LMFDB - Embedded newform 169.4.b.a.168.1

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:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.a.168.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000i q^{2} -7.00000 q^{3} -17.0000 q^{4} -7.00000i q^{5} +35.0000i q^{6} +13.0000i q^{7} +45.0000i q^{8} +22.0000 q^{9} +O(q^{10})$	\(q-5.00000i q^{2} -7.00000 q^{3} -17.0000 q^{4} -7.00000i q^{5} +35.0000i q^{6} +13.0000i q^{7} +45.0000i q^{8} +22.0000 q^{9} -35.0000 q^{10} +26.0000i q^{11} +119.000 q^{12} +65.0000 q^{14} +49.0000i q^{15} +89.0000 q^{16} -77.0000 q^{17} -110.000i q^{18} -126.000i q^{19} +119.000i q^{20} -91.0000i q^{21} +130.000 q^{22} +96.0000 q^{23} -315.000i q^{24} +76.0000 q^{25} +35.0000 q^{27} -221.000i q^{28} -82.0000 q^{29} +245.000 q^{30} +196.000i q^{31} -85.0000i q^{32} -182.000i q^{33} +385.000i q^{34} +91.0000 q^{35} -374.000 q^{36} +131.000i q^{37} -630.000 q^{38} +315.000 q^{40} +336.000i q^{41} -455.000 q^{42} +201.000 q^{43} -442.000i q^{44} -154.000i q^{45} -480.000i q^{46} +105.000i q^{47} -623.000 q^{48} +174.000 q^{49} -380.000i q^{50} +539.000 q^{51} -432.000 q^{53} -175.000i q^{54} +182.000 q^{55} -585.000 q^{56} +882.000i q^{57} +410.000i q^{58} +294.000i q^{59} -833.000i q^{60} -56.0000 q^{61} +980.000 q^{62} +286.000i q^{63} +287.000 q^{64} -910.000 q^{66} +478.000i q^{67} +1309.00 q^{68} -672.000 q^{69} -455.000i q^{70} +9.00000i q^{71} +990.000i q^{72} -98.0000i q^{73} +655.000 q^{74} -532.000 q^{75} +2142.00i q^{76} -338.000 q^{77} +1304.00 q^{79} -623.000i q^{80} -839.000 q^{81} +1680.00 q^{82} -308.000i q^{83} +1547.00i q^{84} +539.000i q^{85} -1005.00i q^{86} +574.000 q^{87} -1170.00 q^{88} +1190.00i q^{89} -770.000 q^{90} -1632.00 q^{92} -1372.00i q^{93} +525.000 q^{94} -882.000 q^{95} +595.000i q^{96} +70.0000i q^{97} -870.000i q^{98} +572.000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} - 34 q^{4} + 44 q^{9}+O(q^{10}) $ 2 * q - 14 * q^3 - 34 * q^4 + 44 * q^9	$ 2 q - 14 q^{3} - 34 q^{4} + 44 q^{9} - 70 q^{10} + 238 q^{12} + 130 q^{14} + 178 q^{16} - 154 q^{17} + 260 q^{22} + 192 q^{23} + 152 q^{25} + 70 q^{27} - 164 q^{29} + 490 q^{30} + 182 q^{35} - 748 q^{36} - 1260 q^{38} + 630 q^{40} - 910 q^{42} + 402 q^{43} - 1246 q^{48} + 348 q^{49} + 1078 q^{51} - 864 q^{53} + 364 q^{55} - 1170 q^{56} - 112 q^{61} + 1960 q^{62} + 574 q^{64} - 1820 q^{66} + 2618 q^{68} - 1344 q^{69} + 1310 q^{74} - 1064 q^{75} - 676 q^{77} + 2608 q^{79} - 1678 q^{81} + 3360 q^{82} + 1148 q^{87} - 2340 q^{88} - 1540 q^{90} - 3264 q^{92} + 1050 q^{94} - 1764 q^{95}+O(q^{100}) $ 2 * q - 14 * q^3 - 34 * q^4 + 44 * q^9 - 70 * q^10 + 238 * q^12 + 130 * q^14 + 178 * q^16 - 154 * q^17 + 260 * q^22 + 192 * q^23 + 152 * q^25 + 70 * q^27 - 164 * q^29 + 490 * q^30 + 182 * q^35 - 748 * q^36 - 1260 * q^38 + 630 * q^40 - 910 * q^42 + 402 * q^43 - 1246 * q^48 + 348 * q^49 + 1078 * q^51 - 864 * q^53 + 364 * q^55 - 1170 * q^56 - 112 * q^61 + 1960 * q^62 + 574 * q^64 - 1820 * q^66 + 2618 * q^68 - 1344 * q^69 + 1310 * q^74 - 1064 * q^75 - 676 * q^77 + 2608 * q^79 - 1678 * q^81 + 3360 * q^82 + 1148 * q^87 - 2340 * q^88 - 1540 * q^90 - 3264 * q^92 + 1050 * q^94 - 1764 * 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 $

$2$	− 5.00000i	− 1.76777i	−0.467707	−	0.883883i	$-0.654920\pi$
$2$	− 5.00000i	− 1.76777i	0.467707	−	0.883883i	$-0.345080\pi$
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	−17.0000	−2.12500
$5$	− 7.00000i	− 0.626099i	−0.949737	−	0.313050i	$-0.898649\pi$
$5$	− 7.00000i	− 0.626099i	0.949737	−	0.313050i	$-0.101351\pi$
$6$	35.0000i	2.38145i
$7$	13.0000i	0.701934i	0.936388	+	0.350967i	$0.114147\pi$
$7$	13.0000i	0.701934i	−0.936388	+	0.350967i	$0.885853\pi$
$8$	45.0000i	1.98874i
$9$	22.0000	0.814815
$10$	−35.0000	−1.10680
$11$	26.0000i	0.712663i	0.934360	+	0.356332i	$0.115973\pi$
$11$	26.0000i	0.712663i	−0.934360	+	0.356332i	$0.884027\pi$
$12$	119.000	2.86270
$13$	0	0
$13$	0	0
$14$	65.0000	1.24086
$15$	49.0000i	0.843450i
$16$	89.0000	1.39062
$17$	−77.0000	−1.09854	−0.549272	−	0.835644i	$-0.685095\pi$
$17$	−77.0000	−1.09854	−0.549272	+	0.835644i	$0.685095\pi$
$18$	− 110.000i	− 1.44040i
$19$	− 126.000i	− 1.52139i	−0.649110	−	0.760694i	$-0.724859\pi$
$19$	− 126.000i	− 1.52139i	0.649110	−	0.760694i	$-0.275141\pi$
$20$	119.000i	1.33046i
$21$	− 91.0000i	− 0.945611i
$22$	130.000	1.25982
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	− 315.000i	− 2.67913i
$25$	76.0000	0.608000
$26$	0	0
$27$	35.0000	0.249472
$28$	− 221.000i	− 1.49161i
$29$	−82.0000	−0.525070	−0.262535	−	0.964923i	$-0.584558\pi$
$29$	−82.0000	−0.525070	−0.262535	+	0.964923i	$0.584558\pi$
$30$	245.000	1.49102
$31$	196.000i	1.13557i	0.823177	+	0.567785i	$0.192199\pi$
$31$	196.000i	1.13557i	−0.823177	+	0.567785i	$0.807801\pi$
$32$	− 85.0000i	− 0.469563i
$33$	− 182.000i	− 0.960065i
$34$	385.000i	1.94197i
$35$	91.0000	0.439480
$36$	−374.000	−1.73148
$37$	131.000i	0.582061i	0.956714	+	0.291031i	$0.0939982\pi$
$37$	131.000i	0.582061i	−0.956714	+	0.291031i	$0.906002\pi$
$38$	−630.000	−2.68946
$39$	0	0
$40$	315.000	1.24515
$41$	336.000i	1.27986i	0.768432	+	0.639932i	$0.221037\pi$
$41$	336.000i	1.27986i	−0.768432	+	0.639932i	$0.778963\pi$
$42$	−455.000	−1.67162
$43$	201.000	0.712842	0.356421	−	0.934325i	$-0.383997\pi$
$43$	201.000	0.712842	0.356421	+	0.934325i	$0.383997\pi$
$44$	− 442.000i	− 1.51441i
$45$	− 154.000i	− 0.510155i
$46$	− 480.000i	− 1.53852i
$47$	105.000i	0.325869i	0.986637	+	0.162934i	$0.0520959\pi$
$47$	105.000i	0.325869i	−0.986637	+	0.162934i	$0.947904\pi$
$48$	−623.000	−1.87338
$49$	174.000	0.507289
$50$	− 380.000i	− 1.07480i
$51$	539.000	1.47990
$52$	0	0
$53$	−432.000	−1.11962	−0.559809	−	0.828622i	$-0.689126\pi$
$53$	−432.000	−1.11962	−0.559809	+	0.828622i	$0.689126\pi$
$54$	− 175.000i	− 0.441009i
$55$	182.000	0.446198
$56$	−585.000	−1.39596
$57$	882.000i	2.04954i
$58$	410.000i	0.928201i
$59$	294.000i	0.648738i	0.945931	+	0.324369i	$0.105152\pi$
$59$	294.000i	0.648738i	−0.945931	+	0.324369i	$0.894848\pi$
$60$	− 833.000i	− 1.79233i
$61$	−56.0000	−0.117542	−0.0587710	−	0.998271i	$-0.518718\pi$
$61$	−56.0000	−0.117542	−0.0587710	+	0.998271i	$0.518718\pi$
$62$	980.000	2.00742
$63$	286.000i	0.571946i
$64$	287.000	0.560547
$65$	0	0
$66$	−910.000	−1.69717
$67$	478.000i	0.871597i	0.900044	+	0.435798i	$0.143534\pi$
$67$	478.000i	0.871597i	−0.900044	+	0.435798i	$0.856466\pi$
$68$	1309.00	2.33441
$69$	−672.000	−1.17245
$70$	− 455.000i	− 0.776899i
$71$	9.00000i	0.0150437i	0.999972	+	0.00752186i	$0.00239430\pi$
$71$	9.00000i	0.0150437i	−0.999972	+	0.00752186i	$0.997606\pi$
$72$	990.000i	1.62045i
$73$	− 98.0000i	− 0.157124i	−0.996909	−	0.0785619i	$-0.974967\pi$
$73$	− 98.0000i	− 0.157124i	0.996909	−	0.0785619i	$-0.0250328\pi$
$74$	655.000	1.02895
$75$	−532.000	−0.819068
$76$	2142.00i	3.23295i
$77$	−338.000	−0.500243
$78$	0	0
$79$	1304.00	1.85711	0.928554	−	0.371198i	$-0.121053\pi$
$79$	1304.00	1.85711	0.928554	+	0.371198i	$0.121053\pi$
$80$	− 623.000i	− 0.870669i
$81$	−839.000	−1.15089
$82$	1680.00	2.26250
$83$	− 308.000i	− 0.407318i	−0.979042	−	0.203659i	$-0.934717\pi$
$83$	− 308.000i	− 0.407318i	0.979042	−	0.203659i	$-0.0652834\pi$
$84$	1547.00i	2.00942i
$85$	539.000i	0.687797i
$86$	− 1005.00i	− 1.26014i
$87$	574.000	0.707348
$88$	−1170.00	−1.41730
$89$	1190.00i	1.41730i	0.705560	+	0.708650i	$0.250696\pi$
$89$	1190.00i	1.41730i	−0.705560	+	0.708650i	$0.749304\pi$
$90$	−770.000	−0.901835
$91$	0	0
$92$	−1632.00	−1.84943
$93$	− 1372.00i	− 1.52978i
$94$	525.000	0.576060
$95$	−882.000	−0.952540
$96$	595.000i	0.632572i
$97$	70.0000i	0.0732724i	0.999329	+	0.0366362i	$0.0116643\pi$
$97$	70.0000i	0.0732724i	−0.999329	+	0.0366362i	$0.988336\pi$
$98$	− 870.000i	− 0.896768i
$99$	572.000i	0.580689i
$100$	−1292.00	−1.29200
$101$	−420.000	−0.413778	−0.206889	−	0.978364i	$-0.566334\pi$
$101$	−420.000	−0.413778	−0.206889	+	0.978364i	$0.566334\pi$
$102$	− 2695.00i	− 2.61613i
$103$	−588.000	−0.562499	−0.281249	−	0.959635i	$-0.590749\pi$
$103$	−588.000	−0.562499	−0.281249	+	0.959635i	$0.590749\pi$
$104$	0	0
$105$	−637.000	−0.592046
$106$	2160.00i	1.97922i
$107$	−684.000	−0.617989	−0.308994	−	0.951064i	$-0.599992\pi$
$107$	−684.000	−0.617989	−0.308994	+	0.951064i	$0.599992\pi$
$108$	−595.000	−0.530129
$109$	373.000i	0.327770i	0.986479	+	0.163885i	$0.0524026\pi$
$109$	373.000i	0.327770i	−0.986479	+	0.163885i	$0.947597\pi$
$110$	− 910.000i	− 0.788774i
$111$	− 917.000i	− 0.784124i
$112$	1157.00i	0.976127i
$113$	−1734.00	−1.44355	−0.721774	−	0.692128i	$-0.756673\pi$
$113$	−1734.00	−1.44355	−0.721774	+	0.692128i	$0.756673\pi$
$114$	4410.00	3.62311
$115$	− 672.000i	− 0.544907i
$116$	1394.00	1.11577
$117$	0	0
$118$	1470.00	1.14682
$119$	− 1001.00i	− 0.771105i
$120$	−2205.00	−1.67740
$121$	655.000	0.492111
$122$	280.000i	0.207787i
$123$	− 2352.00i	− 1.72417i
$124$	− 3332.00i	− 2.41308i
$125$	− 1407.00i	− 1.00677i
$126$	1430.00	1.01107
$127$	−1892.00	−1.32195	−0.660976	−	0.750407i	$-0.729857\pi$
$127$	−1892.00	−1.32195	−0.660976	+	0.750407i	$0.729857\pi$
$128$	− 2115.00i	− 1.46048i
$129$	−1407.00	−0.960306
$130$	0	0
$131$	1435.00	0.957073	0.478536	−	0.878068i	$-0.341167\pi$
$131$	1435.00	0.957073	0.478536	+	0.878068i	$0.341167\pi$
$132$	3094.00i	2.04014i
$133$	1638.00	1.06791
$134$	2390.00	1.54078
$135$	− 245.000i	− 0.156194i
$136$	− 3465.00i	− 2.18472i
$137$	1776.00i	1.10755i	0.832667	+	0.553773i	$0.186813\pi$
$137$	1776.00i	1.10755i	−0.832667	+	0.553773i	$0.813187\pi$
$138$	3360.00i	2.07262i
$139$	−1869.00	−1.14048	−0.570239	−	0.821479i	$-0.693150\pi$
$139$	−1869.00	−1.14048	−0.570239	+	0.821479i	$0.693150\pi$
$140$	−1547.00	−0.933895
$141$	− 735.000i	− 0.438994i
$142$	45.0000	0.0265938
$143$	0	0
$144$	1958.00	1.13310
$145$	574.000i	0.328746i
$146$	−490.000	−0.277758
$147$	−1218.00	−0.683394
$148$	− 2227.00i	− 1.23688i
$149$	2466.00i	1.35586i	0.735128	+	0.677928i	$0.237122\pi$
$149$	2466.00i	1.35586i	−0.735128	+	0.677928i	$0.762878\pi$
$150$	2660.00i	1.44792i
$151$	3323.00i	1.79087i	0.445189	+	0.895437i	$0.353137\pi$
$151$	3323.00i	1.79087i	−0.445189	+	0.895437i	$0.646863\pi$
$152$	5670.00	3.02564
$153$	−1694.00	−0.895110
$154$	1690.00i	0.884312i
$155$	1372.00	0.710979
$156$	0	0
$157$	−2730.00	−1.38776	−0.693878	−	0.720092i	$-0.744099\pi$
$157$	−2730.00	−1.38776	−0.693878	+	0.720092i	$0.744099\pi$
$158$	− 6520.00i	− 3.28293i
$159$	3024.00	1.50829
$160$	−595.000	−0.293993
$161$	1248.00i	0.610908i
$162$	4195.00i	2.03451i
$163$	544.000i	0.261407i	0.991421	+	0.130704i	$0.0417236\pi$
$163$	544.000i	0.261407i	−0.991421	+	0.130704i	$0.958276\pi$
$164$	− 5712.00i	− 2.71971i
$165$	−1274.00	−0.601096
$166$	−1540.00	−0.720043
$167$	− 1624.00i	− 0.752508i	−0.926516	−	0.376254i	$-0.877212\pi$
$167$	− 1624.00i	− 0.752508i	0.926516	−	0.376254i	$-0.122788\pi$
$168$	4095.00	1.88057
$169$	0	0
$170$	2695.00	1.21587
$171$	− 2772.00i	− 1.23965i
$172$	−3417.00	−1.51479
$173$	336.000	0.147662	0.0738312	−	0.997271i	$-0.476477\pi$
$173$	336.000	0.147662	0.0738312	+	0.997271i	$0.476477\pi$
$174$	− 2870.00i	− 1.25043i
$175$	988.000i	0.426776i
$176$	2314.00i	0.991047i
$177$	− 2058.00i	− 0.873948i
$178$	5950.00	2.50546
$179$	3029.00	1.26479	0.632397	−	0.774645i	$-0.282071\pi$
$179$	3029.00	1.26479	0.632397	+	0.774645i	$0.282071\pi$
$180$	2618.00i	1.08408i
$181$	28.0000	0.0114985	0.00574924	−	0.999983i	$-0.498170\pi$
$181$	28.0000	0.0114985	0.00574924	+	0.999983i	$0.498170\pi$
$182$	0	0
$183$	392.000	0.158347
$184$	4320.00i	1.73084i
$185$	917.000	0.364428
$186$	−6860.00	−2.70430
$187$	− 2002.00i	− 0.782892i
$188$	− 1785.00i	− 0.692471i
$189$	455.000i	0.175113i
$190$	4410.00i	1.68387i
$191$	422.000	0.159868	0.0799342	−	0.996800i	$-0.474529\pi$
$191$	422.000	0.159868	0.0799342	+	0.996800i	$0.474529\pi$
$192$	−2009.00	−0.755141
$193$	− 492.000i	− 0.183497i	−0.995782	−	0.0917485i	$-0.970754\pi$
$193$	− 492.000i	− 0.183497i	0.995782	−	0.0917485i	$-0.0292456\pi$
$194$	350.000	0.129529
$195$	0	0
$196$	−2958.00	−1.07799
$197$	2991.00i	1.08173i	0.841111	+	0.540863i	$0.181902\pi$
$197$	2991.00i	1.08173i	−0.841111	+	0.540863i	$0.818098\pi$
$198$	2860.00	1.02652
$199$	70.0000	0.0249355	0.0124678	−	0.999922i	$-0.496031\pi$
$199$	70.0000	0.0249355	0.0124678	+	0.999922i	$0.496031\pi$
$200$	3420.00i	1.20915i
$201$	− 3346.00i	− 1.17417i
$202$	2100.00i	0.731463i
$203$	− 1066.00i	− 0.368564i
$204$	−9163.00	−3.14480
$205$	2352.00	0.801321
$206$	2940.00i	0.994367i
$207$	2112.00	0.709150
$208$	0	0
$209$	3276.00	1.08424
$210$	3185.00i	1.04660i
$211$	2851.00	0.930194	0.465097	−	0.885260i	$-0.346019\pi$
$211$	2851.00	0.930194	0.465097	+	0.885260i	$0.346019\pi$
$212$	7344.00	2.37919
$213$	− 63.0000i	− 0.0202661i
$214$	3420.00i	1.09246i
$215$	− 1407.00i	− 0.446310i
$216$	1575.00i	0.496135i
$217$	−2548.00	−0.797095
$218$	1865.00	0.579421
$219$	686.000i	0.211669i
$220$	−3094.00	−0.948170
$221$	0	0
$222$	−4585.00	−1.38615
$223$	217.000i	0.0651632i	0.999469	+	0.0325816i	$0.0103729\pi$
$223$	217.000i	0.0651632i	−0.999469	+	0.0325816i	$0.989627\pi$
$224$	1105.00	0.329602
$225$	1672.00	0.495407
$226$	8670.00i	2.55186i
$227$	− 2576.00i	− 0.753194i	−0.926377	−	0.376597i	$-0.877094\pi$
$227$	− 2576.00i	− 0.753194i	0.926377	−	0.376597i	$-0.122906\pi$
$228$	− 14994.0i	− 4.35527i
$229$	− 455.000i	− 0.131298i	−0.997843	−	0.0656490i	$-0.979088\pi$
$229$	− 455.000i	− 0.131298i	0.997843	−	0.0656490i	$-0.0209118\pi$
$230$	−3360.00	−0.963269
$231$	2366.00	0.673902
$232$	− 3690.00i	− 1.04423i
$233$	−3061.00	−0.860656	−0.430328	−	0.902673i	$-0.641602\pi$
$233$	−3061.00	−0.860656	−0.430328	+	0.902673i	$0.641602\pi$
$234$	0	0
$235$	735.000	0.204026
$236$	− 4998.00i	− 1.37857i
$237$	−9128.00	−2.50180
$238$	−5005.00	−1.36313
$239$	− 3477.00i	− 0.941039i	−0.882389	−	0.470520i	$-0.844066\pi$
$239$	− 3477.00i	− 0.941039i	0.882389	−	0.470520i	$-0.155934\pi$
$240$	4361.00i	1.17292i
$241$	1610.00i	0.430329i	0.976578	+	0.215164i	$0.0690287\pi$
$241$	1610.00i	0.430329i	−0.976578	+	0.215164i	$0.930971\pi$
$242$	− 3275.00i	− 0.869938i
$243$	4928.00	1.30095
$244$	952.000	0.249777
$245$	− 1218.00i	− 0.317613i
$246$	−11760.0	−3.04793
$247$	0	0
$248$	−8820.00	−2.25835
$249$	2156.00i	0.548719i
$250$	−7035.00	−1.77973
$251$	−1008.00	−0.253484	−0.126742	−	0.991936i	$-0.540452\pi$
$251$	−1008.00	−0.253484	−0.126742	+	0.991936i	$0.540452\pi$
$252$	− 4862.00i	− 1.21539i
$253$	2496.00i	0.620246i
$254$	9460.00i	2.33690i
$255$	− 3773.00i	− 0.926566i
$256$	−8279.00	−2.02124
$257$	−6041.00	−1.46625	−0.733127	−	0.680092i	$-0.761940\pi$
$257$	−6041.00	−1.46625	−0.733127	+	0.680092i	$0.761940\pi$
$258$	7035.00i	1.69760i
$259$	−1703.00	−0.408569
$260$	0	0
$261$	−1804.00	−0.427834
$262$	− 7175.00i	− 1.69188i
$263$	−3708.00	−0.869373	−0.434686	−	0.900582i	$-0.643141\pi$
$263$	−3708.00	−0.869373	−0.434686	+	0.900582i	$0.643141\pi$
$264$	8190.00	1.90932
$265$	3024.00i	0.700992i
$266$	− 8190.00i	− 1.88782i
$267$	− 8330.00i	− 1.90932i
$268$	− 8126.00i	− 1.85214i
$269$	8344.00	1.89124	0.945618	−	0.325278i	$-0.105458\pi$
$269$	8344.00	1.89124	0.945618	+	0.325278i	$0.105458\pi$
$270$	−1225.00	−0.276115
$271$	1617.00i	0.362457i	0.983441	+	0.181228i	$0.0580073\pi$
$271$	1617.00i	0.362457i	−0.983441	+	0.181228i	$0.941993\pi$
$272$	−6853.00	−1.52766
$273$	0	0
$274$	8880.00	1.95788
$275$	1976.00i	0.433299i
$276$	11424.0	2.49146
$277$	3820.00	0.828598	0.414299	−	0.910141i	$-0.364027\pi$
$277$	3820.00	0.828598	0.414299	+	0.910141i	$0.364027\pi$
$278$	9345.00i	2.01610i
$279$	4312.00i	0.925278i
$280$	4095.00i	0.874011i
$281$	6214.00i	1.31920i	0.751615	+	0.659602i	$0.229275\pi$
$281$	6214.00i	1.31920i	−0.751615	+	0.659602i	$0.770725\pi$
$282$	−3675.00	−0.776039
$283$	5292.00	1.11158	0.555789	−	0.831323i	$-0.312416\pi$
$283$	5292.00	1.11158	0.555789	+	0.831323i	$0.312416\pi$
$284$	− 153.000i	− 0.0319679i
$285$	6174.00	1.28321
$286$	0	0
$287$	−4368.00	−0.898379
$288$	− 1870.00i	− 0.382607i
$289$	1016.00	0.206798
$290$	2870.00	0.581146
$291$	− 490.000i	− 0.0987090i
$292$	1666.00i	0.333888i
$293$	903.000i	0.180047i	0.995940	+	0.0900236i	$0.0286942\pi$
$293$	903.000i	0.180047i	−0.995940	+	0.0900236i	$0.971306\pi$
$294$	6090.00i	1.20808i
$295$	2058.00	0.406174
$296$	−5895.00	−1.15757
$297$	910.000i	0.177790i
$298$	12330.0	2.39684
$299$	0	0
$300$	9044.00	1.74052
$301$	2613.00i	0.500368i
$302$	16615.0	3.16585
$303$	2940.00	0.557421
$304$	− 11214.0i	− 2.11568i
$305$	392.000i	0.0735930i
$306$	8470.00i	1.58235i
$307$	− 2114.00i	− 0.393004i	−0.980503	−	0.196502i	$-0.937042\pi$
$307$	− 2114.00i	− 0.393004i	0.980503	−	0.196502i	$-0.0629583\pi$
$308$	5746.00	1.06302
$309$	4116.00	0.757770
$310$	− 6860.00i	− 1.25684i
$311$	−3402.00	−0.620288	−0.310144	−	0.950690i	$-0.600377\pi$
$311$	−3402.00	−0.620288	−0.310144	+	0.950690i	$0.600377\pi$
$312$	0	0
$313$	−10689.0	−1.93028	−0.965141	−	0.261732i	$-0.915706\pi$
$313$	−10689.0	−1.93028	−0.965141	+	0.261732i	$0.915706\pi$
$314$	13650.0i	2.45323i
$315$	2002.00	0.358095
$316$	−22168.0	−3.94635
$317$	− 7054.00i	− 1.24982i	−0.780698	−	0.624909i	$-0.785136\pi$
$317$	− 7054.00i	− 1.24982i	0.780698	−	0.624909i	$-0.214864\pi$
$318$	− 15120.0i	− 2.66631i
$319$	− 2132.00i	− 0.374198i
$320$	− 2009.00i	− 0.350958i
$321$	4788.00	0.832524
$322$	6240.00	1.07994
$323$	9702.00i	1.67131i
$324$	14263.0	2.44564
$325$	0	0
$326$	2720.00	0.462107
$327$	− 2611.00i	− 0.441555i
$328$	−15120.0	−2.54531
$329$	−1365.00	−0.228738
$330$	6370.00i	1.06260i
$331$	9704.00i	1.61142i	0.592310	+	0.805710i	$0.298216\pi$
$331$	9704.00i	1.61142i	−0.592310	+	0.805710i	$0.701784\pi$
$332$	5236.00i	0.865551i
$333$	2882.00i	0.474272i
$334$	−8120.00	−1.33026
$335$	3346.00	0.545706
$336$	− 8099.00i	− 1.31499i
$337$	10449.0	1.68900	0.844500	−	0.535555i	$-0.179897\pi$
$337$	10449.0	1.68900	0.844500	+	0.535555i	$0.179897\pi$
$338$	0	0
$339$	12138.0	1.94468
$340$	− 9163.00i	− 1.46157i
$341$	−5096.00	−0.809278
$342$	−13860.0	−2.19141
$343$	6721.00i	1.05802i
$344$	9045.00i	1.41766i
$345$	4704.00i	0.734072i
$346$	− 1680.00i	− 0.261033i
$347$	−621.000	−0.0960721	−0.0480361	−	0.998846i	$-0.515296\pi$
$347$	−621.000	−0.0960721	−0.0480361	+	0.998846i	$0.515296\pi$
$348$	−9758.00	−1.50311
$349$	− 12481.0i	− 1.91431i	−0.289584	−	0.957153i	$-0.593517\pi$
$349$	− 12481.0i	− 1.91431i	0.289584	−	0.957153i	$-0.406483\pi$
$350$	4940.00	0.754440
$351$	0	0
$352$	2210.00	0.334640
$353$	− 1400.00i	− 0.211089i	−0.994415	−	0.105545i	$-0.966341\pi$
$353$	− 1400.00i	− 0.211089i	0.994415	−	0.105545i	$-0.0336586\pi$
$354$	−10290.0	−1.54494
$355$	63.0000	0.00941885
$356$	− 20230.0i	− 3.01176i
$357$	7007.00i	1.03879i
$358$	− 15145.0i	− 2.23586i
$359$	4968.00i	0.730365i	0.930936	+	0.365182i	$0.118993\pi$
$359$	4968.00i	0.730365i	−0.930936	+	0.365182i	$0.881007\pi$
$360$	6930.00	1.01456
$361$	−9017.00	−1.31462
$362$	− 140.000i	− 0.0203266i
$363$	−4585.00	−0.662948
$364$	0	0
$365$	−686.000	−0.0983750
$366$	− 1960.00i	− 0.279920i
$367$	8722.00	1.24056	0.620279	−	0.784381i	$-0.287019\pi$
$367$	8722.00	1.24056	0.620279	+	0.784381i	$0.287019\pi$
$368$	8544.00	1.21029
$369$	7392.00i	1.04285i
$370$	− 4585.00i	− 0.644224i
$371$	− 5616.00i	− 0.785898i
$372$	23324.0i	3.25079i
$373$	10012.0	1.38982	0.694908	−	0.719098i	$-0.255445\pi$
$373$	10012.0	1.38982	0.694908	+	0.719098i	$0.255445\pi$
$374$	−10010.0	−1.38397
$375$	9849.00i	1.35627i
$376$	−4725.00	−0.648067
$377$	0	0
$378$	2275.00	0.309559
$379$	− 3372.00i	− 0.457013i	−0.973542	−	0.228507i	$-0.926616\pi$
$379$	− 3372.00i	− 0.457013i	0.973542	−	0.228507i	$-0.0733843\pi$
$380$	14994.0	2.02415
$381$	13244.0	1.78087
$382$	− 2110.00i	− 0.282610i
$383$	− 847.000i	− 0.113002i	−0.998403	−	0.0565009i	$-0.982006\pi$
$383$	− 847.000i	− 0.113002i	0.998403	−	0.0565009i	$-0.0179944\pi$
$384$	14805.0i	1.96749i
$385$	2366.00i	0.313201i
$386$	−2460.00	−0.324380
$387$	4422.00	0.580834
$388$	− 1190.00i	− 0.155704i
$389$	−11314.0	−1.47466	−0.737330	−	0.675533i	$-0.763914\pi$
$389$	−11314.0	−1.47466	−0.737330	+	0.675533i	$0.763914\pi$
$390$	0	0
$391$	−7392.00	−0.956086
$392$	7830.00i	1.00886i
$393$	−10045.0	−1.28932
$394$	14955.0	1.91224
$395$	− 9128.00i	− 1.16273i
$396$	− 9724.00i	− 1.23396i
$397$	− 1862.00i	− 0.235393i	−0.993050	−	0.117697i	$-0.962449\pi$
$397$	− 1862.00i	− 0.235393i	0.993050	−	0.117697i	$-0.0375510\pi$
$398$	− 350.000i	− 0.0440802i
$399$	−11466.0	−1.43864
$400$	6764.00	0.845500
$401$	− 6820.00i	− 0.849313i	−0.905355	−	0.424657i	$-0.860395\pi$
$401$	− 6820.00i	− 0.849313i	0.905355	−	0.424657i	$-0.139605\pi$
$402$	−16730.0	−2.07566
$403$	0	0
$404$	7140.00	0.879278
$405$	5873.00i	0.720572i
$406$	−5330.00	−0.651536
$407$	−3406.00	−0.414814
$408$	24255.0i	2.94314i
$409$	− 12992.0i	− 1.57069i	−0.619057	−	0.785346i	$-0.712485\pi$
$409$	− 12992.0i	− 1.57069i	0.619057	−	0.785346i	$-0.287515\pi$
$410$	− 11760.0i	− 1.41655i
$411$	− 12432.0i	− 1.49203i
$412$	9996.00	1.19531
$413$	−3822.00	−0.455371
$414$	− 10560.0i	− 1.25361i
$415$	−2156.00	−0.255021
$416$	0	0
$417$	13083.0	1.53640
$418$	− 16380.0i	− 1.91668i
$419$	−7343.00	−0.856155	−0.428078	−	0.903742i	$-0.640809\pi$
$419$	−7343.00	−0.856155	−0.428078	+	0.903742i	$0.640809\pi$
$420$	10829.0	1.25810
$421$	− 5059.00i	− 0.585655i	−0.956165	−	0.292827i	$-0.905404\pi$
$421$	− 5059.00i	− 0.585655i	0.956165	−	0.292827i	$-0.0945961\pi$
$422$	− 14255.0i	− 1.64437i
$423$	2310.00i	0.265523i
$424$	− 19440.0i	− 2.22663i
$425$	−5852.00	−0.667915
$426$	−315.000	−0.0358258
$427$	− 728.000i	− 0.0825068i
$428$	11628.0	1.31323
$429$	0	0
$430$	−7035.00	−0.788972
$431$	3243.00i	0.362436i	0.983443	+	0.181218i	$0.0580039\pi$
$431$	3243.00i	0.362436i	−0.983443	+	0.181218i	$0.941996\pi$
$432$	3115.00	0.346922
$433$	−11599.0	−1.28733	−0.643663	−	0.765309i	$-0.722586\pi$
$433$	−11599.0	−1.28733	−0.643663	+	0.765309i	$0.722586\pi$
$434$	12740.0i	1.40908i
$435$	− 4018.00i	− 0.442870i
$436$	− 6341.00i	− 0.696511i
$437$	− 12096.0i	− 1.32410i
$438$	3430.00	0.374182
$439$	17374.0	1.88887	0.944437	−	0.328692i	$-0.106608\pi$
$439$	17374.0	1.88887	0.944437	+	0.328692i	$0.106608\pi$
$440$	8190.00i	0.887370i
$441$	3828.00	0.413346
$442$	0	0
$443$	989.000	0.106070	0.0530348	−	0.998593i	$-0.483111\pi$
$443$	989.000	0.106070	0.0530348	+	0.998593i	$0.483111\pi$
$444$	15589.0i	1.66626i
$445$	8330.00	0.887370
$446$	1085.00	0.115193
$447$	− 17262.0i	− 1.82654i
$448$	3731.00i	0.393467i
$449$	14474.0i	1.52131i	0.649154	+	0.760657i	$0.275123\pi$
$449$	14474.0i	1.52131i	−0.649154	+	0.760657i	$0.724877\pi$
$450$	− 8360.00i	− 0.875765i
$451$	−8736.00	−0.912111
$452$	29478.0	3.06754
$453$	− 23261.0i	− 2.41258i
$454$	−12880.0	−1.33147
$455$	0	0
$456$	−39690.0	−4.07600
$457$	− 1594.00i	− 0.163160i	−0.996667	−	0.0815801i	$-0.974003\pi$
$457$	− 1594.00i	− 0.163160i	0.996667	−	0.0815801i	$-0.0259966\pi$
$458$	−2275.00	−0.232104
$459$	−2695.00	−0.274056
$460$	11424.0i	1.15793i
$461$	− 5915.00i	− 0.597590i	−0.954317	−	0.298795i	$-0.903415\pi$
$461$	− 5915.00i	− 0.597590i	0.954317	−	0.298795i	$-0.0965847\pi$
$462$	− 11830.0i	− 1.19130i
$463$	11072.0i	1.11136i	0.831396	+	0.555680i	$0.187542\pi$
$463$	11072.0i	1.11136i	−0.831396	+	0.555680i	$0.812458\pi$
$464$	−7298.00	−0.730175
$465$	−9604.00	−0.957795
$466$	15305.0i	1.52144i
$467$	−1260.00	−0.124852	−0.0624260	−	0.998050i	$-0.519884\pi$
$467$	−1260.00	−0.124852	−0.0624260	+	0.998050i	$0.519884\pi$
$468$	0	0
$469$	−6214.00	−0.611804
$470$	− 3675.00i	− 0.360670i
$471$	19110.0	1.86952
$472$	−13230.0	−1.29017
$473$	5226.00i	0.508016i
$474$	45640.0i	4.42260i
$475$	− 9576.00i	− 0.925004i
$476$	17017.0i	1.63860i
$477$	−9504.00	−0.912281
$478$	−17385.0	−1.66354
$479$	12033.0i	1.14781i	0.818921	+	0.573906i	$0.194572\pi$
$479$	12033.0i	1.14781i	−0.818921	+	0.573906i	$0.805428\pi$
$480$	4165.00	0.396053
$481$	0	0
$482$	8050.00	0.760721
$483$	− 8736.00i	− 0.822985i
$484$	−11135.0	−1.04574
$485$	490.000	0.0458758
$486$	− 24640.0i	− 2.29978i
$487$	− 2280.00i	− 0.212149i	−0.994358	−	0.106075i	$-0.966172\pi$
$487$	− 2280.00i	− 0.212149i	0.994358	−	0.106075i	$-0.0338282\pi$
$488$	− 2520.00i	− 0.233760i
$489$	− 3808.00i	− 0.352155i
$490$	−6090.00	−0.561466
$491$	−16767.0	−1.54111	−0.770554	−	0.637375i	$-0.780020\pi$
$491$	−16767.0	−1.54111	−0.770554	+	0.637375i	$0.780020\pi$
$492$	39984.0i	3.66386i
$493$	6314.00	0.576812
$494$	0	0
$495$	4004.00	0.363569
$496$	17444.0i	1.57915i
$497$	−117.000	−0.0105597
$498$	10780.0	0.970007
$499$	12840.0i	1.15190i	0.817485	+	0.575949i	$0.195367\pi$
$499$	12840.0i	1.15190i	−0.817485	+	0.575949i	$0.804633\pi$
$500$	23919.0i	2.13938i
$501$	11368.0i	1.01374i
$502$	5040.00i	0.448100i
$503$	−2198.00	−0.194839	−0.0974195	−	0.995243i	$-0.531059\pi$
$503$	−2198.00	−0.194839	−0.0974195	+	0.995243i	$0.531059\pi$
$504$	−12870.0	−1.13745
$505$	2940.00i	0.259066i
$506$	12480.0	1.09645
$507$	0	0
$508$	32164.0	2.80915
$509$	− 17066.0i	− 1.48612i	−0.669223	−	0.743062i	$-0.733373\pi$
$509$	− 17066.0i	− 1.48612i	0.669223	−	0.743062i	$-0.266627\pi$
$510$	−18865.0	−1.63795
$511$	1274.00	0.110290
$512$	24475.0i	2.11260i
$513$	− 4410.00i	− 0.379544i
$514$	30205.0i	2.59200i
$515$	4116.00i	0.352180i
$516$	23919.0	2.04065
$517$	−2730.00	−0.232235
$518$	8515.00i	0.722254i
$519$	−2352.00	−0.198924
$520$	0	0
$521$	2583.00	0.217204	0.108602	−	0.994085i	$-0.465363\pi$
$521$	2583.00	0.217204	0.108602	+	0.994085i	$0.465363\pi$
$522$	9020.00i	0.756312i
$523$	18620.0	1.55678	0.778390	−	0.627781i	$-0.216037\pi$
$523$	18620.0	1.55678	0.778390	+	0.627781i	$0.216037\pi$
$524$	−24395.0	−2.03378
$525$	− 6916.00i	− 0.574931i
$526$	18540.0i	1.53685i
$527$	− 15092.0i	− 1.24747i
$528$	− 16198.0i	− 1.33509i
$529$	−2951.00	−0.242541
$530$	15120.0	1.23919
$531$	6468.00i	0.528601i
$532$	−27846.0	−2.26932
$533$	0	0
$534$	−41650.0	−3.37523
$535$	4788.00i	0.386922i
$536$	−21510.0	−1.73338
$537$	−21203.0	−1.70387
$538$	− 41720.0i	− 3.34327i
$539$	4524.00i	0.361526i
$540$	4165.00i	0.331913i
$541$	16833.0i	1.33772i	0.743388	+	0.668861i	$0.233218\pi$
$541$	16833.0i	1.33772i	−0.743388	+	0.668861i	$0.766782\pi$
$542$	8085.00	0.640739
$543$	−196.000	−0.0154902
$544$	6545.00i	0.515836i
$545$	2611.00	0.205216
$546$	0	0
$547$	−8615.00	−0.673402	−0.336701	−	0.941612i	$-0.609311\pi$
$547$	−8615.00	−0.673402	−0.336701	+	0.941612i	$0.609311\pi$
$548$	− 30192.0i	− 2.35354i
$549$	−1232.00	−0.0957750
$550$	9880.00	0.765972
$551$	10332.0i	0.798835i
$552$	− 30240.0i	− 2.33170i
$553$	16952.0i	1.30357i
$554$	− 19100.0i	− 1.46477i
$555$	−6419.00	−0.490939
$556$	31773.0	2.42352
$557$	− 8535.00i	− 0.649263i	−0.945841	−	0.324632i	$-0.894760\pi$
$557$	− 8535.00i	− 0.649263i	0.945841	−	0.324632i	$-0.105240\pi$
$558$	21560.0	1.63568
$559$	0	0
$560$	8099.00	0.611152
$561$	14014.0i	1.05467i
$562$	31070.0	2.33204
$563$	4641.00	0.347415	0.173708	−	0.984797i	$-0.444425\pi$
$563$	4641.00	0.347415	0.173708	+	0.984797i	$0.444425\pi$
$564$	12495.0i	0.932862i
$565$	12138.0i	0.903804i
$566$	− 26460.0i	− 1.96501i
$567$	− 10907.0i	− 0.807850i
$568$	−405.000	−0.0299180
$569$	4793.00	0.353134	0.176567	−	0.984289i	$-0.443501\pi$
$569$	4793.00	0.353134	0.176567	+	0.984289i	$0.443501\pi$
$570$	− 30870.0i	− 2.26842i
$571$	5563.00	0.407713	0.203857	−	0.979001i	$-0.434652\pi$
$571$	5563.00	0.407713	0.203857	+	0.979001i	$0.434652\pi$
$572$	0	0
$573$	−2954.00	−0.215367
$574$	21840.0i	1.58813i
$575$	7296.00	0.529155
$576$	6314.00	0.456742
$577$	24038.0i	1.73434i	0.498011	+	0.867171i	$0.334064\pi$
$577$	24038.0i	1.73434i	−0.498011	+	0.867171i	$0.665936\pi$
$578$	− 5080.00i	− 0.365571i
$579$	3444.00i	0.247198i
$580$	− 9758.00i	− 0.698584i
$581$	4004.00	0.285910
$582$	−2450.00	−0.174494
$583$	− 11232.0i	− 0.797911i
$584$	4410.00	0.312478
$585$	0	0
$586$	4515.00	0.318281
$587$	− 21224.0i	− 1.49235i	−0.665751	−	0.746174i	$-0.731889\pi$
$587$	− 21224.0i	− 1.49235i	0.665751	−	0.746174i	$-0.268111\pi$
$588$	20706.0	1.45221
$589$	24696.0	1.72764
$590$	− 10290.0i	− 0.718021i
$591$	− 20937.0i	− 1.45725i
$592$	11659.0i	0.809429i
$593$	− 4354.00i	− 0.301513i	−0.988571	−	0.150757i	$-0.951829\pi$
$593$	− 4354.00i	− 0.301513i	0.988571	−	0.150757i	$-0.0481710\pi$
$594$	4550.00	0.314291
$595$	−7007.00	−0.482788
$596$	− 41922.0i	− 2.88119i
$597$	−490.000	−0.0335919
$598$	0	0
$599$	7310.00	0.498629	0.249314	−	0.968423i	$-0.419795\pi$
$599$	7310.00	0.498629	0.249314	+	0.968423i	$0.419795\pi$
$600$	− 23940.0i	− 1.62891i
$601$	−7595.00	−0.515485	−0.257743	−	0.966214i	$-0.582979\pi$
$601$	−7595.00	−0.515485	−0.257743	+	0.966214i	$0.582979\pi$
$602$	13065.0	0.884534
$603$	10516.0i	0.710190i
$604$	− 56491.0i	− 3.80561i
$605$	− 4585.00i	− 0.308110i
$606$	− 14700.0i	− 0.985391i
$607$	−826.000	−0.0552328	−0.0276164	−	0.999619i	$-0.508792\pi$
$607$	−826.000	−0.0552328	−0.0276164	+	0.999619i	$0.508792\pi$
$608$	−10710.0	−0.714388
$609$	7462.00i	0.496511i
$610$	1960.00	0.130095
$611$	0	0
$612$	28798.0	1.90211
$613$	14590.0i	0.961312i	0.876909	+	0.480656i	$0.159602\pi$
$613$	14590.0i	0.961312i	−0.876909	+	0.480656i	$0.840398\pi$
$614$	−10570.0	−0.694740
$615$	−16464.0	−1.07950
$616$	− 15210.0i	− 0.994851i
$617$	4888.00i	0.318936i	0.987203	+	0.159468i	$0.0509779\pi$
$617$	4888.00i	0.318936i	−0.987203	+	0.159468i	$0.949022\pi$
$618$	− 20580.0i	− 1.33956i
$619$	11004.0i	0.714520i	0.934005	+	0.357260i	$0.116289\pi$
$619$	11004.0i	0.714520i	−0.934005	+	0.357260i	$0.883711\pi$
$620$	−23324.0	−1.51083
$621$	3360.00	0.217121
$622$	17010.0i	1.09653i
$623$	−15470.0	−0.994851
$624$	0	0
$625$	−349.000	−0.0223360
$626$	53445.0i	3.41229i
$627$	−22932.0	−1.46063
$628$	46410.0	2.94898
$629$	− 10087.0i	− 0.639420i
$630$	− 10010.0i	− 0.633028i
$631$	4975.00i	0.313869i	0.987609	+	0.156935i	$0.0501612\pi$
$631$	4975.00i	0.313869i	−0.987609	+	0.156935i	$0.949839\pi$
$632$	58680.0i	3.69330i
$633$	−19957.0	−1.25311
$634$	−35270.0	−2.20939
$635$	13244.0i	0.827673i
$636$	−51408.0	−3.20513
$637$	0	0
$638$	−10660.0	−0.661494
$639$	198.000i	0.0122578i
$640$	−14805.0	−0.914405
$641$	−3950.00	−0.243394	−0.121697	−	0.992567i	$-0.538834\pi$
$641$	−3950.00	−0.243394	−0.121697	+	0.992567i	$0.538834\pi$
$642$	− 23940.0i	− 1.47171i
$643$	− 3682.00i	− 0.225823i	−0.993605	−	0.112911i	$-0.963982\pi$
$643$	− 3682.00i	− 0.225823i	0.993605	−	0.112911i	$-0.0360176\pi$
$644$	− 21216.0i	− 1.29818i
$645$	9849.00i	0.601247i
$646$	48510.0	2.95449
$647$	−10402.0	−0.632063	−0.316032	−	0.948749i	$-0.602351\pi$
$647$	−10402.0	−0.632063	−0.316032	+	0.948749i	$0.602351\pi$
$648$	− 37755.0i	− 2.28882i
$649$	−7644.00	−0.462332
$650$	0	0
$651$	17836.0	1.07381
$652$	− 9248.00i	− 0.555490i
$653$	−31680.0	−1.89852	−0.949260	−	0.314491i	$-0.898166\pi$
$653$	−31680.0	−1.89852	−0.949260	+	0.314491i	$0.898166\pi$
$654$	−13055.0	−0.780567
$655$	− 10045.0i	− 0.599222i
$656$	29904.0i	1.77981i
$657$	− 2156.00i	− 0.128027i
$658$	6825.00i	0.404356i
$659$	21940.0	1.29691	0.648453	−	0.761255i	$-0.275416\pi$
$659$	21940.0	1.29691	0.648453	+	0.761255i	$0.275416\pi$
$660$	21658.0	1.27733
$661$	31374.0i	1.84615i	0.384616	+	0.923077i	$0.374334\pi$
$661$	31374.0i	1.84615i	−0.384616	+	0.923077i	$0.625666\pi$
$662$	48520.0	2.84862
$663$	0	0
$664$	13860.0	0.810049
$665$	− 11466.0i	− 0.668620i
$666$	14410.0	0.838403
$667$	−7872.00	−0.456979
$668$	27608.0i	1.59908i
$669$	− 1519.00i	− 0.0877847i
$670$	− 16730.0i	− 0.964681i
$671$	− 1456.00i	− 0.0837679i
$672$	−7735.00	−0.444024
$673$	−18013.0	−1.03172	−0.515862	−	0.856672i	$-0.672528\pi$
$673$	−18013.0	−1.03172	−0.515862	+	0.856672i	$0.672528\pi$
$674$	− 52245.0i	− 2.98576i
$675$	2660.00	0.151679
$676$	0	0
$677$	−10640.0	−0.604030	−0.302015	−	0.953303i	$-0.597659\pi$
$677$	−10640.0	−0.604030	−0.302015	+	0.953303i	$0.597659\pi$
$678$	− 60690.0i	− 3.43774i
$679$	−910.000	−0.0514324
$680$	−24255.0	−1.36785
$681$	18032.0i	1.01467i
$682$	25480.0i	1.43062i
$683$	9336.00i	0.523034i	0.965199	+	0.261517i	$0.0842227\pi$
$683$	9336.00i	0.523034i	−0.965199	+	0.261517i	$0.915777\pi$
$684$	47124.0i	2.63426i
$685$	12432.0	0.693434
$686$	33605.0	1.87033
$687$	3185.00i	0.176878i
$688$	17889.0	0.991296
$689$	0	0
$690$	23520.0	1.29767
$691$	4200.00i	0.231224i	0.993294	+	0.115612i	$0.0368829\pi$
$691$	4200.00i	0.231224i	−0.993294	+	0.115612i	$0.963117\pi$
$692$	−5712.00	−0.313783
$693$	−7436.00	−0.407605
$694$	3105.00i	0.169833i
$695$	13083.0i	0.714052i
$696$	25830.0i	1.40673i
$697$	− 25872.0i	− 1.40599i
$698$	−62405.0	−3.38405
$699$	21427.0	1.15943
$700$	− 16796.0i	− 0.906899i
$701$	−9872.00	−0.531898	−0.265949	−	0.963987i	$-0.585685\pi$
$701$	−9872.00	−0.531898	−0.265949	+	0.963987i	$0.585685\pi$
$702$	0	0
$703$	16506.0	0.885541
$704$	7462.00i	0.399481i
$705$	−5145.00	−0.274854
$706$	−7000.00	−0.373156
$707$	− 5460.00i	− 0.290445i
$708$	34986.0i	1.85714i
$709$	− 28450.0i	− 1.50700i	−0.657449	−	0.753499i	$-0.728364\pi$
$709$	− 28450.0i	− 1.50700i	0.657449	−	0.753499i	$-0.271636\pi$
$710$	− 315.000i	− 0.0166503i
$711$	28688.0	1.51320
$712$	−53550.0	−2.81864
$713$	18816.0i	0.988310i
$714$	35035.0	1.83635
$715$	0	0
$716$	−51493.0	−2.68769
$717$	24339.0i	1.26772i
$718$	24840.0	1.29111
$719$	−32718.0	−1.69705	−0.848523	−	0.529159i	$-0.822507\pi$
$719$	−32718.0	−1.69705	−0.848523	+	0.529159i	$0.822507\pi$
$720$	− 13706.0i	− 0.709434i
$721$	− 7644.00i	− 0.394837i
$722$	45085.0i	2.32395i
$723$	− 11270.0i	− 0.579718i
$724$	−476.000	−0.0244343
$725$	−6232.00	−0.319242
$726$	22925.0i	1.17194i
$727$	22834.0	1.16488	0.582439	−	0.812874i	$-0.302099\pi$
$727$	22834.0	1.16488	0.582439	+	0.812874i	$0.302099\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	3430.00i	0.173904i
$731$	−15477.0	−0.783088
$732$	−6664.00	−0.336487
$733$	7875.00i	0.396821i	0.980119	+	0.198410i	$0.0635779\pi$
$733$	7875.00i	0.396821i	−0.980119	+	0.198410i	$0.936422\pi$
$734$	− 43610.0i	− 2.19302i
$735$	8526.00i	0.427872i
$736$	− 8160.00i	− 0.408671i
$737$	−12428.0	−0.621155
$738$	36960.0	1.84352
$739$	2140.00i	0.106524i	0.998581	+	0.0532620i	$0.0169618\pi$
$739$	2140.00i	0.106524i	−0.998581	+	0.0532620i	$0.983038\pi$
$740$	−15589.0	−0.774410
$741$	0	0
$742$	−28080.0	−1.38928
$743$	31971.0i	1.57860i	0.614006	+	0.789302i	$0.289557\pi$
$743$	31971.0i	1.57860i	−0.614006	+	0.789302i	$0.710443\pi$
$744$	61740.0	3.04234
$745$	17262.0	0.848900
$746$	− 50060.0i	− 2.45687i
$747$	− 6776.00i	− 0.331889i
$748$	34034.0i	1.66364i
$749$	− 8892.00i	− 0.433787i
$750$	49245.0	2.39756
$751$	7432.00	0.361115	0.180558	−	0.983564i	$-0.442210\pi$
$751$	7432.00	0.361115	0.180558	+	0.983564i	$0.442210\pi$
$752$	9345.00i	0.453161i
$753$	7056.00	0.341481
$754$	0	0
$755$	23261.0	1.12126
$756$	− 7735.00i	− 0.372115i
$757$	20176.0	0.968704	0.484352	−	0.874873i	$-0.339055\pi$
$757$	20176.0	0.968704	0.484352	+	0.874873i	$0.339055\pi$
$758$	−16860.0	−0.807893
$759$	− 17472.0i	− 0.835564i
$760$	− 39690.0i	− 1.89435i
$761$	9478.00i	0.451481i	0.974187	+	0.225741i	$0.0724802\pi$
$761$	9478.00i	0.451481i	−0.974187	+	0.225741i	$0.927520\pi$
$762$	− 66220.0i	− 3.14816i
$763$	−4849.00	−0.230073
$764$	−7174.00	−0.339720
$765$	11858.0i	0.560427i
$766$	−4235.00	−0.199761
$767$	0	0
$768$	57953.0	2.72292
$769$	− 12096.0i	− 0.567221i	−0.958940	−	0.283610i	$-0.908468\pi$
$769$	− 12096.0i	− 0.567221i	0.958940	−	0.283610i	$-0.0915323\pi$
$770$	11830.0	0.553667
$771$	42287.0	1.97526
$772$	8364.00i	0.389931i
$773$	17941.0i	0.834790i	0.908725	+	0.417395i	$0.137057\pi$
$773$	17941.0i	0.834790i	−0.908725	+	0.417395i	$0.862943\pi$
$774$	− 22110.0i	− 1.02678i
$775$	14896.0i	0.690426i
$776$	−3150.00	−0.145720
$777$	11921.0	0.550403
$778$	56570.0i	2.60685i
$779$	42336.0	1.94717
$780$	0	0
$781$	−234.000	−0.0107211
$782$	36960.0i	1.69014i
$783$	−2870.00	−0.130990
$784$	15486.0	0.705448
$785$	19110.0i	0.868873i
$786$	50225.0i	2.27922i
$787$	− 6664.00i	− 0.301837i	−0.988546	−	0.150919i	$-0.951777\pi$
$787$	− 6664.00i	− 0.301837i	0.988546	−	0.150919i	$-0.0482232\pi$
$788$	− 50847.0i	− 2.29867i
$789$	25956.0	1.17118
$790$	−45640.0	−2.05544
$791$	− 22542.0i	− 1.01328i
$792$	−25740.0	−1.15484
$793$	0	0
$794$	−9310.00	−0.416120
$795$	− 21168.0i	− 0.944342i
$796$	−1190.00	−0.0529880
$797$	1442.00	0.0640882	0.0320441	−	0.999486i	$-0.489798\pi$
$797$	1442.00	0.0640882	0.0320441	+	0.999486i	$0.489798\pi$
$798$	57330.0i	2.54318i
$799$	− 8085.00i	− 0.357981i
$800$	− 6460.00i	− 0.285494i
$801$	26180.0i	1.15484i
$802$	−34100.0	−1.50139
$803$	2548.00	0.111976
$804$	56882.0i	2.49512i
$805$	8736.00	0.382489
$806$	0	0
$807$	−58408.0	−2.54778
$808$	− 18900.0i	− 0.822896i
$809$	30207.0	1.31276	0.656379	−	0.754431i	$-0.272087\pi$
$809$	30207.0	1.31276	0.656379	+	0.754431i	$0.272087\pi$
$810$	29365.0	1.27380
$811$	21140.0i	0.915322i	0.889127	+	0.457661i	$0.151313\pi$
$811$	21140.0i	0.915322i	−0.889127	+	0.457661i	$0.848687\pi$
$812$	18122.0i	0.783199i
$813$	− 11319.0i	− 0.488284i
$814$	17030.0i	0.733294i
$815$	3808.00	0.163667
$816$	47971.0	2.05799
$817$	− 25326.0i	− 1.08451i
$818$	−64960.0	−2.77662
$819$	0	0
$820$	−39984.0	−1.70281
$821$	569.000i	0.0241879i	0.999927	+	0.0120939i	$0.00384971\pi$
$821$	569.000i	0.0241879i	−0.999927	+	0.0120939i	$0.996150\pi$
$822$	−62160.0	−2.63757
$823$	8538.00	0.361623	0.180812	−	0.983518i	$-0.442128\pi$
$823$	8538.00	0.361623	0.180812	+	0.983518i	$0.442128\pi$
$824$	− 26460.0i	− 1.11866i
$825$	− 13832.0i	− 0.583719i
$826$	19110.0i	0.804990i
$827$	32702.0i	1.37504i	0.726164	+	0.687521i	$0.241301\pi$
$827$	32702.0i	1.37504i	−0.726164	+	0.687521i	$0.758699\pi$
$828$	−35904.0	−1.50694
$829$	21154.0	0.886259	0.443130	−	0.896458i	$-0.353868\pi$
$829$	21154.0	0.886259	0.443130	+	0.896458i	$0.353868\pi$
$830$	10780.0i	0.450818i
$831$	−26740.0	−1.11625
$832$	0	0
$833$	−13398.0	−0.557279
$834$	− 65415.0i	− 2.71599i
$835$	−11368.0	−0.471145
$836$	−55692.0	−2.30400
$837$	6860.00i	0.283293i
$838$	36715.0i	1.51348i
$839$	2184.00i	0.0898690i	0.998990	+	0.0449345i	$0.0143079\pi$
$839$	2184.00i	0.0898690i	−0.998990	+	0.0449345i	$0.985692\pi$
$840$	− 28665.0i	− 1.17742i
$841$	−17665.0	−0.724302
$842$	−25295.0	−1.03530
$843$	− 43498.0i	− 1.77717i
$844$	−48467.0	−1.97666
$845$	0	0
$846$	11550.0	0.469382
$847$	8515.00i	0.345430i
$848$	−38448.0	−1.55697
$849$	−37044.0	−1.49746
$850$	29260.0i	1.18072i
$851$	12576.0i	0.506580i
$852$	1071.00i	0.0430656i
$853$	− 36687.0i	− 1.47261i	−0.676648	−	0.736307i	$-0.736568\pi$
$853$	− 36687.0i	− 1.47261i	0.676648	−	0.736307i	$-0.263432\pi$
$854$	−3640.00	−0.145853
$855$	−19404.0	−0.776144
$856$	− 30780.0i	− 1.22902i
$857$	−36806.0	−1.46706	−0.733529	−	0.679658i	$-0.762128\pi$
$857$	−36806.0	−1.46706	−0.733529	+	0.679658i	$0.762128\pi$
$858$	0	0
$859$	4900.00	0.194628	0.0973142	−	0.995254i	$-0.468975\pi$
$859$	4900.00	0.194628	0.0973142	+	0.995254i	$0.468975\pi$
$860$	23919.0i	0.948408i
$861$	30576.0	1.21025
$862$	16215.0	0.640702
$863$	− 13697.0i	− 0.540268i	−0.962823	−	0.270134i	$-0.912932\pi$
$863$	− 13697.0i	− 0.540268i	0.962823	−	0.270134i	$-0.0870680\pi$
$864$	− 2975.00i	− 0.117143i
$865$	− 2352.00i	− 0.0924513i
$866$	57995.0i	2.27569i
$867$	−7112.00	−0.278588
$868$	43316.0	1.69383
$869$	33904.0i	1.32349i
$870$	−20090.0	−0.782891
$871$	0	0
$872$	−16785.0	−0.651848
$873$	1540.00i	0.0597034i
$874$	−60480.0	−2.34069
$875$	18291.0	0.706684
$876$	− 11662.0i	− 0.449797i
$877$	6239.00i	0.240224i	0.992760	+	0.120112i	$0.0383253\pi$
$877$	6239.00i	0.240224i	−0.992760	+	0.120112i	$0.961675\pi$
$878$	− 86870.0i	− 3.33909i
$879$	− 6321.00i	− 0.242551i
$880$	16198.0	0.620494
$881$	−133.000	−0.00508613	−0.00254307	−	0.999997i	$-0.500809\pi$
$881$	−133.000	−0.00508613	−0.00254307	+	0.999997i	$0.500809\pi$
$882$	− 19140.0i	− 0.730700i
$883$	26003.0	0.991020	0.495510	−	0.868602i	$-0.334981\pi$
$883$	26003.0	0.991020	0.495510	+	0.868602i	$0.334981\pi$
$884$	0	0
$885$	−14406.0	−0.547178
$886$	− 4945.00i	− 0.187506i
$887$	−31248.0	−1.18287	−0.591435	−	0.806353i	$-0.701438\pi$
$887$	−31248.0	−1.18287	−0.591435	+	0.806353i	$0.701438\pi$
$888$	41265.0	1.55942
$889$	− 24596.0i	− 0.927923i
$890$	− 41650.0i	− 1.56866i
$891$	− 21814.0i	− 0.820198i
$892$	− 3689.00i	− 0.138472i
$893$	13230.0	0.495773
$894$	−86310.0	−3.22890
$895$	− 21203.0i	− 0.791886i
$896$	27495.0	1.02516
$897$	0	0
$898$	72370.0	2.68933
$899$	− 16072.0i	− 0.596253i
$900$	−28424.0	−1.05274
$901$	33264.0	1.22995
$902$	43680.0i	1.61240i
$903$	− 18291.0i	− 0.674071i
$904$	− 78030.0i	− 2.87084i
$905$	− 196.000i	− 0.00719918i
$906$	−116305.	−4.26487
$907$	38253.0	1.40041	0.700204	−	0.713943i	$-0.253092\pi$
$907$	38253.0	1.40041	0.700204	+	0.713943i	$0.253092\pi$
$908$	43792.0i	1.60054i
$909$	−9240.00	−0.337152
$910$	0	0
$911$	36374.0	1.32286	0.661429	−	0.750007i	$-0.269950\pi$
$911$	36374.0	1.32286	0.661429	+	0.750007i	$0.269950\pi$
$912$	78498.0i	2.85014i
$913$	8008.00	0.290281
$914$	−7970.00	−0.288429
$915$	− 2744.00i	− 0.0991408i
$916$	7735.00i	0.279008i
$917$	18655.0i	0.671802i
$918$	13475.0i	0.484468i
$919$	−27648.0	−0.992408	−0.496204	−	0.868206i	$-0.665273\pi$
$919$	−27648.0	−0.992408	−0.496204	+	0.868206i	$0.665273\pi$
$920$	30240.0	1.08368
$921$	14798.0i	0.529436i
$922$	−29575.0	−1.05640
$923$	0	0
$924$	−40222.0	−1.43204
$925$	9956.00i	0.353893i
$926$	55360.0	1.96462
$927$	−12936.0	−0.458332
$928$	6970.00i	0.246553i
$929$	756.000i	0.0266992i	0.999911	+	0.0133496i	$0.00424944\pi$
$929$	756.000i	0.0266992i	−0.999911	+	0.0133496i	$0.995751\pi$
$930$	48020.0i	1.69316i
$931$	− 21924.0i	− 0.771783i
$932$	52037.0	1.82889
$933$	23814.0	0.835622
$934$	6300.00i	0.220709i
$935$	−14014.0	−0.490168
$936$	0	0
$937$	20846.0	0.726797	0.363399	−	0.931634i	$-0.381616\pi$
$937$	20846.0	0.726797	0.363399	+	0.931634i	$0.381616\pi$
$938$	31070.0i	1.08153i
$939$	74823.0	2.60038
$940$	−12495.0	−0.433555
$941$	− 41321.0i	− 1.43148i	−0.698365	−	0.715742i	$-0.746089\pi$
$941$	− 41321.0i	− 1.43148i	0.698365	−	0.715742i	$-0.253911\pi$
$942$	− 95550.0i	− 3.30487i
$943$	32256.0i	1.11389i
$944$	26166.0i	0.902151i
$945$	3185.00	0.109638
$946$	26130.0	0.898055
$947$	− 54966.0i	− 1.88612i	−0.332624	−	0.943060i	$-0.607934\pi$
$947$	− 54966.0i	− 1.88612i	0.332624	−	0.943060i	$-0.392066\pi$
$948$	155176.	5.31633
$949$	0	0
$950$	−47880.0	−1.63519
$951$	49378.0i	1.68369i
$952$	45045.0	1.53353
$953$	44553.0	1.51439	0.757195	−	0.653189i	$-0.226569\pi$
$953$	44553.0	1.51439	0.757195	+	0.653189i	$0.226569\pi$
$954$	47520.0i	1.61270i
$955$	− 2954.00i	− 0.100093i
$956$	59109.0i	1.99971i
$957$	14924.0i	0.504101i
$958$	60165.0	2.02906
$959$	−23088.0	−0.777425
$960$	14063.0i	0.472793i
$961$	−8625.00	−0.289517
$962$	0	0
$963$	−15048.0	−0.503546
$964$	− 27370.0i	− 0.914448i
$965$	−3444.00	−0.114887
$966$	−43680.0	−1.45485
$967$	− 27907.0i	− 0.928054i	−0.885821	−	0.464027i	$-0.846404\pi$
$967$	− 27907.0i	− 0.928054i	0.885821	−	0.464027i	$-0.153596\pi$
$968$	29475.0i	0.978680i
$969$	− 67914.0i	− 2.25151i
$970$	− 2450.00i	− 0.0810977i
$971$	−16443.0	−0.543441	−0.271720	−	0.962376i	$-0.587593\pi$
$971$	−16443.0	−0.543441	−0.271720	+	0.962376i	$0.587593\pi$
$972$	−83776.0	−2.76452
$973$	− 24297.0i	− 0.800541i
$974$	−11400.0	−0.375030
$975$	0	0
$976$	−4984.00	−0.163457
$977$	− 45414.0i	− 1.48713i	−0.668666	−	0.743563i	$-0.733134\pi$
$977$	− 45414.0i	− 1.48713i	0.668666	−	0.743563i	$-0.266866\pi$
$978$	−19040.0	−0.622528
$979$	−30940.0	−1.01006
$980$	20706.0i	0.674927i
$981$	8206.00i	0.267072i
$982$	83835.0i	2.72432i
$983$	8981.00i	0.291403i	0.989329	+	0.145702i	$0.0465440\pi$
$983$	8981.00i	0.291403i	−0.989329	+	0.145702i	$0.953456\pi$
$984$	105840.	3.42892
$985$	20937.0	0.677267
$986$	− 31570.0i	− 1.01967i
$987$	9555.00	0.308145
$988$	0	0
$989$	19296.0	0.620402
$990$	− 20020.0i	− 0.642704i
$991$	−17414.0	−0.558198	−0.279099	−	0.960262i	$-0.590036\pi$
$991$	−17414.0	−0.558198	−0.279099	+	0.960262i	$0.590036\pi$
$992$	16660.0	0.533221
$993$	− 67928.0i	− 2.17083i
$994$	585.000i	0.0186671i
$995$	− 490.000i	− 0.0156121i
$996$	− 36652.0i	− 1.16603i
$997$	−23702.0	−0.752909	−0.376454	−	0.926435i	$-0.622857\pi$
$997$	−23702.0	−0.752909	−0.376454	+	0.926435i	$0.622857\pi$
$998$	64200.0	2.03629
$999$	4585.00i	0.145208i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.a.168.1		2
13.2	odd	12		169.4.c.e.22.1		2
13.3	even	3		169.4.e.e.147.2		4
13.4	even	6		169.4.e.e.23.2		4
13.5	odd	4		13.4.a.a.1.1	✓	1
13.6	odd	12		169.4.c.e.146.1		2
13.7	odd	12		169.4.c.a.146.1		2
13.8	odd	4		169.4.a.e.1.1		1
13.9	even	3		169.4.e.e.23.1		4
13.10	even	6		169.4.e.e.147.1		4
13.11	odd	12		169.4.c.a.22.1		2
13.12	even	2	inner	169.4.b.a.168.2		2
39.5	even	4		117.4.a.b.1.1		1
39.8	even	4		1521.4.a.a.1.1		1
52.31	even	4		208.4.a.g.1.1		1
65.18	even	4		325.4.b.b.274.2		2
65.44	odd	4		325.4.a.d.1.1		1
65.57	even	4		325.4.b.b.274.1		2
91.83	even	4		637.4.a.a.1.1		1
104.5	odd	4		832.4.a.r.1.1		1
104.83	even	4		832.4.a.a.1.1		1
143.109	even	4		1573.4.a.a.1.1		1
156.83	odd	4		1872.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.a.1.1	✓	1	13.5	odd	4
117.4.a.b.1.1		1	39.5	even	4
169.4.a.e.1.1		1	13.8	odd	4
169.4.b.a.168.1		2	1.1	even	1	trivial
169.4.b.a.168.2		2	13.12	even	2	inner
169.4.c.a.22.1		2	13.11	odd	12
169.4.c.a.146.1		2	13.7	odd	12
169.4.c.e.22.1		2	13.2	odd	12
169.4.c.e.146.1		2	13.6	odd	12
169.4.e.e.23.1		4	13.9	even	3
169.4.e.e.23.2		4	13.4	even	6
169.4.e.e.147.1		4	13.10	even	6
169.4.e.e.147.2		4	13.3	even	3
208.4.a.g.1.1		1	52.31	even	4
325.4.a.d.1.1		1	65.44	odd	4
325.4.b.b.274.1		2	65.57	even	4
325.4.b.b.274.2		2	65.18	even	4
637.4.a.a.1.1		1	91.83	even	4
832.4.a.a.1.1		1	104.83	even	4
832.4.a.r.1.1		1	104.5	odd	4
1521.4.a.a.1.1		1	39.8	even	4
1573.4.a.a.1.1		1	143.109	even	4
1872.4.a.k.1.1		1	156.83	odd	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)

\(f(q)\)	\(=\)	\(q-5.00000i q^{2} -7.00000 q^{3} -17.0000 q^{4} -7.00000i q^{5} +35.0000i q^{6} +13.0000i q^{7} +45.0000i q^{8} +22.0000 q^{9} +O(q^{10})\)	\(q-5.00000i q^{2} -7.00000 q^{3} -17.0000 q^{4} -7.00000i q^{5} +35.0000i q^{6} +13.0000i q^{7} +45.0000i q^{8} +22.0000 q^{9} -35.0000 q^{10} +26.0000i q^{11} +119.000 q^{12} +65.0000 q^{14} +49.0000i q^{15} +89.0000 q^{16} -77.0000 q^{17} -110.000i q^{18} -126.000i q^{19} +119.000i q^{20} -91.0000i q^{21} +130.000 q^{22} +96.0000 q^{23} -315.000i q^{24} +76.0000 q^{25} +35.0000 q^{27} -221.000i q^{28} -82.0000 q^{29} +245.000 q^{30} +196.000i q^{31} -85.0000i q^{32} -182.000i q^{33} +385.000i q^{34} +91.0000 q^{35} -374.000 q^{36} +131.000i q^{37} -630.000 q^{38} +315.000 q^{40} +336.000i q^{41} -455.000 q^{42} +201.000 q^{43} -442.000i q^{44} -154.000i q^{45} -480.000i q^{46} +105.000i q^{47} -623.000 q^{48} +174.000 q^{49} -380.000i q^{50} +539.000 q^{51} -432.000 q^{53} -175.000i q^{54} +182.000 q^{55} -585.000 q^{56} +882.000i q^{57} +410.000i q^{58} +294.000i q^{59} -833.000i q^{60} -56.0000 q^{61} +980.000 q^{62} +286.000i q^{63} +287.000 q^{64} -910.000 q^{66} +478.000i q^{67} +1309.00 q^{68} -672.000 q^{69} -455.000i q^{70} +9.00000i q^{71} +990.000i q^{72} -98.0000i q^{73} +655.000 q^{74} -532.000 q^{75} +2142.00i q^{76} -338.000 q^{77} +1304.00 q^{79} -623.000i q^{80} -839.000 q^{81} +1680.00 q^{82} -308.000i q^{83} +1547.00i q^{84} +539.000i q^{85} -1005.00i q^{86} +574.000 q^{87} -1170.00 q^{88} +1190.00i q^{89} -770.000 q^{90} -1632.00 q^{92} -1372.00i q^{93} +525.000 q^{94} -882.000 q^{95} +595.000i q^{96} +70.0000i q^{97} -870.000i q^{98} +572.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} - 34 q^{4} + 44 q^{9}+O(q^{10}) \) 2 * q - 14 * q^3 - 34 * q^4 + 44 * q^9	\( 2 q - 14 q^{3} - 34 q^{4} + 44 q^{9} - 70 q^{10} + 238 q^{12} + 130 q^{14} + 178 q^{16} - 154 q^{17} + 260 q^{22} + 192 q^{23} + 152 q^{25} + 70 q^{27} - 164 q^{29} + 490 q^{30} + 182 q^{35} - 748 q^{36} - 1260 q^{38} + 630 q^{40} - 910 q^{42} + 402 q^{43} - 1246 q^{48} + 348 q^{49} + 1078 q^{51} - 864 q^{53} + 364 q^{55} - 1170 q^{56} - 112 q^{61} + 1960 q^{62} + 574 q^{64} - 1820 q^{66} + 2618 q^{68} - 1344 q^{69} + 1310 q^{74} - 1064 q^{75} - 676 q^{77} + 2608 q^{79} - 1678 q^{81} + 3360 q^{82} + 1148 q^{87} - 2340 q^{88} - 1540 q^{90} - 3264 q^{92} + 1050 q^{94} - 1764 q^{95}+O(q^{100}) \) 2 * q - 14 * q^3 - 34 * q^4 + 44 * q^9 - 70 * q^10 + 238 * q^12 + 130 * q^14 + 178 * q^16 - 154 * q^17 + 260 * q^22 + 192 * q^23 + 152 * q^25 + 70 * q^27 - 164 * q^29 + 490 * q^30 + 182 * q^35 - 748 * q^36 - 1260 * q^38 + 630 * q^40 - 910 * q^42 + 402 * q^43 - 1246 * q^48 + 348 * q^49 + 1078 * q^51 - 864 * q^53 + 364 * q^55 - 1170 * q^56 - 112 * q^61 + 1960 * q^62 + 574 * q^64 - 1820 * q^66 + 2618 * q^68 - 1344 * q^69 + 1310 * q^74 - 1064 * q^75 - 676 * q^77 + 2608 * q^79 - 1678 * q^81 + 3360 * q^82 + 1148 * q^87 - 2340 * q^88 - 1540 * q^90 - 3264 * q^92 + 1050 * q^94 - 1764 * q^95

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