LMFDB - Embedded newform 169.4.b.b.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:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46410i q^{2} -7.00000 q^{3} -4.00000 q^{4} -13.8564i q^{5} +24.2487i q^{6} -22.5167i q^{7} -13.8564i q^{8} +22.0000 q^{9} +O(q^{10})$	\(q-3.46410i q^{2} -7.00000 q^{3} -4.00000 q^{4} -13.8564i q^{5} +24.2487i q^{6} -22.5167i q^{7} -13.8564i q^{8} +22.0000 q^{9} -48.0000 q^{10} -22.5167i q^{11} +28.0000 q^{12} -78.0000 q^{14} +96.9948i q^{15} -80.0000 q^{16} +27.0000 q^{17} -76.2102i q^{18} +88.3346i q^{19} +55.4256i q^{20} +157.617i q^{21} -78.0000 q^{22} +57.0000 q^{23} +96.9948i q^{24} -67.0000 q^{25} +35.0000 q^{27} +90.0666i q^{28} -69.0000 q^{29} +336.000 q^{30} +72.7461i q^{31} +166.277i q^{32} +157.617i q^{33} -93.5307i q^{34} -312.000 q^{35} -88.0000 q^{36} -39.8372i q^{37} +306.000 q^{38} -192.000 q^{40} -393.176i q^{41} +546.000 q^{42} -85.0000 q^{43} +90.0666i q^{44} -304.841i q^{45} -197.454i q^{46} +342.946i q^{47} +560.000 q^{48} -164.000 q^{49} +232.095i q^{50} -189.000 q^{51} +426.000 q^{53} -121.244i q^{54} -312.000 q^{55} -312.000 q^{56} -618.342i q^{57} +239.023i q^{58} +19.0526i q^{59} -387.979i q^{60} -17.0000 q^{61} +252.000 q^{62} -495.367i q^{63} -64.0000 q^{64} +546.000 q^{66} +164.545i q^{67} -108.000 q^{68} -399.000 q^{69} +1080.80i q^{70} -583.701i q^{71} -304.841i q^{72} -1004.59i q^{73} -138.000 q^{74} +469.000 q^{75} -353.338i q^{76} -507.000 q^{77} -1244.00 q^{79} +1108.51i q^{80} -839.000 q^{81} -1362.00 q^{82} +426.084i q^{83} -630.466i q^{84} -374.123i q^{85} +294.449i q^{86} +483.000 q^{87} -312.000 q^{88} +306.573i q^{89} -1056.00 q^{90} -228.000 q^{92} -509.223i q^{93} +1188.00 q^{94} +1224.00 q^{95} -1163.94i q^{96} -1234.95i q^{97} +568.113i q^{98} -495.367i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} - 8 q^{4} + 44 q^{9}+O(q^{10}) $ 2 * q - 14 * q^3 - 8 * q^4 + 44 * q^9	$ 2 q - 14 q^{3} - 8 q^{4} + 44 q^{9} - 96 q^{10} + 56 q^{12} - 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} - 134 q^{25} + 70 q^{27} - 138 q^{29} + 672 q^{30} - 624 q^{35} - 176 q^{36} + 612 q^{38} - 384 q^{40} + 1092 q^{42} - 170 q^{43} + 1120 q^{48} - 328 q^{49} - 378 q^{51} + 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} - 128 q^{64} + 1092 q^{66} - 216 q^{68} - 798 q^{69} - 276 q^{74} + 938 q^{75} - 1014 q^{77} - 2488 q^{79} - 1678 q^{81} - 2724 q^{82} + 966 q^{87} - 624 q^{88} - 2112 q^{90} - 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100}) $ 2 * q - 14 * q^3 - 8 * q^4 + 44 * q^9 - 96 * q^10 + 56 * q^12 - 156 * q^14 - 160 * q^16 + 54 * q^17 - 156 * q^22 + 114 * q^23 - 134 * q^25 + 70 * q^27 - 138 * q^29 + 672 * q^30 - 624 * q^35 - 176 * q^36 + 612 * q^38 - 384 * q^40 + 1092 * q^42 - 170 * q^43 + 1120 * q^48 - 328 * q^49 - 378 * q^51 + 852 * q^53 - 624 * q^55 - 624 * q^56 - 34 * q^61 + 504 * q^62 - 128 * q^64 + 1092 * q^66 - 216 * q^68 - 798 * q^69 - 276 * q^74 + 938 * q^75 - 1014 * q^77 - 2488 * q^79 - 1678 * q^81 - 2724 * q^82 + 966 * q^87 - 624 * q^88 - 2112 * q^90 - 456 * q^92 + 2376 * q^94 + 2448 * 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 $

$2$	− 3.46410i	− 1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	− 3.46410i	− 1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	−4.00000	−0.500000
$5$	− 13.8564i	− 1.23935i	−0.784857	−	0.619677i	$-0.787263\pi$
$5$	− 13.8564i	− 1.23935i	0.784857	−	0.619677i	$-0.212737\pi$
$6$	24.2487i	1.64992i
$7$	− 22.5167i	− 1.21579i	−0.794019	−	0.607893i	$-0.792015\pi$
$7$	− 22.5167i	− 1.21579i	0.794019	−	0.607893i	$-0.207985\pi$
$8$	− 13.8564i	− 0.612372i
$9$	22.0000	0.814815
$10$	−48.0000	−1.51789
$11$	− 22.5167i	− 0.617184i	−0.951194	−	0.308592i	$-0.900142\pi$
$11$	− 22.5167i	− 0.617184i	0.951194	−	0.308592i	$-0.0998578\pi$
$12$	28.0000	0.673575
$13$	0	0
$13$	0	0
$14$	−78.0000	−1.48903
$15$	96.9948i	1.66960i
$16$	−80.0000	−1.25000
$17$	27.0000	0.385204	0.192602	−	0.981277i	$-0.438307\pi$
$17$	27.0000	0.385204	0.192602	+	0.981277i	$0.438307\pi$
$18$	− 76.2102i	− 0.997940i
$19$	88.3346i	1.06660i	0.845927	+	0.533299i	$0.179048\pi$
$19$	88.3346i	1.06660i	−0.845927	+	0.533299i	$0.820952\pi$
$20$	55.4256i	0.619677i
$21$	157.617i	1.63785i
$22$	−78.0000	−0.755893
$23$	57.0000	0.516753	0.258377	−	0.966044i	$-0.416812\pi$
$23$	57.0000	0.516753	0.258377	+	0.966044i	$0.416812\pi$
$24$	96.9948i	0.824958i
$25$	−67.0000	−0.536000
$26$	0	0
$27$	35.0000	0.249472
$28$	90.0666i	0.607893i
$29$	−69.0000	−0.441827	−0.220913	−	0.975293i	$-0.570904\pi$
$29$	−69.0000	−0.441827	−0.220913	+	0.975293i	$0.570904\pi$
$30$	336.000	2.04483
$31$	72.7461i	0.421471i	0.977543	+	0.210735i	$0.0675858\pi$
$31$	72.7461i	0.421471i	−0.977543	+	0.210735i	$0.932414\pi$
$32$	166.277i	0.918559i
$33$	157.617i	0.831440i
$34$	− 93.5307i	− 0.471776i
$35$	−312.000	−1.50679
$36$	−88.0000	−0.407407
$37$	− 39.8372i	− 0.177005i	−0.996076	−	0.0885026i	$-0.971792\pi$
$37$	− 39.8372i	− 0.177005i	0.996076	−	0.0885026i	$-0.0282081\pi$
$38$	306.000	1.30631
$39$	0	0
$40$	−192.000	−0.758947
$41$	− 393.176i	− 1.49765i	−0.662767	−	0.748826i	$-0.730618\pi$
$41$	− 393.176i	− 1.49765i	0.662767	−	0.748826i	$-0.269382\pi$
$42$	546.000	2.00594
$43$	−85.0000	−0.301451	−0.150725	−	0.988576i	$-0.548161\pi$
$43$	−85.0000	−0.301451	−0.150725	+	0.988576i	$0.548161\pi$
$44$	90.0666i	0.308592i
$45$	− 304.841i	− 1.00984i
$46$	− 197.454i	− 0.632891i
$47$	342.946i	1.06434i	0.846639	+	0.532168i	$0.178623\pi$
$47$	342.946i	1.06434i	−0.846639	+	0.532168i	$0.821377\pi$
$48$	560.000	1.68394
$49$	−164.000	−0.478134
$50$	232.095i	0.656463i
$51$	−189.000	−0.518927
$52$	0	0
$53$	426.000	1.10407	0.552034	−	0.833822i	$-0.313852\pi$
$53$	426.000	1.10407	0.552034	+	0.833822i	$0.313852\pi$
$54$	− 121.244i	− 0.305540i
$55$	−312.000	−0.764910
$56$	−312.000	−0.744513
$57$	− 618.342i	− 1.43687i
$58$	239.023i	0.541125i
$59$	19.0526i	0.0420412i	0.999779	+	0.0210206i	$0.00669156\pi$
$59$	19.0526i	0.0420412i	−0.999779	+	0.0210206i	$0.993308\pi$
$60$	− 387.979i	− 0.834799i
$61$	−17.0000	−0.0356824	−0.0178412	−	0.999841i	$-0.505679\pi$
$61$	−17.0000	−0.0356824	−0.0178412	+	0.999841i	$0.505679\pi$
$62$	252.000	0.516194
$63$	− 495.367i	− 0.990640i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	546.000	1.01830
$67$	164.545i	0.300035i	0.988683	+	0.150018i	$0.0479330\pi$
$67$	164.545i	0.300035i	−0.988683	+	0.150018i	$0.952067\pi$
$68$	−108.000	−0.192602
$69$	−399.000	−0.696144
$70$	1080.80i	1.84543i
$71$	− 583.701i	− 0.975670i	−0.872936	−	0.487835i	$-0.837787\pi$
$71$	− 583.701i	− 0.975670i	0.872936	−	0.487835i	$-0.162213\pi$
$72$	− 304.841i	− 0.498970i
$73$	− 1004.59i	− 1.61066i	−0.592826	−	0.805331i	$-0.701988\pi$
$73$	− 1004.59i	− 1.61066i	0.592826	−	0.805331i	$-0.298012\pi$
$74$	−138.000	−0.216786
$75$	469.000	0.722073
$76$	− 353.338i	− 0.533299i
$77$	−507.000	−0.750364
$78$	0	0
$79$	−1244.00	−1.77166	−0.885829	−	0.464012i	$-0.846409\pi$
$79$	−1244.00	−1.77166	−0.885829	+	0.464012i	$0.846409\pi$
$80$	1108.51i	1.54919i
$81$	−839.000	−1.15089
$82$	−1362.00	−1.83424
$83$	426.084i	0.563480i	0.959491	+	0.281740i	$0.0909116\pi$
$83$	426.084i	0.563480i	−0.959491	+	0.281740i	$0.909088\pi$
$84$	− 630.466i	− 0.818923i
$85$	− 374.123i	− 0.477404i
$86$	294.449i	0.369200i
$87$	483.000	0.595207
$88$	−312.000	−0.377947
$89$	306.573i	0.365131i	0.983194	+	0.182566i	$0.0584402\pi$
$89$	306.573i	0.365131i	−0.983194	+	0.182566i	$0.941560\pi$
$90$	−1056.00	−1.23680
$91$	0	0
$92$	−228.000	−0.258377
$93$	− 509.223i	− 0.567785i
$94$	1188.00	1.30354
$95$	1224.00	1.32189
$96$	− 1163.94i	− 1.23744i
$97$	− 1234.95i	− 1.29268i	−0.763048	−	0.646342i	$-0.776298\pi$
$97$	− 1234.95i	− 1.29268i	0.763048	−	0.646342i	$-0.223702\pi$
$98$	568.113i	0.585592i
$99$	− 495.367i	− 0.502891i
$100$	268.000	0.268000
$101$	1959.00	1.92998	0.964989	−	0.262290i	$-0.0844778\pi$
$101$	1959.00	1.92998	0.964989	+	0.262290i	$0.0844778\pi$
$102$	654.715i	0.635554i
$103$	1856.00	1.77551	0.887753	−	0.460320i	$-0.152265\pi$
$103$	1856.00	1.77551	0.887753	+	0.460320i	$0.152265\pi$
$104$	0	0
$105$	2184.00	2.02987
$106$	− 1475.71i	− 1.35220i
$107$	−255.000	−0.230390	−0.115195	−	0.993343i	$-0.536749\pi$
$107$	−255.000	−0.230390	−0.115195	+	0.993343i	$0.536749\pi$
$108$	−140.000	−0.124736
$109$	609.682i	0.535752i	0.963453	+	0.267876i	$0.0863217\pi$
$109$	609.682i	0.535752i	−0.963453	+	0.267876i	$0.913678\pi$
$110$	1080.80i	0.936820i
$111$	278.860i	0.238453i
$112$	1801.33i	1.51973i
$113$	411.000	0.342156	0.171078	−	0.985257i	$-0.445275\pi$
$113$	411.000	0.342156	0.171078	+	0.985257i	$0.445275\pi$
$114$	−2142.00	−1.75980
$115$	− 789.815i	− 0.640440i
$116$	276.000	0.220913
$117$	0	0
$118$	66.0000	0.0514898
$119$	− 607.950i	− 0.468325i
$120$	1344.00	1.02242
$121$	824.000	0.619083
$122$	58.8897i	0.0437018i
$123$	2752.23i	2.01756i
$124$	− 290.985i	− 0.210735i
$125$	− 803.672i	− 0.575061i
$126$	−1716.00	−1.21328
$127$	−2243.00	−1.56720	−0.783599	−	0.621267i	$-0.786618\pi$
$127$	−2243.00	−1.56720	−0.783599	+	0.621267i	$0.786618\pi$
$128$	1551.92i	1.07165i
$129$	595.000	0.406099
$130$	0	0
$131$	−372.000	−0.248105	−0.124053	−	0.992276i	$-0.539589\pi$
$131$	−372.000	−0.248105	−0.124053	+	0.992276i	$0.539589\pi$
$132$	− 630.466i	− 0.415720i
$133$	1989.00	1.29675
$134$	570.000	0.367466
$135$	− 484.974i	− 0.309185i
$136$	− 374.123i	− 0.235888i
$137$	1189.92i	0.742056i	0.928622	+	0.371028i	$0.120995\pi$
$137$	1189.92i	0.742056i	−0.928622	+	0.371028i	$0.879005\pi$
$138$	1382.18i	0.852599i
$139$	−2545.00	−1.55298	−0.776490	−	0.630130i	$-0.783002\pi$
$139$	−2545.00	−1.55298	−0.776490	+	0.630130i	$0.783002\pi$
$140$	1248.00	0.753395
$141$	− 2400.62i	− 1.43382i
$142$	−2022.00	−1.19495
$143$	0	0
$144$	−1760.00	−1.01852
$145$	956.092i	0.547580i
$146$	−3480.00	−1.97265
$147$	1148.00	0.644119
$148$	159.349i	0.0885026i
$149$	− 1304.23i	− 0.717094i	−0.933512	−	0.358547i	$-0.883272\pi$
$149$	− 1304.23i	− 0.717094i	0.933512	−	0.358547i	$-0.116728\pi$
$150$	− 1624.66i	− 0.884355i
$151$	86.6025i	0.0466729i	0.999728	+	0.0233365i	$0.00742890\pi$
$151$	86.6025i	0.0466729i	−0.999728	+	0.0233365i	$0.992571\pi$
$152$	1224.00	0.653155
$153$	594.000	0.313870
$154$	1756.30i	0.919004i
$155$	1008.00	0.522352
$156$	0	0
$157$	−1534.00	−0.779787	−0.389893	−	0.920860i	$-0.627488\pi$
$157$	−1534.00	−0.779787	−0.389893	+	0.920860i	$0.627488\pi$
$158$	4309.34i	2.16983i
$159$	−2982.00	−1.48735
$160$	2304.00	1.13842
$161$	− 1283.45i	− 0.628261i
$162$	2906.38i	1.40955i
$163$	1633.32i	0.784858i	0.919782	+	0.392429i	$0.128365\pi$
$163$	1633.32i	0.784858i	−0.919782	+	0.392429i	$0.871635\pi$
$164$	1572.70i	0.748826i
$165$	2184.00	1.03045
$166$	1476.00	0.690119
$167$	1626.40i	0.753618i	0.926291	+	0.376809i	$0.122979\pi$
$167$	1626.40i	0.753618i	−0.926291	+	0.376809i	$0.877021\pi$
$168$	2184.00	1.00297
$169$	0	0
$170$	−1296.00	−0.584698
$171$	1943.36i	0.869079i
$172$	340.000	0.150725
$173$	−873.000	−0.383659	−0.191829	−	0.981428i	$-0.561442\pi$
$173$	−873.000	−0.383659	−0.191829	+	0.981428i	$0.561442\pi$
$174$	− 1673.16i	− 0.728977i
$175$	1508.62i	0.651661i
$176$	1801.33i	0.771481i
$177$	− 133.368i	− 0.0566359i
$178$	1062.00	0.447193
$179$	−1287.00	−0.537402	−0.268701	−	0.963224i	$-0.586594\pi$
$179$	−1287.00	−0.537402	−0.268701	+	0.963224i	$0.586594\pi$
$180$	1219.36i	0.504922i
$181$	2.00000	0.000821319 0	0.000410660	−	1.00000i	$-0.499869\pi$
$181$	2.00000	0.000821319 0	0.000410660		1.00000i	$0.499869\pi$
$182$	0	0
$183$	119.000	0.0480696
$184$	− 789.815i	− 0.316445i
$185$	−552.000	−0.219372
$186$	−1764.00	−0.695391
$187$	− 607.950i	− 0.237742i
$188$	− 1371.78i	− 0.532168i
$189$	− 788.083i	− 0.303305i
$190$	− 4240.06i	− 1.61898i
$191$	−2841.00	−1.07627	−0.538135	−	0.842859i	$-0.680871\pi$
$191$	−2841.00	−1.07627	−0.538135	+	0.842859i	$0.680871\pi$
$192$	448.000	0.168394
$193$	− 4245.26i	− 1.58332i	−0.610964	−	0.791659i	$-0.709218\pi$
$193$	− 4245.26i	− 1.58332i	0.610964	−	0.791659i	$-0.290782\pi$
$194$	−4278.00	−1.58321
$195$	0	0
$196$	656.000	0.239067
$197$	2752.23i	0.995371i	0.867357	+	0.497686i	$0.165817\pi$
$197$	2752.23i	0.995371i	−0.867357	+	0.497686i	$0.834183\pi$
$198$	−1716.00	−0.615913
$199$	−1685.00	−0.600234	−0.300117	−	0.953902i	$-0.597026\pi$
$199$	−1685.00	−0.600234	−0.300117	+	0.953902i	$0.597026\pi$
$200$	928.379i	0.328232i
$201$	− 1151.81i	− 0.404192i
$202$	− 6786.18i	− 2.36373i
$203$	1553.65i	0.537167i
$204$	756.000	0.259464
$205$	−5448.00	−1.85612
$206$	− 6429.37i	− 2.17454i
$207$	1254.00	0.421058
$208$	0	0
$209$	1989.00	0.658287
$210$	− 7565.60i	− 2.48608i
$211$	1681.00	0.548459	0.274229	−	0.961664i	$-0.411577\pi$
$211$	1681.00	0.548459	0.274229	+	0.961664i	$0.411577\pi$
$212$	−1704.00	−0.552034
$213$	4085.91i	1.31437i
$214$	883.346i	0.282170i
$215$	1177.79i	0.373604i
$216$	− 484.974i	− 0.152770i
$217$	1638.00	0.512418
$218$	2112.00	0.656159
$219$	7032.13i	2.16980i
$220$	1248.00	0.382455
$221$	0	0
$222$	966.000	0.292044
$223$	− 4096.30i	− 1.23008i	−0.788495	−	0.615042i	$-0.789139\pi$
$223$	− 4096.30i	− 1.23008i	0.788495	−	0.615042i	$-0.210861\pi$
$224$	3744.00	1.11677
$225$	−1474.00	−0.436741
$226$	− 1423.75i	− 0.419054i
$227$	− 438.209i	− 0.128128i	−0.997946	−	0.0640638i	$-0.979594\pi$
$227$	− 438.209i	− 0.128128i	0.997946	−	0.0640638i	$-0.0204061\pi$
$228$	2473.37i	0.718433i
$229$	− 180.133i	− 0.0519805i	−0.999662	−	0.0259903i	$-0.991726\pi$
$229$	− 180.133i	− 0.0519805i	0.999662	−	0.0259903i	$-0.00827389\pi$
$230$	−2736.00	−0.784376
$231$	3549.00	1.01085
$232$	956.092i	0.270563i
$233$	−5778.00	−1.62459	−0.812295	−	0.583247i	$-0.801782\pi$
$233$	−5778.00	−1.62459	−0.812295	+	0.583247i	$0.801782\pi$
$234$	0	0
$235$	4752.00	1.31909
$236$	− 76.2102i	− 0.0210206i
$237$	8708.00	2.38669
$238$	−2106.00	−0.573579
$239$	1860.22i	0.503464i	0.967797	+	0.251732i	$0.0810001\pi$
$239$	1860.22i	0.503464i	−0.967797	+	0.251732i	$0.919000\pi$
$240$	− 7759.59i	− 2.08700i
$241$	− 2059.41i	− 0.550449i	−0.961380	−	0.275224i	$-0.911248\pi$
$241$	− 2059.41i	− 0.550449i	0.961380	−	0.275224i	$-0.0887521\pi$
$242$	− 2854.42i	− 0.758219i
$243$	4928.00	1.30095
$244$	68.0000	0.0178412
$245$	2272.45i	0.592578i
$246$	9534.00	2.47100
$247$	0	0
$248$	1008.00	0.258097
$249$	− 2982.59i	− 0.759093i
$250$	−2784.00	−0.704302
$251$	4491.00	1.12936	0.564680	−	0.825310i	$-0.309000\pi$
$251$	4491.00	1.12936	0.564680	+	0.825310i	$0.309000\pi$
$252$	1981.47i	0.495320i
$253$	− 1283.45i	− 0.318932i
$254$	7769.98i	1.91942i
$255$	2618.86i	0.643135i
$256$	4864.00	1.18750
$257$	5451.00	1.32305	0.661525	−	0.749923i	$-0.269909\pi$
$257$	5451.00	1.32305	0.661525	+	0.749923i	$0.269909\pi$
$258$	− 2061.14i	− 0.497368i
$259$	−897.000	−0.215200
$260$	0	0
$261$	−1518.00	−0.360007
$262$	1288.65i	0.303866i
$263$	−783.000	−0.183581	−0.0917906	−	0.995778i	$-0.529259\pi$
$263$	−783.000	−0.183581	−0.0917906	+	0.995778i	$0.529259\pi$
$264$	2184.00	0.509151
$265$	− 5902.83i	− 1.36833i
$266$	− 6890.10i	− 1.58819i
$267$	− 2146.01i	− 0.491887i
$268$	− 658.179i	− 0.150018i
$269$	−5085.00	−1.15256	−0.576279	−	0.817253i	$-0.695496\pi$
$269$	−5085.00	−1.15256	−0.576279	+	0.817253i	$0.695496\pi$
$270$	−1680.00	−0.378672
$271$	− 1325.02i	− 0.297008i	−0.988912	−	0.148504i	$-0.952554\pi$
$271$	− 1325.02i	− 0.297008i	0.988912	−	0.148504i	$-0.0474458\pi$
$272$	−2160.00	−0.481505
$273$	0	0
$274$	4122.00	0.908829
$275$	1508.62i	0.330811i
$276$	1596.00	0.348072
$277$	−3421.00	−0.742050	−0.371025	−	0.928623i	$-0.620994\pi$
$277$	−3421.00	−0.742050	−0.371025	+	0.928623i	$0.620994\pi$
$278$	8816.14i	1.90200i
$279$	1600.41i	0.343421i
$280$	4323.20i	0.922716i
$281$	− 810.600i	− 0.172087i	−0.996291	−	0.0860433i	$-0.972578\pi$
$281$	− 810.600i	− 0.172087i	0.996291	−	0.0860433i	$-0.0274223\pi$
$282$	−8316.00	−1.75607
$283$	7177.00	1.50752	0.753760	−	0.657149i	$-0.228238\pi$
$283$	7177.00	1.50752	0.753760	+	0.657149i	$0.228238\pi$
$284$	2334.80i	0.487835i
$285$	−8568.00	−1.78079
$286$	0	0
$287$	−8853.00	−1.82082
$288$	3658.09i	0.748455i
$289$	−4184.00	−0.851618
$290$	3312.00	0.670646
$291$	8644.67i	1.74144i
$292$	4018.36i	0.805331i
$293$	− 9313.24i	− 1.85695i	−0.371400	−	0.928473i	$-0.621122\pi$
$293$	− 9313.24i	− 1.85695i	0.371400	−	0.928473i	$-0.378878\pi$
$294$	− 3976.79i	− 0.788881i
$295$	264.000	0.0521040
$296$	−552.000	−0.108393
$297$	− 788.083i	− 0.153970i
$298$	−4518.00	−0.878257
$299$	0	0
$300$	−1876.00	−0.361036
$301$	1913.92i	0.366499i
$302$	300.000	0.0571625
$303$	−13713.0	−2.59997
$304$	− 7066.77i	− 1.33325i
$305$	235.559i	0.0442232i
$306$	− 2057.68i	− 0.384410i
$307$	4777.00i	0.888070i	0.896009	+	0.444035i	$0.146453\pi$
$307$	4777.00i	0.888070i	−0.896009	+	0.444035i	$0.853547\pi$
$308$	2028.00	0.375182
$309$	−12992.0	−2.39187
$310$	− 3491.81i	− 0.639748i
$311$	6192.00	1.12899	0.564495	−	0.825436i	$-0.309071\pi$
$311$	6192.00	1.12899	0.564495	+	0.825436i	$0.309071\pi$
$312$	0	0
$313$	−770.000	−0.139051	−0.0695255	−	0.997580i	$-0.522149\pi$
$313$	−770.000	−0.139051	−0.0695255	+	0.997580i	$0.522149\pi$
$314$	5313.93i	0.955040i
$315$	−6864.00	−1.22775
$316$	4976.00	0.885829
$317$	− 8057.50i	− 1.42762i	−0.700341	−	0.713808i	$-0.746969\pi$
$317$	− 8057.50i	− 1.42762i	0.700341	−	0.713808i	$-0.253031\pi$
$318$	10330.0i	1.82162i
$319$	1553.65i	0.272689i
$320$	886.810i	0.154919i
$321$	1785.00	0.310371
$322$	−4446.00	−0.769459
$323$	2385.03i	0.410857i
$324$	3356.00	0.575446
$325$	0	0
$326$	5658.00	0.961250
$327$	− 4267.77i	− 0.721738i
$328$	−5448.00	−0.917120
$329$	7722.00	1.29400
$330$	− 7565.60i	− 1.26204i
$331$	5277.56i	0.876377i	0.898883	+	0.438189i	$0.144380\pi$
$331$	5277.56i	0.876377i	−0.898883	+	0.438189i	$0.855620\pi$
$332$	− 1704.34i	− 0.281740i
$333$	− 876.418i	− 0.144226i
$334$	5634.00	0.922990
$335$	2280.00	0.371850
$336$	− 12609.3i	− 2.04731i
$337$	8278.00	1.33808	0.669038	−	0.743228i	$-0.266706\pi$
$337$	8278.00	1.33808	0.669038	+	0.743228i	$0.266706\pi$
$338$	0	0
$339$	−2877.00	−0.460936
$340$	1496.49i	0.238702i
$341$	1638.00	0.260125
$342$	6732.00	1.06440
$343$	− 4030.48i	− 0.634477i
$344$	1177.79i	0.184600i
$345$	5528.71i	0.862770i
$346$	3024.16i	0.469884i
$347$	6867.00	1.06236	0.531181	−	0.847258i	$-0.321748\pi$
$347$	6867.00	1.06236	0.531181	+	0.847258i	$0.321748\pi$
$348$	−1932.00	−0.297604
$349$	12153.8i	1.86412i	0.362303	+	0.932060i	$0.381990\pi$
$349$	12153.8i	1.86412i	−0.362303	+	0.932060i	$0.618010\pi$
$350$	5226.00	0.798118
$351$	0	0
$352$	3744.00	0.566920
$353$	5807.57i	0.875653i	0.899059	+	0.437827i	$0.144252\pi$
$353$	5807.57i	0.875653i	−0.899059	+	0.437827i	$0.855748\pi$
$354$	−462.000	−0.0693645
$355$	−8088.00	−1.20920
$356$	− 1226.29i	− 0.182566i
$357$	4255.65i	0.630904i
$358$	4458.30i	0.658180i
$359$	− 1340.61i	− 0.197088i	−0.995133	−	0.0985439i	$-0.968581\pi$
$359$	− 1340.61i	− 0.197088i	0.995133	−	0.0985439i	$-0.0314185\pi$
$360$	−4224.00	−0.618401
$361$	−944.000	−0.137629
$362$	− 6.92820i	− 0.00100591i
$363$	−5768.00	−0.833999
$364$	0	0
$365$	−13920.0	−1.99618
$366$	− 412.228i	− 0.0588730i
$367$	3665.00	0.521285	0.260642	−	0.965435i	$-0.416066\pi$
$367$	3665.00	0.521285	0.260642	+	0.965435i	$0.416066\pi$
$368$	−4560.00	−0.645941
$369$	− 8649.86i	− 1.22031i
$370$	1912.18i	0.268675i
$371$	− 9592.10i	− 1.34231i
$372$	2036.89i	0.283892i
$373$	5371.00	0.745576	0.372788	−	0.927917i	$-0.378402\pi$
$373$	5371.00	0.745576	0.372788	+	0.927917i	$0.378402\pi$
$374$	−2106.00	−0.291173
$375$	5625.70i	0.774693i
$376$	4752.00	0.651770
$377$	0	0
$378$	−2730.00	−0.371471
$379$	− 11509.5i	− 1.55990i	−0.625842	−	0.779950i	$-0.715244\pi$
$379$	− 11509.5i	− 1.55990i	0.625842	−	0.779950i	$-0.284756\pi$
$380$	−4896.00	−0.660946
$381$	15701.0	2.11125
$382$	9841.51i	1.31816i
$383$	− 2419.67i	− 0.322819i	−0.986888	−	0.161409i	$-0.948396\pi$
$383$	− 2419.67i	− 0.322819i	0.986888	−	0.161409i	$-0.0516040\pi$
$384$	− 10863.4i	− 1.44368i
$385$	7025.20i	0.929967i
$386$	−14706.0	−1.93916
$387$	−1870.00	−0.245626
$388$	4939.81i	0.646342i
$389$	−9858.00	−1.28489	−0.642443	−	0.766334i	$-0.722079\pi$
$389$	−9858.00	−1.28489	−0.642443	+	0.766334i	$0.722079\pi$
$390$	0	0
$391$	1539.00	0.199055
$392$	2272.45i	0.292796i
$393$	2604.00	0.334235
$394$	9534.00	1.21908
$395$	17237.4i	2.19571i
$396$	1981.47i	0.251446i
$397$	8720.88i	1.10249i	0.834344	+	0.551245i	$0.185847\pi$
$397$	8720.88i	1.10249i	−0.834344	+	0.551245i	$0.814153\pi$
$398$	5837.01i	0.735133i
$399$	−13923.0	−1.74692
$400$	5360.00	0.670000
$401$	− 7584.65i	− 0.944537i	−0.881455	−	0.472269i	$-0.843435\pi$
$401$	− 7584.65i	− 0.944537i	0.881455	−	0.472269i	$-0.156565\pi$
$402$	−3990.00	−0.495033
$403$	0	0
$404$	−7836.00	−0.964989
$405$	11625.5i	1.42636i
$406$	5382.00	0.657892
$407$	−897.000	−0.109245
$408$	2618.86i	0.317777i
$409$	− 4304.15i	− 0.520358i	−0.965560	−	0.260179i	$-0.916218\pi$
$409$	− 4304.15i	− 0.520358i	0.965560	−	0.260179i	$-0.0837815\pi$
$410$	18872.4i	2.27327i
$411$	− 8329.43i	− 0.999661i
$412$	−7424.00	−0.887753
$413$	429.000	0.0511131
$414$	− 4343.98i	− 0.515689i
$415$	5904.00	0.698352
$416$	0	0
$417$	17815.0	2.09210
$418$	− 6890.10i	− 0.806234i
$419$	5397.00	0.629262	0.314631	−	0.949214i	$-0.398119\pi$
$419$	5397.00	0.629262	0.314631	+	0.949214i	$0.398119\pi$
$420$	−8736.00	−1.01494
$421$	− 7260.76i	− 0.840541i	−0.907399	−	0.420270i	$-0.861935\pi$
$421$	− 7260.76i	− 0.840541i	0.907399	−	0.420270i	$-0.138065\pi$
$422$	− 5823.15i	− 0.671722i
$423$	7544.81i	0.867237i
$424$	− 5902.83i	− 0.676101i
$425$	−1809.00	−0.206469
$426$	14154.0	1.60977
$427$	382.783i	0.0433822i
$428$	1020.00	0.115195
$429$	0	0
$430$	4080.00	0.457570
$431$	− 486.706i	− 0.0543940i	−0.999630	−	0.0271970i	$-0.991342\pi$
$431$	− 486.706i	− 0.0543940i	0.999630	−	0.0271970i	$-0.00865814\pi$
$432$	−2800.00	−0.311840
$433$	12139.0	1.34726	0.673629	−	0.739069i	$-0.264734\pi$
$433$	12139.0	1.34726	0.673629	+	0.739069i	$0.264734\pi$
$434$	− 5674.20i	− 0.627581i
$435$	− 6692.64i	− 0.737673i
$436$	− 2438.73i	− 0.267876i
$437$	5035.07i	0.551167i
$438$	24360.0	2.65746
$439$	461.000	0.0501192	0.0250596	−	0.999686i	$-0.492022\pi$
$439$	461.000	0.0501192	0.0250596	+	0.999686i	$0.492022\pi$
$440$	4323.20i	0.468410i
$441$	−3608.00	−0.389591
$442$	0	0
$443$	12156.0	1.30372	0.651861	−	0.758338i	$-0.273988\pi$
$443$	12156.0	1.30372	0.651861	+	0.758338i	$0.273988\pi$
$444$	− 1115.44i	− 0.119226i
$445$	4248.00	0.452527
$446$	−14190.0	−1.50654
$447$	9129.64i	0.966034i
$448$	1441.07i	0.151973i
$449$	296.181i	0.0311306i	0.999879	+	0.0155653i	$0.00495479\pi$
$449$	296.181i	0.0311306i	−0.999879	+	0.0155653i	$0.995045\pi$
$450$	5106.09i	0.534896i
$451$	−8853.00	−0.924327
$452$	−1644.00	−0.171078
$453$	− 606.218i	− 0.0628755i
$454$	−1518.00	−0.156924
$455$	0	0
$456$	−8568.00	−0.879898
$457$	611.414i	0.0625837i	0.999510	+	0.0312918i	$0.00996213\pi$
$457$	611.414i	0.0625837i	−0.999510	+	0.0312918i	$0.990038\pi$
$458$	−624.000	−0.0636629
$459$	945.000	0.0960977
$460$	3159.26i	0.320220i
$461$	13127.2i	1.32624i	0.748514	+	0.663119i	$0.230767\pi$
$461$	13127.2i	1.32624i	−0.748514	+	0.663119i	$0.769233\pi$
$462$	− 12294.1i	− 1.23804i
$463$	834.848i	0.0837985i	0.999122	+	0.0418992i	$0.0133408\pi$
$463$	834.848i	0.0837985i	−0.999122	+	0.0418992i	$0.986659\pi$
$464$	5520.00	0.552284
$465$	−7056.00	−0.703686
$466$	20015.6i	1.98971i
$467$	14496.0	1.43639	0.718196	−	0.695841i	$-0.244968\pi$
$467$	14496.0	1.43639	0.718196	+	0.695841i	$0.244968\pi$
$468$	0	0
$469$	3705.00	0.364778
$470$	− 16461.4i	− 1.61555i
$471$	10738.0	1.05049
$472$	264.000	0.0257449
$473$	1913.92i	0.186051i
$474$	− 30165.4i	− 2.92309i
$475$	− 5918.42i	− 0.571696i
$476$	2431.80i	0.234162i
$477$	9372.00	0.899611
$478$	6444.00	0.616614
$479$	− 8897.54i	− 0.848725i	−0.905492	−	0.424362i	$-0.860498\pi$
$479$	− 8897.54i	− 0.848725i	0.905492	−	0.424362i	$-0.139502\pi$
$480$	−16128.0	−1.53362
$481$	0	0
$482$	−7134.00	−0.674159
$483$	8984.15i	0.846362i
$484$	−3296.00	−0.309542
$485$	−17112.0	−1.60209
$486$	− 17071.1i	− 1.59333i
$487$	− 4754.48i	− 0.442394i	−0.975229	−	0.221197i	$-0.929004\pi$
$487$	− 4754.48i	− 0.442394i	0.975229	−	0.221197i	$-0.0709964\pi$
$488$	235.559i	0.0218509i
$489$	− 11433.3i	− 1.05732i
$490$	7872.00	0.725757
$491$	−1635.00	−0.150278	−0.0751390	−	0.997173i	$-0.523940\pi$
$491$	−1635.00	−0.150278	−0.0751390	+	0.997173i	$0.523940\pi$
$492$	− 11008.9i	− 1.00878i
$493$	−1863.00	−0.170193
$494$	0	0
$495$	−6864.00	−0.623260
$496$	− 5819.69i	− 0.526838i
$497$	−13143.0	−1.18621
$498$	−10332.0	−0.929695
$499$	14434.9i	1.29498i	0.762074	+	0.647490i	$0.224181\pi$
$499$	14434.9i	1.29498i	−0.762074	+	0.647490i	$0.775819\pi$
$500$	3214.69i	0.287530i
$501$	− 11384.8i	− 1.01524i
$502$	− 15557.3i	− 1.38318i
$503$	12687.0	1.12462	0.562312	−	0.826925i	$-0.309912\pi$
$503$	12687.0	1.12462	0.562312	+	0.826925i	$0.309912\pi$
$504$	−6864.00	−0.606641
$505$	− 27144.7i	− 2.39193i
$506$	−4446.00	−0.390610
$507$	0	0
$508$	8972.00	0.783599
$509$	− 5748.68i	− 0.500600i	−0.968168	−	0.250300i	$-0.919471\pi$
$509$	− 5748.68i	− 0.500600i	0.968168	−	0.250300i	$-0.0805293\pi$
$510$	9072.00	0.787676
$511$	−22620.0	−1.95822
$512$	− 4434.05i	− 0.382733i
$513$	3091.71i	0.266086i
$514$	− 18882.8i	− 1.62040i
$515$	− 25717.5i	− 2.20048i
$516$	−2380.00	−0.203050
$517$	7722.00	0.656892
$518$	3107.30i	0.263565i
$519$	6111.00	0.516846
$520$	0	0
$521$	6054.00	0.509080	0.254540	−	0.967062i	$-0.418076\pi$
$521$	6054.00	0.509080	0.254540	+	0.967062i	$0.418076\pi$
$522$	5258.51i	0.440917i
$523$	−14803.0	−1.23765	−0.618824	−	0.785530i	$-0.712391\pi$
$523$	−14803.0	−1.23765	−0.618824	+	0.785530i	$0.712391\pi$
$524$	1488.00	0.124053
$525$	− 10560.3i	− 0.877885i
$526$	2712.39i	0.224840i
$527$	1964.15i	0.162352i
$528$	− 12609.3i	− 1.03930i
$529$	−8918.00	−0.732966
$530$	−20448.0	−1.67586
$531$	419.156i	0.0342558i
$532$	−7956.00	−0.648377
$533$	0	0
$534$	−7434.00	−0.602436
$535$	3533.38i	0.285536i
$536$	2280.00	0.183733
$537$	9009.00	0.723961
$538$	17615.0i	1.41159i
$539$	3692.73i	0.295097i
$540$	1939.90i	0.154592i
$541$	21470.5i	1.70626i	0.521695	+	0.853132i	$0.325300\pi$
$541$	21470.5i	1.70626i	−0.521695	+	0.853132i	$0.674700\pi$
$542$	−4590.00	−0.363759
$543$	−14.0000	−0.00110644
$544$	4489.48i	0.353832i
$545$	8448.00	0.663986
$546$	0	0
$547$	−13516.0	−1.05649	−0.528247	−	0.849091i	$-0.677151\pi$
$547$	−13516.0	−1.05649	−0.528247	+	0.849091i	$0.677151\pi$
$548$	− 4759.68i	− 0.371028i
$549$	−374.000	−0.0290746
$550$	5226.00	0.405159
$551$	− 6095.09i	− 0.471251i
$552$	5528.71i	0.426300i
$553$	28010.7i	2.15396i
$554$	11850.7i	0.908822i
$555$	3864.00	0.295527
$556$	10180.0	0.776490
$557$	2890.79i	0.219905i	0.993937	+	0.109952i	$0.0350698\pi$
$557$	2890.79i	0.219905i	−0.993937	+	0.109952i	$0.964930\pi$
$558$	5544.00	0.420603
$559$	0	0
$560$	24960.0	1.88349
$561$	4255.65i	0.320274i
$562$	−2808.00	−0.210762
$563$	11583.0	0.867079	0.433539	−	0.901135i	$-0.357265\pi$
$563$	11583.0	0.867079	0.433539	+	0.901135i	$0.357265\pi$
$564$	9602.49i	0.716911i
$565$	− 5694.98i	− 0.424053i
$566$	− 24861.9i	− 1.84633i
$567$	18891.5i	1.39924i
$568$	−8088.00	−0.597473
$569$	12879.0	0.948885	0.474443	−	0.880286i	$-0.342650\pi$
$569$	12879.0	0.948885	0.474443	+	0.880286i	$0.342650\pi$
$570$	29680.4i	2.18101i
$571$	−11636.0	−0.852805	−0.426402	−	0.904534i	$-0.640219\pi$
$571$	−11636.0	−0.852805	−0.426402	+	0.904534i	$0.640219\pi$
$572$	0	0
$573$	19887.0	1.44990
$574$	30667.7i	2.23004i
$575$	−3819.00	−0.276980
$576$	−1408.00	−0.101852
$577$	− 12311.4i	− 0.888269i	−0.895960	−	0.444134i	$-0.853511\pi$
$577$	− 12311.4i	− 0.888269i	0.895960	−	0.444134i	$-0.146489\pi$
$578$	14493.8i	1.04301i
$579$	29716.8i	2.13297i
$580$	− 3824.37i	− 0.273790i
$581$	9594.00	0.685071
$582$	29946.0	2.13282
$583$	− 9592.10i	− 0.681414i
$584$	−13920.0	−0.986325
$585$	0	0
$586$	−32262.0	−2.27428
$587$	− 15645.6i	− 1.10011i	−0.835129	−	0.550054i	$-0.814607\pi$
$587$	− 15645.6i	− 1.10011i	0.835129	−	0.550054i	$-0.185393\pi$
$588$	−4592.00	−0.322059
$589$	−6426.00	−0.449539
$590$	− 914.523i	− 0.0638141i
$591$	− 19265.6i	− 1.34092i
$592$	3186.97i	0.221256i
$593$	− 25821.4i	− 1.78813i	−0.447942	−	0.894063i	$-0.647843\pi$
$593$	− 25821.4i	− 1.78813i	0.447942	−	0.894063i	$-0.352157\pi$
$594$	−2730.00	−0.188575
$595$	−8424.00	−0.580421
$596$	5216.94i	0.358547i
$597$	11795.0	0.808605
$598$	0	0
$599$	1668.00	0.113777	0.0568887	−	0.998381i	$-0.481882\pi$
$599$	1668.00	0.113777	0.0568887	+	0.998381i	$0.481882\pi$
$600$	− 6498.65i	− 0.442177i
$601$	13699.0	0.929773	0.464887	−	0.885370i	$-0.346095\pi$
$601$	13699.0	0.929773	0.464887	+	0.885370i	$0.346095\pi$
$602$	6630.00	0.448868
$603$	3619.99i	0.244473i
$604$	− 346.410i	− 0.0233365i
$605$	− 11417.7i	− 0.767264i
$606$	47503.2i	3.18430i
$607$	−23173.0	−1.54953	−0.774764	−	0.632251i	$-0.782131\pi$
$607$	−23173.0	−1.54953	−0.774764	+	0.632251i	$0.782131\pi$
$608$	−14688.0	−0.979732
$609$	− 10875.5i	− 0.723644i
$610$	816.000	0.0541621
$611$	0	0
$612$	−2376.00	−0.156935
$613$	16615.6i	1.09477i	0.836880	+	0.547387i	$0.184377\pi$
$613$	16615.6i	1.09477i	−0.836880	+	0.547387i	$0.815623\pi$
$614$	16548.0	1.08766
$615$	38136.0	2.50047
$616$	7025.20i	0.459502i
$617$	− 28393.5i	− 1.85264i	−0.376736	−	0.926321i	$-0.622954\pi$
$617$	− 28393.5i	− 1.85264i	0.376736	−	0.926321i	$-0.377046\pi$
$618$	45005.6i	2.92944i
$619$	6245.78i	0.405556i	0.979225	+	0.202778i	$0.0649969\pi$
$619$	6245.78i	0.405556i	−0.979225	+	0.202778i	$0.935003\pi$
$620$	−4032.00	−0.261176
$621$	1995.00	0.128916
$622$	− 21449.7i	− 1.38273i
$623$	6903.00	0.443921
$624$	0	0
$625$	−19511.0	−1.24870
$626$	2667.36i	0.170302i
$627$	−13923.0	−0.886812
$628$	6136.00	0.389893
$629$	− 1075.60i	− 0.0681830i
$630$	23777.6i	1.50369i
$631$	− 22379.8i	− 1.41193i	−0.708247	−	0.705964i	$-0.750514\pi$
$631$	− 22379.8i	− 1.41193i	0.708247	−	0.705964i	$-0.249486\pi$
$632$	17237.4i	1.08491i
$633$	−11767.0	−0.738857
$634$	−27912.0	−1.74847
$635$	31079.9i	1.94231i
$636$	11928.0	0.743673
$637$	0	0
$638$	5382.00	0.333974
$639$	− 12841.4i	− 0.794990i
$640$	21504.0	1.32816
$641$	19827.0	1.22172	0.610858	−	0.791740i	$-0.290825\pi$
$641$	19827.0	1.22172	0.610858	+	0.791740i	$0.290825\pi$
$642$	− 6183.42i	− 0.380125i
$643$	8450.68i	0.518293i	0.965838	+	0.259146i	$0.0834412\pi$
$643$	8450.68i	0.518293i	−0.965838	+	0.259146i	$0.916559\pi$
$644$	5133.80i	0.314130i
$645$	− 8244.56i	− 0.503301i
$646$	8262.00	0.503195
$647$	2949.00	0.179192	0.0895959	−	0.995978i	$-0.471442\pi$
$647$	2949.00	0.179192	0.0895959	+	0.995978i	$0.471442\pi$
$648$	11625.5i	0.704774i
$649$	429.000	0.0259472
$650$	0	0
$651$	−11466.0	−0.690304
$652$	− 6533.30i	− 0.392429i
$653$	12039.0	0.721474	0.360737	−	0.932668i	$-0.382525\pi$
$653$	12039.0	0.721474	0.360737	+	0.932668i	$0.382525\pi$
$654$	−14784.0	−0.883945
$655$	5154.58i	0.307490i
$656$	31454.0i	1.87206i
$657$	− 22101.0i	− 1.31239i
$658$	− 26749.8i	− 1.58483i
$659$	3363.00	0.198792	0.0993960	−	0.995048i	$-0.468309\pi$
$659$	3363.00	0.198792	0.0993960	+	0.995048i	$0.468309\pi$
$660$	−8736.00	−0.515225
$661$	10158.5i	0.597759i	0.954291	+	0.298880i	$0.0966129\pi$
$661$	10158.5i	0.597759i	−0.954291	+	0.298880i	$0.903387\pi$
$662$	18282.0	1.07334
$663$	0	0
$664$	5904.00	0.345060
$665$	− 27560.4i	− 1.60714i
$666$	−3036.00	−0.176641
$667$	−3933.00	−0.228315
$668$	− 6505.58i	− 0.376809i
$669$	28674.1i	1.65711i
$670$	− 7898.15i	− 0.455421i
$671$	382.783i	0.0220226i
$672$	−26208.0	−1.50446
$673$	−18169.0	−1.04066	−0.520329	−	0.853966i	$-0.674191\pi$
$673$	−18169.0	−1.04066	−0.520329	+	0.853966i	$0.674191\pi$
$674$	− 28675.8i	− 1.63880i
$675$	−2345.00	−0.133717
$676$	0	0
$677$	9042.00	0.513312	0.256656	−	0.966503i	$-0.417379\pi$
$677$	9042.00	0.513312	0.256656	+	0.966503i	$0.417379\pi$
$678$	9966.22i	0.564529i
$679$	−27807.0	−1.57163
$680$	−5184.00	−0.292349
$681$	3067.46i	0.172607i
$682$	− 5674.20i	− 0.318587i
$683$	12462.1i	0.698169i	0.937091	+	0.349084i	$0.113507\pi$
$683$	12462.1i	0.698169i	−0.937091	+	0.349084i	$0.886493\pi$
$684$	− 7773.44i	− 0.434540i
$685$	16488.0	0.919670
$686$	−13962.0	−0.777072
$687$	1260.93i	0.0700256i
$688$	6800.00	0.376813
$689$	0	0
$690$	19152.0	1.05667
$691$	− 4318.00i	− 0.237720i	−0.992911	−	0.118860i	$-0.962076\pi$
$691$	− 4318.00i	− 0.237720i	0.992911	−	0.118860i	$-0.0379240\pi$
$692$	3492.00	0.191829
$693$	−11154.0	−0.611408
$694$	− 23788.0i	− 1.30112i
$695$	35264.6i	1.92469i
$696$	− 6692.64i	− 0.364489i
$697$	− 10615.7i	− 0.576901i
$698$	42102.0	2.28307
$699$	40446.0	2.18857
$700$	− 6034.47i	− 0.325830i
$701$	−18270.0	−0.984377	−0.492189	−	0.870489i	$-0.663803\pi$
$701$	−18270.0	−0.984377	−0.492189	+	0.870489i	$0.663803\pi$
$702$	0	0
$703$	3519.00	0.188793
$704$	1441.07i	0.0771481i
$705$	−33264.0	−1.77701
$706$	20118.0	1.07245
$707$	− 44110.1i	− 2.34644i
$708$	533.472i	0.0283179i
$709$	− 1629.86i	− 0.0863338i	−0.999068	−	0.0431669i	$-0.986255\pi$
$709$	− 1629.86i	− 0.0863338i	0.999068	−	0.0431669i	$-0.0137447\pi$
$710$	28017.7i	1.48096i
$711$	−27368.0	−1.44357
$712$	4248.00	0.223596
$713$	4146.53i	0.217796i
$714$	14742.0	0.772697
$715$	0	0
$716$	5148.00	0.268701
$717$	− 13021.6i	− 0.678241i
$718$	−4644.00	−0.241382
$719$	9831.00	0.509923	0.254961	−	0.966951i	$-0.417937\pi$
$719$	9831.00	0.509923	0.254961	+	0.966951i	$0.417937\pi$
$720$	24387.3i	1.26231i
$721$	− 41790.9i	− 2.15863i
$722$	3270.11i	0.168561i
$723$	14415.9i	0.741537i
$724$	−8.00000	−0.000410660 0
$725$	4623.00	0.236819
$726$	19980.9i	1.02144i
$727$	−15464.0	−0.788897	−0.394448	−	0.918918i	$-0.629064\pi$
$727$	−15464.0	−0.788897	−0.394448	+	0.918918i	$0.629064\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	48220.3i	2.44481i
$731$	−2295.00	−0.116120
$732$	−476.000	−0.0240348
$733$	12616.3i	0.635733i	0.948136	+	0.317866i	$0.102966\pi$
$733$	12616.3i	0.635733i	−0.948136	+	0.317866i	$0.897034\pi$
$734$	− 12695.9i	− 0.638441i
$735$	− 15907.2i	− 0.798291i
$736$	9477.78i	0.474668i
$737$	3705.00	0.185177
$738$	−29964.0	−1.49457
$739$	− 16283.0i	− 0.810528i	−0.914200	−	0.405264i	$-0.867180\pi$
$739$	− 16283.0i	− 0.810528i	0.914200	−	0.405264i	$-0.132820\pi$
$740$	2208.00	0.109686
$741$	0	0
$742$	−33228.0	−1.64399
$743$	10806.3i	0.533571i	0.963756	+	0.266786i	$0.0859616\pi$
$743$	10806.3i	0.533571i	−0.963756	+	0.266786i	$0.914038\pi$
$744$	−7056.00	−0.347696
$745$	−18072.0	−0.888734
$746$	− 18605.7i	− 0.913140i
$747$	9373.86i	0.459132i
$748$	2431.80i	0.118871i
$749$	5741.75i	0.280105i
$750$	19488.0	0.948802
$751$	−13615.0	−0.661542	−0.330771	−	0.943711i	$-0.607309\pi$
$751$	−13615.0	−0.661542	−0.330771	+	0.943711i	$0.607309\pi$
$752$	− 27435.7i	− 1.33042i
$753$	−31437.0	−1.52142
$754$	0	0
$755$	1200.00	0.0578443
$756$	3152.33i	0.151652i
$757$	5551.00	0.266519	0.133259	−	0.991081i	$-0.457456\pi$
$757$	5551.00	0.266519	0.133259	+	0.991081i	$0.457456\pi$
$758$	−39870.0	−1.91048
$759$	8984.15i	0.429649i
$760$	− 16960.2i	− 0.809490i
$761$	− 10082.3i	− 0.480265i	−0.970740	−	0.240133i	$-0.922809\pi$
$761$	− 10082.3i	− 0.480265i	0.970740	−	0.240133i	$-0.0771909\pi$
$762$	− 54389.9i	− 2.58574i
$763$	13728.0	0.651359
$764$	11364.0	0.538135
$765$	− 8230.71i	− 0.388996i
$766$	−8382.00	−0.395371
$767$	0	0
$768$	−34048.0	−1.59974
$769$	29758.4i	1.39547i	0.716357	+	0.697733i	$0.245808\pi$
$769$	29758.4i	1.39547i	−0.716357	+	0.697733i	$0.754192\pi$
$770$	24336.0	1.13897
$771$	−38157.0	−1.78235
$772$	16981.0i	0.791659i
$773$	− 27735.3i	− 1.29052i	−0.763964	−	0.645259i	$-0.776749\pi$
$773$	− 27735.3i	− 1.29052i	0.763964	−	0.645259i	$-0.223251\pi$
$774$	6477.87i	0.300830i
$775$	− 4873.99i	− 0.225908i
$776$	−17112.0	−0.791604
$777$	6279.00	0.289907
$778$	34149.1i	1.57366i
$779$	34731.0	1.59739
$780$	0	0
$781$	−13143.0	−0.602168
$782$	− 5331.25i	− 0.243792i
$783$	−2415.00	−0.110224
$784$	13120.0	0.597668
$785$	21255.7i	0.966432i
$786$	− 9020.52i	− 0.409353i
$787$	31549.3i	1.42899i	0.699643	+	0.714493i	$0.253342\pi$
$787$	31549.3i	1.42899i	−0.699643	+	0.714493i	$0.746658\pi$
$788$	− 11008.9i	− 0.497686i
$789$	5481.00	0.247311
$790$	59712.0	2.68919
$791$	− 9254.35i	− 0.415988i
$792$	−6864.00	−0.307957
$793$	0	0
$794$	30210.0	1.35027
$795$	41319.8i	1.84335i
$796$	6740.00	0.300117
$797$	1455.00	0.0646659	0.0323330	−	0.999477i	$-0.489706\pi$
$797$	1455.00	0.0646659	0.0323330	+	0.999477i	$0.489706\pi$
$798$	48230.7i	2.13953i
$799$	9259.54i	0.409986i
$800$	− 11140.6i	− 0.492347i
$801$	6744.61i	0.297514i
$802$	−26274.0	−1.15682
$803$	−22620.0	−0.994075
$804$	4607.26i	0.202096i
$805$	−17784.0	−0.778638
$806$	0	0
$807$	35595.0	1.55267
$808$	− 27144.7i	− 1.18187i
$809$	1659.00	0.0720981	0.0360490	−	0.999350i	$-0.488523\pi$
$809$	1659.00	0.0720981	0.0360490	+	0.999350i	$0.488523\pi$
$810$	40272.0	1.74693
$811$	− 4402.87i	− 0.190636i	−0.995447	−	0.0953180i	$-0.969613\pi$
$811$	− 4402.87i	− 0.190636i	0.995447	−	0.0953180i	$-0.0303868\pi$
$812$	− 6214.60i	− 0.268583i
$813$	9275.13i	0.400114i
$814$	3107.30i	0.133797i
$815$	22632.0	0.972717
$816$	15120.0	0.648659
$817$	− 7508.44i	− 0.321526i
$818$	−14910.0	−0.637306
$819$	0	0
$820$	21792.0	0.928061
$821$	− 28701.8i	− 1.22010i	−0.792364	−	0.610049i	$-0.791150\pi$
$821$	− 28701.8i	− 1.22010i	0.792364	−	0.610049i	$-0.208850\pi$
$822$	−28854.0	−1.22433
$823$	15779.0	0.668313	0.334156	−	0.942518i	$-0.391549\pi$
$823$	15779.0	0.668313	0.334156	+	0.942518i	$0.391549\pi$
$824$	− 25717.5i	− 1.08727i
$825$	− 10560.3i	− 0.445652i
$826$	− 1486.10i	− 0.0626005i
$827$	7354.29i	0.309231i	0.987975	+	0.154615i	$0.0494138\pi$
$827$	7354.29i	0.309231i	−0.987975	+	0.154615i	$0.950586\pi$
$828$	−5016.00	−0.210529
$829$	17371.0	0.727768	0.363884	−	0.931444i	$-0.381450\pi$
$829$	17371.0	0.727768	0.363884	+	0.931444i	$0.381450\pi$
$830$	− 20452.1i	− 0.855303i
$831$	23947.0	0.999654
$832$	0	0
$833$	−4428.00	−0.184179
$834$	− 61713.0i	− 2.56228i
$835$	22536.0	0.934001
$836$	−7956.00	−0.329144
$837$	2546.11i	0.105145i
$838$	− 18695.8i	− 0.770685i
$839$	− 29474.3i	− 1.21283i	−0.795148	−	0.606416i	$-0.792607\pi$
$839$	− 29474.3i	− 1.21283i	0.795148	−	0.606416i	$-0.207393\pi$
$840$	− 30262.4i	− 1.24304i
$841$	−19628.0	−0.804789
$842$	−25152.0	−1.02945
$843$	5674.20i	0.231827i
$844$	−6724.00	−0.274229
$845$	0	0
$846$	26136.0	1.06214
$847$	− 18553.7i	− 0.752673i
$848$	−34080.0	−1.38008
$849$	−50239.0	−2.03086
$850$	6266.56i	0.252872i
$851$	− 2270.72i	− 0.0914680i
$852$	− 16343.6i	− 0.657187i
$853$	− 2909.85i	− 0.116801i	−0.998293	−	0.0584005i	$-0.981400\pi$
$853$	− 2909.85i	− 0.116801i	0.998293	−	0.0584005i	$-0.0186000\pi$
$854$	1326.00	0.0531321
$855$	26928.0	1.07710
$856$	3533.38i	0.141085i
$857$	−5346.00	−0.213087	−0.106544	−	0.994308i	$-0.533978\pi$
$857$	−5346.00	−0.213087	−0.106544	+	0.994308i	$0.533978\pi$
$858$	0	0
$859$	24244.0	0.962974	0.481487	−	0.876453i	$-0.340097\pi$
$859$	24244.0	0.962974	0.481487	+	0.876453i	$0.340097\pi$
$860$	− 4711.18i	− 0.186802i
$861$	61971.0	2.45292
$862$	−1686.00	−0.0666188
$863$	− 32780.8i	− 1.29301i	−0.762908	−	0.646507i	$-0.776229\pi$
$863$	− 32780.8i	− 1.29301i	0.762908	−	0.646507i	$-0.223771\pi$
$864$	5819.69i	0.229155i
$865$	12096.6i	0.475489i
$866$	− 42050.7i	− 1.65005i
$867$	29288.0	1.14726
$868$	−6552.00	−0.256209
$869$	28010.7i	1.09344i
$870$	−23184.0	−0.903461
$871$	0	0
$872$	8448.00	0.328080
$873$	− 27168.9i	− 1.05330i
$874$	17442.0	0.675039
$875$	−18096.0	−0.699150
$876$	− 28128.5i	− 1.08490i
$877$	4543.17i	0.174928i	0.996168	+	0.0874640i	$0.0278763\pi$
$877$	4543.17i	0.174928i	−0.996168	+	0.0874640i	$0.972124\pi$
$878$	− 1596.95i	− 0.0613832i
$879$	65192.7i	2.50159i
$880$	24960.0	0.956138
$881$	−20517.0	−0.784603	−0.392302	−	0.919837i	$-0.628321\pi$
$881$	−20517.0	−0.784603	−0.392302	+	0.919837i	$0.628321\pi$
$882$	12498.5i	0.477149i
$883$	−23852.0	−0.909042	−0.454521	−	0.890736i	$-0.650189\pi$
$883$	−23852.0	−0.909042	−0.454521	+	0.890736i	$0.650189\pi$
$884$	0	0
$885$	−1848.00	−0.0701919
$886$	− 42109.6i	− 1.59673i
$887$	38757.0	1.46712	0.733558	−	0.679626i	$-0.237858\pi$
$887$	38757.0	1.46712	0.733558	+	0.679626i	$0.237858\pi$
$888$	3864.00	0.146022
$889$	50504.9i	1.90538i
$890$	− 14715.5i	− 0.554230i
$891$	18891.5i	0.710312i
$892$	16385.2i	0.615042i
$893$	−30294.0	−1.13522
$894$	31626.0	1.18315
$895$	17833.2i	0.666031i
$896$	34944.0	1.30290
$897$	0	0
$898$	1026.00	0.0381270
$899$	− 5019.48i	− 0.186217i
$900$	5896.00	0.218370
$901$	11502.0	0.425291
$902$	30667.7i	1.13206i
$903$	− 13397.4i	− 0.493730i
$904$	− 5694.98i	− 0.209527i
$905$	− 27.7128i	− 0.00101791i
$906$	−2100.00	−0.0770064
$907$	−39071.0	−1.43035	−0.715177	−	0.698943i	$-0.753654\pi$
$907$	−39071.0	−1.43035	−0.715177	+	0.698943i	$0.753654\pi$
$908$	1752.84i	0.0640638i
$909$	43098.0	1.57257
$910$	0	0
$911$	−53040.0	−1.92897	−0.964486	−	0.264134i	$-0.914914\pi$
$911$	−53040.0	−1.92897	−0.964486	+	0.264134i	$0.914914\pi$
$912$	49467.4i	1.79608i
$913$	9594.00	0.347771
$914$	2118.00	0.0766490
$915$	− 1648.91i	− 0.0595753i
$916$	720.533i	0.0259903i
$917$	8376.20i	0.301643i
$918$	− 3273.58i	− 0.117695i
$919$	367.000	0.0131732	0.00658662	−	0.999978i	$-0.497903\pi$
$919$	367.000	0.0131732	0.00658662	+	0.999978i	$0.497903\pi$
$920$	−10944.0	−0.392188
$921$	− 33439.0i	− 1.19636i
$922$	45474.0	1.62430
$923$	0	0
$924$	−14196.0	−0.505427
$925$	2669.09i	0.0948748i
$926$	2892.00	0.102632
$927$	40832.0	1.44671
$928$	− 11473.1i	− 0.405844i
$929$	29935.0i	1.05720i	0.848872	+	0.528599i	$0.177282\pi$
$929$	29935.0i	1.05720i	−0.848872	+	0.528599i	$0.822718\pi$
$930$	24442.7i	0.861836i
$931$	− 14486.9i	− 0.509976i
$932$	23112.0	0.812295
$933$	−43344.0	−1.52092
$934$	− 50215.6i	− 1.75921i
$935$	−8424.00	−0.294646
$936$	0	0
$937$	42166.0	1.47012	0.735060	−	0.678002i	$-0.237154\pi$
$937$	42166.0	1.47012	0.735060	+	0.678002i	$0.237154\pi$
$938$	− 12834.5i	− 0.446760i
$939$	5390.00	0.187323
$940$	−19008.0	−0.659545
$941$	35022.1i	1.21327i	0.794981	+	0.606635i	$0.207481\pi$
$941$	35022.1i	1.21327i	−0.794981	+	0.606635i	$0.792519\pi$
$942$	− 37197.5i	− 1.28658i
$943$	− 22411.0i	− 0.773916i
$944$	− 1524.20i	− 0.0525515i
$945$	−10920.0	−0.375902
$946$	6630.00	0.227865
$947$	− 2599.81i	− 0.0892106i	−0.999005	−	0.0446053i	$-0.985797\pi$
$947$	− 2599.81i	− 0.0892106i	0.999005	−	0.0446053i	$-0.0142030\pi$
$948$	−34832.0	−1.19334
$949$	0	0
$950$	−20502.0	−0.700182
$951$	56402.5i	1.92321i
$952$	−8424.00	−0.286789
$953$	10623.0	0.361084	0.180542	−	0.983567i	$-0.442215\pi$
$953$	10623.0	0.361084	0.180542	+	0.983567i	$0.442215\pi$
$954$	− 32465.6i	− 1.10179i
$955$	39366.1i	1.33388i
$956$	− 7440.89i	− 0.251732i
$957$	− 10875.5i	− 0.367353i
$958$	−30822.0	−1.03947
$959$	26793.0	0.902180
$960$	− 6207.67i	− 0.208700i
$961$	24499.0	0.822362
$962$	0	0
$963$	−5610.00	−0.187726
$964$	8237.63i	0.275224i
$965$	−58824.0	−1.96229
$966$	31122.0	1.03658
$967$	− 20199.2i	− 0.671729i	−0.941910	−	0.335864i	$-0.890972\pi$
$967$	− 20199.2i	− 0.671729i	0.941910	−	0.335864i	$-0.109028\pi$
$968$	− 11417.7i	− 0.379110i
$969$	− 16695.2i	− 0.553486i
$970$	59277.7i	1.96216i
$971$	−2325.00	−0.0768412	−0.0384206	−	0.999262i	$-0.512233\pi$
$971$	−2325.00	−0.0768412	−0.0384206	+	0.999262i	$0.512233\pi$
$972$	−19712.0	−0.650476
$973$	57304.9i	1.88809i
$974$	−16470.0	−0.541820
$975$	0	0
$976$	1360.00	0.0446030
$977$	32938.4i	1.07860i	0.842113	+	0.539300i	$0.181311\pi$
$977$	32938.4i	1.07860i	−0.842113	+	0.539300i	$0.818689\pi$
$978$	−39606.0	−1.29495
$979$	6903.00	0.225353
$980$	− 9089.80i	− 0.296289i
$981$	13413.0i	0.436538i
$982$	5663.81i	0.184052i
$983$	− 42702.0i	− 1.38554i	−0.721161	−	0.692768i	$-0.756391\pi$
$983$	− 42702.0i	− 1.38554i	0.721161	−	0.692768i	$-0.243609\pi$
$984$	38136.0	1.23550
$985$	38136.0	1.23362
$986$	6453.62i	0.208443i
$987$	−54054.0	−1.74322
$988$	0	0
$989$	−4845.00	−0.155776
$990$	23777.6i	0.763335i
$991$	−4843.00	−0.155240	−0.0776201	−	0.996983i	$-0.524732\pi$
$991$	−4843.00	−0.155240	−0.0776201	+	0.996983i	$0.524732\pi$
$992$	−12096.0	−0.387146
$993$	− 36942.9i	− 1.18061i
$994$	45528.7i	1.45280i
$995$	23348.0i	0.743902i
$996$	11930.4i	0.379546i
$997$	10943.0	0.347611	0.173806	−	0.984780i	$-0.444394\pi$
$997$	10943.0	0.347611	0.173806	+	0.984780i	$0.444394\pi$
$998$	50004.0	1.58602
$999$	− 1394.30i	− 0.0441579i

Twists

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

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

\(f(q)\)	\(=\)	\(q-3.46410i q^{2} -7.00000 q^{3} -4.00000 q^{4} -13.8564i q^{5} +24.2487i q^{6} -22.5167i q^{7} -13.8564i q^{8} +22.0000 q^{9} +O(q^{10})\)	\(q-3.46410i q^{2} -7.00000 q^{3} -4.00000 q^{4} -13.8564i q^{5} +24.2487i q^{6} -22.5167i q^{7} -13.8564i q^{8} +22.0000 q^{9} -48.0000 q^{10} -22.5167i q^{11} +28.0000 q^{12} -78.0000 q^{14} +96.9948i q^{15} -80.0000 q^{16} +27.0000 q^{17} -76.2102i q^{18} +88.3346i q^{19} +55.4256i q^{20} +157.617i q^{21} -78.0000 q^{22} +57.0000 q^{23} +96.9948i q^{24} -67.0000 q^{25} +35.0000 q^{27} +90.0666i q^{28} -69.0000 q^{29} +336.000 q^{30} +72.7461i q^{31} +166.277i q^{32} +157.617i q^{33} -93.5307i q^{34} -312.000 q^{35} -88.0000 q^{36} -39.8372i q^{37} +306.000 q^{38} -192.000 q^{40} -393.176i q^{41} +546.000 q^{42} -85.0000 q^{43} +90.0666i q^{44} -304.841i q^{45} -197.454i q^{46} +342.946i q^{47} +560.000 q^{48} -164.000 q^{49} +232.095i q^{50} -189.000 q^{51} +426.000 q^{53} -121.244i q^{54} -312.000 q^{55} -312.000 q^{56} -618.342i q^{57} +239.023i q^{58} +19.0526i q^{59} -387.979i q^{60} -17.0000 q^{61} +252.000 q^{62} -495.367i q^{63} -64.0000 q^{64} +546.000 q^{66} +164.545i q^{67} -108.000 q^{68} -399.000 q^{69} +1080.80i q^{70} -583.701i q^{71} -304.841i q^{72} -1004.59i q^{73} -138.000 q^{74} +469.000 q^{75} -353.338i q^{76} -507.000 q^{77} -1244.00 q^{79} +1108.51i q^{80} -839.000 q^{81} -1362.00 q^{82} +426.084i q^{83} -630.466i q^{84} -374.123i q^{85} +294.449i q^{86} +483.000 q^{87} -312.000 q^{88} +306.573i q^{89} -1056.00 q^{90} -228.000 q^{92} -509.223i q^{93} +1188.00 q^{94} +1224.00 q^{95} -1163.94i q^{96} -1234.95i q^{97} +568.113i q^{98} -495.367i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} - 8 q^{4} + 44 q^{9}+O(q^{10}) \) 2 * q - 14 * q^3 - 8 * q^4 + 44 * q^9	\( 2 q - 14 q^{3} - 8 q^{4} + 44 q^{9} - 96 q^{10} + 56 q^{12} - 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} - 134 q^{25} + 70 q^{27} - 138 q^{29} + 672 q^{30} - 624 q^{35} - 176 q^{36} + 612 q^{38} - 384 q^{40} + 1092 q^{42} - 170 q^{43} + 1120 q^{48} - 328 q^{49} - 378 q^{51} + 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} - 128 q^{64} + 1092 q^{66} - 216 q^{68} - 798 q^{69} - 276 q^{74} + 938 q^{75} - 1014 q^{77} - 2488 q^{79} - 1678 q^{81} - 2724 q^{82} + 966 q^{87} - 624 q^{88} - 2112 q^{90} - 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100}) \) 2 * q - 14 * q^3 - 8 * q^4 + 44 * q^9 - 96 * q^10 + 56 * q^12 - 156 * q^14 - 160 * q^16 + 54 * q^17 - 156 * q^22 + 114 * q^23 - 134 * q^25 + 70 * q^27 - 138 * q^29 + 672 * q^30 - 624 * q^35 - 176 * q^36 + 612 * q^38 - 384 * q^40 + 1092 * q^42 - 170 * q^43 + 1120 * q^48 - 328 * q^49 - 378 * q^51 + 852 * q^53 - 624 * q^55 - 624 * q^56 - 34 * q^61 + 504 * q^62 - 128 * q^64 + 1092 * q^66 - 216 * q^68 - 798 * q^69 - 276 * q^74 + 938 * q^75 - 1014 * q^77 - 2488 * q^79 - 1678 * q^81 - 2724 * q^82 + 966 * q^87 - 624 * q^88 - 2112 * q^90 - 456 * q^92 + 2376 * q^94 + 2448 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more