LMFDB - Embedded newform 420.2.f.a.209.1

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			209.1
Root			$-1.62968 - 0.586627i$ of defining polynomial
Character	$\chi$	$=$	420.209
Dual form			420.2.f.a.209.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.62968 - 0.586627i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(2.31174 + 1.91203i) q^{9} +O(q^{10})$	$q+(-1.62968 - 0.586627i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(2.31174 + 1.91203i) q^{9} -0.359964i q^{11} -4.48660 q^{13} +(-1.31174 + 3.64408i) q^{15} -7.99190i q^{17} +(-4.31174 - 1.55207i) q^{21} -5.00000 q^{25} +(-2.64575 - 4.47214i) q^{27} -10.7523i q^{29} +(-0.211164 + 0.586627i) q^{33} -5.91608i q^{35} +(7.31174 + 2.63196i) q^{39} +(4.27543 - 5.16920i) q^{45} +12.4640i q^{47} +7.00000 q^{49} +(-4.68826 + 13.0243i) q^{51} -0.804903 q^{55} +(6.11628 + 5.05876i) q^{63} +10.0323i q^{65} -11.8322i q^{71} -10.5830 q^{73} +(8.14842 + 2.93313i) q^{75} -0.952374i q^{77} +15.8704 q^{79} +(1.68826 + 8.84024i) q^{81} +8.94427i q^{83} -17.8704 q^{85} +(-6.30757 + 17.5228i) q^{87} -11.8704 q^{91} +15.0696 q^{97} +(0.688262 - 0.832142i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.62968	−	0.586627i	−0.940898	−	0.338689i
$3$	−1.62968	−	0.586627i	−0.940898	−	0.338689i
$4$		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
$9$	2.31174	+	1.91203i	0.770579	+	0.637344i
$10$		0			0
$11$		−	0.359964i		−	0.108533i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$		−	0.359964i		−	0.108533i	0.998526	−	0.0542666i	$-0.0172821\pi$
$12$		0			0
$13$	−4.48660			−1.24436			−0.622179	−	0.782875i	$-0.713753\pi$
$13$	−4.48660			−1.24436			−0.622179	+	0.782875i	$0.713753\pi$
$14$		0			0
$15$	−1.31174	+	3.64408i	−0.338689	+	0.940898i
$16$		0			0
$17$		−	7.99190i		−	1.93832i	−0.246433	−	0.969160i	$-0.579258\pi$
$17$		−	7.99190i		−	1.93832i	0.246433	−	0.969160i	$-0.420742\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	−4.31174	−	1.55207i	−0.940898	−	0.338689i
$22$		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
$29$		−	10.7523i		−	1.99665i	−0.0578882	−	0.998323i	$-0.518437\pi$
$29$		−	10.7523i		−	1.99665i	0.0578882	−	0.998323i	$-0.481563\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−0.211164	+	0.586627i	−0.0367590	+	0.102119i
$34$		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
$39$	7.31174	+	2.63196i	1.17082	+	0.421451i
$40$		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
$45$	4.27543	−	5.16920i	0.637344	−	0.770579i
$46$		0			0
$47$			12.4640i			1.81807i	0.416724	+	0.909033i	$0.363178\pi$
$47$			12.4640i			1.81807i	−0.416724	+	0.909033i	$0.636822\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−4.68826	+	13.0243i	−0.656488	+	1.82376i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−0.804903			−0.108533
$56$		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
$63$	6.11628	+	5.05876i	0.770579	+	0.637344i
$64$		0			0
$65$			10.0323i			1.24436i
$66$		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
$75$	8.14842	+	2.93313i	0.940898	+	0.338689i
$76$		0			0
$77$		−	0.952374i		−	0.108533i
$78$		0			0
$79$	15.8704			1.78556			0.892781	−	0.450490i	$-0.148751\pi$
$79$	15.8704			1.78556			0.892781	+	0.450490i	$0.148751\pi$
$80$		0			0
$81$	1.68826	+	8.84024i	0.187585	+	0.982248i
$82$		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
$85$	−17.8704			−1.93832
$86$		0			0
$87$	−6.30757	+	17.5228i	−0.676243	+	1.87864i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−11.8704			−1.24436
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	15.0696			1.53009			0.765043	−	0.643979i	$-0.222718\pi$
$97$	15.0696			1.53009			0.765043	+	0.643979i	$0.222718\pi$
$98$		0			0
$99$	0.688262	−	0.832142i	0.0691730	−	0.0836334i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	18.7513			1.84762			0.923810	−	0.382851i	$-0.125058\pi$
$103$	18.7513			1.84762			0.923810	+	0.382851i	$0.125058\pi$
$104$		0			0
$105$	−3.47053	+	9.64134i	−0.338689	+	0.940898i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	9.87043			0.945415			0.472708	−	0.881219i	$-0.343277\pi$
$109$	9.87043			0.945415			0.472708	+	0.881219i	$0.343277\pi$
$110$		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
$117$	−10.3718	−	8.57852i	−0.958877	−	0.793085i
$118$		0			0
$119$		−	21.1446i		−	1.93832i
$120$		0			0
$121$	10.8704			0.988221
$122$		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
$141$	7.31174	−	20.3124i	0.615759	−	1.71062i
$142$		0			0
$143$			1.61501i			0.135054i
$144$		0			0
$145$	−24.0428			−1.99665
$146$		0			0
$147$	−11.4078	−	4.10639i	−0.940898	−	0.338689i
$148$		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
$151$	−23.8704			−1.94255			−0.971274	−	0.237964i	$-0.923520\pi$
$151$	−23.8704			−1.94255			−0.971274	+	0.237964i	$0.923520\pi$
$152$		0			0
$153$	15.2808	−	18.4752i	1.23538	−	1.49363i
$154$		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
$165$	1.31174	+	0.472178i	0.102119	+	0.0367590i
$166$		0			0
$167$			5.42451i			0.419761i	0.977727	+	0.209881i	$0.0673075\pi$
$167$			5.42451i			0.419761i	−0.977727	+	0.209881i	$0.932692\pi$
$168$		0			0
$169$	7.12957			0.548429
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	15.0314i		−	1.14282i	−0.820666	−	0.571409i	$-0.806397\pi$
$173$		−	15.0314i		−	1.14282i	0.820666	−	0.571409i	$-0.193603\pi$
$174$		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
$187$	−2.87679			−0.210372
$188$		0			0
$189$	−7.00000	−	11.8322i	−0.509175	−	0.860663i
$190$		0			0
$191$			20.4246i			1.47788i	0.673774	+	0.738938i	$0.264672\pi$
$191$			20.4246i			1.47788i	−0.673774	+	0.738938i	$0.735328\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	5.88524	−	16.3495i	0.421451	−	1.17082i
$196$		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
$203$		−	28.4478i		−	1.99665i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	3.87043			0.266451			0.133226	−	0.991086i	$-0.457467\pi$
$211$	3.87043			0.266451			0.133226	+	0.991086i	$0.457467\pi$
$212$		0			0
$213$	−6.94106	+	19.2827i	−0.475594	+	1.32123i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	17.2470	+	6.20828i	1.16544	+	0.419516i
$220$		0			0
$221$			35.8564i			2.41197i
$222$		0			0
$223$	−20.3611			−1.36348			−0.681740	−	0.731594i	$-0.738777\pi$
$223$	−20.3611			−1.36348			−0.681740	+	0.731594i	$0.738777\pi$
$224$		0			0
$225$	−11.5587	−	9.56016i	−0.770579	−	0.637344i
$226$		0			0
$227$		−	21.4083i		−	1.42092i	−0.703738	−	0.710460i	$-0.748487\pi$
$227$		−	21.4083i		−	1.42092i	0.703738	−	0.710460i	$-0.251513\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	−0.558688	+	1.55207i	−0.0367590	+	0.102119i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	27.8704			1.81807
$236$		0			0
$237$	−25.8638	−	9.31002i	−1.68003	−	0.604751i
$238$		0			0
$239$			22.5844i			1.46087i	0.682985	+	0.730433i	$0.260682\pi$
$239$			22.5844i			1.46087i	−0.682985	+	0.730433i	$0.739318\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	2.43459	−	15.3972i	0.156179	−	0.987729i
$244$		0			0
$245$		−	15.6525i		−	1.00000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	5.24695	−	14.5763i	0.332512	−	0.923738i
$250$		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
$255$	29.1231	+	10.4833i	1.82376	+	0.656488i
$256$		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
$261$	20.5587	−	24.8564i	1.27255	−	1.53857i
$262$		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
$273$	19.3450	+	6.96351i	1.17082	+	0.421451i
$274$		0			0
$275$			1.79982i			0.108533i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	31.5369i		−	1.88133i	−0.339333	−	0.940666i	$-0.610201\pi$
$281$		−	31.5369i		−	1.88133i	0.339333	−	0.940666i	$-0.389799\pi$
$282$		0			0
$283$	−27.7245			−1.64805			−0.824025	−	0.566553i	$-0.808277\pi$
$283$	−27.7245			−1.64805			−0.824025	+	0.566553i	$0.808277\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−46.8704			−2.75708
$290$		0			0
$291$	−24.5587	−	8.84024i	−1.43966	−	0.518224i
$292$		0			0
$293$			32.9200i			1.92320i	0.274446	+	0.961602i	$0.411505\pi$
$293$			32.9200i			1.92320i	−0.274446	+	0.961602i	$0.588495\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.60981	+	0.952374i	−0.0934104	+	0.0552624i
$298$		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
$307$	11.3879			0.649942			0.324971	−	0.945724i	$-0.394645\pi$
$307$	11.3879			0.649942			0.324971	+	0.945724i	$0.394645\pi$
$308$		0			0
$309$	−30.5587	−	11.0000i	−1.73842	−	0.625769i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	2.87679			0.162606			0.0813030	−	0.996689i	$-0.474092\pi$
$313$	2.87679			0.162606			0.0813030	+	0.996689i	$0.474092\pi$
$314$		0			0
$315$	11.3117	−	13.6764i	0.637344	−	0.770579i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$	−3.87043			−0.216702
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	22.4330			1.24436
$326$		0			0
$327$	−16.0857	−	5.79026i	−0.889540	−	0.320202i
$328$		0			0
$329$			32.9767i			1.81807i
$330$		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
$351$	11.8704	+	20.0647i	0.633596	+	1.07097i
$352$		0			0
$353$			6.08715i			0.323986i	0.986792	+	0.161993i	$0.0517922\pi$
$353$			6.08715i			0.323986i	−0.986792	+	0.161993i	$0.948208\pi$
$354$		0			0
$355$	−26.4575			−1.40422
$356$		0			0
$357$	−12.4040	+	34.4590i	−0.656488	+	1.82376i
$358$		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
$363$	−17.7154	−	6.37688i	−0.929815	−	0.334700i
$364$		0			0
$365$			23.6643i			1.23865i
$366$		0			0
$367$	38.3075			1.99964			0.999818	−	0.0190919i	$-0.00607750\pi$
$367$	38.3075			1.99964			0.999818	+	0.0190919i	$0.00607750\pi$
$368$		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
$375$	6.55869	−	18.2204i	0.338689	−	0.940898i
$376$		0			0
$377$			48.2411i			2.48454i
$378$		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
$385$	−2.12957			−0.108533
$386$		0			0
$387$		0			0
$388$		0			0
$389$			30.8170i			1.56248i	0.624230	+	0.781241i	$0.285413\pi$
$389$			30.8170i			1.56248i	−0.624230	+	0.781241i	$0.714587\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	35.4874i		−	1.78556i
$396$		0			0
$397$	34.6258			1.73782			0.868910	−	0.494971i	$-0.164821\pi$
$397$	34.6258			1.73782			0.868910	+	0.494971i	$0.164821\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	8.59249i		−	0.429088i	−0.976714	−	0.214544i	$-0.931173\pi$
$401$		−	8.59249i		−	0.429088i	0.976714	−	0.214544i	$-0.0688266\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	19.7674	−	3.77507i	0.982248	−	0.187585i
$406$		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
$421$	33.8704			1.65074			0.825372	−	0.564590i	$-0.190966\pi$
$421$	33.8704			1.65074			0.825372	+	0.564590i	$0.190966\pi$
$422$		0			0
$423$	−23.8316	+	28.8136i	−1.15873	+	1.40096i
$424$		0			0
$425$			39.9595i			1.93832i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	0.947410	−	2.63196i	0.0457414	−	0.127072i
$430$		0			0
$431$		−	23.3044i		−	1.12253i	−0.827636	−	0.561266i	$-0.810315\pi$
$431$		−	23.3044i		−	1.12253i	0.827636	−	0.561266i	$-0.189685\pi$
$432$		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
$435$	39.1822	+	14.1042i	1.87864	+	0.676243i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	16.1822	+	13.3842i	0.770579	+	0.637344i
$442$		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
$447$	13.8821	−	38.5654i	0.656602	−	1.82408i
$448$		0			0
$449$		−	12.9121i		−	0.609357i	−0.952455	−	0.304679i	$-0.901451\pi$
$449$		−	12.9121i		−	0.609357i	0.952455	−	0.304679i	$-0.0985491\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	38.9012	+	14.0030i	1.82774	+	0.657920i
$454$		0			0
$455$			26.5431i			1.24436i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	−35.7409	+	21.1446i	−1.66824	+	0.986944i
$460$		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
$467$		−	14.3688i		−	0.664908i	−0.943119	−	0.332454i	$-0.892123\pi$
$467$		−	14.3688i		−	0.664908i	0.943119	−	0.332454i	$-0.107877\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−34.4939	−	12.4166i	−1.58940	−	0.572125i
$472$		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
$485$		−	33.6967i		−	1.53009i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			43.3690i			1.95722i	0.205731	+	0.978609i	$0.434043\pi$
$491$			43.3690i			1.95722i	−0.205731	+	0.978609i	$0.565957\pi$
$492$		0			0
$493$	−85.9310			−3.87014
$494$		0			0
$495$	−1.86073	−	1.53900i	−0.0836334	−	0.0691730i
$496$		0			0
$497$		−	31.3050i		−	1.40422i
$498$		0			0
$499$	−35.8704			−1.60578			−0.802890	−	0.596127i	$-0.796706\pi$
$499$	−35.8704			−1.60578			−0.802890	+	0.596127i	$0.796706\pi$
$500$		0			0
$501$	3.18216	−	8.84024i	0.142169	−	0.394953i
$502$		0			0
$503$		−	1.61501i		−	0.0720099i	−0.999352	−	0.0360049i	$-0.988537\pi$
$503$		−	1.61501i		−	0.0720099i	0.999352	−	0.0360049i	$-0.0114632\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−11.6190	−	4.18240i	−0.516016	−	0.185747i
$508$		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
$515$		−	41.9292i		−	1.84762i
$516$		0			0
$517$	4.48660			0.197320
$518$		0			0
$519$	−8.81784	+	24.4965i	−0.387060	+	1.07528i
$520$		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
$525$	21.5587	+	7.76034i	0.940898	+	0.338689i
$526$		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
$537$	−6.94106	+	19.2827i	−0.299529	+	0.832109i
$538$		0			0
$539$		−	2.51975i		−	0.108533i
$540$		0			0
$541$	6.12957			0.263531			0.131765	−	0.991281i	$-0.457935\pi$
$541$	6.12957			0.263531			0.131765	+	0.991281i	$0.457935\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	22.0709i		−	0.945415i
$546$		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
$553$	41.9892			1.78556
$554$		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
$561$	4.68826	+	1.68760i	0.197939	+	0.0712507i
$562$		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
$567$	4.46672	+	23.3891i	0.187585	+	0.982248i
$568$		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
$573$	11.9816	−	33.2857i	0.500540	−	1.39053i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−11.8500			−0.493322			−0.246661	−	0.969102i	$-0.579333\pi$
$577$	−11.8500			−0.493322			−0.246661	+	0.969102i	$0.579333\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			23.6643i			0.981761i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−19.1822	+	23.1921i	−0.793085	+	0.958877i
$586$		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
$593$		−	41.8642i		−	1.71916i	−0.511003	−	0.859579i	$-0.670726\pi$
$593$		−	41.8642i		−	1.71916i	0.511003	−	0.859579i	$-0.329274\pi$
$594$		0			0
$595$	−47.2807			−1.93832
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	18.9848i		−	0.775698i	−0.921723	−	0.387849i	$-0.873218\pi$
$599$		−	18.9848i		−	0.775698i	0.921723	−	0.387849i	$-0.126782\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	24.3070i		−	0.988221i
$606$		0			0
$607$	−39.9173			−1.62019			−0.810097	−	0.586296i	$-0.800586\pi$
$607$	−39.9173			−1.62019			−0.810097	+	0.586296i	$0.800586\pi$
$608$		0			0
$609$	−16.6883	+	46.3610i	−0.676243	+	1.87864i
$610$		0			0
$611$		−	55.9211i		−	2.26233i
$612$		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
$631$	−8.12957			−0.323633			−0.161817	−	0.986821i	$-0.551735\pi$
$631$	−8.12957			−0.323633			−0.161817	+	0.986821i	$0.551735\pi$
$632$		0			0
$633$	−6.30757	−	2.27050i	−0.250703	−	0.0902441i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−31.4062			−1.24436
$638$		0			0
$639$	22.6235	−	27.3528i	0.894971	−	1.08206i
$640$		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
$643$	30.9441			1.22032			0.610158	−	0.792279i	$-0.291106\pi$
$643$	30.9441			1.22032			0.610158	+	0.792279i	$0.291106\pi$
$644$		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
$657$	−24.4651	−	20.2351i	−0.954476	−	0.789445i
$658$		0			0
$659$			41.2093i			1.60528i	0.596461	+	0.802642i	$0.296573\pi$
$659$			41.2093i			1.60528i	−0.596461	+	0.802642i	$0.703427\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	21.0344	−	58.4347i	0.816907	−	2.26941i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	33.1822	+	11.9444i	1.28290	+	0.461796i
$670$		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
$677$			18.8409i			0.724115i	0.932156	+	0.362058i	$0.117926\pi$
$677$			18.8409i			0.724115i	−0.932156	+	0.362058i	$0.882074\pi$
$678$		0			0
$679$	39.8704			1.53009
$680$		0			0
$681$	−12.5587	+	34.8888i	−0.481250	+	1.33694i
$682$		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
$693$	1.82097	−	2.20164i	0.0691730	−	0.0836334i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			7.87256i			0.297342i	0.988887	+	0.148671i	$0.0474996\pi$
$701$			7.87256i			0.297342i	−0.988887	+	0.148671i	$0.952500\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−45.4200	−	16.3495i	−1.71062	−	0.615759i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−45.6113			−1.71297			−0.856484	−	0.516174i	$-0.827356\pi$
$709$	−45.6113			−1.71297			−0.856484	+	0.516174i	$0.827356\pi$
$710$		0			0
$711$	36.6883	+	30.3448i	1.37592	+	1.13802i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	3.61128			0.135054
$716$		0			0
$717$	13.2486	−	36.8055i	0.494779	−	1.37453i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	49.6113			1.84762
$722$		0			0
$723$		0			0
$724$		0			0
$725$			53.7613i			1.99665i
$726$		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
$733$	7.70621			0.284635			0.142318	−	0.989821i	$-0.454545\pi$
$733$	7.70621			0.284635			0.142318	+	0.989821i	$0.454545\pi$
$734$		0			0
$735$	−9.18216	+	25.5086i	−0.338689	+	0.940898i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−51.6113			−1.89855			−0.949276	−	0.314445i	$-0.898182\pi$
$739$	−51.6113			−1.89855			−0.949276	+	0.314445i	$0.898182\pi$
$740$		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
$747$	−17.1017	+	20.6768i	−0.625720	+	0.756525i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−39.6113			−1.44544			−0.722718	−	0.691143i	$-0.757107\pi$
$751$	−39.6113			−1.44544			−0.722718	+	0.691143i	$0.757107\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			53.3759i			1.94255i
$756$		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
$763$	26.1147			0.945415
$764$		0			0
$765$	−41.3117	−	34.1688i	−1.49363	−	1.23538i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−2.62348	+	7.28817i	−0.0944822	+	0.262477i
$772$		0			0
$773$			46.9990i			1.69044i	0.534421	+	0.845218i	$0.320530\pi$
$773$			46.9990i			1.69044i	−0.534421	+	0.845218i	$0.679470\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−4.25915			−0.152404
$782$		0			0
$783$	−48.0856	+	28.4478i	−1.71844	+	1.01664i
$784$		0			0
$785$		−	47.3286i		−	1.68923i
$786$		0			0
$787$	6.55849			0.233785			0.116892	−	0.993145i	$-0.462707\pi$
$787$	6.55849			0.233785			0.116892	+	0.993145i	$0.462707\pi$
$788$		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
$797$		−	34.8247i		−	1.23355i	−0.787138	−	0.616777i	$-0.788438\pi$
$797$		−	34.8247i		−	1.23355i	0.787138	−	0.616777i	$-0.211562\pi$
$798$		0			0
$799$	99.6113			3.52399
$800$		0			0
$801$		0			0
$802$		0			0
$803$			3.80950i			0.134434i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	54.4813i		−	1.91546i	−0.287670	−	0.957730i	$-0.592880\pi$
$809$		−	54.4813i		−	1.91546i	0.287670	−	0.957730i	$-0.407120\pi$
$810$		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
$819$	−27.4413	−	22.6966i	−0.958877	−	0.793085i
$820$		0			0
$821$		−	52.3215i		−	1.82603i	−0.407923	−	0.913016i	$-0.633747\pi$
$821$		−	52.3215i		−	1.82603i	0.407923	−	0.913016i	$-0.366253\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	1.05582	−	2.93313i	0.0367590	−	0.102119i
$826$		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
$833$		−	55.9433i		−	1.93832i
$834$		0			0
$835$	12.1296			0.419761
$836$		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
$841$	−86.6113			−2.98660
$842$		0			0
$843$	−18.5004	+	51.3951i	−0.637187	+	1.77014i
$844$		0			0
$845$		−	15.9422i		−	0.548429i
$846$		0			0
$847$	28.7604			0.988221
$848$		0			0
$849$	45.1822	+	16.2639i	1.55065	+	0.558177i
$850$		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
$865$	−33.6113			−1.14282
$866$		0			0
$867$	76.3840	+	27.4955i	2.59414	+	0.933795i
$868$		0			0
$869$		−	5.71278i		−	0.193793i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	34.8370	+	28.8136i	1.17905	+	0.975192i
$874$		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
$879$	19.3117	−	53.6491i	0.651369	−	1.80954i
$880$		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
$891$	3.18216	−	0.607713i	0.106606	−	0.0203592i
$892$		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
$913$	3.21961			0.106554
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	31.6113			1.04276			0.521380	−	0.853325i	$-0.325417\pi$
$919$	31.6113			1.04276			0.521380	+	0.853325i	$0.325417\pi$
$920$		0			0
$921$	−18.5587	−	6.68045i	−0.611530	−	0.220128i
$922$		0			0
$923$			53.0862i			1.74735i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	43.3481	+	35.8531i	1.42374	+	1.17757i
$928$		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
$935$			6.43270i			0.210372i
$936$		0			0
$937$	−36.2356			−1.18377			−0.591883	−	0.806024i	$-0.701615\pi$
$937$	−36.2356			−1.18377			−0.591883	+	0.806024i	$0.701615\pi$
$938$		0			0
$939$	−4.68826	−	1.68760i	−0.152996	−	0.0550729i
$940$		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
$949$	47.4817			1.54132
$950$		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
$955$	45.6709			1.47788
$956$		0			0
$957$	6.30757	+	2.27050i	0.203895	+	0.0733947i
$958$		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
$975$	−36.5587	−	13.1598i	−1.17082	−	0.421451i
$976$		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
$981$	22.8178	+	18.8726i	0.728517	+	0.602555i
$982$		0			0
$983$			33.5826i			1.07112i	0.844498	+	0.535559i	$0.179899\pi$
$983$			33.5826i			1.07112i	−0.844498	+	0.535559i	$0.820101\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	19.3450	−	53.7416i	0.615759	−	1.71062i
$988$		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
$993$	−13.0375	−	4.69302i	−0.413732	−	0.148928i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	61.5454			1.94916			0.974581	−	0.224034i	$-0.0719228\pi$
$997$	61.5454			1.94916			0.974581	+	0.224034i	$0.0719228\pi$
$998$		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.209.1	✓	8
3.2	odd	2	inner	420.2.f.a.209.2	yes	8
4.3	odd	2		1680.2.k.d.209.8		8
5.2	odd	4		2100.2.d.j.1301.4		8
5.3	odd	4		2100.2.d.j.1301.5		8
5.4	even	2	inner	420.2.f.a.209.8	yes	8
7.6	odd	2	inner	420.2.f.a.209.8	yes	8
12.11	even	2		1680.2.k.d.209.7		8
15.2	even	4		2100.2.d.j.1301.6		8
15.8	even	4		2100.2.d.j.1301.3		8
15.14	odd	2	inner	420.2.f.a.209.7	yes	8
20.19	odd	2		1680.2.k.d.209.1		8
21.20	even	2	inner	420.2.f.a.209.7	yes	8
28.27	even	2		1680.2.k.d.209.1		8
35.13	even	4		2100.2.d.j.1301.4		8
35.27	even	4		2100.2.d.j.1301.5		8
35.34	odd	2	CM	420.2.f.a.209.1	✓	8
60.59	even	2		1680.2.k.d.209.2		8
84.83	odd	2		1680.2.k.d.209.2		8
105.62	odd	4		2100.2.d.j.1301.3		8
105.83	odd	4		2100.2.d.j.1301.6		8
105.104	even	2	inner	420.2.f.a.209.2	yes	8
140.139	even	2		1680.2.k.d.209.8		8
420.419	odd	2		1680.2.k.d.209.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.f.a.209.1	✓	8	1.1	even	1	trivial
420.2.f.a.209.1	✓	8	35.34	odd	2	CM
420.2.f.a.209.2	yes	8	3.2	odd	2	inner
420.2.f.a.209.2	yes	8	105.104	even	2	inner
420.2.f.a.209.7	yes	8	15.14	odd	2	inner
420.2.f.a.209.7	yes	8	21.20	even	2	inner
420.2.f.a.209.8	yes	8	5.4	even	2	inner
420.2.f.a.209.8	yes	8	7.6	odd	2	inner
1680.2.k.d.209.1		8	20.19	odd	2
1680.2.k.d.209.1		8	28.27	even	2
1680.2.k.d.209.2		8	60.59	even	2
1680.2.k.d.209.2		8	84.83	odd	2
1680.2.k.d.209.7		8	12.11	even	2
1680.2.k.d.209.7		8	420.419	odd	2
1680.2.k.d.209.8		8	4.3	odd	2
1680.2.k.d.209.8		8	140.139	even	2
2100.2.d.j.1301.3		8	15.8	even	4
2100.2.d.j.1301.3		8	105.62	odd	4
2100.2.d.j.1301.4		8	5.2	odd	4
2100.2.d.j.1301.4		8	35.13	even	4
2100.2.d.j.1301.5		8	5.3	odd	4
2100.2.d.j.1301.5		8	35.27	even	4
2100.2.d.j.1301.6		8	15.2	even	4
2100.2.d.j.1301.6		8	105.83	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)

\(f(q)\)	\(=\)	\(q+(-1.62968 - 0.586627i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(2.31174 + 1.91203i) q^{9} +O(q^{10})\)	\(q+(-1.62968 - 0.586627i) q^{3} -2.23607i q^{5} +2.64575 q^{7} +(2.31174 + 1.91203i) q^{9} -0.359964i q^{11} -4.48660 q^{13} +(-1.31174 + 3.64408i) q^{15} -7.99190i q^{17} +(-4.31174 - 1.55207i) q^{21} -5.00000 q^{25} +(-2.64575 - 4.47214i) q^{27} -10.7523i q^{29} +(-0.211164 + 0.586627i) q^{33} -5.91608i q^{35} +(7.31174 + 2.63196i) q^{39} +(4.27543 - 5.16920i) q^{45} +12.4640i q^{47} +7.00000 q^{49} +(-4.68826 + 13.0243i) q^{51} -0.804903 q^{55} +(6.11628 + 5.05876i) q^{63} +10.0323i q^{65} -11.8322i q^{71} -10.5830 q^{73} +(8.14842 + 2.93313i) q^{75} -0.952374i q^{77} +15.8704 q^{79} +(1.68826 + 8.84024i) q^{81} +8.94427i q^{83} -17.8704 q^{85} +(-6.30757 + 17.5228i) q^{87} -11.8704 q^{91} +15.0696 q^{97} +(0.688262 - 0.832142i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9}+O(q^{10}) \) 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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