LMFDB - Embedded newform 4620.2.m.b.1121.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1121,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4620.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.m (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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.22
Character	$\chi$	$=$	4620.1121
Dual form			4620.2.m.b.1121.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.503567 + 1.65723i) q^{3} -1.00000i q^{5} +1.00000i q^{7} +(-2.49284 - 1.66906i) q^{9} +O(q^{10})$	\(q+(-0.503567 + 1.65723i) q^{3} -1.00000i q^{5} +1.00000i q^{7} +(-2.49284 - 1.66906i) q^{9} +(1.97659 + 2.66328i) q^{11} +3.37100i q^{13} +(1.65723 + 0.503567i) q^{15} -6.52370 q^{17} +3.37205i q^{19} +(-1.65723 - 0.503567i) q^{21} +9.40609i q^{23} -1.00000 q^{25} +(4.02133 - 3.29073i) q^{27} +5.14568 q^{29} +4.95504 q^{31} +(-5.40903 + 1.93453i) q^{33} +1.00000 q^{35} -5.91282 q^{37} +(-5.58653 - 1.69752i) q^{39} +5.62026 q^{41} -8.43147i q^{43} +(-1.66906 + 2.49284i) q^{45} -6.74315i q^{47} -1.00000 q^{49} +(3.28512 - 10.8113i) q^{51} +1.58760i q^{53} +(2.66328 - 1.97659i) q^{55} +(-5.58827 - 1.69806i) q^{57} -10.9526i q^{59} +7.32704i q^{61} +(1.66906 - 2.49284i) q^{63} +3.37100 q^{65} +0.0203833 q^{67} +(-15.5881 - 4.73660i) q^{69} -7.58489i q^{71} +11.2643i q^{73} +(0.503567 - 1.65723i) q^{75} +(-2.66328 + 1.97659i) q^{77} +6.40822i q^{79} +(3.42850 + 8.32138i) q^{81} -2.00211 q^{83} +6.52370i q^{85} +(-2.59120 + 8.52759i) q^{87} +11.2528i q^{89} -3.37100 q^{91} +(-2.49520 + 8.21165i) q^{93} +3.37205 q^{95} -7.17338 q^{97} +(-0.482156 - 9.93819i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) $ 48 * q - 4 * q^3 + 6 * q^9	$ 48 q - 4 q^{3} + 6 q^{9} + 6 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{21} - 48 q^{25} - 16 q^{27} - 36 q^{29} - 16 q^{31} - 4 q^{33} + 48 q^{35} - 8 q^{37} + 18 q^{39} - 48 q^{49} + 30 q^{51} + 4 q^{55} - 16 q^{57} + 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} - 4 q^{77} - 22 q^{81} - 20 q^{83} - 44 q^{87} - 12 q^{91} - 28 q^{93} - 8 q^{95} - 56 q^{97} - 12 q^{99}+O(q^{100}) $ 48 * q - 4 * q^3 + 6 * q^9 + 6 * q^11 + 2 * q^15 - 4 * q^17 - 2 * q^21 - 48 * q^25 - 16 * q^27 - 36 * q^29 - 16 * q^31 - 4 * q^33 + 48 * q^35 - 8 * q^37 + 18 * q^39 - 48 * q^49 + 30 * q^51 + 4 * q^55 - 16 * q^57 + 12 * q^65 + 24 * q^67 + 4 * q^69 + 4 * q^75 - 4 * q^77 - 22 * q^81 - 20 * q^83 - 44 * q^87 - 12 * q^91 - 28 * q^93 - 8 * q^95 - 56 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times$.

$n$	$661$	$1541$	$2311$	$2521$	$3697$
$\chi(n)$	$1$	$-1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.503567	+	1.65723i	−0.290735	+	0.956804i
$3$	−0.503567	+	1.65723i	−0.290735	+	0.956804i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−2.49284	−	1.66906i	−0.830947	−	0.556352i
$10$		0			0
$11$	1.97659	+	2.66328i	0.595965	+	0.803010i
$11$	1.97659	+	2.66328i	0.595965	+	0.803010i
$12$		0			0
$13$			3.37100i			0.934946i	0.884007	+	0.467473i	$0.154836\pi$
$13$			3.37100i			0.934946i	−0.884007	+	0.467473i	$0.845164\pi$
$14$		0			0
$15$	1.65723	+	0.503567i	0.427896	+	0.130021i
$16$		0			0
$17$	−6.52370			−1.58223			−0.791115	−	0.611667i	$-0.790499\pi$
$17$	−6.52370			−1.58223			−0.791115	+	0.611667i	$0.790499\pi$
$18$		0			0
$19$			3.37205i			0.773602i	0.922163	+	0.386801i	$0.126420\pi$
$19$			3.37205i			0.773602i	−0.922163	+	0.386801i	$0.873580\pi$
$20$		0			0
$21$	−1.65723	−	0.503567i	−0.361638	−	0.109887i
$22$		0			0
$23$			9.40609i			1.96131i	0.195754	+	0.980653i	$0.437285\pi$
$23$			9.40609i			1.96131i	−0.195754	+	0.980653i	$0.562715\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	4.02133	−	3.29073i	0.773905	−	0.633302i
$28$		0			0
$29$	5.14568			0.955529			0.477765	−	0.878488i	$-0.341447\pi$
$29$	5.14568			0.955529			0.477765	+	0.878488i	$0.341447\pi$
$30$		0			0
$31$	4.95504			0.889951			0.444976	−	0.895543i	$-0.353212\pi$
$31$	4.95504			0.889951			0.444976	+	0.895543i	$0.353212\pi$
$32$		0			0
$33$	−5.40903	+	1.93453i	−0.941591	+	0.336758i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	−5.91282			−0.972062			−0.486031	−	0.873942i	$-0.661556\pi$
$37$	−5.91282			−0.972062			−0.486031	+	0.873942i	$0.661556\pi$
$38$		0			0
$39$	−5.58653	−	1.69752i	−0.894560	−	0.271821i
$40$		0			0
$41$	5.62026			0.877738			0.438869	−	0.898551i	$-0.355379\pi$
$41$	5.62026			0.877738			0.438869	+	0.898551i	$0.355379\pi$
$42$		0			0
$43$		−	8.43147i		−	1.28579i	−0.765955	−	0.642894i	$-0.777734\pi$
$43$		−	8.43147i		−	1.28579i	0.765955	−	0.642894i	$-0.222266\pi$
$44$		0			0
$45$	−1.66906	+	2.49284i	−0.248808	+	0.371611i
$46$		0			0
$47$		−	6.74315i		−	0.983590i	−0.870711	−	0.491795i	$-0.836341\pi$
$47$		−	6.74315i		−	0.983590i	0.870711	−	0.491795i	$-0.163659\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	3.28512	−	10.8113i	0.460009	−	1.51388i
$52$		0			0
$53$			1.58760i			0.218074i	0.994038	+	0.109037i	$0.0347766\pi$
$53$			1.58760i			0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$54$		0			0
$55$	2.66328	−	1.97659i	0.359117	−	0.266524i
$56$		0			0
$57$	−5.58827	−	1.69806i	−0.740185	−	0.224913i
$58$		0			0
$59$		−	10.9526i		−	1.42591i	−0.701212	−	0.712953i	$-0.747357\pi$
$59$		−	10.9526i		−	1.42591i	0.701212	−	0.712953i	$-0.252643\pi$
$60$		0			0
$61$			7.32704i			0.938131i	0.883163	+	0.469066i	$0.155409\pi$
$61$			7.32704i			0.938131i	−0.883163	+	0.469066i	$0.844591\pi$
$62$		0			0
$63$	1.66906	−	2.49284i	0.210281	−	0.314068i
$64$		0			0
$65$	3.37100			0.418121
$66$		0			0
$67$	0.0203833			0.00249022			0.00124511	−	0.999999i	$-0.499604\pi$
$67$	0.0203833			0.00249022			0.00124511	+	0.999999i	$0.499604\pi$
$68$		0			0
$69$	−15.5881	−	4.73660i	−1.87658	−	0.570220i
$70$		0			0
$71$		−	7.58489i		−	0.900161i	−0.892988	−	0.450081i	$-0.851395\pi$
$71$		−	7.58489i		−	0.900161i	0.892988	−	0.450081i	$-0.148605\pi$
$72$		0			0
$73$			11.2643i			1.31838i	0.751975	+	0.659192i	$0.229101\pi$
$73$			11.2643i			1.31838i	−0.751975	+	0.659192i	$0.770899\pi$
$74$		0			0
$75$	0.503567	−	1.65723i	0.0581469	−	0.191361i
$76$		0			0
$77$	−2.66328	+	1.97659i	−0.303509	+	0.225254i
$78$		0			0
$79$			6.40822i			0.720981i	0.932763	+	0.360490i	$0.117391\pi$
$79$			6.40822i			0.720981i	−0.932763	+	0.360490i	$0.882609\pi$
$80$		0			0
$81$	3.42850	+	8.32138i	0.380945	+	0.924598i
$82$		0			0
$83$	−2.00211			−0.219761			−0.109880	−	0.993945i	$-0.535047\pi$
$83$	−2.00211			−0.219761			−0.109880	+	0.993945i	$0.535047\pi$
$84$		0			0
$85$			6.52370i			0.707595i
$86$		0			0
$87$	−2.59120	+	8.52759i	−0.277806	+	0.914254i
$88$		0			0
$89$			11.2528i			1.19280i	0.802689	+	0.596398i	$0.203402\pi$
$89$			11.2528i			1.19280i	−0.802689	+	0.596398i	$0.796598\pi$
$90$		0			0
$91$	−3.37100			−0.353377
$92$		0			0
$93$	−2.49520	+	8.21165i	−0.258740	+	0.851509i
$94$		0			0
$95$	3.37205			0.345965
$96$		0			0
$97$	−7.17338			−0.728347			−0.364173	−	0.931331i	$-0.618648\pi$
$97$	−7.17338			−0.728347			−0.364173	+	0.931331i	$0.618648\pi$
$98$		0			0
$99$	−0.482156	−	9.93819i	−0.0484585	−	0.998825i
$100$		0			0
$101$	−17.0955			−1.70107			−0.850533	−	0.525921i	$-0.823721\pi$
$101$	−17.0955			−1.70107			−0.850533	+	0.525921i	$0.823721\pi$
$102$		0			0
$103$	−3.04232			−0.299769			−0.149884	−	0.988704i	$-0.547890\pi$
$103$	−3.04232			−0.299769			−0.149884	+	0.988704i	$0.547890\pi$
$104$		0			0
$105$	−0.503567	+	1.65723i	−0.0491431	+	0.161729i
$106$		0			0
$107$	−17.5060			−1.69237			−0.846183	−	0.532893i	$-0.821105\pi$
$107$	−17.5060			−1.69237			−0.846183	+	0.532893i	$0.821105\pi$
$108$		0			0
$109$		−	2.42580i		−	0.232349i	−0.993229	−	0.116175i	$-0.962937\pi$
$109$		−	2.42580i		−	0.232349i	0.993229	−	0.116175i	$-0.0370633\pi$
$110$		0			0
$111$	2.97750	−	9.79892i	0.282612	−	0.930072i
$112$		0			0
$113$		−	20.7421i		−	1.95126i	−0.219428	−	0.975629i	$-0.570419\pi$
$113$		−	20.7421i		−	1.95126i	0.219428	−	0.975629i	$-0.429581\pi$
$114$		0			0
$115$	9.40609			0.877123
$116$		0			0
$117$	5.62638	−	8.40336i	0.520159	−	0.776891i
$118$		0			0
$119$		−	6.52370i		−	0.598027i
$120$		0			0
$121$	−3.18617	+	10.5285i	−0.289652	+	0.957132i
$122$		0			0
$123$	−2.83018	+	9.31408i	−0.255189	+	0.839823i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			5.61305i			0.498078i	0.968493	+	0.249039i	$0.0801147\pi$
$127$			5.61305i			0.498078i	−0.968493	+	0.249039i	$0.919885\pi$
$128$		0			0
$129$	13.9729	+	4.24581i	1.23025	+	0.373823i
$130$		0			0
$131$	5.81673			0.508210			0.254105	−	0.967177i	$-0.418219\pi$
$131$	5.81673			0.508210			0.254105	+	0.967177i	$0.418219\pi$
$132$		0			0
$133$	−3.37205			−0.292394
$134$		0			0
$135$	−3.29073	−	4.02133i	−0.283221	−	0.346101i
$136$		0			0
$137$		−	22.0809i		−	1.88650i	−0.332082	−	0.943250i	$-0.607751\pi$
$137$		−	22.0809i		−	1.88650i	0.332082	−	0.943250i	$-0.392249\pi$
$138$		0			0
$139$			19.0637i			1.61696i	0.588524	+	0.808480i	$0.299709\pi$
$139$			19.0637i			1.61696i	−0.588524	+	0.808480i	$0.700291\pi$
$140$		0			0
$141$	11.1750	+	3.39563i	0.941102	+	0.285964i
$142$		0			0
$143$	−8.97793	+	6.66309i	−0.750772	+	0.557195i
$144$		0			0
$145$		−	5.14568i		−	0.427326i
$146$		0			0
$147$	0.503567	−	1.65723i	0.0415335	−	0.136686i
$148$		0			0
$149$	−5.55956			−0.455457			−0.227728	−	0.973725i	$-0.573130\pi$
$149$	−5.55956			−0.455457			−0.227728	+	0.973725i	$0.573130\pi$
$150$		0			0
$151$			13.2743i			1.08025i	0.841585	+	0.540125i	$0.181623\pi$
$151$			13.2743i			1.08025i	−0.841585	+	0.540125i	$0.818377\pi$
$152$		0			0
$153$	16.2625	+	10.8884i	1.31475	+	0.880277i
$154$		0			0
$155$		−	4.95504i		−	0.397998i
$156$		0			0
$157$	−8.46912			−0.675909			−0.337954	−	0.941162i	$-0.609735\pi$
$157$	−8.46912			−0.675909			−0.337954	+	0.941162i	$0.609735\pi$
$158$		0			0
$159$	−2.63102	−	0.799463i	−0.208654	−	0.0634015i
$160$		0			0
$161$	−9.40609			−0.741304
$162$		0			0
$163$	−9.45381			−0.740480			−0.370240	−	0.928936i	$-0.620724\pi$
$163$	−9.45381			−0.740480			−0.370240	+	0.928936i	$0.620724\pi$
$164$		0			0
$165$	1.93453	+	5.40903i	0.150603	+	0.421092i
$166$		0			0
$167$	−24.5982			−1.90346			−0.951732	−	0.306932i	$-0.900698\pi$
$167$	−24.5982			−1.90346			−0.951732	+	0.306932i	$0.900698\pi$
$168$		0			0
$169$	1.63638			0.125875
$170$		0			0
$171$	5.62814	−	8.40599i	0.430395	−	0.642822i
$172$		0			0
$173$	−10.5920			−0.805298			−0.402649	−	0.915354i	$-0.631911\pi$
$173$	−10.5920			−0.805298			−0.402649	+	0.915354i	$0.631911\pi$
$174$		0			0
$175$		−	1.00000i		−	0.0755929i
$176$		0			0
$177$	18.1510	+	5.51537i	1.36431	+	0.414561i
$178$		0			0
$179$			13.8504i			1.03523i	0.855614	+	0.517614i	$0.173180\pi$
$179$			13.8504i			1.03523i	−0.855614	+	0.517614i	$0.826820\pi$
$180$		0			0
$181$	−13.8936			−1.03270			−0.516351	−	0.856377i	$-0.672710\pi$
$181$	−13.8936			−1.03270			−0.516351	+	0.856377i	$0.672710\pi$
$182$		0			0
$183$	−12.1426	−	3.68966i	−0.897608	−	0.272747i
$184$		0			0
$185$			5.91282i			0.434719i
$186$		0			0
$187$	−12.8947	−	17.3745i	−0.942954	−	1.27055i
$188$		0			0
$189$	3.29073	+	4.02133i	0.239366	+	0.292509i
$190$		0			0
$191$		−	2.24354i		−	0.162337i	−0.996700	−	0.0811685i	$-0.974135\pi$
$191$		−	2.24354i		−	0.162337i	0.996700	−	0.0811685i	$-0.0258652\pi$
$192$		0			0
$193$		−	4.77918i		−	0.344013i	−0.985096	−	0.172007i	$-0.944975\pi$
$193$		−	4.77918i		−	0.344013i	0.985096	−	0.172007i	$-0.0550250\pi$
$194$		0			0
$195$	−1.69752	+	5.58653i	−0.121562	+	0.400060i
$196$		0			0
$197$	15.6285			1.11348			0.556741	−	0.830686i	$-0.312052\pi$
$197$	15.6285			1.11348			0.556741	+	0.830686i	$0.312052\pi$
$198$		0			0
$199$	6.24796			0.442906			0.221453	−	0.975171i	$-0.428920\pi$
$199$	6.24796			0.442906			0.221453	+	0.975171i	$0.428920\pi$
$200$		0			0
$201$	−0.0102644	+	0.0337799i	−0.000723993	+	0.00238265i
$202$		0			0
$203$			5.14568i			0.361156i
$204$		0			0
$205$		−	5.62026i		−	0.392536i
$206$		0			0
$207$	15.6993	−	23.4479i	1.09118	−	1.62974i
$208$		0			0
$209$	−8.98073	+	6.66517i	−0.621210	+	0.461040i
$210$		0			0
$211$		−	18.9412i		−	1.30397i	−0.758232	−	0.651984i	$-0.773937\pi$
$211$		−	18.9412i		−	1.30397i	0.758232	−	0.651984i	$-0.226063\pi$
$212$		0			0
$213$	12.5699	+	3.81950i	0.861277	+	0.261708i
$214$		0			0
$215$	−8.43147			−0.575022
$216$		0			0
$217$			4.95504i			0.336370i
$218$		0			0
$219$	−18.6675	−	5.67232i	−1.26143	−	0.383300i
$220$		0			0
$221$		−	21.9914i		−	1.47930i
$222$		0			0
$223$	1.38657			0.0928513			0.0464257	−	0.998922i	$-0.485217\pi$
$223$	1.38657			0.0928513			0.0464257	+	0.998922i	$0.485217\pi$
$224$		0			0
$225$	2.49284	+	1.66906i	0.166189	+	0.111270i
$226$		0			0
$227$	−11.2701			−0.748024			−0.374012	−	0.927424i	$-0.622018\pi$
$227$	−11.2701			−0.748024			−0.374012	+	0.927424i	$0.622018\pi$
$228$		0			0
$229$	−1.33605			−0.0882888			−0.0441444	−	0.999025i	$-0.514056\pi$
$229$	−1.33605			−0.0882888			−0.0441444	+	0.999025i	$0.514056\pi$
$230$		0			0
$231$	−1.93453	−	5.40903i	−0.127283	−	0.355888i
$232$		0			0
$233$	2.38655			0.156348			0.0781740	−	0.996940i	$-0.475091\pi$
$233$	2.38655			0.156348			0.0781740	+	0.996940i	$0.475091\pi$
$234$		0			0
$235$	−6.74315			−0.439875
$236$		0			0
$237$	−10.6199	−	3.22697i	−0.689837	−	0.209614i
$238$		0			0
$239$	6.45696			0.417666			0.208833	−	0.977951i	$-0.433034\pi$
$239$	6.45696			0.417666			0.208833	+	0.977951i	$0.433034\pi$
$240$		0			0
$241$		−	0.373463i		−	0.0240569i	−0.999928	−	0.0120284i	$-0.996171\pi$
$241$		−	0.373463i		−	0.0240569i	0.999928	−	0.0120284i	$-0.00382886\pi$
$242$		0			0
$243$	−15.5169	+	1.49145i	−0.995412	+	0.0956766i
$244$		0			0
$245$			1.00000i			0.0638877i
$246$		0			0
$247$	−11.3672			−0.723276
$248$		0			0
$249$	1.00820	−	3.31797i	0.0638920	−	0.210268i
$250$		0			0
$251$			16.0908i			1.01564i	0.861463	+	0.507820i	$0.169548\pi$
$251$			16.0908i			1.01564i	−0.861463	+	0.507820i	$0.830452\pi$
$252$		0			0
$253$	−25.0511	+	18.5920i	−1.57495	+	1.16887i
$254$		0			0
$255$	−10.8113	−	3.28512i	−0.677029	−	0.205722i
$256$		0			0
$257$		−	9.06590i		−	0.565515i	−0.959191	−	0.282758i	$-0.908751\pi$
$257$		−	9.06590i		−	0.565515i	0.959191	−	0.282758i	$-0.0912492\pi$
$258$		0			0
$259$		−	5.91282i		−	0.367405i
$260$		0			0
$261$	−12.8274	−	8.58843i	−0.793994	−	0.531611i
$262$		0			0
$263$	13.0748			0.806225			0.403112	−	0.915151i	$-0.367928\pi$
$263$	13.0748			0.806225			0.403112	+	0.915151i	$0.367928\pi$
$264$		0			0
$265$	1.58760			0.0975254
$266$		0			0
$267$	−18.6485	−	5.66655i	−1.14127	−	0.346787i
$268$		0			0
$269$		−	6.23199i		−	0.379971i	−0.981787	−	0.189986i	$-0.939156\pi$
$269$		−	6.23199i		−	0.379971i	0.981787	−	0.189986i	$-0.0608441\pi$
$270$		0			0
$271$		−	20.4522i		−	1.24238i	−0.783660	−	0.621190i	$-0.786649\pi$
$271$		−	20.4522i		−	1.24238i	0.783660	−	0.621190i	$-0.213351\pi$
$272$		0			0
$273$	1.69752	−	5.58653i	0.102739	−	0.338112i
$274$		0			0
$275$	−1.97659	−	2.66328i	−0.119193	−	0.160602i
$276$		0			0
$277$			10.7848i			0.647996i	0.946058	+	0.323998i	$0.105027\pi$
$277$			10.7848i			0.647996i	−0.946058	+	0.323998i	$0.894973\pi$
$278$		0			0
$279$	−12.3521	−	8.27024i	−0.739502	−	0.495126i
$280$		0			0
$281$	−0.607065			−0.0362145			−0.0181072	−	0.999836i	$-0.505764\pi$
$281$	−0.607065			−0.0362145			−0.0181072	+	0.999836i	$0.505764\pi$
$282$		0			0
$283$			28.8728i			1.71631i	0.513390	+	0.858155i	$0.328389\pi$
$283$			28.8728i			1.71631i	−0.513390	+	0.858155i	$0.671611\pi$
$284$		0			0
$285$	−1.69806	+	5.58827i	−0.100584	+	0.331021i
$286$		0			0
$287$			5.62026i			0.331754i
$288$		0			0
$289$	25.5587			1.50345
$290$		0			0
$291$	3.61228	−	11.8880i	0.211756	−	0.696885i
$292$		0			0
$293$	−7.10835			−0.415274			−0.207637	−	0.978206i	$-0.566577\pi$
$293$	−7.10835			−0.415274			−0.207637	+	0.978206i	$0.566577\pi$
$294$		0			0
$295$	−10.9526			−0.637685
$296$		0			0
$297$	16.7127	+	4.20550i	0.969768	+	0.244028i
$298$		0			0
$299$	−31.7079			−1.83372
$300$		0			0
$301$	8.43147			0.485982
$302$		0			0
$303$	8.60874	−	28.3312i	0.494559	−	1.62759i
$304$		0			0
$305$	7.32704			0.419545
$306$		0			0
$307$		−	8.17912i		−	0.466807i	−0.972380	−	0.233404i	$-0.925014\pi$
$307$		−	8.17912i		−	0.466807i	0.972380	−	0.233404i	$-0.0749863\pi$
$308$		0			0
$309$	1.53201	−	5.04183i	0.0871531	−	0.286820i
$310$		0			0
$311$		−	12.8864i		−	0.730718i	−0.930867	−	0.365359i	$-0.880946\pi$
$311$		−	12.8864i		−	0.730718i	0.930867	−	0.365359i	$-0.119054\pi$
$312$		0			0
$313$	−6.45708			−0.364976			−0.182488	−	0.983208i	$-0.558415\pi$
$313$	−6.45708			−0.364976			−0.182488	+	0.983208i	$0.558415\pi$
$314$		0			0
$315$	−2.49284	−	1.66906i	−0.140456	−	0.0940407i
$316$		0			0
$317$		−	6.03359i		−	0.338880i	−0.985541	−	0.169440i	$-0.945804\pi$
$317$		−	6.03359i		−	0.338880i	0.985541	−	0.169440i	$-0.0541959\pi$
$318$		0			0
$319$	10.1709	+	13.7044i	0.569462	+	0.767300i
$320$		0			0
$321$	8.81543	−	29.0115i	0.492029	−	1.61926i
$322$		0			0
$323$		−	21.9983i		−	1.22402i
$324$		0			0
$325$		−	3.37100i		−	0.186989i
$326$		0			0
$327$	4.02011	+	1.22155i	0.222313	+	0.0675521i
$328$		0			0
$329$	6.74315			0.371762
$330$		0			0
$331$	1.91680			0.105357			0.0526785	−	0.998612i	$-0.483224\pi$
$331$	1.91680			0.105357			0.0526785	+	0.998612i	$0.483224\pi$
$332$		0			0
$333$	14.7397	+	9.86883i	0.807731	+	0.540808i
$334$		0			0
$335$		−	0.0203833i		−	0.00111366i
$336$		0			0
$337$			16.1461i			0.879535i	0.898112	+	0.439767i	$0.144939\pi$
$337$			16.1461i			0.879535i	−0.898112	+	0.439767i	$0.855061\pi$
$338$		0			0
$339$	34.3746	+	10.4451i	1.86697	+	0.567298i
$340$		0			0
$341$	9.79409	+	13.1967i	0.530380	+	0.714640i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−4.73660	+	15.5881i	−0.255010	+	0.839234i
$346$		0			0
$347$	27.9304			1.49939			0.749693	−	0.661786i	$-0.230201\pi$
$347$	27.9304			1.49939			0.749693	+	0.661786i	$0.230201\pi$
$348$		0			0
$349$			0.853594i			0.0456918i	0.999739	+	0.0228459i	$0.00727272\pi$
$349$			0.853594i			0.0456918i	−0.999739	+	0.0228459i	$0.992727\pi$
$350$		0			0
$351$	11.0931	+	13.5559i	0.592103	+	0.723560i
$352$		0			0
$353$		−	21.5400i		−	1.14646i	−0.819394	−	0.573230i	$-0.805690\pi$
$353$		−	21.5400i		−	1.14646i	0.819394	−	0.573230i	$-0.194310\pi$
$354$		0			0
$355$	−7.58489			−0.402564
$356$		0			0
$357$	10.8113	+	3.28512i	0.572194	+	0.173867i
$358$		0			0
$359$	−25.4227			−1.34176			−0.670879	−	0.741567i	$-0.734083\pi$
$359$	−25.4227			−1.34176			−0.670879	+	0.741567i	$0.734083\pi$
$360$		0			0
$361$	7.62926			0.401540
$362$		0			0
$363$	−15.8436	−	10.5820i	−0.831576	−	0.555411i
$364$		0			0
$365$	11.2643			0.589599
$366$		0			0
$367$	15.6090			0.814781			0.407390	−	0.913254i	$-0.366439\pi$
$367$	15.6090			0.814781			0.407390	+	0.913254i	$0.366439\pi$
$368$		0			0
$369$	−14.0104	−	9.38053i	−0.729353	−	0.488331i
$370$		0			0
$371$	−1.58760			−0.0824240
$372$		0			0
$373$			24.7228i			1.28010i	0.768335	+	0.640048i	$0.221086\pi$
$373$			24.7228i			1.28010i	−0.768335	+	0.640048i	$0.778914\pi$
$374$		0			0
$375$	−1.65723	−	0.503567i	−0.0855791	−	0.0260041i
$376$		0			0
$377$			17.3461i			0.893369i
$378$		0			0
$379$	3.38791			0.174025			0.0870127	−	0.996207i	$-0.472268\pi$
$379$	3.38791			0.174025			0.0870127	+	0.996207i	$0.472268\pi$
$380$		0			0
$381$	−9.30213	−	2.82655i	−0.476563	−	0.144808i
$382$		0			0
$383$		−	24.1520i		−	1.23411i	−0.786920	−	0.617055i	$-0.788326\pi$
$383$		−	24.1520i		−	1.23411i	0.786920	−	0.617055i	$-0.211674\pi$
$384$		0			0
$385$	1.97659	+	2.66328i	0.100736	+	0.135734i
$386$		0			0
$387$	−14.0726	+	21.0183i	−0.715351	+	1.06842i
$388$		0			0
$389$			8.06736i			0.409032i	0.978863	+	0.204516i	$0.0655620\pi$
$389$			8.06736i			0.409032i	−0.978863	+	0.204516i	$0.934438\pi$
$390$		0			0
$391$		−	61.3626i		−	3.10324i
$392$		0			0
$393$	−2.92912	+	9.63968i	−0.147754	+	0.486257i
$394$		0			0
$395$	6.40822			0.322432
$396$		0			0
$397$	−0.252709			−0.0126831			−0.00634156	−	0.999980i	$-0.502019\pi$
$397$	−0.252709			−0.0126831			−0.00634156	+	0.999980i	$0.502019\pi$
$398$		0			0
$399$	1.69806	−	5.58827i	0.0850091	−	0.279764i
$400$		0			0
$401$			15.6756i			0.782802i	0.920220	+	0.391401i	$0.128009\pi$
$401$			15.6756i			0.782802i	−0.920220	+	0.391401i	$0.871991\pi$
$402$		0			0
$403$			16.7034i			0.832057i
$404$		0			0
$405$	8.32138	−	3.42850i	0.413493	−	0.170364i
$406$		0			0
$407$	−11.6872	−	15.7475i	−0.579315	−	0.780576i
$408$		0			0
$409$			13.0047i			0.643040i	0.946903	+	0.321520i	$0.104194\pi$
$409$			13.0047i			0.643040i	−0.946903	+	0.321520i	$0.895806\pi$
$410$		0			0
$411$	36.5932	+	11.1192i	1.80501	+	0.548471i
$412$		0			0
$413$	10.9526			0.538942
$414$		0			0
$415$			2.00211i			0.0982799i
$416$		0			0
$417$	−31.5929	−	9.59984i	−1.54711	−	0.470106i
$418$		0			0
$419$		−	36.7083i		−	1.79332i	−0.442724	−	0.896658i	$-0.645988\pi$
$419$		−	36.7083i		−	1.79332i	0.442724	−	0.896658i	$-0.354012\pi$
$420$		0			0
$421$	34.1268			1.66324			0.831620	−	0.555345i	$-0.187414\pi$
$421$	34.1268			1.66324			0.831620	+	0.555345i	$0.187414\pi$
$422$		0			0
$423$	−11.2547	+	16.8096i	−0.547222	+	0.817311i
$424$		0			0
$425$	6.52370			0.316446
$426$		0			0
$427$	−7.32704			−0.354580
$428$		0			0
$429$	−6.52130	−	18.2338i	−0.314851	−	0.880337i
$430$		0			0
$431$	−37.9132			−1.82622			−0.913108	−	0.407717i	$-0.866325\pi$
$431$	−37.9132			−1.82622			−0.913108	+	0.407717i	$0.866325\pi$
$432$		0			0
$433$	−26.0033			−1.24964			−0.624820	−	0.780769i	$-0.714827\pi$
$433$	−26.0033			−1.24964			−0.624820	+	0.780769i	$0.714827\pi$
$434$		0			0
$435$	8.52759	+	2.59120i	0.408867	+	0.124238i
$436$		0			0
$437$	−31.7178			−1.51727
$438$		0			0
$439$		−	12.6212i		−	0.602379i	−0.953564	−	0.301190i	$-0.902616\pi$
$439$		−	12.6212i		−	0.602379i	0.953564	−	0.301190i	$-0.0973837\pi$
$440$		0			0
$441$	2.49284	+	1.66906i	0.118707	+	0.0794789i
$442$		0			0
$443$		−	19.9097i		−	0.945937i	−0.881079	−	0.472969i	$-0.843182\pi$
$443$		−	19.9097i		−	0.945937i	0.881079	−	0.472969i	$-0.156818\pi$
$444$		0			0
$445$	11.2528			0.533435
$446$		0			0
$447$	2.79961	−	9.21348i	0.132417	−	0.435783i
$448$		0			0
$449$		−	11.1254i		−	0.525042i	−0.964926	−	0.262521i	$-0.915446\pi$
$449$		−	11.1254i		−	0.525042i	0.964926	−	0.262521i	$-0.0845539\pi$
$450$		0			0
$451$	11.1090	+	14.9684i	0.523101	+	0.704832i
$452$		0			0
$453$	−21.9986	−	6.68452i	−1.03359	−	0.314066i
$454$		0			0
$455$			3.37100i			0.158035i
$456$		0			0
$457$		−	7.42509i		−	0.347331i	−0.984805	−	0.173665i	$-0.944439\pi$
$457$		−	7.42509i		−	0.347331i	0.984805	−	0.173665i	$-0.0555611\pi$
$458$		0			0
$459$	−26.2339	+	21.4678i	−1.22450	+	1.00203i
$460$		0			0
$461$	26.4783			1.23322			0.616610	−	0.787269i	$-0.288506\pi$
$461$	26.4783			1.23322			0.616610	+	0.787269i	$0.288506\pi$
$462$		0			0
$463$	28.7438			1.33584			0.667919	−	0.744234i	$-0.267185\pi$
$463$	28.7438			1.33584			0.667919	+	0.744234i	$0.267185\pi$
$464$		0			0
$465$	8.21165	+	2.49520i	0.380806	+	0.115712i
$466$		0			0
$467$		−	7.70769i		−	0.356669i	−0.983970	−	0.178335i	$-0.942929\pi$
$467$		−	7.70769i		−	0.356669i	0.983970	−	0.178335i	$-0.0570710\pi$
$468$		0			0
$469$			0.0203833i			0.000941215i
$470$		0			0
$471$	4.26477	−	14.0353i	0.196510	−	0.646712i
$472$		0			0
$473$	22.4554	−	16.6656i	1.03250	−	0.766284i
$474$		0			0
$475$		−	3.37205i		−	0.154720i
$476$		0			0
$477$	2.64979	−	3.95763i	0.121326	−	0.181207i
$478$		0			0
$479$	−8.64553			−0.395024			−0.197512	−	0.980300i	$-0.563286\pi$
$479$	−8.64553			−0.395024			−0.197512	+	0.980300i	$0.563286\pi$
$480$		0			0
$481$		−	19.9321i		−	0.908826i
$482$		0			0
$483$	4.73660	−	15.5881i	0.215523	−	0.709282i
$484$		0			0
$485$			7.17338i			0.325727i
$486$		0			0
$487$	8.94341			0.405264			0.202632	−	0.979255i	$-0.435050\pi$
$487$	8.94341			0.405264			0.202632	+	0.979255i	$0.435050\pi$
$488$		0			0
$489$	4.76063	−	15.6672i	0.215283	−	0.708494i
$490$		0			0
$491$	13.6863			0.617654			0.308827	−	0.951118i	$-0.400064\pi$
$491$	13.6863			0.617654			0.308827	+	0.951118i	$0.400064\pi$
$492$		0			0
$493$	−33.5689			−1.51187
$494$		0			0
$495$	−9.93819	+	0.482156i	−0.446688	+	0.0216713i
$496$		0			0
$497$	7.58489			0.340229
$498$		0			0
$499$	−34.8680			−1.56090			−0.780452	−	0.625215i	$-0.785011\pi$
$499$	−34.8680			−1.56090			−0.780452	+	0.625215i	$0.785011\pi$
$500$		0			0
$501$	12.3868	−	40.7649i	0.553403	−	1.82124i
$502$		0			0
$503$	−26.9351			−1.20098			−0.600488	−	0.799634i	$-0.705027\pi$
$503$	−26.9351			−1.20098			−0.600488	+	0.799634i	$0.705027\pi$
$504$		0			0
$505$			17.0955i			0.760740i
$506$		0			0
$507$	−0.824026	+	2.71186i	−0.0365963	+	0.120438i
$508$		0			0
$509$			25.8760i			1.14693i	0.819228	+	0.573467i	$0.194402\pi$
$509$			25.8760i			1.14693i	−0.819228	+	0.573467i	$0.805598\pi$
$510$		0			0
$511$	−11.2643			−0.498302
$512$		0			0
$513$	11.0965	+	13.5601i	0.489924	+	0.598694i
$514$		0			0
$515$			3.04232i			0.134061i
$516$		0			0
$517$	17.9589	−	13.3285i	0.789833	−	0.586185i
$518$		0			0
$519$	5.33381	−	17.5535i	0.234128	−	0.770512i
$520$		0			0
$521$			36.8954i			1.61642i	0.588896	+	0.808209i	$0.299563\pi$
$521$			36.8954i			1.61642i	−0.588896	+	0.808209i	$0.700437\pi$
$522$		0			0
$523$			28.7364i			1.25655i	0.777990	+	0.628277i	$0.216239\pi$
$523$			28.7364i			1.25655i	−0.777990	+	0.628277i	$0.783761\pi$
$524$		0			0
$525$	1.65723	+	0.503567i	0.0723276	+	0.0219775i
$526$		0			0
$527$	−32.3252			−1.40811
$528$		0			0
$529$	−65.4746			−2.84672
$530$		0			0
$531$	−18.2805	+	27.3031i	−0.793306	+	1.18485i
$532$		0			0
$533$			18.9459i			0.820638i
$534$		0			0
$535$			17.5060i			0.756849i
$536$		0			0
$537$	−22.9533	−	6.97461i	−0.990510	−	0.300977i
$538$		0			0
$539$	−1.97659	−	2.66328i	−0.0851378	−	0.114716i
$540$		0			0
$541$		−	34.3734i		−	1.47783i	−0.673801	−	0.738913i	$-0.735339\pi$
$541$		−	34.3734i		−	1.47783i	0.673801	−	0.738913i	$-0.264661\pi$
$542$		0			0
$543$	6.99635	−	23.0249i	0.300242	−	0.988093i
$544$		0			0
$545$	−2.42580			−0.103910
$546$		0			0
$547$			22.3462i			0.955454i	0.878508	+	0.477727i	$0.158539\pi$
$547$			22.3462i			0.955454i	−0.878508	+	0.477727i	$0.841461\pi$
$548$		0			0
$549$	12.2292	−	18.2651i	0.521931	−	0.779537i
$550$		0			0
$551$			17.3515i			0.739199i
$552$		0			0
$553$	−6.40822			−0.272505
$554$		0			0
$555$	−9.79892	−	2.97750i	−0.415941	−	0.126388i
$556$		0			0
$557$	3.94278			0.167061			0.0835305	−	0.996505i	$-0.473380\pi$
$557$	3.94278			0.167061			0.0835305	+	0.996505i	$0.473380\pi$
$558$		0			0
$559$	28.4225			1.20214
$560$		0			0
$561$	35.2869	−	12.6203i	1.48981	−	0.532829i
$562$		0			0
$563$	13.2181			0.557076			0.278538	−	0.960425i	$-0.410150\pi$
$563$	13.2181			0.557076			0.278538	+	0.960425i	$0.410150\pi$
$564$		0			0
$565$	−20.7421			−0.872629
$566$		0			0
$567$	−8.32138	+	3.42850i	−0.349465	+	0.143984i
$568$		0			0
$569$	2.23840			0.0938385			0.0469192	−	0.998899i	$-0.485060\pi$
$569$	2.23840			0.0938385			0.0469192	+	0.998899i	$0.485060\pi$
$570$		0			0
$571$			4.93905i			0.206693i	0.994645	+	0.103346i	$0.0329550\pi$
$571$			4.93905i			0.206693i	−0.994645	+	0.103346i	$0.967045\pi$
$572$		0			0
$573$	3.71807	+	1.12977i	0.155325	+	0.0471970i
$574$		0			0
$575$		−	9.40609i		−	0.392261i
$576$		0			0
$577$	18.5009			0.770203			0.385102	−	0.922874i	$-0.374166\pi$
$577$	18.5009			0.770203			0.385102	+	0.922874i	$0.374166\pi$
$578$		0			0
$579$	7.92022	+	2.40664i	0.329153	+	0.100017i
$580$		0			0
$581$		−	2.00211i		−	0.0830617i
$582$		0			0
$583$	−4.22823	+	3.13804i	−0.175115	+	0.129964i
$584$		0			0
$585$	−8.40336	−	5.62638i	−0.347436	−	0.232622i
$586$		0			0
$587$		−	18.5373i		−	0.765117i	−0.923931	−	0.382559i	$-0.875043\pi$
$587$		−	18.5373i		−	0.765117i	0.923931	−	0.382559i	$-0.124957\pi$
$588$		0			0
$589$			16.7086i			0.688468i
$590$		0			0
$591$	−7.86999	+	25.9000i	−0.323728	+	1.06538i
$592$		0			0
$593$	−13.8421			−0.568427			−0.284213	−	0.958761i	$-0.591732\pi$
$593$	−13.8421			−0.568427			−0.284213	+	0.958761i	$0.591732\pi$
$594$		0			0
$595$	−6.52370			−0.267446
$596$		0			0
$597$	−3.14627	+	10.3543i	−0.128768	+	0.423774i
$598$		0			0
$599$		−	12.8029i		−	0.523111i	−0.965188	−	0.261556i	$-0.915765\pi$
$599$		−	12.8029i		−	0.523111i	0.965188	−	0.261556i	$-0.0842355\pi$
$600$		0			0
$601$			27.0576i			1.10370i	0.833942	+	0.551852i	$0.186079\pi$
$601$			27.0576i			1.10370i	−0.833942	+	0.551852i	$0.813921\pi$
$602$		0			0
$603$	−0.0508124	−	0.0340209i	−0.00206924	−	0.00138544i
$604$		0			0
$605$	10.5285	+	3.18617i	0.428043	+	0.129536i
$606$		0			0
$607$			38.7207i			1.57162i	0.618466	+	0.785812i	$0.287755\pi$
$607$			38.7207i			1.57162i	−0.618466	+	0.785812i	$0.712245\pi$
$608$		0			0
$609$	−8.52759	−	2.59120i	−0.345555	−	0.105001i
$610$		0			0
$611$	22.7311			0.919604
$612$		0			0
$613$			1.47477i			0.0595655i	0.999556	+	0.0297827i	$0.00948154\pi$
$613$			1.47477i			0.0595655i	−0.999556	+	0.0297827i	$0.990518\pi$
$614$		0			0
$615$	9.31408	+	2.83018i	0.375580	+	0.114124i
$616$		0			0
$617$			21.0123i			0.845922i	0.906148	+	0.422961i	$0.139009\pi$
$617$			21.0123i			0.845922i	−0.906148	+	0.422961i	$0.860991\pi$
$618$		0			0
$619$	−25.8122			−1.03748			−0.518739	−	0.854932i	$-0.673599\pi$
$619$	−25.8122			−1.03748			−0.518739	+	0.854932i	$0.673599\pi$
$620$		0			0
$621$	30.9529	+	37.8250i	1.24210	+	1.51786i
$622$		0			0
$623$	−11.2528			−0.450835
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−6.52334	−	18.2395i	−0.260517	−	0.728417i
$628$		0			0
$629$	38.5735			1.53803
$630$		0			0
$631$	16.6419			0.662502			0.331251	−	0.943543i	$-0.392529\pi$
$631$	16.6419			0.662502			0.331251	+	0.943543i	$0.392529\pi$
$632$		0			0
$633$	31.3900	+	9.53819i	1.24764	+	0.379109i
$634$		0			0
$635$	5.61305			0.222747
$636$		0			0
$637$		−	3.37100i		−	0.133564i
$638$		0			0
$639$	−12.6596	+	18.9079i	−0.500806	+	0.747986i
$640$		0			0
$641$		−	6.11283i		−	0.241442i	−0.992686	−	0.120721i	$-0.961479\pi$
$641$		−	6.11283i		−	0.241442i	0.992686	−	0.120721i	$-0.0385207\pi$
$642$		0			0
$643$	−46.9445			−1.85131			−0.925655	−	0.378369i	$-0.876485\pi$
$643$	−46.9445			−1.85131			−0.925655	+	0.378369i	$0.876485\pi$
$644$		0			0
$645$	4.24581	−	13.9729i	0.167179	−	0.550183i
$646$		0			0
$647$			37.1982i			1.46241i	0.682158	+	0.731205i	$0.261042\pi$
$647$			37.1982i			1.46241i	−0.682158	+	0.731205i	$0.738958\pi$
$648$		0			0
$649$	29.1699	−	21.6488i	1.14502	−	0.849790i
$650$		0			0
$651$	−8.21165	−	2.49520i	−0.321840	−	0.0977944i
$652$		0			0
$653$			26.9970i			1.05647i	0.849097	+	0.528237i	$0.177147\pi$
$653$			26.9970i			1.05647i	−0.849097	+	0.528237i	$0.822853\pi$
$654$		0			0
$655$		−	5.81673i		−	0.227279i
$656$		0			0
$657$	18.8007	−	28.0800i	0.733485	−	1.09551i
$658$		0			0
$659$	35.6598			1.38911			0.694554	−	0.719440i	$-0.255602\pi$
$659$	35.6598			1.38911			0.694554	+	0.719440i	$0.255602\pi$
$660$		0			0
$661$	−7.00032			−0.272281			−0.136140	−	0.990690i	$-0.543470\pi$
$661$	−7.00032			−0.272281			−0.136140	+	0.990690i	$0.543470\pi$
$662$		0			0
$663$	36.4448	+	11.0741i	1.41540	+	0.430084i
$664$		0			0
$665$			3.37205i			0.130763i
$666$		0			0
$667$			48.4008i			1.87409i
$668$		0			0
$669$	−0.698229	+	2.29786i	−0.0269951	+	0.0888405i
$670$		0			0
$671$	−19.5140	+	14.4826i	−0.753329	+	0.559093i
$672$		0			0
$673$			9.81504i			0.378342i	0.981944	+	0.189171i	$0.0605800\pi$
$673$			9.81504i			0.378342i	−0.981944	+	0.189171i	$0.939420\pi$
$674$		0			0
$675$	−4.02133	+	3.29073i	−0.154781	+	0.126660i
$676$		0			0
$677$	51.5564			1.98148			0.990738	−	0.135790i	$-0.0433572\pi$
$677$	51.5564			1.98148			0.990738	+	0.135790i	$0.0433572\pi$
$678$		0			0
$679$		−	7.17338i		−	0.275289i
$680$		0			0
$681$	5.67526	−	18.6772i	0.217476	−	0.715712i
$682$		0			0
$683$			5.79306i			0.221665i	0.993839	+	0.110833i	$0.0353518\pi$
$683$			5.79306i			0.221665i	−0.993839	+	0.110833i	$0.964648\pi$
$684$		0			0
$685$	−22.0809			−0.843669
$686$		0			0
$687$	0.672792	−	2.21415i	0.0256686	−	0.0844751i
$688$		0			0
$689$	−5.35179			−0.203887
$690$		0			0
$691$	−32.1455			−1.22287			−0.611435	−	0.791294i	$-0.709408\pi$
$691$	−32.1455			−1.22287			−0.611435	+	0.791294i	$0.709408\pi$
$692$		0			0
$693$	9.93819	−	0.482156i	0.377520	−	0.0183156i
$694$		0			0
$695$	19.0637			0.723126
$696$		0			0
$697$	−36.6649			−1.38878
$698$		0			0
$699$	−1.20179	+	3.95506i	−0.0454558	+	0.149594i
$700$		0			0
$701$	14.0196			0.529512			0.264756	−	0.964315i	$-0.414709\pi$
$701$	14.0196			0.529512			0.264756	+	0.964315i	$0.414709\pi$
$702$		0			0
$703$		−	19.9383i		−	0.751989i
$704$		0			0
$705$	3.39563	−	11.1750i	0.127887	−	0.420874i
$706$		0			0
$707$		−	17.0955i		−	0.642943i
$708$		0			0
$709$	28.1119			1.05577			0.527883	−	0.849317i	$-0.322986\pi$
$709$	28.1119			1.05577			0.527883	+	0.849317i	$0.322986\pi$
$710$		0			0
$711$	10.6957	−	15.9747i	0.401119	−	0.599096i
$712$		0			0
$713$			46.6076i			1.74547i
$714$		0			0
$715$	6.66309	+	8.97793i	0.249185	+	0.335755i
$716$		0			0
$717$	−3.25151	+	10.7007i	−0.121430	+	0.399624i
$718$		0			0
$719$			1.75876i			0.0655908i	0.999462	+	0.0327954i	$0.0104410\pi$
$719$			1.75876i			0.0655908i	−0.999462	+	0.0327954i	$0.989559\pi$
$720$		0			0
$721$		−	3.04232i		−	0.113302i
$722$		0			0
$723$	0.618915	+	0.188064i	0.0230177	+	0.00699416i
$724$		0			0
$725$	−5.14568			−0.191106
$726$		0			0
$727$	17.8286			0.661226			0.330613	−	0.943766i	$-0.392745\pi$
$727$	17.8286			0.661226			0.330613	+	0.943766i	$0.392745\pi$
$728$		0			0
$729$	5.34214	−	26.4662i	0.197857	−	0.980231i
$730$		0			0
$731$			55.0044i			2.03441i
$732$		0			0
$733$			42.9535i			1.58652i	0.608880	+	0.793262i	$0.291619\pi$
$733$			42.9535i			1.58652i	−0.608880	+	0.793262i	$0.708381\pi$
$734$		0			0
$735$	−1.65723	−	0.503567i	−0.0611279	−	0.0185744i
$736$		0			0
$737$	0.0402895	+	0.0542866i	0.00148408	+	0.00199967i
$738$		0			0
$739$		−	38.2009i		−	1.40524i	−0.711565	−	0.702621i	$-0.752013\pi$
$739$		−	38.2009i		−	1.40524i	0.711565	−	0.702621i	$-0.247987\pi$
$740$		0			0
$741$	5.72414	−	18.8381i	0.210282	−	0.692033i
$742$		0			0
$743$	−8.93636			−0.327843			−0.163922	−	0.986473i	$-0.552414\pi$
$743$	−8.93636			−0.327843			−0.163922	+	0.986473i	$0.552414\pi$
$744$		0			0
$745$			5.55956i			0.203687i
$746$		0			0
$747$	4.99095	+	3.34164i	0.182609	+	0.122264i
$748$		0			0
$749$		−	17.5060i		−	0.639654i
$750$		0			0
$751$	10.1780			0.371402			0.185701	−	0.982606i	$-0.440544\pi$
$751$	10.1780			0.371402			0.185701	+	0.982606i	$0.440544\pi$
$752$		0			0
$753$	−26.6661	−	8.10278i	−0.971768	−	0.295282i
$754$		0			0
$755$	13.2743			0.483102
$756$		0			0
$757$	18.2116			0.661912			0.330956	−	0.943646i	$-0.392629\pi$
$757$	18.2116			0.661912			0.330956	+	0.943646i	$0.392629\pi$
$758$		0			0
$759$	−18.1964	−	50.8778i	−0.660486	−	1.84675i
$760$		0			0
$761$	−37.9210			−1.37464			−0.687318	−	0.726357i	$-0.741212\pi$
$761$	−37.9210			−1.37464			−0.687318	+	0.726357i	$0.741212\pi$
$762$		0			0
$763$	2.42580			0.0878198
$764$		0			0
$765$	10.8884	−	16.2625i	0.393672	−	0.587974i
$766$		0			0
$767$	36.9212			1.33315
$768$		0			0
$769$		−	32.2685i		−	1.16363i	−0.813320	−	0.581816i	$-0.802342\pi$
$769$		−	32.2685i		−	1.16363i	0.813320	−	0.581816i	$-0.197658\pi$
$770$		0			0
$771$	15.0243	+	4.56529i	0.541087	+	0.164415i
$772$		0			0
$773$			50.7019i			1.82362i	0.410611	+	0.911811i	$0.365315\pi$
$773$			50.7019i			1.82362i	−0.410611	+	0.911811i	$0.634685\pi$
$774$		0			0
$775$	−4.95504			−0.177990
$776$		0			0
$777$	9.79892	+	2.97750i	0.351534	+	0.106817i
$778$		0			0
$779$			18.9518i			0.679019i
$780$		0			0
$781$	20.2007	−	14.9922i	0.722839	−	0.536464i
$782$		0			0
$783$	20.6925	−	16.9331i	0.739489	−	0.605139i
$784$		0			0
$785$			8.46912i			0.302276i
$786$		0			0
$787$		−	32.3757i		−	1.15407i	−0.816720	−	0.577035i	$-0.804210\pi$
$787$		−	32.3757i		−	1.15407i	0.816720	−	0.577035i	$-0.195790\pi$
$788$		0			0
$789$	−6.58402	+	21.6679i	−0.234397	+	0.771399i
$790$		0			0
$791$	20.7421			0.737506
$792$		0			0
$793$	−24.6994			−0.877103
$794$		0			0
$795$	−0.799463	+	2.63102i	−0.0283540	+	0.0933127i
$796$		0			0
$797$		−	37.9914i		−	1.34572i	−0.739768	−	0.672862i	$-0.765065\pi$
$797$		−	37.9914i		−	1.34572i	0.739768	−	0.672862i	$-0.234935\pi$
$798$		0			0
$799$			43.9903i			1.55627i
$800$		0			0
$801$	18.7816	−	28.0515i	0.663615	−	0.991150i
$802$		0			0
$803$	−30.0000	+	22.2649i	−1.05868	+	0.785710i
$804$		0			0
$805$			9.40609i			0.331521i
$806$		0			0
$807$	10.3279	+	3.13823i	0.363558	+	0.110471i
$808$		0			0
$809$	12.6600			0.445103			0.222551	−	0.974921i	$-0.428562\pi$
$809$	12.6600			0.445103			0.222551	+	0.974921i	$0.428562\pi$
$810$		0			0
$811$			23.7103i			0.832582i	0.909232	+	0.416291i	$0.136670\pi$
$811$			23.7103i			0.832582i	−0.909232	+	0.416291i	$0.863330\pi$
$812$		0			0
$813$	33.8940	+	10.2990i	1.18871	+	0.361203i
$814$		0			0
$815$			9.45381i			0.331153i
$816$		0			0
$817$	28.4314			0.994688
$818$		0			0
$819$	8.40336	+	5.62638i	0.293637	+	0.196602i
$820$		0			0
$821$	39.5954			1.38189			0.690945	−	0.722907i	$-0.257195\pi$
$821$	39.5954			1.38189			0.690945	+	0.722907i	$0.257195\pi$
$822$		0			0
$823$	−21.9998			−0.766866			−0.383433	−	0.923569i	$-0.625258\pi$
$823$	−21.9998			−0.766866			−0.383433	+	0.923569i	$0.625258\pi$
$824$		0			0
$825$	5.40903	−	1.93453i	0.188318	−	0.0673517i
$826$		0			0
$827$	6.27532			0.218214			0.109107	−	0.994030i	$-0.465201\pi$
$827$	6.27532			0.218214			0.109107	+	0.994030i	$0.465201\pi$
$828$		0			0
$829$	−19.6078			−0.681007			−0.340503	−	0.940243i	$-0.610598\pi$
$829$	−19.6078			−0.681007			−0.340503	+	0.940243i	$0.610598\pi$
$830$		0			0
$831$	−17.8729	−	5.43087i	−0.620005	−	0.188395i
$832$		0			0
$833$	6.52370			0.226033
$834$		0			0
$835$			24.5982i			0.851255i
$836$		0			0
$837$	19.9258	−	16.3057i	0.688737	−	0.563608i
$838$		0			0
$839$			48.1462i			1.66219i	0.556130	+	0.831095i	$0.312286\pi$
$839$			48.1462i			1.66219i	−0.556130	+	0.831095i	$0.687714\pi$
$840$		0			0
$841$	−2.52195			−0.0869639
$842$		0			0
$843$	0.305698	−	1.00605i	0.0105288	−	0.0346502i
$844$		0			0
$845$		−	1.63638i		−	0.0562931i
$846$		0			0
$847$	−10.5285	−	3.18617i	−0.361762	−	0.109478i
$848$		0			0
$849$	−47.8490	−	14.5394i	−1.64217	−	0.498991i
$850$		0			0
$851$		−	55.6165i		−	1.90651i
$852$		0			0
$853$			44.5842i			1.52653i	0.646084	+	0.763266i	$0.276406\pi$
$853$			44.5842i			1.52653i	−0.646084	+	0.763266i	$0.723594\pi$
$854$		0			0
$855$	−8.40599	−	5.62814i	−0.287479	−	0.192478i
$856$		0			0
$857$	16.7365			0.571707			0.285854	−	0.958273i	$-0.407723\pi$
$857$	16.7365			0.571707			0.285854	+	0.958273i	$0.407723\pi$
$858$		0			0
$859$	31.6210			1.07889			0.539447	−	0.842019i	$-0.318633\pi$
$859$	31.6210			1.07889			0.539447	+	0.842019i	$0.318633\pi$
$860$		0			0
$861$	−9.31408	−	2.83018i	−0.317423	−	0.0964523i
$862$		0			0
$863$		−	5.15759i		−	0.175566i	−0.996140	−	0.0877832i	$-0.972022\pi$
$863$		−	5.15759i		−	0.175566i	0.996140	−	0.0877832i	$-0.0279783\pi$
$864$		0			0
$865$			10.5920i			0.360140i
$866$		0			0
$867$	−12.8705	+	42.3567i	−0.437106	+	1.43851i
$868$		0			0
$869$	−17.0669	+	12.6664i	−0.578955	+	0.429679i
$870$		0			0
$871$			0.0687122i			0.00232822i
$872$		0			0
$873$	17.8821	+	11.9728i	0.605217	+	0.405217i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$		−	15.9299i		−	0.537914i	−0.963152	−	0.268957i	$-0.913321\pi$
$877$		−	15.9299i		−	0.537914i	0.963152	−	0.268957i	$-0.0866789\pi$
$878$		0			0
$879$	3.57953	−	11.7802i	0.120735	−	0.397336i
$880$		0			0
$881$		−	46.5218i		−	1.56736i	−0.621166	−	0.783679i	$-0.713341\pi$
$881$		−	46.5218i		−	1.56736i	0.621166	−	0.783679i	$-0.286659\pi$
$882$		0			0
$883$	3.92372			0.132044			0.0660218	−	0.997818i	$-0.478969\pi$
$883$	3.92372			0.132044			0.0660218	+	0.997818i	$0.478969\pi$
$884$		0			0
$885$	5.51537	−	18.1510i	0.185397	−	0.610139i
$886$		0			0
$887$	52.6501			1.76782			0.883908	−	0.467660i	$-0.154903\pi$
$887$	52.6501			1.76782			0.883908	+	0.467660i	$0.154903\pi$
$888$		0			0
$889$	−5.61305			−0.188256
$890$		0			0
$891$	−15.3855	+	25.5791i	−0.515432	+	0.856930i
$892$		0			0
$893$	22.7383			0.760907
$894$		0			0
$895$	13.8504			0.462968
$896$		0			0
$897$	15.9671	−	52.5474i	0.533125	−	1.75451i
$898$		0			0
$899$	25.4971			0.850374
$900$		0			0
$901$		−	10.3570i		−	0.345043i
$902$		0			0
$903$	−4.24581	+	13.9729i	−0.141292	+	0.464989i
$904$		0			0
$905$			13.8936i			0.461838i
$906$		0			0
$907$	−11.5564			−0.383724			−0.191862	−	0.981422i	$-0.561453\pi$
$907$	−11.5564			−0.383724			−0.191862	+	0.981422i	$0.561453\pi$
$908$		0			0
$909$	42.6164	+	28.5334i	1.41350	+	0.946392i
$910$		0			0
$911$		−	41.1115i		−	1.36208i	−0.732245	−	0.681042i	$-0.761527\pi$
$911$		−	41.1115i		−	1.36208i	0.732245	−	0.681042i	$-0.238473\pi$
$912$		0			0
$913$	−3.95736	−	5.33220i	−0.130970	−	0.176470i
$914$		0			0
$915$	−3.68966	+	12.1426i	−0.121976	+	0.401422i
$916$		0			0
$917$			5.81673i			0.192085i
$918$		0			0
$919$		−	53.6204i		−	1.76877i	−0.466755	−	0.884387i	$-0.654577\pi$
$919$		−	53.6204i		−	1.76877i	0.466755	−	0.884387i	$-0.345423\pi$
$920$		0			0
$921$	13.5547	+	4.11874i	0.446643	+	0.135717i
$922$		0			0
$923$	25.5687			0.841602
$924$		0			0
$925$	5.91282			0.194412
$926$		0			0
$927$	7.58401	+	5.07780i	0.249092	+	0.166777i
$928$		0			0
$929$			47.2026i			1.54867i	0.632779	+	0.774333i	$0.281914\pi$
$929$			47.2026i			1.54867i	−0.632779	+	0.774333i	$0.718086\pi$
$930$		0			0
$931$		−	3.37205i		−	0.110515i
$932$		0			0
$933$	21.3557	+	6.48915i	0.699154	+	0.212445i
$934$		0			0
$935$	−17.3745	+	12.8947i	−0.568206	+	0.421702i
$936$		0			0
$937$			43.0087i			1.40503i	0.711668	+	0.702516i	$0.247940\pi$
$937$			43.0087i			1.40503i	−0.711668	+	0.702516i	$0.752060\pi$
$938$		0			0
$939$	3.25157	−	10.7009i	0.106111	−	0.349210i
$940$		0			0
$941$	−38.8592			−1.26677			−0.633387	−	0.773835i	$-0.718336\pi$
$941$	−38.8592			−1.26677			−0.633387	+	0.773835i	$0.718336\pi$
$942$		0			0
$943$			52.8647i			1.72151i
$944$		0			0
$945$	4.02133	−	3.29073i	0.130814	−	0.107048i
$946$		0			0
$947$			37.1042i			1.20573i	0.797845	+	0.602863i	$0.205973\pi$
$947$			37.1042i			1.20573i	−0.797845	+	0.602863i	$0.794027\pi$
$948$		0			0
$949$	−37.9718			−1.23262
$950$		0			0
$951$	9.99906	+	3.03832i	0.324242	+	0.0985242i
$952$		0			0
$953$	6.61464			0.214269			0.107135	−	0.994245i	$-0.465832\pi$
$953$	6.61464			0.214269			0.107135	+	0.994245i	$0.465832\pi$
$954$		0			0
$955$	−2.24354			−0.0725993
$956$		0			0
$957$	−27.8331	+	9.95448i	−0.899718	+	0.321783i
$958$		0			0
$959$	22.0809			0.713030
$960$		0			0
$961$	−6.44759			−0.207987
$962$		0			0
$963$	43.6396	+	29.2184i	1.40627	+	0.941551i
$964$		0			0
$965$	−4.77918			−0.153847
$966$		0			0
$967$		−	45.8012i		−	1.47287i	−0.676510	−	0.736433i	$-0.736509\pi$
$967$		−	45.8012i		−	1.47287i	0.676510	−	0.736433i	$-0.263491\pi$
$968$		0			0
$969$	36.4562	+	11.0776i	1.17114	+	0.355864i
$970$		0			0
$971$		−	18.6547i		−	0.598659i	−0.954150	−	0.299329i	$-0.903237\pi$
$971$		−	18.6547i		−	0.598659i	0.954150	−	0.299329i	$-0.0967629\pi$
$972$		0			0
$973$	−19.0637			−0.611153
$974$		0			0
$975$	5.58653	+	1.69752i	0.178912	+	0.0543643i
$976$		0			0
$977$			38.7856i			1.24086i	0.784262	+	0.620430i	$0.213042\pi$
$977$			38.7856i			1.24086i	−0.784262	+	0.620430i	$0.786958\pi$
$978$		0			0
$979$	−29.9694	+	22.2422i	−0.957828	+	0.710865i
$980$		0			0
$981$	−4.04880	+	6.04713i	−0.129268	+	0.193070i
$982$		0			0
$983$			53.2948i			1.69984i	0.526911	+	0.849920i	$0.323350\pi$
$983$			53.2948i			1.69984i	−0.526911	+	0.849920i	$0.676650\pi$
$984$		0			0
$985$		−	15.6285i		−	0.497964i
$986$		0			0
$987$	−3.39563	+	11.1750i	−0.108084	+	0.355703i
$988$		0			0
$989$	79.3072			2.52182
$990$		0			0
$991$	50.1334			1.59254			0.796270	−	0.604942i	$-0.206804\pi$
$991$	50.1334			1.59254			0.796270	+	0.604942i	$0.206804\pi$
$992$		0			0
$993$	−0.965238	+	3.17659i	−0.0306309	+	0.100806i
$994$		0			0
$995$		−	6.24796i		−	0.198074i
$996$		0			0
$997$		−	35.6026i		−	1.12755i	−0.825930	−	0.563773i	$-0.809349\pi$
$997$		−	35.6026i		−	1.12755i	0.825930	−	0.563773i	$-0.190651\pi$
$998$		0			0
$999$	−23.7774	+	19.4575i	−0.752283	+	0.615609i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.m.b.1121.22	yes	48
3.2	odd	2		4620.2.m.a.1121.21	✓	48
11.10	odd	2		4620.2.m.a.1121.22	yes	48
33.32	even	2	inner	4620.2.m.b.1121.21	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.m.a.1121.21	✓	48	3.2	odd	2
4620.2.m.a.1121.22	yes	48	11.10	odd	2
4620.2.m.b.1121.21	yes	48	33.32	even	2	inner
4620.2.m.b.1121.22	yes	48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.503567 + 1.65723i) q^{3} -1.00000i q^{5} +1.00000i q^{7} +(-2.49284 - 1.66906i) q^{9} +O(q^{10})\)	\(q+(-0.503567 + 1.65723i) q^{3} -1.00000i q^{5} +1.00000i q^{7} +(-2.49284 - 1.66906i) q^{9} +(1.97659 + 2.66328i) q^{11} +3.37100i q^{13} +(1.65723 + 0.503567i) q^{15} -6.52370 q^{17} +3.37205i q^{19} +(-1.65723 - 0.503567i) q^{21} +9.40609i q^{23} -1.00000 q^{25} +(4.02133 - 3.29073i) q^{27} +5.14568 q^{29} +4.95504 q^{31} +(-5.40903 + 1.93453i) q^{33} +1.00000 q^{35} -5.91282 q^{37} +(-5.58653 - 1.69752i) q^{39} +5.62026 q^{41} -8.43147i q^{43} +(-1.66906 + 2.49284i) q^{45} -6.74315i q^{47} -1.00000 q^{49} +(3.28512 - 10.8113i) q^{51} +1.58760i q^{53} +(2.66328 - 1.97659i) q^{55} +(-5.58827 - 1.69806i) q^{57} -10.9526i q^{59} +7.32704i q^{61} +(1.66906 - 2.49284i) q^{63} +3.37100 q^{65} +0.0203833 q^{67} +(-15.5881 - 4.73660i) q^{69} -7.58489i q^{71} +11.2643i q^{73} +(0.503567 - 1.65723i) q^{75} +(-2.66328 + 1.97659i) q^{77} +6.40822i q^{79} +(3.42850 + 8.32138i) q^{81} -2.00211 q^{83} +6.52370i q^{85} +(-2.59120 + 8.52759i) q^{87} +11.2528i q^{89} -3.37100 q^{91} +(-2.49520 + 8.21165i) q^{93} +3.37205 q^{95} -7.17338 q^{97} +(-0.482156 - 9.93819i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) \) 48 * q - 4 * q^3 + 6 * q^9	\( 48 q - 4 q^{3} + 6 q^{9} + 6 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{21} - 48 q^{25} - 16 q^{27} - 36 q^{29} - 16 q^{31} - 4 q^{33} + 48 q^{35} - 8 q^{37} + 18 q^{39} - 48 q^{49} + 30 q^{51} + 4 q^{55} - 16 q^{57} + 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} - 4 q^{77} - 22 q^{81} - 20 q^{83} - 44 q^{87} - 12 q^{91} - 28 q^{93} - 8 q^{95} - 56 q^{97} - 12 q^{99}+O(q^{100}) \) 48 * q - 4 * q^3 + 6 * q^9 + 6 * q^11 + 2 * q^15 - 4 * q^17 - 2 * q^21 - 48 * q^25 - 16 * q^27 - 36 * q^29 - 16 * q^31 - 4 * q^33 + 48 * q^35 - 8 * q^37 + 18 * q^39 - 48 * q^49 + 30 * q^51 + 4 * q^55 - 16 * q^57 + 12 * q^65 + 24 * q^67 + 4 * q^69 + 4 * q^75 - 4 * q^77 - 22 * q^81 - 20 * q^83 - 44 * q^87 - 12 * q^91 - 28 * q^93 - 8 * q^95 - 56 * q^97 - 12 * q^99

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