LMFDB - Embedded newform 61.2.b.a.60.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [61,2,Mod(60,61)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("61.60"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(61, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	61.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.487087452330$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.29952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 13 $ x^4 + 8*x^2 + 13
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			60.1
Root			$-2.39417i$ of defining polynomial
Character	$\chi$	$=$	61.60
Dual form			61.2.b.a.60.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.39417i q^{2} -2.73205 q^{3} -3.73205 q^{4} +1.73205 q^{5} +6.54099i q^{6} -4.14682i q^{7} +4.14682i q^{8} +4.46410 q^{9} -4.14682i q^{10} +2.39417i q^{11} +10.1962 q^{12} +3.00000 q^{13} -9.92820 q^{14} -4.73205 q^{15} +2.46410 q^{16} -1.75265i q^{17} -10.6878i q^{18} +4.73205 q^{19} -6.46410 q^{20} +11.3293i q^{21} +5.73205 q^{22} +5.89948i q^{23} -11.3293i q^{24} -2.00000 q^{25} -7.18251i q^{26} -4.00000 q^{27} +15.4762i q^{28} +1.75265i q^{29} +11.3293i q^{30} -3.03569i q^{31} +2.39417i q^{32} -6.54099i q^{33} -4.19615 q^{34} -7.18251i q^{35} -16.6603 q^{36} +3.03569i q^{37} -11.3293i q^{38} -8.19615 q^{39} +7.18251i q^{40} +0.464102 q^{41} +27.1244 q^{42} -11.3293i q^{43} -8.93516i q^{44} +7.73205 q^{45} +14.1244 q^{46} -0.928203 q^{47} -6.73205 q^{48} -10.1962 q^{49} +4.78834i q^{50} +4.78834i q^{51} -11.1962 q^{52} +1.28303i q^{53} +9.57668i q^{54} +4.14682i q^{55} +17.1962 q^{56} -12.9282 q^{57} +4.19615 q^{58} +8.93516i q^{59} +17.6603 q^{60} +(7.19615 - 3.03569i) q^{61} -7.26795 q^{62} -18.5118i q^{63} +10.6603 q^{64} +5.19615 q^{65} -15.6603 q^{66} +10.2182i q^{67} +6.54099i q^{68} -16.1177i q^{69} -17.1962 q^{70} +6.54099i q^{71} +18.5118i q^{72} -1.19615 q^{73} +7.26795 q^{74} +5.46410 q^{75} -17.6603 q^{76} +9.92820 q^{77} +19.6230i q^{78} +9.40479i q^{79} +4.26795 q^{80} -2.46410 q^{81} -1.11114i q^{82} +9.46410 q^{83} -42.2817i q^{84} -3.03569i q^{85} -27.1244 q^{86} -4.78834i q^{87} -9.92820 q^{88} -11.7990i q^{89} -18.5118i q^{90} -12.4405i q^{91} -22.0172i q^{92} +8.29365i q^{93} +2.22228i q^{94} +8.19615 q^{95} -6.54099i q^{96} -3.46410 q^{97} +24.4113i q^{98} +10.6878i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{4} + 4 q^{9} + 20 q^{12} + 12 q^{13} - 12 q^{14} - 12 q^{15} - 4 q^{16} + 12 q^{19} - 12 q^{20} + 16 q^{22} - 8 q^{25} - 16 q^{27} + 4 q^{34} - 32 q^{36} - 12 q^{39} - 12 q^{41} + 60 q^{42}+ \cdots + 12 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^4 + 4 * q^9 + 20 * q^12 + 12 * q^13 - 12 * q^14 - 12 * q^15 - 4 * q^16 + 12 * q^19 - 12 * q^20 + 16 * q^22 - 8 * q^25 - 16 * q^27 + 4 * q^34 - 32 * q^36 - 12 * q^39 - 12 * q^41 + 60 * q^42 + 24 * q^45 + 8 * q^46 + 24 * q^47 - 20 * q^48 - 20 * q^49 - 24 * q^52 + 48 * q^56 - 24 * q^57 - 4 * q^58 + 36 * q^60 + 8 * q^61 - 36 * q^62 + 8 * q^64 - 28 * q^66 - 48 * q^70 + 16 * q^73 + 36 * q^74 + 8 * q^75 - 36 * q^76 + 12 * q^77 + 24 * q^80 + 4 * q^81 + 24 * q^83 - 60 * q^86 - 12 * q^88 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/61\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		−	2.39417i		−	1.69293i	−0.532441	−	0.846467i	$-0.678725\pi$
$2$		−	2.39417i		−	1.69293i	0.532441	−	0.846467i	$-0.321275\pi$
$3$	−2.73205			−1.57735			−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205			−1.57735			−0.788675	+	0.614810i	$0.789233\pi$
$4$	−3.73205			−1.86603
$5$	1.73205			0.774597			0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205			0.774597			0.387298	+	0.921954i	$0.373408\pi$
$6$			6.54099i			2.67035i
$7$		−	4.14682i		−	1.56735i	−0.621170	−	0.783676i	$-0.713342\pi$
$7$		−	4.14682i		−	1.56735i	0.621170	−	0.783676i	$-0.286658\pi$
$8$			4.14682i			1.46612i
$9$	4.46410			1.48803
$10$		−	4.14682i		−	1.31134i
$11$			2.39417i			0.721869i	0.932591	+	0.360935i	$0.117542\pi$
$11$			2.39417i			0.721869i	−0.932591	+	0.360935i	$0.882458\pi$
$12$	10.1962			2.94338
$13$	3.00000			0.832050			0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000			0.832050			0.416025	+	0.909353i	$0.363423\pi$
$14$	−9.92820			−2.65342
$15$	−4.73205			−1.22181
$16$	2.46410			0.616025
$17$		−	1.75265i		−	0.425081i	−0.977152	−	0.212541i	$-0.931826\pi$
$17$		−	1.75265i		−	0.425081i	0.977152	−	0.212541i	$-0.0681738\pi$
$18$		−	10.6878i		−	2.51914i
$19$	4.73205			1.08561			0.542803	−	0.839860i	$-0.317363\pi$
$19$	4.73205			1.08561			0.542803	+	0.839860i	$0.317363\pi$
$20$	−6.46410			−1.44542
$21$			11.3293i			2.47226i
$22$	5.73205			1.22208
$23$			5.89948i			1.23013i	0.788478	+	0.615063i	$0.210869\pi$
$23$			5.89948i			1.23013i	−0.788478	+	0.615063i	$0.789131\pi$
$24$		−	11.3293i		−	2.31259i
$25$	−2.00000			−0.400000
$26$		−	7.18251i		−	1.40861i
$27$	−4.00000			−0.769800
$28$			15.4762i			2.92472i
$29$			1.75265i			0.325460i	0.986671	+	0.162730i	$0.0520299\pi$
$29$			1.75265i			0.325460i	−0.986671	+	0.162730i	$0.947970\pi$
$30$			11.3293i			2.06844i
$31$		−	3.03569i		−	0.545225i	−0.962124	−	0.272613i	$-0.912112\pi$
$31$		−	3.03569i		−	0.545225i	0.962124	−	0.272613i	$-0.0878877\pi$
$32$			2.39417i			0.423233i
$33$		−	6.54099i		−	1.13864i
$34$	−4.19615			−0.719634
$35$		−	7.18251i		−	1.21407i
$36$	−16.6603			−2.77671
$37$			3.03569i			0.499064i	0.968367	+	0.249532i	$0.0802767\pi$
$37$			3.03569i			0.499064i	−0.968367	+	0.249532i	$0.919723\pi$
$38$		−	11.3293i		−	1.83786i
$39$	−8.19615			−1.31243
$40$			7.18251i			1.13565i
$41$	0.464102			0.0724805			0.0362402	−	0.999343i	$-0.488462\pi$
$41$	0.464102			0.0724805			0.0362402	+	0.999343i	$0.488462\pi$
$42$	27.1244			4.18538
$43$		−	11.3293i		−	1.72771i	−0.503743	−	0.863854i	$-0.668044\pi$
$43$		−	11.3293i		−	1.72771i	0.503743	−	0.863854i	$-0.331956\pi$
$44$		−	8.93516i		−	1.34703i
$45$	7.73205			1.15263
$46$	14.1244			2.08252
$47$	−0.928203			−0.135392			−0.0676962	−	0.997706i	$-0.521565\pi$
$47$	−0.928203			−0.135392			−0.0676962	+	0.997706i	$0.521565\pi$
$48$	−6.73205			−0.971688
$49$	−10.1962			−1.45659
$50$			4.78834i			0.677174i
$51$			4.78834i			0.670502i
$52$	−11.1962			−1.55263
$53$			1.28303i			0.176238i	0.996110	+	0.0881190i	$0.0280856\pi$
$53$			1.28303i			0.176238i	−0.996110	+	0.0881190i	$0.971914\pi$
$54$			9.57668i			1.30322i
$55$			4.14682i			0.559158i
$56$	17.1962			2.29793
$57$	−12.9282			−1.71238
$58$	4.19615			0.550982
$59$			8.93516i			1.16326i	0.813454	+	0.581630i	$0.197585\pi$
$59$			8.93516i			1.16326i	−0.813454	+	0.581630i	$0.802415\pi$
$60$	17.6603			2.27993
$61$	7.19615	−	3.03569i	0.921373	−	0.388680i
$61$	7.19615	−	3.03569i	0.921373	−	0.388680i
$62$	−7.26795			−0.923030
$63$		−	18.5118i		−	2.33227i
$64$	10.6603			1.33253
$65$	5.19615			0.644503
$66$	−15.6603			−1.92764
$67$			10.2182i			1.24835i	0.781284	+	0.624176i	$0.214565\pi$
$67$			10.2182i			1.24835i	−0.781284	+	0.624176i	$0.785435\pi$
$68$			6.54099i			0.793212i
$69$		−	16.1177i		−	1.94034i
$70$	−17.1962			−2.05533
$71$			6.54099i			0.776273i	0.921602	+	0.388137i	$0.126881\pi$
$71$			6.54099i			0.776273i	−0.921602	+	0.388137i	$0.873119\pi$
$72$			18.5118i			2.18164i
$73$	−1.19615			−0.139999			−0.0699995	−	0.997547i	$-0.522300\pi$
$73$	−1.19615			−0.139999			−0.0699995	+	0.997547i	$0.522300\pi$
$74$	7.26795			0.844882
$75$	5.46410			0.630940
$76$	−17.6603			−2.02577
$77$	9.92820			1.13142
$78$			19.6230i			2.22187i
$79$			9.40479i			1.05812i	0.848584	+	0.529061i	$0.177456\pi$
$79$			9.40479i			1.05812i	−0.848584	+	0.529061i	$0.822544\pi$
$80$	4.26795			0.477171
$81$	−2.46410			−0.273789
$82$		−	1.11114i		−	0.122705i
$83$	9.46410			1.03882			0.519410	−	0.854525i	$-0.326152\pi$
$83$	9.46410			1.03882			0.519410	+	0.854525i	$0.326152\pi$
$84$		−	42.2817i		−	4.61331i
$85$		−	3.03569i		−	0.329266i
$86$	−27.1244			−2.92489
$87$		−	4.78834i		−	0.513364i
$88$	−9.92820			−1.05835
$89$		−	11.7990i		−	1.25069i	−0.780350	−	0.625343i	$-0.784959\pi$
$89$		−	11.7990i		−	1.25069i	0.780350	−	0.625343i	$-0.215041\pi$
$90$		−	18.5118i		−	1.95132i
$91$		−	12.4405i		−	1.30412i
$92$		−	22.0172i		−	2.29545i
$93$			8.29365i			0.860011i
$94$			2.22228i			0.229210i
$95$	8.19615			0.840907
$96$		−	6.54099i		−	0.667587i
$97$	−3.46410			−0.351726			−0.175863	−	0.984415i	$-0.556272\pi$
$97$	−3.46410			−0.351726			−0.175863	+	0.984415i	$0.556272\pi$
$98$			24.4113i			2.46592i
$99$			10.6878i			1.07417i
$100$	7.46410			0.746410
$101$			7.82403i			0.778520i	0.921128	+	0.389260i	$0.127269\pi$
$101$			7.82403i			0.778520i	−0.921128	+	0.389260i	$0.872731\pi$
$102$	11.4641			1.13512
$103$	−9.12436			−0.899049			−0.449525	−	0.893268i	$-0.648407\pi$
$103$	−9.12436			−0.899049			−0.449525	+	0.893268i	$0.648407\pi$
$104$			12.4405i			1.21989i
$105$			19.6230i			1.91501i
$106$	3.07180			0.298359
$107$	−15.4641			−1.49497			−0.747486	−	0.664278i	$-0.768739\pi$
$107$	−15.4641			−1.49497			−0.747486	+	0.664278i	$0.768739\pi$
$108$	14.9282			1.43647
$109$	−11.0000			−1.05361			−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000			−1.05361			−0.526804	+	0.849987i	$0.676610\pi$
$110$	9.92820			0.946617
$111$		−	8.29365i		−	0.787198i
$112$		−	10.2182i		−	0.965529i
$113$	−2.07180			−0.194898			−0.0974491	−	0.995241i	$-0.531068\pi$
$113$	−2.07180			−0.194898			−0.0974491	+	0.995241i	$0.531068\pi$
$114$			30.9523i			2.89895i
$115$			10.2182i			0.952852i
$116$		−	6.54099i		−	0.607316i
$117$	13.3923			1.23812
$118$	21.3923			1.96932
$119$	−7.26795			−0.666252
$120$		−	19.6230i		−	1.79133i
$121$	5.26795			0.478904
$122$	−7.26795	−	17.2288i	−0.658009	−	1.55982i
$123$	−1.26795			−0.114327
$124$			11.3293i			1.01740i
$125$	−12.1244			−1.08444
$126$	−44.3205			−3.94838
$127$	12.5885			1.11704			0.558522	−	0.829489i	$-0.311368\pi$
$127$	12.5885			1.11704			0.558522	+	0.829489i	$0.311368\pi$
$128$		−	20.7341i		−	1.83265i
$129$			30.9523i			2.72520i
$130$		−	12.4405i		−	1.09110i
$131$	11.6603			1.01876			0.509381	−	0.860541i	$-0.329875\pi$
$131$	11.6603			1.01876			0.509381	+	0.860541i	$0.329875\pi$
$132$			24.4113i			2.12473i
$133$		−	19.6230i		−	1.70153i
$134$	24.4641			2.11338
$135$	−6.92820			−0.596285
$136$	7.26795			0.623222
$137$	1.73205			0.147979			0.0739895	−	0.997259i	$-0.476427\pi$
$137$	1.73205			0.147979			0.0739895	+	0.997259i	$0.476427\pi$
$138$	−38.5885			−3.28487
$139$		−	4.14682i		−	0.351729i	−0.984414	−	0.175865i	$-0.943728\pi$
$139$		−	4.14682i		−	0.351729i	0.984414	−	0.175865i	$-0.0562721\pi$
$140$			26.8055i			2.26548i
$141$	2.53590			0.213561
$142$	15.6603			1.31418
$143$			7.18251i			0.600632i
$144$	11.0000			0.916667
$145$			3.03569i			0.252100i
$146$			2.86379i			0.237009i
$147$	27.8564			2.29756
$148$		−	11.3293i		−	0.931266i
$149$	−6.46410			−0.529560			−0.264780	−	0.964309i	$-0.585299\pi$
$149$	−6.46410			−0.529560			−0.264780	+	0.964309i	$0.585299\pi$
$150$		−	13.0820i		−	1.06814i
$151$		−	12.4405i		−	1.01239i	−0.862419	−	0.506196i	$-0.831051\pi$
$151$		−	12.4405i		−	1.01239i	0.862419	−	0.506196i	$-0.168949\pi$
$152$			19.6230i			1.59163i
$153$		−	7.82403i		−	0.632535i
$154$		−	23.7698i		−	1.91543i
$155$		−	5.25796i		−	0.422330i
$156$	30.5885			2.44904
$157$		−	9.10706i		−	0.726822i	−0.931629	−	0.363411i	$-0.881612\pi$
$157$		−	9.10706i		−	0.726822i	0.931629	−	0.363411i	$-0.118388\pi$
$158$	22.5167			1.79133
$159$		−	3.50531i		−	0.277989i
$160$			4.14682i			0.327835i
$161$	24.4641			1.92804
$162$			5.89948i			0.463507i
$163$	−15.1244			−1.18463			−0.592315	−	0.805706i	$-0.701786\pi$
$163$	−15.1244			−1.18463			−0.592315	+	0.805706i	$0.701786\pi$
$164$	−1.73205			−0.135250
$165$		−	11.3293i		−	0.881988i
$166$		−	22.6587i		−	1.75865i
$167$	−24.5885			−1.90271			−0.951356	−	0.308094i	$-0.900309\pi$
$167$	−24.5885			−1.90271			−0.951356	+	0.308094i	$0.900309\pi$
$168$	−46.9808			−3.62464
$169$	−4.00000			−0.307692
$170$	−7.26795			−0.557426
$171$	21.1244			1.61542
$172$			42.2817i			3.22395i
$173$			9.57668i			0.728102i	0.931379	+	0.364051i	$0.118607\pi$
$173$			9.57668i			0.728102i	−0.931379	+	0.364051i	$0.881393\pi$
$174$	−11.4641			−0.869091
$175$			8.29365i			0.626941i
$176$			5.89948i			0.444690i
$177$		−	24.4113i		−	1.83487i
$178$	−28.2487			−2.11733
$179$	22.3923			1.67368			0.836840	−	0.547448i	$-0.184401\pi$
$179$	22.3923			1.67368			0.836840	+	0.547448i	$0.184401\pi$
$180$	−28.8564			−2.15083
$181$			22.6587i			1.68421i	0.539317	+	0.842103i	$0.318682\pi$
$181$			22.6587i			1.68421i	−0.539317	+	0.842103i	$0.681318\pi$
$182$	−29.7846			−2.20778
$183$	−19.6603	+	8.29365i	−1.45333	+	0.613084i
$184$	−24.4641			−1.80352
$185$			5.25796i			0.386573i
$186$	19.8564			1.45594
$187$	4.19615			0.306853
$188$	3.46410			0.252646
$189$			16.5873i			1.20655i
$190$		−	19.6230i		−	1.42360i
$191$		−	23.3002i		−	1.68594i	−0.537959	−	0.842971i	$-0.680804\pi$
$191$		−	23.3002i		−	1.68594i	0.537959	−	0.842971i	$-0.319196\pi$
$192$	−29.1244			−2.10187
$193$		−	6.07137i		−	0.437027i	−0.975834	−	0.218513i	$-0.929879\pi$
$193$		−	6.07137i		−	0.437027i	0.975834	−	0.218513i	$-0.0701208\pi$
$194$			8.29365i			0.595449i
$195$	−14.1962			−1.01661
$196$	38.0526			2.71804
$197$	−4.26795			−0.304079			−0.152039	−	0.988374i	$-0.548584\pi$
$197$	−4.26795			−0.304079			−0.152039	+	0.988374i	$0.548584\pi$
$198$	25.5885			1.81849
$199$	2.00000			0.141776			0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000			0.141776			0.0708881	+	0.997484i	$0.477417\pi$
$200$		−	8.29365i		−	0.586450i
$201$		−	27.9166i		−	1.96909i
$202$	18.7321			1.31798
$203$	7.26795			0.510110
$204$		−	17.8703i		−	1.25117i
$205$	0.803848			0.0561432
$206$			21.8453i			1.52203i
$207$			26.3359i			1.83047i
$208$	7.39230			0.512564
$209$			11.3293i			0.783666i
$210$	46.9808			3.24198
$211$			19.6230i			1.35090i	0.737405	+	0.675451i	$0.236051\pi$
$211$			19.6230i			1.35090i	−0.737405	+	0.675451i	$0.763949\pi$
$212$		−	4.78834i		−	0.328865i
$213$		−	17.8703i		−	1.22445i
$214$			37.0237i			2.53089i
$215$		−	19.6230i		−	1.33828i
$216$		−	16.5873i		−	1.12862i
$217$	−12.5885			−0.854560
$218$			26.3359i			1.78369i
$219$	3.26795			0.220828
$220$		−	15.4762i		−	1.04340i
$221$		−	5.25796i		−	0.353689i
$222$	−19.8564			−1.33267
$223$			1.11114i			0.0744073i	0.999308	+	0.0372037i	$0.0118450\pi$
$223$			1.11114i			0.0744073i	−0.999308	+	0.0372037i	$0.988155\pi$
$224$	9.92820			0.663356
$225$	−8.92820			−0.595214
$226$			4.96023i			0.329950i
$227$			11.9709i			0.794533i	0.917703	+	0.397267i	$0.130041\pi$
$227$			11.9709i			0.794533i	−0.917703	+	0.397267i	$0.869959\pi$
$228$	48.2487			3.19535
$229$	−16.2679			−1.07502			−0.537508	−	0.843259i	$-0.680634\pi$
$229$	−16.2679			−1.07502			−0.537508	+	0.843259i	$0.680634\pi$
$230$	24.4641			1.61312
$231$	−27.1244			−1.78465
$232$	−7.26795			−0.477164
$233$		−	12.6124i		−	0.826264i	−0.910671	−	0.413132i	$-0.864435\pi$
$233$		−	12.6124i		−	0.826264i	0.910671	−	0.413132i	$-0.135565\pi$
$234$		−	32.0635i		−	2.09605i
$235$	−1.60770			−0.104874
$236$		−	33.3465i		−	2.17067i
$237$		−	25.6944i		−	1.66903i
$238$			17.4007i			1.12792i
$239$	13.2679			0.858232			0.429116	−	0.903249i	$-0.358825\pi$
$239$	13.2679			0.858232			0.429116	+	0.903249i	$0.358825\pi$
$240$	−11.6603			−0.752666
$241$	19.0526			1.22728			0.613642	−	0.789585i	$-0.289704\pi$
$241$	19.0526			1.22728			0.613642	+	0.789585i	$0.289704\pi$
$242$		−	12.6124i		−	0.810754i
$243$	18.7321			1.20166
$244$	−26.8564	+	11.3293i	−1.71931	+	0.725286i
$245$	−17.6603			−1.12827
$246$			3.03569i			0.193548i
$247$	14.1962			0.903280
$248$	12.5885			0.799368
$249$	−25.8564			−1.63858
$250$			29.0278i			1.83588i
$251$		−	14.8346i		−	0.936354i	−0.883635	−	0.468177i	$-0.844911\pi$
$251$		−	14.8346i		−	0.936354i	0.883635	−	0.468177i	$-0.155089\pi$
$252$			69.0871i			4.35208i
$253$	−14.1244			−0.887991
$254$		−	30.1389i		−	1.89108i
$255$			8.29365i			0.519368i
$256$	−28.3205			−1.77003
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$	74.1051			4.61358
$259$	12.5885			0.782209
$260$	−19.3923			−1.20266
$261$			7.82403i			0.484295i
$262$		−	27.9166i		−	1.72470i
$263$	−20.1962			−1.24535			−0.622674	−	0.782481i	$-0.713954\pi$
$263$	−20.1962			−1.24535			−0.622674	+	0.782481i	$0.713954\pi$
$264$	27.1244			1.66939
$265$			2.22228i			0.136513i
$266$	−46.9808			−2.88058
$267$			32.2354i			1.97277i
$268$		−	38.1348i		−	2.32946i
$269$	−2.53590			−0.154616			−0.0773082	−	0.997007i	$-0.524633\pi$
$269$	−2.53590			−0.154616			−0.0773082	+	0.997007i	$0.524633\pi$
$270$			16.5873i			1.00947i
$271$	6.19615			0.376389			0.188195	−	0.982132i	$-0.439736\pi$
$271$	6.19615			0.376389			0.188195	+	0.982132i	$0.439736\pi$
$272$		−	4.31872i		−	0.261861i
$273$			33.9880i			2.05705i
$274$		−	4.14682i		−	0.250519i
$275$		−	4.78834i		−	0.288748i
$276$			60.1520i			3.62072i
$277$			19.6230i			1.17903i	0.807757	+	0.589515i	$0.200681\pi$
$277$			19.6230i			1.17903i	−0.807757	+	0.589515i	$0.799319\pi$
$278$	−9.92820			−0.595454
$279$		−	13.5516i		−	0.811314i
$280$	29.7846			1.77997
$281$			5.72758i			0.341679i	0.985299	+	0.170840i	$0.0546480\pi$
$281$			5.72758i			0.341679i	−0.985299	+	0.170840i	$0.945352\pi$
$282$		−	6.07137i		−	0.361545i
$283$	14.5885			0.867194			0.433597	−	0.901107i	$-0.357244\pi$
$283$	14.5885			0.867194			0.433597	+	0.901107i	$0.357244\pi$
$284$		−	24.4113i		−	1.44855i
$285$	−22.3923			−1.32641
$286$	17.1962			1.01683
$287$		−	1.92455i		−	0.113602i
$288$			10.6878i			0.629786i
$289$	13.9282			0.819306
$290$	7.26795			0.426789
$291$	9.46410			0.554795
$292$	4.46410			0.261242
$293$	−1.85641			−0.108452			−0.0542262	−	0.998529i	$-0.517269\pi$
$293$	−1.85641			−0.108452			−0.0542262	+	0.998529i	$0.517269\pi$
$294$		−	66.6930i		−	3.88961i
$295$			15.4762i			0.901057i
$296$	−12.5885			−0.731689
$297$		−	9.57668i		−	0.555695i
$298$			15.4762i			0.896510i
$299$			17.6984i			1.02353i
$300$	−20.3923			−1.17735
$301$	−46.9808			−2.70793
$302$	−29.7846			−1.71391
$303$		−	21.3756i		−	1.22800i
$304$	11.6603			0.668761
$305$	12.4641	−	5.25796i	0.713692	−	0.301070i
$306$	−18.7321			−1.07084
$307$		−	1.11114i		−	0.0634160i	−0.999497	−	0.0317080i	$-0.989905\pi$
$307$		−	1.11114i		−	0.0634160i	0.999497	−	0.0317080i	$-0.0100947\pi$
$308$	−37.0526			−2.11127
$309$	24.9282			1.41812
$310$	−12.5885			−0.714976
$311$		−	31.1242i		−	1.76489i	−0.470414	−	0.882446i	$-0.655895\pi$
$311$		−	31.1242i		−	1.76489i	0.470414	−	0.882446i	$-0.344105\pi$
$312$		−	33.9880i		−	1.92419i
$313$			22.6587i			1.28074i	0.768065	+	0.640372i	$0.221220\pi$
$313$			22.6587i			1.28074i	−0.768065	+	0.640372i	$0.778780\pi$
$314$	−21.8038			−1.23046
$315$		−	32.0635i		−	1.80657i
$316$		−	35.0991i		−	1.97448i
$317$	5.07180			0.284860			0.142430	−	0.989805i	$-0.454508\pi$
$317$	5.07180			0.284860			0.142430	+	0.989805i	$0.454508\pi$
$318$	−8.39230			−0.470617
$319$	−4.19615			−0.234939
$320$	18.4641			1.03217
$321$	42.2487			2.35809
$322$		−	58.5712i		−	3.26405i
$323$		−	8.29365i		−	0.461471i
$324$	9.19615			0.510897
$325$	−6.00000			−0.332820
$326$			36.2103i			2.00550i
$327$	30.0526			1.66191
$328$			1.92455i			0.106265i
$329$			3.84910i			0.212208i
$330$	−27.1244			−1.49315
$331$			30.6546i			1.68493i	0.538752	+	0.842464i	$0.318896\pi$
$331$			30.6546i			1.68493i	−0.538752	+	0.842464i	$0.681104\pi$
$332$	−35.3205			−1.93846
$333$			13.5516i			0.742624i
$334$			58.8690i			3.22117i
$335$			17.6984i			0.966969i
$336$			27.9166i			1.52298i
$337$		−	14.3650i		−	0.782513i	−0.920282	−	0.391256i	$-0.872041\pi$
$337$		−	14.3650i		−	0.782513i	0.920282	−	0.391256i	$-0.127959\pi$
$338$			9.57668i			0.520903i
$339$	5.66025			0.307423
$340$			11.3293i			0.614419i
$341$	7.26795			0.393582
$342$		−	50.5753i		−	2.73480i
$343$			13.2539i			0.715642i
$344$	46.9808			2.53303
$345$		−	27.9166i		−	1.50298i
$346$	22.9282			1.23263
$347$	−25.5167			−1.36981			−0.684903	−	0.728634i	$-0.740155\pi$
$347$	−25.5167			−1.36981			−0.684903	+	0.728634i	$0.740155\pi$
$348$			17.8703i			0.957950i
$349$		−	30.1389i		−	1.61330i	−0.591030	−	0.806649i	$-0.701279\pi$
$349$		−	30.1389i		−	1.61330i	0.591030	−	0.806649i	$-0.298721\pi$
$350$	19.8564			1.06137
$351$	−12.0000			−0.640513
$352$	−5.73205			−0.305519
$353$	25.9808			1.38282			0.691408	−	0.722464i	$-0.256991\pi$
$353$	25.9808			1.38282			0.691408	+	0.722464i	$0.256991\pi$
$354$	−58.4449			−3.10631
$355$			11.3293i			0.601299i
$356$			44.0343i			2.33381i
$357$	19.8564			1.05091
$358$		−	53.6110i		−	2.83343i
$359$		−	18.6837i		−	0.986090i	−0.870004	−	0.493045i	$-0.835884\pi$
$359$		−	18.6837i		−	0.986090i	0.870004	−	0.493045i	$-0.164116\pi$
$360$			32.0635i			1.68989i
$361$	3.39230			0.178542
$362$	54.2487			2.85125
$363$	−14.3923			−0.755400
$364$			46.4285i			2.43351i
$365$	−2.07180			−0.108443
$366$	19.8564	+	47.0700i	1.03791	+	2.46039i
$367$	11.0718			0.577943			0.288972	−	0.957338i	$-0.406687\pi$
$367$	11.0718			0.577943			0.288972	+	0.957338i	$0.406687\pi$
$368$			14.5369i			0.757789i
$369$	2.07180			0.107853
$370$	12.5885			0.654443
$371$	5.32051			0.276227
$372$		−	30.9523i		−	1.60480i
$373$		−	11.3293i		−	0.586611i	−0.956019	−	0.293305i	$-0.905245\pi$
$373$		−	11.3293i		−	0.586611i	0.956019	−	0.293305i	$-0.0947553\pi$
$374$		−	10.0463i		−	0.519482i
$375$	33.1244			1.71053
$376$		−	3.84910i		−	0.198502i
$377$			5.25796i			0.270799i
$378$	39.7128			2.04261
$379$	−10.5885			−0.543893			−0.271946	−	0.962312i	$-0.587667\pi$
$379$	−10.5885			−0.543893			−0.271946	+	0.962312i	$0.587667\pi$
$380$	−30.5885			−1.56915
$381$	−34.3923			−1.76197
$382$	−55.7846			−2.85419
$383$			18.9815i			0.969908i	0.874540	+	0.484954i	$0.161164\pi$
$383$			18.9815i			0.969908i	−0.874540	+	0.484954i	$0.838836\pi$
$384$			56.6467i			2.89074i
$385$	17.1962			0.876397
$386$	−14.5359			−0.739858
$387$		−	50.5753i		−	2.57089i
$388$	12.9282			0.656330
$389$			7.35440i			0.372883i	0.982466	+	0.186442i	$0.0596955\pi$
$389$			7.35440i			0.372883i	−0.982466	+	0.186442i	$0.940305\pi$
$390$			33.9880i			1.72105i
$391$	10.3397			0.522903
$392$		−	42.2817i		−	2.13555i
$393$	−31.8564			−1.60694
$394$			10.2182i			0.514785i
$395$			16.2896i			0.819617i
$396$		−	39.8875i		−	2.00442i
$397$		−	7.48024i		−	0.375422i	−0.982224	−	0.187711i	$-0.939893\pi$
$397$		−	7.48024i		−	0.375422i	0.982224	−	0.187711i	$-0.0601069\pi$
$398$		−	4.78834i		−	0.240018i
$399$			53.6110i			2.68391i
$400$	−4.92820			−0.246410
$401$			10.8597i			0.542308i	0.962536	+	0.271154i	$0.0874053\pi$
$401$			10.8597i			0.542308i	−0.962536	+	0.271154i	$0.912595\pi$
$402$	−66.8372			−3.33353
$403$		−	9.10706i		−	0.453655i
$404$		−	29.1997i		−	1.45274i
$405$	−4.26795			−0.212076
$406$		−	17.4007i		−	0.863583i
$407$	−7.26795			−0.360259
$408$	−19.8564			−0.983039
$409$		−	8.29365i		−	0.410095i	−0.978752	−	0.205047i	$-0.934265\pi$
$409$		−	8.29365i		−	0.410095i	0.978752	−	0.205047i	$-0.0657348\pi$
$410$		−	1.92455i		−	0.0950466i
$411$	−4.73205			−0.233415
$412$	34.0526			1.67765
$413$	37.0526			1.82324
$414$	63.0526			3.09886
$415$	16.3923			0.804667
$416$			7.18251i			0.352152i
$417$			11.3293i			0.554800i
$418$	27.1244			1.32670
$419$		−	16.1177i		−	0.787400i	−0.919239	−	0.393700i	$-0.871195\pi$
$419$		−	16.1177i		−	0.787400i	0.919239	−	0.393700i	$-0.128805\pi$
$420$		−	73.2340i		−	3.57345i
$421$			11.3293i			0.552158i	0.961135	+	0.276079i	$0.0890351\pi$
$421$			11.3293i			0.552158i	−0.961135	+	0.276079i	$0.910965\pi$
$422$	46.9808			2.28699
$423$	−4.14359			−0.201468
$424$	−5.32051			−0.258387
$425$			3.50531i			0.170032i
$426$	−42.7846			−2.07292
$427$	−12.5885	−	29.8412i	−0.609198	−	1.44412i
$428$	57.7128			2.78965
$429$		−	19.6230i		−	0.947407i
$430$	−46.9808			−2.26561
$431$	−36.2487			−1.74604			−0.873019	−	0.487685i	$-0.837841\pi$
$431$	−36.2487			−1.74604			−0.873019	+	0.487685i	$0.837841\pi$
$432$	−9.85641			−0.474217
$433$		−	24.8809i		−	1.19570i	−0.801607	−	0.597851i	$-0.796021\pi$
$433$		−	24.8809i		−	1.19570i	0.801607	−	0.597851i	$-0.203979\pi$
$434$			30.1389i			1.44671i
$435$		−	8.29365i		−	0.397650i
$436$	41.0526			1.96606
$437$			27.9166i			1.33543i
$438$		−	7.82403i		−	0.373846i
$439$	−1.80385			−0.0860929			−0.0430465	−	0.999073i	$-0.513706\pi$
$439$	−1.80385			−0.0860929			−0.0430465	+	0.999073i	$0.513706\pi$
$440$	−17.1962			−0.819794
$441$	−45.5167			−2.16746
$442$	−12.5885			−0.598772
$443$	−24.2487			−1.15209			−0.576046	−	0.817418i	$-0.695405\pi$
$443$	−24.2487			−1.15209			−0.576046	+	0.817418i	$0.695405\pi$
$444$			30.9523i			1.46893i
$445$		−	20.4364i		−	0.968778i
$446$	2.66025			0.125967
$447$	17.6603			0.835301
$448$		−	44.2062i		−	2.08855i
$449$	0.803848			0.0379359			0.0189680	−	0.999820i	$-0.493962\pi$
$449$	0.803848			0.0379359			0.0189680	+	0.999820i	$0.493962\pi$
$450$			21.3756i			1.00766i
$451$			1.11114i			0.0523215i
$452$	7.73205			0.363685
$453$			33.9880i			1.59690i
$454$	28.6603			1.34509
$455$		−	21.5475i		−	1.01016i
$456$		−	53.6110i		−	2.51056i
$457$			21.8453i			1.02188i	0.859617	+	0.510939i	$0.170702\pi$
$457$			21.8453i			1.02188i	−0.859617	+	0.510939i	$0.829298\pi$
$458$			38.9482i			1.81993i
$459$			7.01062i			0.327228i
$460$		−	38.1348i		−	1.77805i
$461$	−8.53590			−0.397556			−0.198778	−	0.980045i	$-0.563697\pi$
$461$	−8.53590			−0.397556			−0.198778	+	0.980045i	$0.563697\pi$
$462$			64.9403i			3.02130i
$463$	6.92820			0.321981			0.160990	−	0.986956i	$-0.448531\pi$
$463$	6.92820			0.321981			0.160990	+	0.986956i	$0.448531\pi$
$464$			4.31872i			0.200491i
$465$			14.3650i			0.666162i
$466$	−30.1962			−1.39881
$467$			24.2394i			1.12167i	0.827929	+	0.560834i	$0.189519\pi$
$467$			24.2394i			1.12167i	−0.827929	+	0.560834i	$0.810481\pi$
$468$	−49.9808			−2.31036
$469$	42.3731			1.95661
$470$			3.84910i			0.177546i
$471$			24.8809i			1.14645i
$472$	−37.0526			−1.70548
$473$	27.1244			1.24718
$474$	−61.5167			−2.82555
$475$	−9.46410			−0.434243
$476$	27.1244			1.24324
$477$			5.72758i			0.262248i
$478$		−	31.7657i		−	1.45293i
$479$	−37.1769			−1.69866			−0.849328	−	0.527865i	$-0.822993\pi$
$479$	−37.1769			−1.69866			−0.849328	+	0.527865i	$0.822993\pi$
$480$		−	11.3293i		−	0.517111i
$481$			9.10706i			0.415246i
$482$		−	45.6151i		−	2.07771i
$483$	−66.8372			−3.04120
$484$	−19.6603			−0.893648
$485$	−6.00000			−0.272446
$486$		−	44.8477i		−	2.03433i
$487$	38.9808			1.76639			0.883193	−	0.469009i	$-0.155389\pi$
$487$	38.9808			1.76639			0.883193	+	0.469009i	$0.155389\pi$
$488$	12.5885	+	29.8412i	0.569853	+	1.35085i
$489$	41.3205			1.86858
$490$			42.2817i			1.91009i
$491$	−15.8038			−0.713218			−0.356609	−	0.934254i	$-0.616067\pi$
$491$	−15.8038			−0.713218			−0.356609	+	0.934254i	$0.616067\pi$
$492$	4.73205			0.213337
$493$	3.07180			0.138347
$494$		−	33.9880i		−	1.52919i
$495$			18.5118i			0.832046i
$496$		−	7.48024i		−	0.335873i
$497$	27.1244			1.21669
$498$			61.9046i			2.77401i
$499$			0.297729i			0.0133282i	0.999978	+	0.00666408i	$0.00212126\pi$
$499$			0.297729i			0.0133282i	−0.999978	+	0.00666408i	$0.997879\pi$
$500$	45.2487			2.02358
$501$	67.1769			3.00124
$502$	−35.5167			−1.58519
$503$	−18.5885			−0.828818			−0.414409	−	0.910091i	$-0.636012\pi$
$503$	−18.5885			−0.828818			−0.414409	+	0.910091i	$0.636012\pi$
$504$	76.7654			3.41940
$505$			13.5516i			0.603039i
$506$			33.8161i			1.50331i
$507$	10.9282			0.485339
$508$	−46.9808			−2.08443
$509$		−	32.7050i		−	1.44962i	−0.688948	−	0.724811i	$-0.741927\pi$
$509$		−	32.7050i		−	1.44962i	0.688948	−	0.724811i	$-0.258073\pi$
$510$	19.8564			0.879256
$511$			4.96023i			0.219428i
$512$			26.3359i			1.16389i
$513$	−18.9282			−0.835701
$514$			14.3650i			0.633614i
$515$	−15.8038			−0.696401
$516$		−	115.516i		−	5.08529i
$517$		−	2.22228i		−	0.0977356i
$518$		−	30.1389i		−	1.32423i
$519$		−	26.1640i		−	1.14847i
$520$			21.5475i			0.944922i
$521$			35.2710i			1.54525i	0.634861	+	0.772626i	$0.281057\pi$
$521$			35.2710i			1.54525i	−0.634861	+	0.772626i	$0.718943\pi$
$522$	18.7321			0.819880
$523$		−	44.2062i		−	1.93300i	−0.256662	−	0.966501i	$-0.582623\pi$
$523$		−	44.2062i		−	1.93300i	0.256662	−	0.966501i	$-0.417377\pi$
$524$	−43.5167			−1.90103
$525$		−	22.6587i		−	0.988905i
$526$			48.3530i			2.10829i
$527$	−5.32051			−0.231765
$528$		−	16.1177i		−	0.701432i
$529$	−11.8038			−0.513211
$530$	5.32051			0.231108
$531$			39.8875i			1.73097i
$532$			73.2340i			3.17510i
$533$	1.39230			0.0603074
$534$	77.1769			3.33977
$535$	−26.7846			−1.15800
$536$	−42.3731			−1.83024
$537$	−61.1769			−2.63998
$538$			6.07137i			0.261755i
$539$		−	24.4113i		−	1.05147i
$540$	25.8564			1.11268
$541$			27.1032i			1.16526i	0.812738	+	0.582629i	$0.197976\pi$
$541$			27.1032i			1.16526i	−0.812738	+	0.582629i	$0.802024\pi$
$542$		−	14.8346i		−	0.637202i
$543$		−	61.9046i		−	2.65658i
$544$	4.19615			0.179909
$545$	−19.0526			−0.816122
$546$	81.3731			3.48245
$547$		−	19.9207i		−	0.851748i	−0.904782	−	0.425874i	$-0.859967\pi$
$547$		−	19.9207i		−	0.851748i	0.904782	−	0.425874i	$-0.140033\pi$
$548$	−6.46410			−0.276133
$549$	32.1244	−	13.5516i	1.37103	−	0.578369i
$550$	−11.4641			−0.488831
$551$			8.29365i			0.353321i
$552$	66.8372			2.84478
$553$	39.0000			1.65845
$554$	46.9808			1.99602
$555$		−	14.3650i		−	0.609761i
$556$			15.4762i			0.656335i
$557$			29.6693i			1.25713i	0.777758	+	0.628564i	$0.216357\pi$
$557$			29.6693i			1.25713i	−0.777758	+	0.628564i	$0.783643\pi$
$558$	−32.4449			−1.37350
$559$		−	33.9880i		−	1.43754i
$560$		−	17.6984i		−	0.747895i
$561$	−11.4641			−0.484015
$562$	13.7128			0.578440
$563$	22.0526			0.929405			0.464702	−	0.885467i	$-0.346161\pi$
$563$	22.0526			0.929405			0.464702	+	0.885467i	$0.346161\pi$
$564$	−9.46410			−0.398511
$565$	−3.58846			−0.150968
$566$		−	34.9272i		−	1.46810i
$567$			10.2182i			0.429124i
$568$	−27.1244			−1.13811
$569$	−11.1962			−0.469367			−0.234684	−	0.972072i	$-0.575405\pi$
$569$	−11.1962			−0.469367			−0.234684	+	0.972072i	$0.575405\pi$
$570$			53.6110i			2.24552i
$571$	−40.3923			−1.69037			−0.845183	−	0.534478i	$-0.820508\pi$
$571$	−40.3923			−1.69037			−0.845183	+	0.534478i	$0.820508\pi$
$572$		−	26.8055i		−	1.12079i
$573$			63.6573i			2.65932i
$574$	−4.60770			−0.192321
$575$		−	11.7990i		−	0.492051i
$576$	47.5885			1.98285
$577$			15.1784i			0.631886i	0.948778	+	0.315943i	$0.102321\pi$
$577$			15.1784i			0.631886i	−0.948778	+	0.315943i	$0.897679\pi$
$578$		−	33.3465i		−	1.38703i
$579$			16.5873i			0.689345i
$580$		−	11.3293i		−	0.470425i
$581$		−	39.2460i		−	1.62820i
$582$		−	22.6587i		−	0.939232i
$583$	−3.07180			−0.127221
$584$		−	4.96023i		−	0.205256i
$585$	23.1962			0.959043
$586$			4.44455i			0.183603i
$587$			26.6336i			1.09929i	0.835400	+	0.549643i	$0.185236\pi$
$587$			26.6336i			1.09929i	−0.835400	+	0.549643i	$0.814764\pi$
$588$	−103.962			−4.28730
$589$		−	14.3650i		−	0.591900i
$590$	37.0526			1.52543
$591$	11.6603			0.479639
$592$			7.48024i			0.307436i
$593$		−	23.1283i		−	0.949765i	−0.880049	−	0.474883i	$-0.842491\pi$
$593$		−	23.1283i		−	0.949765i	0.880049	−	0.474883i	$-0.157509\pi$
$594$	−22.9282			−0.940756
$595$	−12.5885			−0.516076
$596$	24.1244			0.988172
$597$	−5.46410			−0.223631
$598$	42.3731			1.73276
$599$		−	31.5938i		−	1.29089i	−0.763807	−	0.645445i	$-0.776672\pi$
$599$		−	31.5938i		−	1.29089i	0.763807	−	0.645445i	$-0.223328\pi$
$600$			22.6587i			0.925036i
$601$	−3.00000			−0.122373			−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000			−0.122373			−0.0611863	+	0.998126i	$0.519488\pi$
$602$			112.480i			4.58434i
$603$			45.6151i			1.85759i
$604$			46.4285i			1.88915i
$605$	9.12436			0.370958
$606$	−51.1769			−2.07892
$607$	6.78461			0.275379			0.137689	−	0.990475i	$-0.456032\pi$
$607$	6.78461			0.275379			0.137689	+	0.990475i	$0.456032\pi$
$608$			11.3293i			0.459465i
$609$	−19.8564			−0.804622
$610$	−12.5885	−	29.8412i	−0.509692	−	1.20823i
$611$	−2.78461			−0.112653
$612$			29.1997i			1.18033i
$613$	24.9282			1.00684			0.503420	−	0.864042i	$-0.332075\pi$
$613$	24.9282			1.00684			0.503420	+	0.864042i	$0.332075\pi$
$614$	−2.66025			−0.107359
$615$	−2.19615			−0.0885574
$616$			41.1705i			1.65881i
$617$		−	11.7990i		−	0.475008i	−0.971387	−	0.237504i	$-0.923671\pi$
$617$		−	11.7990i		−	0.475008i	0.971387	−	0.237504i	$-0.0763293\pi$
$618$		−	59.6824i		−	2.40078i
$619$	7.60770			0.305779			0.152890	−	0.988243i	$-0.451142\pi$
$619$	7.60770			0.305779			0.152890	+	0.988243i	$0.451142\pi$
$620$			19.6230i			0.788078i
$621$		−	23.5979i		−	0.946952i
$622$	−74.5167			−2.98785
$623$	−48.9282			−1.96027
$624$	−20.1962			−0.808493
$625$	−11.0000			−0.440000
$626$	54.2487			2.16821
$627$		−	30.9523i		−	1.23612i
$628$			33.9880i			1.35627i
$629$	5.32051			0.212143
$630$	−76.7654			−3.05841
$631$		−	7.18251i		−	0.285931i	−0.989728	−	0.142966i	$-0.954336\pi$
$631$		−	7.18251i		−	0.285931i	0.989728	−	0.142966i	$-0.0456638\pi$
$632$	−39.0000			−1.55134
$633$		−	53.6110i		−	2.13085i
$634$		−	12.1427i		−	0.482250i
$635$	21.8038			0.865259
$636$			13.0820i			0.518735i
$637$	−30.5885			−1.21196
$638$			10.0463i			0.397737i
$639$			29.1997i			1.15512i
$640$		−	35.9126i		−	1.41957i
$641$		−	13.4258i		−	0.530286i	−0.964209	−	0.265143i	$-0.914581\pi$
$641$		−	13.4258i		−	0.530286i	0.964209	−	0.265143i	$-0.0854192\pi$
$642$		−	101.151i		−	3.99210i
$643$		−	25.6944i		−	1.01329i	−0.862156	−	0.506643i	$-0.830886\pi$
$643$		−	25.6944i		−	1.01329i	0.862156	−	0.506643i	$-0.169114\pi$
$644$	−91.3013			−3.59777
$645$			53.6110i			2.11093i
$646$	−19.8564			−0.781240
$647$			11.5012i			0.452160i	0.974109	+	0.226080i	$0.0725911\pi$
$647$			11.5012i			0.452160i	−0.974109	+	0.226080i	$0.927409\pi$
$648$		−	10.2182i		−	0.401409i
$649$	−21.3923			−0.839721
$650$			14.3650i			0.563442i
$651$	34.3923			1.34794
$652$	56.4449			2.21055
$653$		−	1.28303i		−	0.0502089i	−0.999685	−	0.0251045i	$-0.992008\pi$
$653$		−	1.28303i		−	0.0502089i	0.999685	−	0.0251045i	$-0.00799184\pi$
$654$		−	71.9509i		−	2.81350i
$655$	20.1962			0.789129
$656$	1.14359			0.0446498
$657$	−5.33975			−0.208323
$658$	9.21539			0.359253
$659$	28.7321			1.11924			0.559621	−	0.828749i	$-0.310947\pi$
$659$	28.7321			1.11924			0.559621	+	0.828749i	$0.310947\pi$
$660$			42.2817i			1.64581i
$661$		−	27.9166i		−	1.08583i	−0.839787	−	0.542916i	$-0.817320\pi$
$661$		−	27.9166i		−	1.08583i	0.839787	−	0.542916i	$-0.182680\pi$
$662$	73.3923			2.85247
$663$			14.3650i			0.557891i
$664$			39.2460i			1.52304i
$665$		−	33.9880i		−	1.31800i
$666$	32.4449			1.25721
$667$	−10.3397			−0.400357
$668$	91.7654			3.55051
$669$		−	3.03569i		−	0.117366i
$670$	42.3731			1.63701
$671$	7.26795	+	17.2288i	0.280576	+	0.665111i
$672$	−27.1244			−1.04634
$673$			19.6230i			0.756410i	0.925722	+	0.378205i	$0.123459\pi$
$673$			19.6230i			0.756410i	−0.925722	+	0.378205i	$0.876541\pi$
$674$	−34.3923			−1.32474
$675$	8.00000			0.307920
$676$	14.9282			0.574162
$677$			37.9629i			1.45903i	0.683963	+	0.729517i	$0.260255\pi$
$677$			37.9629i			1.45903i	−0.683963	+	0.729517i	$0.739745\pi$
$678$		−	13.5516i		−	0.520446i
$679$			14.3650i			0.551279i
$680$	12.5885			0.482745
$681$		−	32.7050i		−	1.25326i
$682$		−	17.4007i		−	0.666308i
$683$	−4.73205			−0.181067			−0.0905334	−	0.995893i	$-0.528857\pi$
$683$	−4.73205			−0.181067			−0.0905334	+	0.995893i	$0.528857\pi$
$684$	−78.8372			−3.01441
$685$	3.00000			0.114624
$686$	31.7321			1.21154
$687$	44.4449			1.69568
$688$		−	27.9166i		−	1.06431i
$689$			3.84910i			0.146639i
$690$	−66.8372			−2.54445
$691$	45.4641			1.72954			0.864768	−	0.502172i	$-0.167465\pi$
$691$	45.4641			1.72954			0.864768	+	0.502172i	$0.167465\pi$
$692$		−	35.7407i		−	1.35866i
$693$	44.3205			1.68360
$694$			61.0912i			2.31899i
$695$		−	7.18251i		−	0.272448i
$696$	19.8564			0.752655
$697$		−	0.813410i		−	0.0308101i
$698$	−72.1577			−2.73121
$699$			34.4576i			1.30331i
$700$		−	30.9523i		−	1.16989i
$701$		−	46.2566i		−	1.74709i	−0.486746	−	0.873544i	$-0.661816\pi$
$701$		−	46.2566i		−	1.74709i	0.486746	−	0.873544i	$-0.338184\pi$
$702$			28.7300i			1.08435i
$703$			14.3650i			0.541787i
$704$			25.5225i			0.961914i
$705$	4.39230			0.165424
$706$		−	62.2024i		−	2.34102i
$707$	32.4449			1.22021
$708$			91.1043i			3.42391i
$709$		−	5.25796i		−	0.197467i	−0.995114	−	0.0987335i	$-0.968521\pi$
$709$		−	5.25796i		−	0.197467i	0.995114	−	0.0987335i	$-0.0314791\pi$
$710$	27.1244			1.01796
$711$			41.9839i			1.57452i
$712$	48.9282			1.83366
$713$	17.9090			0.670696
$714$		−	47.5396i		−	1.77913i
$715$			12.4405i			0.465247i
$716$	−83.5692			−3.12313
$717$	−36.2487			−1.35373
$718$	−44.7321			−1.66939
$719$	−17.6603			−0.658616			−0.329308	−	0.944222i	$-0.606815\pi$
$719$	−17.6603			−0.658616			−0.329308	+	0.944222i	$0.606815\pi$
$720$	19.0526			0.710047
$721$			37.8371i			1.40913i
$722$		−	8.12176i		−	0.302260i
$723$	−52.0526			−1.93586
$724$		−	84.5633i		−	3.14277i
$725$		−	3.50531i		−	0.130184i
$726$			34.4576i			1.27884i
$727$	21.7128			0.805284			0.402642	−	0.915358i	$-0.368092\pi$
$727$	21.7128			0.805284			0.402642	+	0.915358i	$0.368092\pi$
$728$	51.5885			1.91200
$729$	−43.7846			−1.62165
$730$			4.96023i			0.183586i
$731$	−19.8564			−0.734416
$732$	73.3731	−	30.9523i	2.71195	−	1.14403i
$733$	15.3923			0.568528			0.284264	−	0.958746i	$-0.408251\pi$
$733$	15.3923			0.568528			0.284264	+	0.958746i	$0.408251\pi$
$734$		−	26.5078i		−	0.978419i
$735$	48.2487			1.77968
$736$	−14.1244			−0.520631
$737$	−24.4641			−0.901147
$738$		−	4.96023i		−	0.182589i
$739$			9.40479i			0.345961i	0.984925	+	0.172980i	$0.0553397\pi$
$739$			9.40479i			0.345961i	−0.984925	+	0.172980i	$0.944660\pi$
$740$		−	19.6230i		−	0.721355i
$741$	−38.7846			−1.42479
$742$		−	12.7382i		−	0.467634i
$743$			12.7843i			0.469009i	0.972115	+	0.234505i	$0.0753468\pi$
$743$			12.7843i			0.469009i	−0.972115	+	0.234505i	$0.924653\pi$
$744$	−34.3923			−1.26088
$745$	−11.1962			−0.410195
$746$	−27.1244			−0.993093
$747$	42.2487			1.54580
$748$	−15.6603			−0.572596
$749$			64.1269i			2.34315i
$750$		−	79.3053i		−	2.89582i
$751$	−36.3923			−1.32797			−0.663987	−	0.747744i	$-0.731137\pi$
$751$	−36.3923			−1.32797			−0.663987	+	0.747744i	$0.731137\pi$
$752$	−2.28719			−0.0834051
$753$			40.5290i			1.47696i
$754$	12.5885			0.458445
$755$		−	21.5475i		−	0.784195i
$756$		−	61.9046i		−	2.25145i
$757$	4.41154			0.160340			0.0801701	−	0.996781i	$-0.474454\pi$
$757$	4.41154			0.160340			0.0801701	+	0.996781i	$0.474454\pi$
$758$			25.3506i			0.920774i
$759$	38.5885			1.40067
$760$			33.9880i			1.23287i
$761$			10.9855i			0.398226i	0.979977	+	0.199113i	$0.0638060\pi$
$761$			10.9855i			0.398226i	−0.979977	+	0.199113i	$0.936194\pi$
$762$			82.3410i			2.98290i
$763$			45.6151i			1.65138i
$764$			86.9575i			3.14601i
$765$		−	13.5516i		−	0.489960i
$766$	45.4449			1.64199
$767$			26.8055i			0.967890i
$768$	77.3731			2.79196
$769$			9.10706i			0.328409i	0.986426	+	0.164204i	$0.0525057\pi$
$769$			9.10706i			0.328409i	−0.986426	+	0.164204i	$0.947494\pi$
$770$		−	41.1705i		−	1.48368i
$771$	16.3923			0.590354
$772$			22.6587i			0.815503i
$773$	−26.5359			−0.954430			−0.477215	−	0.878787i	$-0.658354\pi$
$773$	−26.5359			−0.954430			−0.477215	+	0.878787i	$0.658354\pi$
$774$	−121.086			−4.35234
$775$			6.07137i			0.218090i
$776$		−	14.3650i		−	0.515674i
$777$	−34.3923			−1.23382
$778$	17.6077			0.631266
$779$	2.19615			0.0786853
$780$	52.9808			1.89702
$781$	−15.6603			−0.560368
$782$		−	24.7551i		−	0.885241i
$783$		−	7.01062i		−	0.250539i
$784$	−25.1244			−0.897298
$785$		−	15.7739i		−	0.562994i
$786$			76.2697i			2.72045i
$787$			6.88478i			0.245416i	0.992443	+	0.122708i	$0.0391579\pi$
$787$			6.88478i			0.245416i	−0.992443	+	0.122708i	$0.960842\pi$
$788$	15.9282			0.567419
$789$	55.1769			1.96435
$790$	39.0000			1.38756
$791$			8.59138i			0.305474i
$792$	−44.3205			−1.57486
$793$	21.5885	−	9.10706i	0.766629	−	0.323401i
$794$	−17.9090			−0.635565
$795$		−	6.07137i		−	0.215329i
$796$	−7.46410			−0.264558
$797$	−4.51666			−0.159988			−0.0799942	−	0.996795i	$-0.525490\pi$
$797$	−4.51666			−0.159988			−0.0799942	+	0.996795i	$0.525490\pi$
$798$	128.354			4.54368
$799$			1.62682i			0.0575527i
$800$		−	4.78834i		−	0.169293i
$801$		−	52.6717i		−	1.86106i
$802$	26.0000			0.918092
$803$		−	2.86379i		−	0.101061i
$804$			104.186i			3.67437i
$805$	42.3731			1.49345
$806$	−21.8038			−0.768008
$807$	6.92820			0.243884
$808$	−32.4449			−1.14141
$809$	7.48334			0.263100			0.131550	−	0.991310i	$-0.458005\pi$
$809$	7.48334			0.263100			0.131550	+	0.991310i	$0.458005\pi$
$810$			10.2182i			0.359031i
$811$		−	53.3133i		−	1.87208i	−0.351891	−	0.936041i	$-0.614461\pi$
$811$		−	53.3133i		−	1.87208i	0.351891	−	0.936041i	$-0.385539\pi$
$812$	−27.1244			−0.951878
$813$	−16.9282			−0.593698
$814$			17.4007i			0.609894i
$815$	−26.1962			−0.917611
$816$			11.7990i			0.413046i
$817$		−	53.6110i		−	1.87561i
$818$	−19.8564			−0.694263
$819$		−	55.5355i		−	1.94057i
$820$	−3.00000			−0.104765
$821$		−	2.56606i		−	0.0895562i	−0.998997	−	0.0447781i	$-0.985742\pi$
$821$		−	2.56606i		−	0.0895562i	0.998997	−	0.0447781i	$-0.0142581\pi$
$822$			11.3293i			0.395156i
$823$			17.4007i			0.606551i	0.952903	+	0.303275i	$0.0980802\pi$
$823$			17.4007i			0.606551i	−0.952903	+	0.303275i	$0.901920\pi$
$824$		−	37.8371i		−	1.31812i
$825$			13.0820i			0.455456i
$826$		−	88.7101i		−	3.08662i
$827$	4.98076			0.173198			0.0865990	−	0.996243i	$-0.472400\pi$
$827$	4.98076			0.173198			0.0865990	+	0.996243i	$0.472400\pi$
$828$		−	98.2868i		−	3.41570i
$829$	−6.17691			−0.214533			−0.107267	−	0.994230i	$-0.534210\pi$
$829$	−6.17691			−0.214533			−0.107267	+	0.994230i	$0.534210\pi$
$830$		−	39.2460i		−	1.36225i
$831$		−	53.6110i		−	1.85974i
$832$	31.9808			1.10873
$833$			17.8703i			0.619170i
$834$	27.1244			0.939240
$835$	−42.5885			−1.47383
$836$		−	42.2817i		−	1.46234i
$837$			12.1427i			0.419715i
$838$	−38.5885			−1.33302
$839$	−18.0000			−0.621429			−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000			−0.621429			−0.310715	+	0.950503i	$0.600568\pi$
$840$	−81.3731			−2.80764
$841$	25.9282			0.894076
$842$	27.1244			0.934767
$843$		−	15.6481i		−	0.538948i
$844$		−	73.2340i		−	2.52082i
$845$	−6.92820			−0.238337
$846$			9.92047i			0.341073i
$847$		−	21.8453i		−	0.750612i
$848$			3.16152i			0.108567i
$849$	−39.8564			−1.36787
$850$	8.39230			0.287854
$851$	−17.9090			−0.613911
$852$			66.6930i			2.28486i
$853$	−31.5885			−1.08157			−0.540784	−	0.841161i	$-0.681872\pi$
$853$	−31.5885			−1.08157			−0.540784	+	0.841161i	$0.681872\pi$
$854$	−71.4449	+	30.1389i	−2.44479	+	1.03133i
$855$	36.5885			1.25130
$856$		−	64.1269i		−	2.19181i
$857$	1.14359			0.0390644			0.0195322	−	0.999809i	$-0.493782\pi$
$857$	1.14359			0.0390644			0.0195322	+	0.999809i	$0.493782\pi$
$858$	−46.9808			−1.60390
$859$	−32.9808			−1.12529			−0.562645	−	0.826699i	$-0.690216\pi$
$859$	−32.9808			−1.12529			−0.562645	+	0.826699i	$0.690216\pi$
$860$			73.2340i			2.49726i
$861$			5.25796i			0.179191i
$862$			86.7856i			2.95593i
$863$	23.0718			0.785373			0.392687	−	0.919672i	$-0.371546\pi$
$863$	23.0718			0.785373			0.392687	+	0.919672i	$0.371546\pi$
$864$		−	9.57668i		−	0.325805i
$865$			16.5873i			0.563985i
$866$	−59.5692			−2.02424
$867$	−38.0526			−1.29233
$868$	46.9808			1.59463
$869$	−22.5167			−0.763825
$870$	−19.8564			−0.673195
$871$			30.6546i			1.03869i
$872$		−	45.6151i		−	1.54472i
$873$	−15.4641			−0.523381
$874$	66.8372			2.26080
$875$			50.2776i			1.69969i
$876$	−12.1962			−0.412070
$877$			12.7382i			0.430139i	0.976599	+	0.215069i	$0.0689977\pi$
$877$			12.7382i			0.430139i	−0.976599	+	0.215069i	$0.931002\pi$
$878$			4.31872i			0.145750i
$879$	5.07180			0.171067
$880$			10.2182i			0.344455i
$881$	43.0526			1.45048			0.725239	−	0.688497i	$-0.241729\pi$
$881$	43.0526			1.45048			0.725239	+	0.688497i	$0.241729\pi$
$882$			108.975i			3.66937i
$883$			8.59138i			0.289123i	0.989496	+	0.144561i	$0.0461771\pi$
$883$			8.59138i			0.289123i	−0.989496	+	0.144561i	$0.953823\pi$
$884$			19.6230i			0.659992i
$885$		−	42.2817i		−	1.42128i
$886$			58.0555i			1.95041i
$887$		−	30.4827i		−	1.02351i	−0.859132	−	0.511754i	$-0.828996\pi$
$887$		−	30.4827i		−	1.02351i	0.859132	−	0.511754i	$-0.171004\pi$
$888$	34.3923			1.15413
$889$		−	52.2021i		−	1.75080i
$890$	−48.9282			−1.64008
$891$		−	5.89948i		−	0.197640i
$892$		−	4.14682i		−	0.138846i
$893$	−4.39230			−0.146983
$894$		−	42.2817i		−	1.41411i
$895$	38.7846			1.29643
$896$	−85.9808			−2.87242
$897$		−	48.3530i		−	1.61446i
$898$		−	1.92455i		−	0.0642230i
$899$	5.32051			0.177449
$900$	33.3205			1.11068
$901$	2.24871			0.0749154
$902$	2.66025			0.0885768
$903$	128.354			4.27135
$904$		−	8.59138i		−	0.285745i
$905$			39.2460i			1.30458i
$906$	81.3731			2.70344
$907$			15.1784i			0.503992i	0.967728	+	0.251996i	$0.0810869\pi$
$907$			15.1784i			0.503992i	−0.967728	+	0.251996i	$0.918913\pi$
$908$		−	44.6758i		−	1.48262i
$909$			34.9272i			1.15846i
$910$	−51.5885			−1.71014
$911$	54.8372			1.81684			0.908418	−	0.418063i	$-0.137291\pi$
$911$	54.8372			1.81684			0.908418	+	0.418063i	$0.137291\pi$
$912$	−31.8564			−1.05487
$913$			22.6587i			0.749893i
$914$	52.3013			1.72997
$915$	−34.0526	+	14.3650i	−1.12574	+	0.474893i
$916$	60.7128			2.00601
$917$		−	48.3530i		−	1.59676i
$918$	16.7846			0.553975
$919$	−26.0000			−0.857661			−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000			−0.857661			−0.428830	+	0.903385i	$0.641074\pi$
$920$	−42.3731			−1.39700
$921$			3.03569i			0.100029i
$922$			20.4364i			0.673037i
$923$			19.6230i			0.645898i
$924$	101.229			3.33021
$925$		−	6.07137i		−	0.199625i
$926$		−	16.5873i		−	0.545092i
$927$	−40.7321			−1.33782
$928$	−4.19615			−0.137745
$929$	−41.7846			−1.37091			−0.685454	−	0.728116i	$-0.740396\pi$
$929$	−41.7846			−1.37091			−0.685454	+	0.728116i	$0.740396\pi$
$930$	34.3923			1.12777
$931$	−48.2487			−1.58129
$932$			47.0700i			1.54183i
$933$			85.0329i			2.78385i
$934$	58.0333			1.89891
$935$	7.26795			0.237687
$936$			55.5355i			1.81524i
$937$	−35.1051			−1.14683			−0.573417	−	0.819264i	$-0.694383\pi$
$937$	−35.1051			−1.14683			−0.573417	+	0.819264i	$0.694383\pi$
$938$		−	101.448i		−	3.31241i
$939$		−	61.9046i		−	2.02018i
$940$	6.00000			0.195698
$941$		−	1.75265i		−	0.0571349i	−0.999592	−	0.0285674i	$-0.990905\pi$
$941$		−	1.75265i		−	0.0571349i	0.999592	−	0.0285674i	$-0.00909454\pi$
$942$	59.5692			1.94087
$943$			2.73796i			0.0891602i
$944$			22.0172i			0.716597i
$945$			28.7300i			0.934588i
$946$		−	64.9403i		−	2.11139i
$947$		−	22.0172i		−	0.715461i	−0.933825	−	0.357731i	$-0.883551\pi$
$947$		−	22.0172i		−	0.715461i	0.933825	−	0.357731i	$-0.116449\pi$
$948$			95.8926i			3.11445i
$949$	−3.58846			−0.116486
$950$			22.6587i			0.735144i
$951$	−13.8564			−0.449325
$952$		−	30.1389i		−	0.976808i
$953$			25.3506i			0.821185i	0.911819	+	0.410593i	$0.134678\pi$
$953$			25.3506i			0.821185i	−0.911819	+	0.410593i	$0.865322\pi$
$954$	13.7128			0.443969
$955$		−	40.3571i		−	1.30593i
$956$	−49.5167			−1.60148
$957$	11.4641			0.370582
$958$			89.0079i			2.87571i
$959$		−	7.18251i		−	0.231935i
$960$	−50.4449			−1.62810
$961$	21.7846			0.702729
$962$	21.8038			0.702984
$963$	−69.0333			−2.22457
$964$	−71.1051			−2.29014
$965$		−	10.5159i		−	0.338520i
$966$			160.020i			5.14854i
$967$	−31.3731			−1.00889			−0.504445	−	0.863444i	$-0.668303\pi$
$967$	−31.3731			−1.00889			−0.504445	+	0.863444i	$0.668303\pi$
$968$			21.8453i			0.702133i
$969$			22.6587i			0.727901i
$970$			14.3650i			0.461233i
$971$	16.1436			0.518073			0.259036	−	0.965868i	$-0.416595\pi$
$971$	16.1436			0.518073			0.259036	+	0.965868i	$0.416595\pi$
$972$	−69.9090			−2.24233
$973$	−17.1962			−0.551283
$974$		−	93.3266i		−	2.99038i
$975$	16.3923			0.524974
$976$	17.7321	−	7.48024i	0.567589	−	0.239437i
$977$	34.6410			1.10826			0.554132	−	0.832429i	$-0.313050\pi$
$977$	34.6410			1.10826			0.554132	+	0.832429i	$0.313050\pi$
$978$		−	98.9283i		−	3.16338i
$979$	28.2487			0.902833
$980$	65.9090			2.10538
$981$	−49.1051			−1.56781
$982$			37.8371i			1.20743i
$983$		−	20.5622i		−	0.655833i	−0.944707	−	0.327917i	$-0.893653\pi$
$983$		−	20.5622i		−	0.655833i	0.944707	−	0.327917i	$-0.106347\pi$
$984$		−	5.25796i		−	0.167618i
$985$	−7.39230			−0.235538
$986$		−	7.35440i		−	0.234212i
$987$		−	10.5159i		−	0.334726i
$988$	−52.9808			−1.68554
$989$	66.8372			2.12530
$990$	44.3205			1.40860
$991$	−1.60770			−0.0510701			−0.0255351	−	0.999674i	$-0.508129\pi$
$991$	−1.60770			−0.0510701			−0.0255351	+	0.999674i	$0.508129\pi$
$992$	7.26795			0.230758
$993$		−	83.7499i		−	2.65772i
$994$		−	64.9403i		−	2.05978i
$995$	3.46410			0.109819
$996$	96.4974			3.05764
$997$			13.5516i			0.429184i	0.976704	+	0.214592i	$0.0688422\pi$
$997$			13.5516i			0.429184i	−0.976704	+	0.214592i	$0.931158\pi$
$998$	0.712813			0.0225637
$999$		−	12.1427i		−	0.384179i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.2.b.a.60.1	✓	4
3.2	odd	2		549.2.c.c.487.4		4
4.3	odd	2		976.2.h.d.609.4		4
5.2	odd	4		1525.2.d.c.1524.8		8
5.3	odd	4		1525.2.d.c.1524.1		8
5.4	even	2		1525.2.c.b.426.4		4
61.11	odd	4		3721.2.a.f.1.4		4
61.50	odd	4		3721.2.a.f.1.1		4
61.60	even	2	inner	61.2.b.a.60.4	yes	4
183.182	odd	2		549.2.c.c.487.1		4
244.243	odd	2		976.2.h.d.609.3		4
305.182	odd	4		1525.2.d.c.1524.2		8
305.243	odd	4		1525.2.d.c.1524.7		8
305.304	even	2		1525.2.c.b.426.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.b.a.60.1	✓	4	1.1	even	1	trivial
61.2.b.a.60.4	yes	4	61.60	even	2	inner
549.2.c.c.487.1		4	183.182	odd	2
549.2.c.c.487.4		4	3.2	odd	2
976.2.h.d.609.3		4	244.243	odd	2
976.2.h.d.609.4		4	4.3	odd	2
1525.2.c.b.426.1		4	305.304	even	2
1525.2.c.b.426.4		4	5.4	even	2
1525.2.d.c.1524.1		8	5.3	odd	4
1525.2.d.c.1524.2		8	305.182	odd	4
1525.2.d.c.1524.7		8	305.243	odd	4
1525.2.d.c.1524.8		8	5.2	odd	4
3721.2.a.f.1.1		4	61.50	odd	4
3721.2.a.f.1.4		4	61.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	61.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.39417i q^{2} -2.73205 q^{3} -3.73205 q^{4} +1.73205 q^{5} +6.54099i q^{6} -4.14682i q^{7} +4.14682i q^{8} +4.46410 q^{9} -4.14682i q^{10} +2.39417i q^{11} +10.1962 q^{12} +3.00000 q^{13} -9.92820 q^{14} -4.73205 q^{15} +2.46410 q^{16} -1.75265i q^{17} -10.6878i q^{18} +4.73205 q^{19} -6.46410 q^{20} +11.3293i q^{21} +5.73205 q^{22} +5.89948i q^{23} -11.3293i q^{24} -2.00000 q^{25} -7.18251i q^{26} -4.00000 q^{27} +15.4762i q^{28} +1.75265i q^{29} +11.3293i q^{30} -3.03569i q^{31} +2.39417i q^{32} -6.54099i q^{33} -4.19615 q^{34} -7.18251i q^{35} -16.6603 q^{36} +3.03569i q^{37} -11.3293i q^{38} -8.19615 q^{39} +7.18251i q^{40} +0.464102 q^{41} +27.1244 q^{42} -11.3293i q^{43} -8.93516i q^{44} +7.73205 q^{45} +14.1244 q^{46} -0.928203 q^{47} -6.73205 q^{48} -10.1962 q^{49} +4.78834i q^{50} +4.78834i q^{51} -11.1962 q^{52} +1.28303i q^{53} +9.57668i q^{54} +4.14682i q^{55} +17.1962 q^{56} -12.9282 q^{57} +4.19615 q^{58} +8.93516i q^{59} +17.6603 q^{60} +(7.19615 - 3.03569i) q^{61} -7.26795 q^{62} -18.5118i q^{63} +10.6603 q^{64} +5.19615 q^{65} -15.6603 q^{66} +10.2182i q^{67} +6.54099i q^{68} -16.1177i q^{69} -17.1962 q^{70} +6.54099i q^{71} +18.5118i q^{72} -1.19615 q^{73} +7.26795 q^{74} +5.46410 q^{75} -17.6603 q^{76} +9.92820 q^{77} +19.6230i q^{78} +9.40479i q^{79} +4.26795 q^{80} -2.46410 q^{81} -1.11114i q^{82} +9.46410 q^{83} -42.2817i q^{84} -3.03569i q^{85} -27.1244 q^{86} -4.78834i q^{87} -9.92820 q^{88} -11.7990i q^{89} -18.5118i q^{90} -12.4405i q^{91} -22.0172i q^{92} +8.29365i q^{93} +2.22228i q^{94} +8.19615 q^{95} -6.54099i q^{96} -3.46410 q^{97} +24.4113i q^{98} +10.6878i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{4} + 4 q^{9} + 20 q^{12} + 12 q^{13} - 12 q^{14} - 12 q^{15} - 4 q^{16} + 12 q^{19} - 12 q^{20} + 16 q^{22} - 8 q^{25} - 16 q^{27} + 4 q^{34} - 32 q^{36} - 12 q^{39} - 12 q^{41} + 60 q^{42}+ \cdots + 12 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^4 + 4 * q^9 + 20 * q^12 + 12 * q^13 - 12 * q^14 - 12 * q^15 - 4 * q^16 + 12 * q^19 - 12 * q^20 + 16 * q^22 - 8 * q^25 - 16 * q^27 + 4 * q^34 - 32 * q^36 - 12 * q^39 - 12 * q^41 + 60 * q^42 + 24 * q^45 + 8 * q^46 + 24 * q^47 - 20 * q^48 - 20 * q^49 - 24 * q^52 + 48 * q^56 - 24 * q^57 - 4 * q^58 + 36 * q^60 + 8 * q^61 - 36 * q^62 + 8 * q^64 - 28 * q^66 - 48 * q^70 + 16 * q^73 + 36 * q^74 + 8 * q^75 - 36 * q^76 + 12 * q^77 + 24 * q^80 + 4 * q^81 + 24 * q^83 - 60 * q^86 - 12 * q^88 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more