LMFDB - Embedded newform 1680.2.k.b.209.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(209,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1680.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.k (of order $2$, degree $1$, not 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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 9 $ x^4 - x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			209.1
Root			$1.32288 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	1680.209
Dual form			1680.2.k.b.209.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32288 - 1.11803i) q^{3} +2.23607i q^{5} -2.64575 q^{7} +(0.500000 + 2.95804i) q^{9} +5.91608i q^{11} -2.64575 q^{13} +(2.50000 - 2.95804i) q^{15} -2.23607i q^{17} +(3.50000 + 2.95804i) q^{21} -5.00000 q^{25} +(2.64575 - 4.47214i) q^{27} -5.91608i q^{29} +(6.61438 - 7.82624i) q^{33} -5.91608i q^{35} +(3.50000 + 2.95804i) q^{39} +(-6.61438 + 1.11803i) q^{45} -11.1803i q^{47} +7.00000 q^{49} +(-2.50000 + 2.95804i) q^{51} -13.2288 q^{55} +(-1.32288 - 7.82624i) q^{63} -5.91608i q^{65} -11.8322i q^{71} -10.5830 q^{73} +(6.61438 + 5.59017i) q^{75} -15.6525i q^{77} +1.00000 q^{79} +(-8.50000 + 2.95804i) q^{81} +8.94427i q^{83} +5.00000 q^{85} +(-6.61438 + 7.82624i) q^{87} +7.00000 q^{91} -18.5203 q^{97} +(-17.5000 + 2.95804i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9} + 10 q^{15} + 14 q^{21} - 20 q^{25} + 14 q^{39} + 28 q^{49} - 10 q^{51} + 4 q^{79} - 34 q^{81} + 20 q^{85} + 28 q^{91} - 70 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 + 10 * q^15 + 14 * q^21 - 20 * q^25 + 14 * q^39 + 28 * q^49 - 10 * q^51 + 4 * q^79 - 34 * q^81 + 20 * q^85 + 28 * q^91 - 70 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.32288	−	1.11803i	−0.763763	−	0.645497i
$3$	−1.32288	−	1.11803i	−0.763763	−	0.645497i
$4$		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
$9$	0.500000	+	2.95804i	0.166667	+	0.986013i
$10$		0			0
$11$			5.91608i			1.78377i	0.452267	+	0.891883i	$0.350615\pi$
$11$			5.91608i			1.78377i	−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$	−2.64575			−0.733799			−0.366900	−	0.930261i	$-0.619581\pi$
$13$	−2.64575			−0.733799			−0.366900	+	0.930261i	$0.619581\pi$
$14$		0			0
$15$	2.50000	−	2.95804i	0.645497	−	0.763763i
$16$		0			0
$17$		−	2.23607i		−	0.542326i	−0.962533	−	0.271163i	$-0.912592\pi$
$17$		−	2.23607i		−	0.542326i	0.962533	−	0.271163i	$-0.0874083\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	3.50000	+	2.95804i	0.763763	+	0.645497i
$22$		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
$29$		−	5.91608i		−	1.09859i	−0.835629	−	0.549294i	$-0.814897\pi$
$29$		−	5.91608i		−	1.09859i	0.835629	−	0.549294i	$-0.185103\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	6.61438	−	7.82624i	1.15142	−	1.36237i
$34$		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
$39$	3.50000	+	2.95804i	0.560449	+	0.473665i
$40$		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
$45$	−6.61438	+	1.11803i	−0.986013	+	0.166667i
$46$		0			0
$47$		−	11.1803i		−	1.63082i	−0.578884	−	0.815410i	$-0.696511\pi$
$47$		−	11.1803i		−	1.63082i	0.578884	−	0.815410i	$-0.303489\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−2.50000	+	2.95804i	−0.350070	+	0.414208i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−13.2288			−1.78377
$56$		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
$63$	−1.32288	−	7.82624i	−0.166667	−	0.986013i
$64$		0			0
$65$		−	5.91608i		−	0.733799i
$66$		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
$75$	6.61438	+	5.59017i	0.763763	+	0.645497i
$76$		0			0
$77$		−	15.6525i		−	1.78377i
$78$		0			0
$79$	1.00000			0.112509			0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000			0.112509			0.0562544	+	0.998416i	$0.482084\pi$
$80$		0			0
$81$	−8.50000	+	2.95804i	−0.944444	+	0.328671i
$82$		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
$85$	5.00000			0.542326
$86$		0			0
$87$	−6.61438	+	7.82624i	−0.709136	+	0.839061i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	7.00000			0.733799
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18.5203			−1.88045			−0.940224	−	0.340557i	$-0.889384\pi$
$97$	−18.5203			−1.88045			−0.940224	+	0.340557i	$0.889384\pi$
$98$		0			0
$99$	−17.5000	+	2.95804i	−1.75882	+	0.297294i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	2.64575			0.260694			0.130347	−	0.991468i	$-0.458391\pi$
$103$	2.64575			0.260694			0.130347	+	0.991468i	$0.458391\pi$
$104$		0			0
$105$	−6.61438	+	7.82624i	−0.645497	+	0.763763i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$		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
$117$	−1.32288	−	7.82624i	−0.122300	−	0.723536i
$118$		0			0
$119$			5.91608i			0.542326i
$120$		0			0
$121$	−24.0000			−2.18182
$122$		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
$141$	−12.5000	+	14.7902i	−1.05269	+	1.24556i
$142$		0			0
$143$		−	15.6525i		−	1.30893i
$144$		0			0
$145$	13.2288			1.09859
$146$		0			0
$147$	−9.26013	−	7.82624i	−0.763763	−	0.645497i
$148$		0			0
$149$		−	23.6643i		−	1.93866i	−0.245770	−	0.969328i	$-0.579041\pi$
$149$		−	23.6643i		−	1.93866i	0.245770	−	0.969328i	$-0.420959\pi$
$150$		0			0
$151$	−17.0000			−1.38344			−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000			−1.38344			−0.691720	+	0.722166i	$0.743147\pi$
$152$		0			0
$153$	6.61438	−	1.11803i	0.534741	−	0.0903877i
$154$		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
$165$	17.5000	+	14.7902i	1.36237	+	1.15142i
$166$		0			0
$167$		−	24.5967i		−	1.90335i	−0.307102	−	0.951677i	$-0.599359\pi$
$167$		−	24.5967i		−	1.90335i	0.307102	−	0.951677i	$-0.400641\pi$
$168$		0			0
$169$	−6.00000			−0.461538
$170$		0			0
$171$		0			0
$172$		0			0
$173$			11.1803i			0.850026i	0.905187	+	0.425013i	$0.139730\pi$
$173$			11.1803i			0.850026i	−0.905187	+	0.425013i	$0.860270\pi$
$174$		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
$187$	13.2288			0.967382
$188$		0			0
$189$	−7.00000	+	11.8322i	−0.509175	+	0.860663i
$190$		0			0
$191$			5.91608i			0.428073i	0.976826	+	0.214036i	$0.0686611\pi$
$191$			5.91608i			0.428073i	−0.976826	+	0.214036i	$0.931339\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	−6.61438	+	7.82624i	−0.473665	+	0.560449i
$196$		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
$203$			15.6525i			1.09859i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−23.0000			−1.58339			−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000			−1.58339			−0.791693	+	0.610920i	$0.790800\pi$
$212$		0			0
$213$	−13.2288	+	15.6525i	−0.906419	+	1.07249i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	14.0000	+	11.8322i	0.946032	+	0.799543i
$220$		0			0
$221$			5.91608i			0.397959i
$222$		0			0
$223$	−29.1033			−1.94890			−0.974449	−	0.224607i	$-0.927890\pi$
$223$	−29.1033			−1.94890			−0.974449	+	0.224607i	$0.927890\pi$
$224$		0			0
$225$	−2.50000	−	14.7902i	−0.166667	−	0.986013i
$226$		0			0
$227$			29.0689i			1.92937i	0.263407	+	0.964685i	$0.415154\pi$
$227$			29.0689i			1.92937i	−0.263407	+	0.964685i	$0.584846\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	−17.5000	+	20.7063i	−1.15142	+	1.36237i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	25.0000			1.63082
$236$		0			0
$237$	−1.32288	−	1.11803i	−0.0859300	−	0.0726241i
$238$		0			0
$239$		−	29.5804i		−	1.91340i	−0.291081	−	0.956698i	$-0.594015\pi$
$239$		−	29.5804i		−	1.91340i	0.291081	−	0.956698i	$-0.405985\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	14.5516	+	5.59017i	0.933488	+	0.358610i
$244$		0			0
$245$			15.6525i			1.00000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	10.0000	−	11.8322i	0.633724	−	0.749833i
$250$		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
$255$	−6.61438	−	5.59017i	−0.414208	−	0.350070i
$256$		0			0
$257$			4.47214i			0.278964i	0.990225	+	0.139482i	$0.0445438\pi$
$257$			4.47214i			0.278964i	−0.990225	+	0.139482i	$0.955456\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	17.5000	−	2.95804i	1.08322	−	0.183098i
$262$		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
$273$	−9.26013	−	7.82624i	−0.560449	−	0.473665i
$274$		0			0
$275$		−	29.5804i		−	1.78377i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	5.91608i		−	0.352924i	−0.984307	−	0.176462i	$-0.943535\pi$
$281$		−	5.91608i		−	0.352924i	0.984307	−	0.176462i	$-0.0564652\pi$
$282$		0			0
$283$	2.64575			0.157274			0.0786368	−	0.996903i	$-0.474943\pi$
$283$	2.64575			0.157274			0.0786368	+	0.996903i	$0.474943\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	12.0000			0.705882
$290$		0			0
$291$	24.5000	+	20.7063i	1.43622	+	1.21382i
$292$		0			0
$293$			24.5967i			1.43696i	0.695549	+	0.718479i	$0.255161\pi$
$293$			24.5967i			1.43696i	−0.695549	+	0.718479i	$0.744839\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	26.4575	+	15.6525i	1.53522	+	0.908249i
$298$		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
$307$	34.3948			1.96301			0.981507	−	0.191429i	$-0.0613121\pi$
$307$	34.3948			1.96301			0.981507	+	0.191429i	$0.0613121\pi$
$308$		0			0
$309$	−3.50000	−	2.95804i	−0.199108	−	0.168277i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	29.1033			1.64501			0.822507	−	0.568755i	$-0.192575\pi$
$313$	29.1033			1.64501			0.822507	+	0.568755i	$0.192575\pi$
$314$		0			0
$315$	17.5000	−	2.95804i	0.986013	−	0.166667i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$	35.0000			1.95962
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	13.2288			0.733799
$326$		0			0
$327$	−14.5516	−	12.2984i	−0.804707	−	0.680102i
$328$		0			0
$329$			29.5804i			1.63082i
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		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
$351$	−7.00000	+	11.8322i	−0.373632	+	0.631554i
$352$		0			0
$353$		−	29.0689i		−	1.54718i	−0.633686	−	0.773590i	$-0.718459\pi$
$353$		−	29.0689i		−	1.54718i	0.633686	−	0.773590i	$-0.281541\pi$
$354$		0			0
$355$	26.4575			1.40422
$356$		0			0
$357$	6.61438	−	7.82624i	0.350070	−	0.414208i
$358$		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
$363$	31.7490	+	26.8328i	1.66639	+	1.40836i
$364$		0			0
$365$		−	23.6643i		−	1.23865i
$366$		0			0
$367$	18.5203			0.966750			0.483375	−	0.875413i	$-0.339411\pi$
$367$	18.5203			0.966750			0.483375	+	0.875413i	$0.339411\pi$
$368$		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
$375$	−12.5000	+	14.7902i	−0.645497	+	0.763763i
$376$		0			0
$377$			15.6525i			0.806144i
$378$		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
$383$			35.7771i			1.82812i	0.405575	+	0.914062i	$0.367071\pi$
$383$			35.7771i			1.82812i	−0.405575	+	0.914062i	$0.632929\pi$
$384$		0			0
$385$	35.0000			1.78377
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	5.91608i		−	0.299957i	−0.988689	−	0.149979i	$-0.952080\pi$
$389$		−	5.91608i		−	0.299957i	0.988689	−	0.149979i	$-0.0479205\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			2.23607i			0.112509i
$396$		0			0
$397$	−34.3948			−1.72622			−0.863112	−	0.505013i	$-0.831488\pi$
$397$	−34.3948			−1.72622			−0.863112	+	0.505013i	$0.831488\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			29.5804i			1.47717i	0.674158	+	0.738587i	$0.264507\pi$
$401$			29.5804i			1.47717i	−0.674158	+	0.738587i	$0.735493\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−6.61438	−	19.0066i	−0.328671	−	0.944444i
$406$		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
$421$	−37.0000			−1.80327			−0.901635	−	0.432498i	$-0.857632\pi$
$421$	−37.0000			−1.80327			−0.901635	+	0.432498i	$0.857632\pi$
$422$		0			0
$423$	33.0719	−	5.59017i	1.60801	−	0.271803i
$424$		0			0
$425$			11.1803i			0.542326i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−17.5000	+	20.7063i	−0.844908	+	0.999709i
$430$		0			0
$431$			41.4126i			1.99477i	0.0722525	+	0.997386i	$0.476981\pi$
$431$			41.4126i			1.99477i	−0.0722525	+	0.997386i	$0.523019\pi$
$432$		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
$435$	−17.5000	−	14.7902i	−0.839061	−	0.709136i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	3.50000	+	20.7063i	0.166667	+	0.986013i
$442$		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
$447$	−26.4575	+	31.3050i	−1.25140	+	1.48067i
$448$		0			0
$449$		−	41.4126i		−	1.95438i	−0.212368	−	0.977190i	$-0.568118\pi$
$449$		−	41.4126i		−	1.95438i	0.212368	−	0.977190i	$-0.431882\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	22.4889	+	19.0066i	1.05662	+	0.893007i
$454$		0			0
$455$			15.6525i			0.733799i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	−10.0000	−	5.91608i	−0.466760	−	0.276139i
$460$		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
$467$			42.4853i			1.96598i	0.183646	+	0.982992i	$0.441210\pi$
$467$			42.4853i			1.96598i	−0.183646	+	0.982992i	$0.558790\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−28.0000	−	23.6643i	−1.29017	−	1.09039i
$472$		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
$485$		−	41.4126i		−	1.88045i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.5804i		−	1.33494i	−0.744635	−	0.667472i	$-0.767376\pi$
$491$		−	29.5804i		−	1.33494i	0.744635	−	0.667472i	$-0.232624\pi$
$492$		0			0
$493$	−13.2288			−0.595793
$494$		0			0
$495$	−6.61438	−	39.1312i	−0.297294	−	1.75882i
$496$		0			0
$497$			31.3050i			1.40422i
$498$		0			0
$499$	−41.0000			−1.83541			−0.917706	−	0.397260i	$-0.869961\pi$
$499$	−41.0000			−1.83541			−0.917706	+	0.397260i	$0.869961\pi$
$500$		0			0
$501$	−27.5000	+	32.5384i	−1.22861	+	1.45371i
$502$		0			0
$503$		−	38.0132i		−	1.69492i	−0.530857	−	0.847461i	$-0.678130\pi$
$503$		−	38.0132i		−	1.69492i	0.530857	−	0.847461i	$-0.321870\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	7.93725	+	6.70820i	0.352506	+	0.297922i
$508$		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
$515$			5.91608i			0.260694i
$516$		0			0
$517$	66.1438			2.90900
$518$		0			0
$519$	12.5000	−	14.7902i	0.548689	−	0.649218i
$520$		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	$-0.800443\pi$
$523$	−37.0405			−1.61967			−0.809834	+	0.586659i	$0.800443\pi$
$524$		0			0
$525$	−17.5000	−	14.7902i	−0.763763	−	0.645497i
$526$		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
$537$	−13.2288	+	15.6525i	−0.570863	+	0.675454i
$538$		0			0
$539$			41.4126i			1.78377i
$540$		0			0
$541$	−43.0000			−1.84871			−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000			−1.84871			−0.924357	+	0.381528i	$0.875398\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			24.5967i			1.05361i
$546$		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
$553$	−2.64575			−0.112509
$554$		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
$561$	−17.5000	−	14.7902i	−0.738851	−	0.624443i
$562$		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
$567$	22.4889	−	7.82624i	0.944444	−	0.328671i
$568$		0			0
$569$			47.3286i			1.98412i	0.125767	+	0.992060i	$0.459861\pi$
$569$			47.3286i			1.98412i	−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$	6.61438	−	7.82624i	0.276320	−	0.326946i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−34.3948			−1.43187			−0.715936	−	0.698165i	$-0.754000\pi$
$577$	−34.3948			−1.43187			−0.715936	+	0.698165i	$0.754000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	23.6643i		−	0.981761i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	17.5000	−	2.95804i	0.723536	−	0.122300i
$586$		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
$593$		−	42.4853i		−	1.74466i	−0.488916	−	0.872331i	$-0.662608\pi$
$593$		−	42.4853i		−	1.74466i	0.488916	−	0.872331i	$-0.337392\pi$
$594$		0			0
$595$	−13.2288			−0.542326
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	29.5804i		−	1.20862i	−0.796748	−	0.604311i	$-0.793448\pi$
$599$		−	29.5804i		−	1.20862i	0.796748	−	0.604311i	$-0.206552\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	53.6656i		−	2.18182i
$606$		0			0
$607$	−44.9778			−1.82559			−0.912796	−	0.408416i	$-0.866081\pi$
$607$	−44.9778			−1.82559			−0.912796	+	0.408416i	$0.866081\pi$
$608$		0			0
$609$	17.5000	−	20.7063i	0.709136	−	0.839061i
$610$		0			0
$611$			29.5804i			1.19669i
$612$		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
$631$	−47.0000			−1.87104			−0.935520	−	0.353273i	$-0.885069\pi$
$631$	−47.0000			−1.87104			−0.935520	+	0.353273i	$0.885069\pi$
$632$		0			0
$633$	30.4261	+	25.7148i	1.20933	+	1.02207i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−18.5203			−0.733799
$638$		0			0
$639$	35.0000	−	5.91608i	1.38458	−	0.234036i
$640$		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
$643$	50.2693			1.98243			0.991213	−	0.132273i	$-0.0422275\pi$
$643$	50.2693			1.98243			0.991213	+	0.132273i	$0.0422275\pi$
$644$		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
$657$	−5.29150	−	31.3050i	−0.206441	−	1.22132i
$658$		0			0
$659$			5.91608i			0.230458i	0.993339	+	0.115229i	$0.0367601\pi$
$659$			5.91608i			0.230458i	−0.993339	+	0.115229i	$0.963240\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	6.61438	−	7.82624i	0.256881	−	0.303946i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	38.5000	+	32.5384i	1.48850	+	1.25801i
$670$		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
$677$			51.4296i			1.97660i	0.152527	+	0.988299i	$0.451259\pi$
$677$			51.4296i			1.97660i	−0.152527	+	0.988299i	$0.548741\pi$
$678$		0			0
$679$	49.0000			1.88045
$680$		0			0
$681$	32.5000	−	38.4545i	1.24540	−	1.47358i
$682$		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
$693$	46.3006	−	7.82624i	1.75882	−	0.297294i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	41.4126i		−	1.56413i	−0.623196	−	0.782065i	$-0.714166\pi$
$701$		−	41.4126i		−	1.56413i	0.623196	−	0.782065i	$-0.285834\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−33.0719	−	27.9508i	−1.24556	−	1.05269i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.00000			−0.0375558			−0.0187779	−	0.999824i	$-0.505978\pi$
$709$	−1.00000			−0.0375558			−0.0187779	+	0.999824i	$0.505978\pi$
$710$		0			0
$711$	0.500000	+	2.95804i	0.0187515	+	0.110935i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	35.0000			1.30893
$716$		0			0
$717$	−33.0719	+	39.1312i	−1.23509	+	1.46138i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−7.00000			−0.260694
$722$		0			0
$723$		0			0
$724$		0			0
$725$			29.5804i			1.09859i
$726$		0			0
$727$	−5.29150			−0.196251			−0.0981255	−	0.995174i	$-0.531285\pi$
$727$	−5.29150			−0.196251			−0.0981255	+	0.995174i	$0.531285\pi$
$728$		0			0
$729$	−13.0000	−	23.6643i	−0.481481	−	0.876456i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−50.2693			−1.85674			−0.928369	−	0.371660i	$-0.878789\pi$
$733$	−50.2693			−1.85674			−0.928369	+	0.371660i	$0.878789\pi$
$734$		0			0
$735$	17.5000	−	20.7063i	0.645497	−	0.763763i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−11.0000			−0.404642			−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000			−0.404642			−0.202321	+	0.979319i	$0.564848\pi$
$740$		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
$747$	−26.4575	+	4.47214i	−0.968030	+	0.163627i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	13.0000			0.474377			0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000			0.474377			0.237188	+	0.971464i	$0.423774\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	38.0132i		−	1.38344i
$756$		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
$763$	−29.1033			−1.05361
$764$		0			0
$765$	2.50000	+	14.7902i	0.0903877	+	0.534741i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	5.00000	−	5.91608i	0.180071	−	0.213062i
$772$		0			0
$773$		−	2.23607i		−	0.0804258i	−0.999191	−	0.0402129i	$-0.987196\pi$
$773$		−	2.23607i		−	0.0804258i	0.999191	−	0.0402129i	$-0.0128036\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	70.0000			2.50480
$782$		0			0
$783$	−26.4575	−	15.6525i	−0.945514	−	0.559374i
$784$		0			0
$785$			47.3286i			1.68923i
$786$		0			0
$787$	−44.9778			−1.60328			−0.801642	−	0.597804i	$-0.796040\pi$
$787$	−44.9778			−1.60328			−0.801642	+	0.597804i	$0.796040\pi$
$788$		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
$797$		−	55.9017i		−	1.98014i	−0.140576	−	0.990070i	$-0.544895\pi$
$797$		−	55.9017i		−	1.98014i	0.140576	−	0.990070i	$-0.455105\pi$
$798$		0			0
$799$	−25.0000			−0.884436
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	62.6099i		−	2.20946i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	41.4126i		−	1.45599i	−0.685583	−	0.727994i	$-0.740453\pi$
$809$		−	41.4126i		−	1.45599i	0.685583	−	0.727994i	$-0.259547\pi$
$810$		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
$819$	3.50000	+	20.7063i	0.122300	+	0.723536i
$820$		0			0
$821$		−	5.91608i		−	0.206473i	−0.994657	−	0.103236i	$-0.967080\pi$
$821$		−	5.91608i		−	0.206473i	0.994657	−	0.103236i	$-0.0329198\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−33.0719	+	39.1312i	−1.15142	+	1.36237i
$826$		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
$833$		−	15.6525i		−	0.542326i
$834$		0			0
$835$	55.0000			1.90335
$836$		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
$841$	−6.00000			−0.206897
$842$		0			0
$843$	−6.61438	+	7.82624i	−0.227811	+	0.269550i
$844$		0			0
$845$		−	13.4164i		−	0.461538i
$846$		0			0
$847$	63.4980			2.18182
$848$		0			0
$849$	−3.50000	−	2.95804i	−0.120120	−	0.101520i
$850$		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	$-0.682432\pi$
$857$		−	49.1935i		−	1.68042i	0.542263	−	0.840209i	$-0.317568\pi$
$858$		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
$865$	−25.0000			−0.850026
$866$		0			0
$867$	−15.8745	−	13.4164i	−0.539127	−	0.455645i
$868$		0			0
$869$			5.91608i			0.200689i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−9.26013	−	54.7837i	−0.313408	−	1.85415i
$874$		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
$879$	27.5000	−	32.5384i	0.927552	−	1.09749i
$880$		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
$891$	−17.5000	−	50.2867i	−0.586272	−	1.68467i
$892$		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
$913$	−52.9150			−1.75123
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−29.0000			−0.956622			−0.478311	−	0.878191i	$-0.658751\pi$
$919$	−29.0000			−0.956622			−0.478311	+	0.878191i	$0.658751\pi$
$920$		0			0
$921$	−45.5000	−	38.4545i	−1.49928	−	1.26712i
$922$		0			0
$923$			31.3050i			1.03042i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.32288	+	7.82624i	0.0434489	+	0.257047i
$928$		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
$935$			29.5804i			0.967382i
$936$		0			0
$937$	60.8523			1.98796			0.993979	−	0.109574i	$-0.0349486\pi$
$937$	60.8523			1.98796			0.993979	+	0.109574i	$0.0349486\pi$
$938$		0			0
$939$	−38.5000	−	32.5384i	−1.25640	−	1.06185i
$940$		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
$949$	28.0000			0.908918
$950$		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
$955$	−13.2288			−0.428073
$956$		0			0
$957$	−46.3006	−	39.1312i	−1.49669	−	1.26493i
$958$		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
$975$	−17.5000	−	14.7902i	−0.560449	−	0.473665i
$976$		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
$981$	5.50000	+	32.5384i	0.175601	+	1.03887i
$982$		0			0
$983$			29.0689i			0.927153i	0.886057	+	0.463577i	$0.153434\pi$
$983$			29.0689i			0.927153i	−0.886057	+	0.463577i	$0.846566\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	33.0719	−	39.1312i	1.05269	−	1.24556i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$	10.5830	+	8.94427i	0.335842	+	0.283838i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−18.5203			−0.586542			−0.293271	−	0.956029i	$-0.594744\pi$
$997$	−18.5203			−0.586542			−0.293271	+	0.956029i	$0.594744\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.k.b.209.1		4
3.2	odd	2	inner	1680.2.k.b.209.2		4
4.3	odd	2		105.2.g.b.104.4	yes	4
5.4	even	2	inner	1680.2.k.b.209.4		4
7.6	odd	2	inner	1680.2.k.b.209.4		4
12.11	even	2		105.2.g.b.104.3	yes	4
15.14	odd	2	inner	1680.2.k.b.209.3		4
20.3	even	4		525.2.b.f.251.2		4
20.7	even	4		525.2.b.f.251.3		4
20.19	odd	2		105.2.g.b.104.1	✓	4
21.20	even	2	inner	1680.2.k.b.209.3		4
28.3	even	6		735.2.p.b.509.2		8
28.11	odd	6		735.2.p.b.509.3		8
28.19	even	6		735.2.p.b.374.4		8
28.23	odd	6		735.2.p.b.374.1		8
28.27	even	2		105.2.g.b.104.1	✓	4
35.34	odd	2	CM	1680.2.k.b.209.1		4
60.23	odd	4		525.2.b.f.251.4		4
60.47	odd	4		525.2.b.f.251.1		4
60.59	even	2		105.2.g.b.104.2	yes	4
84.11	even	6		735.2.p.b.509.1		8
84.23	even	6		735.2.p.b.374.3		8
84.47	odd	6		735.2.p.b.374.2		8
84.59	odd	6		735.2.p.b.509.4		8
84.83	odd	2		105.2.g.b.104.2	yes	4
105.104	even	2	inner	1680.2.k.b.209.2		4
140.19	even	6		735.2.p.b.374.1		8
140.27	odd	4		525.2.b.f.251.2		4
140.39	odd	6		735.2.p.b.509.2		8
140.59	even	6		735.2.p.b.509.3		8
140.79	odd	6		735.2.p.b.374.4		8
140.83	odd	4		525.2.b.f.251.3		4
140.139	even	2		105.2.g.b.104.4	yes	4
420.59	odd	6		735.2.p.b.509.1		8
420.83	even	4		525.2.b.f.251.1		4
420.167	even	4		525.2.b.f.251.4		4
420.179	even	6		735.2.p.b.509.4		8
420.299	odd	6		735.2.p.b.374.3		8
420.359	even	6		735.2.p.b.374.2		8
420.419	odd	2		105.2.g.b.104.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.g.b.104.1	✓	4	20.19	odd	2
105.2.g.b.104.1	✓	4	28.27	even	2
105.2.g.b.104.2	yes	4	60.59	even	2
105.2.g.b.104.2	yes	4	84.83	odd	2
105.2.g.b.104.3	yes	4	12.11	even	2
105.2.g.b.104.3	yes	4	420.419	odd	2
105.2.g.b.104.4	yes	4	4.3	odd	2
105.2.g.b.104.4	yes	4	140.139	even	2
525.2.b.f.251.1		4	60.47	odd	4
525.2.b.f.251.1		4	420.83	even	4
525.2.b.f.251.2		4	20.3	even	4
525.2.b.f.251.2		4	140.27	odd	4
525.2.b.f.251.3		4	20.7	even	4
525.2.b.f.251.3		4	140.83	odd	4
525.2.b.f.251.4		4	60.23	odd	4
525.2.b.f.251.4		4	420.167	even	4
735.2.p.b.374.1		8	28.23	odd	6
735.2.p.b.374.1		8	140.19	even	6
735.2.p.b.374.2		8	84.47	odd	6
735.2.p.b.374.2		8	420.359	even	6
735.2.p.b.374.3		8	84.23	even	6
735.2.p.b.374.3		8	420.299	odd	6
735.2.p.b.374.4		8	28.19	even	6
735.2.p.b.374.4		8	140.79	odd	6
735.2.p.b.509.1		8	84.11	even	6
735.2.p.b.509.1		8	420.59	odd	6
735.2.p.b.509.2		8	28.3	even	6
735.2.p.b.509.2		8	140.39	odd	6
735.2.p.b.509.3		8	28.11	odd	6
735.2.p.b.509.3		8	140.59	even	6
735.2.p.b.509.4		8	84.59	odd	6
735.2.p.b.509.4		8	420.179	even	6
1680.2.k.b.209.1		4	1.1	even	1	trivial
1680.2.k.b.209.1		4	35.34	odd	2	CM
1680.2.k.b.209.2		4	3.2	odd	2	inner
1680.2.k.b.209.2		4	105.104	even	2	inner
1680.2.k.b.209.3		4	15.14	odd	2	inner
1680.2.k.b.209.3		4	21.20	even	2	inner
1680.2.k.b.209.4		4	5.4	even	2	inner
1680.2.k.b.209.4		4	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.32288 - 1.11803i) q^{3} +2.23607i q^{5} -2.64575 q^{7} +(0.500000 + 2.95804i) q^{9} +5.91608i q^{11} -2.64575 q^{13} +(2.50000 - 2.95804i) q^{15} -2.23607i q^{17} +(3.50000 + 2.95804i) q^{21} -5.00000 q^{25} +(2.64575 - 4.47214i) q^{27} -5.91608i q^{29} +(6.61438 - 7.82624i) q^{33} -5.91608i q^{35} +(3.50000 + 2.95804i) q^{39} +(-6.61438 + 1.11803i) q^{45} -11.1803i q^{47} +7.00000 q^{49} +(-2.50000 + 2.95804i) q^{51} -13.2288 q^{55} +(-1.32288 - 7.82624i) q^{63} -5.91608i q^{65} -11.8322i q^{71} -10.5830 q^{73} +(6.61438 + 5.59017i) q^{75} -15.6525i q^{77} +1.00000 q^{79} +(-8.50000 + 2.95804i) q^{81} +8.94427i q^{83} +5.00000 q^{85} +(-6.61438 + 7.82624i) q^{87} +7.00000 q^{91} -18.5203 q^{97} +(-17.5000 + 2.95804i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9} + 10 q^{15} + 14 q^{21} - 20 q^{25} + 14 q^{39} + 28 q^{49} - 10 q^{51} + 4 q^{79} - 34 q^{81} + 20 q^{85} + 28 q^{91} - 70 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 + 10 * q^15 + 14 * q^21 - 20 * q^25 + 14 * q^39 + 28 * q^49 - 10 * q^51 + 4 * q^79 - 34 * q^81 + 20 * q^85 + 28 * q^91 - 70 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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