LMFDB - Embedded newform 169.4.b.g.168.13

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 108 x^{16} + 4636 x^{14} + 101999 x^{12} + 1237806 x^{10} + 8358937 x^{8} + 30682857 x^{6} + \cdots + 16777216 $ x^18 + 108x^16 + 4636x^14 + 101999x^12 + 1237806x^10 + 8358937x^8 + 30682857x^6 + 58141952x^4 + 52035584x^2 + 16777216
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.13
Root			$1.22799i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.g.168.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22799i q^{2} -9.74867 q^{3} +3.03607 q^{4} -8.20685i q^{5} -21.7199i q^{6} -8.35495i q^{7} +24.5882i q^{8} +68.0366 q^{9} +O(q^{10})$	\(q+2.22799i q^{2} -9.74867 q^{3} +3.03607 q^{4} -8.20685i q^{5} -21.7199i q^{6} -8.35495i q^{7} +24.5882i q^{8} +68.0366 q^{9} +18.2848 q^{10} +9.69898i q^{11} -29.5976 q^{12} +18.6147 q^{14} +80.0059i q^{15} -30.4937 q^{16} -44.6219 q^{17} +151.585i q^{18} +87.7418i q^{19} -24.9166i q^{20} +81.4497i q^{21} -21.6092 q^{22} -107.053 q^{23} -239.703i q^{24} +57.6475 q^{25} -400.052 q^{27} -25.3662i q^{28} -14.0430 q^{29} -178.252 q^{30} +171.090i q^{31} +128.766i q^{32} -94.5522i q^{33} -99.4170i q^{34} -68.5678 q^{35} +206.564 q^{36} +413.954i q^{37} -195.488 q^{38} +201.792 q^{40} +258.282i q^{41} -181.469 q^{42} -61.0718 q^{43} +29.4468i q^{44} -558.367i q^{45} -238.512i q^{46} +68.7115i q^{47} +297.273 q^{48} +273.195 q^{49} +128.438i q^{50} +435.004 q^{51} +328.701 q^{53} -891.312i q^{54} +79.5981 q^{55} +205.433 q^{56} -855.367i q^{57} -31.2876i q^{58} -147.144i q^{59} +242.904i q^{60} -97.8083 q^{61} -381.186 q^{62} -568.442i q^{63} -530.839 q^{64} +210.661 q^{66} +677.597i q^{67} -135.475 q^{68} +1043.62 q^{69} -152.768i q^{70} +786.767i q^{71} +1672.90i q^{72} -997.675i q^{73} -922.285 q^{74} -561.987 q^{75} +266.390i q^{76} +81.0345 q^{77} +383.897 q^{79} +250.258i q^{80} +2062.99 q^{81} -575.448 q^{82} +519.718i q^{83} +247.287i q^{84} +366.205i q^{85} -136.067i q^{86} +136.901 q^{87} -238.481 q^{88} -683.222i q^{89} +1244.03 q^{90} -325.019 q^{92} -1667.90i q^{93} -153.088 q^{94} +720.085 q^{95} -1255.30i q^{96} -347.294i q^{97} +608.675i q^{98} +659.886i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9}+O(q^{10}) $ 18 * q + 2 * q^3 - 74 * q^4 + 132 * q^9	$ 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9} + 294 q^{10} - 78 q^{12} - 294 q^{14} + 538 q^{16} + 110 q^{17} + 680 q^{22} + 408 q^{23} - 614 q^{25} - 1336 q^{27} + 560 q^{29} - 1042 q^{30} + 40 q^{35} + 1818 q^{36} + 1478 q^{38} + 26 q^{40} - 8 q^{42} + 1066 q^{43} - 264 q^{48} - 806 q^{49} - 940 q^{51} - 556 q^{53} - 500 q^{55} - 500 q^{56} - 272 q^{61} - 4070 q^{62} - 568 q^{64} + 6558 q^{66} - 3072 q^{68} + 4100 q^{69} - 3980 q^{74} - 4786 q^{75} + 1436 q^{77} + 824 q^{79} - 1670 q^{81} - 5514 q^{82} - 1572 q^{87} + 1272 q^{88} + 2560 q^{90} + 8020 q^{92} - 5062 q^{94} + 3228 q^{95}+O(q^{100}) $ 18 * q + 2 * q^3 - 74 * q^4 + 132 * q^9 + 294 * q^10 - 78 * q^12 - 294 * q^14 + 538 * q^16 + 110 * q^17 + 680 * q^22 + 408 * q^23 - 614 * q^25 - 1336 * q^27 + 560 * q^29 - 1042 * q^30 + 40 * q^35 + 1818 * q^36 + 1478 * q^38 + 26 * q^40 - 8 * q^42 + 1066 * q^43 - 264 * q^48 - 806 * q^49 - 940 * q^51 - 556 * q^53 - 500 * q^55 - 500 * q^56 - 272 * q^61 - 4070 * q^62 - 568 * q^64 + 6558 * q^66 - 3072 * q^68 + 4100 * q^69 - 3980 * q^74 - 4786 * q^75 + 1436 * q^77 + 824 * q^79 - 1670 * q^81 - 5514 * q^82 - 1572 * q^87 + 1272 * q^88 + 2560 * q^90 + 8020 * q^92 - 5062 * q^94 + 3228 * q^95

Display 100 coefficients 10 coefficients

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 $

$2$	2.22799i	0.787713i	0.919172	+	0.393856i	$0.128859\pi$
$2$	2.22799i	0.787713i	−0.919172	+	0.393856i	$0.871141\pi$
$3$	−9.74867	−1.87613	−0.938066	−	0.346455i	$-0.887385\pi$
$3$	−9.74867	−1.87613	−0.938066	+	0.346455i	$0.887385\pi$
$4$	3.03607	0.379509
$5$	− 8.20685i	− 0.734043i	−0.930212	−	0.367022i	$-0.880377\pi$
$5$	− 8.20685i	− 0.734043i	0.930212	−	0.367022i	$-0.119623\pi$
$6$	− 21.7199i	− 1.47785i
$7$	− 8.35495i	− 0.451125i	−0.974229	−	0.225562i	$-0.927578\pi$
$7$	− 8.35495i	− 0.451125i	0.974229	−	0.225562i	$-0.0724220\pi$
$8$	24.5882i	1.08666i
$9$	68.0366	2.51987
$10$	18.2848	0.578215
$11$	9.69898i	0.265850i	0.991126	+	0.132925i	$0.0424370\pi$
$11$	9.69898i	0.265850i	−0.991126	+	0.132925i	$0.957563\pi$
$12$	−29.5976	−0.712009
$13$	0	0
$13$	0	0
$14$	18.6147	0.355357
$15$	80.0059i	1.37716i
$16$	−30.4937	−0.476465
$17$	−44.6219	−0.636612	−0.318306	−	0.947988i	$-0.603114\pi$
$17$	−44.6219	−0.636612	−0.318306	+	0.947988i	$0.603114\pi$
$18$	151.585i	1.98494i
$19$	87.7418i	1.05944i	0.848173	+	0.529720i	$0.177703\pi$
$19$	87.7418i	1.05944i	−0.848173	+	0.529720i	$0.822297\pi$
$20$	− 24.9166i	− 0.278576i
$21$	81.4497i	0.846370i
$22$	−21.6092	−0.209414
$23$	−107.053	−0.970522	−0.485261	−	0.874369i	$-0.661275\pi$
$23$	−107.053	−0.970522	−0.485261	+	0.874369i	$0.661275\pi$
$24$	− 239.703i	− 2.03871i
$25$	57.6475	0.461180
$26$	0	0
$27$	−400.052	−2.85149
$28$	− 25.3662i	− 0.171206i
$29$	−14.0430	−0.0899214	−0.0449607	−	0.998989i	$-0.514316\pi$
$29$	−14.0430	−0.0899214	−0.0449607	+	0.998989i	$0.514316\pi$
$30$	−178.252	−1.08481
$31$	171.090i	0.991247i	0.868537	+	0.495624i	$0.165060\pi$
$31$	171.090i	0.991247i	−0.868537	+	0.495624i	$0.834940\pi$
$32$	128.766i	0.711339i
$33$	− 94.5522i	− 0.498770i
$34$	− 99.4170i	− 0.501467i
$35$	−68.5678	−0.331145
$36$	206.564	0.956314
$37$	413.954i	1.83929i	0.392754	+	0.919643i	$0.371522\pi$
$37$	413.954i	1.83929i	−0.392754	+	0.919643i	$0.628478\pi$
$38$	−195.488	−0.834534
$39$	0	0
$40$	201.792	0.797653
$41$	258.282i	0.983825i	0.870645	+	0.491912i	$0.163702\pi$
$41$	258.282i	0.983825i	−0.870645	+	0.491912i	$0.836298\pi$
$42$	−181.469	−0.666697
$43$	−61.0718	−0.216590	−0.108295	−	0.994119i	$-0.534539\pi$
$43$	−61.0718	−0.216590	−0.108295	+	0.994119i	$0.534539\pi$
$44$	29.4468i	0.100892i
$45$	− 558.367i	− 1.84970i
$46$	− 238.512i	− 0.764492i
$47$	68.7115i	0.213247i	0.994299	+	0.106623i	$0.0340039\pi$
$47$	68.7115i	0.213247i	−0.994299	+	0.106623i	$0.965996\pi$
$48$	297.273	0.893911
$49$	273.195	0.796486
$50$	128.438i	0.363278i
$51$	435.004	1.19437
$52$	0	0
$53$	328.701	0.851896	0.425948	−	0.904748i	$-0.359941\pi$
$53$	328.701	0.851896	0.425948	+	0.904748i	$0.359941\pi$
$54$	− 891.312i	− 2.24615i
$55$	79.5981	0.195146
$56$	205.433	0.490218
$57$	− 855.367i	− 1.98765i
$58$	− 31.2876i	− 0.0708322i
$59$	− 147.144i	− 0.324687i	−0.986734	−	0.162344i	$-0.948095\pi$
$59$	− 147.144i	− 0.324687i	0.986734	−	0.162344i	$-0.0519053\pi$
$60$	242.904i	0.522645i
$61$	−97.8083	−0.205296	−0.102648	−	0.994718i	$-0.532732\pi$
$61$	−97.8083	−0.205296	−0.102648	+	0.994718i	$0.532732\pi$
$62$	−381.186	−0.780818
$63$	− 568.442i	− 1.13678i
$64$	−530.839	−1.03680
$65$	0	0
$66$	210.661	0.392888
$67$	677.597i	1.23555i	0.786356	+	0.617774i	$0.211965\pi$
$67$	677.597i	1.23555i	−0.786356	+	0.617774i	$0.788035\pi$
$68$	−135.475	−0.241600
$69$	1043.62	1.82083
$70$	− 152.768i	− 0.260847i
$71$	786.767i	1.31510i	0.753411	+	0.657549i	$0.228407\pi$
$71$	786.767i	1.31510i	−0.753411	+	0.657549i	$0.771593\pi$
$72$	1672.90i	2.73824i
$73$	− 997.675i	− 1.59958i	−0.600282	−	0.799788i	$-0.704945\pi$
$73$	− 997.675i	− 1.59958i	0.600282	−	0.799788i	$-0.295055\pi$
$74$	−922.285	−1.44883
$75$	−561.987	−0.865236
$76$	266.390i	0.402067i
$77$	81.0345	0.119932
$78$	0	0
$79$	383.897	0.546731	0.273366	−	0.961910i	$-0.411863\pi$
$79$	383.897	0.546731	0.273366	+	0.961910i	$0.411863\pi$
$80$	250.258i	0.349746i
$81$	2062.99	2.82989
$82$	−575.448	−0.774971
$83$	519.718i	0.687307i	0.939096	+	0.343654i	$0.111665\pi$
$83$	519.718i	0.687307i	−0.939096	+	0.343654i	$0.888335\pi$
$84$	247.287i	0.321205i
$85$	366.205i	0.467301i
$86$	− 136.067i	− 0.170610i
$87$	136.901	0.168704
$88$	−238.481	−0.288888
$89$	− 683.222i	− 0.813723i	−0.913490	−	0.406862i	$-0.866623\pi$
$89$	− 683.222i	− 0.813723i	0.913490	−	0.406862i	$-0.133377\pi$
$90$	1244.03	1.45703
$91$	0	0
$92$	−325.019	−0.368321
$93$	− 1667.90i	− 1.85971i
$94$	−153.088	−0.167977
$95$	720.085	0.777675
$96$	− 1255.30i	− 1.33457i
$97$	− 347.294i	− 0.363530i	−0.983342	−	0.181765i	$-0.941819\pi$
$97$	− 347.294i	− 0.363530i	0.983342	−	0.181765i	$-0.0581809\pi$
$98$	608.675i	0.627402i
$99$	659.886i	0.669909i
$100$	175.022	0.175022
$101$	−554.794	−0.546575	−0.273288	−	0.961932i	$-0.588111\pi$
$101$	−554.794	−0.546575	−0.273288	+	0.961932i	$0.588111\pi$
$102$	969.184i	0.940819i
$103$	−1137.13	−1.08781	−0.543905	−	0.839147i	$-0.683055\pi$
$103$	−1137.13	−1.08781	−0.543905	+	0.839147i	$0.683055\pi$
$104$	0	0
$105$	668.445	0.621272
$106$	732.341i	0.671050i
$107$	−1556.14	−1.40596	−0.702980	−	0.711210i	$-0.748147\pi$
$107$	−1556.14	−1.40596	−0.702980	+	0.711210i	$0.748147\pi$
$108$	−1214.59	−1.08216
$109$	− 71.6448i	− 0.0629572i	−0.999504	−	0.0314786i	$-0.989978\pi$
$109$	− 71.6448i	− 0.0629572i	0.999504	−	0.0314786i	$-0.0100216\pi$
$110$	177.344i	0.153719i
$111$	− 4035.50i	− 3.45075i
$112$	254.774i	0.214945i
$113$	−980.872	−0.816573	−0.408286	−	0.912854i	$-0.633874\pi$
$113$	−980.872	−0.816573	−0.408286	+	0.912854i	$0.633874\pi$
$114$	1905.75	1.56570
$115$	878.565i	0.712405i
$116$	−42.6355	−0.0341259
$117$	0	0
$118$	327.836	0.255760
$119$	372.814i	0.287191i
$120$	−1967.20	−1.49650
$121$	1236.93	0.929324
$122$	− 217.916i	− 0.161714i
$123$	− 2517.90i	− 1.84579i
$124$	519.441i	0.376187i
$125$	− 1498.96i	− 1.07257i
$126$	1266.48	0.895455
$127$	2177.31	1.52130	0.760649	−	0.649164i	$-0.224881\pi$
$127$	2177.31	1.52130	0.760649	+	0.649164i	$0.224881\pi$
$128$	− 152.575i	− 0.105358i
$129$	595.369	0.406351
$130$	0	0
$131$	−1919.81	−1.28041	−0.640207	−	0.768202i	$-0.721152\pi$
$131$	−1919.81	−1.28041	−0.640207	+	0.768202i	$0.721152\pi$
$132$	− 287.067i	− 0.189288i
$133$	733.079	0.477940
$134$	−1509.68	−0.973257
$135$	3283.17i	2.09311i
$136$	− 1097.17i	− 0.691778i
$137$	744.013i	0.463980i	0.972718	+	0.231990i	$0.0745237\pi$
$137$	744.013i	0.463980i	−0.972718	+	0.231990i	$0.925476\pi$
$138$	2325.17i	1.43429i
$139$	2820.00	1.72078	0.860392	−	0.509633i	$-0.170219\pi$
$139$	2820.00	1.72078	0.860392	+	0.509633i	$0.170219\pi$
$140$	−208.177	−0.125672
$141$	− 669.846i	− 0.400080i
$142$	−1752.91	−1.03592
$143$	0	0
$144$	−2074.69	−1.20063
$145$	115.249i	0.0660062i
$146$	2222.81	1.26001
$147$	−2663.29	−1.49431
$148$	1256.79i	0.698025i
$149$	2894.60i	1.59151i	0.605619	+	0.795755i	$0.292926\pi$
$149$	2894.60i	1.59151i	−0.605619	+	0.795755i	$0.707074\pi$
$150$	− 1252.10i	− 0.681557i
$151$	− 494.004i	− 0.266235i	−0.991100	−	0.133118i	$-0.957501\pi$
$151$	− 494.004i	− 0.266235i	0.991100	−	0.133118i	$-0.0424988\pi$
$152$	−2157.42	−1.15125
$153$	−3035.92	−1.60418
$154$	180.544i	0.0944717i
$155$	1404.11	0.727618
$156$	0	0
$157$	50.7450	0.0257955	0.0128977	−	0.999917i	$-0.495894\pi$
$157$	50.7450	0.0257955	0.0128977	+	0.999917i	$0.495894\pi$
$158$	855.318i	0.430667i
$159$	−3204.39	−1.59827
$160$	1056.77	0.522154
$161$	894.419i	0.437826i
$162$	4596.32i	2.22914i
$163$	751.801i	0.361261i	0.983551	+	0.180631i	$0.0578139\pi$
$163$	751.801i	0.361261i	−0.983551	+	0.180631i	$0.942186\pi$
$164$	784.161i	0.373370i
$165$	−775.976	−0.366119
$166$	−1157.93	−0.541401
$167$	− 2974.32i	− 1.37820i	−0.724664	−	0.689102i	$-0.758005\pi$
$167$	− 2974.32i	− 1.37820i	0.724664	−	0.689102i	$-0.241995\pi$
$168$	−2002.70	−0.919714
$169$	0	0
$170$	−815.901	−0.368099
$171$	5969.66i	2.66966i
$172$	−185.418	−0.0821976
$173$	−1633.74	−0.717982	−0.358991	−	0.933341i	$-0.616879\pi$
$173$	−1633.74	−0.717982	−0.358991	+	0.933341i	$0.616879\pi$
$174$	305.013i	0.132891i
$175$	− 481.642i	− 0.208050i
$176$	− 295.758i	− 0.126668i
$177$	1434.46i	0.609156i
$178$	1522.21	0.640980
$179$	−3392.65	−1.41664	−0.708320	−	0.705892i	$-0.750546\pi$
$179$	−3392.65	−1.41664	−0.708320	+	0.705892i	$0.750546\pi$
$180$	− 1695.24i	− 0.701976i
$181$	3801.07	1.56095	0.780473	−	0.625190i	$-0.214979\pi$
$181$	3801.07	1.56095	0.780473	+	0.625190i	$0.214979\pi$
$182$	0	0
$183$	953.501	0.385163
$184$	− 2632.23i	− 1.05462i
$185$	3397.26	1.35012
$186$	3716.06	1.46492
$187$	− 432.787i	− 0.169243i
$188$	208.613i	0.0809290i
$189$	3342.42i	1.28638i
$190$	1604.34i	0.612584i
$191$	1265.51	0.479419	0.239710	−	0.970845i	$-0.422948\pi$
$191$	1265.51	0.479419	0.239710	+	0.970845i	$0.422948\pi$
$192$	5174.98	1.94517
$193$	2212.32i	0.825109i	0.910933	+	0.412555i	$0.135363\pi$
$193$	2212.32i	0.825109i	−0.910933	+	0.412555i	$0.864637\pi$
$194$	773.767	0.286357
$195$	0	0
$196$	829.438	0.302273
$197$	− 4357.25i	− 1.57585i	−0.615774	−	0.787923i	$-0.711157\pi$
$197$	− 4357.25i	− 1.57585i	0.615774	−	0.787923i	$-0.288843\pi$
$198$	−1470.22	−0.527696
$199$	−1090.63	−0.388507	−0.194254	−	0.980951i	$-0.562228\pi$
$199$	−1090.63	−0.388507	−0.194254	+	0.980951i	$0.562228\pi$
$200$	1417.45i	0.501145i
$201$	− 6605.67i	− 2.31805i
$202$	− 1236.08i	− 0.430544i
$203$	117.329i	0.0405658i
$204$	1320.70	0.453273
$205$	2119.68	0.722170
$206$	− 2533.50i	− 0.856881i
$207$	−7283.49	−2.44559
$208$	0	0
$209$	−851.007	−0.281652
$210$	1489.29i	0.489384i
$211$	−727.448	−0.237344	−0.118672	−	0.992934i	$-0.537864\pi$
$211$	−727.448	−0.237344	−0.118672	+	0.992934i	$0.537864\pi$
$212$	997.958	0.323302
$213$	− 7669.93i	− 2.46730i
$214$	− 3467.06i	− 1.10749i
$215$	501.207i	0.158986i
$216$	− 9836.58i	− 3.09859i
$217$	1429.45	0.447176
$218$	159.624	0.0495922
$219$	9726.01i	3.00102i
$220$	241.665	0.0740595
$221$	0	0
$222$	8991.05	2.71820
$223$	− 1015.45i	− 0.304931i	−0.988309	−	0.152465i	$-0.951279\pi$
$223$	− 1015.45i	− 0.304931i	0.988309	−	0.152465i	$-0.0487212\pi$
$224$	1075.83	0.320903
$225$	3922.14	1.16212
$226$	− 2185.37i	− 0.643225i
$227$	5292.59i	1.54749i	0.633494	+	0.773747i	$0.281620\pi$
$227$	5292.59i	1.54749i	−0.633494	+	0.773747i	$0.718380\pi$
$228$	− 2596.95i	− 0.754330i
$229$	3010.03i	0.868597i	0.900769	+	0.434298i	$0.143004\pi$
$229$	3010.03i	0.868597i	−0.900769	+	0.434298i	$0.856996\pi$
$230$	−1957.43	−0.561171
$231$	−789.979	−0.225008
$232$	− 345.292i	− 0.0977136i
$233$	2373.96	0.667482	0.333741	−	0.942665i	$-0.391689\pi$
$233$	2373.96	0.667482	0.333741	+	0.942665i	$0.391689\pi$
$234$	0	0
$235$	563.905	0.156532
$236$	− 446.740i	− 0.123222i
$237$	−3742.49	−1.02574
$238$	−830.624	−0.226224
$239$	− 783.439i	− 0.212035i	−0.994364	−	0.106018i	$-0.966190\pi$
$239$	− 783.439i	− 0.212035i	0.994364	−	0.106018i	$-0.0338100\pi$
$240$	− 2439.68i	− 0.656169i
$241$	− 3517.40i	− 0.940148i	−0.882627	−	0.470074i	$-0.844227\pi$
$241$	− 3517.40i	− 0.940148i	0.882627	−	0.470074i	$-0.155773\pi$
$242$	2755.86i	0.732040i
$243$	−9310.02	−2.45777
$244$	−296.953	−0.0779117
$245$	− 2242.07i	− 0.584656i
$246$	5609.86	1.45395
$247$	0	0
$248$	−4206.80	−1.07715
$249$	− 5066.56i	− 1.28948i
$250$	3339.67	0.844877
$251$	−1416.96	−0.356326	−0.178163	−	0.984001i	$-0.557015\pi$
$251$	−1416.96	−0.356326	−0.178163	+	0.984001i	$0.557015\pi$
$252$	− 1725.83i	− 0.431417i
$253$	− 1038.30i	− 0.258013i
$254$	4851.01i	1.19835i
$255$	− 3570.02i	− 0.876718i
$256$	−3906.78	−0.953804
$257$	1482.63	0.359861	0.179930	−	0.983679i	$-0.442413\pi$
$257$	1482.63	0.359861	0.179930	+	0.983679i	$0.442413\pi$
$258$	1326.47i	0.320088i
$259$	3458.56	0.829748
$260$	0	0
$261$	−955.438	−0.226591
$262$	− 4277.31i	− 1.00860i
$263$	7229.64	1.69505	0.847526	−	0.530753i	$-0.178091\pi$
$263$	7229.64	1.69505	0.847526	+	0.530753i	$0.178091\pi$
$264$	2324.87	0.541992
$265$	− 2697.60i	− 0.625329i
$266$	1633.29i	0.376479i
$267$	6660.50i	1.52665i
$268$	2057.23i	0.468901i
$269$	−494.904	−0.112174	−0.0560870	−	0.998426i	$-0.517862\pi$
$269$	−494.904	−0.112174	−0.0560870	+	0.998426i	$0.517862\pi$
$270$	−7314.87	−1.64877
$271$	− 4205.58i	− 0.942696i	−0.881947	−	0.471348i	$-0.843768\pi$
$271$	− 4205.58i	− 0.942696i	0.881947	−	0.471348i	$-0.156232\pi$
$272$	1360.69	0.303323
$273$	0	0
$274$	−1657.65	−0.365483
$275$	559.123i	0.122605i
$276$	3168.50	0.691020
$277$	−4208.55	−0.912878	−0.456439	−	0.889755i	$-0.650875\pi$
$277$	−4208.55	−0.912878	−0.456439	+	0.889755i	$0.650875\pi$
$278$	6282.92i	1.35548i
$279$	11640.4i	2.49782i
$280$	− 1685.96i	− 0.359841i
$281$	− 4740.83i	− 1.00646i	−0.864153	−	0.503228i	$-0.832145\pi$
$281$	− 4740.83i	− 1.00646i	0.864153	−	0.503228i	$-0.167855\pi$
$282$	1492.41	0.315148
$283$	3742.82	0.786176	0.393088	−	0.919501i	$-0.371407\pi$
$283$	3742.82	0.786176	0.393088	+	0.919501i	$0.371407\pi$
$284$	2388.68i	0.499091i
$285$	−7019.87	−1.45902
$286$	0	0
$287$	2157.93	0.443828
$288$	8760.81i	1.79249i
$289$	−2921.89	−0.594726
$290$	−256.773	−0.0519939
$291$	3385.66i	0.682030i
$292$	− 3029.01i	− 0.607053i
$293$	5254.97i	1.04778i	0.851787	+	0.523889i	$0.175519\pi$
$293$	5254.97i	1.04778i	−0.851787	+	0.523889i	$0.824481\pi$
$294$	− 5933.77i	− 1.17709i
$295$	−1207.59	−0.238334
$296$	−10178.4	−1.99867
$297$	− 3880.10i	− 0.758069i
$298$	−6449.14	−1.25365
$299$	0	0
$300$	−1706.23	−0.328364
$301$	510.251i	0.0977090i
$302$	1100.64	0.209717
$303$	5408.51	1.02545
$304$	− 2675.58i	− 0.504786i
$305$	802.698i	0.150696i
$306$	− 6764.00i	− 1.26363i
$307$	− 252.464i	− 0.0469344i	−0.999725	−	0.0234672i	$-0.992529\pi$
$307$	− 252.464i	− 0.0469344i	0.999725	−	0.0234672i	$-0.00747053\pi$
$308$	246.026	0.0455151
$309$	11085.5	2.04088
$310$	3128.34i	0.573154i
$311$	−2561.20	−0.466986	−0.233493	−	0.972359i	$-0.575016\pi$
$311$	−2561.20	−0.466986	−0.233493	+	0.972359i	$0.575016\pi$
$312$	0	0
$313$	−695.893	−0.125668	−0.0628342	−	0.998024i	$-0.520014\pi$
$313$	−695.893	−0.125668	−0.0628342	+	0.998024i	$0.520014\pi$
$314$	113.059i	0.0203194i
$315$	−4665.12	−0.834444
$316$	1165.54	0.207489
$317$	5747.37i	1.01831i	0.860675	+	0.509155i	$0.170042\pi$
$317$	5747.37i	1.01831i	−0.860675	+	0.509155i	$0.829958\pi$
$318$	− 7139.35i	− 1.25898i
$319$	− 136.203i	− 0.0239056i
$320$	4356.52i	0.761053i
$321$	15170.3	2.63777
$322$	−1992.75	−0.344881
$323$	− 3915.21i	− 0.674452i
$324$	6263.39	1.07397
$325$	0	0
$326$	−1675.00	−0.284570
$327$	698.442i	0.118116i
$328$	−6350.69	−1.06908
$329$	574.081	0.0962010
$330$	− 1728.87i	− 0.288397i
$331$	− 4244.76i	− 0.704874i	−0.935836	−	0.352437i	$-0.885353\pi$
$331$	− 4244.76i	− 0.704874i	0.935836	−	0.352437i	$-0.114647\pi$
$332$	1577.90i	0.260839i
$333$	28164.0i	4.63477i
$334$	6626.76	1.08563
$335$	5560.94	0.906946
$336$	− 2483.70i	− 0.403265i
$337$	7122.49	1.15130	0.575648	−	0.817698i	$-0.304750\pi$
$337$	7122.49	1.15130	0.575648	+	0.817698i	$0.304750\pi$
$338$	0	0
$339$	9562.20	1.53200
$340$	1111.82i	0.177345i
$341$	−1659.40	−0.263523
$342$	−13300.3	−2.10292
$343$	− 5148.28i	− 0.810440i
$344$	− 1501.65i	− 0.235359i
$345$	− 8564.84i	− 1.33657i
$346$	− 3639.95i	− 0.565563i
$347$	−3367.90	−0.521033	−0.260517	−	0.965469i	$-0.583893\pi$
$347$	−3367.90	−0.521033	−0.260517	+	0.965469i	$0.583893\pi$
$348$	415.640	0.0640248
$349$	− 6079.71i	− 0.932491i	−0.884655	−	0.466246i	$-0.845606\pi$
$349$	− 6079.71i	− 0.932491i	0.884655	−	0.466246i	$-0.154394\pi$
$350$	1073.09	0.163884
$351$	0	0
$352$	−1248.90	−0.189110
$353$	3230.99i	0.487162i	0.969881	+	0.243581i	$0.0783222\pi$
$353$	3230.99i	0.487162i	−0.969881	+	0.243581i	$0.921678\pi$
$354$	−3195.96	−0.479840
$355$	6456.88	0.965340
$356$	− 2074.31i	− 0.308815i
$357$	− 3634.44i	− 0.538809i
$358$	− 7558.78i	− 1.11590i
$359$	− 5345.81i	− 0.785908i	−0.919558	−	0.392954i	$-0.871453\pi$
$359$	− 5345.81i	− 0.785908i	0.919558	−	0.392954i	$-0.128547\pi$
$360$	13729.2	2.00999
$361$	−839.632	−0.122413
$362$	8468.73i	1.22958i
$363$	−12058.4	−1.74353
$364$	0	0
$365$	−8187.78	−1.17416
$366$	2124.39i	0.303398i
$367$	−12171.3	−1.73117	−0.865583	−	0.500765i	$-0.833052\pi$
$367$	−12171.3	−1.73117	−0.865583	+	0.500765i	$0.833052\pi$
$368$	3264.43	0.462419
$369$	17572.6i	2.47911i
$370$	7569.05i	1.06350i
$371$	− 2746.28i	− 0.384312i
$372$	− 5063.86i	− 0.705776i
$373$	−4463.95	−0.619664	−0.309832	−	0.950791i	$-0.600273\pi$
$373$	−4463.95	−0.619664	−0.309832	+	0.950791i	$0.600273\pi$
$374$	964.244	0.133315
$375$	14612.9i	2.01228i
$376$	−1689.49	−0.231726
$377$	0	0
$378$	−7446.87	−1.01330
$379$	5369.06i	0.727679i	0.931462	+	0.363839i	$0.118534\pi$
$379$	5369.06i	0.727679i	−0.931462	+	0.363839i	$0.881466\pi$
$380$	2186.23	0.295134
$381$	−21225.9	−2.85416
$382$	2819.54i	0.377645i
$383$	195.084i	0.0260269i	0.999915	+	0.0130135i	$0.00414243\pi$
$383$	195.084i	0.0260269i	−0.999915	+	0.0130135i	$0.995858\pi$
$384$	1487.40i	0.197665i
$385$	− 665.038i	− 0.0880351i
$386$	−4929.02	−0.649949
$387$	−4155.12	−0.545779
$388$	− 1054.41i	− 0.137963i
$389$	−9120.52	−1.18876	−0.594381	−	0.804183i	$-0.702603\pi$
$389$	−9120.52	−1.18876	−0.594381	+	0.804183i	$0.702603\pi$
$390$	0	0
$391$	4776.89	0.617845
$392$	6717.38i	0.865507i
$393$	18715.6	2.40223
$394$	9707.91	1.24131
$395$	− 3150.59i	− 0.401325i
$396$	2003.46i	0.254236i
$397$	6613.46i	0.836071i	0.908431	+	0.418036i	$0.137281\pi$
$397$	6613.46i	0.836071i	−0.908431	+	0.418036i	$0.862719\pi$
$398$	− 2429.92i	− 0.306032i
$399$	−7146.54	−0.896678
$400$	−1757.89	−0.219736
$401$	− 8589.26i	− 1.06964i	−0.844965	−	0.534822i	$-0.820379\pi$
$401$	− 8589.26i	− 1.06964i	0.844965	−	0.534822i	$-0.179621\pi$
$402$	14717.4	1.82596
$403$	0	0
$404$	−1684.39	−0.207430
$405$	− 16930.7i	− 2.07726i
$406$	−261.407	−0.0319542
$407$	−4014.93	−0.488975
$408$	10696.0i	1.29787i
$409$	7515.59i	0.908612i	0.890846	+	0.454306i	$0.150113\pi$
$409$	7515.59i	0.908612i	−0.890846	+	0.454306i	$0.849887\pi$
$410$	4722.62i	0.568863i
$411$	− 7253.14i	− 0.870489i
$412$	−3452.39	−0.412833
$413$	−1229.38	−0.146474
$414$	− 16227.5i	− 1.92642i
$415$	4265.25	0.504513
$416$	0	0
$417$	−27491.2	−3.22842
$418$	− 1896.03i	− 0.221861i
$419$	−4557.33	−0.531361	−0.265680	−	0.964061i	$-0.585597\pi$
$419$	−4557.33	−0.531361	−0.265680	+	0.964061i	$0.585597\pi$
$420$	2029.45	0.235778
$421$	2225.19i	0.257599i	0.991671	+	0.128800i	$0.0411124\pi$
$421$	2225.19i	0.257599i	−0.991671	+	0.128800i	$0.958888\pi$
$422$	− 1620.75i	− 0.186959i
$423$	4674.90i	0.537355i
$424$	8082.17i	0.925719i
$425$	−2572.34	−0.293593
$426$	17088.5	1.94352
$427$	817.183i	0.0926142i
$428$	−4724.54	−0.533574
$429$	0	0
$430$	−1116.68	−0.125235
$431$	396.028i	0.0442599i	0.999755	+	0.0221299i	$0.00704475\pi$
$431$	396.028i	0.0442599i	−0.999755	+	0.0221299i	$0.992955\pi$
$432$	12199.1	1.35863
$433$	−5188.06	−0.575802	−0.287901	−	0.957660i	$-0.592957\pi$
$433$	−5188.06	−0.575802	−0.287901	+	0.957660i	$0.592957\pi$
$434$	3184.79i	0.352246i
$435$	− 1123.52i	− 0.123836i
$436$	− 217.519i	− 0.0238928i
$437$	− 9392.99i	− 1.02821i
$438$	−21669.4	−2.36394
$439$	−11329.9	−1.23176	−0.615882	−	0.787838i	$-0.711200\pi$
$439$	−11329.9	−1.23176	−0.615882	+	0.787838i	$0.711200\pi$
$440$	1957.18i	0.212056i
$441$	18587.2	2.00705
$442$	0	0
$443$	15625.2	1.67579	0.837897	−	0.545828i	$-0.183785\pi$
$443$	15625.2	1.67579	0.837897	+	0.545828i	$0.183785\pi$
$444$	− 12252.1i	− 1.30959i
$445$	−5607.10	−0.597308
$446$	2262.41	0.240198
$447$	− 28218.5i	− 2.98588i
$448$	4435.14i	0.467724i
$449$	− 13686.6i	− 1.43855i	−0.694725	−	0.719276i	$-0.744474\pi$
$449$	− 13686.6i	− 1.43855i	0.694725	−	0.719276i	$-0.255526\pi$
$450$	8738.49i	0.915414i
$451$	−2505.07	−0.261550
$452$	−2978.00	−0.309896
$453$	4815.89i	0.499492i
$454$	−11791.8	−1.21898
$455$	0	0
$456$	21031.9	2.15989
$457$	12673.1i	1.29720i	0.761127	+	0.648602i	$0.224646\pi$
$457$	12673.1i	1.29720i	−0.761127	+	0.648602i	$0.775354\pi$
$458$	−6706.32	−0.684205
$459$	17851.1	1.81529
$460$	2667.38i	0.270364i
$461$	9154.85i	0.924911i	0.886642	+	0.462456i	$0.153032\pi$
$461$	9154.85i	0.924911i	−0.886642	+	0.462456i	$0.846968\pi$
$462$	− 1760.06i	− 0.177241i
$463$	− 6910.59i	− 0.693655i	−0.937929	−	0.346827i	$-0.887259\pi$
$463$	− 6910.59i	− 0.693655i	0.937929	−	0.346827i	$-0.112741\pi$
$464$	428.223	0.0428443
$465$	−13688.2	−1.36511
$466$	5289.16i	0.525784i
$467$	−2920.19	−0.289359	−0.144679	−	0.989479i	$-0.546215\pi$
$467$	−2920.19	−0.289359	−0.144679	+	0.989479i	$0.546215\pi$
$468$	0	0
$469$	5661.29	0.557386
$470$	1256.37i	0.123303i
$471$	−494.696	−0.0483958
$472$	3618.02	0.352823
$473$	− 592.334i	− 0.0575804i
$474$	− 8338.21i	− 0.807989i
$475$	5058.10i	0.488593i
$476$	1131.89i	0.108992i
$477$	22363.7	2.14667
$478$	1745.49	0.167023
$479$	17088.6i	1.63006i	0.579419	+	0.815030i	$0.303280\pi$
$479$	17088.6i	1.63006i	−0.579419	+	0.815030i	$0.696720\pi$
$480$	−10302.1	−0.979630
$481$	0	0
$482$	7836.73	0.740567
$483$	− 8719.39i	− 0.821421i
$484$	3755.40	0.352686
$485$	−2850.19	−0.266846
$486$	− 20742.6i	− 1.93602i
$487$	− 13813.0i	− 1.28527i	−0.766174	−	0.642634i	$-0.777842\pi$
$487$	− 13813.0i	− 1.28527i	0.766174	−	0.642634i	$-0.222158\pi$
$488$	− 2404.93i	− 0.223086i
$489$	− 7329.06i	− 0.677774i
$490$	4995.31	0.460541
$491$	11848.4	1.08903	0.544513	−	0.838752i	$-0.316714\pi$
$491$	11848.4	1.08903	0.544513	+	0.838752i	$0.316714\pi$
$492$	− 7644.53i	− 0.700492i
$493$	626.625	0.0572450
$494$	0	0
$495$	5415.59	0.491743
$496$	− 5217.17i	− 0.472294i
$497$	6573.40	0.593274
$498$	11288.2	1.01574
$499$	− 5224.46i	− 0.468696i	−0.972153	−	0.234348i	$-0.924705\pi$
$499$	− 5224.46i	− 0.468696i	0.972153	−	0.234348i	$-0.0752955\pi$
$500$	− 4550.95i	− 0.407049i
$501$	28995.7i	2.58569i
$502$	− 3156.97i	− 0.280682i
$503$	7884.39	0.698902	0.349451	−	0.936955i	$-0.386368\pi$
$503$	7884.39	0.698902	0.349451	+	0.936955i	$0.386368\pi$
$504$	13977.0	1.23529
$505$	4553.12i	0.401210i
$506$	2313.32	0.203241
$507$	0	0
$508$	6610.45	0.577345
$509$	− 4026.56i	− 0.350637i	−0.984512	−	0.175318i	$-0.943905\pi$
$509$	− 4026.56i	− 0.350637i	0.984512	−	0.175318i	$-0.0560954\pi$
$510$	7953.95	0.690602
$511$	−8335.53	−0.721609
$512$	− 9924.86i	− 0.856681i
$513$	− 35101.3i	− 3.02098i
$514$	3303.29i	0.283467i
$515$	9332.23i	0.798499i
$516$	1807.58	0.154214
$517$	−666.432	−0.0566918
$518$	7705.64i	0.653603i
$519$	15926.8	1.34703
$520$	0	0
$521$	6196.12	0.521030	0.260515	−	0.965470i	$-0.416108\pi$
$521$	6196.12	0.521030	0.260515	+	0.965470i	$0.416108\pi$
$522$	− 2128.70i	− 0.178488i
$523$	7899.39	0.660452	0.330226	−	0.943902i	$-0.392875\pi$
$523$	7899.39	0.660452	0.330226	+	0.943902i	$0.392875\pi$
$524$	−5828.67	−0.485928
$525$	4695.37i	0.390329i
$526$	16107.6i	1.33521i
$527$	− 7634.36i	− 0.631040i
$528$	2883.25i	0.237646i
$529$	−706.752	−0.0580876
$530$	6010.22	0.492580
$531$	− 10011.2i	− 0.818171i
$532$	2225.68	0.181382
$533$	0	0
$534$	−14839.5	−1.20256
$535$	12771.0i	1.03203i
$536$	−16660.9	−1.34262
$537$	33073.8	2.65780
$538$	− 1102.64i	− 0.0883609i
$539$	2649.71i	0.211746i
$540$	9967.94i	0.794355i
$541$	6146.22i	0.488441i	0.969720	+	0.244220i	$0.0785320\pi$
$541$	6146.22i	0.488441i	−0.969720	+	0.244220i	$0.921468\pi$
$542$	9369.98	0.742574
$543$	−37055.4	−2.92854
$544$	− 5745.79i	− 0.452847i
$545$	−587.979	−0.0462133
$546$	0	0
$547$	5555.49	0.434252	0.217126	−	0.976144i	$-0.430332\pi$
$547$	5555.49	0.434252	0.217126	+	0.976144i	$0.430332\pi$
$548$	2258.87i	0.176084i
$549$	−6654.54	−0.517321
$550$	−1245.72	−0.0965775
$551$	− 1232.16i	− 0.0952663i
$552$	25660.8i	1.97861i
$553$	− 3207.44i	− 0.246644i
$554$	− 9376.59i	− 0.719085i
$555$	−33118.8	−2.53300
$556$	8561.70	0.653052
$557$	− 7580.71i	− 0.576669i	−0.957530	−	0.288335i	$-0.906898\pi$
$557$	− 7580.71i	− 0.576669i	0.957530	−	0.288335i	$-0.0931016\pi$
$558$	−25934.6	−1.96756
$559$	0	0
$560$	2090.89	0.157779
$561$	4219.10i	0.317523i
$562$	10562.5	0.792799
$563$	13594.6	1.01766	0.508831	−	0.860866i	$-0.330078\pi$
$563$	13594.6	1.01766	0.508831	+	0.860866i	$0.330078\pi$
$564$	− 2033.70i	− 0.151834i
$565$	8049.88i	0.599400i
$566$	8338.97i	0.619281i
$567$	− 17236.2i	− 1.27664i
$568$	−19345.2	−1.42906
$569$	−5650.14	−0.416285	−0.208143	−	0.978099i	$-0.566742\pi$
$569$	−5650.14	−0.416285	−0.208143	+	0.978099i	$0.566742\pi$
$570$	− 15640.2i	− 1.14929i
$571$	−6297.53	−0.461547	−0.230773	−	0.973008i	$-0.574126\pi$
$571$	−6297.53	−0.461547	−0.230773	+	0.973008i	$0.574126\pi$
$572$	0	0
$573$	−12337.0	−0.899454
$574$	4807.84i	0.349609i
$575$	−6171.32	−0.447586
$576$	−36116.5	−2.61259
$577$	17838.9i	1.28707i	0.765415	+	0.643537i	$0.222534\pi$
$577$	17838.9i	1.28707i	−0.765415	+	0.643537i	$0.777466\pi$
$578$	− 6509.93i	− 0.468473i
$579$	− 21567.2i	− 1.54801i
$580$	349.903i	0.0250499i
$581$	4342.22	0.310061
$582$	−7543.20	−0.537243
$583$	3188.06i	0.226477i
$584$	24531.1	1.73819
$585$	0	0
$586$	−11708.0	−0.825348
$587$	− 455.091i	− 0.0319993i	−0.999872	−	0.0159997i	$-0.994907\pi$
$587$	− 455.091i	− 0.0319993i	0.999872	−	0.0159997i	$-0.00509307\pi$
$588$	−8085.92	−0.567105
$589$	−15011.7	−1.05017
$590$	− 2690.50i	− 0.187739i
$591$	42477.4i	2.95649i
$592$	− 12623.0i	− 0.876355i
$593$	16240.6i	1.12466i	0.826913	+	0.562330i	$0.190095\pi$
$593$	16240.6i	1.12466i	−0.826913	+	0.562330i	$0.809905\pi$
$594$	8644.82	0.597140
$595$	3059.63	0.210811
$596$	8788.21i	0.603992i
$597$	10632.2	0.728891
$598$	0	0
$599$	−6704.05	−0.457296	−0.228648	−	0.973509i	$-0.573430\pi$
$599$	−6704.05	−0.457296	−0.228648	+	0.973509i	$0.573430\pi$
$600$	− 13818.3i	− 0.940214i
$601$	26413.2	1.79271	0.896354	−	0.443339i	$-0.146206\pi$
$601$	26413.2	1.79271	0.896354	+	0.443339i	$0.146206\pi$
$602$	−1136.83	−0.0769666
$603$	46101.4i	3.11343i
$604$	− 1499.83i	− 0.101039i
$605$	− 10151.3i	− 0.682164i
$606$	12050.1i	0.807758i
$607$	−3326.37	−0.222427	−0.111214	−	0.993797i	$-0.535474\pi$
$607$	−3326.37	−0.222427	−0.111214	+	0.993797i	$0.535474\pi$
$608$	−11298.2	−0.753621
$609$	− 1143.80i	− 0.0761068i
$610$	−1788.40	−0.118705
$611$	0	0
$612$	−9217.27	−0.608801
$613$	− 26258.7i	− 1.73014i	−0.501647	−	0.865072i	$-0.667272\pi$
$613$	− 26258.7i	− 1.73014i	0.501647	−	0.865072i	$-0.332728\pi$
$614$	562.486	0.0369708
$615$	−20664.1	−1.35489
$616$	1992.50i	0.130325i
$617$	− 27352.9i	− 1.78474i	−0.451304	−	0.892370i	$-0.649041\pi$
$617$	− 27352.9i	− 1.78474i	0.451304	−	0.892370i	$-0.350959\pi$
$618$	24698.3i	1.60762i
$619$	− 13056.5i	− 0.847795i	−0.905710	−	0.423897i	$-0.860662\pi$
$619$	− 13056.5i	− 0.847795i	0.905710	−	0.423897i	$-0.139338\pi$
$620$	4262.98	0.276137
$621$	42826.6	2.76743
$622$	− 5706.33i	− 0.367851i
$623$	−5708.28	−0.367091
$624$	0	0
$625$	−5095.82	−0.326132
$626$	− 1550.44i	− 0.0989905i
$627$	8296.19	0.528417
$628$	154.065	0.00978961
$629$	− 18471.4i	− 1.17091i
$630$	− 10393.8i	− 0.657302i
$631$	6552.91i	0.413419i	0.978402	+	0.206709i	$0.0662755\pi$
$631$	6552.91i	0.413419i	−0.978402	+	0.206709i	$0.933725\pi$
$632$	9439.35i	0.594109i
$633$	7091.66	0.445289
$634$	−12805.1	−0.802137
$635$	− 17868.8i	− 1.11670i
$636$	−9728.76	−0.606557
$637$	0	0
$638$	303.458	0.0188308
$639$	53528.9i	3.31388i
$640$	−1252.16	−0.0773373
$641$	5764.77	0.355218	0.177609	−	0.984101i	$-0.443164\pi$
$641$	5764.77	0.355218	0.177609	+	0.984101i	$0.443164\pi$
$642$	33799.2i	2.07780i
$643$	− 12279.5i	− 0.753120i	−0.926392	−	0.376560i	$-0.877107\pi$
$643$	− 12279.5i	− 0.753120i	0.926392	−	0.376560i	$-0.122893\pi$
$644$	2715.52i	0.166159i
$645$	− 4886.10i	− 0.298279i
$646$	8723.04	0.531274
$647$	29024.3	1.76362	0.881810	−	0.471605i	$-0.156325\pi$
$647$	29024.3	1.76362	0.881810	+	0.471605i	$0.156325\pi$
$648$	50725.3i	3.07512i
$649$	1427.15	0.0863182
$650$	0	0
$651$	−13935.2	−0.838962
$652$	2282.52i	0.137102i
$653$	−17167.1	−1.02879	−0.514396	−	0.857552i	$-0.671984\pi$
$653$	−17167.1	−1.02879	−0.514396	+	0.857552i	$0.671984\pi$
$654$	−1556.12	−0.0930415
$655$	15755.6i	0.939880i
$656$	− 7875.97i	− 0.468758i
$657$	− 67878.5i	− 4.03073i
$658$	1279.05i	0.0757787i
$659$	18313.0	1.08251	0.541254	−	0.840859i	$-0.317950\pi$
$659$	18313.0	1.08251	0.541254	+	0.840859i	$0.317950\pi$
$660$	−2355.92	−0.138945
$661$	6885.30i	0.405155i	0.979266	+	0.202577i	$0.0649317\pi$
$661$	6885.30i	0.405155i	−0.979266	+	0.202577i	$0.935068\pi$
$662$	9457.28	0.555238
$663$	0	0
$664$	−12779.0	−0.746867
$665$	− 6016.27i	− 0.350828i
$666$	−62749.1	−3.65087
$667$	1503.34	0.0872706
$668$	− 9030.25i	− 0.523040i
$669$	9899.29i	0.572090i
$670$	12389.7i	0.714413i
$671$	− 948.641i	− 0.0545781i
$672$	−10488.0	−0.602056
$673$	−6238.26	−0.357307	−0.178653	−	0.983912i	$-0.557174\pi$
$673$	−6238.26	−0.357307	−0.178653	+	0.983912i	$0.557174\pi$
$674$	15868.8i	0.906890i
$675$	−23062.0	−1.31505
$676$	0	0
$677$	25482.4	1.44663	0.723316	−	0.690517i	$-0.242617\pi$
$677$	25482.4	1.44663	0.723316	+	0.690517i	$0.242617\pi$
$678$	21304.5i	1.20678i
$679$	−2901.62	−0.163997
$680$	−9004.34	−0.507795
$681$	− 51595.7i	− 2.90331i
$682$	− 3697.12i	− 0.207581i
$683$	− 27426.5i	− 1.53652i	−0.640136	−	0.768261i	$-0.721122\pi$
$683$	− 27426.5i	− 1.53652i	0.640136	−	0.768261i	$-0.278878\pi$
$684$	18124.3i	1.01316i
$685$	6106.00	0.340582
$686$	11470.3	0.638394
$687$	− 29343.8i	− 1.62960i
$688$	1862.31	0.103197
$689$	0	0
$690$	19082.4	1.05283
$691$	− 13886.3i	− 0.764486i	−0.924062	−	0.382243i	$-0.875152\pi$
$691$	− 13886.3i	− 0.764486i	0.924062	−	0.382243i	$-0.124848\pi$
$692$	−4960.14	−0.272480
$693$	5513.31	0.302213
$694$	− 7503.65i	− 0.410425i
$695$	− 23143.3i	− 1.26313i
$696$	3366.14i	0.183324i
$697$	− 11525.0i	− 0.626314i
$698$	13545.5	0.734535
$699$	−23143.0	−1.25229
$700$	− 1462.30i	− 0.0789567i
$701$	−15744.4	−0.848301	−0.424151	−	0.905592i	$-0.639427\pi$
$701$	−15744.4	−0.848301	−0.424151	+	0.905592i	$0.639427\pi$
$702$	0	0
$703$	−36321.1	−1.94861
$704$	− 5148.60i	− 0.275632i
$705$	−5497.33	−0.293676
$706$	−7198.61	−0.383744
$707$	4635.28i	0.246574i
$708$	4355.12i	0.231180i
$709$	28689.1i	1.51967i	0.650118	+	0.759833i	$0.274719\pi$
$709$	28689.1i	1.51967i	−0.650118	+	0.759833i	$0.725281\pi$
$710$	14385.9i	0.760410i
$711$	26119.0	1.37769
$712$	16799.2	0.884237
$713$	− 18315.6i	− 0.962027i
$714$	8097.48	0.424427
$715$	0	0
$716$	−10300.3	−0.537627
$717$	7637.49i	0.397807i
$718$	11910.4	0.619070
$719$	12619.1	0.654538	0.327269	−	0.944931i	$-0.393872\pi$
$719$	12619.1	0.654538	0.327269	+	0.944931i	$0.393872\pi$
$720$	17026.7i	0.881315i
$721$	9500.63i	0.490738i
$722$	− 1870.69i	− 0.0964265i
$723$	34290.0i	1.76384i
$724$	11540.3	0.592392
$725$	−809.544	−0.0414700
$726$	− 26866.0i	− 1.37340i
$727$	12644.5	0.645061	0.322531	−	0.946559i	$-0.395466\pi$
$727$	12644.5	0.645061	0.322531	+	0.946559i	$0.395466\pi$
$728$	0	0
$729$	35059.5	1.78121
$730$	− 18242.3i	− 0.924900i
$731$	2725.14	0.137884
$732$	2894.89	0.146173
$733$	19109.0i	0.962903i	0.876473	+	0.481451i	$0.159890\pi$
$733$	19109.0i	0.962903i	−0.876473	+	0.481451i	$0.840110\pi$
$734$	− 27117.6i	− 1.36366i
$735$	21857.2i	1.09689i
$736$	− 13784.7i	− 0.690370i
$737$	−6572.01	−0.328471
$738$	−39151.6	−1.95283
$739$	14093.6i	0.701543i	0.936461	+	0.350771i	$0.114081\pi$
$739$	14093.6i	0.701543i	−0.936461	+	0.350771i	$0.885919\pi$
$740$	10314.3	0.512381
$741$	0	0
$742$	6118.67	0.302727
$743$	− 17587.4i	− 0.868397i	−0.900817	−	0.434198i	$-0.857032\pi$
$743$	− 17587.4i	− 0.868397i	0.900817	−	0.434198i	$-0.142968\pi$
$744$	41010.7	2.02087
$745$	23755.6	1.16824
$746$	− 9945.63i	− 0.488117i
$747$	35359.9i	1.73193i
$748$	− 1313.97i	− 0.0642293i
$749$	13001.5i	0.634263i
$750$	−32557.3	−1.58510
$751$	16587.0	0.805948	0.402974	−	0.915211i	$-0.367977\pi$
$751$	16587.0	0.805948	0.402974	+	0.915211i	$0.367977\pi$
$752$	− 2095.27i	− 0.101605i
$753$	13813.5	0.668514
$754$	0	0
$755$	−4054.22	−0.195428
$756$	10147.8i	0.488191i
$757$	33819.5	1.62376	0.811882	−	0.583822i	$-0.198443\pi$
$757$	33819.5	1.62376	0.811882	+	0.583822i	$0.198443\pi$
$758$	−11962.2	−0.573202
$759$	10122.1i	0.484068i
$760$	17705.6i	0.845066i
$761$	28131.2i	1.34002i	0.742351	+	0.670011i	$0.233710\pi$
$761$	28131.2i	1.34002i	−0.742351	+	0.670011i	$0.766290\pi$
$762$	− 47290.9i	− 2.24825i
$763$	−598.589	−0.0284015
$764$	3842.18	0.181944
$765$	24915.4i	1.17754i
$766$	−434.645	−0.0205018
$767$	0	0
$768$	38085.9	1.78946
$769$	23735.0i	1.11301i	0.830844	+	0.556506i	$0.187858\pi$
$769$	23735.0i	1.11301i	−0.830844	+	0.556506i	$0.812142\pi$
$770$	1481.70	0.0693463
$771$	−14453.7	−0.675146
$772$	6716.75i	0.313136i
$773$	9536.25i	0.443719i	0.975079	+	0.221860i	$0.0712127\pi$
$773$	9536.25i	0.443719i	−0.975079	+	0.221860i	$0.928787\pi$
$774$	− 9257.55i	− 0.429917i
$775$	9862.92i	0.457144i
$776$	8539.34	0.395032
$777$	−33716.4	−1.55672
$778$	− 20320.4i	− 0.936403i
$779$	−22662.1	−1.04230
$780$	0	0
$781$	−7630.84	−0.349619
$782$	10642.8i	0.486685i
$783$	5617.94	0.256410
$784$	−8330.73	−0.379498
$785$	− 416.457i	− 0.0189350i
$786$	41698.1i	1.89227i
$787$	17696.2i	0.801527i	0.916182	+	0.400763i	$0.131255\pi$
$787$	17696.2i	0.801527i	−0.916182	+	0.400763i	$0.868745\pi$
$788$	− 13228.9i	− 0.598047i
$789$	−70479.4	−3.18014
$790$	7019.47	0.316128
$791$	8195.14i	0.368376i
$792$	−16225.4	−0.727961
$793$	0	0
$794$	−14734.7	−0.658584
$795$	26298.0i	1.17320i
$796$	−3311.24	−0.147442
$797$	32566.7	1.44739	0.723696	−	0.690118i	$-0.242442\pi$
$797$	32566.7	1.44739	0.723696	+	0.690118i	$0.242442\pi$
$798$	− 15922.4i	− 0.706325i
$799$	− 3066.04i	− 0.135755i
$800$	7423.05i	0.328056i
$801$	− 46484.1i	− 2.05048i
$802$	19136.8	0.842572
$803$	9676.44	0.425248
$804$	− 20055.3i	− 0.879721i
$805$	7340.36	0.321384
$806$	0	0
$807$	4824.66	0.210453
$808$	− 13641.4i	− 0.593940i
$809$	11622.3	0.505089	0.252544	−	0.967585i	$-0.418733\pi$
$809$	11622.3	0.505089	0.252544	+	0.967585i	$0.418733\pi$
$810$	37721.3	1.63629
$811$	− 6494.39i	− 0.281195i	−0.990067	−	0.140597i	$-0.955098\pi$
$811$	− 6494.39i	− 0.281195i	0.990067	−	0.140597i	$-0.0449023\pi$
$812$	356.218i	0.0153951i
$813$	40998.8i	1.76862i
$814$	− 8945.22i	− 0.385172i
$815$	6169.92	0.265181
$816$	−13264.9	−0.569074
$817$	− 5358.55i	− 0.229464i
$818$	−16744.7	−0.715725
$819$	0	0
$820$	6435.49	0.274070
$821$	− 34897.8i	− 1.48349i	−0.670685	−	0.741743i	$-0.734000\pi$
$821$	− 34897.8i	− 1.48349i	0.670685	−	0.741743i	$-0.266000\pi$
$822$	16159.9	0.685695
$823$	−1502.61	−0.0636426	−0.0318213	−	0.999494i	$-0.510131\pi$
$823$	−1502.61	−0.0636426	−0.0318213	+	0.999494i	$0.510131\pi$
$824$	− 27959.9i	− 1.18208i
$825$	− 5450.70i	− 0.230023i
$826$	− 2739.05i	− 0.115380i
$827$	27887.8i	1.17262i	0.810088	+	0.586308i	$0.199419\pi$
$827$	27887.8i	1.17262i	−0.810088	+	0.586308i	$0.800581\pi$
$828$	−22113.2	−0.928124
$829$	−30843.7	−1.29221	−0.646107	−	0.763247i	$-0.723604\pi$
$829$	−30843.7	−1.29221	−0.646107	+	0.763247i	$0.723604\pi$
$830$	9502.93i	0.397412i
$831$	41027.8	1.71268
$832$	0	0
$833$	−12190.5	−0.507053
$834$	− 61250.1i	− 2.54307i
$835$	−24409.8	−1.01166
$836$	−2583.71	−0.106890
$837$	− 68445.0i	− 2.82653i
$838$	− 10153.7i	− 0.418560i
$839$	− 36598.8i	− 1.50599i	−0.658023	−	0.752997i	$-0.728607\pi$
$839$	− 36598.8i	− 1.50599i	0.658023	−	0.752997i	$-0.271393\pi$
$840$	16435.9i	0.675110i
$841$	−24191.8	−0.991914
$842$	−4957.71	−0.202914
$843$	46216.8i	1.88825i
$844$	−2208.58	−0.0900741
$845$	0	0
$846$	−10415.6	−0.423282
$847$	− 10334.5i	− 0.419241i
$848$	−10023.3	−0.405898
$849$	−36487.6	−1.47497
$850$	− 5731.15i	− 0.231267i
$851$	− 44314.8i	− 1.78507i
$852$	− 23286.4i	− 0.936362i
$853$	21578.4i	0.866155i	0.901357	+	0.433077i	$0.142572\pi$
$853$	21578.4i	0.866155i	−0.901357	+	0.433077i	$0.857428\pi$
$854$	−1820.67	−0.0729534
$855$	48992.1	1.95964
$856$	− 38262.7i	− 1.52779i
$857$	31199.6	1.24359	0.621795	−	0.783180i	$-0.286404\pi$
$857$	31199.6	1.24359	0.621795	+	0.783180i	$0.286404\pi$
$858$	0	0
$859$	8035.71	0.319179	0.159590	−	0.987183i	$-0.448983\pi$
$859$	8035.71	0.319179	0.159590	+	0.987183i	$0.448983\pi$
$860$	1521.70i	0.0603366i
$861$	−21037.0	−0.832680
$862$	−882.346	−0.0348641
$863$	8741.47i	0.344801i	0.985027	+	0.172400i	$0.0551523\pi$
$863$	8741.47i	0.344801i	−0.985027	+	0.172400i	$0.944848\pi$
$864$	− 51513.2i	− 2.02837i
$865$	13407.9i	0.527030i
$866$	− 11558.9i	− 0.453566i
$867$	28484.5	1.11578
$868$	4339.90	0.169707
$869$	3723.41i	0.145349i
$870$	2503.20	0.0975475
$871$	0	0
$872$	1761.62	0.0684128
$873$	− 23628.7i	− 0.916049i
$874$	20927.5	0.809934
$875$	−12523.7	−0.483863
$876$	29528.8i	1.13891i
$877$	1744.13i	0.0671552i	0.999436	+	0.0335776i	$0.0106901\pi$
$877$	1744.13i	0.0671552i	−0.999436	+	0.0335776i	$0.989310\pi$
$878$	− 25242.8i	− 0.970276i
$879$	− 51229.0i	− 1.96577i
$880$	−2427.24	−0.0929800
$881$	5688.81	0.217549	0.108775	−	0.994066i	$-0.465307\pi$
$881$	5688.81	0.217549	0.108775	+	0.994066i	$0.465307\pi$
$882$	41412.2i	1.58098i
$883$	3940.14	0.150165	0.0750827	−	0.997177i	$-0.476078\pi$
$883$	3940.14	0.150165	0.0750827	+	0.997177i	$0.476078\pi$
$884$	0	0
$885$	11772.4	0.447147
$886$	34812.8i	1.32004i
$887$	−36880.4	−1.39608	−0.698039	−	0.716060i	$-0.745944\pi$
$887$	−36880.4	−1.39608	−0.698039	+	0.716060i	$0.745944\pi$
$888$	99225.8	3.74978
$889$	− 18191.3i	− 0.686295i
$890$	− 12492.6i	− 0.470507i
$891$	20008.9i	0.752328i
$892$	− 3082.97i	− 0.115724i
$893$	−6028.88	−0.225922
$894$	62870.5	2.35202
$895$	27843.0i	1.03987i
$896$	−1274.75	−0.0475296
$897$	0	0
$898$	30493.5	1.13317
$899$	− 2402.62i	− 0.0891343i
$900$	11907.9	0.441033
$901$	−14667.2	−0.542327
$902$	− 5581.26i	− 0.206026i
$903$	− 4974.27i	− 0.183315i
$904$	− 24117.9i	− 0.887334i
$905$	− 31194.8i	− 1.14580i
$906$	−10729.7	−0.393457
$907$	−17912.0	−0.655741	−0.327871	−	0.944723i	$-0.606331\pi$
$907$	−17912.0	−0.655741	−0.327871	+	0.944723i	$0.606331\pi$
$908$	16068.7i	0.587288i
$909$	−37746.3	−1.37730
$910$	0	0
$911$	51246.0	1.86373	0.931864	−	0.362807i	$-0.118181\pi$
$911$	51246.0	1.86373	0.931864	+	0.362807i	$0.118181\pi$
$912$	26083.3i	0.947045i
$913$	−5040.74	−0.182721
$914$	−28235.5	−1.02182
$915$	− 7825.24i	− 0.282726i
$916$	9138.67i	0.329640i
$917$	16039.9i	0.577627i
$918$	39772.0i	1.42993i
$919$	25496.5	0.915182	0.457591	−	0.889163i	$-0.348712\pi$
$919$	25496.5	0.915182	0.457591	+	0.889163i	$0.348712\pi$
$920$	−21602.4	−0.774140
$921$	2461.19i	0.0880552i
$922$	−20396.9	−0.728564
$923$	0	0
$924$	−2398.43	−0.0853924
$925$	23863.4i	0.848243i
$926$	15396.7	0.546401
$927$	−77366.2	−2.74114
$928$	− 1808.26i	− 0.0639646i
$929$	45048.6i	1.59095i	0.605983	+	0.795477i	$0.292780\pi$
$929$	45048.6i	1.59095i	−0.605983	+	0.795477i	$0.707220\pi$
$930$	− 30497.2i	− 1.07531i
$931$	23970.6i	0.843830i
$932$	7207.51	0.253315
$933$	24968.3	0.876127
$934$	− 6506.16i	− 0.227931i
$935$	−3551.82	−0.124232
$936$	0	0
$937$	−2280.50	−0.0795099	−0.0397550	−	0.999209i	$-0.512658\pi$
$937$	−2280.50	−0.0795099	−0.0397550	+	0.999209i	$0.512658\pi$
$938$	12613.3i	0.439060i
$939$	6784.03	0.235770
$940$	1712.06	0.0594054
$941$	31174.8i	1.07999i	0.841669	+	0.539994i	$0.181573\pi$
$941$	31174.8i	1.07999i	−0.841669	+	0.539994i	$0.818427\pi$
$942$	− 1102.18i	− 0.0381220i
$943$	− 27649.7i	− 0.954823i
$944$	4486.98i	0.154702i
$945$	27430.7	0.944256
$946$	1319.71	0.0453568
$947$	28394.0i	0.974318i	0.873313	+	0.487159i	$0.161967\pi$
$947$	28394.0i	0.974318i	−0.873313	+	0.487159i	$0.838033\pi$
$948$	−11362.4	−0.389277
$949$	0	0
$950$	−11269.4	−0.384871
$951$	− 56029.2i	− 1.91049i
$952$	−9166.83	−0.312078
$953$	33916.8	1.15286	0.576429	−	0.817147i	$-0.304446\pi$
$953$	33916.8	1.15286	0.576429	+	0.817147i	$0.304446\pi$
$954$	49826.0i	1.69096i
$955$	− 10385.9i	− 0.351915i
$956$	− 2378.57i	− 0.0804693i
$957$	1327.80i	0.0448501i
$958$	−38073.2	−1.28402
$959$	6216.19	0.209313
$960$	− 42470.3i	− 1.42784i
$961$	519.228	0.0174290
$962$	0	0
$963$	−105874.	−3.54284
$964$	− 10679.1i	− 0.356794i
$965$	18156.2	0.605666
$966$	19426.7	0.647044
$967$	− 10792.9i	− 0.358920i	−0.983765	−	0.179460i	$-0.942565\pi$
$967$	− 10792.9i	− 0.358920i	0.983765	−	0.179460i	$-0.0574350\pi$
$968$	30413.9i	1.00986i
$969$	38168.1i	1.26536i
$970$	− 6350.19i	− 0.210198i
$971$	7431.13	0.245599	0.122799	−	0.992432i	$-0.460813\pi$
$971$	7431.13	0.245599	0.122799	+	0.992432i	$0.460813\pi$
$972$	−28265.8	−0.932744
$973$	− 23560.9i	− 0.776288i
$974$	30775.1	1.01242
$975$	0	0
$976$	2982.54	0.0978164
$977$	12422.6i	0.406790i	0.979097	+	0.203395i	$0.0651976\pi$
$977$	12422.6i	0.406790i	−0.979097	+	0.203395i	$0.934802\pi$
$978$	16329.1	0.533891
$979$	6626.56	0.216329
$980$	− 6807.08i	− 0.221882i
$981$	− 4874.47i	− 0.158644i
$982$	26398.2i	0.857840i
$983$	38791.6i	1.25866i	0.777140	+	0.629328i	$0.216670\pi$
$983$	38791.6i	1.25866i	−0.777140	+	0.629328i	$0.783330\pi$
$984$	61910.8	2.00574
$985$	−35759.3	−1.15674
$986$	1396.11i	0.0450926i
$987$	−5596.53	−0.180486
$988$	0	0
$989$	6537.89	0.210205
$990$	12065.9i	0.387352i
$991$	−1066.51	−0.0341863	−0.0170932	−	0.999854i	$-0.505441\pi$
$991$	−1066.51	−0.0341863	−0.0170932	+	0.999854i	$0.505441\pi$
$992$	−22030.6	−0.705113
$993$	41380.8i	1.32244i
$994$	14645.4i	0.467329i
$995$	8950.66i	0.285181i
$996$	− 15382.4i	− 0.489369i
$997$	−30651.6	−0.973665	−0.486833	−	0.873495i	$-0.661848\pi$
$997$	−30651.6	−0.973665	−0.486833	+	0.873495i	$0.661848\pi$
$998$	11640.0	0.369198
$999$	− 165603.i	− 5.24470i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.g.168.13		18
13.2	odd	12		169.4.c.k.22.4		18
13.3	even	3		169.4.e.h.147.6		36
13.4	even	6		169.4.e.h.23.6		36
13.5	odd	4		169.4.a.l.1.6	yes	9
13.6	odd	12		169.4.c.k.146.4		18
13.7	odd	12		169.4.c.l.146.6		18
13.8	odd	4		169.4.a.k.1.4	✓	9
13.9	even	3		169.4.e.h.23.13		36
13.10	even	6		169.4.e.h.147.13		36
13.11	odd	12		169.4.c.l.22.6		18
13.12	even	2	inner	169.4.b.g.168.6		18
39.5	even	4		1521.4.a.bg.1.4		9
39.8	even	4		1521.4.a.bh.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.4	✓	9	13.8	odd	4
169.4.a.l.1.6	yes	9	13.5	odd	4
169.4.b.g.168.6		18	13.12	even	2	inner
169.4.b.g.168.13		18	1.1	even	1	trivial
169.4.c.k.22.4		18	13.2	odd	12
169.4.c.k.146.4		18	13.6	odd	12
169.4.c.l.22.6		18	13.11	odd	12
169.4.c.l.146.6		18	13.7	odd	12
169.4.e.h.23.6		36	13.4	even	6
169.4.e.h.23.13		36	13.9	even	3
169.4.e.h.147.6		36	13.3	even	3
169.4.e.h.147.13		36	13.10	even	6
1521.4.a.bg.1.4		9	39.5	even	4
1521.4.a.bh.1.6		9	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)

\(f(q)\)	\(=\)	\(q+2.22799i q^{2} -9.74867 q^{3} +3.03607 q^{4} -8.20685i q^{5} -21.7199i q^{6} -8.35495i q^{7} +24.5882i q^{8} +68.0366 q^{9} +O(q^{10})\)	\(q+2.22799i q^{2} -9.74867 q^{3} +3.03607 q^{4} -8.20685i q^{5} -21.7199i q^{6} -8.35495i q^{7} +24.5882i q^{8} +68.0366 q^{9} +18.2848 q^{10} +9.69898i q^{11} -29.5976 q^{12} +18.6147 q^{14} +80.0059i q^{15} -30.4937 q^{16} -44.6219 q^{17} +151.585i q^{18} +87.7418i q^{19} -24.9166i q^{20} +81.4497i q^{21} -21.6092 q^{22} -107.053 q^{23} -239.703i q^{24} +57.6475 q^{25} -400.052 q^{27} -25.3662i q^{28} -14.0430 q^{29} -178.252 q^{30} +171.090i q^{31} +128.766i q^{32} -94.5522i q^{33} -99.4170i q^{34} -68.5678 q^{35} +206.564 q^{36} +413.954i q^{37} -195.488 q^{38} +201.792 q^{40} +258.282i q^{41} -181.469 q^{42} -61.0718 q^{43} +29.4468i q^{44} -558.367i q^{45} -238.512i q^{46} +68.7115i q^{47} +297.273 q^{48} +273.195 q^{49} +128.438i q^{50} +435.004 q^{51} +328.701 q^{53} -891.312i q^{54} +79.5981 q^{55} +205.433 q^{56} -855.367i q^{57} -31.2876i q^{58} -147.144i q^{59} +242.904i q^{60} -97.8083 q^{61} -381.186 q^{62} -568.442i q^{63} -530.839 q^{64} +210.661 q^{66} +677.597i q^{67} -135.475 q^{68} +1043.62 q^{69} -152.768i q^{70} +786.767i q^{71} +1672.90i q^{72} -997.675i q^{73} -922.285 q^{74} -561.987 q^{75} +266.390i q^{76} +81.0345 q^{77} +383.897 q^{79} +250.258i q^{80} +2062.99 q^{81} -575.448 q^{82} +519.718i q^{83} +247.287i q^{84} +366.205i q^{85} -136.067i q^{86} +136.901 q^{87} -238.481 q^{88} -683.222i q^{89} +1244.03 q^{90} -325.019 q^{92} -1667.90i q^{93} -153.088 q^{94} +720.085 q^{95} -1255.30i q^{96} -347.294i q^{97} +608.675i q^{98} +659.886i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9}+O(q^{10}) \) 18 * q + 2 * q^3 - 74 * q^4 + 132 * q^9	\( 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9} + 294 q^{10} - 78 q^{12} - 294 q^{14} + 538 q^{16} + 110 q^{17} + 680 q^{22} + 408 q^{23} - 614 q^{25} - 1336 q^{27} + 560 q^{29} - 1042 q^{30} + 40 q^{35} + 1818 q^{36} + 1478 q^{38} + 26 q^{40} - 8 q^{42} + 1066 q^{43} - 264 q^{48} - 806 q^{49} - 940 q^{51} - 556 q^{53} - 500 q^{55} - 500 q^{56} - 272 q^{61} - 4070 q^{62} - 568 q^{64} + 6558 q^{66} - 3072 q^{68} + 4100 q^{69} - 3980 q^{74} - 4786 q^{75} + 1436 q^{77} + 824 q^{79} - 1670 q^{81} - 5514 q^{82} - 1572 q^{87} + 1272 q^{88} + 2560 q^{90} + 8020 q^{92} - 5062 q^{94} + 3228 q^{95}+O(q^{100}) \) 18 * q + 2 * q^3 - 74 * q^4 + 132 * q^9 + 294 * q^10 - 78 * q^12 - 294 * q^14 + 538 * q^16 + 110 * q^17 + 680 * q^22 + 408 * q^23 - 614 * q^25 - 1336 * q^27 + 560 * q^29 - 1042 * q^30 + 40 * q^35 + 1818 * q^36 + 1478 * q^38 + 26 * q^40 - 8 * q^42 + 1066 * q^43 - 264 * q^48 - 806 * q^49 - 940 * q^51 - 556 * q^53 - 500 * q^55 - 500 * q^56 - 272 * q^61 - 4070 * q^62 - 568 * q^64 + 6558 * q^66 - 3072 * q^68 + 4100 * q^69 - 3980 * q^74 - 4786 * q^75 + 1436 * q^77 + 824 * q^79 - 1670 * q^81 - 5514 * q^82 - 1572 * q^87 + 1272 * q^88 + 2560 * q^90 + 8020 * q^92 - 5062 * q^94 + 3228 * q^95

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