LMFDB - Embedded newform 420.2.f.a.209.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(209,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("420.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.31116960000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 $ x^8 + x^6 - 8x^4 + 9x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			209.6
Root			$0.306808 + 1.70466i$ of defining polynomial
Character	$\chi$	$=$	420.209
Dual form			420.2.f.a.209.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.306808 + 1.70466i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(-2.81174 + 1.04601i) q^{9} +O(q^{10})$	$q+(0.306808 + 1.70466i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(-2.81174 + 1.04601i) q^{9} -5.55612i q^{11} +7.13235 q^{13} +(3.81174 - 0.686044i) q^{15} +5.75583i q^{17} +(0.811738 + 4.51011i) q^{21} -5.00000 q^{25} +(-2.64575 - 4.47214i) q^{27} +4.83619i q^{29} +(9.47129 - 1.70466i) q^{33} -5.91608i q^{35} +(2.18826 + 12.1582i) q^{39} +(2.33894 + 6.28724i) q^{45} -1.28369i q^{47} +7.00000 q^{49} +(-9.81174 + 1.76593i) q^{51} -12.4239 q^{55} +(-7.43916 + 2.76748i) q^{63} -15.9484i q^{65} -11.8322i q^{71} -10.5830 q^{73} +(-1.53404 - 8.52330i) q^{75} -14.7001i q^{77} -14.8704 q^{79} +(6.81174 - 5.88220i) q^{81} +8.94427i q^{83} +12.8704 q^{85} +(-8.24406 + 1.48378i) q^{87} +18.8704 q^{91} +3.45065 q^{97} +(5.81174 + 15.6223i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{9}+O(q^{10}) $ 8 * q - 2 * q^9	$ 8 q - 2 q^{9} + 10 q^{15} - 14 q^{21} - 40 q^{25} + 38 q^{39} + 56 q^{49} - 58 q^{51} + 4 q^{79} + 34 q^{81} - 20 q^{85} + 28 q^{91} + 26 q^{99}+O(q^{100}) $ 8 * q - 2 * q^9 + 10 * q^15 - 14 * q^21 - 40 * q^25 + 38 * q^39 + 56 * q^49 - 58 * q^51 + 4 * q^79 + 34 * q^81 - 20 * q^85 + 28 * q^91 + 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	0.306808	+	1.70466i	0.177136	+	0.984186i
$3$	0.306808	+	1.70466i	0.177136	+	0.984186i
$4$		0			0
$5$		−	2.23607i		−	1.00000i
$5$		−	2.23607i		−	1.00000i
$6$		0			0
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$		0			0
$9$	−2.81174	+	1.04601i	−0.937246	+	0.348669i
$10$		0			0
$11$		−	5.55612i		−	1.67523i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$		−	5.55612i		−	1.67523i	0.546259	−	0.837616i	$-0.316051\pi$
$12$		0			0
$13$	7.13235			1.97816			0.989079	−	0.147386i	$-0.0470859\pi$
$13$	7.13235			1.97816			0.989079	+	0.147386i	$0.0470859\pi$
$14$		0			0
$15$	3.81174	−	0.686044i	0.984186	−	0.177136i
$16$		0			0
$17$			5.75583i			1.39599i	0.716101	+	0.697997i	$0.245925\pi$
$17$			5.75583i			1.39599i	−0.716101	+	0.697997i	$0.754075\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	0.811738	+	4.51011i	0.177136	+	0.984186i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$	−2.64575	−	4.47214i	−0.509175	−	0.860663i
$28$		0			0
$29$			4.83619i			0.898058i	0.893517	+	0.449029i	$0.148230\pi$
$29$			4.83619i			0.898058i	−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	9.47129	−	1.70466i	1.64874	−	0.296743i
$34$		0			0
$35$		−	5.91608i		−	1.00000i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	2.18826	+	12.1582i	0.350402	+	1.94688i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	2.33894	+	6.28724i	0.348669	+	0.937246i
$46$		0			0
$47$		−	1.28369i		−	0.187246i	−0.995608	−	0.0936230i	$-0.970155\pi$
$47$		−	1.28369i		−	0.187246i	0.995608	−	0.0936230i	$-0.0298448\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−9.81174	+	1.76593i	−1.37392	+	0.247280i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−12.4239			−1.67523
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−7.43916	+	2.76748i	−0.937246	+	0.348669i
$64$		0			0
$65$		−	15.9484i		−	1.97816i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.8322i		−	1.40422i	−0.712069	−	0.702109i	$-0.752242\pi$
$71$		−	11.8322i		−	1.40422i	0.712069	−	0.702109i	$-0.247758\pi$
$72$		0			0
$73$	−10.5830			−1.23865			−0.619324	−	0.785136i	$-0.712593\pi$
$73$	−10.5830			−1.23865			−0.619324	+	0.785136i	$0.712593\pi$
$74$		0			0
$75$	−1.53404	−	8.52330i	−0.177136	−	0.984186i
$76$		0			0
$77$		−	14.7001i		−	1.67523i
$78$		0			0
$79$	−14.8704			−1.67305			−0.836527	−	0.547926i	$-0.815418\pi$
$79$	−14.8704			−1.67305			−0.836527	+	0.547926i	$0.815418\pi$
$80$		0			0
$81$	6.81174	−	5.88220i	0.756860	−	0.653577i
$82$		0			0
$83$			8.94427i			0.981761i	0.871227	+	0.490881i	$0.163325\pi$
$83$			8.94427i			0.981761i	−0.871227	+	0.490881i	$0.836675\pi$
$84$		0			0
$85$	12.8704			1.39599
$86$		0			0
$87$	−8.24406	+	1.48378i	−0.883856	+	0.159078i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	18.8704			1.97816
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.45065			0.350361			0.175180	−	0.984536i	$-0.443949\pi$
$97$	3.45065			0.350361			0.175180	+	0.984536i	$0.443949\pi$
$98$		0			0
$99$	5.81174	+	15.6223i	0.584102	+	1.57010i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−16.1055			−1.58693			−0.793463	−	0.608618i	$-0.791724\pi$
$103$	−16.1055			−1.58693			−0.793463	+	0.608618i	$0.791724\pi$
$104$		0			0
$105$	10.0849	−	1.81510i	0.984186	−	0.177136i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−20.8704			−1.99902			−0.999512	−	0.0312328i	$-0.990057\pi$
$109$	−20.8704			−1.99902			−0.999512	+	0.0312328i	$0.990057\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−20.0543	+	7.46049i	−1.85402	+	0.689723i
$118$		0			0
$119$			15.2285i			1.39599i
$120$		0			0
$121$	−19.8704			−1.80640
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−10.0000	+	5.91608i	−0.860663	+	0.509175i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	2.18826	−	0.393847i	0.184285	−	0.0331679i
$142$		0			0
$143$		−	39.6282i		−	3.31387i
$144$		0			0
$145$	10.8140			0.898058
$146$		0			0
$147$	2.14766	+	11.9326i	0.177136	+	0.984186i
$148$		0			0
$149$			23.6643i			1.93866i	0.245770	+	0.969328i	$0.420959\pi$
$149$			23.6643i			1.93866i	−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	6.87043			0.559107			0.279554	−	0.960130i	$-0.409814\pi$
$151$	6.87043			0.559107			0.279554	+	0.960130i	$0.409814\pi$
$152$		0			0
$153$	−6.02064	−	16.1839i	−0.486740	−	1.30839i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	21.1660			1.68923			0.844616	−	0.535373i	$-0.179829\pi$
$157$	21.1660			1.68923			0.844616	+	0.535373i	$0.179829\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	−3.81174	−	21.1785i	−0.296743	−	1.64874i
$166$		0			0
$167$			19.1722i			1.48359i	0.670625	+	0.741796i	$0.266026\pi$
$167$			19.1722i			1.48359i	−0.670625	+	0.741796i	$0.733974\pi$
$168$		0			0
$169$	37.8704			2.91311
$170$		0			0
$171$		0			0
$172$		0			0
$173$			26.2118i			1.99284i	0.0845218	+	0.996422i	$0.473064\pi$
$173$			26.2118i			1.99284i	−0.0845218	+	0.996422i	$0.526936\pi$
$174$		0			0
$175$	−13.2288			−1.00000
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	11.8322i		−	0.884377i	−0.896922	−	0.442189i	$-0.854202\pi$
$179$		−	11.8322i		−	0.884377i	0.896922	−	0.442189i	$-0.145798\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	31.9801			2.33861
$188$		0			0
$189$	−7.00000	−	11.8322i	−0.509175	−	0.860663i
$190$		0			0
$191$		−	26.3407i		−	1.90595i	−0.303052	−	0.952974i	$-0.598006\pi$
$191$		−	26.3407i		−	1.90595i	0.303052	−	0.952974i	$-0.401994\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	27.1866	−	4.89310i	1.94688	−	0.350402i
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			12.7954i			0.898058i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−26.8704			−1.84984			−0.924918	−	0.380166i	$-0.875867\pi$
$211$	−26.8704			−1.84984			−0.924918	+	0.380166i	$0.875867\pi$
$212$		0			0
$213$	20.1698	−	3.63020i	1.38201	−	0.248737i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−3.24695	−	18.0404i	−0.219409	−	1.21906i
$220$		0			0
$221$			41.0526i			2.76150i
$222$		0			0
$223$	−8.74216			−0.585418			−0.292709	−	0.956202i	$-0.594557\pi$
$223$	−8.74216			−0.585418			−0.292709	+	0.956202i	$0.594557\pi$
$224$		0			0
$225$	14.0587	−	5.23004i	0.937246	−	0.348669i
$226$		0			0
$227$		−	7.66058i		−	0.508450i	−0.967145	−	0.254225i	$-0.918180\pi$
$227$		−	7.66058i		−	0.508450i	0.967145	−	0.254225i	$-0.0818204\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	25.0587	−	4.51011i	1.64874	−	0.296743i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−2.87043			−0.187246
$236$		0			0
$237$	−4.56237	−	25.3490i	−0.296358	−	1.64660i
$238$		0			0
$239$			6.99597i			0.452532i	0.974066	+	0.226266i	$0.0726518\pi$
$239$			6.99597i			0.452532i	−0.974066	+	0.226266i	$0.927348\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	12.1170	+	9.80700i	0.777309	+	0.629119i
$244$		0			0
$245$		−	15.6525i		−	1.00000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−15.2470	+	2.74417i	−0.966236	+	0.173905i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	3.94875	+	21.9397i	0.247280	+	1.37392i
$256$		0			0
$257$		−	4.47214i		−	0.278964i	−0.990225	−	0.139482i	$-0.955456\pi$
$257$		−	4.47214i		−	0.278964i	0.990225	−	0.139482i	$-0.0445438\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−5.05869	−	13.5981i	−0.313125	−	0.841701i
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	5.78960	+	32.1677i	0.350402	+	1.94688i
$274$		0			0
$275$			27.7806i			1.67523i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			25.6208i			1.52841i	0.644974	+	0.764204i	$0.276868\pi$
$281$			25.6208i			1.52841i	−0.644974	+	0.764204i	$0.723132\pi$
$282$		0			0
$283$	30.3703			1.80532			0.902662	−	0.430350i	$-0.141610\pi$
$283$	30.3703			1.80532			0.902662	+	0.430350i	$0.141610\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.1296			−0.948798
$290$		0			0
$291$	1.05869	+	5.88220i	0.0620614	+	0.344820i
$292$		0			0
$293$		−	8.32322i		−	0.486248i	−0.969995	−	0.243124i	$-0.921828\pi$
$293$		−	8.32322i		−	0.486248i	0.969995	−	0.243124i	$-0.0781721\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−24.8477	+	14.7001i	−1.44181	+	0.852986i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	23.0069			1.31307			0.656535	−	0.754295i	$-0.272021\pi$
$307$	23.0069			1.31307			0.656535	+	0.754295i	$0.272021\pi$
$308$		0			0
$309$	−4.94131	−	27.4545i	−0.281101	−	1.56183i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−31.9801			−1.80762			−0.903810	−	0.427934i	$-0.859241\pi$
$313$	−31.9801			−1.80762			−0.903810	+	0.427934i	$0.859241\pi$
$314$		0			0
$315$	6.18826	+	16.6345i	0.348669	+	0.937246i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$	26.8704			1.50446
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−35.6618			−1.97816
$326$		0			0
$327$	−6.40321	−	35.5770i	−0.354099	−	1.96741i
$328$		0			0
$329$		−	3.39633i		−	0.187246i
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	18.5203			1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	−18.8704	−	31.8968i	−1.00723	−	1.70253i
$352$		0			0
$353$		−	35.1560i		−	1.87117i	−0.353106	−	0.935583i	$-0.614874\pi$
$353$		−	35.1560i		−	1.87117i	0.353106	−	0.935583i	$-0.385126\pi$
$354$		0			0
$355$	−26.4575			−1.40422
$356$		0			0
$357$	−25.9594	+	4.67222i	−1.37392	+	0.247280i
$358$		0			0
$359$		−	11.8322i		−	0.624477i	−0.950004	−	0.312239i	$-0.898921\pi$
$359$		−	11.8322i		−	0.624477i	0.950004	−	0.312239i	$-0.101079\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	−6.09641	−	33.8723i	−0.319978	−	1.77784i
$364$		0			0
$365$			23.6643i			1.23865i
$366$		0			0
$367$	−19.7872			−1.03289			−0.516443	−	0.856322i	$-0.672744\pi$
$367$	−19.7872			−1.03289			−0.516443	+	0.856322i	$0.672744\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−19.0587	+	3.43022i	−0.984186	+	0.177136i
$376$		0			0
$377$			34.4934i			1.77650i
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			35.7771i			1.82812i	0.405575	+	0.914062i	$0.367071\pi$
$383$			35.7771i			1.82812i	−0.405575	+	0.914062i	$0.632929\pi$
$384$		0			0
$385$	−32.8704			−1.67523
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	36.7330i		−	1.86244i	−0.364459	−	0.931219i	$-0.618746\pi$
$389$		−	36.7330i		−	1.86244i	0.364459	−	0.931219i	$-0.381254\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			33.2513i			1.67305i
$396$		0			0
$397$	−0.231042			−0.0115956			−0.00579782	−	0.999983i	$-0.501846\pi$
$397$	−0.231042			−0.0115956			−0.00579782	+	0.999983i	$0.501846\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			38.1729i			1.90626i	0.302556	+	0.953131i	$0.402160\pi$
$401$			38.1729i			1.90626i	−0.302556	+	0.953131i	$0.597840\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−13.1530	−	15.2315i	−0.653577	−	0.756860i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	20.0000			0.981761
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	3.12957			0.152526			0.0762630	−	0.997088i	$-0.475701\pi$
$421$	3.12957			0.152526			0.0762630	+	0.997088i	$0.475701\pi$
$422$		0			0
$423$	1.34275	+	3.60941i	0.0652869	+	0.175496i
$424$		0			0
$425$		−	28.7791i		−	1.39599i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	67.5526	−	12.1582i	3.26147	−	0.587005i
$430$		0			0
$431$		−	18.1082i		−	0.872241i	−0.899888	−	0.436121i	$-0.856352\pi$
$431$		−	18.1082i		−	0.872241i	0.899888	−	0.436121i	$-0.143648\pi$
$432$		0			0
$433$	−10.5830			−0.508587			−0.254293	−	0.967127i	$-0.581843\pi$
$433$	−10.5830			−0.508587			−0.254293	+	0.967127i	$0.581843\pi$
$434$		0			0
$435$	3.31784	+	18.4343i	0.159078	+	0.883856i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−19.6822	+	7.32205i	−0.937246	+	0.348669i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−40.3396	+	7.26040i	−1.90800	+	0.343405i
$448$		0			0
$449$		−	28.5005i		−	1.34502i	−0.740087	−	0.672511i	$-0.765216\pi$
$449$		−	28.5005i		−	1.34502i	0.740087	−	0.672511i	$-0.234784\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.10790	+	11.7117i	0.0990379	+	0.550266i
$454$		0			0
$455$		−	42.1956i		−	1.97816i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	25.7409	−	15.2285i	1.20148	−	0.710805i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	28.1165i		−	1.30108i	−0.759473	−	0.650538i	$-0.774543\pi$
$467$		−	28.1165i		−	1.30108i	0.759473	−	0.650538i	$-0.225457\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	6.49390	+	36.0809i	0.299223	+	1.66252i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	7.71590i		−	0.350361i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	13.7886i		−	0.622273i	−0.950365	−	0.311136i	$-0.899290\pi$
$491$		−	13.7886i		−	0.622273i	0.950365	−	0.311136i	$-0.100710\pi$
$492$		0			0
$493$	−27.8363			−1.25368
$494$		0			0
$495$	34.9326	−	12.9954i	1.57010	−	0.584102i
$496$		0			0
$497$		−	31.3050i		−	1.40422i
$498$		0			0
$499$	−5.12957			−0.229631			−0.114816	−	0.993387i	$-0.536628\pi$
$499$	−5.12957			−0.229631			−0.114816	+	0.993387i	$0.536628\pi$
$500$		0			0
$501$	−32.6822	+	5.88220i	−1.46013	+	0.262797i
$502$		0			0
$503$			39.6282i			1.76693i	0.468495	+	0.883466i	$0.344797\pi$
$503$			39.6282i			1.76693i	−0.468495	+	0.883466i	$0.655203\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	11.6190	+	64.5562i	0.516016	+	2.86704i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−28.0000			−1.23865
$512$		0			0
$513$		0			0
$514$		0			0
$515$			36.0131i			1.58693i
$516$		0			0
$517$	−7.13235			−0.313680
$518$		0			0
$519$	−44.6822	+	8.04198i	−1.96133	+	0.353004i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	37.0405			1.61967			0.809834	−	0.586659i	$-0.199557\pi$
$523$	37.0405			1.61967			0.809834	+	0.586659i	$0.199557\pi$
$524$		0			0
$525$	−4.05869	−	22.5505i	−0.177136	−	0.984186i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	20.1698	−	3.63020i	0.870392	−	0.156655i
$538$		0			0
$539$		−	38.8928i		−	1.67523i
$540$		0			0
$541$	36.8704			1.58518			0.792592	−	0.609753i	$-0.208731\pi$
$541$	36.8704			1.58518			0.792592	+	0.609753i	$0.208731\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			46.6677i			1.99902i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−39.3434			−1.67305
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	9.81174	+	54.5152i	0.414252	+	2.30163i
$562$		0			0
$563$		−	44.7214i		−	1.88478i	−0.334515	−	0.942390i	$-0.608573\pi$
$563$		−	44.7214i		−	1.88478i	0.334515	−	0.942390i	$-0.391427\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	18.0222	−	15.5628i	0.756860	−	0.653577i
$568$		0			0
$569$		−	47.3286i		−	1.98412i	−0.125767	−	0.992060i	$-0.540139\pi$
$569$		−	47.3286i		−	1.98412i	0.125767	−	0.992060i	$-0.459861\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$		0			0
$573$	44.9020	−	8.08155i	1.87581	−	0.337611i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	46.2448			1.92519			0.962597	−	0.270936i	$-0.0873333\pi$
$577$	46.2448			1.92519			0.962597	+	0.270936i	$0.0873333\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			23.6643i			0.981761i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	16.6822	+	44.8428i	0.689723	+	1.85402i
$586$		0			0
$587$			8.94427i			0.369170i	0.982817	+	0.184585i	$0.0590940\pi$
$587$			8.94427i			0.369170i	−0.982817	+	0.184585i	$0.940906\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	0.621055i		−	0.0255037i	−0.999919	−	0.0127518i	$-0.995941\pi$
$593$		−	0.621055i		−	0.0255037i	0.999919	−	0.0127518i	$-0.00405915\pi$
$594$		0			0
$595$	34.0519			1.39599
$596$		0			0
$597$		0			0
$598$		0			0
$599$			48.5652i			1.98432i	0.124975	+	0.992160i	$0.460115\pi$
$599$			48.5652i			1.98432i	−0.124975	+	0.992160i	$0.539885\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			44.4316i			1.80640i
$606$		0			0
$607$	−5.06046			−0.205398			−0.102699	−	0.994712i	$-0.532748\pi$
$607$	−5.06046			−0.205398			−0.102699	+	0.994712i	$0.532748\pi$
$608$		0			0
$609$	−21.8117	+	3.92572i	−0.883856	+	0.159078i
$610$		0			0
$611$		−	9.15575i		−	0.370402i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−38.8704			−1.54741			−0.773704	−	0.633548i	$-0.781598\pi$
$631$	−38.8704			−1.54741			−0.773704	+	0.633548i	$0.781598\pi$
$632$		0			0
$633$	−8.24406	−	45.8050i	−0.327672	−	1.82058i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	49.9265			1.97816
$638$		0			0
$639$	12.3765	+	33.2689i	0.489608	+	1.31610i
$640$		0			0
$641$		−	47.3286i		−	1.86937i	−0.355479	−	0.934684i	$-0.615682\pi$
$641$		−	47.3286i		−	1.86937i	0.355479	−	0.934684i	$-0.384318\pi$
$642$		0			0
$643$	19.3252			0.762110			0.381055	−	0.924552i	$-0.375561\pi$
$643$	19.3252			0.762110			0.381055	+	0.924552i	$0.375561\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	17.8885i		−	0.703271i	−0.936137	−	0.351636i	$-0.885626\pi$
$647$		−	17.8885i		−	0.703271i	0.936137	−	0.351636i	$-0.114374\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	29.7566	−	11.0699i	1.16092	−	0.431878i
$658$		0			0
$659$		−	47.1253i		−	1.83574i	−0.396878	−	0.917871i	$-0.629907\pi$
$659$		−	47.1253i		−	1.83574i	0.396878	−	0.917871i	$-0.370093\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	−69.9808	+	12.5953i	−2.71783	+	0.489160i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2.68216	−	14.9024i	−0.103698	−	0.576161i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	13.2288	+	22.3607i	0.509175	+	0.860663i
$676$		0			0
$677$			32.5886i			1.25248i	0.779629	+	0.626242i	$0.215408\pi$
$677$			32.5886i			1.25248i	−0.779629	+	0.626242i	$0.784592\pi$
$678$		0			0
$679$	9.12957			0.350361
$680$		0			0
$681$	13.0587	−	2.35033i	0.500410	−	0.0900647i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	15.3764	+	41.3328i	0.584102	+	1.57010i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	49.2851i		−	1.86147i	−0.365690	−	0.930737i	$-0.619167\pi$
$701$		−	49.2851i		−	1.86147i	0.365690	−	0.930737i	$-0.380833\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−0.880670	−	4.89310i	−0.0331679	−	0.184285i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	46.6113			1.75052			0.875262	−	0.483650i	$-0.160689\pi$
$709$	46.6113			1.75052			0.875262	+	0.483650i	$0.160689\pi$
$710$		0			0
$711$	41.8117	−	15.5546i	1.56806	−	0.583342i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−88.6113			−3.31387
$716$		0			0
$717$	−11.9258	+	2.14642i	−0.445376	+	0.0801595i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−42.6113			−1.58693
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	24.1809i		−	0.898058i
$726$		0			0
$727$	5.29150			0.196251			0.0981255	−	0.995174i	$-0.468715\pi$
$727$	5.29150			0.196251			0.0981255	+	0.995174i	$0.468715\pi$
$728$		0			0
$729$	−13.0000	+	23.6643i	−0.481481	+	0.876456i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	42.5631			1.57210			0.786051	−	0.618161i	$-0.212122\pi$
$733$	42.5631			1.57210			0.786051	+	0.618161i	$0.212122\pi$
$734$		0			0
$735$	26.6822	−	4.80230i	0.984186	−	0.177136i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	40.6113			1.49391			0.746955	−	0.664875i	$-0.231515\pi$
$739$	40.6113			1.49391			0.746955	+	0.664875i	$0.231515\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	52.9150			1.93866
$746$		0			0
$747$	−9.35577	−	25.1489i	−0.342310	−	0.920152i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	52.6113			1.91981			0.959906	−	0.280321i	$-0.0904408\pi$
$751$	52.6113			1.91981			0.959906	+	0.280321i	$0.0904408\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	15.3627i		−	0.559107i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−55.2180			−1.99902
$764$		0			0
$765$	−36.1883	+	13.4626i	−1.30839	+	0.486740i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	7.62348	−	1.37209i	0.274553	−	0.0494145i
$772$		0			0
$773$		−	49.2351i		−	1.77086i	−0.464770	−	0.885431i	$-0.653863\pi$
$773$		−	49.2351i		−	1.77086i	0.464770	−	0.885431i	$-0.346137\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−65.7409			−2.35239
$782$		0			0
$783$	21.6281	−	12.7954i	0.772925	−	0.457269i
$784$		0			0
$785$		−	47.3286i		−	1.68923i
$786$		0			0
$787$	−51.5363			−1.83707			−0.918535	−	0.395340i	$-0.870627\pi$
$787$	−51.5363			−1.83707			−0.918535	+	0.395340i	$0.870627\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	21.0770i		−	0.746585i	−0.927714	−	0.373293i	$-0.878229\pi$
$797$		−	21.0770i		−	0.746585i	0.927714	−	0.373293i	$-0.121771\pi$
$798$		0			0
$799$	7.38872			0.261394
$800$		0			0
$801$		0			0
$802$		0			0
$803$			58.8004i			2.07502i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			13.0687i			0.459471i	0.973253	+	0.229736i	$0.0737862\pi$
$809$			13.0687i			0.459471i	−0.973253	+	0.229736i	$0.926214\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−53.0587	+	19.7386i	−1.85402	+	0.689723i
$820$		0			0
$821$			46.4054i			1.61956i	0.586734	+	0.809780i	$0.300414\pi$
$821$			46.4054i			1.61956i	−0.586734	+	0.809780i	$0.699586\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−47.3565	+	8.52330i	−1.64874	+	0.296743i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			40.2908i			1.39599i
$834$		0			0
$835$	42.8704			1.48359
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	5.61128			0.193492
$842$		0			0
$843$	−43.6748	+	7.86067i	−1.50424	+	0.270736i
$844$		0			0
$845$		−	84.6808i		−	2.91311i
$846$		0			0
$847$	−52.5722			−1.80640
$848$		0			0
$849$	9.31784	+	51.7710i	0.319787	+	1.77678i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−42.3320			−1.44942			−0.724710	−	0.689054i	$-0.758026\pi$
$853$	−42.3320			−1.44942			−0.724710	+	0.689054i	$0.758026\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			49.1935i			1.68042i	0.542263	+	0.840209i	$0.317568\pi$
$857$			49.1935i			1.68042i	−0.542263	+	0.840209i	$0.682432\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	58.6113			1.99284
$866$		0			0
$867$	−4.94868	−	27.4955i	−0.168066	−	0.933795i
$868$		0			0
$869$			82.6218i			2.80275i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−9.70234	+	3.60941i	−0.328374	+	0.122160i
$874$		0			0
$875$			29.5804i			1.00000i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	14.1883	−	2.55363i	0.478558	−	0.0861318i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			35.7771i			1.20128i	0.799521	+	0.600639i	$0.205087\pi$
$887$			35.7771i			1.20128i	−0.799521	+	0.600639i	$0.794913\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−32.6822	−	37.8468i	−1.09489	−	1.26792i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−26.4575			−0.884377
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			59.1608i			1.96008i	0.198789	+	0.980042i	$0.436299\pi$
$911$			59.1608i			1.96008i	−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$	49.6954			1.64468
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−60.6113			−1.99938			−0.999691	−	0.0248659i	$-0.992084\pi$
$919$	−60.6113			−1.99938			−0.999691	+	0.0248659i	$0.992084\pi$
$920$		0			0
$921$	7.05869	+	39.2189i	0.232592	+	1.29231i
$922$		0			0
$923$		−	84.3911i		−	2.77777i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	45.2846	−	16.8465i	1.48734	−	0.553312i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	71.5096i		−	2.33861i
$936$		0			0
$937$	−24.6167			−0.804191			−0.402096	−	0.915598i	$-0.631718\pi$
$937$	−24.6167			−0.804191			−0.402096	+	0.915598i	$0.631718\pi$
$938$		0			0
$939$	−9.81174	−	54.5152i	−0.320194	−	1.77903i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−26.4575	+	15.6525i	−0.860663	+	0.509175i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	−75.4817			−2.45024
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−58.8997			−1.90595
$956$		0			0
$957$	8.24406	+	45.8050i	0.266493	+	1.48066i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−10.9413	−	60.7912i	−0.350402	−	1.94688i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	58.6822	−	21.8306i	1.87358	−	0.696998i
$982$		0			0
$983$		−	62.6515i		−	1.99827i	−0.0415592	−	0.999136i	$-0.513233\pi$
$983$		−	62.6515i		−	1.99827i	0.0415592	−	0.999136i	$-0.486767\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	5.78960	−	1.04202i	0.184285	−	0.0331679i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−52.0000			−1.65183			−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000			−1.65183			−0.825917	+	0.563791i	$0.809342\pi$
$992$		0			0
$993$	2.45446	+	13.6373i	0.0778901	+	0.432766i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−43.0251			−1.36262			−0.681310	−	0.731995i	$-0.738589\pi$
$997$	−43.0251			−1.36262			−0.681310	+	0.731995i	$0.738589\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.f.a.209.6	yes	8
3.2	odd	2	inner	420.2.f.a.209.5	yes	8
4.3	odd	2		1680.2.k.d.209.3		8
5.2	odd	4		2100.2.d.j.1301.7		8
5.3	odd	4		2100.2.d.j.1301.2		8
5.4	even	2	inner	420.2.f.a.209.3	✓	8
7.6	odd	2	inner	420.2.f.a.209.3	✓	8
12.11	even	2		1680.2.k.d.209.4		8
15.2	even	4		2100.2.d.j.1301.1		8
15.8	even	4		2100.2.d.j.1301.8		8
15.14	odd	2	inner	420.2.f.a.209.4	yes	8
20.19	odd	2		1680.2.k.d.209.6		8
21.20	even	2	inner	420.2.f.a.209.4	yes	8
28.27	even	2		1680.2.k.d.209.6		8
35.13	even	4		2100.2.d.j.1301.7		8
35.27	even	4		2100.2.d.j.1301.2		8
35.34	odd	2	CM	420.2.f.a.209.6	yes	8
60.59	even	2		1680.2.k.d.209.5		8
84.83	odd	2		1680.2.k.d.209.5		8
105.62	odd	4		2100.2.d.j.1301.8		8
105.83	odd	4		2100.2.d.j.1301.1		8
105.104	even	2	inner	420.2.f.a.209.5	yes	8
140.139	even	2		1680.2.k.d.209.3		8
420.419	odd	2		1680.2.k.d.209.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.f.a.209.3	✓	8	5.4	even	2	inner
420.2.f.a.209.3	✓	8	7.6	odd	2	inner
420.2.f.a.209.4	yes	8	15.14	odd	2	inner
420.2.f.a.209.4	yes	8	21.20	even	2	inner
420.2.f.a.209.5	yes	8	3.2	odd	2	inner
420.2.f.a.209.5	yes	8	105.104	even	2	inner
420.2.f.a.209.6	yes	8	1.1	even	1	trivial
420.2.f.a.209.6	yes	8	35.34	odd	2	CM
1680.2.k.d.209.3		8	4.3	odd	2
1680.2.k.d.209.3		8	140.139	even	2
1680.2.k.d.209.4		8	12.11	even	2
1680.2.k.d.209.4		8	420.419	odd	2
1680.2.k.d.209.5		8	60.59	even	2
1680.2.k.d.209.5		8	84.83	odd	2
1680.2.k.d.209.6		8	20.19	odd	2
1680.2.k.d.209.6		8	28.27	even	2
2100.2.d.j.1301.1		8	15.2	even	4
2100.2.d.j.1301.1		8	105.83	odd	4
2100.2.d.j.1301.2		8	5.3	odd	4
2100.2.d.j.1301.2		8	35.27	even	4
2100.2.d.j.1301.7		8	5.2	odd	4
2100.2.d.j.1301.7		8	35.13	even	4
2100.2.d.j.1301.8		8	15.8	even	4
2100.2.d.j.1301.8		8	105.62	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.306808 + 1.70466i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(-2.81174 + 1.04601i) q^{9} +O(q^{10})\)	\(q+(0.306808 + 1.70466i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(-2.81174 + 1.04601i) q^{9} -5.55612i q^{11} +7.13235 q^{13} +(3.81174 - 0.686044i) q^{15} +5.75583i q^{17} +(0.811738 + 4.51011i) q^{21} -5.00000 q^{25} +(-2.64575 - 4.47214i) q^{27} +4.83619i q^{29} +(9.47129 - 1.70466i) q^{33} -5.91608i q^{35} +(2.18826 + 12.1582i) q^{39} +(2.33894 + 6.28724i) q^{45} -1.28369i q^{47} +7.00000 q^{49} +(-9.81174 + 1.76593i) q^{51} -12.4239 q^{55} +(-7.43916 + 2.76748i) q^{63} -15.9484i q^{65} -11.8322i q^{71} -10.5830 q^{73} +(-1.53404 - 8.52330i) q^{75} -14.7001i q^{77} -14.8704 q^{79} +(6.81174 - 5.88220i) q^{81} +8.94427i q^{83} +12.8704 q^{85} +(-8.24406 + 1.48378i) q^{87} +18.8704 q^{91} +3.45065 q^{97} +(5.81174 + 15.6223i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9}+O(q^{10}) \) 8 * q - 2 * q^9	\( 8 q - 2 q^{9} + 10 q^{15} - 14 q^{21} - 40 q^{25} + 38 q^{39} + 56 q^{49} - 58 q^{51} + 4 q^{79} + 34 q^{81} - 20 q^{85} + 28 q^{91} + 26 q^{99}+O(q^{100}) \) 8 * q - 2 * q^9 + 10 * q^15 - 14 * q^21 - 40 * q^25 + 38 * q^39 + 56 * q^49 - 58 * q^51 + 4 * q^79 + 34 * q^81 - 20 * q^85 + 28 * q^91 + 26 * q^99

Introduction

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