17384.6i	1.13432i	0.823607	+	0.567162i	$0.191959\pi$
$617$	17384.6i	1.13432i	−0.823607	+	0.567162i	$0.808041\pi$
$618$	630.466i	0.0410373i
$619$	8209.92i	0.533093i	0.963822	+	0.266547i	$0.0858826\pi$
$619$	8209.92i	0.533093i	−0.963822	+	0.266547i	$0.914117\pi$
$620$	1350.00	0.0874473
$621$	7800.00	0.504031
$622$	8511.30i	0.548669i
$623$	13536.0	0.870479
$624$	0	0
$625$	14509.0	0.928576
$626$	− 897.202i	− 0.0572834i
$627$	−3168.00	−0.201783
$628$	6055.00	0.384747
$629$	− 16819.9i	− 1.06622i
$630$	956.092i	0.0604629i
$631$	− 12865.7i	− 0.811687i	−0.913943	−	0.405843i	$-0.866978\pi$
$631$	− 12865.7i	− 0.811687i	0.913943	−	0.405843i	$-0.133022\pi$
$632$	28731.3i	1.80834i
$633$	2084.00	0.130856
$634$	6783.00	0.424901
$635$	495.367i	0.0309575i
$636$	930.000	0.0579825
$637$	0	0
$638$	3384.00	0.209990
$639$	24300.7i	1.50441i
$640$	−1599.00	−0.0987594
$641$	6201.00	0.382098	0.191049	−	0.981581i	$-0.438811\pi$
$641$	6201.00	0.382098	0.191049	+	0.981581i	$0.438811\pi$
$642$	− 5216.94i	− 0.320710i
$643$	16821.7i	1.03170i	0.856679	+	0.515849i	$0.172524\pi$
$643$	16821.7i	1.03170i	−0.856679	+	0.515849i	$0.827476\pi$
$644$	− 5404.00i	− 0.330664i
$645$	360.267i	0.0219930i
$646$	−23166.0	−1.41092
$647$	−13494.0	−0.819944	−0.409972	−	0.912098i	$-0.634462\pi$
$647$	−13494.0	−0.819944	−0.409972	+	0.912098i	$0.634462\pi$
$648$	9479.51i	0.574677i
$649$	−3936.00	−0.238061
$650$	0	0
$651$	4320.00	0.260083
$652$	5022.95i	0.301708i
$653$	−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
$655$	− 3419.07i	− 0.203960i
$656$	271.932i	0.0161847i
$657$	10556.8i	0.626883i
$658$	7233.04i	0.428531i
$659$	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
$661$	− 11852.4i	− 0.697437i	−0.937228	−	0.348718i	$-0.886617\pi$
$661$	− 11852.4i	− 0.697437i	0.937228	−	0.348718i	$-0.113383\pi$
$662$	−12912.0	−0.758065
$663$	0	0
$664$	−17784.0	−1.03939
$665$	− 2743.57i	− 0.159986i
$666$	−5727.00	−0.333208
$667$	10998.0	0.638447
$668$	4572.61i	0.264850i
$669$	− 4815.10i	− 0.278270i
$670$	2359.05i	0.136027i
$671$	2009.18i	0.115594i
$672$	−5040.00	−0.289319
$673$	8021.00	0.459416	0.229708	−	0.973260i	$-0.426223\pi$
$673$	8021.00	0.459416	0.229708	+	0.973260i	$0.426223\pi$
$674$	− 6192.08i	− 0.353873i
$675$	−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	$0.710426\pi$
$678$	− 2379.84i	− 0.134804i
$679$	2784.00	0.157349
$680$	−4563.00	−0.257328
$681$	− 4815.10i	− 0.270947i
$682$	3741.23i	0.210057i
$683$	26538.5i	1.48677i	0.668861	+	0.743387i	$0.266782\pi$
$683$	26538.5i	1.48677i	−0.668861	+	0.743387i	$0.733218\pi$
$684$	− 13146.3i	− 0.734883i
$685$	−1467.00	−0.0818266
$686$	−11856.0	−0.659860
$687$	− 5016.02i	− 0.278563i
$688$	104.000	0.00576303
$689$	0	0
$690$	468.000	0.0258210
$691$	− 831.384i	− 0.0457704i	−0.999738	−	0.0228852i	$-0.992715\pi$
$691$	− 831.384i	− 0.0457704i	0.999738	−	0.0228852i	$-0.00728522\pi$
$692$	12870.0	0.707000
$693$	4416.00	0.242063
$694$	− 12065.5i	− 0.659941i
$695$	408.764i	0.0223098i
$696$	− 6349.70i	− 0.345811i
$697$	31816.0i	1.72901i
$698$	11520.0	0.624697
$699$	−11700.0	−0.633097
$700$	8452.41i	0.456387i
$701$	30186.0	1.62640	0.813202	−	0.581981i	$-0.197722\pi$
$701$	30186.0	1.62640	0.813202	+	0.581981i	$0.197722\pi$
$702$	0	0
$703$	16434.0	0.881679
$704$	− 4253.92i	− 0.227735i
$705$	1044.00	0.0557721
$706$	−9753.00	−0.519914
$707$	5944.40i	0.316212i
$708$	2840.56i	0.150784i
$709$	11880.1i	0.629292i	0.949209	+	0.314646i	$0.101886\pi$
$709$	11880.1i	0.629292i	−0.949209	+	0.314646i	$0.898114\pi$
$710$	3169.65i	0.167542i
$711$	−29348.0	−1.54801
$712$	21996.0	1.15777
$713$	12159.0i	0.638651i
$714$	−5616.00	−0.294361
$715$	0	0
$716$	18720.0	0.977094
$717$	10766.4i	0.560780i
$718$	−12348.0	−0.641815
$719$	18408.0	0.954802	0.477401	−	0.878686i	$-0.341579\pi$
$719$	18408.0	0.954802	0.477401	+	0.878686i	$0.341579\pi$
$720$	− 39.8372i	− 0.00206201i
$721$	2521.87i	0.130262i
$722$	− 10754.3i	− 0.554340i
$723$	− 9834.58i	− 0.505881i
$724$	−3185.00	−0.163494
$725$	−17202.0	−0.881195
$726$	3945.61i	0.201702i
$727$	21112.0	1.07703	0.538515	−	0.842616i	$-0.318986\pi$
$727$	21112.0	1.07703	0.538515	+	0.842616i	$0.318986\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	1376.98i	0.0698142i
$731$	12168.0	0.615663
$732$	1450.00	0.0732152
$733$	23959.5i	1.20732i	0.797243	+	0.603658i	$0.206291\pi$
$733$	23959.5i	1.20732i	−0.797243	+	0.603658i	$0.793709\pi$
$734$	3.46410i	0 0.000174199i
$735$	523.079i	0.0262504i
$736$	− 14185.5i	− 0.710441i
$737$	10896.0	0.544585
$738$	10833.0	0.540336
$739$	3166.19i	0.157605i	0.996890	+	0.0788025i	$0.0251097\pi$
$739$	3166.19i	0.157605i	−0.996890	+	0.0788025i	$0.974890\pi$
$740$	1245.00	0.0618474
$741$	0	0
$742$	−2232.00	−0.110430
$743$	− 30103.0i	− 1.48637i	−0.669086	−	0.743185i	$-0.733314\pi$
$743$	− 30103.0i	− 1.48637i	0.669086	−	0.743185i	$-0.266686\pi$
$744$	7020.00	0.345922
$745$	−81.0000	−0.00398337
$746$	6060.45i	0.297438i
$747$	− 18165.7i	− 0.889759i
$748$	8106.00i	0.396236i
$749$	− 20867.7i	− 1.01801i
$750$	−1482.00	−0.0721533
$751$	28496.0	1.38460	0.692299	−	0.721610i	$-0.256598\pi$
$751$	28496.0	1.38460	0.692299	+	0.721610i	$0.256598\pi$
$752$	− 301.377i	− 0.0146145i
$753$	−7956.00	−0.385037
$754$	0	0
$755$	−3066.00	−0.147792
$756$	− 6928.20i	− 0.333302i
$757$	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
$759$	− 2161.60i	− 0.103374i
$760$	− 4458.30i	− 0.212789i
$761$	− 41326.7i	− 1.96858i	−0.176547	−	0.984292i	$-0.556493\pi$
$761$	− 41326.7i	− 1.96858i	0.176547	−	0.984292i	$-0.443507\pi$
$762$	990.733i	0.0471004i
$763$	21504.0	1.02031
$764$	−12990.0	−0.615133
$765$	− 4660.95i	− 0.220284i
$766$	−12756.0	−0.601688
$767$	0	0
$768$	−8110.00	−0.381047
$769$	− 14071.2i	− 0.659844i	−0.944008	−	0.329922i	$-0.892978\pi$
$769$	− 14071.2i	− 0.659844i	0.944008	−	0.329922i	$-0.107022\pi$
$770$	576.000	0.0269579
$771$	4134.00	0.193103
$772$	5585.86i	0.260414i
$773$	− 200.918i	− 0.00934866i	−0.999989	−	0.00467433i	$-0.998512\pi$
$773$	− 200.918i	− 0.00934866i	0.999989	−	0.00467433i	$-0.00148789\pi$
$774$	− 4143.07i	− 0.192402i
$775$	− 19017.9i	− 0.881476i
$776$	4524.00	0.209281
$777$	3984.00	0.183945
$778$	− 2094.05i	− 0.0964978i
$779$	−31086.0	−1.42975
$780$	0	0
$781$	14640.0	0.670756
$782$	− 15806.7i	− 0.722821i
$783$	14100.0	0.643541
$784$	151.000	0.00687864
$785$	2097.51i	0.0953675i
$786$	− 6838.14i	− 0.310316i
$787$	6903.95i	0.312706i	0.987701	+	0.156353i	$0.0499737\pi$
$787$	6903.95i	0.312706i	−0.987701	+	0.156353i	$0.950026\pi$
$788$	10253.7i	0.463546i
$789$	−4104.00	−0.185179
$790$	−3828.00	−0.172398
$791$	− 9519.35i	− 0.427900i
$792$	7176.00	0.321955
$793$	0	0
$794$	20256.0	0.905363
$795$	322.161i	0.0143722i
$796$	−12610.0	−0.561494
$797$	−31278.0	−1.39012	−0.695059	−	0.718953i	$-0.744622\pi$
$797$	−31278.0	−1.39012	−0.695059	+	0.718953i	$0.744622\pi$
$798$	− 5487.14i	− 0.243412i
$799$	− 35261.1i	− 1.56126i
$800$	22187.6i	0.980561i
$801$	22468.2i	0.991103i
$802$	5163.00	0.227322
$803$	6360.00	0.279501
$804$	− 7863.51i	− 0.344931i
$805$	1872.00	0.0819619
$806$	0	0
$807$	6660.00	0.290512
$808$	9659.65i	0.420576i
$809$	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
$811$	− 14026.1i	− 0.607305i	−0.952783	−	0.303653i	$-0.901794\pi$
$811$	− 14026.1i	− 0.607305i	0.952783	−	0.303653i	$-0.0982062\pi$
$812$	− 9768.77i	− 0.422188i
$813$	5611.84i	0.242086i
$814$	3450.25i	0.148564i
$815$	−1740.00	−0.0747847
$816$	234.000	0.0100388
$817$	11888.8i	0.509102i
$818$	75.0000	0.00320576
$819$	0	0
$820$	−2355.00	−0.100293
$821$	− 8036.72i	− 0.341636i	−0.985303	−	0.170818i	$-0.945359\pi$
$821$	− 8036.72i	− 0.341636i	0.985303	−	0.170818i	$-0.0546410\pi$
$822$	−2934.00	−0.124495
$823$	−40300.0	−1.70689	−0.853445	−	0.521184i	$-0.825491\pi$
$823$	−40300.0	−1.70689	−0.853445	+	0.521184i	$0.825491\pi$
$824$	4098.03i	0.173255i
$825$	3380.96i	0.142679i
$826$	− 6817.35i	− 0.287174i
$827$	− 39525.4i	− 1.66195i	−0.556310	−	0.830975i	$-0.687783\pi$
$827$	− 39525.4i	− 1.66195i	0.556310	−	0.830975i	$-0.312217\pi$
$828$	8970.00	0.376484
$829$	−12311.0	−0.515776	−0.257888	−	0.966175i	$-0.583027\pi$
$829$	−12311.0	−0.515776	−0.257888	+	0.966175i	$0.583027\pi$
$830$	− 2369.45i	− 0.0990899i
$831$	754.000	0.0314753
$832$	0	0
$833$	17667.0	0.734844
$834$	817.528i	0.0339433i
$835$	−1584.00	−0.0656486
$836$	−7920.00	−0.327654
$837$	15588.5i	0.643747i
$838$	− 16388.7i	− 0.675581i
$839$	− 21467.0i	− 0.883343i	−0.897177	−	0.441671i	$-0.854386\pi$
$839$	− 21467.0i	− 0.883343i	0.897177	−	0.441671i	$-0.145614\pi$
$840$	− 1080.80i	− 0.0443942i
$841$	−4508.00	−0.184837
$842$	12243.0	0.501095
$843$	72.7461i	0.00297214i
$844$	5210.00	0.212483
$845$	0	0
$846$	−12006.0	−0.487913
$847$	15782.4i	0.640249i
$848$	93.0000	0.00376608
$849$	−14248.0	−0.575960
$850$	24723.3i	0.997649i
$851$	11213.3i	0.451688i
$852$	− 10565.5i	− 0.424846i
$853$	− 774.227i	− 0.0310774i	−0.999879	−	0.0155387i	$-0.995054\pi$
$853$	− 774.227i	− 0.0310774i	0.999879	−	0.0155387i	$-0.00494632\pi$
$854$	−3480.00	−0.139442
$855$	4554.00	0.182156
$856$	− 33910.1i	− 1.35400i
$857$	13923.0	0.554960	0.277480	−	0.960731i	$-0.410501\pi$
$857$	13923.0	0.554960	0.277480	+	0.960731i	$0.410501\pi$
$858$	0	0
$859$	−22358.0	−0.888062	−0.444031	−	0.896011i	$-0.646452\pi$
$859$	−22358.0	−0.888062	−0.444031	+	0.896011i	$0.646452\pi$
$860$	900.666i	0.0357122i
$861$	−7536.00	−0.298288
$862$	17196.0	0.679464
$863$	2230.88i	0.0879955i	0.999032	+	0.0439977i	$0.0140094\pi$
$863$	2230.88i	0.0879955i	−0.999032	+	0.0439977i	$0.985991\pi$
$864$	− 18186.5i	− 0.716109i
$865$	4458.30i	0.175245i
$866$	− 11461.0i	− 0.449723i
$867$	17552.0	0.687540
$868$	10800.0	0.422322
$869$	17680.8i	0.690195i
$870$	846.000	0.0329679
$871$	0	0
$872$	34944.0	1.35706
$873$	4621.11i	0.179153i
$874$	15444.0	0.597713
$875$	−5928.00	−0.229032
$876$	− 4589.93i	− 0.177031i
$877$	16754.1i	0.645093i	0.946554	+	0.322547i	$0.104539\pi$
$877$	16754.1i	0.645093i	−0.946554	+	0.322547i	$0.895461\pi$
$878$	24227.9i	0.931268i
$879$	16645.0i	0.638706i
$880$	−24.0000	−0.000919363 0
$881$	17355.0	0.663683	0.331842	−	0.943335i	$-0.392330\pi$
$881$	17355.0	0.663683	0.331842	+	0.943335i	$0.392330\pi$
$882$	− 6015.41i	− 0.229648i
$883$	−46982.0	−1.79057	−0.895283	−	0.445497i	$-0.853027\pi$
$883$	−46982.0	−1.79057	−0.895283	+	0.445497i	$0.853027\pi$
$884$	0	0
$885$	−984.000	−0.0373749
$886$	3471.03i	0.131616i
$887$	−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
$889$	3962.93i	0.149508i
$890$	2930.63i	0.110376i
$891$	5833.55i	0.219339i
$892$	− 12037.8i	− 0.451854i
$893$	34452.0	1.29103
$894$	−162.000	−0.00606050
$895$	6484.80i	0.242193i
$896$	−12792.0	−0.476954
$897$	0	0
$898$	15732.0	0.584614
$899$	21979.7i	0.815423i
$900$	−14030.0	−0.519630
$901$	10881.0	0.402329
$902$	− 6526.37i	− 0.240914i
$903$	2882.13i	0.106214i
$904$	− 15468.9i	− 0.569126i
$905$	− 1103.32i	− 0.0405254i
$906$	−6132.00	−0.224859
$907$	−30836.0	−1.12888	−0.564439	−	0.825475i	$-0.690908\pi$
$907$	−30836.0	−1.12888	−0.564439	+	0.825475i	$0.690908\pi$
$908$	− 12037.8i	− 0.439964i
$909$	−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	$0.666556\pi$
$912$	228.631i	0.00830123i
$913$	−10944.0	−0.396707
$914$	−4371.00	−0.158184
$915$	502.295i	0.0181479i
$916$	− 12540.0i	− 0.452331i
$917$	− 27352.5i	− 0.985017i
$918$	− 20265.0i	− 0.728589i
$919$	−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
$921$	4440.98i	0.158887i
$922$	33927.0	1.21185
$923$	0	0
$924$	−1920.00	−0.0683586
$925$	− 17538.7i	− 0.623427i
$926$	14952.0	0.530619
$927$	−4186.00	−0.148313
$928$	− 25643.0i	− 0.907083i
$929$	− 6978.43i	− 0.246453i	−0.992379	−	0.123227i	$-0.960676\pi$
$929$	− 6978.43i	− 0.246453i	0.992379	−	0.123227i	$-0.0393242\pi$
$930$	935.307i	0.0329784i
$931$	17261.6i	0.607655i
$932$	−29250.0	−1.02802
$933$	9828.00	0.344860
$934$	9457.00i	0.331309i
$935$	−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	$0.733939\pi$
$938$	18872.4i	0.656937i
$939$	−1036.00	−0.0360049
$940$	2610.00	0.0905626
$941$	− 4884.38i	− 0.169210i	−0.996415	−	0.0846049i	$-0.973037\pi$
$941$	− 4884.38i	− 0.169210i	0.996415	−	0.0846049i	$-0.0269628\pi$
$942$	4195.03i	0.145097i
$943$	− 21210.7i	− 0.732466i
$944$	284.056i	0.00979369i
$945$	2400.00	0.0826159
$946$	−2496.00	−0.0857843
$947$	21765.0i	0.746849i	0.927661	+	0.373424i	$0.121816\pi$
$947$	21765.0i	0.746849i	−0.927661	+	0.373424i	$0.878184\pi$
$948$	12760.0	0.437158
$949$	0	0
$950$	−24156.0	−0.824973
$951$	− 7832.33i	− 0.267067i
$952$	−36504.0	−1.24275
$953$	6474.00	0.220056	0.110028	−	0.993928i	$-0.464906\pi$
$953$	6474.00	0.220056	0.110028	+	0.993928i	$0.464906\pi$
$954$	− 3704.86i	− 0.125733i
$955$	− 4499.87i	− 0.152474i
$956$	26916.1i	0.910594i
$957$	− 3907.51i	− 0.131987i
$958$	4422.00	0.149132
$959$	−11736.0	−0.395177
$960$	− 1063.48i	− 0.0357538i
$961$	5491.00	0.184317
$962$	0	0
$963$	34638.0	1.15908
$964$	− 24586.5i	− 0.821449i
$965$	−1935.00	−0.0645491
$966$	3744.00	0.124701
$967$	− 7541.35i	− 0.250789i	−0.992107	−	0.125395i	$-0.959980\pi$
$967$	− 7541.35i	− 0.250789i	0.992107	−	0.125395i	$-0.0400197\pi$
$968$	25646.5i	0.851559i
$969$	26749.8i	0.886819i
$970$	602.754i	0.0199518i
$971$	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
$973$	3270.11i	0.107744i
$974$	18756.0	0.617024
$975$	0	0
$976$	145.000	0.00475547
$977$	− 25216.9i	− 0.825753i	−0.910787	−	0.412877i	$-0.864524\pi$
$977$	− 25216.9i	− 0.825753i	0.910787	−	0.412877i	$-0.135476\pi$
$978$	−3480.00	−0.113781
$979$	13536.0	0.441892
$980$	1307.70i	0.0426254i
$981$	35694.1i	1.16170i
$982$	19724.6i	0.640975i
$983$	56440.6i	1.83131i	0.401967	+	0.915654i	$0.368327\pi$
$983$	56440.6i	1.83131i	−0.401967	+	0.915654i	$0.631673\pi$
$984$	−12246.0	−0.396736
$985$	−3552.00	−0.114900
$986$	− 28573.6i	− 0.922891i
$987$	8352.00	0.269349
$988$	0	0
$989$	−8112.00	−0.260816
$990$	956.092i	0.0306935i
$991$	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
$993$	14909.5i	0.476474i
$994$	25357.2i	0.809137i
$995$	− 4368.23i	− 0.139178i
$996$	7898.15i	0.251268i
$997$	−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
$999$	14376.0i	0.455292i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.e.b.4.1	✓	2	13.10	even	6
13.4.e.b.10.1	yes	2	13.9	even	3
117.4.q.a.10.1		2	39.35	odd	6
117.4.q.a.82.1		2	39.23	odd	6
169.4.a.i.1.1		2	13.8	odd	4
169.4.a.i.1.2		2	13.5	odd	4
169.4.b.d.168.1		2	13.12	even	2	inner
169.4.b.d.168.2		2	1.1	even	1	trivial
169.4.c.h.22.1		4	13.2	odd	12
169.4.c.h.22.2		4	13.11	odd	12
169.4.c.h.146.1		4	13.6	odd	12
169.4.c.h.146.2		4	13.7	odd	12
169.4.e.a.23.1		2	13.4	even	6
169.4.e.a.147.1		2	13.3	even	3
208.4.w.b.17.1		2	52.23	odd	6
208.4.w.b.49.1		2	52.35	odd	6
1521.4.a.o.1.1		2	39.5	even	4
1521.4.a.o.1.2		2	39.8	even	4

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.b (of order \(2\), degree \(1\), not minimal)

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

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