LMFDB - Embedded newform 4620.2.h.e.1849.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4620,2,Mod(1849,4620)] 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("4620.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4620, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [14,0,0,0,-4,0,0,0,-14,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} + 12 x^{12} - 32 x^{11} + 86 x^{10} - 220 x^{9} + 585 x^{8} - 1536 x^{7} + \cdots + 78125 $ x^14 - 4x^13 + 12x^12 - 32x^11 + 86x^10 - 220x^9 + 585x^8 - 1536x^7 + 2925x^6 - 5500x^5 + 10750x^4 - 20000x^3 + 37500x^2 - 62500*x + 78125
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.11
Root			$0.151739 - 2.23091i$ of defining polynomial
Character	$\chi$	$=$	4620.1849
Dual form			4620.2.h.e.1849.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.151739 - 2.23091i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} -3.38902i q^{13} +(2.23091 - 0.151739i) q^{15} +4.81652i q^{17} -2.96123 q^{19} -1.00000 q^{21} -2.03848i q^{23} +(-4.95395 + 0.677032i) q^{25} -1.00000i q^{27} +7.32176 q^{29} -4.29904 q^{31} +1.00000i q^{33} +(2.23091 - 0.151739i) q^{35} -6.82236i q^{37} +3.38902 q^{39} +7.50965 q^{41} +7.04690i q^{43} +(0.151739 + 2.23091i) q^{45} -10.0772i q^{47} -1.00000 q^{49} -4.81652 q^{51} -2.21412i q^{53} +(-0.151739 - 2.23091i) q^{55} -2.96123i q^{57} -3.12508 q^{59} +9.01516 q^{61} -1.00000i q^{63} +(-7.56061 + 0.514246i) q^{65} -7.88349i q^{67} +2.03848 q^{69} -12.3116 q^{71} -1.06878i q^{73} +(-0.677032 - 4.95395i) q^{75} +1.00000i q^{77} +5.81097 q^{79} +1.00000 q^{81} -5.76593i q^{83} +(10.7452 - 0.730852i) q^{85} +7.32176i q^{87} -8.49297 q^{89} +3.38902 q^{91} -4.29904i q^{93} +(0.449334 + 6.60626i) q^{95} -2.53311i q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{5} - 14 q^{9} + 14 q^{11} + 6 q^{19} - 14 q^{21} - 8 q^{25} + 38 q^{29} + 4 q^{31} + 8 q^{39} - 12 q^{41} + 4 q^{45} - 14 q^{49} + 14 q^{51} - 4 q^{55} + 18 q^{59} - 54 q^{61} - 2 q^{65} + 26 q^{69}+ \cdots - 14 q^{99}+O(q^{100}) $ 14 * q - 4 * q^5 - 14 * q^9 + 14 * q^11 + 6 * q^19 - 14 * q^21 - 8 * q^25 + 38 * q^29 + 4 * q^31 + 8 * q^39 - 12 * q^41 + 4 * q^45 - 14 * q^49 + 14 * q^51 - 4 * q^55 + 18 * q^59 - 54 * q^61 - 2 * q^65 + 26 * q^69 - 24 * q^71 + 48 * q^79 + 14 * q^81 + 12 * q^85 + 46 * q^89 + 8 * q^91 - 72 * q^95 - 14 * q^99

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−0.151739	−	2.23091i	−0.0678596	−	0.997695i
$5$	−0.151739	−	2.23091i	−0.0678596	−	0.997695i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$		−	3.38902i		−	0.939946i	−0.882681	−	0.469973i	$-0.844264\pi$
$13$		−	3.38902i		−	0.939946i	0.882681	−	0.469973i	$-0.155736\pi$
$14$		0			0
$15$	2.23091	−	0.151739i	0.576019	−	0.0391788i
$16$		0			0
$17$			4.81652i			1.16818i	0.811690	+	0.584089i	$0.198548\pi$
$17$			4.81652i			1.16818i	−0.811690	+	0.584089i	$0.801452\pi$
$18$		0			0
$19$	−2.96123			−0.679354			−0.339677	−	0.940542i	$-0.610318\pi$
$19$	−2.96123			−0.679354			−0.339677	+	0.940542i	$0.610318\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$		−	2.03848i		−	0.425052i	−0.977155	−	0.212526i	$-0.931831\pi$
$23$		−	2.03848i		−	0.425052i	0.977155	−	0.212526i	$-0.0681689\pi$
$24$		0			0
$25$	−4.95395	+	0.677032i	−0.990790	+	0.135406i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	7.32176			1.35962			0.679809	−	0.733389i	$-0.262063\pi$
$29$	7.32176			1.35962			0.679809	+	0.733389i	$0.262063\pi$
$30$		0			0
$31$	−4.29904			−0.772131			−0.386065	−	0.922471i	$-0.626166\pi$
$31$	−4.29904			−0.772131			−0.386065	+	0.922471i	$0.626166\pi$
$32$		0			0
$33$			1.00000i			0.174078i
$34$		0			0
$35$	2.23091	−	0.151739i	0.377093	−	0.0256485i
$36$		0			0
$37$		−	6.82236i		−	1.12159i	−0.827955	−	0.560794i	$-0.810496\pi$
$37$		−	6.82236i		−	1.12159i	0.827955	−	0.560794i	$-0.189504\pi$
$38$		0			0
$39$	3.38902			0.542678
$40$		0			0
$41$	7.50965			1.17281			0.586405	−	0.810018i	$-0.300543\pi$
$41$	7.50965			1.17281			0.586405	+	0.810018i	$0.300543\pi$
$42$		0			0
$43$			7.04690i			1.07464i	0.843378	+	0.537321i	$0.180564\pi$
$43$			7.04690i			1.07464i	−0.843378	+	0.537321i	$0.819436\pi$
$44$		0			0
$45$	0.151739	+	2.23091i	0.0226199	+	0.332565i
$46$		0			0
$47$		−	10.0772i		−	1.46991i	−0.678115	−	0.734956i	$-0.737203\pi$
$47$		−	10.0772i		−	1.46991i	0.678115	−	0.734956i	$-0.262797\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−4.81652			−0.674448
$52$		0			0
$53$		−	2.21412i		−	0.304133i	−0.988370	−	0.152067i	$-0.951407\pi$
$53$		−	2.21412i		−	0.304133i	0.988370	−	0.152067i	$-0.0485929\pi$
$54$		0			0
$55$	−0.151739	−	2.23091i	−0.0204604	−	0.300816i
$56$		0			0
$57$		−	2.96123i		−	0.392225i
$58$		0			0
$59$	−3.12508			−0.406851			−0.203426	−	0.979090i	$-0.565208\pi$
$59$	−3.12508			−0.406851			−0.203426	+	0.979090i	$0.565208\pi$
$60$		0			0
$61$	9.01516			1.15427			0.577136	−	0.816648i	$-0.304170\pi$
$61$	9.01516			1.15427			0.577136	+	0.816648i	$0.304170\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−7.56061	+	0.514246i	−0.937779	+	0.0637844i
$66$		0			0
$67$		−	7.88349i		−	0.963121i	−0.876413	−	0.481561i	$-0.840070\pi$
$67$		−	7.88349i		−	0.963121i	0.876413	−	0.481561i	$-0.159930\pi$
$68$		0			0
$69$	2.03848			0.245404
$70$		0			0
$71$	−12.3116			−1.46112			−0.730560	−	0.682849i	$-0.760741\pi$
$71$	−12.3116			−1.46112			−0.730560	+	0.682849i	$0.760741\pi$
$72$		0			0
$73$		−	1.06878i		−	0.125091i	−0.998042	−	0.0625457i	$-0.980078\pi$
$73$		−	1.06878i		−	0.125091i	0.998042	−	0.0625457i	$-0.0199219\pi$
$74$		0			0
$75$	−0.677032	−	4.95395i	−0.0781769	−	0.572033i
$76$		0			0
$77$			1.00000i			0.113961i
$78$		0			0
$79$	5.81097			0.653785			0.326893	−	0.945061i	$-0.393998\pi$
$79$	5.81097			0.653785			0.326893	+	0.945061i	$0.393998\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	5.76593i		−	0.632893i	−0.948610	−	0.316447i	$-0.897510\pi$
$83$		−	5.76593i		−	0.632893i	0.948610	−	0.316447i	$-0.102490\pi$
$84$		0			0
$85$	10.7452	−	0.730852i	1.16548	−	0.0792721i
$86$		0			0
$87$			7.32176i			0.784975i
$88$		0			0
$89$	−8.49297			−0.900253			−0.450127	−	0.892965i	$-0.648621\pi$
$89$	−8.49297			−0.900253			−0.450127	+	0.892965i	$0.648621\pi$
$90$		0			0
$91$	3.38902			0.355266
$92$		0			0
$93$		−	4.29904i		−	0.445790i
$94$		0			0
$95$	0.449334	+	6.60626i	0.0461007	+	0.677788i
$96$		0			0
$97$		−	2.53311i		−	0.257198i	−0.991697	−	0.128599i	$-0.958952\pi$
$97$		−	2.53311i		−	0.257198i	0.991697	−	0.128599i	$-0.0410480\pi$
$98$		0			0
$99$	−1.00000			−0.100504
$100$		0			0
$101$	−18.8305			−1.87370			−0.936852	−	0.349726i	$-0.886275\pi$
$101$	−18.8305			−1.87370			−0.936852	+	0.349726i	$0.886275\pi$
$102$		0			0
$103$			0.191906i			0.0189090i	0.999955	+	0.00945451i	$0.00300951\pi$
$103$			0.191906i			0.0189090i	−0.999955	+	0.00945451i	$0.996990\pi$
$104$		0			0
$105$	0.151739	+	2.23091i	0.0148082	+	0.217715i
$106$		0			0
$107$		−	13.0381i		−	1.26044i	−0.776416	−	0.630221i	$-0.782964\pi$
$107$		−	13.0381i		−	1.26044i	0.776416	−	0.630221i	$-0.217036\pi$
$108$		0			0
$109$	13.7451			1.31654			0.658272	−	0.752780i	$-0.271288\pi$
$109$	13.7451			1.31654			0.658272	+	0.752780i	$0.271288\pi$
$110$		0			0
$111$	6.82236			0.647549
$112$		0			0
$113$			12.3484i			1.16164i	0.814033	+	0.580819i	$0.197268\pi$
$113$			12.3484i			1.16164i	−0.814033	+	0.580819i	$0.802732\pi$
$114$		0			0
$115$	−4.54766	+	0.309316i	−0.424072	+	0.0288438i
$116$		0			0
$117$			3.38902i			0.313315i
$118$		0			0
$119$	−4.81652			−0.441530
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$			7.50965i			0.677122i
$124$		0			0
$125$	2.26211	+	10.9491i	0.202329	+	0.979318i
$126$		0			0
$127$		−	17.1239i		−	1.51950i	−0.650213	−	0.759752i	$-0.725321\pi$
$127$		−	17.1239i		−	1.51950i	0.650213	−	0.759752i	$-0.274679\pi$
$128$		0			0
$129$	−7.04690			−0.620445
$130$		0			0
$131$	−20.8886			−1.82505			−0.912523	−	0.409025i	$-0.865869\pi$
$131$	−20.8886			−1.82505			−0.912523	+	0.409025i	$0.865869\pi$
$132$		0			0
$133$		−	2.96123i		−	0.256772i
$134$		0			0
$135$	−2.23091	+	0.151739i	−0.192006	+	0.0130596i
$136$		0			0
$137$		−	12.9479i		−	1.10621i	−0.833110	−	0.553107i	$-0.813442\pi$
$137$		−	12.9479i		−	1.10621i	0.833110	−	0.553107i	$-0.186558\pi$
$138$		0			0
$139$	−9.33171			−0.791505			−0.395753	−	0.918357i	$-0.629516\pi$
$139$	−9.33171			−0.791505			−0.395753	+	0.918357i	$0.629516\pi$
$140$		0			0
$141$	10.0772			0.848654
$142$		0			0
$143$		−	3.38902i		−	0.283404i
$144$		0			0
$145$	−1.11100	−	16.3342i	−0.0922631	−	1.35648i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	24.2248			1.98458			0.992288	−	0.123951i	$-0.0395566\pi$
$149$	24.2248			1.98458			0.992288	+	0.123951i	$0.0395566\pi$
$150$		0			0
$151$	−10.1828			−0.828662			−0.414331	−	0.910126i	$-0.635984\pi$
$151$	−10.1828			−0.828662			−0.414331	+	0.910126i	$0.635984\pi$
$152$		0			0
$153$		−	4.81652i		−	0.389392i
$154$		0			0
$155$	0.652331	+	9.59079i	0.0523965	+	0.770351i
$156$		0			0
$157$		−	10.8219i		−	0.863678i	−0.901951	−	0.431839i	$-0.857865\pi$
$157$		−	10.8219i		−	0.863678i	0.901951	−	0.431839i	$-0.142135\pi$
$158$		0			0
$159$	2.21412			0.175591
$160$		0			0
$161$	2.03848			0.160654
$162$		0			0
$163$		−	12.9680i		−	1.01573i	−0.861437	−	0.507865i	$-0.830435\pi$
$163$		−	12.9680i		−	1.01573i	0.861437	−	0.507865i	$-0.169565\pi$
$164$		0			0
$165$	2.23091	−	0.151739i	0.173676	−	0.0118128i
$166$		0			0
$167$		−	9.42488i		−	0.729319i	−0.931141	−	0.364659i	$-0.881185\pi$
$167$		−	9.42488i		−	0.729319i	0.931141	−	0.364659i	$-0.118815\pi$
$168$		0			0
$169$	1.51453			0.116502
$170$		0			0
$171$	2.96123			0.226451
$172$		0			0
$173$		−	12.5524i		−	0.954342i	−0.878810	−	0.477171i	$-0.841662\pi$
$173$		−	12.5524i		−	0.954342i	0.878810	−	0.477171i	$-0.158338\pi$
$174$		0			0
$175$	−0.677032	−	4.95395i	−0.0511788	−	0.374483i
$176$		0			0
$177$		−	3.12508i		−	0.234896i
$178$		0			0
$179$	−4.77271			−0.356729			−0.178365	−	0.983964i	$-0.557081\pi$
$179$	−4.77271			−0.356729			−0.178365	+	0.983964i	$0.557081\pi$
$180$		0			0
$181$	−7.11292			−0.528699			−0.264350	−	0.964427i	$-0.585157\pi$
$181$	−7.11292			−0.528699			−0.264350	+	0.964427i	$0.585157\pi$
$182$		0			0
$183$			9.01516i			0.666420i
$184$		0			0
$185$	−15.2201	+	1.03522i	−1.11900	+	0.0761106i
$186$		0			0
$187$			4.81652i			0.352219i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	−14.6340			−1.05888			−0.529439	−	0.848348i	$-0.677597\pi$
$191$	−14.6340			−1.05888			−0.529439	+	0.848348i	$0.677597\pi$
$192$		0			0
$193$		−	13.6330i		−	0.981328i	−0.871349	−	0.490664i	$-0.836754\pi$
$193$		−	13.6330i		−	0.981328i	0.871349	−	0.490664i	$-0.163246\pi$
$194$		0			0
$195$	−0.514246	−	7.56061i	−0.0368259	−	0.541427i
$196$		0			0
$197$		−	0.330782i		−	0.0235672i	−0.999931	−	0.0117836i	$-0.996249\pi$
$197$		−	0.330782i		−	0.0235672i	0.999931	−	0.0117836i	$-0.00375093\pi$
$198$		0			0
$199$	20.1131			1.42578			0.712889	−	0.701277i	$-0.247387\pi$
$199$	20.1131			1.42578			0.712889	+	0.701277i	$0.247387\pi$
$200$		0			0
$201$	7.88349			0.556058
$202$		0			0
$203$			7.32176i			0.513887i
$204$		0			0
$205$	−1.13950	−	16.7534i	−0.0795864	−	1.17011i
$206$		0			0
$207$			2.03848i			0.141684i
$208$		0			0
$209$	−2.96123			−0.204833
$210$		0			0
$211$	−5.52882			−0.380619			−0.190310	−	0.981724i	$-0.560949\pi$
$211$	−5.52882			−0.380619			−0.190310	+	0.981724i	$0.560949\pi$
$212$		0			0
$213$		−	12.3116i		−	0.843578i
$214$		0			0
$215$	15.7210	−	1.06929i	1.07216	−	0.0729248i
$216$		0			0
$217$		−	4.29904i		−	0.291838i
$218$		0			0
$219$	1.06878			0.0722216
$220$		0			0
$221$	16.3233			1.09802
$222$		0			0
$223$		−	13.1597i		−	0.881242i	−0.897693	−	0.440621i	$-0.854758\pi$
$223$		−	13.1597i		−	0.881242i	0.897693	−	0.440621i	$-0.145242\pi$
$224$		0			0
$225$	4.95395	−	0.677032i	0.330263	−	0.0451355i
$226$		0			0
$227$			12.8202i			0.850908i	0.904980	+	0.425454i	$0.139886\pi$
$227$			12.8202i			0.850908i	−0.904980	+	0.425454i	$0.860114\pi$
$228$		0			0
$229$	28.0550			1.85393			0.926964	−	0.375150i	$-0.122409\pi$
$229$	28.0550			1.85393			0.926964	+	0.375150i	$0.122409\pi$
$230$		0			0
$231$	−1.00000			−0.0657952
$232$		0			0
$233$		−	26.1136i		−	1.71076i	−0.518003	−	0.855379i	$-0.673324\pi$
$233$		−	26.1136i		−	1.71076i	0.518003	−	0.855379i	$-0.326676\pi$
$234$		0			0
$235$	−22.4814	+	1.52910i	−1.46652	+	0.0997477i
$236$		0			0
$237$			5.81097i			0.377463i
$238$		0			0
$239$	−20.3797			−1.31825			−0.659126	−	0.752033i	$-0.729074\pi$
$239$	−20.3797			−1.31825			−0.659126	+	0.752033i	$0.729074\pi$
$240$		0			0
$241$	−4.99845			−0.321979			−0.160989	−	0.986956i	$-0.551468\pi$
$241$	−4.99845			−0.321979			−0.160989	+	0.986956i	$0.551468\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	0.151739	+	2.23091i	0.00969423	+	0.142528i
$246$		0			0
$247$			10.0357i			0.638555i
$248$		0			0
$249$	5.76593			0.365401
$250$		0			0
$251$	2.67141			0.168618			0.0843091	−	0.996440i	$-0.473132\pi$
$251$	2.67141			0.168618			0.0843091	+	0.996440i	$0.473132\pi$
$252$		0			0
$253$		−	2.03848i		−	0.128158i
$254$		0			0
$255$	0.730852	+	10.7452i	0.0457678	+	0.672893i
$256$		0			0
$257$			28.8879i			1.80198i	0.433844	+	0.900988i	$0.357157\pi$
$257$			28.8879i			1.80198i	−0.433844	+	0.900988i	$0.642843\pi$
$258$		0			0
$259$	6.82236			0.423921
$260$		0			0
$261$	−7.32176			−0.453206
$262$		0			0
$263$		−	16.2179i		−	1.00004i	−0.866014	−	0.500019i	$-0.833326\pi$
$263$		−	16.2179i		−	1.00004i	0.866014	−	0.500019i	$-0.166674\pi$
$264$		0			0
$265$	−4.93952	+	0.335968i	−0.303432	+	0.0206384i
$266$		0			0
$267$		−	8.49297i		−	0.519762i
$268$		0			0
$269$	15.1952			0.926470			0.463235	−	0.886235i	$-0.346689\pi$
$269$	15.1952			0.926470			0.463235	+	0.886235i	$0.346689\pi$
$270$		0			0
$271$	5.71049			0.346888			0.173444	−	0.984844i	$-0.444510\pi$
$271$	5.71049			0.346888			0.173444	+	0.984844i	$0.444510\pi$
$272$		0			0
$273$			3.38902i			0.205113i
$274$		0			0
$275$	−4.95395	+	0.677032i	−0.298734	+	0.0408266i
$276$		0			0
$277$			12.6726i			0.761420i	0.924694	+	0.380710i	$0.124320\pi$
$277$			12.6726i			0.761420i	−0.924694	+	0.380710i	$0.875680\pi$
$278$		0			0
$279$	4.29904			0.257377
$280$		0			0
$281$	4.97868			0.297003			0.148502	−	0.988912i	$-0.452555\pi$
$281$	4.97868			0.297003			0.148502	+	0.988912i	$0.452555\pi$
$282$		0			0
$283$		−	0.549902i		−	0.0326883i	−0.999866	−	0.0163442i	$-0.994797\pi$
$283$		−	0.549902i		−	0.0326883i	0.999866	−	0.0163442i	$-0.00520274\pi$
$284$		0			0
$285$	−6.60626	+	0.449334i	−0.391321	+	0.0266162i
$286$		0			0
$287$			7.50965i			0.443280i
$288$		0			0
$289$	−6.19885			−0.364639
$290$		0			0
$291$	2.53311			0.148493
$292$		0			0
$293$		−	19.5587i		−	1.14263i	−0.820729	−	0.571317i	$-0.806433\pi$
$293$		−	19.5587i		−	1.14263i	0.820729	−	0.571317i	$-0.193567\pi$
$294$		0			0
$295$	0.474196	+	6.97179i	0.0276088	+	0.405914i
$296$		0			0
$297$		−	1.00000i		−	0.0580259i
$298$		0			0
$299$	−6.90844			−0.399525
$300$		0			0
$301$	−7.04690			−0.406177
$302$		0			0
$303$		−	18.8305i		−	1.08178i
$304$		0			0
$305$	−1.36795	−	20.1120i	−0.0783285	−	1.15161i
$306$		0			0
$307$		−	10.7762i		−	0.615031i	−0.951543	−	0.307515i	$-0.900502\pi$
$307$		−	10.7762i		−	0.615031i	0.951543	−	0.307515i	$-0.0994975\pi$
$308$		0			0
$309$	−0.191906			−0.0109171
$310$		0			0
$311$	−18.3169			−1.03866			−0.519329	−	0.854574i	$-0.673818\pi$
$311$	−18.3169			−1.03866			−0.519329	+	0.854574i	$0.673818\pi$
$312$		0			0
$313$			0.863241i			0.0487933i	0.999702	+	0.0243966i	$0.00776646\pi$
$313$			0.863241i			0.0487933i	−0.999702	+	0.0243966i	$0.992234\pi$
$314$		0			0
$315$	−2.23091	+	0.151739i	−0.125698	+	0.00854951i
$316$		0			0
$317$			2.76184i			0.155121i	0.996988	+	0.0775603i	$0.0247130\pi$
$317$			2.76184i			0.155121i	−0.996988	+	0.0775603i	$0.975287\pi$
$318$		0			0
$319$	7.32176			0.409940
$320$		0			0
$321$	13.0381			0.727716
$322$		0			0
$323$		−	14.2628i		−	0.793606i
$324$		0			0
$325$	2.29448	+	16.7890i	0.127275	+	0.931289i
$326$		0			0
$327$			13.7451i			0.760107i
$328$		0			0
$329$	10.0772			0.555574
$330$		0			0
$331$	32.7451			1.79983			0.899916	−	0.436064i	$-0.143628\pi$
$331$	32.7451			1.79983			0.899916	+	0.436064i	$0.143628\pi$
$332$		0			0
$333$			6.82236i			0.373863i
$334$		0			0
$335$	−17.5874	+	1.19623i	−0.960901	+	0.0653570i
$336$		0			0
$337$			9.07475i			0.494333i	0.968973	+	0.247167i	$0.0794995\pi$
$337$			9.07475i			0.494333i	−0.968973	+	0.247167i	$0.920500\pi$
$338$		0			0
$339$	−12.3484			−0.670672
$340$		0			0
$341$	−4.29904			−0.232806
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−0.309316	−	4.54766i	−0.0166530	−	0.244838i
$346$		0			0
$347$			29.9947i			1.61020i	0.593138	+	0.805100i	$0.297889\pi$
$347$			29.9947i			1.61020i	−0.593138	+	0.805100i	$0.702111\pi$
$348$		0			0
$349$	8.98598			0.481009			0.240504	−	0.970648i	$-0.422687\pi$
$349$	8.98598			0.481009			0.240504	+	0.970648i	$0.422687\pi$
$350$		0			0
$351$	−3.38902			−0.180893
$352$		0			0
$353$		−	10.4164i		−	0.554408i	−0.960811	−	0.277204i	$-0.910592\pi$
$353$		−	10.4164i		−	0.554408i	0.960811	−	0.277204i	$-0.0894078\pi$
$354$		0			0
$355$	1.86815	+	27.4661i	0.0991510	+	1.45775i
$356$		0			0
$357$		−	4.81652i		−	0.254917i
$358$		0			0
$359$	−17.4454			−0.920735			−0.460367	−	0.887728i	$-0.652282\pi$
$359$	−17.4454			−0.920735			−0.460367	+	0.887728i	$0.652282\pi$
$360$		0			0
$361$	−10.2311			−0.538479
$362$		0			0
$363$			1.00000i			0.0524864i
$364$		0			0
$365$	−2.38436	+	0.162176i	−0.124803	+	0.00848866i
$366$		0			0
$367$		−	3.53828i		−	0.184697i	−0.995727	−	0.0923484i	$-0.970563\pi$
$367$		−	3.53828i		−	0.184697i	0.995727	−	0.0923484i	$-0.0294374\pi$
$368$		0			0
$369$	−7.50965			−0.390937
$370$		0			0
$371$	2.21412			0.114952
$372$		0			0
$373$		−	14.0779i		−	0.728925i	−0.931218	−	0.364463i	$-0.881253\pi$
$373$		−	14.0779i		−	0.728925i	0.931218	−	0.364463i	$-0.118747\pi$
$374$		0			0
$375$	−10.9491	+	2.26211i	−0.565409	+	0.116815i
$376$		0			0
$377$		−	24.8136i		−	1.27797i
$378$		0			0
$379$	32.0475			1.64617			0.823085	−	0.567918i	$-0.192251\pi$
$379$	32.0475			1.64617			0.823085	+	0.567918i	$0.192251\pi$
$380$		0			0
$381$	17.1239			0.877286
$382$		0			0
$383$			22.3390i			1.14147i	0.821135	+	0.570734i	$0.193341\pi$
$383$			22.3390i			1.14147i	−0.821135	+	0.570734i	$0.806659\pi$
$384$		0			0
$385$	2.23091	−	0.151739i	0.113698	−	0.00773332i
$386$		0			0
$387$		−	7.04690i		−	0.358214i
$388$		0			0
$389$	−3.74503			−0.189881			−0.0949404	−	0.995483i	$-0.530266\pi$
$389$	−3.74503			−0.189881			−0.0949404	+	0.995483i	$0.530266\pi$
$390$		0			0
$391$	9.81836			0.496536
$392$		0			0
$393$		−	20.8886i		−	1.05369i
$394$		0			0
$395$	−0.881750	−	12.9638i	−0.0443656	−	0.652278i
$396$		0			0
$397$		−	32.7965i		−	1.64601i	−0.568036	−	0.823004i	$-0.692297\pi$
$397$		−	32.7965i		−	1.64601i	0.568036	−	0.823004i	$-0.307703\pi$
$398$		0			0
$399$	2.96123			0.148247
$400$		0			0
$401$	−23.5449			−1.17578			−0.587889	−	0.808942i	$-0.700041\pi$
$401$	−23.5449			−1.17578			−0.587889	+	0.808942i	$0.700041\pi$
$402$		0			0
$403$			14.5695i			0.725761i
$404$		0			0
$405$	−0.151739	−	2.23091i	−0.00753996	−	0.110855i
$406$		0			0
$407$		−	6.82236i		−	0.338172i
$408$		0			0
$409$	−17.2005			−0.850509			−0.425254	−	0.905074i	$-0.639815\pi$
$409$	−17.2005			−0.850509			−0.425254	+	0.905074i	$0.639815\pi$
$410$		0			0
$411$	12.9479			0.638673
$412$		0			0
$413$		−	3.12508i		−	0.153775i
$414$		0			0
$415$	−12.8633	+	0.874915i	−0.631434	+	0.0429479i
$416$		0			0
$417$		−	9.33171i		−	0.456976i
$418$		0			0
$419$	−13.8627			−0.677236			−0.338618	−	0.940924i	$-0.609959\pi$
$419$	−13.8627			−0.677236			−0.338618	+	0.940924i	$0.609959\pi$
$420$		0			0
$421$	6.27285			0.305720			0.152860	−	0.988248i	$-0.451152\pi$
$421$	6.27285			0.305720			0.152860	+	0.988248i	$0.451152\pi$
$422$		0			0
$423$			10.0772i			0.489971i
$424$		0			0
$425$	−3.26094	−	23.8608i	−0.158179	−	1.15742i
$426$		0			0
$427$			9.01516i			0.436274i
$428$		0			0
$429$	3.38902			0.163624
$430$		0			0
$431$	19.6629			0.947129			0.473565	−	0.880759i	$-0.342967\pi$
$431$	19.6629			0.947129			0.473565	+	0.880759i	$0.342967\pi$
$432$		0			0
$433$		−	10.8292i		−	0.520420i	−0.965552	−	0.260210i	$-0.916208\pi$
$433$		−	10.8292i		−	0.520420i	0.965552	−	0.260210i	$-0.0837918\pi$
$434$		0			0
$435$	16.3342	−	1.11100i	0.783166	−	0.0532681i
$436$		0			0
$437$			6.03640i			0.288760i
$438$		0			0
$439$	8.12495			0.387783			0.193891	−	0.981023i	$-0.437889\pi$
$439$	8.12495			0.387783			0.193891	+	0.981023i	$0.437889\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			22.8005i			1.08328i	0.840610	+	0.541641i	$0.182197\pi$
$443$			22.8005i			1.08328i	−0.840610	+	0.541641i	$0.817803\pi$
$444$		0			0
$445$	1.28871	+	18.9471i	0.0610909	+	0.898178i
$446$		0			0
$447$			24.2248i			1.14580i
$448$		0			0
$449$	15.8095			0.746095			0.373047	−	0.927812i	$-0.378313\pi$
$449$	15.8095			0.746095			0.373047	+	0.927812i	$0.378313\pi$
$450$		0			0
$451$	7.50965			0.353615
$452$		0			0
$453$		−	10.1828i		−	0.478428i
$454$		0			0
$455$	−0.514246	−	7.56061i	−0.0241082	−	0.354447i
$456$		0			0
$457$			12.6199i			0.590334i	0.955446	+	0.295167i	$0.0953753\pi$
$457$			12.6199i			0.590334i	−0.955446	+	0.295167i	$0.904625\pi$
$458$		0			0
$459$	4.81652			0.224816
$460$		0			0
$461$	−2.77071			−0.129045			−0.0645224	−	0.997916i	$-0.520552\pi$
$461$	−2.77071			−0.129045			−0.0645224	+	0.997916i	$0.520552\pi$
$462$		0			0
$463$		−	34.1613i		−	1.58761i	−0.608172	−	0.793805i	$-0.708097\pi$
$463$		−	34.1613i		−	1.58761i	0.608172	−	0.793805i	$-0.291903\pi$
$464$		0			0
$465$	−9.59079	+	0.652331i	−0.444762	+	0.0302511i
$466$		0			0
$467$		−	19.8327i		−	0.917750i	−0.888501	−	0.458875i	$-0.848253\pi$
$467$		−	19.8327i		−	0.917750i	0.888501	−	0.458875i	$-0.151747\pi$
$468$		0			0
$469$	7.88349			0.364026
$470$		0			0
$471$	10.8219			0.498645
$472$		0			0
$473$			7.04690i			0.324017i
$474$		0			0
$475$	14.6698	−	2.00485i	0.673097	−	0.0919888i
$476$		0			0
$477$			2.21412i			0.101378i
$478$		0			0
$479$	10.4343			0.476757			0.238379	−	0.971172i	$-0.423384\pi$
$479$	10.4343			0.476757			0.238379	+	0.971172i	$0.423384\pi$
$480$		0			0
$481$	−23.1211			−1.05423
$482$		0			0
$483$			2.03848i			0.0927538i
$484$		0			0
$485$	−5.65115	+	0.384371i	−0.256605	+	0.0174534i
$486$		0			0
$487$			10.8586i			0.492051i	0.969263	+	0.246026i	$0.0791247\pi$
$487$			10.8586i			0.492051i	−0.969263	+	0.246026i	$0.920875\pi$
$488$		0			0
$489$	12.9680			0.586432
$490$		0			0
$491$	28.5156			1.28689			0.643446	−	0.765491i	$-0.277504\pi$
$491$	28.5156			1.28689			0.643446	+	0.765491i	$0.277504\pi$
$492$		0			0
$493$			35.2654i			1.58827i
$494$		0			0
$495$	0.151739	+	2.23091i	0.00682015	+	0.100272i
$496$		0			0
$497$		−	12.3116i		−	0.552251i
$498$		0			0
$499$	18.8229			0.842628			0.421314	−	0.906915i	$-0.361569\pi$
$499$	18.8229			0.842628			0.421314	+	0.906915i	$0.361569\pi$
$500$		0			0
$501$	9.42488			0.421072
$502$		0			0
$503$			16.8546i			0.751508i	0.926719	+	0.375754i	$0.122616\pi$
$503$			16.8546i			0.751508i	−0.926719	+	0.375754i	$0.877384\pi$
$504$		0			0
$505$	2.85731	+	42.0092i	0.127149	+	1.86938i
$506$		0			0
$507$			1.51453i			0.0672627i
$508$		0			0
$509$	−31.1638			−1.38131			−0.690655	−	0.723184i	$-0.742678\pi$
$509$	−31.1638			−1.38131			−0.690655	+	0.723184i	$0.742678\pi$
$510$		0			0
$511$	1.06878			0.0472801
$512$		0			0
$513$			2.96123i			0.130742i
$514$		0			0
$515$	0.428125	−	0.0291195i	0.0188654	−	0.00128316i
$516$		0			0
$517$		−	10.0772i		−	0.443195i
$518$		0			0
$519$	12.5524			0.550990
$520$		0			0
$521$	−13.9556			−0.611405			−0.305702	−	0.952127i	$-0.598891\pi$
$521$	−13.9556			−0.611405			−0.305702	+	0.952127i	$0.598891\pi$
$522$		0			0
$523$			12.9192i			0.564916i	0.959280	+	0.282458i	$0.0911498\pi$
$523$			12.9192i			0.564916i	−0.959280	+	0.282458i	$0.908850\pi$
$524$		0			0
$525$	4.95395	−	0.677032i	0.216208	−	0.0295481i
$526$		0			0
$527$		−	20.7064i		−	0.901985i
$528$		0			0
$529$	18.8446			0.819331
$530$		0			0
$531$	3.12508			0.135617
$532$		0			0
$533$		−	25.4504i		−	1.10238i
$534$		0			0
$535$	−29.0869	+	1.97839i	−1.25754	+	0.0855331i
$536$		0			0
$537$		−	4.77271i		−	0.205958i
$538$		0			0
$539$	−1.00000			−0.0430730
$540$		0			0
$541$	−7.41829			−0.318937			−0.159469	−	0.987203i	$-0.550978\pi$
$541$	−7.41829			−0.318937			−0.159469	+	0.987203i	$0.550978\pi$
$542$		0			0
$543$		−	7.11292i		−	0.305245i
$544$		0			0
$545$	−2.08567	−	30.6642i	−0.0893401	−	1.31351i
$546$		0			0
$547$		−	17.9007i		−	0.765379i	−0.923877	−	0.382690i	$-0.874998\pi$
$547$		−	17.9007i		−	0.765379i	0.923877	−	0.382690i	$-0.125002\pi$
$548$		0			0
$549$	−9.01516			−0.384758
$550$		0			0
$551$	−21.6815			−0.923661
$552$		0			0
$553$			5.81097i			0.247108i
$554$		0			0
$555$	−1.03522	−	15.2201i	−0.0439425	−	0.646057i
$556$		0			0
$557$		−	6.19444i		−	0.262467i	−0.991351	−	0.131233i	$-0.958106\pi$
$557$		−	6.19444i		−	0.262467i	0.991351	−	0.131233i	$-0.0418937\pi$
$558$		0			0
$559$	23.8821			1.01011
$560$		0			0
$561$	−4.81652			−0.203354
$562$		0			0
$563$		−	20.7997i		−	0.876603i	−0.898828	−	0.438301i	$-0.855580\pi$
$563$		−	20.7997i		−	0.876603i	0.898828	−	0.438301i	$-0.144420\pi$
$564$		0			0
$565$	27.5482	−	1.87373i	1.15896	−	0.0788283i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−3.40101			−0.142578			−0.0712889	−	0.997456i	$-0.522711\pi$
$569$	−3.40101			−0.142578			−0.0712889	+	0.997456i	$0.522711\pi$
$570$		0			0
$571$	21.7365			0.909644			0.454822	−	0.890582i	$-0.349703\pi$
$571$	21.7365			0.909644			0.454822	+	0.890582i	$0.349703\pi$
$572$		0			0
$573$		−	14.6340i		−	0.611343i
$574$		0			0
$575$	1.38011	+	10.0985i	0.0575547	+	0.421137i
$576$		0			0
$577$			4.37074i			0.181956i	0.995853	+	0.0909781i	$0.0289993\pi$
$577$			4.37074i			0.181956i	−0.995853	+	0.0909781i	$0.971001\pi$
$578$		0			0
$579$	13.6330			0.566570
$580$		0			0
$581$	5.76593			0.239211
$582$		0			0
$583$		−	2.21412i		−	0.0916997i
$584$		0			0
$585$	7.56061	−	0.514246i	0.312593	−	0.0212615i
$586$		0			0
$587$			25.8142i			1.06547i	0.846283	+	0.532733i	$0.178835\pi$
$587$			25.8142i			1.06547i	−0.846283	+	0.532733i	$0.821165\pi$
$588$		0			0
$589$	12.7305			0.524550
$590$		0			0
$591$	0.330782			0.0136066
$592$		0			0
$593$		−	33.8908i		−	1.39173i	−0.718174	−	0.695864i	$-0.755022\pi$
$593$		−	33.8908i		−	1.39173i	0.718174	−	0.695864i	$-0.244978\pi$
$594$		0			0
$595$	0.730852	+	10.7452i	0.0299620	+	0.440512i
$596$		0			0
$597$			20.1131i			0.823173i
$598$		0			0
$599$	33.2287			1.35769			0.678843	−	0.734283i	$-0.262481\pi$
$599$	33.2287			1.35769			0.678843	+	0.734283i	$0.262481\pi$
$600$		0			0
$601$	16.3536			0.667076			0.333538	−	0.942737i	$-0.391758\pi$
$601$	16.3536			0.667076			0.333538	+	0.942737i	$0.391758\pi$
$602$		0			0
$603$			7.88349i			0.321040i
$604$		0			0
$605$	−0.151739	−	2.23091i	−0.00616906	−	0.0906995i
$606$		0			0
$607$			31.9496i			1.29680i	0.761301	+	0.648398i	$0.224561\pi$
$607$			31.9496i			1.29680i	−0.761301	+	0.648398i	$0.775439\pi$
$608$		0			0
$609$	−7.32176			−0.296693
$610$		0			0
$611$	−34.1519			−1.38164
$612$		0			0
$613$			1.93598i			0.0781935i	0.999235	+	0.0390968i	$0.0124481\pi$
$613$			1.93598i			0.0781935i	−0.999235	+	0.0390968i	$0.987552\pi$
$614$		0			0
$615$	16.7534	−	1.13950i	0.675561	−	0.0459492i
$616$		0			0
$617$			16.8681i			0.679086i	0.940591	+	0.339543i	$0.110272\pi$
$617$			16.8681i			0.679086i	−0.940591	+	0.339543i	$0.889728\pi$
$618$		0			0
$619$	−10.3947			−0.417798			−0.208899	−	0.977937i	$-0.566988\pi$
$619$	−10.3947			−0.417798			−0.208899	+	0.977937i	$0.566988\pi$
$620$		0			0
$621$	−2.03848			−0.0818012
$622$		0			0
$623$		−	8.49297i		−	0.340264i
$624$		0			0
$625$	24.0833	−	6.70797i	0.963330	−	0.268319i
$626$		0			0
$627$		−	2.96123i		−	0.118260i
$628$		0			0
$629$	32.8600			1.31021
$630$		0			0
$631$	−25.0492			−0.997192			−0.498596	−	0.866835i	$-0.666151\pi$
$631$	−25.0492			−0.997192			−0.498596	+	0.866835i	$0.666151\pi$
$632$		0			0
$633$		−	5.52882i		−	0.219751i
$634$		0			0
$635$	−38.2020	+	2.59836i	−1.51600	+	0.103113i
$636$		0			0
$637$			3.38902i			0.134278i
$638$		0			0
$639$	12.3116			0.487040
$640$		0			0
$641$	27.0840			1.06975			0.534877	−	0.844930i	$-0.320358\pi$
$641$	27.0840			1.06975			0.534877	+	0.844930i	$0.320358\pi$
$642$		0			0
$643$			46.7234i			1.84259i	0.388861	+	0.921296i	$0.372869\pi$
$643$			46.7234i			1.84259i	−0.388861	+	0.921296i	$0.627131\pi$
$644$		0			0
$645$	1.06929	+	15.7210i	0.0421032	+	0.619015i
$646$		0			0
$647$		−	8.15802i		−	0.320725i	−0.987058	−	0.160362i	$-0.948734\pi$
$647$		−	8.15802i		−	0.320725i	0.987058	−	0.160362i	$-0.0512663\pi$
$648$		0			0
$649$	−3.12508			−0.122670
$650$		0			0
$651$	4.29904			0.168493
$652$		0			0
$653$			45.7326i			1.78965i	0.446412	+	0.894827i	$0.352701\pi$
$653$			45.7326i			1.78965i	−0.446412	+	0.894827i	$0.647299\pi$
$654$		0			0
$655$	3.16961	+	46.6007i	0.123847	+	1.82084i
$656$		0			0
$657$			1.06878i			0.0416972i
$658$		0			0
$659$	−14.9934			−0.584061			−0.292030	−	0.956409i	$-0.594331\pi$
$659$	−14.9934			−0.584061			−0.292030	+	0.956409i	$0.594331\pi$
$660$		0			0
$661$	−8.73743			−0.339847			−0.169923	−	0.985457i	$-0.554352\pi$
$661$	−8.73743			−0.339847			−0.169923	+	0.985457i	$0.554352\pi$
$662$		0			0
$663$			16.3233i			0.633944i
$664$		0			0
$665$	−6.60626	+	0.449334i	−0.256180	+	0.0174244i
$666$		0			0
$667$		−	14.9252i		−	0.577907i
$668$		0			0
$669$	13.1597			0.508785
$670$		0			0
$671$	9.01516			0.348026
$672$		0			0
$673$			13.2701i			0.511526i	0.966740	+	0.255763i	$0.0823266\pi$
$673$			13.2701i			0.511526i	−0.966740	+	0.255763i	$0.917673\pi$
$674$		0			0
$675$	0.677032	+	4.95395i	0.0260590	+	0.190678i
$676$		0			0
$677$			8.89896i			0.342015i	0.985270	+	0.171007i	$0.0547022\pi$
$677$			8.89896i			0.342015i	−0.985270	+	0.171007i	$0.945298\pi$
$678$		0			0
$679$	2.53311			0.0972118
$680$		0			0
$681$	−12.8202			−0.491272
$682$		0			0
$683$		−	15.4081i		−	0.589573i	−0.955563	−	0.294787i	$-0.904751\pi$
$683$		−	15.4081i		−	0.589573i	0.955563	−	0.294787i	$-0.0952486\pi$
$684$		0			0
$685$	−28.8856	+	1.96470i	−1.10366	+	0.0750673i
$686$		0			0
$687$			28.0550i			1.07037i
$688$		0			0
$689$	−7.50372			−0.285869
$690$		0			0
$691$	−39.1277			−1.48849			−0.744244	−	0.667908i	$-0.767190\pi$
$691$	−39.1277			−1.48849			−0.744244	+	0.667908i	$0.767190\pi$
$692$		0			0
$693$		−	1.00000i		−	0.0379869i
$694$		0			0
$695$	1.41598	+	20.8182i	0.0537113	+	0.789681i
$696$		0			0
$697$			36.1704i			1.37005i
$698$		0			0
$699$	26.1136			0.987707
$700$		0			0
$701$	−22.4261			−0.847023			−0.423511	−	0.905891i	$-0.639203\pi$
$701$	−22.4261			−0.847023			−0.423511	+	0.905891i	$0.639203\pi$
$702$		0			0
$703$			20.2026i			0.761955i
$704$		0			0
$705$	−1.52910	−	22.4814i	−0.0575893	−	0.846698i
$706$		0			0
$707$		−	18.8305i		−	0.708193i
$708$		0			0
$709$	−23.7547			−0.892127			−0.446063	−	0.895001i	$-0.647174\pi$
$709$	−23.7547			−0.892127			−0.446063	+	0.895001i	$0.647174\pi$
$710$		0			0
$711$	−5.81097			−0.217928
$712$		0			0
$713$			8.76349i			0.328195i
$714$		0			0
$715$	−7.56061	+	0.514246i	−0.282751	+	0.0192317i
$716$		0			0
$717$		−	20.3797i		−	0.761093i
$718$		0			0
$719$	−12.8015			−0.477417			−0.238709	−	0.971091i	$-0.576724\pi$
$719$	−12.8015			−0.477417			−0.238709	+	0.971091i	$0.576724\pi$
$720$		0			0
$721$	−0.191906			−0.00714694
$722$		0			0
$723$		−	4.99845i		−	0.185894i
$724$		0			0
$725$	−36.2717	+	4.95707i	−1.34710	+	0.184101i
$726$		0			0
$727$		−	50.6012i		−	1.87670i	−0.345692	−	0.938348i	$-0.612356\pi$
$727$		−	50.6012i		−	1.87670i	0.345692	−	0.938348i	$-0.387644\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−33.9415			−1.25537
$732$		0			0
$733$			41.1552i			1.52010i	0.649863	+	0.760052i	$0.274826\pi$
$733$			41.1552i			1.52010i	−0.649863	+	0.760052i	$0.725174\pi$
$734$		0			0
$735$	−2.23091	+	0.151739i	−0.0822885	+	0.00559697i
$736$		0			0
$737$		−	7.88349i		−	0.290392i
$738$		0			0
$739$	50.0765			1.84210			0.921048	−	0.389450i	$-0.127335\pi$
$739$	50.0765			1.84210			0.921048	+	0.389450i	$0.127335\pi$
$740$		0			0
$741$	−10.0357			−0.368670
$742$		0			0
$743$			28.8228i			1.05741i	0.848806	+	0.528704i	$0.177322\pi$
$743$			28.8228i			1.05741i	−0.848806	+	0.528704i	$0.822678\pi$
$744$		0			0
$745$	−3.67585	−	54.0435i	−0.134673	−	1.98000i
$746$		0			0
$747$			5.76593i			0.210964i
$748$		0			0
$749$	13.0381			0.476402
$750$		0			0
$751$	−12.1268			−0.442513			−0.221257	−	0.975216i	$-0.571016\pi$
$751$	−12.1268			−0.442513			−0.221257	+	0.975216i	$0.571016\pi$
$752$		0			0
$753$			2.67141i			0.0973517i
$754$		0			0
$755$	1.54512	+	22.7169i	0.0562327	+	0.826751i
$756$		0			0
$757$		−	8.63104i		−	0.313700i	−0.987622	−	0.156850i	$-0.949866\pi$
$757$		−	8.63104i		−	0.313700i	0.987622	−	0.156850i	$-0.0501340\pi$
$758$		0			0
$759$	2.03848			0.0739920
$760$		0			0
$761$	5.78831			0.209826			0.104913	−	0.994481i	$-0.466544\pi$
$761$	5.78831			0.209826			0.104913	+	0.994481i	$0.466544\pi$
$762$		0			0
$763$			13.7451i			0.497607i
$764$		0			0
$765$	−10.7452	+	0.730852i	−0.388495	+	0.0264240i
$766$		0			0
$767$			10.5910i			0.382418i
$768$		0			0
$769$	−43.9440			−1.58466			−0.792331	−	0.610091i	$-0.791133\pi$
$769$	−43.9440			−1.58466			−0.792331	+	0.610091i	$0.791133\pi$
$770$		0			0
$771$	−28.8879			−1.04037
$772$		0			0
$773$		−	8.30909i		−	0.298857i	−0.988773	−	0.149429i	$-0.952257\pi$
$773$		−	8.30909i		−	0.298857i	0.988773	−	0.149429i	$-0.0477434\pi$
$774$		0			0
$775$	21.2972	−	2.91059i	0.765019	−	0.104551i
$776$		0			0
$777$			6.82236i			0.244751i
$778$		0			0
$779$	−22.2378			−0.796752
$780$		0			0
$781$	−12.3116			−0.440544
$782$		0			0
$783$		−	7.32176i		−	0.261658i
$784$		0			0
$785$	−24.1426	+	1.64209i	−0.861687	+	0.0586089i
$786$		0			0
$787$			13.1652i			0.469288i	0.972081	+	0.234644i	$0.0753924\pi$
$787$			13.1652i			0.469288i	−0.972081	+	0.234644i	$0.924608\pi$
$788$		0			0
$789$	16.2179			0.577372
$790$		0			0
$791$	−12.3484			−0.439058
$792$		0			0
$793$		−	30.5526i		−	1.08495i
$794$		0			0
$795$	−0.335968	−	4.93952i	−0.0119156	−	0.175187i
$796$		0			0
$797$			1.24357i			0.0440495i	0.999757	+	0.0220248i	$0.00701127\pi$
$797$			1.24357i			0.0440495i	−0.999757	+	0.0220248i	$0.992989\pi$
$798$		0			0
$799$	48.5371			1.71712
$800$		0			0
$801$	8.49297			0.300084
$802$		0			0
$803$		−	1.06878i		−	0.0377165i
$804$		0			0
$805$	−0.309316	−	4.54766i	−0.0109019	−	0.160284i
$806$		0			0
$807$			15.1952i			0.534898i
$808$		0			0
$809$	−40.9417			−1.43943			−0.719716	−	0.694268i	$-0.755728\pi$
$809$	−40.9417			−1.43943			−0.719716	+	0.694268i	$0.755728\pi$
$810$		0			0
$811$	−43.9307			−1.54262			−0.771308	−	0.636463i	$-0.780397\pi$
$811$	−43.9307			−1.54262			−0.771308	+	0.636463i	$0.780397\pi$
$812$		0			0
$813$			5.71049i			0.200276i
$814$		0			0
$815$	−28.9304	+	1.96774i	−1.01339	+	0.0689270i
$816$		0			0
$817$		−	20.8675i		−	0.730062i
$818$		0			0
$819$	−3.38902			−0.118422
$820$		0			0
$821$	−39.7766			−1.38821			−0.694106	−	0.719872i	$-0.744200\pi$
$821$	−39.7766			−1.38821			−0.694106	+	0.719872i	$0.744200\pi$
$822$		0			0
$823$		−	38.4688i		−	1.34094i	−0.741938	−	0.670468i	$-0.766093\pi$
$823$		−	38.4688i		−	1.34094i	0.741938	−	0.670468i	$-0.233907\pi$
$824$		0			0
$825$	−0.677032	−	4.95395i	−0.0235712	−	0.172474i
$826$		0			0
$827$			46.7428i			1.62541i	0.582678	+	0.812703i	$0.302005\pi$
$827$			46.7428i			1.62541i	−0.582678	+	0.812703i	$0.697995\pi$
$828$		0			0
$829$	−47.1386			−1.63719			−0.818595	−	0.574371i	$-0.805247\pi$
$829$	−47.1386			−1.63719			−0.818595	+	0.574371i	$0.805247\pi$
$830$		0			0
$831$	−12.6726			−0.439606
$832$		0			0
$833$		−	4.81652i		−	0.166882i
$834$		0			0
$835$	−21.0261	+	1.43012i	−0.727638	+	0.0494913i
$836$		0			0
$837$			4.29904i			0.148597i
$838$		0			0
$839$	37.9005			1.30847			0.654235	−	0.756291i	$-0.272991\pi$
$839$	37.9005			1.30847			0.654235	+	0.756291i	$0.272991\pi$
$840$		0			0
$841$	24.6082			0.848559
$842$		0			0
$843$			4.97868i			0.171475i
$844$		0			0
$845$	−0.229813	−	3.37879i	−0.00790581	−	0.116234i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	0.549902			0.0188726
$850$		0			0
$851$	−13.9072			−0.476733
$852$		0			0
$853$			0.240696i			0.00824129i	0.999992	+	0.00412064i	$0.00131165\pi$
$853$			0.240696i			0.00824129i	−0.999992	+	0.00412064i	$0.998688\pi$
$854$		0			0
$855$	−0.449334	−	6.60626i	−0.0153669	−	0.225929i
$856$		0			0
$857$			22.8751i			0.781399i	0.920518	+	0.390699i	$0.127767\pi$
$857$			22.8751i			0.781399i	−0.920518	+	0.390699i	$0.872233\pi$
$858$		0			0
$859$	−28.8705			−0.985047			−0.492524	−	0.870299i	$-0.663925\pi$
$859$	−28.8705			−0.985047			−0.492524	+	0.870299i	$0.663925\pi$
$860$		0			0
$861$	−7.50965			−0.255928
$862$		0			0
$863$			32.7619i			1.11523i	0.830100	+	0.557615i	$0.188283\pi$
$863$			32.7619i			1.11523i	−0.830100	+	0.557615i	$0.811717\pi$
$864$		0			0
$865$	−28.0033	+	1.90469i	−0.952142	+	0.0647613i
$866$		0			0
$867$		−	6.19885i		−	0.210524i
$868$		0			0
$869$	5.81097			0.197124
$870$		0			0
$871$	−26.7173			−0.905281
$872$		0			0
$873$			2.53311i			0.0857327i
$874$		0			0
$875$	−10.9491	+	2.26211i	−0.370147	+	0.0764731i
$876$		0			0
$877$			0.0942057i			0.00318110i	0.999999	+	0.00159055i	$0.000506288\pi$
$877$			0.0942057i			0.00318110i	−0.999999	+	0.00159055i	$0.999494\pi$
$878$		0			0
$879$	19.5587			0.659700
$880$		0			0
$881$	28.1290			0.947690			0.473845	−	0.880608i	$-0.342866\pi$
$881$	28.1290			0.947690			0.473845	+	0.880608i	$0.342866\pi$
$882$		0			0
$883$		−	3.71466i		−	0.125008i	−0.998045	−	0.0625042i	$-0.980091\pi$
$883$		−	3.71466i		−	0.125008i	0.998045	−	0.0625042i	$-0.0199087\pi$
$884$		0			0
$885$	−6.97179	+	0.474196i	−0.234354	+	0.0159399i
$886$		0			0
$887$		−	13.5731i		−	0.455739i	−0.973692	−	0.227870i	$-0.926824\pi$
$887$		−	13.5731i		−	0.455739i	0.973692	−	0.227870i	$-0.0731760\pi$
$888$		0			0
$889$	17.1239			0.574318
$890$		0			0
$891$	1.00000			0.0335013
$892$		0			0
$893$			29.8410i			0.998590i
$894$		0			0
$895$	0.724205	+	10.6475i	0.0242075	+	0.355907i
$896$		0			0
$897$		−	6.90844i		−	0.230666i
$898$		0			0
$899$	−31.4766			−1.04980
$900$		0			0
$901$	10.6644			0.355282
$902$		0			0
$903$		−	7.04690i		−	0.234506i
$904$		0			0
$905$	1.07931	+	15.8683i	0.0358773	+	0.527480i
$906$		0			0
$907$		−	16.9317i		−	0.562207i	−0.959677	−	0.281104i	$-0.909300\pi$
$907$		−	16.9317i		−	0.562207i	0.959677	−	0.281104i	$-0.0907005\pi$
$908$		0			0
$909$	18.8305			0.624568
$910$		0			0
$911$	−2.22474			−0.0737088			−0.0368544	−	0.999321i	$-0.511734\pi$
$911$	−2.22474			−0.0737088			−0.0368544	+	0.999321i	$0.511734\pi$
$912$		0			0
$913$		−	5.76593i		−	0.190825i
$914$		0			0
$915$	20.1120	−	1.36795i	0.664884	−	0.0452230i
$916$		0			0
$917$		−	20.8886i		−	0.689803i
$918$		0			0
$919$	13.9917			0.461544			0.230772	−	0.973008i	$-0.425875\pi$
$919$	13.9917			0.461544			0.230772	+	0.973008i	$0.425875\pi$
$920$		0			0
$921$	10.7762			0.355088
$922$		0			0
$923$			41.7243i			1.37337i
$924$		0			0
$925$	4.61895	+	33.7976i	0.151870	+	1.11126i
$926$		0			0
$927$		−	0.191906i		−	0.00630301i
$928$		0			0
$929$	11.0149			0.361387			0.180694	−	0.983539i	$-0.442166\pi$
$929$	11.0149			0.361387			0.180694	+	0.983539i	$0.442166\pi$
$930$		0			0
$931$	2.96123			0.0970505
$932$		0			0
$933$		−	18.3169i		−	0.599670i
$934$		0			0
$935$	10.7452	−	0.730852i	0.351407	−	0.0239014i
$936$		0			0
$937$		−	16.2565i		−	0.531078i	−0.964100	−	0.265539i	$-0.914450\pi$
$937$		−	16.2565i		−	0.531078i	0.964100	−	0.265539i	$-0.0855499\pi$
$938$		0			0
$939$	−0.863241			−0.0281708
$940$		0			0
$941$	20.5336			0.669376			0.334688	−	0.942329i	$-0.391369\pi$
$941$	20.5336			0.669376			0.334688	+	0.942329i	$0.391369\pi$
$942$		0			0
$943$		−	15.3082i		−	0.498505i
$944$		0			0
$945$	−0.151739	−	2.23091i	−0.00493606	−	0.0725716i
$946$		0			0
$947$			30.7115i			0.997989i	0.866605	+	0.498995i	$0.166297\pi$
$947$			30.7115i			0.997989i	−0.866605	+	0.498995i	$0.833703\pi$
$948$		0			0
$949$	−3.62213			−0.117579
$950$		0			0
$951$	−2.76184			−0.0895589
$952$		0			0
$953$		−	5.72512i		−	0.185455i	−0.995692	−	0.0927274i	$-0.970441\pi$
$953$		−	5.72512i		−	0.185455i	0.995692	−	0.0927274i	$-0.0295585\pi$
$954$		0			0
$955$	2.22054	+	32.6471i	0.0718550	+	1.05644i
$956$		0			0
$957$			7.32176i			0.236679i
$958$		0			0
$959$	12.9479			0.418110
$960$		0			0
$961$	−12.5182			−0.403814
$962$		0			0
$963$			13.0381i			0.420147i
$964$		0			0
$965$	−30.4141	+	2.06866i	−0.979065	+	0.0665925i
$966$		0			0
$967$			44.1493i			1.41975i	0.704329	+	0.709873i	$0.251248\pi$
$967$			44.1493i			1.41975i	−0.704329	+	0.709873i	$0.748752\pi$
$968$		0			0
$969$	14.2628			0.458188
$970$		0			0
$971$	−36.4494			−1.16972			−0.584858	−	0.811136i	$-0.698850\pi$
$971$	−36.4494			−1.16972			−0.584858	+	0.811136i	$0.698850\pi$
$972$		0			0
$973$		−	9.33171i		−	0.299161i
$974$		0			0
$975$	−16.7890	+	2.29448i	−0.537680	+	0.0734820i
$976$		0			0
$977$			25.1541i			0.804751i	0.915475	+	0.402376i	$0.131815\pi$
$977$			25.1541i			0.804751i	−0.915475	+	0.402376i	$0.868185\pi$
$978$		0			0
$979$	−8.49297			−0.271437
$980$		0			0
$981$	−13.7451			−0.438848
$982$		0			0
$983$			28.4471i			0.907321i	0.891175	+	0.453660i	$0.149882\pi$
$983$			28.4471i			0.907321i	−0.891175	+	0.453660i	$0.850118\pi$
$984$		0			0
$985$	−0.737946	+	0.0501925i	−0.0235129	+	0.00159926i
$986$		0			0
$987$			10.0772i			0.320761i
$988$		0			0
$989$	14.3649			0.456778
$990$		0			0
$991$	23.9184			0.759794			0.379897	−	0.925029i	$-0.375959\pi$
$991$	23.9184			0.759794			0.379897	+	0.925029i	$0.375959\pi$
$992$		0			0
$993$			32.7451i			1.03913i
$994$		0			0
$995$	−3.05193	−	44.8705i	−0.0967527	−	1.42249i
$996$		0			0
$997$			47.9664i			1.51911i	0.650443	+	0.759555i	$0.274583\pi$
$997$			47.9664i			1.51911i	−0.650443	+	0.759555i	$0.725417\pi$
$998$		0			0
$999$	−6.82236			−0.215850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.h.e.1849.11	yes	14
5.4	even	2	inner	4620.2.h.e.1849.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.h.e.1849.4	✓	14	5.4	even	2	inner
4620.2.h.e.1849.11	yes	14	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.151739 - 2.23091i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} -3.38902i q^{13} +(2.23091 - 0.151739i) q^{15} +4.81652i q^{17} -2.96123 q^{19} -1.00000 q^{21} -2.03848i q^{23} +(-4.95395 + 0.677032i) q^{25} -1.00000i q^{27} +7.32176 q^{29} -4.29904 q^{31} +1.00000i q^{33} +(2.23091 - 0.151739i) q^{35} -6.82236i q^{37} +3.38902 q^{39} +7.50965 q^{41} +7.04690i q^{43} +(0.151739 + 2.23091i) q^{45} -10.0772i q^{47} -1.00000 q^{49} -4.81652 q^{51} -2.21412i q^{53} +(-0.151739 - 2.23091i) q^{55} -2.96123i q^{57} -3.12508 q^{59} +9.01516 q^{61} -1.00000i q^{63} +(-7.56061 + 0.514246i) q^{65} -7.88349i q^{67} +2.03848 q^{69} -12.3116 q^{71} -1.06878i q^{73} +(-0.677032 - 4.95395i) q^{75} +1.00000i q^{77} +5.81097 q^{79} +1.00000 q^{81} -5.76593i q^{83} +(10.7452 - 0.730852i) q^{85} +7.32176i q^{87} -8.49297 q^{89} +3.38902 q^{91} -4.29904i q^{93} +(0.449334 + 6.60626i) q^{95} -2.53311i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{5} - 14 q^{9} + 14 q^{11} + 6 q^{19} - 14 q^{21} - 8 q^{25} + 38 q^{29} + 4 q^{31} + 8 q^{39} - 12 q^{41} + 4 q^{45} - 14 q^{49} + 14 q^{51} - 4 q^{55} + 18 q^{59} - 54 q^{61} - 2 q^{65} + 26 q^{69}+ \cdots - 14 q^{99}+O(q^{100}) \) 14 * q - 4 * q^5 - 14 * q^9 + 14 * q^11 + 6 * q^19 - 14 * q^21 - 8 * q^25 + 38 * q^29 + 4 * q^31 + 8 * q^39 - 12 * q^41 + 4 * q^45 - 14 * q^49 + 14 * q^51 - 4 * q^55 + 18 * q^59 - 54 * q^61 - 2 * q^65 + 26 * q^69 - 24 * q^71 + 48 * q^79 + 14 * q^81 + 12 * q^85 + 46 * q^89 + 8 * q^91 - 72 * q^95 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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