LMFDB - Embedded newform 2100.2.d.j.1301.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1301,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1301");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.d (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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.121550625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 $ x^8 - x^7 - 4x^6 - 9x^5 + 23x^4 + 18x^3 - 16x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1301.3
Root			$-1.44918 + 1.77086i$ of defining polynomial
Character	$\chi$	$=$	2100.1301
Dual form			2100.2.d.j.1301.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.586627 - 1.62968i) q^{3} -2.64575i q^{7} +(-2.31174 + 1.91203i) q^{9} +O(q^{10})$	$q+(-0.586627 - 1.62968i) q^{3} -2.64575i q^{7} +(-2.31174 + 1.91203i) q^{9} +0.359964i q^{11} -4.48660i q^{13} +7.99190 q^{17} +(-4.31174 + 1.55207i) q^{21} +(4.47214 + 2.64575i) q^{27} -10.7523i q^{29} +(0.586627 - 0.211164i) q^{33} +(-7.31174 + 2.63196i) q^{39} -12.4640 q^{47} -7.00000 q^{49} +(-4.68826 - 13.0243i) q^{51} +(5.05876 + 6.11628i) q^{63} +11.8322i q^{71} -10.5830i q^{73} +0.952374 q^{77} -15.8704 q^{79} +(1.68826 - 8.84024i) q^{81} +8.94427 q^{83} +(-17.5228 + 6.30757i) q^{87} -11.8704 q^{91} -15.0696i q^{97} +(-0.688262 - 0.832142i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{9}+O(q^{10}) $ 8 * q + 2 * q^9	$ 8 q + 2 q^{9} - 14 q^{21} - 38 q^{39} - 56 q^{49} - 58 q^{51} - 4 q^{79} + 34 q^{81} + 28 q^{91} - 26 q^{99}+O(q^{100}) $ 8 * q + 2 * q^9 - 14 * q^21 - 38 * q^39 - 56 * q^49 - 58 * q^51 - 4 * q^79 + 34 * q^81 + 28 * q^91 - 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	−0.586627	−	1.62968i	−0.338689	−	0.940898i
$3$	−0.586627	−	1.62968i	−0.338689	−	0.940898i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$		0			0
$9$	−2.31174	+	1.91203i	−0.770579	+	0.637344i
$10$		0			0
$11$			0.359964i			0.108533i	0.998526	+	0.0542666i	$0.0172821\pi$
$11$			0.359964i			0.108533i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$		0			0
$13$		−	4.48660i		−	1.24436i	−0.782875	−	0.622179i	$-0.786247\pi$
$13$		−	4.48660i		−	1.24436i	0.782875	−	0.622179i	$-0.213753\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.99190			1.93832			0.969160	−	0.246433i	$-0.0792584\pi$
$17$	7.99190			1.93832			0.969160	+	0.246433i	$0.0792584\pi$
$18$		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		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.47214	+	2.64575i	0.860663	+	0.509175i
$28$		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
$33$	0.586627	−	0.211164i	0.102119	−	0.0367590i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		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			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12.4640			−1.81807			−0.909033	−	0.416724i	$-0.863178\pi$
$47$	−12.4640			−1.81807			−0.909033	+	0.416724i	$0.863178\pi$
$48$		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		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		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
$63$	5.05876	+	6.11628i	0.637344	+	0.770579i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.8322i			1.40422i	0.712069	+	0.702109i	$0.247758\pi$
$71$			11.8322i			1.40422i	−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$		−	10.5830i		−	1.23865i	−0.785136	−	0.619324i	$-0.787407\pi$
$73$		−	10.5830i		−	1.23865i	0.785136	−	0.619324i	$-0.212593\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.952374			0.108533
$78$		0			0
$79$	−15.8704			−1.78556			−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−15.8704			−1.78556			−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$	1.68826	−	8.84024i	0.187585	−	0.982248i
$82$		0			0
$83$	8.94427			0.981761			0.490881	−	0.871227i	$-0.336675\pi$
$83$	8.94427			0.981761			0.490881	+	0.871227i	$0.336675\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−17.5228	+	6.30757i	−1.87864	+	0.676243i
$88$		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
$97$		−	15.0696i		−	1.53009i	−0.643979	−	0.765043i	$-0.722718\pi$
$97$		−	15.0696i		−	1.53009i	0.643979	−	0.765043i	$-0.277282\pi$
$98$		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
$103$			18.7513i			1.84762i	0.382851	+	0.923810i	$0.374942\pi$
$103$			18.7513i			1.84762i	−0.382851	+	0.923810i	$0.625058\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−9.87043			−0.945415			−0.472708	−	0.881219i	$-0.656723\pi$
$109$	−9.87043			−0.945415			−0.472708	+	0.881219i	$0.656723\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	8.57852	+	10.3718i	0.793085	+	0.958877i
$118$		0			0
$119$		−	21.1446i		−	1.93832i
$120$		0			0
$121$	10.8704			0.988221
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		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
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		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
$143$	1.61501			0.135054
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.10639	+	11.4078i	0.338689	+	0.940898i
$148$		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
$153$	−18.4752	+	15.2808i	−1.49363	+	1.23538i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	21.1660i		−	1.68923i	−0.535373	−	0.844616i	$-0.679829\pi$
$157$		−	21.1660i		−	1.68923i	0.535373	−	0.844616i	$-0.320171\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.42451			−0.419761			−0.209881	−	0.977727i	$-0.567308\pi$
$167$	−5.42451			−0.419761			−0.209881	+	0.977727i	$0.567308\pi$
$168$		0			0
$169$	−7.12957			−0.548429
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−15.0314			−1.14282			−0.571409	−	0.820666i	$-0.693603\pi$
$173$	−15.0314			−1.14282			−0.571409	+	0.820666i	$0.693603\pi$
$174$		0			0
$175$		0			0
$176$		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
$187$			2.87679i			0.210372i
$188$		0			0
$189$	7.00000	−	11.8322i	0.509175	−	0.860663i
$190$		0			0
$191$		−	20.4246i		−	1.47788i	−0.673774	−	0.738938i	$-0.735328\pi$
$191$		−	20.4246i		−	1.47788i	0.673774	−	0.738938i	$-0.264672\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−28.4478			−1.99665
$204$		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
$213$	19.2827	−	6.94106i	1.32123	−	0.475594i
$214$		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
$223$		−	20.3611i		−	1.36348i	−0.731594	−	0.681740i	$-0.761223\pi$
$223$		−	20.3611i		−	1.36348i	0.731594	−	0.681740i	$-0.238777\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	21.4083			1.42092			0.710460	−	0.703738i	$-0.248487\pi$
$227$	21.4083			1.42092			0.710460	+	0.703738i	$0.248487\pi$
$228$		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		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	9.31002	+	25.8638i	0.604751	+	1.68003i
$238$		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
$243$	−15.3972	+	2.43459i	−0.987729	+	0.156179i
$244$		0			0
$245$		0			0
$246$		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
$255$		0			0
$256$		0			0
$257$	4.47214			0.278964			0.139482	−	0.990225i	$-0.455456\pi$
$257$	4.47214			0.278964			0.139482	+	0.990225i	$0.455456\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	20.5587	+	24.8564i	1.27255	+	1.53857i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		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
$273$	6.96351	+	19.3450i	0.421451	+	1.17082i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			31.5369i			1.88133i	0.339333	+	0.940666i	$0.389799\pi$
$281$			31.5369i			1.88133i	−0.339333	+	0.940666i	$0.610201\pi$
$282$		0			0
$283$		−	27.7245i		−	1.64805i	−0.566553	−	0.824025i	$-0.691723\pi$
$283$		−	27.7245i		−	1.64805i	0.566553	−	0.824025i	$-0.308277\pi$
$284$		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
$293$	32.9200			1.92320			0.961602	−	0.274446i	$-0.0884946\pi$
$293$	32.9200			1.92320			0.961602	+	0.274446i	$0.0884946\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.952374	+	1.60981i	−0.0552624	+	0.0934104i
$298$		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
$307$		−	11.3879i		−	0.649942i	−0.945724	−	0.324971i	$-0.894645\pi$
$307$		−	11.3879i		−	0.649942i	0.945724	−	0.324971i	$-0.105355\pi$
$308$		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
$313$			2.87679i			0.162606i	0.996689	+	0.0813030i	$0.0259081\pi$
$313$			2.87679i			0.162606i	−0.996689	+	0.0813030i	$0.974092\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$	3.87043			0.216702
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	5.79026	+	16.0857i	0.320202	+	0.889540i
$328$		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			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			18.5203i			1.00000i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		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
$353$	6.08715			0.323986			0.161993	−	0.986792i	$-0.448208\pi$
$353$	6.08715			0.323986			0.161993	+	0.986792i	$0.448208\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−34.4590	+	12.4040i	−1.82376	+	0.656488i
$358$		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
$363$	−6.37688	−	17.7154i	−0.334700	−	0.929815i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	38.3075i		−	1.99964i	−0.0190919	−	0.999818i	$-0.506077\pi$
$367$		−	38.3075i		−	1.99964i	0.0190919	−	0.999818i	$-0.493923\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−48.2411			−2.48454
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	35.7771			1.82812			0.914062	−	0.405575i	$-0.132929\pi$
$383$	35.7771			1.82812			0.914062	+	0.405575i	$0.132929\pi$
$384$		0			0
$385$		0			0
$386$		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
$395$		0			0
$396$		0			0
$397$		−	34.6258i		−	1.73782i	−0.494971	−	0.868910i	$-0.664821\pi$
$397$		−	34.6258i		−	1.73782i	0.494971	−	0.868910i	$-0.335179\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			8.59249i			0.429088i	0.976714	+	0.214544i	$0.0688266\pi$
$401$			8.59249i			0.429088i	−0.976714	+	0.214544i	$0.931173\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		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
$415$		0			0
$416$		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
$423$	28.8136	−	23.8316i	1.40096	−	1.15873i
$424$		0			0
$425$		0			0
$426$		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	$0.189685\pi$
$431$			23.3044i			1.12253i	−0.827636	+	0.561266i	$0.810315\pi$
$432$		0			0
$433$		−	10.5830i		−	0.508587i	−0.967127	−	0.254293i	$-0.918157\pi$
$433$		−	10.5830i		−	0.508587i	0.967127	−	0.254293i	$-0.0818429\pi$
$434$		0			0
$435$		0			0
$436$		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		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	38.5654	−	13.8821i	1.82408	−	0.656602i
$448$		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
$453$	14.0030	+	38.9012i	0.657920	+	1.82774i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		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			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.3688			0.664908			0.332454	−	0.943119i	$-0.392123\pi$
$467$	14.3688			0.664908			0.332454	+	0.943119i	$0.392123\pi$
$468$		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
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	43.3690i		−	1.95722i	−0.205731	−	0.978609i	$-0.565957\pi$
$491$		−	43.3690i		−	1.95722i	0.205731	−	0.978609i	$-0.434043\pi$
$492$		0			0
$493$		−	85.9310i		−	3.87014i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	31.3050			1.40422
$498$		0			0
$499$	35.8704			1.60578			0.802890	−	0.596127i	$-0.203294\pi$
$499$	35.8704			1.60578			0.802890	+	0.596127i	$0.203294\pi$
$500$		0			0
$501$	3.18216	+	8.84024i	0.142169	+	0.394953i
$502$		0			0
$503$	−1.61501			−0.0720099			−0.0360049	−	0.999352i	$-0.511463\pi$
$503$	−1.61501			−0.0720099			−0.0360049	+	0.999352i	$0.511463\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.18240	+	11.6190i	0.185747	+	0.516016i
$508$		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
$515$		0			0
$516$		0			0
$517$		−	4.48660i		−	0.197320i
$518$		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
$523$			37.0405i			1.61967i	0.586659	+	0.809834i	$0.300443\pi$
$523$			37.0405i			1.61967i	−0.586659	+	0.809834i	$0.699557\pi$
$524$		0			0
$525$		0			0
$526$		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
$537$	−19.2827	+	6.94106i	−0.832109	+	0.299529i
$538$		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
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			41.9892i			1.78556i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	4.68826	−	1.68760i	0.197939	−	0.0712507i
$562$		0			0
$563$	−44.7214			−1.88478			−0.942390	−	0.334515i	$-0.891427\pi$
$563$	−44.7214			−1.88478			−0.942390	+	0.334515i	$0.891427\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.3891	−	4.46672i	−0.982248	−	0.187585i
$568$		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
$573$	−33.2857	+	11.9816i	−1.39053	+	0.500540i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			11.8500i			0.493322i	0.969102	+	0.246661i	$0.0793334\pi$
$577$			11.8500i			0.493322i	−0.969102	+	0.246661i	$0.920667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	23.6643i		−	0.981761i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.94427			−0.369170			−0.184585	−	0.982817i	$-0.559094\pi$
$587$	−8.94427			−0.369170			−0.184585	+	0.982817i	$0.559094\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−41.8642			−1.71916			−0.859579	−	0.511003i	$-0.829274\pi$
$593$	−41.8642			−1.71916			−0.859579	+	0.511003i	$0.829274\pi$
$594$		0			0
$595$		0			0
$596$		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
$605$		0			0
$606$		0			0
$607$			39.9173i			1.62019i	0.586296	+	0.810097i	$0.300586\pi$
$607$			39.9173i			1.62019i	−0.586296	+	0.810097i	$0.699414\pi$
$608$		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			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		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
$625$		0			0
$626$		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
$633$	−2.27050	−	6.30757i	−0.0902441	−	0.250703i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			31.4062i			1.24436i
$638$		0			0
$639$	−22.6235	−	27.3528i	−0.894971	−	1.08206i
$640$		0			0
$641$			47.3286i			1.86937i	0.355479	+	0.934684i	$0.384318\pi$
$641$			47.3286i			1.86937i	−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$			30.9441i			1.22032i	0.792279	+	0.610158i	$0.208894\pi$
$643$			30.9441i			1.22032i	−0.792279	+	0.610158i	$0.791106\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.8885			0.703271			0.351636	−	0.936137i	$-0.385626\pi$
$647$	17.8885			0.703271			0.351636	+	0.936137i	$0.385626\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	20.2351	+	24.4651i	0.789445	+	0.954476i
$658$		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
$663$	−58.4347	+	21.0344i	−2.26941	+	0.816907i
$664$		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			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.8409			−0.724115			−0.362058	−	0.932156i	$-0.617926\pi$
$677$	−18.8409			−0.724115			−0.362058	+	0.932156i	$0.617926\pi$
$678$		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		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		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
$693$	−2.20164	+	1.82097i	−0.0836334	+	0.0691730i
$694$		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	$-0.952500\pi$
$701$		−	7.87256i		−	0.297342i	0.988887	−	0.148671i	$-0.0474996\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	45.6113			1.71297			0.856484	−	0.516174i	$-0.172644\pi$
$709$	45.6113			1.71297			0.856484	+	0.516174i	$0.172644\pi$
$710$		0			0
$711$	36.6883	−	30.3448i	1.37592	−	1.13802i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	36.8055	−	13.2486i	1.37453	−	0.494779i
$718$		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
$725$		0			0
$726$		0			0
$727$		−	5.29150i		−	0.196251i	−0.995174	−	0.0981255i	$-0.968715\pi$
$727$		−	5.29150i		−	0.196251i	0.995174	−	0.0981255i	$-0.0312847\pi$
$728$		0			0
$729$	13.0000	+	23.6643i	0.481481	+	0.876456i
$730$		0			0
$731$		0			0
$732$		0			0
$733$			7.70621i			0.284635i	0.989821	+	0.142318i	$0.0454555\pi$
$733$			7.70621i			0.284635i	−0.989821	+	0.142318i	$0.954545\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	51.6113			1.89855			0.949276	−	0.314445i	$-0.101818\pi$
$739$	51.6113			1.89855			0.949276	+	0.314445i	$0.101818\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−20.6768	+	17.1017i	−0.756525	+	0.625720i
$748$		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
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		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
$763$			26.1147i			0.945415i
$764$		0			0
$765$		0			0
$766$		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
$773$	46.9990			1.69044			0.845218	−	0.534421i	$-0.179470\pi$
$773$	46.9990			1.69044			0.845218	+	0.534421i	$0.179470\pi$
$774$		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
$783$	28.4478	−	48.0856i	1.01664	−	1.71844i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	6.55849i		−	0.233785i	−0.993145	−	0.116892i	$-0.962707\pi$
$787$		−	6.55849i		−	0.233785i	0.993145	−	0.116892i	$-0.0372933\pi$
$788$		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
$797$	34.8247			1.23355			0.616777	−	0.787138i	$-0.288438\pi$
$797$	34.8247			1.23355			0.616777	+	0.787138i	$0.288438\pi$
$798$		0			0
$799$	−99.6113			−3.52399
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.80950			0.134434
$804$		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	$0.366253\pi$
$821$			52.3215i			1.82603i	−0.407923	+	0.913016i	$0.633747\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−55.9433			−1.93832
$834$		0			0
$835$		0			0
$836$		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
$843$	51.3951	−	18.5004i	1.77014	−	0.637187i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	28.7604i		−	0.988221i
$848$		0			0
$849$	−45.1822	+	16.2639i	−1.55065	+	0.558177i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	42.3320i		−	1.44942i	−0.689054	−	0.724710i	$-0.741974\pi$
$853$		−	42.3320i		−	1.44942i	0.689054	−	0.724710i	$-0.258026\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−49.1935			−1.68042			−0.840209	−	0.542263i	$-0.817568\pi$
$857$	−49.1935			−1.68042			−0.840209	+	0.542263i	$0.817568\pi$
$858$		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		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−27.4955	−	76.3840i	−0.933795	−	2.59414i
$868$		0			0
$869$		−	5.71278i		−	0.193793i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	28.8136	+	34.8370i	0.975192	+	1.17905i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		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			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−35.7771			−1.20128			−0.600639	−	0.799521i	$-0.705087\pi$
$887$	−35.7771			−1.20128			−0.600639	+	0.799521i	$0.705087\pi$
$888$		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
$895$		0			0
$896$		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			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	59.1608i		−	1.96008i	−0.198789	−	0.980042i	$-0.563701\pi$
$911$		−	59.1608i		−	1.96008i	0.198789	−	0.980042i	$-0.436299\pi$
$912$		0			0
$913$			3.21961i			0.106554i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−31.6113			−1.04276			−0.521380	−	0.853325i	$-0.674583\pi$
$919$	−31.6113			−1.04276			−0.521380	+	0.853325i	$0.674583\pi$
$920$		0			0
$921$	−18.5587	+	6.68045i	−0.611530	+	0.220128i
$922$		0			0
$923$	53.0862			1.74735
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−35.8531	−	43.3481i	−1.17757	−	1.42374i
$928$		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
$935$		0			0
$936$		0			0
$937$			36.2356i			1.18377i	0.806024	+	0.591883i	$0.201615\pi$
$937$			36.2356i			1.18377i	−0.806024	+	0.591883i	$0.798385\pi$
$938$		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
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	−47.4817			−1.54132
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2.27050	−	6.30757i	−0.0733947	−	0.203895i
$958$		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			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		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
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	22.8178	−	18.8726i	0.728517	−	0.602555i
$982$		0			0
$983$	33.5826			1.07112			0.535559	−	0.844498i	$-0.320101\pi$
$983$	33.5826			1.07112			0.535559	+	0.844498i	$0.320101\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	53.7416	−	19.3450i	1.71062	−	0.615759i
$988$		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
$993$	−4.69302	−	13.0375i	−0.148928	−	0.413732i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	61.5454i		−	1.94916i	−0.224034	−	0.974581i	$-0.571923\pi$
$997$		−	61.5454i		−	1.94916i	0.224034	−	0.974581i	$-0.428077\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.d.j.1301.3		8
3.2	odd	2	inner	2100.2.d.j.1301.5		8
5.2	odd	4		420.2.f.a.209.2	yes	8
5.3	odd	4		420.2.f.a.209.7	yes	8
5.4	even	2	inner	2100.2.d.j.1301.6		8
7.6	odd	2	inner	2100.2.d.j.1301.6		8
15.2	even	4		420.2.f.a.209.1	✓	8
15.8	even	4		420.2.f.a.209.8	yes	8
15.14	odd	2	inner	2100.2.d.j.1301.4		8
20.3	even	4		1680.2.k.d.209.2		8
20.7	even	4		1680.2.k.d.209.7		8
21.20	even	2	inner	2100.2.d.j.1301.4		8
35.13	even	4		420.2.f.a.209.2	yes	8
35.27	even	4		420.2.f.a.209.7	yes	8
35.34	odd	2	CM	2100.2.d.j.1301.3		8
60.23	odd	4		1680.2.k.d.209.1		8
60.47	odd	4		1680.2.k.d.209.8		8
105.62	odd	4		420.2.f.a.209.8	yes	8
105.83	odd	4		420.2.f.a.209.1	✓	8
105.104	even	2	inner	2100.2.d.j.1301.5		8
140.27	odd	4		1680.2.k.d.209.2		8
140.83	odd	4		1680.2.k.d.209.7		8
420.83	even	4		1680.2.k.d.209.8		8
420.167	even	4		1680.2.k.d.209.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.f.a.209.1	✓	8	15.2	even	4
420.2.f.a.209.1	✓	8	105.83	odd	4
420.2.f.a.209.2	yes	8	5.2	odd	4
420.2.f.a.209.2	yes	8	35.13	even	4
420.2.f.a.209.7	yes	8	5.3	odd	4
420.2.f.a.209.7	yes	8	35.27	even	4
420.2.f.a.209.8	yes	8	15.8	even	4
420.2.f.a.209.8	yes	8	105.62	odd	4
1680.2.k.d.209.1		8	60.23	odd	4
1680.2.k.d.209.1		8	420.167	even	4
1680.2.k.d.209.2		8	20.3	even	4
1680.2.k.d.209.2		8	140.27	odd	4
1680.2.k.d.209.7		8	20.7	even	4
1680.2.k.d.209.7		8	140.83	odd	4
1680.2.k.d.209.8		8	60.47	odd	4
1680.2.k.d.209.8		8	420.83	even	4
2100.2.d.j.1301.3		8	1.1	even	1	trivial
2100.2.d.j.1301.3		8	35.34	odd	2	CM
2100.2.d.j.1301.4		8	15.14	odd	2	inner
2100.2.d.j.1301.4		8	21.20	even	2	inner
2100.2.d.j.1301.5		8	3.2	odd	2	inner
2100.2.d.j.1301.5		8	105.104	even	2	inner
2100.2.d.j.1301.6		8	5.4	even	2	inner
2100.2.d.j.1301.6		8	7.6	odd	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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.586627 - 1.62968i) q^{3} -2.64575i q^{7} +(-2.31174 + 1.91203i) q^{9} +O(q^{10})\)	\(q+(-0.586627 - 1.62968i) q^{3} -2.64575i q^{7} +(-2.31174 + 1.91203i) q^{9} +0.359964i q^{11} -4.48660i q^{13} +7.99190 q^{17} +(-4.31174 + 1.55207i) q^{21} +(4.47214 + 2.64575i) q^{27} -10.7523i q^{29} +(0.586627 - 0.211164i) q^{33} +(-7.31174 + 2.63196i) q^{39} -12.4640 q^{47} -7.00000 q^{49} +(-4.68826 - 13.0243i) q^{51} +(5.05876 + 6.11628i) q^{63} +11.8322i q^{71} -10.5830i q^{73} +0.952374 q^{77} -15.8704 q^{79} +(1.68826 - 8.84024i) q^{81} +8.94427 q^{83} +(-17.5228 + 6.30757i) q^{87} -11.8704 q^{91} -15.0696i q^{97} +(-0.688262 - 0.832142i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{9}+O(q^{10}) \) 8 * q + 2 * q^9	\( 8 q + 2 q^{9} - 14 q^{21} - 38 q^{39} - 56 q^{49} - 58 q^{51} - 4 q^{79} + 34 q^{81} + 28 q^{91} - 26 q^{99}+O(q^{100}) \) 8 * q + 2 * q^9 - 14 * q^21 - 38 * q^39 - 56 * q^49 - 58 * q^51 - 4 * q^79 + 34 * q^81 + 28 * q^91 - 26 * q^99

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