LMFDB - Embedded newform 169.4.b.c.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.c.168.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} +2.00000 q^{3} -8.00000 q^{4} -17.0000i q^{5} -8.00000i q^{6} +20.0000i q^{7} -23.0000 q^{9} +O(q^{10})$	\(q-4.00000i q^{2} +2.00000 q^{3} -8.00000 q^{4} -17.0000i q^{5} -8.00000i q^{6} +20.0000i q^{7} -23.0000 q^{9} -68.0000 q^{10} -32.0000i q^{11} -16.0000 q^{12} +80.0000 q^{14} -34.0000i q^{15} -64.0000 q^{16} +13.0000 q^{17} +92.0000i q^{18} -30.0000i q^{19} +136.000i q^{20} +40.0000i q^{21} -128.000 q^{22} -78.0000 q^{23} -164.000 q^{25} -100.000 q^{27} -160.000i q^{28} +197.000 q^{29} -136.000 q^{30} +74.0000i q^{31} +256.000i q^{32} -64.0000i q^{33} -52.0000i q^{34} +340.000 q^{35} +184.000 q^{36} -227.000i q^{37} -120.000 q^{38} +165.000i q^{41} +160.000 q^{42} +156.000 q^{43} +256.000i q^{44} +391.000i q^{45} +312.000i q^{46} -162.000i q^{47} -128.000 q^{48} -57.0000 q^{49} +656.000i q^{50} +26.0000 q^{51} +93.0000 q^{53} +400.000i q^{54} -544.000 q^{55} -60.0000i q^{57} -788.000i q^{58} -864.000i q^{59} +272.000i q^{60} +145.000 q^{61} +296.000 q^{62} -460.000i q^{63} +512.000 q^{64} -256.000 q^{66} -862.000i q^{67} -104.000 q^{68} -156.000 q^{69} -1360.00i q^{70} -654.000i q^{71} +215.000i q^{73} -908.000 q^{74} -328.000 q^{75} +240.000i q^{76} +640.000 q^{77} -76.0000 q^{79} +1088.00i q^{80} +421.000 q^{81} +660.000 q^{82} -628.000i q^{83} -320.000i q^{84} -221.000i q^{85} -624.000i q^{86} +394.000 q^{87} -266.000i q^{89} +1564.00 q^{90} +624.000 q^{92} +148.000i q^{93} -648.000 q^{94} -510.000 q^{95} +512.000i q^{96} -238.000i q^{97} +228.000i q^{98} +736.000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 16 q^{4} - 46 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 16 * q^4 - 46 * q^9	$ 2 q + 4 q^{3} - 16 q^{4} - 46 q^{9} - 136 q^{10} - 32 q^{12} + 160 q^{14} - 128 q^{16} + 26 q^{17} - 256 q^{22} - 156 q^{23} - 328 q^{25} - 200 q^{27} + 394 q^{29} - 272 q^{30} + 680 q^{35} + 368 q^{36} - 240 q^{38} + 320 q^{42} + 312 q^{43} - 256 q^{48} - 114 q^{49} + 52 q^{51} + 186 q^{53} - 1088 q^{55} + 290 q^{61} + 592 q^{62} + 1024 q^{64} - 512 q^{66} - 208 q^{68} - 312 q^{69} - 1816 q^{74} - 656 q^{75} + 1280 q^{77} - 152 q^{79} + 842 q^{81} + 1320 q^{82} + 788 q^{87} + 3128 q^{90} + 1248 q^{92} - 1296 q^{94} - 1020 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 - 16 * q^4 - 46 * q^9 - 136 * q^10 - 32 * q^12 + 160 * q^14 - 128 * q^16 + 26 * q^17 - 256 * q^22 - 156 * q^23 - 328 * q^25 - 200 * q^27 + 394 * q^29 - 272 * q^30 + 680 * q^35 + 368 * q^36 - 240 * q^38 + 320 * q^42 + 312 * q^43 - 256 * q^48 - 114 * q^49 + 52 * q^51 + 186 * q^53 - 1088 * q^55 + 290 * q^61 + 592 * q^62 + 1024 * q^64 - 512 * q^66 - 208 * q^68 - 312 * q^69 - 1816 * q^74 - 656 * q^75 + 1280 * q^77 - 152 * q^79 + 842 * q^81 + 1320 * q^82 + 788 * q^87 + 3128 * q^90 + 1248 * q^92 - 1296 * q^94 - 1020 * 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 $

$2$	− 4.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 4.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	−8.00000	−1.00000
$5$	− 17.0000i	− 1.52053i	−0.649615	−	0.760263i	$-0.725070\pi$
$5$	− 17.0000i	− 1.52053i	0.649615	−	0.760263i	$-0.274930\pi$
$6$	− 8.00000i	− 0.544331i
$7$	20.0000i	1.07990i	0.841698	+	0.539949i	$0.181557\pi$
$7$	20.0000i	1.07990i	−0.841698	+	0.539949i	$0.818443\pi$
$8$	0	0
$9$	−23.0000	−0.851852
$10$	−68.0000	−2.15035
$11$	− 32.0000i	− 0.877124i	−0.898701	−	0.438562i	$-0.855488\pi$
$11$	− 32.0000i	− 0.877124i	0.898701	−	0.438562i	$-0.144512\pi$
$12$	−16.0000	−0.384900
$13$	0	0
$13$	0	0
$14$	80.0000	1.52721
$15$	− 34.0000i	− 0.585251i
$16$	−64.0000	−1.00000
$17$	13.0000	0.185468	0.0927342	−	0.995691i	$-0.470439\pi$
$17$	13.0000	0.185468	0.0927342	+	0.995691i	$0.470439\pi$
$18$	92.0000i	1.20470i
$19$	− 30.0000i	− 0.362235i	−0.983461	−	0.181118i	$-0.942029\pi$
$19$	− 30.0000i	− 0.362235i	0.983461	−	0.181118i	$-0.0579715\pi$
$20$	136.000i	1.52053i
$21$	40.0000i	0.415653i
$22$	−128.000	−1.24044
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	0	0
$25$	−164.000	−1.31200
$26$	0	0
$27$	−100.000	−0.712778
$28$	− 160.000i	− 1.07990i
$29$	197.000	1.26145	0.630724	−	0.776007i	$-0.282758\pi$
$29$	197.000	1.26145	0.630724	+	0.776007i	$0.282758\pi$
$30$	−136.000	−0.827670
$31$	74.0000i	0.428735i	0.976753	+	0.214368i	$0.0687691\pi$
$31$	74.0000i	0.428735i	−0.976753	+	0.214368i	$0.931231\pi$
$32$	256.000i	1.41421i
$33$	− 64.0000i	− 0.337605i
$34$	− 52.0000i	− 0.262292i
$35$	340.000	1.64201
$36$	184.000	0.851852
$37$	− 227.000i	− 1.00861i	−0.863526	−	0.504305i	$-0.831749\pi$
$37$	− 227.000i	− 1.00861i	0.863526	−	0.504305i	$-0.168251\pi$
$38$	−120.000	−0.512278
$39$	0	0
$40$	0	0
$41$	165.000i	0.628504i	0.949340	+	0.314252i	$0.101754\pi$
$41$	165.000i	0.628504i	−0.949340	+	0.314252i	$0.898246\pi$
$42$	160.000	0.587822
$43$	156.000	0.553251	0.276625	−	0.960978i	$-0.410784\pi$
$43$	156.000	0.553251	0.276625	+	0.960978i	$0.410784\pi$
$44$	256.000i	0.877124i
$45$	391.000i	1.29526i
$46$	312.000i	1.00004i
$47$	− 162.000i	− 0.502769i	−0.967887	−	0.251384i	$-0.919114\pi$
$47$	− 162.000i	− 0.502769i	0.967887	−	0.251384i	$-0.0808858\pi$
$48$	−128.000	−0.384900
$49$	−57.0000	−0.166181
$50$	656.000i	1.85545i
$51$	26.0000	0.0713868
$52$	0	0
$53$	93.0000	0.241029	0.120514	−	0.992712i	$-0.461546\pi$
$53$	93.0000	0.241029	0.120514	+	0.992712i	$0.461546\pi$
$54$	400.000i	1.00802i
$55$	−544.000	−1.33369
$56$	0	0
$57$	− 60.0000i	− 0.139424i
$58$	− 788.000i	− 1.78396i
$59$	− 864.000i	− 1.90650i	−0.302190	−	0.953248i	$-0.597718\pi$
$59$	− 864.000i	− 1.90650i	0.302190	−	0.953248i	$-0.402282\pi$
$60$	272.000i	0.585251i
$61$	145.000	0.304350	0.152175	−	0.988354i	$-0.451372\pi$
$61$	145.000	0.304350	0.152175	+	0.988354i	$0.451372\pi$
$62$	296.000	0.606323
$63$	− 460.000i	− 0.919914i
$64$	512.000	1.00000
$65$	0	0
$66$	−256.000	−0.477446
$67$	− 862.000i	− 1.57179i	−0.618359	−	0.785896i	$-0.712202\pi$
$67$	− 862.000i	− 1.57179i	0.618359	−	0.785896i	$-0.287798\pi$
$68$	−104.000	−0.185468
$69$	−156.000	−0.272177
$70$	− 1360.00i	− 2.32216i
$71$	− 654.000i	− 1.09318i	−0.837402	−	0.546588i	$-0.815926\pi$
$71$	− 654.000i	− 1.09318i	0.837402	−	0.546588i	$-0.184074\pi$
$72$	0	0
$73$	215.000i	0.344710i	0.985035	+	0.172355i	$0.0551377\pi$
$73$	215.000i	0.344710i	−0.985035	+	0.172355i	$0.944862\pi$
$74$	−908.000	−1.42639
$75$	−328.000	−0.504989
$76$	240.000i	0.362235i
$77$	640.000	0.947205
$78$	0	0
$79$	−76.0000	−0.108236	−0.0541182	−	0.998535i	$-0.517235\pi$
$79$	−76.0000	−0.108236	−0.0541182	+	0.998535i	$0.517235\pi$
$80$	1088.00i	1.52053i
$81$	421.000	0.577503
$82$	660.000	0.888839
$83$	− 628.000i	− 0.830505i	−0.909706	−	0.415253i	$-0.863693\pi$
$83$	− 628.000i	− 0.830505i	0.909706	−	0.415253i	$-0.136307\pi$
$84$	− 320.000i	− 0.415653i
$85$	− 221.000i	− 0.282010i
$86$	− 624.000i	− 0.782415i
$87$	394.000	0.485531
$88$	0	0
$89$	− 266.000i	− 0.316808i	−0.987374	−	0.158404i	$-0.949365\pi$
$89$	− 266.000i	− 0.316808i	0.987374	−	0.158404i	$-0.0506349\pi$
$90$	1564.00	1.83178
$91$	0	0
$92$	624.000	0.707136
$93$	148.000i	0.165020i
$94$	−648.000	−0.711022
$95$	−510.000	−0.550788
$96$	512.000i	0.544331i
$97$	− 238.000i	− 0.249126i	−0.992212	−	0.124563i	$-0.960247\pi$
$97$	− 238.000i	− 0.249126i	0.992212	−	0.124563i	$-0.0397529\pi$
$98$	228.000i	0.235015i
$99$	736.000i	0.747180i
$100$	1312.00	1.31200
$101$	819.000	0.806867	0.403433	−	0.915009i	$-0.367817\pi$
$101$	819.000	0.806867	0.403433	+	0.915009i	$0.367817\pi$
$102$	− 104.000i	− 0.100956i
$103$	−1638.00	−1.56696	−0.783480	−	0.621417i	$-0.786557\pi$
$103$	−1638.00	−1.56696	−0.783480	+	0.621417i	$0.786557\pi$
$104$	0	0
$105$	680.000	0.632011
$106$	− 372.000i	− 0.340866i
$107$	522.000	0.471623	0.235811	−	0.971799i	$-0.424225\pi$
$107$	522.000	0.471623	0.235811	+	0.971799i	$0.424225\pi$
$108$	800.000	0.712778
$109$	1634.00i	1.43586i	0.696115	+	0.717930i	$0.254910\pi$
$109$	1634.00i	1.43586i	−0.696115	+	0.717930i	$0.745090\pi$
$110$	2176.00i	1.88612i
$111$	− 454.000i	− 0.388214i
$112$	− 1280.00i	− 1.07990i
$113$	327.000	0.272226	0.136113	−	0.990693i	$-0.456539\pi$
$113$	327.000	0.272226	0.136113	+	0.990693i	$0.456539\pi$
$114$	−240.000	−0.197176
$115$	1326.00i	1.07522i
$116$	−1576.00	−1.26145
$117$	0	0
$118$	−3456.00	−2.69619
$119$	260.000i	0.200287i
$120$	0	0
$121$	307.000	0.230654
$122$	− 580.000i	− 0.430416i
$123$	330.000i	0.241911i
$124$	− 592.000i	− 0.428735i
$125$	663.000i	0.474404i
$126$	−1840.00	−1.30095
$127$	2158.00	1.50781	0.753904	−	0.656985i	$-0.228169\pi$
$127$	2158.00	1.50781	0.753904	+	0.656985i	$0.228169\pi$
$128$	0	0
$129$	312.000	0.212946
$130$	0	0
$131$	730.000	0.486873	0.243437	−	0.969917i	$-0.421725\pi$
$131$	730.000	0.486873	0.243437	+	0.969917i	$0.421725\pi$
$132$	512.000i	0.337605i
$133$	600.000	0.391177
$134$	−3448.00	−2.22285
$135$	1700.00i	1.08380i
$136$	0	0
$137$	1671.00i	1.04207i	0.853536	+	0.521033i	$0.174453\pi$
$137$	1671.00i	1.04207i	−0.853536	+	0.521033i	$0.825547\pi$
$138$	624.000i	0.384916i
$139$	912.000	0.556510	0.278255	−	0.960507i	$-0.410244\pi$
$139$	912.000	0.556510	0.278255	+	0.960507i	$0.410244\pi$
$140$	−2720.00	−1.64201
$141$	− 324.000i	− 0.193516i
$142$	−2616.00	−1.54598
$143$	0	0
$144$	1472.00	0.851852
$145$	− 3349.00i	− 1.91806i
$146$	860.000	0.487494
$147$	−114.000	−0.0639630
$148$	1816.00i	1.00861i
$149$	2115.00i	1.16287i	0.813593	+	0.581435i	$0.197508\pi$
$149$	2115.00i	1.16287i	−0.813593	+	0.581435i	$0.802492\pi$
$150$	1312.00i	0.714162i
$151$	514.000i	0.277011i	0.990362	+	0.138506i	$0.0442299\pi$
$151$	514.000i	0.277011i	−0.990362	+	0.138506i	$0.955770\pi$
$152$	0	0
$153$	−299.000	−0.157992
$154$	− 2560.00i	− 1.33955i
$155$	1258.00	0.651903
$156$	0	0
$157$	2901.00	1.47468	0.737341	−	0.675521i	$-0.236081\pi$
$157$	2901.00	1.47468	0.737341	+	0.675521i	$0.236081\pi$
$158$	304.000i	0.153069i
$159$	186.000	0.0927721
$160$	4352.00	2.15035
$161$	− 1560.00i	− 0.763635i
$162$	− 1684.00i	− 0.816713i
$163$	2360.00i	1.13405i	0.823702	+	0.567023i	$0.191905\pi$
$163$	2360.00i	1.13405i	−0.823702	+	0.567023i	$0.808095\pi$
$164$	− 1320.00i	− 0.628504i
$165$	−1088.00	−0.513337
$166$	−2512.00	−1.17451
$167$	280.000i	0.129743i	0.997894	+	0.0648714i	$0.0206637\pi$
$167$	280.000i	0.129743i	−0.997894	+	0.0648714i	$0.979336\pi$
$168$	0	0
$169$	0	0
$170$	−884.000	−0.398822
$171$	690.000i	0.308571i
$172$	−1248.00	−0.553251
$173$	−1326.00	−0.582739	−0.291370	−	0.956611i	$-0.594111\pi$
$173$	−1326.00	−0.582739	−0.291370	+	0.956611i	$0.594111\pi$
$174$	− 1576.00i	− 0.686645i
$175$	− 3280.00i	− 1.41683i
$176$	2048.00i	0.877124i
$177$	− 1728.00i	− 0.733810i
$178$	−1064.00	−0.448035
$179$	−4264.00	−1.78048	−0.890241	−	0.455490i	$-0.849464\pi$
$179$	−4264.00	−1.78048	−0.890241	+	0.455490i	$0.849464\pi$
$180$	− 3128.00i	− 1.29526i
$181$	403.000	0.165496	0.0827479	−	0.996571i	$-0.473630\pi$
$181$	403.000	0.165496	0.0827479	+	0.996571i	$0.473630\pi$
$182$	0	0
$183$	290.000	0.117144
$184$	0	0
$185$	−3859.00	−1.53362
$186$	592.000	0.233374
$187$	− 416.000i	− 0.162679i
$188$	1296.00i	0.502769i
$189$	− 2000.00i	− 0.769728i
$190$	2040.00i	0.778932i
$191$	−1246.00	−0.472028	−0.236014	−	0.971750i	$-0.575841\pi$
$191$	−1246.00	−0.472028	−0.236014	+	0.971750i	$0.575841\pi$
$192$	1024.00	0.384900
$193$	267.000i	0.0995807i	0.998760	+	0.0497904i	$0.0158553\pi$
$193$	267.000i	0.0995807i	−0.998760	+	0.0497904i	$0.984145\pi$
$194$	−952.000	−0.352318
$195$	0	0
$196$	456.000	0.166181
$197$	− 1278.00i	− 0.462202i	−0.972930	−	0.231101i	$-0.925767\pi$
$197$	− 1278.00i	− 0.462202i	0.972930	−	0.231101i	$-0.0742327\pi$
$198$	2944.00	1.05667
$199$	−4238.00	−1.50967	−0.754834	−	0.655916i	$-0.772283\pi$
$199$	−4238.00	−1.50967	−0.754834	+	0.655916i	$0.772283\pi$
$200$	0	0
$201$	− 1724.00i	− 0.604983i
$202$	− 3276.00i	− 1.14108i
$203$	3940.00i	1.36224i
$204$	−208.000	−0.0713868
$205$	2805.00	0.955657
$206$	6552.00i	2.21602i
$207$	1794.00	0.602375
$208$	0	0
$209$	−960.000	−0.317725
$210$	− 2720.00i	− 0.893799i
$211$	3070.00	1.00165	0.500823	−	0.865549i	$-0.333031\pi$
$211$	3070.00	1.00165	0.500823	+	0.865549i	$0.333031\pi$
$212$	−744.000	−0.241029
$213$	− 1308.00i	− 0.420764i
$214$	− 2088.00i	− 0.666975i
$215$	− 2652.00i	− 0.841232i
$216$	0	0
$217$	−1480.00	−0.462991
$218$	6536.00	2.03061
$219$	430.000i	0.132679i
$220$	4352.00	1.33369
$221$	0	0
$222$	−1816.00	−0.549018
$223$	5378.00i	1.61497i	0.589891	+	0.807483i	$0.299171\pi$
$223$	5378.00i	1.61497i	−0.589891	+	0.807483i	$0.700829\pi$
$224$	−5120.00	−1.52721
$225$	3772.00	1.11763
$226$	− 1308.00i	− 0.384986i
$227$	3974.00i	1.16195i	0.813920	+	0.580977i	$0.197329\pi$
$227$	3974.00i	1.16195i	−0.813920	+	0.580977i	$0.802671\pi$
$228$	480.000i	0.139424i
$229$	− 6298.00i	− 1.81740i	−0.417455	−	0.908698i	$-0.637078\pi$
$229$	− 6298.00i	− 1.81740i	0.417455	−	0.908698i	$-0.362922\pi$
$230$	5304.00	1.52059
$231$	1280.00	0.364579
$232$	0	0
$233$	−4030.00	−1.13311	−0.566554	−	0.824025i	$-0.691724\pi$
$233$	−4030.00	−1.13311	−0.566554	+	0.824025i	$0.691724\pi$
$234$	0	0
$235$	−2754.00	−0.764473
$236$	6912.00i	1.90650i
$237$	−152.000	−0.0416602
$238$	1040.00	0.283249
$239$	984.000i	0.266317i	0.991095	+	0.133158i	$0.0425119\pi$
$239$	984.000i	0.266317i	−0.991095	+	0.133158i	$0.957488\pi$
$240$	2176.00i	0.585251i
$241$	943.000i	0.252050i	0.992027	+	0.126025i	$0.0402219\pi$
$241$	943.000i	0.252050i	−0.992027	+	0.126025i	$0.959778\pi$
$242$	− 1228.00i	− 0.326194i
$243$	3542.00	0.935059
$244$	−1160.00	−0.304350
$245$	969.000i	0.252682i
$246$	1320.00	0.342114
$247$	0	0
$248$	0	0
$249$	− 1256.00i	− 0.319662i
$250$	2652.00	0.670909
$251$	2730.00	0.686518	0.343259	−	0.939241i	$-0.388469\pi$
$251$	2730.00	0.686518	0.343259	+	0.939241i	$0.388469\pi$
$252$	3680.00i	0.919914i
$253$	2496.00i	0.620246i
$254$	− 8632.00i	− 2.13236i
$255$	− 442.000i	− 0.108546i
$256$	4096.00	1.00000
$257$	1885.00	0.457522	0.228761	−	0.973483i	$-0.426533\pi$
$257$	1885.00	0.457522	0.228761	+	0.973483i	$0.426533\pi$
$258$	− 1248.00i	− 0.301151i
$259$	4540.00	1.08920
$260$	0	0
$261$	−4531.00	−1.07457
$262$	− 2920.00i	− 0.688543i
$263$	4032.00	0.945338	0.472669	−	0.881240i	$-0.343291\pi$
$263$	4032.00	0.945338	0.472669	+	0.881240i	$0.343291\pi$
$264$	0	0
$265$	− 1581.00i	− 0.366491i
$266$	− 2400.00i	− 0.553208i
$267$	− 532.000i	− 0.121940i
$268$	6896.00i	1.57179i
$269$	4006.00	0.907993	0.453997	−	0.891003i	$-0.349998\pi$
$269$	4006.00	0.907993	0.453997	+	0.891003i	$0.349998\pi$
$270$	6800.00	1.53272
$271$	− 4296.00i	− 0.962965i	−0.876456	−	0.481482i	$-0.840099\pi$
$271$	− 4296.00i	− 0.962965i	0.876456	−	0.481482i	$-0.159901\pi$
$272$	−832.000	−0.185468
$273$	0	0
$274$	6684.00	1.47371
$275$	5248.00i	1.15079i
$276$	1248.00	0.272177
$277$	5551.00	1.20407	0.602035	−	0.798470i	$-0.294357\pi$
$277$	5551.00	1.20407	0.602035	+	0.798470i	$0.294357\pi$
$278$	− 3648.00i	− 0.787023i
$279$	− 1702.00i	− 0.365219i
$280$	0	0
$281$	− 5557.00i	− 1.17973i	−0.807504	−	0.589863i	$-0.799182\pi$
$281$	− 5557.00i	− 1.17973i	0.807504	−	0.589863i	$-0.200818\pi$
$282$	−1296.00	−0.273673
$283$	−3120.00	−0.655352	−0.327676	−	0.944790i	$-0.606266\pi$
$283$	−3120.00	−0.655352	−0.327676	+	0.944790i	$0.606266\pi$
$284$	5232.00i	1.09318i
$285$	−1020.00	−0.211999
$286$	0	0
$287$	−3300.00	−0.678721
$288$	− 5888.00i	− 1.20470i
$289$	−4744.00	−0.965601
$290$	−13396.0	−2.71255
$291$	− 476.000i	− 0.0958887i
$292$	− 1720.00i	− 0.344710i
$293$	8301.00i	1.65512i	0.561379	+	0.827559i	$0.310271\pi$
$293$	8301.00i	1.65512i	−0.561379	+	0.827559i	$0.689729\pi$
$294$	456.000i	0.0904573i
$295$	−14688.0	−2.89888
$296$	0	0
$297$	3200.00i	0.625195i
$298$	8460.00	1.64455
$299$	0	0
$300$	2624.00	0.504989
$301$	3120.00i	0.597455i
$302$	2056.00	0.391753
$303$	1638.00	0.310563
$304$	1920.00i	0.362235i
$305$	− 2465.00i	− 0.462772i
$306$	1196.00i	0.223434i
$307$	8678.00i	1.61329i	0.591037	+	0.806644i	$0.298719\pi$
$307$	8678.00i	1.61329i	−0.591037	+	0.806644i	$0.701281\pi$
$308$	−5120.00	−0.947205
$309$	−3276.00	−0.603123
$310$	− 5032.00i	− 0.921930i
$311$	−8658.00	−1.57862	−0.789309	−	0.613996i	$-0.789561\pi$
$311$	−8658.00	−1.57862	−0.789309	+	0.613996i	$0.789561\pi$
$312$	0	0
$313$	−5250.00	−0.948075	−0.474038	−	0.880505i	$-0.657204\pi$
$313$	−5250.00	−0.948075	−0.474038	+	0.880505i	$0.657204\pi$
$314$	− 11604.0i	− 2.08551i
$315$	−7820.00	−1.39875
$316$	608.000	0.108236
$317$	− 6413.00i	− 1.13625i	−0.822944	−	0.568123i	$-0.807670\pi$
$317$	− 6413.00i	− 1.13625i	0.822944	−	0.568123i	$-0.192330\pi$
$318$	− 744.000i	− 0.131200i
$319$	− 6304.00i	− 1.10645i
$320$	− 8704.00i	− 1.52053i
$321$	1044.00	0.181528
$322$	−6240.00	−1.07994
$323$	− 390.000i	− 0.0671832i
$324$	−3368.00	−0.577503
$325$	0	0
$326$	9440.00	1.60378
$327$	3268.00i	0.552663i
$328$	0	0
$329$	3240.00	0.542939
$330$	4352.00i	0.725969i
$331$	− 3488.00i	− 0.579208i	−0.957147	−	0.289604i	$-0.906476\pi$
$331$	− 3488.00i	− 0.579208i	0.957147	−	0.289604i	$-0.0935236\pi$
$332$	5024.00i	0.830505i
$333$	5221.00i	0.859186i
$334$	1120.00	0.183484
$335$	−14654.0	−2.38995
$336$	− 2560.00i	− 0.415653i
$337$	1833.00	0.296290	0.148145	−	0.988966i	$-0.452670\pi$
$337$	1833.00	0.296290	0.148145	+	0.988966i	$0.452670\pi$
$338$	0	0
$339$	654.000	0.104780
$340$	1768.00i	0.282010i
$341$	2368.00	0.376054
$342$	2760.00	0.436385
$343$	5720.00i	0.900440i
$344$	0	0
$345$	2652.00i	0.413852i
$346$	5304.00i	0.824118i
$347$	7230.00	1.11852	0.559260	−	0.828992i	$-0.311085\pi$
$347$	7230.00	1.11852	0.559260	+	0.828992i	$0.311085\pi$
$348$	−3152.00	−0.485531
$349$	− 5258.00i	− 0.806459i	−0.915099	−	0.403230i	$-0.867888\pi$
$349$	− 5258.00i	− 0.806459i	0.915099	−	0.403230i	$-0.132112\pi$
$350$	−13120.0	−2.00370
$351$	0	0
$352$	8192.00	1.24044
$353$	− 3163.00i	− 0.476911i	−0.971153	−	0.238455i	$-0.923359\pi$
$353$	− 3163.00i	− 0.476911i	0.971153	−	0.238455i	$-0.0766411\pi$
$354$	−6912.00	−1.03776
$355$	−11118.0	−1.66220
$356$	2128.00i	0.316808i
$357$	520.000i	0.0770905i
$358$	17056.0i	2.51798i
$359$	− 10068.0i	− 1.48014i	−0.672532	−	0.740068i	$-0.734793\pi$
$359$	− 10068.0i	− 1.48014i	0.672532	−	0.740068i	$-0.265207\pi$
$360$	0	0
$361$	5959.00	0.868786
$362$	− 1612.00i	− 0.234047i
$363$	614.000	0.0887786
$364$	0	0
$365$	3655.00	0.524141
$366$	− 1160.00i	− 0.165667i
$367$	7438.00	1.05793	0.528965	−	0.848644i	$-0.322580\pi$
$367$	7438.00	1.05793	0.528965	+	0.848644i	$0.322580\pi$
$368$	4992.00	0.707136
$369$	− 3795.00i	− 0.535392i
$370$	15436.0i	2.16886i
$371$	1860.00i	0.260287i
$372$	− 1184.00i	− 0.165020i
$373$	−9683.00	−1.34415	−0.672073	−	0.740485i	$-0.734596\pi$
$373$	−9683.00	−1.34415	−0.672073	+	0.740485i	$0.734596\pi$
$374$	−1664.00	−0.230063
$375$	1326.00i	0.182598i
$376$	0	0
$377$	0	0
$378$	−8000.00	−1.08856
$379$	1062.00i	0.143935i	0.997407	+	0.0719674i	$0.0229278\pi$
$379$	1062.00i	0.143935i	−0.997407	+	0.0719674i	$0.977072\pi$
$380$	4080.00	0.550788
$381$	4316.00	0.580355
$382$	4984.00i	0.667549i
$383$	3532.00i	0.471219i	0.971848	+	0.235609i	$0.0757086\pi$
$383$	3532.00i	0.471219i	−0.971848	+	0.235609i	$0.924291\pi$
$384$	0	0
$385$	− 10880.0i	− 1.44025i
$386$	1068.00	0.140828
$387$	−3588.00	−0.471288
$388$	1904.00i	0.249126i
$389$	11063.0	1.44194	0.720972	−	0.692964i	$-0.243696\pi$
$389$	11063.0	1.44194	0.720972	+	0.692964i	$0.243696\pi$
$390$	0	0
$391$	−1014.00	−0.131151
$392$	0	0
$393$	1460.00	0.187398
$394$	−5112.00	−0.653652
$395$	1292.00i	0.164576i
$396$	− 5888.00i	− 0.747180i
$397$	− 5986.00i	− 0.756747i	−0.925653	−	0.378374i	$-0.876483\pi$
$397$	− 5986.00i	− 0.756747i	0.925653	−	0.378374i	$-0.123517\pi$
$398$	16952.0i	2.13499i
$399$	1200.00	0.150564
$400$	10496.0	1.31200
$401$	5935.00i	0.739102i	0.929211	+	0.369551i	$0.120488\pi$
$401$	5935.00i	0.739102i	−0.929211	+	0.369551i	$0.879512\pi$
$402$	−6896.00	−0.855575
$403$	0	0
$404$	−6552.00	−0.806867
$405$	− 7157.00i	− 0.878109i
$406$	15760.0	1.92649
$407$	−7264.00	−0.884676
$408$	0	0
$409$	15089.0i	1.82421i	0.409954	+	0.912106i	$0.365545\pi$
$409$	15089.0i	1.82421i	−0.409954	+	0.912106i	$0.634455\pi$
$410$	− 11220.0i	− 1.35150i
$411$	3342.00i	0.401092i
$412$	13104.0	1.56696
$413$	17280.0	2.05882
$414$	− 7176.00i	− 0.851887i
$415$	−10676.0	−1.26281
$416$	0	0
$417$	1824.00	0.214201
$418$	3840.00i	0.449331i
$419$	−10814.0	−1.26086	−0.630428	−	0.776248i	$-0.717120\pi$
$419$	−10814.0	−1.26086	−0.630428	+	0.776248i	$0.717120\pi$
$420$	−5440.00	−0.632011
$421$	6535.00i	0.756524i	0.925699	+	0.378262i	$0.123478\pi$
$421$	6535.00i	0.756524i	−0.925699	+	0.378262i	$0.876522\pi$
$422$	− 12280.0i	− 1.41654i
$423$	3726.00i	0.428284i
$424$	0	0
$425$	−2132.00	−0.243335
$426$	−5232.00	−0.595050
$427$	2900.00i	0.328667i
$428$	−4176.00	−0.471623
$429$	0	0
$430$	−10608.0	−1.18968
$431$	− 1980.00i	− 0.221284i	−0.993860	−	0.110642i	$-0.964709\pi$
$431$	− 1980.00i	− 0.221284i	0.993860	−	0.110642i	$-0.0352906\pi$
$432$	6400.00	0.712778
$433$	6929.00	0.769022	0.384511	−	0.923120i	$-0.374370\pi$
$433$	6929.00	0.769022	0.384511	+	0.923120i	$0.374370\pi$
$434$	5920.00i	0.654767i
$435$	− 6698.00i	− 0.738263i
$436$	− 13072.0i	− 1.43586i
$437$	2340.00i	0.256150i
$438$	1720.00	0.187636
$439$	4576.00	0.497496	0.248748	−	0.968568i	$-0.419981\pi$
$439$	4576.00	0.497496	0.248748	+	0.968568i	$0.419981\pi$
$440$	0	0
$441$	1311.00	0.141561
$442$	0	0
$443$	−8812.00	−0.945081	−0.472540	−	0.881309i	$-0.656663\pi$
$443$	−8812.00	−0.945081	−0.472540	+	0.881309i	$0.656663\pi$
$444$	3632.00i	0.388214i
$445$	−4522.00	−0.481715
$446$	21512.0	2.28391
$447$	4230.00i	0.447589i
$448$	10240.0i	1.07990i
$449$	1918.00i	0.201595i	0.994907	+	0.100797i	$0.0321394\pi$
$449$	1918.00i	0.201595i	−0.994907	+	0.100797i	$0.967861\pi$
$450$	− 15088.0i	− 1.58057i
$451$	5280.00	0.551276
$452$	−2616.00	−0.272226
$453$	1028.00i	0.106622i
$454$	15896.0	1.64325
$455$	0	0
$456$	0	0
$457$	11761.0i	1.20384i	0.798555	+	0.601922i	$0.205598\pi$
$457$	11761.0i	1.20384i	−0.798555	+	0.601922i	$0.794402\pi$
$458$	−25192.0	−2.57019
$459$	−1300.00	−0.132198
$460$	− 10608.0i	− 1.07522i
$461$	− 901.000i	− 0.0910277i	−0.998964	−	0.0455138i	$-0.985507\pi$
$461$	− 901.000i	− 0.0910277i	0.998964	−	0.0455138i	$-0.0144925\pi$
$462$	− 5120.00i	− 0.515593i
$463$	1372.00i	0.137715i	0.997626	+	0.0688577i	$0.0219354\pi$
$463$	1372.00i	0.137715i	−0.997626	+	0.0688577i	$0.978065\pi$
$464$	−12608.0	−1.26145
$465$	2516.00	0.250918
$466$	16120.0i	1.60246i
$467$	6396.00	0.633772	0.316886	−	0.948464i	$-0.397363\pi$
$467$	6396.00	0.633772	0.316886	+	0.948464i	$0.397363\pi$
$468$	0	0
$469$	17240.0	1.69738
$470$	11016.0i	1.08113i
$471$	5802.00	0.567605
$472$	0	0
$473$	− 4992.00i	− 0.485269i
$474$	608.000i	0.0589164i
$475$	4920.00i	0.475253i
$476$	− 2080.00i	− 0.200287i
$477$	−2139.00	−0.205321
$478$	3936.00	0.376629
$479$	3270.00i	0.311921i	0.987763	+	0.155960i	$0.0498472\pi$
$479$	3270.00i	0.311921i	−0.987763	+	0.155960i	$0.950153\pi$
$480$	8704.00	0.827670
$481$	0	0
$482$	3772.00	0.356452
$483$	− 3120.00i	− 0.293923i
$484$	−2456.00	−0.230654
$485$	−4046.00	−0.378803
$486$	− 14168.0i	− 1.32237i
$487$	− 19920.0i	− 1.85351i	−0.375661	−	0.926757i	$-0.622584\pi$
$487$	− 19920.0i	− 1.85351i	0.375661	−	0.926757i	$-0.377416\pi$
$488$	0	0
$489$	4720.00i	0.436494i
$490$	3876.00	0.357347
$491$	−6552.00	−0.602215	−0.301108	−	0.953590i	$-0.597356\pi$
$491$	−6552.00	−0.602215	−0.301108	+	0.953590i	$0.597356\pi$
$492$	− 2640.00i	− 0.241911i
$493$	2561.00	0.233959
$494$	0	0
$495$	12512.0	1.13611
$496$	− 4736.00i	− 0.428735i
$497$	13080.0	1.18052
$498$	−5024.00	−0.452070
$499$	− 1746.00i	− 0.156637i	−0.996928	−	0.0783183i	$-0.975045\pi$
$499$	− 1746.00i	− 0.156637i	0.996928	−	0.0783183i	$-0.0249551\pi$
$500$	− 5304.00i	− 0.474404i
$501$	560.000i	0.0499380i
$502$	− 10920.0i	− 0.970883i
$503$	14692.0	1.30235	0.651177	−	0.758926i	$-0.274276\pi$
$503$	14692.0	1.30235	0.651177	+	0.758926i	$0.274276\pi$
$504$	0	0
$505$	− 13923.0i	− 1.22686i
$506$	9984.00	0.877160
$507$	0	0
$508$	−17264.0	−1.50781
$509$	− 8077.00i	− 0.703353i	−0.936122	−	0.351677i	$-0.885612\pi$
$509$	− 8077.00i	− 0.703353i	0.936122	−	0.351677i	$-0.114388\pi$
$510$	−1768.00	−0.153507
$511$	−4300.00	−0.372252
$512$	− 16384.0i	− 1.41421i
$513$	3000.00i	0.258193i
$514$	− 7540.00i	− 0.647033i
$515$	27846.0i	2.38260i
$516$	−2496.00	−0.212946
$517$	−5184.00	−0.440990
$518$	− 18160.0i	− 1.54036i
$519$	−2652.00	−0.224296
$520$	0	0
$521$	11247.0	0.945758	0.472879	−	0.881127i	$-0.343215\pi$
$521$	11247.0	0.945758	0.472879	+	0.881127i	$0.343215\pi$
$522$	18124.0i	1.51967i
$523$	2732.00	0.228417	0.114208	−	0.993457i	$-0.463567\pi$
$523$	2732.00	0.228417	0.114208	+	0.993457i	$0.463567\pi$
$524$	−5840.00	−0.486873
$525$	− 6560.00i	− 0.545337i
$526$	− 16128.0i	− 1.33691i
$527$	962.000i	0.0795168i
$528$	4096.00i	0.337605i
$529$	−6083.00	−0.499959
$530$	−6324.00	−0.518296
$531$	19872.0i	1.62405i
$532$	−4800.00	−0.391177
$533$	0	0
$534$	−2128.00	−0.172449
$535$	− 8874.00i	− 0.717115i
$536$	0	0
$537$	−8528.00	−0.685308
$538$	− 16024.0i	− 1.28410i
$539$	1824.00i	0.145761i
$540$	− 13600.0i	− 1.08380i
$541$	− 18375.0i	− 1.46026i	−0.683306	−	0.730132i	$-0.739458\pi$
$541$	− 18375.0i	− 1.46026i	0.683306	−	0.730132i	$-0.260542\pi$
$542$	−17184.0	−1.36184
$543$	806.000	0.0636994
$544$	3328.00i	0.262292i
$545$	27778.0	2.18326
$546$	0	0
$547$	−10346.0	−0.808708	−0.404354	−	0.914603i	$-0.632504\pi$
$547$	−10346.0	−0.808708	−0.404354	+	0.914603i	$0.632504\pi$
$548$	− 13368.0i	− 1.04207i
$549$	−3335.00	−0.259261
$550$	20992.0	1.62746
$551$	− 5910.00i	− 0.456941i
$552$	0	0
$553$	− 1520.00i	− 0.116884i
$554$	− 22204.0i	− 1.70281i
$555$	−7718.00	−0.590290
$556$	−7296.00	−0.556510
$557$	345.000i	0.0262444i	0.999914	+	0.0131222i	$0.00417704\pi$
$557$	345.000i	0.0262444i	−0.999914	+	0.0131222i	$0.995823\pi$
$558$	−6808.00	−0.516498
$559$	0	0
$560$	−21760.0	−1.64201
$561$	− 832.000i	− 0.0626151i
$562$	−22228.0	−1.66838
$563$	8580.00	0.642280	0.321140	−	0.947032i	$-0.395934\pi$
$563$	8580.00	0.642280	0.321140	+	0.947032i	$0.395934\pi$
$564$	2592.00i	0.193516i
$565$	− 5559.00i	− 0.413927i
$566$	12480.0i	0.926808i
$567$	8420.00i	0.623645i
$568$	0	0
$569$	19682.0	1.45011	0.725055	−	0.688691i	$-0.241814\pi$
$569$	19682.0	1.45011	0.725055	+	0.688691i	$0.241814\pi$
$570$	4080.00i	0.299811i
$571$	−26624.0	−1.95128	−0.975639	−	0.219382i	$-0.929596\pi$
$571$	−26624.0	−1.95128	−0.975639	+	0.219382i	$0.929596\pi$
$572$	0	0
$573$	−2492.00	−0.181684
$574$	13200.0i	0.959856i
$575$	12792.0	0.927762
$576$	−11776.0	−0.851852
$577$	14101.0i	1.01739i	0.860948	+	0.508694i	$0.169871\pi$
$577$	14101.0i	1.01739i	−0.860948	+	0.508694i	$0.830129\pi$
$578$	18976.0i	1.36557i
$579$	534.000i	0.0383286i
$580$	26792.0i	1.91806i
$581$	12560.0	0.896862
$582$	−1904.00	−0.135607
$583$	− 2976.00i	− 0.211412i
$584$	0	0
$585$	0	0
$586$	33204.0	2.34069
$587$	− 1408.00i	− 0.0990023i	−0.998774	−	0.0495012i	$-0.984237\pi$
$587$	− 1408.00i	− 0.0990023i	0.998774	−	0.0495012i	$-0.0157632\pi$
$588$	912.000	0.0639630
$589$	2220.00	0.155303
$590$	58752.0i	4.09963i
$591$	− 2556.00i	− 0.177902i
$592$	14528.0i	1.00861i
$593$	− 1241.00i	− 0.0859389i	−0.999076	−	0.0429694i	$-0.986318\pi$
$593$	− 1241.00i	− 0.0859389i	0.999076	−	0.0429694i	$-0.0136818\pi$
$594$	12800.0	0.884159
$595$	4420.00	0.304542
$596$	− 16920.0i	− 1.16287i
$597$	−8476.00	−0.581071
$598$	0	0
$599$	11078.0	0.755651	0.377825	−	0.925877i	$-0.376672\pi$
$599$	11078.0	0.755651	0.377825	+	0.925877i	$0.376672\pi$
$600$	0	0
$601$	−13817.0	−0.937782	−0.468891	−	0.883256i	$-0.655346\pi$
$601$	−13817.0	−0.937782	−0.468891	+	0.883256i	$0.655346\pi$
$602$	12480.0	0.844928
$603$	19826.0i	1.33893i
$604$	− 4112.00i	− 0.277011i
$605$	− 5219.00i	− 0.350715i
$606$	− 6552.00i	− 0.439203i
$607$	8270.00	0.552997	0.276498	−	0.961014i	$-0.410826\pi$
$607$	8270.00	0.552997	0.276498	+	0.961014i	$0.410826\pi$
$608$	7680.00	0.512278
$609$	7880.00i	0.524325i
$610$	−9860.00	−0.654459
$611$	0	0
$612$	2392.00	0.157992
$613$	− 22273.0i	− 1.46753i	−0.679402	−	0.733767i	$-0.737761\pi$
$613$	− 22273.0i	− 1.46753i	0.679402	−	0.733767i	$-0.262239\pi$
$614$	34712.0	2.28153
$615$	5610.00	0.367833
$616$	0	0
$617$	18989.0i	1.23901i	0.784993	+	0.619504i	$0.212666\pi$
$617$	18989.0i	1.23901i	−0.784993	+	0.619504i	$0.787334\pi$
$618$	13104.0i	0.852945i
$619$	72.0000i	0.00467516i	0.999997	+	0.00233758i	$0.000744076\pi$
$619$	72.0000i	0.00467516i	−0.999997	+	0.00233758i	$0.999256\pi$
$620$	−10064.0	−0.651903
$621$	7800.00	0.504031
$622$	34632.0i	2.23250i
$623$	5320.00	0.342121
$624$	0	0
$625$	−9229.00	−0.590656
$626$	21000.0i	1.34078i
$627$	−1920.00	−0.122293
$628$	−23208.0	−1.47468
$629$	− 2951.00i	− 0.187065i
$630$	31280.0i	1.97813i
$631$	− 23380.0i	− 1.47503i	−0.675331	−	0.737514i	$-0.735999\pi$
$631$	− 23380.0i	− 1.47503i	0.675331	−	0.737514i	$-0.264001\pi$
$632$	0	0
$633$	6140.00	0.385534
$634$	−25652.0	−1.60689
$635$	− 36686.0i	− 2.29266i
$636$	−1488.00	−0.0927721
$637$	0	0
$638$	−25216.0	−1.56475
$639$	15042.0i	0.931224i
$640$	0	0
$641$	−6383.00	−0.393313	−0.196656	−	0.980472i	$-0.563008\pi$
$641$	−6383.00	−0.393313	−0.196656	+	0.980472i	$0.563008\pi$
$642$	− 4176.00i	− 0.256719i
$643$	17104.0i	1.04901i	0.851406	+	0.524507i	$0.175750\pi$
$643$	17104.0i	1.04901i	−0.851406	+	0.524507i	$0.824250\pi$
$644$	12480.0i	0.763635i
$645$	− 5304.00i	− 0.323790i
$646$	−1560.00	−0.0950114
$647$	−6994.00	−0.424981	−0.212490	−	0.977163i	$-0.568157\pi$
$647$	−6994.00	−0.424981	−0.212490	+	0.977163i	$0.568157\pi$
$648$	0	0
$649$	−27648.0	−1.67223
$650$	0	0
$651$	−2960.00	−0.178205
$652$	− 18880.0i	− 1.13405i
$653$	−5250.00	−0.314622	−0.157311	−	0.987549i	$-0.550283\pi$
$653$	−5250.00	−0.314622	−0.157311	+	0.987549i	$0.550283\pi$
$654$	13072.0	0.781584
$655$	− 12410.0i	− 0.740304i
$656$	− 10560.0i	− 0.628504i
$657$	− 4945.00i	− 0.293642i
$658$	− 12960.0i	− 0.767832i
$659$	−4340.00	−0.256544	−0.128272	−	0.991739i	$-0.540943\pi$
$659$	−4340.00	−0.256544	−0.128272	+	0.991739i	$0.540943\pi$
$660$	8704.00	0.513337
$661$	− 4179.00i	− 0.245907i	−0.992412	−	0.122953i	$-0.960763\pi$
$661$	− 4179.00i	− 0.245907i	0.992412	−	0.122953i	$-0.0392365\pi$
$662$	−13952.0	−0.819124
$663$	0	0
$664$	0	0
$665$	− 10200.0i	− 0.594796i
$666$	20884.0	1.21507
$667$	−15366.0	−0.892015
$668$	− 2240.00i	− 0.129743i
$669$	10756.0i	0.621601i
$670$	58616.0i	3.37990i
$671$	− 4640.00i	− 0.266953i
$672$	−10240.0	−0.587822
$673$	−22867.0	−1.30974	−0.654872	−	0.755740i	$-0.727278\pi$
$673$	−22867.0	−1.30974	−0.654872	+	0.755740i	$0.727278\pi$
$674$	− 7332.00i	− 0.419018i
$675$	16400.0	0.935165
$676$	0	0
$677$	5410.00	0.307124	0.153562	−	0.988139i	$-0.450925\pi$
$677$	5410.00	0.307124	0.153562	+	0.988139i	$0.450925\pi$
$678$	− 2616.00i	− 0.148181i
$679$	4760.00	0.269031
$680$	0	0
$681$	7948.00i	0.447236i
$682$	− 9472.00i	− 0.531821i
$683$	− 13578.0i	− 0.760685i	−0.924846	−	0.380342i	$-0.875806\pi$
$683$	− 13578.0i	− 0.760685i	0.924846	−	0.380342i	$-0.124194\pi$
$684$	− 5520.00i	− 0.308571i
$685$	28407.0	1.58449
$686$	22880.0	1.27341
$687$	− 12596.0i	− 0.699516i
$688$	−9984.00	−0.553251
$689$	0	0
$690$	10608.0	0.585275
$691$	− 12744.0i	− 0.701599i	−0.936451	−	0.350799i	$-0.885910\pi$
$691$	− 12744.0i	− 0.701599i	0.936451	−	0.350799i	$-0.114090\pi$
$692$	10608.0	0.582739
$693$	−14720.0	−0.806878
$694$	− 28920.0i	− 1.58183i
$695$	− 15504.0i	− 0.846187i
$696$	0	0
$697$	2145.00i	0.116568i
$698$	−21032.0	−1.14051
$699$	−8060.00	−0.436133
$700$	26240.0i	1.41683i
$701$	−16406.0	−0.883946	−0.441973	−	0.897028i	$-0.645721\pi$
$701$	−16406.0	−0.883946	−0.441973	+	0.897028i	$0.645721\pi$
$702$	0	0
$703$	−6810.00	−0.365354
$704$	− 16384.0i	− 0.877124i
$705$	−5508.00	−0.294246
$706$	−12652.0	−0.674454
$707$	16380.0i	0.871334i
$708$	13824.0i	0.733810i
$709$	709.000i	0.0375558i	0.999824	+	0.0187779i	$0.00597754\pi$
$709$	709.000i	0.0375558i	−0.999824	+	0.0187779i	$0.994022\pi$
$710$	44472.0i	2.35071i
$711$	1748.00	0.0922013
$712$	0	0
$713$	− 5772.00i	− 0.303174i
$714$	2080.00	0.109022
$715$	0	0
$716$	34112.0	1.78048
$717$	1968.00i	0.102505i
$718$	−40272.0	−2.09323
$719$	7644.00	0.396486	0.198243	−	0.980153i	$-0.436477\pi$
$719$	7644.00	0.396486	0.198243	+	0.980153i	$0.436477\pi$
$720$	− 25024.0i	− 1.29526i
$721$	− 32760.0i	− 1.69216i
$722$	− 23836.0i	− 1.22865i
$723$	1886.00i	0.0970140i
$724$	−3224.00	−0.165496
$725$	−32308.0	−1.65502
$726$	− 2456.00i	− 0.125552i
$727$	15808.0	0.806446	0.403223	−	0.915102i	$-0.367890\pi$
$727$	15808.0	0.806446	0.403223	+	0.915102i	$0.367890\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	− 14620.0i	− 0.741247i
$731$	2028.00	0.102611
$732$	−2320.00	−0.117144
$733$	2583.00i	0.130157i	0.997880	+	0.0650786i	$0.0207298\pi$
$733$	2583.00i	0.130157i	−0.997880	+	0.0650786i	$0.979270\pi$
$734$	− 29752.0i	− 1.49614i
$735$	1938.00i	0.0972574i
$736$	− 19968.0i	− 1.00004i
$737$	−27584.0	−1.37866
$738$	−15180.0	−0.757159
$739$	4076.00i	0.202893i	0.994841	+	0.101447i	$0.0323471\pi$
$739$	4076.00i	0.202893i	−0.994841	+	0.101447i	$0.967653\pi$
$740$	30872.0	1.53362
$741$	0	0
$742$	7440.00	0.368101
$743$	34056.0i	1.68155i	0.541383	+	0.840776i	$0.317901\pi$
$743$	34056.0i	1.68155i	−0.541383	+	0.840776i	$0.682099\pi$
$744$	0	0
$745$	35955.0	1.76817
$746$	38732.0i	1.90091i
$747$	14444.0i	0.707468i
$748$	3328.00i	0.162679i
$749$	10440.0i	0.509305i
$750$	5304.00	0.258233
$751$	364.000	0.0176865	0.00884324	−	0.999961i	$-0.497185\pi$
$751$	364.000	0.0176865	0.00884324	+	0.999961i	$0.497185\pi$
$752$	10368.0i	0.502769i
$753$	5460.00	0.264241
$754$	0	0
$755$	8738.00	0.421203
$756$	16000.0i	0.769728i
$757$	−6914.00	−0.331960	−0.165980	−	0.986129i	$-0.553079\pi$
$757$	−6914.00	−0.331960	−0.165980	+	0.986129i	$0.553079\pi$
$758$	4248.00	0.203554
$759$	4992.00i	0.238733i
$760$	0	0
$761$	13982.0i	0.666028i	0.942922	+	0.333014i	$0.108066\pi$
$761$	13982.0i	0.666028i	−0.942922	+	0.333014i	$0.891934\pi$
$762$	− 17264.0i	− 0.820746i
$763$	−32680.0	−1.55058
$764$	9968.00	0.472028
$765$	5083.00i	0.240230i
$766$	14128.0	0.666404
$767$	0	0
$768$	8192.00	0.384900
$769$	18066.0i	0.847174i	0.905855	+	0.423587i	$0.139229\pi$
$769$	18066.0i	0.847174i	−0.905855	+	0.423587i	$0.860771\pi$
$770$	−43520.0	−2.03682
$771$	3770.00	0.176100
$772$	− 2136.00i	− 0.0995807i
$773$	− 14434.0i	− 0.671610i	−0.941931	−	0.335805i	$-0.890992\pi$
$773$	− 14434.0i	− 0.671610i	0.941931	−	0.335805i	$-0.109008\pi$
$774$	14352.0i	0.666501i
$775$	− 12136.0i	− 0.562501i
$776$	0	0
$777$	9080.00	0.419232
$778$	− 44252.0i	− 2.03922i
$779$	4950.00	0.227666
$780$	0	0
$781$	−20928.0	−0.958851
$782$	4056.00i	0.185476i
$783$	−19700.0	−0.899132
$784$	3648.00	0.166181
$785$	− 49317.0i	− 2.24229i
$786$	− 5840.00i	− 0.265020i
$787$	− 15398.0i	− 0.697433i	−0.937228	−	0.348716i	$-0.886618\pi$
$787$	− 15398.0i	− 0.697433i	0.937228	−	0.348716i	$-0.113382\pi$
$788$	10224.0i	0.462202i
$789$	8064.00	0.363861
$790$	5168.00	0.232746
$791$	6540.00i	0.293977i
$792$	0	0
$793$	0	0
$794$	−23944.0	−1.07020
$795$	− 3162.00i	− 0.141062i
$796$	33904.0	1.50967
$797$	36842.0	1.63740	0.818702	−	0.574219i	$-0.194694\pi$
$797$	36842.0	1.63740	0.818702	+	0.574219i	$0.194694\pi$
$798$	− 4800.00i	− 0.212930i
$799$	− 2106.00i	− 0.0932477i
$800$	− 41984.0i	− 1.85545i
$801$	6118.00i	0.269874i
$802$	23740.0	1.04525
$803$	6880.00	0.302354
$804$	13792.0i	0.604983i
$805$	−26520.0	−1.16113
$806$	0	0
$807$	8012.00	0.349487
$808$	0	0
$809$	41511.0	1.80402	0.902008	−	0.431719i	$-0.142093\pi$
$809$	41511.0	1.80402	0.902008	+	0.431719i	$0.142093\pi$
$810$	−28628.0	−1.24183
$811$	− 23066.0i	− 0.998714i	−0.866396	−	0.499357i	$-0.833570\pi$
$811$	− 23066.0i	− 0.998714i	0.866396	−	0.499357i	$-0.166430\pi$
$812$	− 31520.0i	− 1.36224i
$813$	− 8592.00i	− 0.370645i
$814$	29056.0i	1.25112i
$815$	40120.0	1.72435
$816$	−1664.00	−0.0713868
$817$	− 4680.00i	− 0.200407i
$818$	60356.0	2.57983
$819$	0	0
$820$	−22440.0	−0.955657
$821$	− 28838.0i	− 1.22589i	−0.790127	−	0.612943i	$-0.789985\pi$
$821$	− 28838.0i	− 1.22589i	0.790127	−	0.612943i	$-0.210015\pi$
$822$	13368.0	0.567229
$823$	−27456.0	−1.16289	−0.581443	−	0.813587i	$-0.697512\pi$
$823$	−27456.0	−1.16289	−0.581443	+	0.813587i	$0.697512\pi$
$824$	0	0
$825$	10496.0i	0.442938i
$826$	− 69120.0i	− 2.91161i
$827$	− 33572.0i	− 1.41162i	−0.708399	−	0.705812i	$-0.750582\pi$
$827$	− 33572.0i	− 1.41162i	0.708399	−	0.705812i	$-0.249418\pi$
$828$	−14352.0	−0.602375
$829$	45799.0	1.91878	0.959388	−	0.282090i	$-0.0910278\pi$
$829$	45799.0	1.91878	0.959388	+	0.282090i	$0.0910278\pi$
$830$	42704.0i	1.78588i
$831$	11102.0	0.463447
$832$	0	0
$833$	−741.000	−0.0308213
$834$	− 7296.00i	− 0.302925i
$835$	4760.00	0.197277
$836$	7680.00	0.317725
$837$	− 7400.00i	− 0.305593i
$838$	43256.0i	1.78312i
$839$	32286.0i	1.32853i	0.747497	+	0.664265i	$0.231255\pi$
$839$	32286.0i	1.32853i	−0.747497	+	0.664265i	$0.768745\pi$
$840$	0	0
$841$	14420.0	0.591250
$842$	26140.0	1.06989
$843$	− 11114.0i	− 0.454077i
$844$	−24560.0	−1.00165
$845$	0	0
$846$	14904.0	0.605686
$847$	6140.00i	0.249083i
$848$	−5952.00	−0.241029
$849$	−6240.00	−0.252245
$850$	8528.00i	0.344127i
$851$	17706.0i	0.713224i
$852$	10464.0i	0.420764i
$853$	20937.0i	0.840409i	0.907429	+	0.420205i	$0.138042\pi$
$853$	20937.0i	0.840409i	−0.907429	+	0.420205i	$0.861958\pi$
$854$	11600.0	0.464805
$855$	11730.0	0.469190
$856$	0	0
$857$	7189.00	0.286548	0.143274	−	0.989683i	$-0.454237\pi$
$857$	7189.00	0.286548	0.143274	+	0.989683i	$0.454237\pi$
$858$	0	0
$859$	−32498.0	−1.29082	−0.645412	−	0.763835i	$-0.723314\pi$
$859$	−32498.0	−1.29082	−0.645412	+	0.763835i	$0.723314\pi$
$860$	21216.0i	0.841232i
$861$	−6600.00	−0.261240
$862$	−7920.00	−0.312942
$863$	− 8428.00i	− 0.332436i	−0.986089	−	0.166218i	$-0.946844\pi$
$863$	− 8428.00i	− 0.332436i	0.986089	−	0.166218i	$-0.0531556\pi$
$864$	− 25600.0i	− 1.00802i
$865$	22542.0i	0.886071i
$866$	− 27716.0i	− 1.08756i
$867$	−9488.00	−0.371660
$868$	11840.0	0.462991
$869$	2432.00i	0.0949367i
$870$	−26792.0	−1.04406
$871$	0	0
$872$	0	0
$873$	5474.00i	0.212219i
$874$	9360.00	0.362250
$875$	−13260.0	−0.512308
$876$	− 3440.00i	− 0.132679i
$877$	6847.00i	0.263634i	0.991274	+	0.131817i	$0.0420811\pi$
$877$	6847.00i	0.263634i	−0.991274	+	0.131817i	$0.957919\pi$
$878$	− 18304.0i	− 0.703565i
$879$	16602.0i	0.637055i
$880$	34816.0	1.33369
$881$	−29731.0	−1.13696	−0.568481	−	0.822697i	$-0.692469\pi$
$881$	−29731.0	−1.13696	−0.568481	+	0.822697i	$0.692469\pi$
$882$	− 5244.00i	− 0.200198i
$883$	23738.0	0.904697	0.452348	−	0.891841i	$-0.350586\pi$
$883$	23738.0	0.904697	0.452348	+	0.891841i	$0.350586\pi$
$884$	0	0
$885$	−29376.0	−1.11578
$886$	35248.0i	1.33655i
$887$	27588.0	1.04432	0.522161	−	0.852847i	$-0.325126\pi$
$887$	27588.0	1.04432	0.522161	+	0.852847i	$0.325126\pi$
$888$	0	0
$889$	43160.0i	1.62828i
$890$	18088.0i	0.681248i
$891$	− 13472.0i	− 0.506542i
$892$	− 43024.0i	− 1.61497i
$893$	−4860.00	−0.182121
$894$	16920.0	0.632986
$895$	72488.0i	2.70727i
$896$	0	0
$897$	0	0
$898$	7672.00	0.285098
$899$	14578.0i	0.540827i
$900$	−30176.0	−1.11763
$901$	1209.00	0.0447033
$902$	− 21120.0i	− 0.779622i
$903$	6240.00i	0.229960i
$904$	0	0
$905$	− 6851.00i	− 0.251641i
$906$	4112.00	0.150786
$907$	37128.0	1.35922	0.679611	−	0.733572i	$-0.262148\pi$
$907$	37128.0	1.35922	0.679611	+	0.733572i	$0.262148\pi$
$908$	− 31792.0i	− 1.16195i
$909$	−18837.0	−0.687331
$910$	0	0
$911$	20516.0	0.746131	0.373066	−	0.927805i	$-0.378307\pi$
$911$	20516.0	0.746131	0.373066	+	0.927805i	$0.378307\pi$
$912$	3840.00i	0.139424i
$913$	−20096.0	−0.728456
$914$	47044.0	1.70249
$915$	− 4930.00i	− 0.178121i
$916$	50384.0i	1.81740i
$917$	14600.0i	0.525774i
$918$	5200.00i	0.186956i
$919$	−21006.0	−0.753998	−0.376999	−	0.926214i	$-0.623044\pi$
$919$	−21006.0	−0.753998	−0.376999	+	0.926214i	$0.623044\pi$
$920$	0	0
$921$	17356.0i	0.620955i
$922$	−3604.00	−0.128733
$923$	0	0
$924$	−10240.0	−0.364579
$925$	37228.0i	1.32330i
$926$	5488.00	0.194759
$927$	37674.0	1.33482
$928$	50432.0i	1.78396i
$929$	− 20427.0i	− 0.721408i	−0.932680	−	0.360704i	$-0.882536\pi$
$929$	− 20427.0i	− 0.721408i	0.932680	−	0.360704i	$-0.117464\pi$
$930$	− 10064.0i	− 0.354851i
$931$	1710.00i	0.0601965i
$932$	32240.0	1.13311
$933$	−17316.0	−0.607610
$934$	− 25584.0i	− 0.896289i
$935$	−7072.00	−0.247357
$936$	0	0
$937$	33191.0	1.15721	0.578603	−	0.815609i	$-0.303598\pi$
$937$	33191.0	1.15721	0.578603	+	0.815609i	$0.303598\pi$
$938$	− 68960.0i	− 2.40045i
$939$	−10500.0	−0.364914
$940$	22032.0	0.764473
$941$	36422.0i	1.26177i	0.775877	+	0.630884i	$0.217308\pi$
$941$	36422.0i	1.26177i	−0.775877	+	0.630884i	$0.782692\pi$
$942$	− 23208.0i	− 0.802715i
$943$	− 12870.0i	− 0.444438i
$944$	55296.0i	1.90650i
$945$	−34000.0	−1.17039
$946$	−19968.0	−0.686275
$947$	− 39630.0i	− 1.35988i	−0.733270	−	0.679938i	$-0.762007\pi$
$947$	− 39630.0i	− 1.35988i	0.733270	−	0.679938i	$-0.237993\pi$
$948$	1216.00	0.0416602
$949$	0	0
$950$	19680.0	0.672109
$951$	− 12826.0i	− 0.437341i
$952$	0	0
$953$	−57642.0	−1.95929	−0.979647	−	0.200727i	$-0.935670\pi$
$953$	−57642.0	−1.95929	−0.979647	+	0.200727i	$0.935670\pi$
$954$	8556.00i	0.290368i
$955$	21182.0i	0.717731i
$956$	− 7872.00i	− 0.266317i
$957$	− 12608.0i	− 0.425871i
$958$	13080.0	0.441123
$959$	−33420.0	−1.12533
$960$	− 17408.0i	− 0.585251i
$961$	24315.0	0.816186
$962$	0	0
$963$	−12006.0	−0.401753
$964$	− 7544.00i	− 0.252050i
$965$	4539.00	0.151415
$966$	−12480.0	−0.415670
$967$	− 2162.00i	− 0.0718979i	−0.999354	−	0.0359489i	$-0.988555\pi$
$967$	− 2162.00i	− 0.0718979i	0.999354	−	0.0359489i	$-0.0114454\pi$
$968$	0	0
$969$	− 780.000i	− 0.0258588i
$970$	16184.0i	0.535708i
$971$	−19758.0	−0.653001	−0.326501	−	0.945197i	$-0.605870\pi$
$971$	−19758.0	−0.653001	−0.326501	+	0.945197i	$0.605870\pi$
$972$	−28336.0	−0.935059
$973$	18240.0i	0.600974i
$974$	−79680.0	−2.62126
$975$	0	0
$976$	−9280.00	−0.304350
$977$	12489.0i	0.408965i	0.978870	+	0.204482i	$0.0655511\pi$
$977$	12489.0i	0.408965i	−0.978870	+	0.204482i	$0.934449\pi$
$978$	18880.0	0.617296
$979$	−8512.00	−0.277880
$980$	− 7752.00i	− 0.252682i
$981$	− 37582.0i	− 1.22314i
$982$	26208.0i	0.851661i
$983$	− 28658.0i	− 0.929856i	−0.885349	−	0.464928i	$-0.846080\pi$
$983$	− 28658.0i	− 0.929856i	0.885349	−	0.464928i	$-0.153920\pi$
$984$	0	0
$985$	−21726.0	−0.702790
$986$	− 10244.0i	− 0.330868i
$987$	6480.00	0.208977
$988$	0	0
$989$	−12168.0	−0.391223
$990$	− 50048.0i	− 1.60670i
$991$	−42794.0	−1.37174	−0.685871	−	0.727723i	$-0.740579\pi$
$991$	−42794.0	−1.37174	−0.685871	+	0.727723i	$0.740579\pi$
$992$	−18944.0	−0.606323
$993$	− 6976.00i	− 0.222937i
$994$	− 52320.0i	− 1.66951i
$995$	72046.0i	2.29549i
$996$	10048.0i	0.319662i
$997$	−52583.0	−1.67033	−0.835166	−	0.549998i	$-0.814628\pi$
$997$	−52583.0	−1.67033	−0.835166	+	0.549998i	$0.814628\pi$
$998$	−6984.00	−0.221518
$999$	22700.0i	0.718915i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.c.168.1		2
13.2	odd	12		169.4.c.d.22.1		2
13.3	even	3		169.4.e.c.147.2		4
13.4	even	6		169.4.e.c.23.2		4
13.5	odd	4		169.4.a.a.1.1		1
13.6	odd	12		169.4.c.d.146.1		2
13.7	odd	12		13.4.c.a.3.1	✓	2
13.8	odd	4		169.4.a.d.1.1		1
13.9	even	3		169.4.e.c.23.1		4
13.10	even	6		169.4.e.c.147.1		4
13.11	odd	12		13.4.c.a.9.1	yes	2
13.12	even	2	inner	169.4.b.c.168.2		2
39.5	even	4		1521.4.a.k.1.1		1
39.8	even	4		1521.4.a.b.1.1		1
39.11	even	12		117.4.g.c.100.1		2
39.20	even	12		117.4.g.c.55.1		2
52.7	even	12		208.4.i.b.81.1		2
52.11	even	12		208.4.i.b.113.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.a.3.1	✓	2	13.7	odd	12
13.4.c.a.9.1	yes	2	13.11	odd	12
117.4.g.c.55.1		2	39.20	even	12
117.4.g.c.100.1		2	39.11	even	12
169.4.a.a.1.1		1	13.5	odd	4
169.4.a.d.1.1		1	13.8	odd	4
169.4.b.c.168.1		2	1.1	even	1	trivial
169.4.b.c.168.2		2	13.12	even	2	inner
169.4.c.d.22.1		2	13.2	odd	12
169.4.c.d.146.1		2	13.6	odd	12
169.4.e.c.23.1		4	13.9	even	3
169.4.e.c.23.2		4	13.4	even	6
169.4.e.c.147.1		4	13.10	even	6
169.4.e.c.147.2		4	13.3	even	3
208.4.i.b.81.1		2	52.7	even	12
208.4.i.b.113.1		2	52.11	even	12
1521.4.a.b.1.1		1	39.8	even	4
1521.4.a.k.1.1		1	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)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +2.00000 q^{3} -8.00000 q^{4} -17.0000i q^{5} -8.00000i q^{6} +20.0000i q^{7} -23.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} +2.00000 q^{3} -8.00000 q^{4} -17.0000i q^{5} -8.00000i q^{6} +20.0000i q^{7} -23.0000 q^{9} -68.0000 q^{10} -32.0000i q^{11} -16.0000 q^{12} +80.0000 q^{14} -34.0000i q^{15} -64.0000 q^{16} +13.0000 q^{17} +92.0000i q^{18} -30.0000i q^{19} +136.000i q^{20} +40.0000i q^{21} -128.000 q^{22} -78.0000 q^{23} -164.000 q^{25} -100.000 q^{27} -160.000i q^{28} +197.000 q^{29} -136.000 q^{30} +74.0000i q^{31} +256.000i q^{32} -64.0000i q^{33} -52.0000i q^{34} +340.000 q^{35} +184.000 q^{36} -227.000i q^{37} -120.000 q^{38} +165.000i q^{41} +160.000 q^{42} +156.000 q^{43} +256.000i q^{44} +391.000i q^{45} +312.000i q^{46} -162.000i q^{47} -128.000 q^{48} -57.0000 q^{49} +656.000i q^{50} +26.0000 q^{51} +93.0000 q^{53} +400.000i q^{54} -544.000 q^{55} -60.0000i q^{57} -788.000i q^{58} -864.000i q^{59} +272.000i q^{60} +145.000 q^{61} +296.000 q^{62} -460.000i q^{63} +512.000 q^{64} -256.000 q^{66} -862.000i q^{67} -104.000 q^{68} -156.000 q^{69} -1360.00i q^{70} -654.000i q^{71} +215.000i q^{73} -908.000 q^{74} -328.000 q^{75} +240.000i q^{76} +640.000 q^{77} -76.0000 q^{79} +1088.00i q^{80} +421.000 q^{81} +660.000 q^{82} -628.000i q^{83} -320.000i q^{84} -221.000i q^{85} -624.000i q^{86} +394.000 q^{87} -266.000i q^{89} +1564.00 q^{90} +624.000 q^{92} +148.000i q^{93} -648.000 q^{94} -510.000 q^{95} +512.000i q^{96} -238.000i q^{97} +228.000i q^{98} +736.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 16 q^{4} - 46 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 16 * q^4 - 46 * q^9	\( 2 q + 4 q^{3} - 16 q^{4} - 46 q^{9} - 136 q^{10} - 32 q^{12} + 160 q^{14} - 128 q^{16} + 26 q^{17} - 256 q^{22} - 156 q^{23} - 328 q^{25} - 200 q^{27} + 394 q^{29} - 272 q^{30} + 680 q^{35} + 368 q^{36} - 240 q^{38} + 320 q^{42} + 312 q^{43} - 256 q^{48} - 114 q^{49} + 52 q^{51} + 186 q^{53} - 1088 q^{55} + 290 q^{61} + 592 q^{62} + 1024 q^{64} - 512 q^{66} - 208 q^{68} - 312 q^{69} - 1816 q^{74} - 656 q^{75} + 1280 q^{77} - 152 q^{79} + 842 q^{81} + 1320 q^{82} + 788 q^{87} + 3128 q^{90} + 1248 q^{92} - 1296 q^{94} - 1020 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 - 16 * q^4 - 46 * q^9 - 136 * q^10 - 32 * q^12 + 160 * q^14 - 128 * q^16 + 26 * q^17 - 256 * q^22 - 156 * q^23 - 328 * q^25 - 200 * q^27 + 394 * q^29 - 272 * q^30 + 680 * q^35 + 368 * q^36 - 240 * q^38 + 320 * q^42 + 312 * q^43 - 256 * q^48 - 114 * q^49 + 52 * q^51 + 186 * q^53 - 1088 * q^55 + 290 * q^61 + 592 * q^62 + 1024 * q^64 - 512 * q^66 - 208 * q^68 - 312 * q^69 - 1816 * q^74 - 656 * q^75 + 1280 * q^77 - 152 * q^79 + 842 * q^81 + 1320 * q^82 + 788 * q^87 + 3128 * q^90 + 1248 * q^92 - 1296 * q^94 - 1020 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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