LMFDB - Embedded newform 4020.2.g.b.1609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.g (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.2
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.07549 - 0.832066i) q^{5} -0.944795i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-2.07549 - 0.832066i) q^{5} -0.944795i q^{7} -1.00000 q^{9} -5.62597 q^{11} +4.06787i q^{13} +(-0.832066 + 2.07549i) q^{15} +2.26053i q^{17} -2.66869 q^{19} -0.944795 q^{21} -4.72171i q^{23} +(3.61533 + 3.45389i) q^{25} +1.00000i q^{27} -3.13766 q^{29} +3.52550 q^{31} +5.62597i q^{33} +(-0.786131 + 1.96091i) q^{35} +7.10768i q^{37} +4.06787 q^{39} +2.05104 q^{41} -1.23412i q^{43} +(2.07549 + 0.832066i) q^{45} -10.2135i q^{47} +6.10736 q^{49} +2.26053 q^{51} +1.98447i q^{53} +(11.6767 + 4.68118i) q^{55} +2.66869i q^{57} +3.82526 q^{59} -4.66924 q^{61} +0.944795i q^{63} +(3.38473 - 8.44283i) q^{65} +1.00000i q^{67} -4.72171 q^{69} +10.9005 q^{71} +3.65326i q^{73} +(3.45389 - 3.61533i) q^{75} +5.31539i q^{77} -4.89735 q^{79} +1.00000 q^{81} +0.492793i q^{83} +(1.88091 - 4.69171i) q^{85} +3.13766i q^{87} +5.50725 q^{89} +3.84330 q^{91} -3.52550i q^{93} +(5.53884 + 2.22053i) q^{95} -8.86639i q^{97} +5.62597 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 - 24 * q^9	$ 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) $ 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	−2.07549	−	0.832066i	−0.928188	−	0.372111i
$5$	−2.07549	−	0.832066i	−0.928188	−	0.372111i
$6$		0			0
$7$		−	0.944795i		−	0.357099i	−0.983931	−	0.178549i	$-0.942860\pi$
$7$		−	0.944795i		−	0.357099i	0.983931	−	0.178549i	$-0.0571404\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.62597			−1.69630			−0.848148	−	0.529760i	$-0.822282\pi$
$11$	−5.62597			−1.69630			−0.848148	+	0.529760i	$0.822282\pi$
$12$		0			0
$13$			4.06787i			1.12822i	0.825698	+	0.564112i	$0.190781\pi$
$13$			4.06787i			1.12822i	−0.825698	+	0.564112i	$0.809219\pi$
$14$		0			0
$15$	−0.832066	+	2.07549i	−0.214838	+	0.535890i
$16$		0			0
$17$			2.26053i			0.548259i	0.961693	+	0.274129i	$0.0883896\pi$
$17$			2.26053i			0.548259i	−0.961693	+	0.274129i	$0.911610\pi$
$18$		0			0
$19$	−2.66869			−0.612239			−0.306120	−	0.951993i	$-0.599031\pi$
$19$	−2.66869			−0.612239			−0.306120	+	0.951993i	$0.599031\pi$
$20$		0			0
$21$	−0.944795			−0.206171
$22$		0			0
$23$		−	4.72171i		−	0.984545i	−0.870441	−	0.492273i	$-0.836166\pi$
$23$		−	4.72171i		−	0.984545i	0.870441	−	0.492273i	$-0.163834\pi$
$24$		0			0
$25$	3.61533	+	3.45389i	0.723067	+	0.690778i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−3.13766			−0.582649			−0.291325	−	0.956624i	$-0.594096\pi$
$29$	−3.13766			−0.582649			−0.291325	+	0.956624i	$0.594096\pi$
$30$		0			0
$31$	3.52550			0.633198			0.316599	−	0.948560i	$-0.397459\pi$
$31$	3.52550			0.633198			0.316599	+	0.948560i	$0.397459\pi$
$32$		0			0
$33$			5.62597i			0.979356i
$34$		0			0
$35$	−0.786131	+	1.96091i	−0.132880	+	0.331455i
$36$		0			0
$37$			7.10768i			1.16849i	0.811576	+	0.584247i	$0.198610\pi$
$37$			7.10768i			1.16849i	−0.811576	+	0.584247i	$0.801390\pi$
$38$		0			0
$39$	4.06787			0.651380
$40$		0			0
$41$	2.05104			0.320319			0.160159	−	0.987091i	$-0.448799\pi$
$41$	2.05104			0.320319			0.160159	+	0.987091i	$0.448799\pi$
$42$		0			0
$43$		−	1.23412i		−	0.188202i	−0.995563	−	0.0941009i	$-0.970002\pi$
$43$		−	1.23412i		−	0.188202i	0.995563	−	0.0941009i	$-0.0299976\pi$
$44$		0			0
$45$	2.07549	+	0.832066i	0.309396	+	0.124037i
$46$		0			0
$47$		−	10.2135i		−	1.48979i	−0.667182	−	0.744895i	$-0.732500\pi$
$47$		−	10.2135i		−	1.48979i	0.667182	−	0.744895i	$-0.267500\pi$
$48$		0			0
$49$	6.10736			0.872480
$50$		0			0
$51$	2.26053			0.316537
$52$		0			0
$53$			1.98447i			0.272588i	0.990668	+	0.136294i	$0.0435192\pi$
$53$			1.98447i			0.272588i	−0.990668	+	0.136294i	$0.956481\pi$
$54$		0			0
$55$	11.6767	+	4.68118i	1.57448	+	0.631210i
$56$		0			0
$57$			2.66869i			0.353477i
$58$		0			0
$59$	3.82526			0.498006			0.249003	−	0.968503i	$-0.419897\pi$
$59$	3.82526			0.498006			0.249003	+	0.968503i	$0.419897\pi$
$60$		0			0
$61$	−4.66924			−0.597835			−0.298918	−	0.954279i	$-0.596626\pi$
$61$	−4.66924			−0.597835			−0.298918	+	0.954279i	$0.596626\pi$
$62$		0			0
$63$			0.944795i			0.119033i
$64$		0			0
$65$	3.38473	−	8.44283i	0.419825	−	1.04720i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	−4.72171			−0.568427
$70$		0			0
$71$	10.9005			1.29365			0.646825	−	0.762638i	$-0.276096\pi$
$71$	10.9005			1.29365			0.646825	+	0.762638i	$0.276096\pi$
$72$		0			0
$73$			3.65326i			0.427582i	0.976879	+	0.213791i	$0.0685811\pi$
$73$			3.65326i			0.427582i	−0.976879	+	0.213791i	$0.931419\pi$
$74$		0			0
$75$	3.45389	−	3.61533i	0.398821	−	0.417463i
$76$		0			0
$77$			5.31539i			0.605745i
$78$		0			0
$79$	−4.89735			−0.550995			−0.275498	−	0.961302i	$-0.588843\pi$
$79$	−4.89735			−0.550995			−0.275498	+	0.961302i	$0.588843\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			0.492793i			0.0540911i	0.999634	+	0.0270456i	$0.00860992\pi$
$83$			0.492793i			0.0540911i	−0.999634	+	0.0270456i	$0.991390\pi$
$84$		0			0
$85$	1.88091	−	4.69171i	0.204013	−	0.508887i
$86$		0			0
$87$			3.13766i			0.336393i
$88$		0			0
$89$	5.50725			0.583768			0.291884	−	0.956454i	$-0.405718\pi$
$89$	5.50725			0.583768			0.291884	+	0.956454i	$0.405718\pi$
$90$		0			0
$91$	3.84330			0.402887
$92$		0			0
$93$		−	3.52550i		−	0.365577i
$94$		0			0
$95$	5.53884	+	2.22053i	0.568273	+	0.227821i
$96$		0			0
$97$		−	8.86639i		−	0.900246i	−0.892967	−	0.450123i	$-0.851380\pi$
$97$		−	8.86639i		−	0.900246i	0.892967	−	0.450123i	$-0.148620\pi$
$98$		0			0
$99$	5.62597			0.565432
$100$		0			0
$101$	−1.40463			−0.139766			−0.0698830	−	0.997555i	$-0.522263\pi$
$101$	−1.40463			−0.139766			−0.0698830	+	0.997555i	$0.522263\pi$
$102$		0			0
$103$			7.03174i			0.692858i	0.938076	+	0.346429i	$0.112606\pi$
$103$			7.03174i			0.692858i	−0.938076	+	0.346429i	$0.887394\pi$
$104$		0			0
$105$	1.96091	+	0.786131i	0.191366	+	0.0767185i
$106$		0			0
$107$		−	1.44335i		−	0.139534i	−0.997563	−	0.0697668i	$-0.977774\pi$
$107$		−	1.44335i		−	0.139534i	0.997563	−	0.0697668i	$-0.0222255\pi$
$108$		0			0
$109$	10.6818			1.02313			0.511564	−	0.859245i	$-0.329066\pi$
$109$	10.6818			1.02313			0.511564	+	0.859245i	$0.329066\pi$
$110$		0			0
$111$	7.10768			0.674631
$112$		0			0
$113$		−	12.2043i		−	1.14808i	−0.818826	−	0.574042i	$-0.805375\pi$
$113$		−	12.2043i		−	1.14808i	0.818826	−	0.574042i	$-0.194625\pi$
$114$		0			0
$115$	−3.92877	+	9.79988i	−0.366360	+	0.913843i
$116$		0			0
$117$		−	4.06787i		−	0.376075i
$118$		0			0
$119$	2.13574			0.195783
$120$		0			0
$121$	20.6516			1.87742
$122$		0			0
$123$		−	2.05104i		−	0.184936i
$124$		0			0
$125$	−4.62973	−	10.1767i	−0.414096	−	0.910233i
$126$		0			0
$127$		−	4.58183i		−	0.406572i	−0.979119	−	0.203286i	$-0.934838\pi$
$127$		−	4.58183i		−	0.406572i	0.979119	−	0.203286i	$-0.0651621\pi$
$128$		0			0
$129$	−1.23412			−0.108658
$130$		0			0
$131$	−6.51837			−0.569512			−0.284756	−	0.958600i	$-0.591913\pi$
$131$	−6.51837			−0.569512			−0.284756	+	0.958600i	$0.591913\pi$
$132$		0			0
$133$			2.52136i			0.218630i
$134$		0			0
$135$	0.832066	−	2.07549i	0.0716128	−	0.178630i
$136$		0			0
$137$			7.95588i			0.679717i	0.940477	+	0.339858i	$0.110379\pi$
$137$			7.95588i			0.679717i	−0.940477	+	0.339858i	$0.889621\pi$
$138$		0			0
$139$	5.46540			0.463569			0.231785	−	0.972767i	$-0.425544\pi$
$139$	5.46540			0.463569			0.231785	+	0.972767i	$0.425544\pi$
$140$		0			0
$141$	−10.2135			−0.860131
$142$		0			0
$143$		−	22.8857i		−	1.91380i
$144$		0			0
$145$	6.51219	+	2.61074i	0.540808	+	0.216810i
$146$		0			0
$147$		−	6.10736i		−	0.503727i
$148$		0			0
$149$	5.12519			0.419872			0.209936	−	0.977715i	$-0.432675\pi$
$149$	5.12519			0.419872			0.209936	+	0.977715i	$0.432675\pi$
$150$		0			0
$151$	−9.40466			−0.765340			−0.382670	−	0.923885i	$-0.624995\pi$
$151$	−9.40466			−0.765340			−0.382670	+	0.923885i	$0.624995\pi$
$152$		0			0
$153$		−	2.26053i		−	0.182753i
$154$		0			0
$155$	−7.31714	−	2.93344i	−0.587727	−	0.235620i
$156$		0			0
$157$			5.70586i			0.455377i	0.973734	+	0.227689i	$0.0731168\pi$
$157$			5.70586i			0.455377i	−0.973734	+	0.227689i	$0.926883\pi$
$158$		0			0
$159$	1.98447			0.157379
$160$		0			0
$161$	−4.46105			−0.351580
$162$		0			0
$163$		−	1.93601i		−	0.151640i	−0.997122	−	0.0758199i	$-0.975843\pi$
$163$		−	1.93601i		−	0.151640i	0.997122	−	0.0758199i	$-0.0241574\pi$
$164$		0			0
$165$	4.68118	−	11.6767i	0.364429	−	0.909027i
$166$		0			0
$167$		−	12.2167i		−	0.945357i	−0.881235	−	0.472679i	$-0.843287\pi$
$167$		−	12.2167i		−	0.945357i	0.881235	−	0.472679i	$-0.156713\pi$
$168$		0			0
$169$	−3.54756			−0.272889
$170$		0			0
$171$	2.66869			0.204080
$172$		0			0
$173$		−	4.46381i		−	0.339377i	−0.985498	−	0.169689i	$-0.945724\pi$
$173$		−	4.46381i		−	0.339377i	0.985498	−	0.169689i	$-0.0542762\pi$
$174$		0			0
$175$	3.26322	−	3.41575i	0.246676	−	0.258206i
$176$		0			0
$177$		−	3.82526i		−	0.287524i
$178$		0			0
$179$	−4.84000			−0.361759			−0.180879	−	0.983505i	$-0.557894\pi$
$179$	−4.84000			−0.361759			−0.180879	+	0.983505i	$0.557894\pi$
$180$		0			0
$181$	−0.565550			−0.0420370			−0.0210185	−	0.999779i	$-0.506691\pi$
$181$	−0.565550			−0.0420370			−0.0210185	+	0.999779i	$0.506691\pi$
$182$		0			0
$183$			4.66924i			0.345160i
$184$		0			0
$185$	5.91405	−	14.7519i	0.434810	−	1.08458i
$186$		0			0
$187$		−	12.7177i		−	0.930009i
$188$		0			0
$189$	0.944795			0.0687237
$190$		0			0
$191$	10.8176			0.782736			0.391368	−	0.920234i	$-0.372002\pi$
$191$	10.8176			0.782736			0.391368	+	0.920234i	$0.372002\pi$
$192$		0			0
$193$			7.05850i			0.508082i	0.967193	+	0.254041i	$0.0817598\pi$
$193$			7.05850i			0.508082i	−0.967193	+	0.254041i	$0.918240\pi$
$194$		0			0
$195$	−8.44283	−	3.38473i	−0.604604	−	0.242386i
$196$		0			0
$197$			3.26050i			0.232301i	0.993232	+	0.116150i	$0.0370555\pi$
$197$			3.26050i			0.232301i	−0.993232	+	0.116150i	$0.962945\pi$
$198$		0			0
$199$	−18.5333			−1.31379			−0.656897	−	0.753980i	$-0.728131\pi$
$199$	−18.5333			−1.31379			−0.656897	+	0.753980i	$0.728131\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$			2.96445i			0.208063i
$204$		0			0
$205$	−4.25692	−	1.70660i	−0.297316	−	0.119194i
$206$		0			0
$207$			4.72171i			0.328182i
$208$		0			0
$209$	15.0140			1.03854
$210$		0			0
$211$	21.9329			1.50992			0.754961	−	0.655770i	$-0.227656\pi$
$211$	21.9329			1.50992			0.754961	+	0.655770i	$0.227656\pi$
$212$		0			0
$213$		−	10.9005i		−	0.746889i
$214$		0			0
$215$	−1.02687	+	2.56141i	−0.0700320	+	0.174687i
$216$		0			0
$217$		−	3.33087i		−	0.226114i
$218$		0			0
$219$	3.65326			0.246864
$220$		0			0
$221$	−9.19553			−0.618559
$222$		0			0
$223$		−	12.8704i		−	0.861866i	−0.902384	−	0.430933i	$-0.858184\pi$
$223$		−	12.8704i		−	0.861866i	0.902384	−	0.430933i	$-0.141816\pi$
$224$		0			0
$225$	−3.61533	−	3.45389i	−0.241022	−	0.230259i
$226$		0			0
$227$			14.2999i			0.949118i	0.880224	+	0.474559i	$0.157392\pi$
$227$			14.2999i			0.949118i	−0.880224	+	0.474559i	$0.842608\pi$
$228$		0			0
$229$	8.59891			0.568232			0.284116	−	0.958790i	$-0.408300\pi$
$229$	8.59891			0.568232			0.284116	+	0.958790i	$0.408300\pi$
$230$		0			0
$231$	5.31539			0.349727
$232$		0			0
$233$		−	4.45199i		−	0.291660i	−0.989310	−	0.145830i	$-0.953415\pi$
$233$		−	4.45199i		−	0.291660i	0.989310	−	0.145830i	$-0.0465852\pi$
$234$		0			0
$235$	−8.49829	+	21.1980i	−0.554367	+	1.38281i
$236$		0			0
$237$			4.89735i			0.318117i
$238$		0			0
$239$	27.8668			1.80255			0.901277	−	0.433244i	$-0.142631\pi$
$239$	27.8668			1.80255			0.901277	+	0.433244i	$0.142631\pi$
$240$		0			0
$241$	13.6816			0.881307			0.440654	−	0.897677i	$-0.354747\pi$
$241$	13.6816			0.881307			0.440654	+	0.897677i	$0.354747\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−12.6758	−	5.08173i	−0.809826	−	0.324660i
$246$		0			0
$247$		−	10.8559i		−	0.690743i
$248$		0			0
$249$	0.492793			0.0312295
$250$		0			0
$251$	12.3171			0.777448			0.388724	−	0.921354i	$-0.372916\pi$
$251$	12.3171			0.777448			0.388724	+	0.921354i	$0.372916\pi$
$252$		0			0
$253$			26.5642i			1.67008i
$254$		0			0
$255$	−4.69171	−	1.88091i	−0.293806	−	0.117787i
$256$		0			0
$257$			20.5865i			1.28415i	0.766643	+	0.642074i	$0.221926\pi$
$257$			20.5865i			1.28415i	−0.766643	+	0.642074i	$0.778074\pi$
$258$		0			0
$259$	6.71530			0.417268
$260$		0			0
$261$	3.13766			0.194216
$262$		0			0
$263$			25.5931i			1.57814i	0.614304	+	0.789069i	$0.289437\pi$
$263$			25.5931i			1.57814i	−0.614304	+	0.789069i	$0.710563\pi$
$264$		0			0
$265$	1.65121	−	4.11875i	0.101433	−	0.253013i
$266$		0			0
$267$		−	5.50725i		−	0.337038i
$268$		0			0
$269$	28.8421			1.75854			0.879268	−	0.476327i	$-0.158032\pi$
$269$	28.8421			1.75854			0.879268	+	0.476327i	$0.158032\pi$
$270$		0			0
$271$	13.9155			0.845307			0.422653	−	0.906291i	$-0.361099\pi$
$271$	13.9155			0.845307			0.422653	+	0.906291i	$0.361099\pi$
$272$		0			0
$273$		−	3.84330i		−	0.232607i
$274$		0			0
$275$	−20.3398	−	19.4315i	−1.22653	−	1.17176i
$276$		0			0
$277$		−	1.48744i		−	0.0893716i	−0.999001	−	0.0446858i	$-0.985771\pi$
$277$		−	1.48744i		−	0.0893716i	0.999001	−	0.0446858i	$-0.0142287\pi$
$278$		0			0
$279$	−3.52550			−0.211066
$280$		0			0
$281$	−16.7850			−1.00131			−0.500655	−	0.865647i	$-0.666907\pi$
$281$	−16.7850			−1.00131			−0.500655	+	0.865647i	$0.666907\pi$
$282$		0			0
$283$			19.7035i			1.17125i	0.810581	+	0.585627i	$0.199151\pi$
$283$			19.7035i			1.17125i	−0.810581	+	0.585627i	$0.800849\pi$
$284$		0			0
$285$	2.22053	−	5.53884i	0.131533	−	0.328093i
$286$		0			0
$287$		−	1.93781i		−	0.114385i
$288$		0			0
$289$	11.8900			0.699412
$290$		0			0
$291$	−8.86639			−0.519757
$292$		0			0
$293$		−	0.740560i		−	0.0432640i	−0.999766	−	0.0216320i	$-0.993114\pi$
$293$		−	0.740560i		−	0.0432640i	0.999766	−	0.0216320i	$-0.00688621\pi$
$294$		0			0
$295$	−7.93929	−	3.18286i	−0.462243	−	0.185314i
$296$		0			0
$297$		−	5.62597i		−	0.326452i
$298$		0			0
$299$	19.2073			1.11079
$300$		0			0
$301$	−1.16599			−0.0672067
$302$		0			0
$303$			1.40463i			0.0806940i
$304$		0			0
$305$	9.69097	+	3.88511i	0.554903	+	0.222461i
$306$		0			0
$307$		−	3.63931i		−	0.207706i	−0.994593	−	0.103853i	$-0.966883\pi$
$307$		−	3.63931i		−	0.207706i	0.994593	−	0.103853i	$-0.0331172\pi$
$308$		0			0
$309$	7.03174			0.400022
$310$		0			0
$311$	−20.5426			−1.16487			−0.582433	−	0.812879i	$-0.697899\pi$
$311$	−20.5426			−1.16487			−0.582433	+	0.812879i	$0.697899\pi$
$312$		0			0
$313$			9.66705i			0.546414i	0.961955	+	0.273207i	$0.0880844\pi$
$313$			9.66705i			0.546414i	−0.961955	+	0.273207i	$0.911916\pi$
$314$		0			0
$315$	0.786131	−	1.96091i	0.0442935	−	0.110485i
$316$		0			0
$317$			23.7539i			1.33415i	0.744991	+	0.667075i	$0.232454\pi$
$317$			23.7539i			1.33415i	−0.744991	+	0.667075i	$0.767546\pi$
$318$		0			0
$319$	17.6524			0.988345
$320$		0			0
$321$	−1.44335			−0.0805598
$322$		0			0
$323$		−	6.03265i		−	0.335666i
$324$		0			0
$325$	−14.0500	+	14.7067i	−0.779353	+	0.815781i
$326$		0			0
$327$		−	10.6818i		−	0.590703i
$328$		0			0
$329$	−9.64965			−0.532002
$330$		0			0
$331$	17.4135			0.957133			0.478567	−	0.878051i	$-0.341156\pi$
$331$	17.4135			0.957133			0.478567	+	0.878051i	$0.341156\pi$
$332$		0			0
$333$		−	7.10768i		−	0.389498i
$334$		0			0
$335$	0.832066	−	2.07549i	0.0454606	−	0.113396i
$336$		0			0
$337$		−	22.7260i		−	1.23796i	−0.785405	−	0.618982i	$-0.787545\pi$
$337$		−	22.7260i		−	1.23796i	0.785405	−	0.618982i	$-0.212455\pi$
$338$		0			0
$339$	−12.2043			−0.662847
$340$		0			0
$341$	−19.8343			−1.07409
$342$		0			0
$343$		−	12.3838i		−	0.668661i
$344$		0			0
$345$	9.79988	+	3.92877i	0.527608	+	0.211518i
$346$		0			0
$347$		−	10.1844i		−	0.546725i	−0.961911	−	0.273363i	$-0.911864\pi$
$347$		−	10.1844i		−	0.546725i	0.961911	−	0.273363i	$-0.0881359\pi$
$348$		0			0
$349$	27.7086			1.48321			0.741603	−	0.670839i	$-0.234066\pi$
$349$	27.7086			1.48321			0.741603	+	0.670839i	$0.234066\pi$
$350$		0			0
$351$	−4.06787			−0.217127
$352$		0			0
$353$		−	25.9273i		−	1.37997i	−0.723823	−	0.689986i	$-0.757617\pi$
$353$		−	25.9273i		−	1.37997i	0.723823	−	0.689986i	$-0.242383\pi$
$354$		0			0
$355$	−22.6239	−	9.06992i	−1.20075	−	0.481381i
$356$		0			0
$357$		−	2.13574i		−	0.113035i
$358$		0			0
$359$	15.2107			0.802790			0.401395	−	0.915905i	$-0.368525\pi$
$359$	15.2107			0.802790			0.401395	+	0.915905i	$0.368525\pi$
$360$		0			0
$361$	−11.8781			−0.625163
$362$		0			0
$363$		−	20.6516i		−	1.08393i
$364$		0			0
$365$	3.03975	−	7.58231i	0.159108	−	0.396876i
$366$		0			0
$367$			5.49651i			0.286915i	0.989656	+	0.143458i	$0.0458221\pi$
$367$			5.49651i			0.286915i	−0.989656	+	0.143458i	$0.954178\pi$
$368$		0			0
$369$	−2.05104			−0.106773
$370$		0			0
$371$	1.87492			0.0973409
$372$		0			0
$373$			15.0190i			0.777654i	0.921311	+	0.388827i	$0.127120\pi$
$373$			15.0190i			0.777654i	−0.921311	+	0.388827i	$0.872880\pi$
$374$		0			0
$375$	−10.1767	+	4.62973i	−0.525523	+	0.239078i
$376$		0			0
$377$		−	12.7636i		−	0.657359i
$378$		0			0
$379$	−20.2137			−1.03831			−0.519153	−	0.854681i	$-0.673752\pi$
$379$	−20.2137			−1.03831			−0.519153	+	0.854681i	$0.673752\pi$
$380$		0			0
$381$	−4.58183			−0.234734
$382$		0			0
$383$			24.5484i			1.25437i	0.778872	+	0.627183i	$0.215792\pi$
$383$			24.5484i			1.25437i	−0.778872	+	0.627183i	$0.784208\pi$
$384$		0			0
$385$	4.42275	−	11.0321i	0.225404	−	0.562245i
$386$		0			0
$387$			1.23412i			0.0627339i
$388$		0			0
$389$	19.8976			1.00885			0.504424	−	0.863456i	$-0.331705\pi$
$389$	19.8976			1.00885			0.504424	+	0.863456i	$0.331705\pi$
$390$		0			0
$391$	10.6736			0.539785
$392$		0			0
$393$			6.51837i			0.328808i
$394$		0			0
$395$	10.1644	+	4.07492i	0.511427	+	0.205031i
$396$		0			0
$397$		−	13.3151i		−	0.668267i	−0.942526	−	0.334133i	$-0.891556\pi$
$397$		−	13.3151i		−	0.668267i	0.942526	−	0.334133i	$-0.108444\pi$
$398$		0			0
$399$	2.52136			0.126226
$400$		0			0
$401$	−36.1802			−1.80675			−0.903377	−	0.428848i	$-0.858920\pi$
$401$	−36.1802			−1.80675			−0.903377	+	0.428848i	$0.858920\pi$
$402$		0			0
$403$			14.3413i			0.714389i
$404$		0			0
$405$	−2.07549	−	0.832066i	−0.103132	−	0.0413457i
$406$		0			0
$407$		−	39.9876i		−	1.98211i
$408$		0			0
$409$	15.4724			0.765059			0.382529	−	0.923943i	$-0.375053\pi$
$409$	15.4724			0.765059			0.382529	+	0.923943i	$0.375053\pi$
$410$		0			0
$411$	7.95588			0.392435
$412$		0			0
$413$		−	3.61408i		−	0.177837i
$414$		0			0
$415$	0.410036	−	1.02279i	0.0201279	−	0.0502067i
$416$		0			0
$417$		−	5.46540i		−	0.267642i
$418$		0			0
$419$	20.2751			0.990503			0.495252	−	0.868750i	$-0.335076\pi$
$419$	20.2751			0.990503			0.495252	+	0.868750i	$0.335076\pi$
$420$		0			0
$421$	−31.6200			−1.54106			−0.770532	−	0.637402i	$-0.780009\pi$
$421$	−31.6200			−1.54106			−0.770532	+	0.637402i	$0.780009\pi$
$422$		0			0
$423$			10.2135i			0.496597i
$424$		0			0
$425$	−7.80762	+	8.17257i	−0.378725	+	0.396428i
$426$		0			0
$427$			4.41147i			0.213486i
$428$		0			0
$429$	−22.8857			−1.10493
$430$		0			0
$431$	11.9341			0.574844			0.287422	−	0.957804i	$-0.407202\pi$
$431$	11.9341			0.574844			0.287422	+	0.957804i	$0.407202\pi$
$432$		0			0
$433$			28.2739i			1.35876i	0.733787	+	0.679379i	$0.237751\pi$
$433$			28.2739i			1.35876i	−0.733787	+	0.679379i	$0.762249\pi$
$434$		0			0
$435$	2.61074	−	6.51219i	0.125175	−	0.312236i
$436$		0			0
$437$			12.6008i			0.602777i
$438$		0			0
$439$	35.4798			1.69336			0.846679	−	0.532104i	$-0.178599\pi$
$439$	35.4798			1.69336			0.846679	+	0.532104i	$0.178599\pi$
$440$		0			0
$441$	−6.10736			−0.290827
$442$		0			0
$443$			20.1397i			0.956866i	0.878124	+	0.478433i	$0.158795\pi$
$443$			20.1397i			0.956866i	−0.878124	+	0.478433i	$0.841205\pi$
$444$		0			0
$445$	−11.4303	−	4.58240i	−0.541846	−	0.217226i
$446$		0			0
$447$		−	5.12519i		−	0.242413i
$448$		0			0
$449$	−6.66364			−0.314476			−0.157238	−	0.987561i	$-0.550259\pi$
$449$	−6.66364			−0.314476			−0.157238	+	0.987561i	$0.550259\pi$
$450$		0			0
$451$	−11.5391			−0.543355
$452$		0			0
$453$			9.40466i			0.441870i
$454$		0			0
$455$	−7.97674	−	3.19788i	−0.373955	−	0.149919i
$456$		0			0
$457$		−	23.0563i		−	1.07853i	−0.842137	−	0.539264i	$-0.818702\pi$
$457$		−	23.0563i		−	1.07853i	0.842137	−	0.539264i	$-0.181298\pi$
$458$		0			0
$459$	−2.26053			−0.105512
$460$		0			0
$461$	22.7329			1.05878			0.529388	−	0.848380i	$-0.322422\pi$
$461$	22.7329			1.05878			0.529388	+	0.848380i	$0.322422\pi$
$462$		0			0
$463$			28.5730i			1.32790i	0.747777	+	0.663949i	$0.231121\pi$
$463$			28.5730i			1.32790i	−0.747777	+	0.663949i	$0.768879\pi$
$464$		0			0
$465$	−2.93344	+	7.31714i	−0.136035	+	0.339324i
$466$		0			0
$467$			7.34202i			0.339748i	0.985466	+	0.169874i	$0.0543361\pi$
$467$			7.34202i			0.339748i	−0.985466	+	0.169874i	$0.945664\pi$
$468$		0			0
$469$	0.944795			0.0436266
$470$		0			0
$471$	5.70586			0.262912
$472$		0			0
$473$			6.94314i			0.319246i
$474$		0			0
$475$	−9.64821	−	9.21736i	−0.442690	−	0.422922i
$476$		0			0
$477$		−	1.98447i		−	0.0908627i
$478$		0			0
$479$	4.44621			0.203152			0.101576	−	0.994828i	$-0.467611\pi$
$479$	4.44621			0.203152			0.101576	+	0.994828i	$0.467611\pi$
$480$		0			0
$481$	−28.9131			−1.31832
$482$		0			0
$483$			4.46105i			0.202985i
$484$		0			0
$485$	−7.37742	+	18.4021i	−0.334991	+	0.835598i
$486$		0			0
$487$			13.7060i			0.621080i	0.950560	+	0.310540i	$0.100510\pi$
$487$			13.7060i			0.621080i	−0.950560	+	0.310540i	$0.899490\pi$
$488$		0			0
$489$	−1.93601			−0.0875493
$490$		0			0
$491$	−26.0535			−1.17578			−0.587889	−	0.808942i	$-0.700041\pi$
$491$	−26.0535			−1.17578			−0.587889	+	0.808942i	$0.700041\pi$
$492$		0			0
$493$		−	7.09277i		−	0.319442i
$494$		0			0
$495$	−11.6767	−	4.68118i	−0.524827	−	0.210403i
$496$		0			0
$497$		−	10.2987i		−	0.461961i
$498$		0			0
$499$	37.6675			1.68623			0.843115	−	0.537734i	$-0.180719\pi$
$499$	37.6675			1.68623			0.843115	+	0.537734i	$0.180719\pi$
$500$		0			0
$501$	−12.2167			−0.545802
$502$		0			0
$503$		−	39.2627i		−	1.75064i	−0.483547	−	0.875318i	$-0.660652\pi$
$503$		−	39.2627i		−	1.75064i	0.483547	−	0.875318i	$-0.339348\pi$
$504$		0			0
$505$	2.91530	+	1.16875i	0.129729	+	0.0520085i
$506$		0			0
$507$			3.54756i			0.157553i
$508$		0			0
$509$	−2.87514			−0.127438			−0.0637192	−	0.997968i	$-0.520296\pi$
$509$	−2.87514			−0.127438			−0.0637192	+	0.997968i	$0.520296\pi$
$510$		0			0
$511$	3.45158			0.152689
$512$		0			0
$513$		−	2.66869i		−	0.117826i
$514$		0			0
$515$	5.85087	−	14.5943i	0.257820	−	0.643103i
$516$		0			0
$517$			57.4608i			2.52712i
$518$		0			0
$519$	−4.46381			−0.195940
$520$		0			0
$521$	−13.2709			−0.581410			−0.290705	−	0.956813i	$-0.593890\pi$
$521$	−13.2709			−0.581410			−0.290705	+	0.956813i	$0.593890\pi$
$522$		0			0
$523$		−	23.0795i		−	1.00920i	−0.863354	−	0.504599i	$-0.831640\pi$
$523$		−	23.0795i		−	1.00920i	0.863354	−	0.504599i	$-0.168360\pi$
$524$		0			0
$525$	−3.41575	−	3.26322i	−0.149075	−	0.142419i
$526$		0			0
$527$			7.96948i			0.347156i
$528$		0			0
$529$	0.705430			0.0306709
$530$		0			0
$531$	−3.82526			−0.166002
$532$		0			0
$533$			8.34337i			0.361391i
$534$		0			0
$535$	−1.20096	+	2.99566i	−0.0519220	+	0.129513i
$536$		0			0
$537$			4.84000i			0.208862i
$538$		0			0
$539$	−34.3599			−1.47998
$540$		0			0
$541$	−8.88466			−0.381981			−0.190991	−	0.981592i	$-0.561170\pi$
$541$	−8.88466			−0.381981			−0.190991	+	0.981592i	$0.561170\pi$
$542$		0			0
$543$			0.565550i			0.0242701i
$544$		0			0
$545$	−22.1699	−	8.88792i	−0.949654	−	0.380717i
$546$		0			0
$547$		−	37.1004i		−	1.58630i	−0.609027	−	0.793150i	$-0.708440\pi$
$547$		−	37.1004i		−	1.58630i	0.609027	−	0.793150i	$-0.291560\pi$
$548$		0			0
$549$	4.66924			0.199278
$550$		0			0
$551$	8.37345			0.356721
$552$		0			0
$553$			4.62699i			0.196760i
$554$		0			0
$555$	−14.7519	−	5.91405i	−0.626184	−	0.251038i
$556$		0			0
$557$		−	17.4506i		−	0.739404i	−0.929150	−	0.369702i	$-0.879460\pi$
$557$		−	17.4506i		−	0.739404i	0.929150	−	0.369702i	$-0.120540\pi$
$558$		0			0
$559$	5.02025			0.212334
$560$		0			0
$561$	−12.7177			−0.536941
$562$		0			0
$563$			27.4364i			1.15631i	0.815928	+	0.578154i	$0.196227\pi$
$563$			27.4364i			1.15631i	−0.815928	+	0.578154i	$0.803773\pi$
$564$		0			0
$565$	−10.1548	+	25.3299i	−0.427215	+	1.06564i
$566$		0			0
$567$		−	0.944795i		−	0.0396776i
$568$		0			0
$569$	−27.8154			−1.16608			−0.583040	−	0.812443i	$-0.698137\pi$
$569$	−27.8154			−1.16608			−0.583040	+	0.812443i	$0.698137\pi$
$570$		0			0
$571$	−0.616545			−0.0258016			−0.0129008	−	0.999917i	$-0.504107\pi$
$571$	−0.616545			−0.0258016			−0.0129008	+	0.999917i	$0.504107\pi$
$572$		0			0
$573$		−	10.8176i		−	0.451913i
$574$		0			0
$575$	16.3083	−	17.0706i	0.680102	−	0.711892i
$576$		0			0
$577$			29.2479i			1.21761i	0.793321	+	0.608803i	$0.208350\pi$
$577$			29.2479i			1.21761i	−0.793321	+	0.608803i	$0.791650\pi$
$578$		0			0
$579$	7.05850			0.293341
$580$		0			0
$581$	0.465589			0.0193159
$582$		0			0
$583$		−	11.1646i		−	0.462390i
$584$		0			0
$585$	−3.38473	+	8.44283i	−0.139942	+	0.349068i
$586$		0			0
$587$			7.16381i			0.295682i	0.989011	+	0.147841i	$0.0472324\pi$
$587$			7.16381i			0.295682i	−0.989011	+	0.147841i	$0.952768\pi$
$588$		0			0
$589$	−9.40845			−0.387669
$590$		0			0
$591$	3.26050			0.134119
$592$		0			0
$593$			10.1719i			0.417709i	0.977947	+	0.208854i	$0.0669734\pi$
$593$			10.1719i			0.417709i	−0.977947	+	0.208854i	$0.933027\pi$
$594$		0			0
$595$	−4.43270	−	1.77707i	−0.181723	−	0.0728528i
$596$		0			0
$597$			18.5333i			0.758519i
$598$		0			0
$599$	32.3966			1.32369			0.661845	−	0.749641i	$-0.269774\pi$
$599$	32.3966			1.32369			0.661845	+	0.749641i	$0.269774\pi$
$600$		0			0
$601$	12.7344			0.519448			0.259724	−	0.965683i	$-0.416368\pi$
$601$	12.7344			0.519448			0.259724	+	0.965683i	$0.416368\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	−42.8622	−	17.1835i	−1.74260	−	0.698608i
$606$		0			0
$607$			28.4924i			1.15647i	0.815870	+	0.578236i	$0.196259\pi$
$607$			28.4924i			1.15647i	−0.815870	+	0.578236i	$0.803741\pi$
$608$		0			0
$609$	2.96445			0.120125
$610$		0			0
$611$	41.5471			1.68082
$612$		0			0
$613$			44.5028i			1.79745i	0.438510	+	0.898726i	$0.355507\pi$
$613$			44.5028i			1.79745i	−0.438510	+	0.898726i	$0.644493\pi$
$614$		0			0
$615$	−1.70660	+	4.25692i	−0.0688168	+	0.171656i
$616$		0			0
$617$			13.3596i			0.537837i	0.963163	+	0.268918i	$0.0866662\pi$
$617$			13.3596i			0.537837i	−0.963163	+	0.268918i	$0.913334\pi$
$618$		0			0
$619$	22.0514			0.886322			0.443161	−	0.896442i	$-0.353857\pi$
$619$	22.0514			0.886322			0.443161	+	0.896442i	$0.353857\pi$
$620$		0			0
$621$	4.72171			0.189476
$622$		0			0
$623$		−	5.20322i		−	0.208463i
$624$		0			0
$625$	1.14128	+	24.9739i	0.0456510	+	0.998957i
$626$		0			0
$627$		−	15.0140i		−	0.599601i
$628$		0			0
$629$	−16.0671			−0.640637
$630$		0			0
$631$	−12.6594			−0.503964			−0.251982	−	0.967732i	$-0.581082\pi$
$631$	−12.6594			−0.503964			−0.251982	+	0.967732i	$0.581082\pi$
$632$		0			0
$633$		−	21.9329i		−	0.871754i
$634$		0			0
$635$	−3.81238	+	9.50955i	−0.151290	+	0.377375i
$636$		0			0
$637$			24.8440i			0.984353i
$638$		0			0
$639$	−10.9005			−0.431217
$640$		0			0
$641$	−36.3562			−1.43598			−0.717991	−	0.696053i	$-0.754938\pi$
$641$	−36.3562			−1.43598			−0.717991	+	0.696053i	$0.754938\pi$
$642$		0			0
$643$			11.4413i			0.451202i	0.974220	+	0.225601i	$0.0724345\pi$
$643$			11.4413i			0.451202i	−0.974220	+	0.225601i	$0.927565\pi$
$644$		0			0
$645$	2.56141	+	1.02687i	0.100855	+	0.0404330i
$646$		0			0
$647$		−	1.81130i		−	0.0712097i	−0.999366	−	0.0356048i	$-0.988664\pi$
$647$		−	1.81130i		−	0.0712097i	0.999366	−	0.0356048i	$-0.0113358\pi$
$648$		0			0
$649$	−21.5208			−0.844765
$650$		0			0
$651$	−3.33087			−0.130547
$652$		0			0
$653$			12.5319i			0.490412i	0.969471	+	0.245206i	$0.0788557\pi$
$653$			12.5319i			0.490412i	−0.969471	+	0.245206i	$0.921144\pi$
$654$		0			0
$655$	13.5288	+	5.42371i	0.528615	+	0.211922i
$656$		0			0
$657$		−	3.65326i		−	0.142527i
$658$		0			0
$659$	−19.9105			−0.775603			−0.387801	−	0.921743i	$-0.626765\pi$
$659$	−19.9105			−0.775603			−0.387801	+	0.921743i	$0.626765\pi$
$660$		0			0
$661$	8.88231			0.345482			0.172741	−	0.984967i	$-0.444738\pi$
$661$	8.88231			0.345482			0.172741	+	0.984967i	$0.444738\pi$
$662$		0			0
$663$			9.19553i			0.357125i
$664$		0			0
$665$	2.09794	−	5.23307i	0.0813546	−	0.202930i
$666$		0			0
$667$			14.8151i			0.573644i
$668$		0			0
$669$	−12.8704			−0.497599
$670$		0			0
$671$	26.2690			1.01410
$672$		0			0
$673$			46.1256i			1.77801i	0.457896	+	0.889006i	$0.348603\pi$
$673$			46.1256i			1.77801i	−0.457896	+	0.889006i	$0.651397\pi$
$674$		0			0
$675$	−3.45389	+	3.61533i	−0.132940	+	0.139154i
$676$		0			0
$677$		−	15.7952i		−	0.607058i	−0.952822	−	0.303529i	$-0.901835\pi$
$677$		−	15.7952i		−	0.607058i	0.952822	−	0.303529i	$-0.0981649\pi$
$678$		0			0
$679$	−8.37692			−0.321477
$680$		0			0
$681$	14.2999			0.547974
$682$		0			0
$683$			4.41692i			0.169009i	0.996423	+	0.0845044i	$0.0269307\pi$
$683$			4.41692i			0.169009i	−0.996423	+	0.0845044i	$0.973069\pi$
$684$		0			0
$685$	6.61981	−	16.5124i	0.252930	−	0.630905i
$686$		0			0
$687$		−	8.59891i		−	0.328069i
$688$		0			0
$689$	−8.07257			−0.307540
$690$		0			0
$691$	−39.3975			−1.49875			−0.749376	−	0.662145i	$-0.769646\pi$
$691$	−39.3975			−1.49875			−0.749376	+	0.662145i	$0.769646\pi$
$692$		0			0
$693$		−	5.31539i		−	0.201915i
$694$		0			0
$695$	−11.3434	−	4.54757i	−0.430280	−	0.172499i
$696$		0			0
$697$			4.63644i			0.175618i
$698$		0			0
$699$	−4.45199			−0.168390
$700$		0			0
$701$	1.71171			0.0646504			0.0323252	−	0.999477i	$-0.489709\pi$
$701$	1.71171			0.0646504			0.0323252	+	0.999477i	$0.489709\pi$
$702$		0			0
$703$		−	18.9682i		−	0.715399i
$704$		0			0
$705$	21.1980	+	8.49829i	0.798363	+	0.320064i
$706$		0			0
$707$			1.32709i			0.0499103i
$708$		0			0
$709$	0.136750			0.00513576			0.00256788	−	0.999997i	$-0.499183\pi$
$709$	0.136750			0.00513576			0.00256788	+	0.999997i	$0.499183\pi$
$710$		0			0
$711$	4.89735			0.183665
$712$		0			0
$713$		−	16.6464i		−	0.623412i
$714$		0			0
$715$	−19.0424	+	47.4991i	−0.712146	+	1.77637i
$716$		0			0
$717$		−	27.8668i		−	1.04070i
$718$		0			0
$719$	−0.725853			−0.0270698			−0.0135349	−	0.999908i	$-0.504308\pi$
$719$	−0.725853			−0.0270698			−0.0135349	+	0.999908i	$0.504308\pi$
$720$		0			0
$721$	6.64355			0.247419
$722$		0			0
$723$		−	13.6816i		−	0.508823i
$724$		0			0
$725$	−11.3437	−	10.8371i	−0.421294	−	0.402481i
$726$		0			0
$727$		−	10.1439i		−	0.376218i	−0.982148	−	0.188109i	$-0.939764\pi$
$727$		−	10.1439i		−	0.376218i	0.982148	−	0.188109i	$-0.0602358\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	2.78977			0.103183
$732$		0			0
$733$			11.1209i			0.410759i	0.978682	+	0.205379i	$0.0658428\pi$
$733$			11.1209i			0.410759i	−0.978682	+	0.205379i	$0.934157\pi$
$734$		0			0
$735$	−5.08173	+	12.6758i	−0.187442	+	0.467553i
$736$		0			0
$737$		−	5.62597i		−	0.207235i
$738$		0			0
$739$	45.1847			1.66214			0.831072	−	0.556164i	$-0.187728\pi$
$739$	45.1847			1.66214			0.831072	+	0.556164i	$0.187728\pi$
$740$		0			0
$741$	−10.8559			−0.398801
$742$		0			0
$743$		−	16.0467i		−	0.588697i	−0.955698	−	0.294348i	$-0.904897\pi$
$743$		−	16.0467i		−	0.588697i	0.955698	−	0.294348i	$-0.0951026\pi$
$744$		0			0
$745$	−10.6373	−	4.26449i	−0.389720	−	0.156239i
$746$		0			0
$747$		−	0.492793i		−	0.0180304i
$748$		0			0
$749$	−1.36367			−0.0498273
$750$		0			0
$751$	47.7676			1.74307			0.871533	−	0.490338i	$-0.163126\pi$
$751$	47.7676			1.74307			0.871533	+	0.490338i	$0.163126\pi$
$752$		0			0
$753$		−	12.3171i		−	0.448860i
$754$		0			0
$755$	19.5193	+	7.82529i	0.710380	+	0.284792i
$756$		0			0
$757$		−	16.5034i		−	0.599825i	−0.953967	−	0.299913i	$-0.903042\pi$
$757$		−	16.5034i		−	0.599825i	0.953967	−	0.299913i	$-0.0969576\pi$
$758$		0			0
$759$	26.5642			0.964221
$760$		0			0
$761$	−15.6638			−0.567811			−0.283906	−	0.958852i	$-0.591630\pi$
$761$	−15.6638			−0.567811			−0.283906	+	0.958852i	$0.591630\pi$
$762$		0			0
$763$		−	10.0921i		−	0.365357i
$764$		0			0
$765$	−1.88091	+	4.69171i	−0.0680044	+	0.169629i
$766$		0			0
$767$			15.5606i			0.561862i
$768$		0			0
$769$	35.1807			1.26865			0.634324	−	0.773068i	$-0.281279\pi$
$769$	35.1807			1.26865			0.634324	+	0.773068i	$0.281279\pi$
$770$		0			0
$771$	20.5865			0.741403
$772$		0			0
$773$		−	39.3072i		−	1.41378i	−0.707322	−	0.706891i	$-0.750097\pi$
$773$		−	39.3072i		−	1.41378i	0.707322	−	0.706891i	$-0.249903\pi$
$774$		0			0
$775$	12.7458	+	12.1767i	0.457844	+	0.437399i
$776$		0			0
$777$		−	6.71530i		−	0.240910i
$778$		0			0
$779$	−5.47359			−0.196112
$780$		0			0
$781$	−61.3259			−2.19441
$782$		0			0
$783$		−	3.13766i		−	0.112131i
$784$		0			0
$785$	4.74765	−	11.8425i	0.169451	−	0.422676i
$786$		0			0
$787$		−	5.93299i		−	0.211488i	−0.994393	−	0.105744i	$-0.966278\pi$
$787$		−	5.93299i		−	0.211488i	0.994393	−	0.105744i	$-0.0337224\pi$
$788$		0			0
$789$	25.5931			0.911139
$790$		0			0
$791$	−11.5306			−0.409980
$792$		0			0
$793$		−	18.9939i		−	0.674492i
$794$		0			0
$795$	−4.11875	−	1.65121i	−0.146077	−	0.0585624i
$796$		0			0
$797$			5.50568i			0.195021i	0.995234	+	0.0975105i	$0.0310880\pi$
$797$			5.50568i			0.195021i	−0.995234	+	0.0975105i	$0.968912\pi$
$798$		0			0
$799$	23.0879			0.816790
$800$		0			0
$801$	−5.50725			−0.194589
$802$		0			0
$803$		−	20.5531i		−	0.725305i
$804$		0			0
$805$	9.25887	+	3.71189i	0.326332	+	0.130827i
$806$		0			0
$807$		−	28.8421i		−	1.01529i
$808$		0			0
$809$	16.4532			0.578465			0.289232	−	0.957259i	$-0.406600\pi$
$809$	16.4532			0.578465			0.289232	+	0.957259i	$0.406600\pi$
$810$		0			0
$811$	−13.4121			−0.470963			−0.235482	−	0.971879i	$-0.575667\pi$
$811$	−13.4121			−0.470963			−0.235482	+	0.971879i	$0.575667\pi$
$812$		0			0
$813$		−	13.9155i		−	0.488038i
$814$		0			0
$815$	−1.61088	+	4.01817i	−0.0564268	+	0.140750i
$816$		0			0
$817$			3.29349i			0.115225i
$818$		0			0
$819$	−3.84330			−0.134296
$820$		0			0
$821$	−11.3339			−0.395557			−0.197778	−	0.980247i	$-0.563373\pi$
$821$	−11.3339			−0.395557			−0.197778	+	0.980247i	$0.563373\pi$
$822$		0			0
$823$		−	53.4387i		−	1.86275i	−0.364055	−	0.931377i	$-0.618608\pi$
$823$		−	53.4387i		−	1.86275i	0.364055	−	0.931377i	$-0.381392\pi$
$824$		0			0
$825$	−19.4315	+	20.3398i	−0.676518	+	0.708140i
$826$		0			0
$827$			31.7522i			1.10413i	0.833801	+	0.552066i	$0.186160\pi$
$827$			31.7522i			1.10413i	−0.833801	+	0.552066i	$0.813840\pi$
$828$		0			0
$829$	23.1665			0.804606			0.402303	−	0.915507i	$-0.368210\pi$
$829$	23.1665			0.804606			0.402303	+	0.915507i	$0.368210\pi$
$830$		0			0
$831$	−1.48744			−0.0515987
$832$		0			0
$833$			13.8059i			0.478345i
$834$		0			0
$835$	−10.1651	+	25.3557i	−0.351778	+	0.877469i
$836$		0			0
$837$			3.52550i			0.121859i
$838$		0			0
$839$	−2.49404			−0.0861039			−0.0430519	−	0.999073i	$-0.513708\pi$
$839$	−2.49404			−0.0861039			−0.0430519	+	0.999073i	$0.513708\pi$
$840$		0			0
$841$	−19.1551			−0.660520
$842$		0			0
$843$			16.7850i			0.578107i
$844$		0			0
$845$	7.36294	+	2.95180i	0.253293	+	0.101545i
$846$		0			0
$847$		−	19.5115i		−	0.670424i
$848$		0			0
$849$	19.7035			0.676223
$850$		0			0
$851$	33.5604			1.15044
$852$		0			0
$853$			6.67269i			0.228469i	0.993454	+	0.114234i	$0.0364415\pi$
$853$			6.67269i			0.228469i	−0.993454	+	0.114234i	$0.963559\pi$
$854$		0			0
$855$	−5.53884	−	2.22053i	−0.189424	−	0.0759404i
$856$		0			0
$857$			32.9938i			1.12705i	0.826100	+	0.563523i	$0.190555\pi$
$857$			32.9938i			1.12705i	−0.826100	+	0.563523i	$0.809445\pi$
$858$		0			0
$859$	−51.1657			−1.74575			−0.872876	−	0.487942i	$-0.837748\pi$
$859$	−51.1657			−1.74575			−0.872876	+	0.487942i	$0.837748\pi$
$860$		0			0
$861$	−1.93781			−0.0660405
$862$		0			0
$863$			0.139238i			0.00473970i	0.999997	+	0.00236985i	$0.000754347\pi$
$863$			0.139238i			0.00473970i	−0.999997	+	0.00236985i	$0.999246\pi$
$864$		0			0
$865$	−3.71419	+	9.26461i	−0.126286	+	0.315006i
$866$		0			0
$867$		−	11.8900i		−	0.403806i
$868$		0			0
$869$	27.5524			0.934651
$870$		0			0
$871$	−4.06787			−0.137834
$872$		0			0
$873$			8.86639i			0.300082i
$874$		0			0
$875$	−9.61491	+	4.37415i	−0.325043	+	0.147873i
$876$		0			0
$877$		−	8.64235i		−	0.291831i	−0.989297	−	0.145916i	$-0.953387\pi$
$877$		−	8.64235i		−	0.291831i	0.989297	−	0.145916i	$-0.0466128\pi$
$878$		0			0
$879$	−0.740560			−0.0249785
$880$		0			0
$881$	−14.8644			−0.500794			−0.250397	−	0.968143i	$-0.580561\pi$
$881$	−14.8644			−0.500794			−0.250397	+	0.968143i	$0.580561\pi$
$882$		0			0
$883$		−	33.7990i		−	1.13743i	−0.822535	−	0.568714i	$-0.807441\pi$
$883$		−	33.7990i		−	1.13743i	0.822535	−	0.568714i	$-0.192559\pi$
$884$		0			0
$885$	−3.18286	+	7.93929i	−0.106991	+	0.266876i
$886$		0			0
$887$			44.6184i			1.49814i	0.662491	+	0.749070i	$0.269499\pi$
$887$			44.6184i			1.49814i	−0.662491	+	0.749070i	$0.730501\pi$
$888$		0			0
$889$	−4.32889			−0.145186
$890$		0			0
$891$	−5.62597			−0.188477
$892$		0			0
$893$			27.2566i			0.912108i
$894$		0			0
$895$	10.0454	+	4.02720i	0.335780	+	0.134614i
$896$		0			0
$897$		−	19.2073i		−	0.641313i
$898$		0			0
$899$	−11.0618			−0.368932
$900$		0			0
$901$	−4.48595			−0.149449
$902$		0			0
$903$			1.16599i			0.0388018i
$904$		0			0
$905$	1.17379	+	0.470575i	0.0390182	+	0.0156424i
$906$		0			0
$907$		−	26.1728i		−	0.869052i	−0.900659	−	0.434526i	$-0.856916\pi$
$907$		−	26.1728i		−	0.869052i	0.900659	−	0.434526i	$-0.143084\pi$
$908$		0			0
$909$	1.40463			0.0465887
$910$		0			0
$911$	33.4840			1.10937			0.554687	−	0.832059i	$-0.312838\pi$
$911$	33.4840			1.10937			0.554687	+	0.832059i	$0.312838\pi$
$912$		0			0
$913$		−	2.77244i		−	0.0917545i
$914$		0			0
$915$	3.88511	−	9.69097i	0.128438	−	0.320374i
$916$		0			0
$917$			6.15852i			0.203372i
$918$		0			0
$919$	3.46095			0.114166			0.0570832	−	0.998369i	$-0.481820\pi$
$919$	3.46095			0.114166			0.0570832	+	0.998369i	$0.481820\pi$
$920$		0			0
$921$	−3.63931			−0.119919
$922$		0			0
$923$			44.3418i			1.45953i
$924$		0			0
$925$	−24.5491	+	25.6966i	−0.807171	+	0.844900i
$926$		0			0
$927$		−	7.03174i		−	0.230953i
$928$		0			0
$929$	−57.2092			−1.87697			−0.938486	−	0.345316i	$-0.887772\pi$
$929$	−57.2092			−1.87697			−0.938486	+	0.345316i	$0.887772\pi$
$930$		0			0
$931$	−16.2987			−0.534167
$932$		0			0
$933$			20.5426i			0.672536i
$934$		0			0
$935$	−10.5819	+	26.3954i	−0.346066	+	0.863223i
$936$		0			0
$937$			38.4291i			1.25542i	0.778446	+	0.627712i	$0.216008\pi$
$937$			38.4291i			1.25542i	−0.778446	+	0.627712i	$0.783992\pi$
$938$		0			0
$939$	9.66705			0.315472
$940$		0			0
$941$	−8.66786			−0.282564			−0.141282	−	0.989969i	$-0.545122\pi$
$941$	−8.66786			−0.282564			−0.141282	+	0.989969i	$0.545122\pi$
$942$		0			0
$943$		−	9.68442i		−	0.315368i
$944$		0			0
$945$	−1.96091	−	0.786131i	−0.0637885	−	0.0255728i
$946$		0			0
$947$			7.24664i			0.235484i	0.993044	+	0.117742i	$0.0375656\pi$
$947$			7.24664i			0.235484i	−0.993044	+	0.117742i	$0.962434\pi$
$948$		0			0
$949$	−14.8610			−0.482408
$950$		0			0
$951$	23.7539			0.770272
$952$		0			0
$953$		−	19.7093i		−	0.638445i	−0.947680	−	0.319223i	$-0.896578\pi$
$953$		−	19.7093i		−	0.638445i	0.947680	−	0.319223i	$-0.103422\pi$
$954$		0			0
$955$	−22.4519	−	9.00098i	−0.726526	−	0.291265i
$956$		0			0
$957$		−	17.6524i		−	0.570621i
$958$		0			0
$959$	7.51667			0.242726
$960$		0			0
$961$	−18.5709			−0.599061
$962$		0			0
$963$			1.44335i			0.0465112i
$964$		0			0
$965$	5.87313	−	14.6499i	0.189063	−	0.471596i
$966$		0			0
$967$			36.6173i			1.17753i	0.808304	+	0.588766i	$0.200386\pi$
$967$			36.6173i			1.17753i	−0.808304	+	0.588766i	$0.799614\pi$
$968$		0			0
$969$	−6.03265			−0.193797
$970$		0			0
$971$	0.955869			0.0306753			0.0153376	−	0.999882i	$-0.495118\pi$
$971$	0.955869			0.0306753			0.0153376	+	0.999882i	$0.495118\pi$
$972$		0			0
$973$		−	5.16368i		−	0.165540i
$974$		0			0
$975$	14.7067	+	14.0500i	0.470992	+	0.449959i
$976$		0			0
$977$		−	35.2714i		−	1.12843i	−0.825628	−	0.564215i	$-0.809179\pi$
$977$		−	35.2714i		−	1.12843i	0.825628	−	0.564215i	$-0.190821\pi$
$978$		0			0
$979$	−30.9837			−0.990242
$980$		0			0
$981$	−10.6818			−0.341042
$982$		0			0
$983$		−	58.2400i		−	1.85757i	−0.370620	−	0.928784i	$-0.620855\pi$
$983$		−	58.2400i		−	1.85757i	0.370620	−	0.928784i	$-0.379145\pi$
$984$		0			0
$985$	2.71295	−	6.76714i	0.0864418	−	0.215619i
$986$		0			0
$987$			9.64965i			0.307152i
$988$		0			0
$989$	−5.82717			−0.185293
$990$		0			0
$991$	18.3179			0.581889			0.290944	−	0.956740i	$-0.406031\pi$
$991$	18.3179			0.581889			0.290944	+	0.956740i	$0.406031\pi$
$992$		0			0
$993$		−	17.4135i		−	0.552601i
$994$		0			0
$995$	38.4658	+	15.4210i	1.21945	+	0.488877i
$996$		0			0
$997$			42.6682i			1.35132i	0.737215	+	0.675658i	$0.236141\pi$
$997$			42.6682i			1.35132i	−0.737215	+	0.675658i	$0.763859\pi$
$998$		0			0
$999$	−7.10768			−0.224877

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.2	✓	24
5.4	even	2	inner	4020.2.g.b.1609.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.2	✓	24	1.1	even	1	trivial
4020.2.g.b.1609.14	yes	24	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.07549 - 0.832066i) q^{5} -0.944795i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-2.07549 - 0.832066i) q^{5} -0.944795i q^{7} -1.00000 q^{9} -5.62597 q^{11} +4.06787i q^{13} +(-0.832066 + 2.07549i) q^{15} +2.26053i q^{17} -2.66869 q^{19} -0.944795 q^{21} -4.72171i q^{23} +(3.61533 + 3.45389i) q^{25} +1.00000i q^{27} -3.13766 q^{29} +3.52550 q^{31} +5.62597i q^{33} +(-0.786131 + 1.96091i) q^{35} +7.10768i q^{37} +4.06787 q^{39} +2.05104 q^{41} -1.23412i q^{43} +(2.07549 + 0.832066i) q^{45} -10.2135i q^{47} +6.10736 q^{49} +2.26053 q^{51} +1.98447i q^{53} +(11.6767 + 4.68118i) q^{55} +2.66869i q^{57} +3.82526 q^{59} -4.66924 q^{61} +0.944795i q^{63} +(3.38473 - 8.44283i) q^{65} +1.00000i q^{67} -4.72171 q^{69} +10.9005 q^{71} +3.65326i q^{73} +(3.45389 - 3.61533i) q^{75} +5.31539i q^{77} -4.89735 q^{79} +1.00000 q^{81} +0.492793i q^{83} +(1.88091 - 4.69171i) q^{85} +3.13766i q^{87} +5.50725 q^{89} +3.84330 q^{91} -3.52550i q^{93} +(5.53884 + 2.22053i) q^{95} -8.86639i q^{97} +5.62597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 24 * q - 4 * q^5 - 24 * q^9	\( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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