LMFDB - Embedded newform 121.3.b.a.120.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,3,Mod(120,121)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("121.120");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	121.b (of order $2$, degree $1$, 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:	$3.29701119876$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			120.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	121.120
Dual form			121.3.b.a.120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.00000 q^{3} +2.00000 q^{4} -7.00000 q^{5} -1.41421i q^{6} +7.07107i q^{7} +8.48528i q^{8} -8.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} -1.00000 q^{3} +2.00000 q^{4} -7.00000 q^{5} -1.41421i q^{6} +7.07107i q^{7} +8.48528i q^{8} -8.00000 q^{9} -9.89949i q^{10} -2.00000 q^{12} +16.9706i q^{13} -10.0000 q^{14} +7.00000 q^{15} -4.00000 q^{16} -4.24264i q^{17} -11.3137i q^{18} -16.9706i q^{19} -14.0000 q^{20} -7.07107i q^{21} -9.00000 q^{23} -8.48528i q^{24} +24.0000 q^{25} -24.0000 q^{26} +17.0000 q^{27} +14.1421i q^{28} +22.6274i q^{29} +9.89949i q^{30} +49.0000 q^{31} +28.2843i q^{32} +6.00000 q^{34} -49.4975i q^{35} -16.0000 q^{36} +17.0000 q^{37} +24.0000 q^{38} -16.9706i q^{39} -59.3970i q^{40} -16.9706i q^{41} +10.0000 q^{42} +46.6690i q^{43} +56.0000 q^{45} -12.7279i q^{46} +32.0000 q^{47} +4.00000 q^{48} -1.00000 q^{49} +33.9411i q^{50} +4.24264i q^{51} +33.9411i q^{52} +16.0000 q^{53} +24.0416i q^{54} -60.0000 q^{56} +16.9706i q^{57} -32.0000 q^{58} -71.0000 q^{59} +14.0000 q^{60} +11.3137i q^{61} +69.2965i q^{62} -56.5685i q^{63} -56.0000 q^{64} -118.794i q^{65} -31.0000 q^{67} -8.48528i q^{68} +9.00000 q^{69} +70.0000 q^{70} -73.0000 q^{71} -67.8823i q^{72} -39.5980i q^{73} +24.0416i q^{74} -24.0000 q^{75} -33.9411i q^{76} +24.0000 q^{78} +156.978i q^{79} +28.0000 q^{80} +55.0000 q^{81} +24.0000 q^{82} -35.3553i q^{83} -14.1421i q^{84} +29.6985i q^{85} -66.0000 q^{86} -22.6274i q^{87} -9.00000 q^{89} +79.1960i q^{90} -120.000 q^{91} -18.0000 q^{92} -49.0000 q^{93} +45.2548i q^{94} +118.794i q^{95} -28.2843i q^{96} -17.0000 q^{97} -1.41421i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{4} - 14 q^{5} - 16 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^4 - 14 * q^5 - 16 * q^9	$ 2 q - 2 q^{3} + 4 q^{4} - 14 q^{5} - 16 q^{9} - 4 q^{12} - 20 q^{14} + 14 q^{15} - 8 q^{16} - 28 q^{20} - 18 q^{23} + 48 q^{25} - 48 q^{26} + 34 q^{27} + 98 q^{31} + 12 q^{34} - 32 q^{36} + 34 q^{37} + 48 q^{38} + 20 q^{42} + 112 q^{45} + 64 q^{47} + 8 q^{48} - 2 q^{49} + 32 q^{53} - 120 q^{56} - 64 q^{58} - 142 q^{59} + 28 q^{60} - 112 q^{64} - 62 q^{67} + 18 q^{69} + 140 q^{70} - 146 q^{71} - 48 q^{75} + 48 q^{78} + 56 q^{80} + 110 q^{81} + 48 q^{82} - 132 q^{86} - 18 q^{89} - 240 q^{91} - 36 q^{92} - 98 q^{93} - 34 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^4 - 14 * q^5 - 16 * q^9 - 4 * q^12 - 20 * q^14 + 14 * q^15 - 8 * q^16 - 28 * q^20 - 18 * q^23 + 48 * q^25 - 48 * q^26 + 34 * q^27 + 98 * q^31 + 12 * q^34 - 32 * q^36 + 34 * q^37 + 48 * q^38 + 20 * q^42 + 112 * q^45 + 64 * q^47 + 8 * q^48 - 2 * q^49 + 32 * q^53 - 120 * q^56 - 64 * q^58 - 142 * q^59 + 28 * q^60 - 112 * q^64 - 62 * q^67 + 18 * q^69 + 140 * q^70 - 146 * q^71 - 48 * q^75 + 48 * q^78 + 56 * q^80 + 110 * q^81 + 48 * q^82 - 132 * q^86 - 18 * q^89 - 240 * q^91 - 36 * q^92 - 98 * q^93 - 34 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.41421i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.41421i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	−1.00000	−0.333333	−0.166667	−	0.986013i	$-0.553300\pi$
$3$	−1.00000	−0.333333	−0.166667	+	0.986013i	$0.553300\pi$
$4$	2.00000	0.500000
$5$	−7.00000	−1.40000	−0.700000	−	0.714143i	$-0.746817\pi$
$5$	−7.00000	−1.40000	−0.700000	+	0.714143i	$0.746817\pi$
$6$	− 1.41421i	− 0.235702i
$7$	7.07107i	1.01015i	0.863075	+	0.505076i	$0.168536\pi$
$7$	7.07107i	1.01015i	−0.863075	+	0.505076i	$0.831464\pi$
$8$	8.48528i	1.06066i
$9$	−8.00000	−0.888889
$10$	− 9.89949i	− 0.989949i
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.166667
$13$	16.9706i	1.30543i	0.757604	+	0.652714i	$0.226370\pi$
$13$	16.9706i	1.30543i	−0.757604	+	0.652714i	$0.773630\pi$
$14$	−10.0000	−0.714286
$15$	7.00000	0.466667
$16$	−4.00000	−0.250000
$17$	− 4.24264i	− 0.249567i	−0.992184	−	0.124784i	$-0.960176\pi$
$17$	− 4.24264i	− 0.249567i	0.992184	−	0.124784i	$-0.0398236\pi$
$18$	− 11.3137i	− 0.628539i
$19$	− 16.9706i	− 0.893188i	−0.894737	−	0.446594i	$-0.852637\pi$
$19$	− 16.9706i	− 0.893188i	0.894737	−	0.446594i	$-0.147363\pi$
$20$	−14.0000	−0.700000
$21$	− 7.07107i	− 0.336718i
$22$	0	0
$23$	−9.00000	−0.391304	−0.195652	−	0.980673i	$-0.562682\pi$
$23$	−9.00000	−0.391304	−0.195652	+	0.980673i	$0.562682\pi$
$24$	− 8.48528i	− 0.353553i
$25$	24.0000	0.960000
$26$	−24.0000	−0.923077
$27$	17.0000	0.629630
$28$	14.1421i	0.505076i
$29$	22.6274i	0.780256i	0.920761	+	0.390128i	$0.127569\pi$
$29$	22.6274i	0.780256i	−0.920761	+	0.390128i	$0.872431\pi$
$30$	9.89949i	0.329983i
$31$	49.0000	1.58065	0.790323	−	0.612691i	$-0.209913\pi$
$31$	49.0000	1.58065	0.790323	+	0.612691i	$0.209913\pi$
$32$	28.2843i	0.883883i
$33$	0	0
$34$	6.00000	0.176471
$35$	− 49.4975i	− 1.41421i
$36$	−16.0000	−0.444444
$37$	17.0000	0.459459	0.229730	−	0.973254i	$-0.426216\pi$
$37$	17.0000	0.459459	0.229730	+	0.973254i	$0.426216\pi$
$38$	24.0000	0.631579
$39$	− 16.9706i	− 0.435143i
$40$	− 59.3970i	− 1.48492i
$41$	− 16.9706i	− 0.413916i	−0.978350	−	0.206958i	$-0.933644\pi$
$41$	− 16.9706i	− 0.413916i	0.978350	−	0.206958i	$-0.0663564\pi$
$42$	10.0000	0.238095
$43$	46.6690i	1.08533i	0.839950	+	0.542663i	$0.182584\pi$
$43$	46.6690i	1.08533i	−0.839950	+	0.542663i	$0.817416\pi$
$44$	0	0
$45$	56.0000	1.24444
$46$	− 12.7279i	− 0.276694i
$47$	32.0000	0.680851	0.340426	−	0.940271i	$-0.389429\pi$
$47$	32.0000	0.680851	0.340426	+	0.940271i	$0.389429\pi$
$48$	4.00000	0.0833333
$49$	−1.00000	−0.0204082
$50$	33.9411i	0.678823i
$51$	4.24264i	0.0831890i
$52$	33.9411i	0.652714i
$53$	16.0000	0.301887	0.150943	−	0.988542i	$-0.451769\pi$
$53$	16.0000	0.301887	0.150943	+	0.988542i	$0.451769\pi$
$54$	24.0416i	0.445215i
$55$	0	0
$56$	−60.0000	−1.07143
$57$	16.9706i	0.297729i
$58$	−32.0000	−0.551724
$59$	−71.0000	−1.20339	−0.601695	−	0.798726i	$-0.705508\pi$
$59$	−71.0000	−1.20339	−0.601695	+	0.798726i	$0.705508\pi$
$60$	14.0000	0.233333
$61$	11.3137i	0.185471i	0.995691	+	0.0927353i	$0.0295610\pi$
$61$	11.3137i	0.185471i	−0.995691	+	0.0927353i	$0.970439\pi$
$62$	69.2965i	1.11768i
$63$	− 56.5685i	− 0.897913i
$64$	−56.0000	−0.875000
$65$	− 118.794i	− 1.82760i
$66$	0	0
$67$	−31.0000	−0.462687	−0.231343	−	0.972872i	$-0.574312\pi$
$67$	−31.0000	−0.462687	−0.231343	+	0.972872i	$0.574312\pi$
$68$	− 8.48528i	− 0.124784i
$69$	9.00000	0.130435
$70$	70.0000	1.00000
$71$	−73.0000	−1.02817	−0.514085	−	0.857740i	$-0.671868\pi$
$71$	−73.0000	−1.02817	−0.514085	+	0.857740i	$0.671868\pi$
$72$	− 67.8823i	− 0.942809i
$73$	− 39.5980i	− 0.542438i	−0.962518	−	0.271219i	$-0.912573\pi$
$73$	− 39.5980i	− 0.542438i	0.962518	−	0.271219i	$-0.0874268\pi$
$74$	24.0416i	0.324887i
$75$	−24.0000	−0.320000
$76$	− 33.9411i	− 0.446594i
$77$	0	0
$78$	24.0000	0.307692
$79$	156.978i	1.98706i	0.113572	+	0.993530i	$0.463771\pi$
$79$	156.978i	1.98706i	−0.113572	+	0.993530i	$0.536229\pi$
$80$	28.0000	0.350000
$81$	55.0000	0.679012
$82$	24.0000	0.292683
$83$	− 35.3553i	− 0.425968i	−0.977056	−	0.212984i	$-0.931682\pi$
$83$	− 35.3553i	− 0.425968i	0.977056	−	0.212984i	$-0.0683182\pi$
$84$	− 14.1421i	− 0.168359i
$85$	29.6985i	0.349394i
$86$	−66.0000	−0.767442
$87$	− 22.6274i	− 0.260085i
$88$	0	0
$89$	−9.00000	−0.101124	−0.0505618	−	0.998721i	$-0.516101\pi$
$89$	−9.00000	−0.101124	−0.0505618	+	0.998721i	$0.516101\pi$
$90$	79.1960i	0.879955i
$91$	−120.000	−1.31868
$92$	−18.0000	−0.195652
$93$	−49.0000	−0.526882
$94$	45.2548i	0.481434i
$95$	118.794i	1.25046i
$96$	− 28.2843i	− 0.294628i
$97$	−17.0000	−0.175258	−0.0876289	−	0.996153i	$-0.527929\pi$
$97$	−17.0000	−0.175258	−0.0876289	+	0.996153i	$0.527929\pi$
$98$	− 1.41421i	− 0.0144308i
$99$	0	0
$100$	48.0000	0.480000
$101$	− 154.149i	− 1.52623i	−0.646262	−	0.763115i	$-0.723669\pi$
$101$	− 154.149i	− 1.52623i	0.646262	−	0.763115i	$-0.276331\pi$
$102$	−6.00000	−0.0588235
$103$	−16.0000	−0.155340	−0.0776699	−	0.996979i	$-0.524748\pi$
$103$	−16.0000	−0.155340	−0.0776699	+	0.996979i	$0.524748\pi$
$104$	−144.000	−1.38462
$105$	49.4975i	0.471405i
$106$	22.6274i	0.213466i
$107$	185.262i	1.73142i	0.500546	+	0.865710i	$0.333133\pi$
$107$	185.262i	1.73142i	−0.500546	+	0.865710i	$0.666867\pi$
$108$	34.0000	0.314815
$109$	46.6690i	0.428156i	0.976817	+	0.214078i	$0.0686747\pi$
$109$	46.6690i	0.428156i	−0.976817	+	0.214078i	$0.931325\pi$
$110$	0	0
$111$	−17.0000	−0.153153
$112$	− 28.2843i	− 0.252538i
$113$	65.0000	0.575221	0.287611	−	0.957747i	$-0.407139\pi$
$113$	65.0000	0.575221	0.287611	+	0.957747i	$0.407139\pi$
$114$	−24.0000	−0.210526
$115$	63.0000	0.547826
$116$	45.2548i	0.390128i
$117$	− 135.765i	− 1.16038i
$118$	− 100.409i	− 0.850925i
$119$	30.0000	0.252101
$120$	59.3970i	0.494975i
$121$	0	0
$122$	−16.0000	−0.131148
$123$	16.9706i	0.137972i
$124$	98.0000	0.790323
$125$	7.00000	0.0560000
$126$	80.0000	0.634921
$127$	− 175.362i	− 1.38081i	−0.723425	−	0.690403i	$-0.757433\pi$
$127$	− 175.362i	− 1.38081i	0.723425	−	0.690403i	$-0.242567\pi$
$128$	33.9411i	0.265165i
$129$	− 46.6690i	− 0.361776i
$130$	168.000	1.29231
$131$	140.007i	1.06876i	0.845245	+	0.534378i	$0.179454\pi$
$131$	140.007i	1.06876i	−0.845245	+	0.534378i	$0.820546\pi$
$132$	0	0
$133$	120.000	0.902256
$134$	− 43.8406i	− 0.327169i
$135$	−119.000	−0.881481
$136$	36.0000	0.264706
$137$	257.000	1.87591	0.937956	−	0.346754i	$-0.112716\pi$
$137$	257.000	1.87591	0.937956	+	0.346754i	$0.112716\pi$
$138$	12.7279i	0.0922313i
$139$	− 86.2670i	− 0.620626i	−0.950634	−	0.310313i	$-0.899566\pi$
$139$	− 86.2670i	− 0.620626i	0.950634	−	0.310313i	$-0.100434\pi$
$140$	− 98.9949i	− 0.707107i
$141$	−32.0000	−0.226950
$142$	− 103.238i	− 0.727025i
$143$	0	0
$144$	32.0000	0.222222
$145$	− 158.392i	− 1.09236i
$146$	56.0000	0.383562
$147$	1.00000	0.00680272
$148$	34.0000	0.229730
$149$	275.772i	1.85082i	0.378972	+	0.925408i	$0.376278\pi$
$149$	275.772i	1.85082i	−0.378972	+	0.925408i	$0.623722\pi$
$150$	− 33.9411i	− 0.226274i
$151$	− 156.978i	− 1.03959i	−0.854292	−	0.519794i	$-0.826009\pi$
$151$	− 156.978i	− 1.03959i	0.854292	−	0.519794i	$-0.173991\pi$
$152$	144.000	0.947368
$153$	33.9411i	0.221837i
$154$	0	0
$155$	−343.000	−2.21290
$156$	− 33.9411i	− 0.217571i
$157$	175.000	1.11465	0.557325	−	0.830295i	$-0.311828\pi$
$157$	175.000	1.11465	0.557325	+	0.830295i	$0.311828\pi$
$158$	−222.000	−1.40506
$159$	−16.0000	−0.100629
$160$	− 197.990i	− 1.23744i
$161$	− 63.6396i	− 0.395277i
$162$	77.7817i	0.480134i
$163$	−160.000	−0.981595	−0.490798	−	0.871274i	$-0.663295\pi$
$163$	−160.000	−0.981595	−0.490798	+	0.871274i	$0.663295\pi$
$164$	− 33.9411i	− 0.206958i
$165$	0	0
$166$	50.0000	0.301205
$167$	16.9706i	0.101620i	0.998708	+	0.0508101i	$0.0161803\pi$
$167$	16.9706i	0.101620i	−0.998708	+	0.0508101i	$0.983820\pi$
$168$	60.0000	0.357143
$169$	−119.000	−0.704142
$170$	−42.0000	−0.247059
$171$	135.765i	0.793944i
$172$	93.3381i	0.542663i
$173$	123.037i	0.711194i	0.934639	+	0.355597i	$0.115722\pi$
$173$	123.037i	0.711194i	−0.934639	+	0.355597i	$0.884278\pi$
$174$	32.0000	0.183908
$175$	169.706i	0.969746i
$176$	0	0
$177$	71.0000	0.401130
$178$	− 12.7279i	− 0.0715052i
$179$	−199.000	−1.11173	−0.555866	−	0.831272i	$-0.687613\pi$
$179$	−199.000	−1.11173	−0.555866	+	0.831272i	$0.687613\pi$
$180$	112.000	0.622222
$181$	−73.0000	−0.403315	−0.201657	−	0.979456i	$-0.564633\pi$
$181$	−73.0000	−0.403315	−0.201657	+	0.979456i	$0.564633\pi$
$182$	− 169.706i	− 0.932449i
$183$	− 11.3137i	− 0.0618235i
$184$	− 76.3675i	− 0.415041i
$185$	−119.000	−0.643243
$186$	− 69.2965i	− 0.372562i
$187$	0	0
$188$	64.0000	0.340426
$189$	120.208i	0.636022i
$190$	−168.000	−0.884211
$191$	215.000	1.12565	0.562827	−	0.826575i	$-0.309714\pi$
$191$	215.000	1.12565	0.562827	+	0.826575i	$0.309714\pi$
$192$	56.0000	0.291667
$193$	135.765i	0.703443i	0.936105	+	0.351722i	$0.114404\pi$
$193$	135.765i	0.703443i	−0.936105	+	0.351722i	$0.885596\pi$
$194$	− 24.0416i	− 0.123926i
$195$	118.794i	0.609200i
$196$	−2.00000	−0.0102041
$197$	− 202.233i	− 1.02656i	−0.858221	−	0.513281i	$-0.828430\pi$
$197$	− 202.233i	− 1.02656i	0.858221	−	0.513281i	$-0.171570\pi$
$198$	0	0
$199$	200.000	1.00503	0.502513	−	0.864570i	$-0.332409\pi$
$199$	200.000	1.00503	0.502513	+	0.864570i	$0.332409\pi$
$200$	203.647i	1.01823i
$201$	31.0000	0.154229
$202$	218.000	1.07921
$203$	−160.000	−0.788177
$204$	8.48528i	0.0415945i
$205$	118.794i	0.579483i
$206$	− 22.6274i	− 0.109842i
$207$	72.0000	0.347826
$208$	− 67.8823i	− 0.326357i
$209$	0	0
$210$	−70.0000	−0.333333
$211$	79.1960i	0.375336i	0.982232	+	0.187668i	$0.0600930\pi$
$211$	79.1960i	0.375336i	−0.982232	+	0.187668i	$0.939907\pi$
$212$	32.0000	0.150943
$213$	73.0000	0.342723
$214$	−262.000	−1.22430
$215$	− 326.683i	− 1.51946i
$216$	144.250i	0.667823i
$217$	346.482i	1.59669i
$218$	−66.0000	−0.302752
$219$	39.5980i	0.180813i
$220$	0	0
$221$	72.0000	0.325792
$222$	− 24.0416i	− 0.108296i
$223$	−111.000	−0.497758	−0.248879	−	0.968535i	$-0.580062\pi$
$223$	−111.000	−0.497758	−0.248879	+	0.968535i	$0.580062\pi$
$224$	−200.000	−0.892857
$225$	−192.000	−0.853333
$226$	91.9239i	0.406743i
$227$	131.522i	0.579391i	0.957119	+	0.289696i	$0.0935541\pi$
$227$	131.522i	0.579391i	−0.957119	+	0.289696i	$0.906446\pi$
$228$	33.9411i	0.148865i
$229$	−303.000	−1.32314	−0.661572	−	0.749882i	$-0.730110\pi$
$229$	−303.000	−1.32314	−0.661572	+	0.749882i	$0.730110\pi$
$230$	89.0955i	0.387372i
$231$	0	0
$232$	−192.000	−0.827586
$233$	79.1960i	0.339897i	0.985453	+	0.169948i	$0.0543601\pi$
$233$	79.1960i	0.339897i	−0.985453	+	0.169948i	$0.945640\pi$
$234$	192.000	0.820513
$235$	−224.000	−0.953191
$236$	−142.000	−0.601695
$237$	− 156.978i	− 0.662353i
$238$	42.4264i	0.178262i
$239$	− 281.428i	− 1.17753i	−0.808306	−	0.588763i	$-0.799615\pi$
$239$	− 281.428i	− 1.17753i	0.808306	−	0.588763i	$-0.200385\pi$
$240$	−28.0000	−0.116667
$241$	− 373.352i	− 1.54918i	−0.632464	−	0.774590i	$-0.717956\pi$
$241$	− 373.352i	− 1.54918i	0.632464	−	0.774590i	$-0.282044\pi$
$242$	0	0
$243$	−208.000	−0.855967
$244$	22.6274i	0.0927353i
$245$	7.00000	0.0285714
$246$	−24.0000	−0.0975610
$247$	288.000	1.16599
$248$	415.779i	1.67653i
$249$	35.3553i	0.141989i
$250$	9.89949i	0.0395980i
$251$	225.000	0.896414	0.448207	−	0.893930i	$-0.352063\pi$
$251$	225.000	0.896414	0.448207	+	0.893930i	$0.352063\pi$
$252$	− 113.137i	− 0.448957i
$253$	0	0
$254$	248.000	0.976378
$255$	− 29.6985i	− 0.116465i
$256$	−272.000	−1.06250
$257$	424.000	1.64981	0.824903	−	0.565275i	$-0.191230\pi$
$257$	424.000	1.64981	0.824903	+	0.565275i	$0.191230\pi$
$258$	66.0000	0.255814
$259$	120.208i	0.464124i
$260$	− 237.588i	− 0.913800i
$261$	− 181.019i	− 0.693561i
$262$	−198.000	−0.755725
$263$	140.007i	0.532347i	0.963925	+	0.266173i	$0.0857593\pi$
$263$	140.007i	0.532347i	−0.963925	+	0.266173i	$0.914241\pi$
$264$	0	0
$265$	−112.000	−0.422642
$266$	169.706i	0.637991i
$267$	9.00000	0.0337079
$268$	−62.0000	−0.231343
$269$	136.000	0.505576	0.252788	−	0.967522i	$-0.418652\pi$
$269$	136.000	0.505576	0.252788	+	0.967522i	$0.418652\pi$
$270$	− 168.291i	− 0.623302i
$271$	− 288.500i	− 1.06457i	−0.846564	−	0.532287i	$-0.821333\pi$
$271$	− 288.500i	− 1.06457i	0.846564	−	0.532287i	$-0.178667\pi$
$272$	16.9706i	0.0623918i
$273$	120.000	0.439560
$274$	363.453i	1.32647i
$275$	0	0
$276$	18.0000	0.0652174
$277$	− 123.037i	− 0.444175i	−0.975027	−	0.222088i	$-0.928713\pi$
$277$	− 123.037i	− 0.444175i	0.975027	−	0.222088i	$-0.0712871\pi$
$278$	122.000	0.438849
$279$	−392.000	−1.40502
$280$	420.000	1.50000
$281$	− 284.257i	− 1.01159i	−0.862654	−	0.505795i	$-0.831199\pi$
$281$	− 284.257i	− 1.01159i	0.862654	−	0.505795i	$-0.168801\pi$
$282$	− 45.2548i	− 0.160478i
$283$	− 79.1960i	− 0.279844i	−0.990163	−	0.139922i	$-0.955315\pi$
$283$	− 79.1960i	− 0.279844i	0.990163	−	0.139922i	$-0.0446852\pi$
$284$	−146.000	−0.514085
$285$	− 118.794i	− 0.416821i
$286$	0	0
$287$	120.000	0.418118
$288$	− 226.274i	− 0.785674i
$289$	271.000	0.937716
$290$	224.000	0.772414
$291$	17.0000	0.0584192
$292$	− 79.1960i	− 0.271219i
$293$	− 164.049i	− 0.559893i	−0.960016	−	0.279947i	$-0.909683\pi$
$293$	− 164.049i	− 0.559893i	0.960016	−	0.279947i	$-0.0903168\pi$
$294$	1.41421i	0.00481025i
$295$	497.000	1.68475
$296$	144.250i	0.487330i
$297$	0	0
$298$	−390.000	−1.30872
$299$	− 152.735i	− 0.510820i
$300$	−48.0000	−0.160000
$301$	−330.000	−1.09635
$302$	222.000	0.735099
$303$	154.149i	0.508743i
$304$	67.8823i	0.223297i
$305$	− 79.1960i	− 0.259659i
$306$	−48.0000	−0.156863
$307$	186.676i	0.608066i	0.952662	+	0.304033i	$0.0983333\pi$
$307$	186.676i	0.608066i	−0.952662	+	0.304033i	$0.901667\pi$
$308$	0	0
$309$	16.0000	0.0517799
$310$	− 485.075i	− 1.56476i
$311$	142.000	0.456592	0.228296	−	0.973592i	$-0.426685\pi$
$311$	142.000	0.456592	0.228296	+	0.973592i	$0.426685\pi$
$312$	144.000	0.461538
$313$	−447.000	−1.42812	−0.714058	−	0.700087i	$-0.753145\pi$
$313$	−447.000	−1.42812	−0.714058	+	0.700087i	$0.753145\pi$
$314$	247.487i	0.788176i
$315$	395.980i	1.25708i
$316$	313.955i	0.993530i
$317$	423.000	1.33438	0.667192	−	0.744885i	$-0.267496\pi$
$317$	423.000	1.33438	0.667192	+	0.744885i	$0.267496\pi$
$318$	− 22.6274i	− 0.0711554i
$319$	0	0
$320$	392.000	1.22500
$321$	− 185.262i	− 0.577140i
$322$	90.0000	0.279503
$323$	−72.0000	−0.222910
$324$	110.000	0.339506
$325$	407.294i	1.25321i
$326$	− 226.274i	− 0.694093i
$327$	− 46.6690i	− 0.142719i
$328$	144.000	0.439024
$329$	226.274i	0.687763i
$330$	0	0
$331$	145.000	0.438066	0.219033	−	0.975717i	$-0.429710\pi$
$331$	145.000	0.438066	0.219033	+	0.975717i	$0.429710\pi$
$332$	− 70.7107i	− 0.212984i
$333$	−136.000	−0.408408
$334$	−24.0000	−0.0718563
$335$	217.000	0.647761
$336$	28.2843i	0.0841794i
$337$	255.973i	0.759563i	0.925076	+	0.379781i	$0.124001\pi$
$337$	255.973i	0.759563i	−0.925076	+	0.379781i	$0.875999\pi$
$338$	− 168.291i	− 0.497904i
$339$	−65.0000	−0.191740
$340$	59.3970i	0.174697i
$341$	0	0
$342$	−192.000	−0.561404
$343$	339.411i	0.989537i
$344$	−396.000	−1.15116
$345$	−63.0000	−0.182609
$346$	−174.000	−0.502890
$347$	− 548.715i	− 1.58131i	−0.612261	−	0.790655i	$-0.709740\pi$
$347$	− 548.715i	− 1.58131i	0.612261	−	0.790655i	$-0.290260\pi$
$348$	− 45.2548i	− 0.130043i
$349$	− 436.992i	− 1.25213i	−0.779772	−	0.626063i	$-0.784665\pi$
$349$	− 436.992i	− 1.25213i	0.779772	−	0.626063i	$-0.215335\pi$
$350$	−240.000	−0.685714
$351$	288.500i	0.821936i
$352$	0	0
$353$	585.000	1.65722	0.828612	−	0.559823i	$-0.189131\pi$
$353$	585.000	1.65722	0.828612	+	0.559823i	$0.189131\pi$
$354$	100.409i	0.283642i
$355$	511.000	1.43944
$356$	−18.0000	−0.0505618
$357$	−30.0000	−0.0840336
$358$	− 281.428i	− 0.786113i
$359$	− 412.950i	− 1.15028i	−0.818055	−	0.575140i	$-0.804948\pi$
$359$	− 412.950i	− 1.15028i	0.818055	−	0.575140i	$-0.195052\pi$
$360$	475.176i	1.31993i
$361$	73.0000	0.202216
$362$	− 103.238i	− 0.285187i
$363$	0	0
$364$	−240.000	−0.659341
$365$	277.186i	0.759413i
$366$	16.0000	0.0437158
$367$	545.000	1.48501	0.742507	−	0.669839i	$-0.233637\pi$
$367$	545.000	1.48501	0.742507	+	0.669839i	$0.233637\pi$
$368$	36.0000	0.0978261
$369$	135.765i	0.367925i
$370$	− 168.291i	− 0.454842i
$371$	113.137i	0.304952i
$372$	−98.0000	−0.263441
$373$	− 233.345i	− 0.625590i	−0.949821	−	0.312795i	$-0.898735\pi$
$373$	− 233.345i	− 0.625590i	0.949821	−	0.312795i	$-0.101265\pi$
$374$	0	0
$375$	−7.00000	−0.0186667
$376$	271.529i	0.722152i
$377$	−384.000	−1.01857
$378$	−170.000	−0.449735
$379$	−447.000	−1.17942	−0.589710	−	0.807615i	$-0.700758\pi$
$379$	−447.000	−1.17942	−0.589710	+	0.807615i	$0.700758\pi$
$380$	237.588i	0.625231i
$381$	175.362i	0.460269i
$382$	304.056i	0.795958i
$383$	−545.000	−1.42298	−0.711488	−	0.702698i	$-0.751979\pi$
$383$	−545.000	−1.42298	−0.711488	+	0.702698i	$0.751979\pi$
$384$	− 33.9411i	− 0.0883883i
$385$	0	0
$386$	−192.000	−0.497409
$387$	− 373.352i	− 0.964735i
$388$	−34.0000	−0.0876289
$389$	215.000	0.552699	0.276350	−	0.961057i	$-0.410875\pi$
$389$	215.000	0.552699	0.276350	+	0.961057i	$0.410875\pi$
$390$	−168.000	−0.430769
$391$	38.1838i	0.0976567i
$392$	− 8.48528i	− 0.0216461i
$393$	− 140.007i	− 0.356252i
$394$	286.000	0.725888
$395$	− 1098.84i	− 2.78188i
$396$	0	0
$397$	−592.000	−1.49118	−0.745592	−	0.666403i	$-0.767833\pi$
$397$	−592.000	−1.49118	−0.745592	+	0.666403i	$0.767833\pi$
$398$	282.843i	0.710660i
$399$	−120.000	−0.300752
$400$	−96.0000	−0.240000
$401$	488.000	1.21696	0.608479	−	0.793570i	$-0.291780\pi$
$401$	488.000	1.21696	0.608479	+	0.793570i	$0.291780\pi$
$402$	43.8406i	0.109056i
$403$	831.558i	2.06342i
$404$	− 308.299i	− 0.763115i
$405$	−385.000	−0.950617
$406$	− 226.274i	− 0.557326i
$407$	0	0
$408$	−36.0000	−0.0882353
$409$	156.978i	0.383809i	0.981414	+	0.191904i	$0.0614663\pi$
$409$	156.978i	0.383809i	−0.981414	+	0.191904i	$0.938534\pi$
$410$	−168.000	−0.409756
$411$	−257.000	−0.625304
$412$	−32.0000	−0.0776699
$413$	− 502.046i	− 1.21561i
$414$	101.823i	0.245950i
$415$	247.487i	0.596355i
$416$	−480.000	−1.15385
$417$	86.2670i	0.206875i
$418$	0	0
$419$	−328.000	−0.782816	−0.391408	−	0.920217i	$-0.628012\pi$
$419$	−328.000	−0.782816	−0.391408	+	0.920217i	$0.628012\pi$
$420$	98.9949i	0.235702i
$421$	208.000	0.494062	0.247031	−	0.969008i	$-0.420545\pi$
$421$	208.000	0.494062	0.247031	+	0.969008i	$0.420545\pi$
$422$	−112.000	−0.265403
$423$	−256.000	−0.605201
$424$	135.765i	0.320199i
$425$	− 101.823i	− 0.239584i
$426$	103.238i	0.242342i
$427$	−80.0000	−0.187354
$428$	370.524i	0.865710i
$429$	0	0
$430$	462.000	1.07442
$431$	561.443i	1.30265i	0.758798	+	0.651326i	$0.225787\pi$
$431$	561.443i	1.30265i	−0.758798	+	0.651326i	$0.774213\pi$
$432$	−68.0000	−0.157407
$433$	39.0000	0.0900693	0.0450346	−	0.998985i	$-0.485660\pi$
$433$	39.0000	0.0900693	0.0450346	+	0.998985i	$0.485660\pi$
$434$	−490.000	−1.12903
$435$	158.392i	0.364119i
$436$	93.3381i	0.214078i
$437$	152.735i	0.349508i
$438$	−56.0000	−0.127854
$439$	− 248.902i	− 0.566974i	−0.958976	−	0.283487i	$-0.908509\pi$
$439$	− 248.902i	− 0.566974i	0.958976	−	0.283487i	$-0.0914913\pi$
$440$	0	0
$441$	8.00000	0.0181406
$442$	101.823i	0.230370i
$443$	175.000	0.395034	0.197517	−	0.980299i	$-0.436712\pi$
$443$	175.000	0.395034	0.197517	+	0.980299i	$0.436712\pi$
$444$	−34.0000	−0.0765766
$445$	63.0000	0.141573
$446$	− 156.978i	− 0.351968i
$447$	− 275.772i	− 0.616939i
$448$	− 395.980i	− 0.883883i
$449$	313.000	0.697105	0.348552	−	0.937289i	$-0.386673\pi$
$449$	313.000	0.697105	0.348552	+	0.937289i	$0.386673\pi$
$450$	− 271.529i	− 0.603398i
$451$	0	0
$452$	130.000	0.287611
$453$	156.978i	0.346529i
$454$	−186.000	−0.409692
$455$	840.000	1.84615
$456$	−144.000	−0.315789
$457$	11.3137i	0.0247565i	0.999923	+	0.0123782i	$0.00394022\pi$
$457$	11.3137i	0.0247565i	−0.999923	+	0.0123782i	$0.996060\pi$
$458$	− 428.507i	− 0.935604i
$459$	− 72.1249i	− 0.157135i
$460$	126.000	0.273913
$461$	871.156i	1.88971i	0.327491	+	0.944854i	$0.393797\pi$
$461$	871.156i	1.88971i	−0.327491	+	0.944854i	$0.606203\pi$
$462$	0	0
$463$	321.000	0.693305	0.346652	−	0.937994i	$-0.387318\pi$
$463$	321.000	0.693305	0.346652	+	0.937994i	$0.387318\pi$
$464$	− 90.5097i	− 0.195064i
$465$	343.000	0.737634
$466$	−112.000	−0.240343
$467$	−161.000	−0.344754	−0.172377	−	0.985031i	$-0.555145\pi$
$467$	−161.000	−0.344754	−0.172377	+	0.985031i	$0.555145\pi$
$468$	− 271.529i	− 0.580190i
$469$	− 219.203i	− 0.467384i
$470$	− 316.784i	− 0.674008i
$471$	−175.000	−0.371550
$472$	− 602.455i	− 1.27639i
$473$	0	0
$474$	222.000	0.468354
$475$	− 407.294i	− 0.857460i
$476$	60.0000	0.126050
$477$	−128.000	−0.268344
$478$	398.000	0.832636
$479$	446.891i	0.932968i	0.884530	+	0.466484i	$0.154479\pi$
$479$	446.891i	0.932968i	−0.884530	+	0.466484i	$0.845521\pi$
$480$	197.990i	0.412479i
$481$	288.500i	0.599791i
$482$	528.000	1.09544
$483$	63.6396i	0.131759i
$484$	0	0
$485$	119.000	0.245361
$486$	− 294.156i	− 0.605260i
$487$	−727.000	−1.49281	−0.746407	−	0.665490i	$-0.768223\pi$
$487$	−727.000	−1.49281	−0.746407	+	0.665490i	$0.768223\pi$
$488$	−96.0000	−0.196721
$489$	160.000	0.327198
$490$	9.89949i	0.0202031i
$491$	520.431i	1.05994i	0.848016	+	0.529970i	$0.177797\pi$
$491$	520.431i	1.05994i	−0.848016	+	0.529970i	$0.822203\pi$
$492$	33.9411i	0.0689860i
$493$	96.0000	0.194726
$494$	407.294i	0.824481i
$495$	0	0
$496$	−196.000	−0.395161
$497$	− 516.188i	− 1.03861i
$498$	−50.0000	−0.100402
$499$	−368.000	−0.737475	−0.368737	−	0.929534i	$-0.620210\pi$
$499$	−368.000	−0.737475	−0.368737	+	0.929534i	$0.620210\pi$
$500$	14.0000	0.0280000
$501$	− 16.9706i	− 0.0338734i
$502$	318.198i	0.633861i
$503$	465.276i	0.925003i	0.886619	+	0.462501i	$0.153048\pi$
$503$	465.276i	0.925003i	−0.886619	+	0.462501i	$0.846952\pi$
$504$	480.000	0.952381
$505$	1079.04i	2.13672i
$506$	0	0
$507$	119.000	0.234714
$508$	− 350.725i	− 0.690403i
$509$	−111.000	−0.218075	−0.109037	−	0.994038i	$-0.534777\pi$
$509$	−111.000	−0.218075	−0.109037	+	0.994038i	$0.534777\pi$
$510$	42.0000	0.0823529
$511$	280.000	0.547945
$512$	− 248.902i	− 0.486136i
$513$	− 288.500i	− 0.562377i
$514$	599.627i	1.16659i
$515$	112.000	0.217476
$516$	− 93.3381i	− 0.180888i
$517$	0	0
$518$	−170.000	−0.328185
$519$	− 123.037i	− 0.237065i
$520$	1008.00	1.93846
$521$	−511.000	−0.980806	−0.490403	−	0.871496i	$-0.663150\pi$
$521$	−511.000	−0.980806	−0.490403	+	0.871496i	$0.663150\pi$
$522$	256.000	0.490421
$523$	695.793i	1.33039i	0.746670	+	0.665194i	$0.231651\pi$
$523$	695.793i	1.33039i	−0.746670	+	0.665194i	$0.768349\pi$
$524$	280.014i	0.534378i
$525$	− 169.706i	− 0.323249i
$526$	−198.000	−0.376426
$527$	− 207.889i	− 0.394477i
$528$	0	0
$529$	−448.000	−0.846881
$530$	− 158.392i	− 0.298853i
$531$	568.000	1.06968
$532$	240.000	0.451128
$533$	288.000	0.540338
$534$	12.7279i	0.0238351i
$535$	− 1296.83i	− 2.42399i
$536$	− 263.044i	− 0.490753i
$537$	199.000	0.370577
$538$	192.333i	0.357496i
$539$	0	0
$540$	−238.000	−0.440741
$541$	− 231.931i	− 0.428708i	−0.976756	−	0.214354i	$-0.931235\pi$
$541$	− 231.931i	− 0.428708i	0.976756	−	0.214354i	$-0.0687646\pi$
$542$	408.000	0.752768
$543$	73.0000	0.134438
$544$	120.000	0.220588
$545$	− 326.683i	− 0.599419i
$546$	169.706i	0.310816i
$547$	− 701.450i	− 1.28236i	−0.767391	−	0.641179i	$-0.778446\pi$
$547$	− 701.450i	− 1.28236i	0.767391	−	0.641179i	$-0.221554\pi$
$548$	514.000	0.937956
$549$	− 90.5097i	− 0.164863i
$550$	0	0
$551$	384.000	0.696915
$552$	76.3675i	0.138347i
$553$	−1110.00	−2.00723
$554$	174.000	0.314079
$555$	119.000	0.214414
$556$	− 172.534i	− 0.310313i
$557$	− 210.718i	− 0.378308i	−0.981947	−	0.189154i	$-0.939425\pi$
$557$	− 210.718i	− 0.378308i	0.981947	−	0.189154i	$-0.0605746\pi$
$558$	− 554.372i	− 0.993498i
$559$	−792.000	−1.41682
$560$	197.990i	0.353553i
$561$	0	0
$562$	402.000	0.715302
$563$	− 543.058i	− 0.964579i	−0.876012	−	0.482290i	$-0.839805\pi$
$563$	− 543.058i	− 0.964579i	0.876012	−	0.482290i	$-0.160195\pi$
$564$	−64.0000	−0.113475
$565$	−455.000	−0.805310
$566$	112.000	0.197880
$567$	388.909i	0.685906i
$568$	− 619.426i	− 1.09054i
$569$	− 79.1960i	− 0.139184i	−0.997576	−	0.0695922i	$-0.977830\pi$
$569$	− 79.1960i	− 0.139184i	0.997576	−	0.0695922i	$-0.0221698\pi$
$570$	168.000	0.294737
$571$	124.451i	0.217952i	0.994044	+	0.108976i	$0.0347572\pi$
$571$	124.451i	0.217952i	−0.994044	+	0.108976i	$0.965243\pi$
$572$	0	0
$573$	−215.000	−0.375218
$574$	169.706i	0.295654i
$575$	−216.000	−0.375652
$576$	448.000	0.777778
$577$	103.000	0.178510	0.0892548	−	0.996009i	$-0.471551\pi$
$577$	103.000	0.178510	0.0892548	+	0.996009i	$0.471551\pi$
$578$	383.252i	0.663066i
$579$	− 135.765i	− 0.234481i
$580$	− 316.784i	− 0.546179i
$581$	250.000	0.430293
$582$	24.0416i	0.0413086i
$583$	0	0
$584$	336.000	0.575342
$585$	950.352i	1.62453i
$586$	232.000	0.395904
$587$	−632.000	−1.07666	−0.538330	−	0.842734i	$-0.680945\pi$
$587$	−632.000	−1.07666	−0.538330	+	0.842734i	$0.680945\pi$
$588$	2.00000	0.00340136
$589$	− 831.558i	− 1.41181i
$590$	702.864i	1.19130i
$591$	202.233i	0.342187i
$592$	−68.0000	−0.114865
$593$	− 497.803i	− 0.839466i	−0.907648	−	0.419733i	$-0.862124\pi$
$593$	− 497.803i	− 0.839466i	0.907648	−	0.419733i	$-0.137876\pi$
$594$	0	0
$595$	−210.000	−0.352941
$596$	551.543i	0.925408i
$597$	−200.000	−0.335008
$598$	216.000	0.361204
$599$	312.000	0.520868	0.260434	−	0.965492i	$-0.416134\pi$
$599$	312.000	0.520868	0.260434	+	0.965492i	$0.416134\pi$
$600$	− 203.647i	− 0.339411i
$601$	411.536i	0.684752i	0.939563	+	0.342376i	$0.111232\pi$
$601$	411.536i	0.684752i	−0.939563	+	0.342376i	$0.888768\pi$
$602$	− 466.690i	− 0.775233i
$603$	248.000	0.411277
$604$	− 313.955i	− 0.519794i
$605$	0	0
$606$	−218.000	−0.359736
$607$	− 123.037i	− 0.202696i	−0.994851	−	0.101348i	$-0.967684\pi$
$607$	− 123.037i	− 0.202696i	0.994851	−	0.101348i	$-0.0323156\pi$
$608$	480.000	0.789474
$609$	160.000	0.262726
$610$	112.000	0.183607
$611$	543.058i	0.888802i
$612$	67.8823i	0.110919i
$613$	543.058i	0.885902i	0.896546	+	0.442951i	$0.146068\pi$
$613$	543.058i	0.885902i	−0.896546	+	0.442951i	$0.853932\pi$
$614$	−264.000	−0.429967
$615$	− 118.794i	− 0.193161i
$616$	0	0
$617$	−1120.00	−1.81524	−0.907618	−	0.419798i	$-0.862101\pi$
$617$	−1120.00	−1.81524	−0.907618	+	0.419798i	$0.862101\pi$
$618$	22.6274i	0.0366139i
$619$	703.000	1.13570	0.567851	−	0.823131i	$-0.307775\pi$
$619$	703.000	1.13570	0.567851	+	0.823131i	$0.307775\pi$
$620$	−686.000	−1.10645
$621$	−153.000	−0.246377
$622$	200.818i	0.322859i
$623$	− 63.6396i	− 0.102150i
$624$	67.8823i	0.108786i
$625$	−649.000	−1.03840
$626$	− 632.153i	− 1.00983i
$627$	0	0
$628$	350.000	0.557325
$629$	− 72.1249i	− 0.114666i
$630$	−560.000	−0.888889
$631$	391.000	0.619651	0.309826	−	0.950793i	$-0.399729\pi$
$631$	391.000	0.619651	0.309826	+	0.950793i	$0.399729\pi$
$632$	−1332.00	−2.10759
$633$	− 79.1960i	− 0.125112i
$634$	598.212i	0.943553i
$635$	1227.54i	1.93313i
$636$	−32.0000	−0.0503145
$637$	− 16.9706i	− 0.0266414i
$638$	0	0
$639$	584.000	0.913928
$640$	− 237.588i	− 0.371231i
$641$	−23.0000	−0.0358814	−0.0179407	−	0.999839i	$-0.505711\pi$
$641$	−23.0000	−0.0358814	−0.0179407	+	0.999839i	$0.505711\pi$
$642$	262.000	0.408100
$643$	−447.000	−0.695179	−0.347589	−	0.937647i	$-0.613000\pi$
$643$	−447.000	−0.695179	−0.347589	+	0.937647i	$0.613000\pi$
$644$	− 127.279i	− 0.197639i
$645$	326.683i	0.506486i
$646$	− 101.823i	− 0.157621i
$647$	−479.000	−0.740340	−0.370170	−	0.928964i	$-0.620701\pi$
$647$	−479.000	−0.740340	−0.370170	+	0.928964i	$0.620701\pi$
$648$	466.690i	0.720201i
$649$	0	0
$650$	−576.000	−0.886154
$651$	− 346.482i	− 0.532231i
$652$	−320.000	−0.490798
$653$	721.000	1.10413	0.552067	−	0.833800i	$-0.313839\pi$
$653$	721.000	1.10413	0.552067	+	0.833800i	$0.313839\pi$
$654$	66.0000	0.100917
$655$	− 980.050i	− 1.49626i
$656$	67.8823i	0.103479i
$657$	316.784i	0.482167i
$658$	−320.000	−0.486322
$659$	− 700.036i	− 1.06227i	−0.847287	−	0.531135i	$-0.821766\pi$
$659$	− 700.036i	− 1.06227i	0.847287	−	0.531135i	$-0.178234\pi$
$660$	0	0
$661$	607.000	0.918306	0.459153	−	0.888357i	$-0.348153\pi$
$661$	607.000	0.918306	0.459153	+	0.888357i	$0.348153\pi$
$662$	205.061i	0.309760i
$663$	−72.0000	−0.108597
$664$	300.000	0.451807
$665$	−840.000	−1.26316
$666$	− 192.333i	− 0.288788i
$667$	− 203.647i	− 0.305317i
$668$	33.9411i	0.0508101i
$669$	111.000	0.165919
$670$	306.884i	0.458036i
$671$	0	0
$672$	200.000	0.297619
$673$	997.021i	1.48146i	0.671805	+	0.740729i	$0.265519\pi$
$673$	997.021i	1.48146i	−0.671805	+	0.740729i	$0.734481\pi$
$674$	−362.000	−0.537092
$675$	408.000	0.604444
$676$	−238.000	−0.352071
$677$	773.575i	1.14265i	0.820724	+	0.571326i	$0.193571\pi$
$677$	773.575i	1.14265i	−0.820724	+	0.571326i	$0.806429\pi$
$678$	− 91.9239i	− 0.135581i
$679$	− 120.208i	− 0.177037i
$680$	−252.000	−0.370588
$681$	− 131.522i	− 0.193130i
$682$	0	0
$683$	−218.000	−0.319180	−0.159590	−	0.987183i	$-0.551017\pi$
$683$	−218.000	−0.319180	−0.159590	+	0.987183i	$0.551017\pi$
$684$	271.529i	0.396972i
$685$	−1799.00	−2.62628
$686$	−480.000	−0.699708
$687$	303.000	0.441048
$688$	− 186.676i	− 0.271332i
$689$	271.529i	0.394091i
$690$	− 89.0955i	− 0.129124i
$691$	863.000	1.24891	0.624457	−	0.781059i	$-0.285320\pi$
$691$	863.000	1.24891	0.624457	+	0.781059i	$0.285320\pi$
$692$	246.073i	0.355597i
$693$	0	0
$694$	776.000	1.11816
$695$	603.869i	0.868877i
$696$	192.000	0.275862
$697$	−72.0000	−0.103300
$698$	618.000	0.885387
$699$	− 79.1960i	− 0.113299i
$700$	339.411i	0.484873i
$701$	403.051i	0.574966i	0.957786	+	0.287483i	$0.0928184\pi$
$701$	403.051i	0.574966i	−0.957786	+	0.287483i	$0.907182\pi$
$702$	−408.000	−0.581197
$703$	− 288.500i	− 0.410383i
$704$	0	0
$705$	224.000	0.317730
$706$	827.315i	1.17183i
$707$	1090.00	1.54173
$708$	142.000	0.200565
$709$	−623.000	−0.878702	−0.439351	−	0.898315i	$-0.644792\pi$
$709$	−623.000	−0.878702	−0.439351	+	0.898315i	$0.644792\pi$
$710$	722.663i	1.01784i
$711$	− 1255.82i	− 1.76628i
$712$	− 76.3675i	− 0.107258i
$713$	−441.000	−0.618513
$714$	− 42.4264i	− 0.0594207i
$715$	0	0
$716$	−398.000	−0.555866
$717$	281.428i	0.392508i
$718$	584.000	0.813370
$719$	281.000	0.390821	0.195410	−	0.980722i	$-0.437396\pi$
$719$	281.000	0.390821	0.195410	+	0.980722i	$0.437396\pi$
$720$	−224.000	−0.311111
$721$	− 113.137i	− 0.156917i
$722$	103.238i	0.142988i
$723$	373.352i	0.516393i
$724$	−146.000	−0.201657
$725$	543.058i	0.749046i
$726$	0	0
$727$	1223.00	1.68226	0.841128	−	0.540836i	$-0.181892\pi$
$727$	1223.00	1.68226	0.841128	+	0.540836i	$0.181892\pi$
$728$	− 1018.23i	− 1.39867i
$729$	−287.000	−0.393690
$730$	−392.000	−0.536986
$731$	198.000	0.270862
$732$	− 22.6274i	− 0.0309118i
$733$	893.783i	1.21935i	0.792652	+	0.609675i	$0.208700\pi$
$733$	893.783i	1.21935i	−0.792652	+	0.609675i	$0.791300\pi$
$734$	770.746i	1.05006i
$735$	−7.00000	−0.00952381
$736$	− 254.558i	− 0.345867i
$737$	0	0
$738$	−192.000	−0.260163
$739$	− 963.079i	− 1.30322i	−0.758554	−	0.651610i	$-0.774094\pi$
$739$	− 963.079i	− 1.30322i	0.758554	−	0.651610i	$-0.225906\pi$
$740$	−238.000	−0.321622
$741$	−288.000	−0.388664
$742$	−160.000	−0.215633
$743$	11.3137i	0.0152271i	0.999971	+	0.00761353i	$0.00242349\pi$
$743$	11.3137i	0.0152271i	−0.999971	+	0.00761353i	$0.997577\pi$
$744$	− 415.779i	− 0.558842i
$745$	− 1930.40i	− 2.59114i
$746$	330.000	0.442359
$747$	282.843i	0.378638i
$748$	0	0
$749$	−1310.00	−1.74900
$750$	− 9.89949i	− 0.0131993i
$751$	527.000	0.701731	0.350866	−	0.936426i	$-0.385887\pi$
$751$	527.000	0.701731	0.350866	+	0.936426i	$0.385887\pi$
$752$	−128.000	−0.170213
$753$	−225.000	−0.298805
$754$	− 543.058i	− 0.720236i
$755$	1098.84i	1.45542i
$756$	240.416i	0.318011i
$757$	38.0000	0.0501982	0.0250991	−	0.999685i	$-0.492010\pi$
$757$	38.0000	0.0501982	0.0250991	+	0.999685i	$0.492010\pi$
$758$	− 632.153i	− 0.833976i
$759$	0	0
$760$	−1008.00	−1.32632
$761$	− 1227.54i	− 1.61306i	−0.591194	−	0.806529i	$-0.701343\pi$
$761$	− 1227.54i	− 1.61306i	0.591194	−	0.806529i	$-0.298657\pi$
$762$	−248.000	−0.325459
$763$	−330.000	−0.432503
$764$	430.000	0.562827
$765$	− 237.588i	− 0.310572i
$766$	− 770.746i	− 1.00620i
$767$	− 1204.91i	− 1.57094i
$768$	272.000	0.354167
$769$	− 311.127i	− 0.404586i	−0.979325	−	0.202293i	$-0.935161\pi$
$769$	− 311.127i	− 0.404586i	0.979325	−	0.202293i	$-0.0648394\pi$
$770$	0	0
$771$	−424.000	−0.549935
$772$	271.529i	0.351722i
$773$	1176.00	1.52135	0.760673	−	0.649136i	$-0.224869\pi$
$773$	1176.00	1.52135	0.760673	+	0.649136i	$0.224869\pi$
$774$	528.000	0.682171
$775$	1176.00	1.51742
$776$	− 144.250i	− 0.185889i
$777$	− 120.208i	− 0.154708i
$778$	304.056i	0.390817i
$779$	−288.000	−0.369705
$780$	237.588i	0.304600i
$781$	0	0
$782$	−54.0000	−0.0690537
$783$	384.666i	0.491272i
$784$	4.00000	0.00510204
$785$	−1225.00	−1.56051
$786$	198.000	0.251908
$787$	1255.82i	1.59571i	0.602851	+	0.797854i	$0.294031\pi$
$787$	1255.82i	1.59571i	−0.602851	+	0.797854i	$0.705969\pi$
$788$	− 404.465i	− 0.513281i
$789$	− 140.007i	− 0.177449i
$790$	1554.00	1.96709
$791$	459.619i	0.581061i
$792$	0	0
$793$	−192.000	−0.242119
$794$	− 837.214i	− 1.05443i
$795$	112.000	0.140881
$796$	400.000	0.502513
$797$	191.000	0.239649	0.119824	−	0.992795i	$-0.461767\pi$
$797$	191.000	0.239649	0.119824	+	0.992795i	$0.461767\pi$
$798$	− 169.706i	− 0.212664i
$799$	− 135.765i	− 0.169918i
$800$	678.823i	0.848528i
$801$	72.0000	0.0898876
$802$	690.136i	0.860519i
$803$	0	0
$804$	62.0000	0.0771144
$805$	445.477i	0.553388i
$806$	−1176.00	−1.45906
$807$	−136.000	−0.168525
$808$	1308.00	1.61881
$809$	− 548.715i	− 0.678263i	−0.940739	−	0.339132i	$-0.889867\pi$
$809$	− 548.715i	− 0.678263i	0.940739	−	0.339132i	$-0.110133\pi$
$810$	− 544.472i	− 0.672188i
$811$	− 405.879i	− 0.500468i	−0.968185	−	0.250234i	$-0.919492\pi$
$811$	− 405.879i	− 0.500468i	0.968185	−	0.250234i	$-0.0805075\pi$
$812$	−320.000	−0.394089
$813$	288.500i	0.354858i
$814$	0	0
$815$	1120.00	1.37423
$816$	− 16.9706i	− 0.0207973i
$817$	792.000	0.969400
$818$	−222.000	−0.271394
$819$	960.000	1.17216
$820$	237.588i	0.289741i
$821$	− 55.1543i	− 0.0671795i	−0.999436	−	0.0335897i	$-0.989306\pi$
$821$	− 55.1543i	− 0.0671795i	0.999436	−	0.0335897i	$-0.0106940\pi$
$822$	− 363.453i	− 0.442157i
$823$	687.000	0.834751	0.417375	−	0.908734i	$-0.362950\pi$
$823$	687.000	0.834751	0.417375	+	0.908734i	$0.362950\pi$
$824$	− 135.765i	− 0.164763i
$825$	0	0
$826$	710.000	0.859564
$827$	16.9706i	0.0205206i	0.999947	+	0.0102603i	$0.00326602\pi$
$827$	16.9706i	0.0205206i	−0.999947	+	0.0102603i	$0.996734\pi$
$828$	144.000	0.173913
$829$	985.000	1.18818	0.594089	−	0.804399i	$-0.297513\pi$
$829$	985.000	1.18818	0.594089	+	0.804399i	$0.297513\pi$
$830$	−350.000	−0.421687
$831$	123.037i	0.148058i
$832$	− 950.352i	− 1.14225i
$833$	4.24264i	0.00509321i
$834$	−122.000	−0.146283
$835$	− 118.794i	− 0.142268i
$836$	0	0
$837$	833.000	0.995221
$838$	− 463.862i	− 0.553535i
$839$	87.0000	0.103695	0.0518474	−	0.998655i	$-0.483489\pi$
$839$	87.0000	0.103695	0.0518474	+	0.998655i	$0.483489\pi$
$840$	−420.000	−0.500000
$841$	329.000	0.391201
$842$	294.156i	0.349354i
$843$	284.257i	0.337197i
$844$	158.392i	0.187668i
$845$	833.000	0.985799
$846$	− 362.039i	− 0.427942i
$847$	0	0
$848$	−64.0000	−0.0754717
$849$	79.1960i	0.0932815i
$850$	144.000	0.169412
$851$	−153.000	−0.179788
$852$	146.000	0.171362
$853$	− 533.159i	− 0.625039i	−0.949911	−	0.312520i	$-0.898827\pi$
$853$	− 533.159i	− 0.625039i	0.949911	−	0.312520i	$-0.101173\pi$
$854$	− 113.137i	− 0.132479i
$855$	− 950.352i	− 1.11152i
$856$	−1572.00	−1.83645
$857$	1368.96i	1.59738i	0.601740	+	0.798692i	$0.294475\pi$
$857$	1368.96i	1.59738i	−0.601740	+	0.798692i	$0.705525\pi$
$858$	0	0
$859$	−977.000	−1.13737	−0.568685	−	0.822556i	$-0.692547\pi$
$859$	−977.000	−1.13737	−0.568685	+	0.822556i	$0.692547\pi$
$860$	− 653.367i	− 0.759729i
$861$	−120.000	−0.139373
$862$	−794.000	−0.921114
$863$	−1272.00	−1.47393	−0.736964	−	0.675932i	$-0.763741\pi$
$863$	−1272.00	−1.47393	−0.736964	+	0.675932i	$0.763741\pi$
$864$	480.833i	0.556519i
$865$	− 861.256i	− 0.995672i
$866$	55.1543i	0.0636886i
$867$	−271.000	−0.312572
$868$	692.965i	0.798346i
$869$	0	0
$870$	−224.000	−0.257471
$871$	− 526.087i	− 0.604004i
$872$	−396.000	−0.454128
$873$	136.000	0.155785
$874$	−216.000	−0.247140
$875$	49.4975i	0.0565685i
$876$	79.1960i	0.0904063i
$877$	605.283i	0.690175i	0.938571	+	0.345087i	$0.112151\pi$
$877$	605.283i	0.690175i	−0.938571	+	0.345087i	$0.887849\pi$
$878$	352.000	0.400911
$879$	164.049i	0.186631i
$880$	0	0
$881$	−295.000	−0.334847	−0.167423	−	0.985885i	$-0.553545\pi$
$881$	−295.000	−0.334847	−0.167423	+	0.985885i	$0.553545\pi$
$882$	11.3137i	0.0128273i
$883$	−584.000	−0.661382	−0.330691	−	0.943739i	$-0.607282\pi$
$883$	−584.000	−0.661382	−0.330691	+	0.943739i	$0.607282\pi$
$884$	144.000	0.162896
$885$	−497.000	−0.561582
$886$	247.487i	0.279331i
$887$	971.565i	1.09534i	0.836695	+	0.547669i	$0.184485\pi$
$887$	971.565i	1.09534i	−0.836695	+	0.547669i	$0.815515\pi$
$888$	− 144.250i	− 0.162443i
$889$	1240.00	1.39483
$890$	89.0955i	0.100107i
$891$	0	0
$892$	−222.000	−0.248879
$893$	− 543.058i	− 0.608128i
$894$	390.000	0.436242
$895$	1393.00	1.55642
$896$	−240.000	−0.267857
$897$	152.735i	0.170273i
$898$	442.649i	0.492927i
$899$	1108.74i	1.23331i
$900$	−384.000	−0.426667
$901$	− 67.8823i	− 0.0753410i
$902$	0	0
$903$	330.000	0.365449
$904$	551.543i	0.610114i
$905$	511.000	0.564641
$906$	−222.000	−0.245033
$907$	−1536.00	−1.69350	−0.846748	−	0.531995i	$-0.821443\pi$
$907$	−1536.00	−1.69350	−0.846748	+	0.531995i	$0.821443\pi$
$908$	263.044i	0.289696i
$909$	1233.19i	1.35665i
$910$	1187.94i	1.30543i
$911$	830.000	0.911087	0.455543	−	0.890214i	$-0.349445\pi$
$911$	830.000	0.911087	0.455543	+	0.890214i	$0.349445\pi$
$912$	− 67.8823i	− 0.0744323i
$913$	0	0
$914$	−16.0000	−0.0175055
$915$	79.1960i	0.0865530i
$916$	−606.000	−0.661572
$917$	−990.000	−1.07961
$918$	102.000	0.111111
$919$	− 564.271i	− 0.614006i	−0.951709	−	0.307003i	$-0.900674\pi$
$919$	− 564.271i	− 0.614006i	0.951709	−	0.307003i	$-0.0993261\pi$
$920$	534.573i	0.581057i
$921$	− 186.676i	− 0.202689i
$922$	−1232.00	−1.33623
$923$	− 1238.85i	− 1.34220i
$924$	0	0
$925$	408.000	0.441081
$926$	453.963i	0.490240i
$927$	128.000	0.138080
$928$	−640.000	−0.689655
$929$	1544.00	1.66200	0.831001	−	0.556271i	$-0.187768\pi$
$929$	1544.00	1.66200	0.831001	+	0.556271i	$0.187768\pi$
$930$	485.075i	0.521586i
$931$	16.9706i	0.0182283i
$932$	158.392i	0.169948i
$933$	−142.000	−0.152197
$934$	− 227.688i	− 0.243778i
$935$	0	0
$936$	1152.00	1.23077
$937$	888.126i	0.947840i	0.880568	+	0.473920i	$0.157161\pi$
$937$	888.126i	0.947840i	−0.880568	+	0.473920i	$0.842839\pi$
$938$	310.000	0.330490
$939$	447.000	0.476038
$940$	−448.000	−0.476596
$941$	− 813.173i	− 0.864158i	−0.901836	−	0.432079i	$-0.857780\pi$
$941$	− 813.173i	− 0.864158i	0.901836	−	0.432079i	$-0.142220\pi$
$942$	− 247.487i	− 0.262725i
$943$	152.735i	0.161967i
$944$	284.000	0.300847
$945$	− 841.457i	− 0.890431i
$946$	0	0
$947$	145.000	0.153115	0.0765576	−	0.997065i	$-0.475607\pi$
$947$	145.000	0.153115	0.0765576	+	0.997065i	$0.475607\pi$
$948$	− 313.955i	− 0.331177i
$949$	672.000	0.708114
$950$	576.000	0.606316
$951$	−423.000	−0.444795
$952$	254.558i	0.267393i
$953$	629.325i	0.660362i	0.943918	+	0.330181i	$0.107110\pi$
$953$	629.325i	0.660362i	−0.943918	+	0.330181i	$0.892890\pi$
$954$	− 181.019i	− 0.189748i
$955$	−1505.00	−1.57592
$956$	− 562.857i	− 0.588763i
$957$	0	0
$958$	−632.000	−0.659708
$959$	1817.26i	1.89496i
$960$	−392.000	−0.408333
$961$	1440.00	1.49844
$962$	−408.000	−0.424116
$963$	− 1482.10i	− 1.53904i
$964$	− 746.705i	− 0.774590i
$965$	− 950.352i	− 0.984820i
$966$	−90.0000	−0.0931677
$967$	− 373.352i	− 0.386093i	−0.981190	−	0.193047i	$-0.938163\pi$
$967$	− 373.352i	− 0.386093i	0.981190	−	0.193047i	$-0.0618369\pi$
$968$	0	0
$969$	72.0000	0.0743034
$970$	168.291i	0.173496i
$971$	−1695.00	−1.74562	−0.872812	−	0.488057i	$-0.837706\pi$
$971$	−1695.00	−1.74562	−0.872812	+	0.488057i	$0.837706\pi$
$972$	−416.000	−0.427984
$973$	610.000	0.626927
$974$	− 1028.13i	− 1.05558i
$975$	− 407.294i	− 0.417737i
$976$	− 45.2548i	− 0.0463677i
$977$	−369.000	−0.377687	−0.188843	−	0.982007i	$-0.560474\pi$
$977$	−369.000	−0.377687	−0.188843	+	0.982007i	$0.560474\pi$
$978$	226.274i	0.231364i
$979$	0	0
$980$	14.0000	0.0142857
$981$	− 373.352i	− 0.380583i
$982$	−736.000	−0.749491
$983$	457.000	0.464903	0.232452	−	0.972608i	$-0.425325\pi$
$983$	457.000	0.464903	0.232452	+	0.972608i	$0.425325\pi$
$984$	−144.000	−0.146341
$985$	1415.63i	1.43719i
$986$	135.765i	0.137692i
$987$	− 226.274i	− 0.229254i
$988$	576.000	0.582996
$989$	− 420.021i	− 0.424693i
$990$	0	0
$991$	−1032.00	−1.04137	−0.520686	−	0.853748i	$-0.674324\pi$
$991$	−1032.00	−1.04137	−0.520686	+	0.853748i	$0.674324\pi$
$992$	1385.93i	1.39711i
$993$	−145.000	−0.146022
$994$	730.000	0.734406
$995$	−1400.00	−1.40704
$996$	70.7107i	0.0709947i
$997$	− 708.521i	− 0.710653i	−0.934742	−	0.355326i	$-0.884370\pi$
$997$	− 708.521i	− 0.710653i	0.934742	−	0.355326i	$-0.115630\pi$
$998$	− 520.431i	− 0.521474i
$999$	289.000	0.289289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.3.b.a.120.2	yes	2
3.2	odd	2		1089.3.c.a.604.1		2
11.2	odd	10		121.3.d.e.40.1		8
11.3	even	5		121.3.d.e.112.2		8
11.4	even	5		121.3.d.e.94.1		8
11.5	even	5		121.3.d.e.118.1		8
11.6	odd	10		121.3.d.e.118.2		8
11.7	odd	10		121.3.d.e.94.2		8
11.8	odd	10		121.3.d.e.112.1		8
11.9	even	5		121.3.d.e.40.2		8
11.10	odd	2	inner	121.3.b.a.120.1	✓	2
33.32	even	2		1089.3.c.a.604.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.b.a.120.1	✓	2	11.10	odd	2	inner
121.3.b.a.120.2	yes	2	1.1	even	1	trivial
121.3.d.e.40.1		8	11.2	odd	10
121.3.d.e.40.2		8	11.9	even	5
121.3.d.e.94.1		8	11.4	even	5
121.3.d.e.94.2		8	11.7	odd	10
121.3.d.e.112.1		8	11.8	odd	10
121.3.d.e.112.2		8	11.3	even	5
121.3.d.e.118.1		8	11.5	even	5
121.3.d.e.118.2		8	11.6	odd	10
1089.3.c.a.604.1		2	3.2	odd	2
1089.3.c.a.604.2		2	33.32	even	2

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:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	121.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.00000 q^{3} +2.00000 q^{4} -7.00000 q^{5} -1.41421i q^{6} +7.07107i q^{7} +8.48528i q^{8} -8.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} -1.00000 q^{3} +2.00000 q^{4} -7.00000 q^{5} -1.41421i q^{6} +7.07107i q^{7} +8.48528i q^{8} -8.00000 q^{9} -9.89949i q^{10} -2.00000 q^{12} +16.9706i q^{13} -10.0000 q^{14} +7.00000 q^{15} -4.00000 q^{16} -4.24264i q^{17} -11.3137i q^{18} -16.9706i q^{19} -14.0000 q^{20} -7.07107i q^{21} -9.00000 q^{23} -8.48528i q^{24} +24.0000 q^{25} -24.0000 q^{26} +17.0000 q^{27} +14.1421i q^{28} +22.6274i q^{29} +9.89949i q^{30} +49.0000 q^{31} +28.2843i q^{32} +6.00000 q^{34} -49.4975i q^{35} -16.0000 q^{36} +17.0000 q^{37} +24.0000 q^{38} -16.9706i q^{39} -59.3970i q^{40} -16.9706i q^{41} +10.0000 q^{42} +46.6690i q^{43} +56.0000 q^{45} -12.7279i q^{46} +32.0000 q^{47} +4.00000 q^{48} -1.00000 q^{49} +33.9411i q^{50} +4.24264i q^{51} +33.9411i q^{52} +16.0000 q^{53} +24.0416i q^{54} -60.0000 q^{56} +16.9706i q^{57} -32.0000 q^{58} -71.0000 q^{59} +14.0000 q^{60} +11.3137i q^{61} +69.2965i q^{62} -56.5685i q^{63} -56.0000 q^{64} -118.794i q^{65} -31.0000 q^{67} -8.48528i q^{68} +9.00000 q^{69} +70.0000 q^{70} -73.0000 q^{71} -67.8823i q^{72} -39.5980i q^{73} +24.0416i q^{74} -24.0000 q^{75} -33.9411i q^{76} +24.0000 q^{78} +156.978i q^{79} +28.0000 q^{80} +55.0000 q^{81} +24.0000 q^{82} -35.3553i q^{83} -14.1421i q^{84} +29.6985i q^{85} -66.0000 q^{86} -22.6274i q^{87} -9.00000 q^{89} +79.1960i q^{90} -120.000 q^{91} -18.0000 q^{92} -49.0000 q^{93} +45.2548i q^{94} +118.794i q^{95} -28.2843i q^{96} -17.0000 q^{97} -1.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{4} - 14 q^{5} - 16 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^4 - 14 * q^5 - 16 * q^9	\( 2 q - 2 q^{3} + 4 q^{4} - 14 q^{5} - 16 q^{9} - 4 q^{12} - 20 q^{14} + 14 q^{15} - 8 q^{16} - 28 q^{20} - 18 q^{23} + 48 q^{25} - 48 q^{26} + 34 q^{27} + 98 q^{31} + 12 q^{34} - 32 q^{36} + 34 q^{37} + 48 q^{38} + 20 q^{42} + 112 q^{45} + 64 q^{47} + 8 q^{48} - 2 q^{49} + 32 q^{53} - 120 q^{56} - 64 q^{58} - 142 q^{59} + 28 q^{60} - 112 q^{64} - 62 q^{67} + 18 q^{69} + 140 q^{70} - 146 q^{71} - 48 q^{75} + 48 q^{78} + 56 q^{80} + 110 q^{81} + 48 q^{82} - 132 q^{86} - 18 q^{89} - 240 q^{91} - 36 q^{92} - 98 q^{93} - 34 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^4 - 14 * q^5 - 16 * q^9 - 4 * q^12 - 20 * q^14 + 14 * q^15 - 8 * q^16 - 28 * q^20 - 18 * q^23 + 48 * q^25 - 48 * q^26 + 34 * q^27 + 98 * q^31 + 12 * q^34 - 32 * q^36 + 34 * q^37 + 48 * q^38 + 20 * q^42 + 112 * q^45 + 64 * q^47 + 8 * q^48 - 2 * q^49 + 32 * q^53 - 120 * q^56 - 64 * q^58 - 142 * q^59 + 28 * q^60 - 112 * q^64 - 62 * q^67 + 18 * q^69 + 140 * q^70 - 146 * q^71 - 48 * q^75 + 48 * q^78 + 56 * q^80 + 110 * q^81 + 48 * q^82 - 132 * q^86 - 18 * q^89 - 240 * q^91 - 36 * q^92 - 98 * q^93 - 34 * q^97

Introduction

L-functions

Modular forms

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Database

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