LMFDB - Embedded newform 1759.1.b.c.1758.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1759,1,Mod(1758,1759)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1759, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1759.1758");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1759 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1759.b (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:	yes
Analytic conductor:	$0.877855357221$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{54})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1 $ x^9 - 9x^7 + 27x^5 - 30x^3 + 9x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} - \cdots)$

Embedding invariants

Embedding label			1758.5
Root			$0.573606$ of defining polynomial
Character	$\chi$	$=$	1759.1758

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.347296 q^{2} -0.879385 q^{4} -0.573606 q^{5} -0.652704 q^{8} +1.00000 q^{9} -0.199211 q^{10} -1.37248 q^{11} +0.792160 q^{13} +0.652704 q^{16} +1.94609 q^{17} +0.347296 q^{18} +0.504421 q^{20} -0.476658 q^{22} -0.116290 q^{23} -0.670976 q^{25} +0.275114 q^{26} +1.53209 q^{31} +0.879385 q^{32} +0.675870 q^{34} -0.879385 q^{36} +0.374395 q^{40} +1.19432 q^{41} +1.19432 q^{43} +1.20694 q^{44} -0.573606 q^{45} -0.0403870 q^{46} -1.98648 q^{47} +1.00000 q^{49} -0.233027 q^{50} -0.696613 q^{52} +1.78727 q^{53} +0.787265 q^{55} +0.532089 q^{62} -0.347296 q^{64} -0.454388 q^{65} -1.71136 q^{68} -1.67098 q^{71} -0.652704 q^{72} -0.374395 q^{80} +1.00000 q^{81} +0.414782 q^{82} -1.11629 q^{85} +0.414782 q^{86} +0.895825 q^{88} -1.00000 q^{89} -0.199211 q^{90} +0.102263 q^{92} -0.689896 q^{94} +0.347296 q^{98} -1.37248 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{4} - 9 q^{8} + 9 q^{9} + 9 q^{16} + 9 q^{25} - 9 q^{32} + 9 q^{36} + 9 q^{49} - 9 q^{55} - 9 q^{62} - 9 q^{72} + 9 q^{81} - 9 q^{85} - 9 q^{89}+O(q^{100}) $ 9 * q + 9 * q^4 - 9 * q^8 + 9 * q^9 + 9 * q^16 + 9 * q^25 - 9 * q^32 + 9 * q^36 + 9 * q^49 - 9 * q^55 - 9 * q^62 - 9 * q^72 + 9 * q^81 - 9 * q^85 - 9 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1759\mathbb{Z}\right)^\times$.

$n$	$6$
$\chi(n)$	$-1$

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 $

$2$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$2$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−0.879385	−0.879385
$5$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$5$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−0.652704	−0.652704
$9$	1.00000	1.00000
$10$	−0.199211	−0.199211
$11$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$11$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$12$	0	0
$13$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$13$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$14$	0	0
$15$	0	0
$16$	0.652704	0.652704
$17$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$17$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$18$	0.347296	0.347296
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0.504421	0.504421
$21$	0	0
$22$	−0.476658	−0.476658
$23$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$23$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$24$	0	0
$25$	−0.670976	−0.670976
$26$	0.275114	0.275114
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$31$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$32$	0.879385	0.879385
$33$	0	0
$34$	0.675870	0.675870
$35$	0	0
$36$	−0.879385	−0.879385
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0.374395	0.374395
$41$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$41$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$42$	0	0
$43$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$43$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$44$	1.20694	1.20694
$45$	−0.573606	−0.573606
$46$	−0.0403870	−0.0403870
$47$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$47$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	−0.233027	−0.233027
$51$	0	0
$52$	−0.696613	−0.696613
$53$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$53$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$54$	0	0
$55$	0.787265	0.787265
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0.532089	0.532089
$63$	0	0
$64$	−0.347296	−0.347296
$65$	−0.454388	−0.454388
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.71136	−1.71136
$69$	0	0
$70$	0	0
$71$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$71$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$72$	−0.652704	−0.652704
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.374395	−0.374395
$81$	1.00000	1.00000
$82$	0.414782	0.414782
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−1.11629	−1.11629
$86$	0.414782	0.414782
$87$	0	0
$88$	0.895825	0.895825
$89$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$90$	−0.199211	−0.199211
$91$	0	0
$92$	0.102263	0.102263
$93$	0	0
$94$	−0.689896	−0.689896
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.347296	0.347296
$99$	−1.37248	−1.37248
$100$	0.590046	0.590046
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−0.517045	−0.517045
$105$	0	0
$106$	0.620711	0.620711
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$109$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$110$	0.273414	0.273414
$111$	0	0
$112$	0	0
$113$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$113$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$114$	0	0
$115$	0.0667045	0.0667045
$116$	0	0
$117$	0.792160	0.792160
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0.883710	0.883710
$122$	0	0
$123$	0	0
$124$	−1.34730	−1.34730
$125$	0.958482	0.958482
$126$	0	0
$127$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$127$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	−0.157807	−0.157807
$131$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$131$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−1.27022	−1.27022
$137$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$137$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	−0.580324	−0.580324
$143$	−1.08723	−1.08723
$144$	0.652704	0.652704
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$149$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	1.94609	1.94609
$154$	0	0
$155$	−0.878816	−0.878816
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−0.504421	−0.504421
$161$	0	0
$162$	0.347296	0.347296
$163$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$163$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$164$	−1.05026	−1.05026
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−0.372483	−0.372483
$170$	−0.387683	−0.387683
$171$	0	0
$172$	−1.05026	−1.05026
$173$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$173$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$174$	0	0
$175$	0	0
$176$	−0.895825	−0.895825
$177$	0	0
$178$	−0.347296	−0.347296
$179$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$180$	0.504421	0.504421
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0.0759027	0.0759027
$185$	0	0
$186$	0	0
$187$	−2.67098	−2.67098
$188$	1.74688	1.74688
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−0.879385	−0.879385
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−0.476658	−0.476658
$199$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$199$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$200$	0.437948	0.437948
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.685068	−0.685068
$206$	0	0
$207$	−0.116290	−0.116290
$208$	0.517045	0.517045
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−1.57169	−1.57169
$213$	0	0
$214$	0	0
$215$	−0.685068	−0.685068
$216$	0	0
$217$	0	0
$218$	−0.689896	−0.689896
$219$	0	0
$220$	−0.692309	−0.692309
$221$	1.54161	1.54161
$222$	0	0
$223$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$223$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$224$	0	0
$225$	−0.670976	−0.670976
$226$	−0.652704	−0.652704
$227$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$227$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0.0231662	0.0231662
$231$	0	0
$232$	0	0
$233$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$233$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$234$	0.275114	0.275114
$235$	1.13946	1.13946
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$239$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$240$	0	0
$241$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$241$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$242$	0.306909	0.306909
$243$	0	0
$244$	0	0
$245$	−0.573606	−0.573606
$246$	0	0
$247$	0	0
$248$	−1.00000	−1.00000
$249$	0	0
$250$	0.332877	0.332877
$251$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$251$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$252$	0	0
$253$	0.159606	0.159606
$254$	0.675870	0.675870
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0.399582	0.399582
$261$	0	0
$262$	0.620711	0.620711
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−1.02519	−1.02519
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$271$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$272$	1.27022	1.27022
$273$	0	0
$274$	0.620711	0.620711
$275$	0.920903	0.920903
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	1.53209	1.53209
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.46943	1.46943
$285$	0	0
$286$	−0.377590	−0.377590
$287$	0	0
$288$	0.879385	0.879385
$289$	2.78727	2.78727
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$293$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−0.652704	−0.652704
$299$	−0.0921200	−0.0921200
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0.675870	0.675870
$307$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$307$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$308$	0	0
$309$	0	0
$310$	−0.305210	−0.305210
$311$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$311$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$312$	0	0
$313$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$313$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0.199211	0.199211
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.879385	−0.879385
$325$	−0.531520	−0.531520
$326$	−0.689896	−0.689896
$327$	0	0
$328$	−0.779535	−0.779535
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$338$	−0.129362	−0.129362
$339$	0	0
$340$	0.981649	0.981649
$341$	−2.10277	−2.10277
$342$	0	0
$343$	0	0
$344$	−0.779535	−0.779535
$345$	0	0
$346$	−0.476658	−0.476658
$347$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$347$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$348$	0	0
$349$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$350$	0	0
$351$	0	0
$352$	−1.20694	−1.20694
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0.958482	0.958482
$356$	0.879385	0.879385
$357$	0	0
$358$	−0.347296	−0.347296
$359$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$359$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$360$	0.374395	0.374395
$361$	1.00000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−0.0759027	−0.0759027
$369$	1.19432	1.19432
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$373$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$374$	−0.927620	−0.927620
$375$	0	0
$376$	1.29658	1.29658
$377$	0	0
$378$	0	0
$379$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$379$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.19432	1.19432
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	−0.226310	−0.226310
$392$	−0.652704	−0.652704
$393$	0	0
$394$	0	0
$395$	0	0
$396$	1.20694	1.20694
$397$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$397$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$398$	0.414782	0.414782
$399$	0	0
$400$	−0.437948	−0.437948
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	1.21366	1.21366
$404$	0	0
$405$	−0.573606	−0.573606
$406$	0	0
$407$	0	0
$408$	0	0
$409$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$409$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$410$	−0.237922	−0.237922
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−0.0403870	−0.0403870
$415$	0	0
$416$	0.696613	0.696613
$417$	0	0
$418$	0	0
$419$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$419$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$420$	0	0
$421$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$421$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$422$	0	0
$423$	−1.98648	−1.98648
$424$	−1.16655	−1.16655
$425$	−1.30578	−1.30578
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−0.237922	−0.237922
$431$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$431$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$432$	0	0
$433$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$433$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$434$	0	0
$435$	0	0
$436$	1.74688	1.74688
$437$	0	0
$438$	0	0
$439$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$439$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$440$	−0.513851	−0.513851
$441$	1.00000	1.00000
$442$	0.535397	0.535397
$443$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$443$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$444$	0	0
$445$	0.573606	0.573606
$446$	0.120615	0.120615
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−0.233027	−0.233027
$451$	−1.63918	−1.63918
$452$	1.65270	1.65270
$453$	0	0
$454$	0.532089	0.532089
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	−0.0586590	−0.0586590
$461$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$461$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−0.0403870	−0.0403870
$467$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$467$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$468$	−0.696613	−0.696613
$469$	0	0
$470$	0.395729	0.395729
$471$	0	0
$472$	0	0
$473$	−1.63918	−1.63918
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.78727	1.78727
$478$	0.675870	0.675870
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0.120615	0.120615
$483$	0	0
$484$	−0.777122	−0.777122
$485$	0	0
$486$	0	0
$487$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$487$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$488$	0	0
$489$	0	0
$490$	−0.199211	−0.199211
$491$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$491$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.787265	0.787265
$496$	1.00000	1.00000
$497$	0	0
$498$	0	0
$499$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$499$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$500$	−0.842875	−0.842875
$501$	0	0
$502$	−0.689896	−0.689896
$503$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$503$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$504$	0	0
$505$	0	0
$506$	0.0554304	0.0554304
$507$	0	0
$508$	−1.71136	−1.71136
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.72641	2.72641
$518$	0	0
$519$	0	0
$520$	0.296581	0.296581
$521$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.57169	−1.57169
$525$	0	0
$526$	0	0
$527$	2.98158	2.98158
$528$	0	0
$529$	−0.986477	−0.986477
$530$	−0.356044	−0.356044
$531$	0	0
$532$	0	0
$533$	0.946090	0.946090
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.37248	−1.37248
$540$	0	0
$541$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$541$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$542$	0.275114	0.275114
$543$	0	0
$544$	1.71136	1.71136
$545$	1.13946	1.13946
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−1.57169	−1.57169
$549$	0	0
$550$	0.319826	0.319826
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$557$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$558$	0.532089	0.532089
$559$	0.946090	0.946090
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	1.07803	1.07803
$566$	0	0
$567$	0	0
$568$	1.09065	1.09065
$569$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$569$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$570$	0	0
$571$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$571$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$572$	0.956090	0.956090
$573$	0	0
$574$	0	0
$575$	0.0780275	0.0780275
$576$	−0.347296	−0.347296
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0.968007	0.968007
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.45299	−2.45299
$584$	0	0
$585$	−0.454388	−0.454388
$586$	−0.580324	−0.580324
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	1.65270	1.65270
$597$	0	0
$598$	−0.0319929	−0.0319929
$599$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$599$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.506902	−0.506902
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.57361	−1.57361
$612$	−1.71136	−1.71136
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−0.0403870	−0.0403870
$615$	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			$\pi$
$620$	0.772818	0.772818
$621$	0	0
$622$	0.120615	0.120615
$623$	0	0
$624$	0	0
$625$	0.121184	0.121184
$626$	−0.652704	−0.652704
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$631$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.11629	−1.11629
$636$	0	0
$637$	0.792160	0.792160
$638$	0	0
$639$	−1.67098	−1.67098
$640$	0.573606	0.573606
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$647$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$648$	−0.652704	−0.652704
$649$	0	0
$650$	−0.184595	−0.184595
$651$	0	0
$652$	1.74688	1.74688
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−1.02519	−1.02519
$656$	0.779535	0.779535
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$661$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	−0.347296	−0.347296
$675$	0	0
$676$	0.327556	0.327556
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0.728606	0.728606
$681$	0	0
$682$	−0.730283	−0.730283
$683$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$683$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$684$	0	0
$685$	−1.02519	−1.02519
$686$	0	0
$687$	0	0
$688$	0.779535	0.779535
$689$	1.41580	1.41580
$690$	0	0
$691$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$691$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$692$	1.20694	1.20694
$693$	0	0
$694$	0.532089	0.532089
$695$	0	0
$696$	0	0
$697$	2.32425	2.32425
$698$	−0.347296	−0.347296
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0.476658	0.476658
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0.332877	0.332877
$711$	0	0
$712$	0.652704	0.652704
$713$	−0.178166	−0.178166
$714$	0	0
$715$	0.623640	0.623640
$716$	0.879385	0.879385
$717$	0	0
$718$	−0.199211	−0.199211
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−0.374395	−0.374395
$721$	0	0
$722$	0.347296	0.347296
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$727$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	2.32425	2.32425
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	−0.102263	−0.102263
$737$	0	0
$738$	0.414782	0.414782
$739$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$739$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	1.07803	1.07803
$746$	0.120615	0.120615
$747$	0	0
$748$	2.34882	2.34882
$749$	0	0
$750$	0	0
$751$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$751$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$752$	−1.29658	−1.29658
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−0.0403870	−0.0403870
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.11629	−1.11629
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$773$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$774$	0.414782	0.414782
$775$	−1.02799	−1.02799
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	2.29339	2.29339
$782$	−0.0785967	−0.0785967
$783$	0	0
$784$	0.652704	0.652704
$785$	0	0
$786$	0	0
$787$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$787$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0.895825	0.895825
$793$	0	0
$794$	−0.689896	−0.689896
$795$	0	0
$796$	−1.05026	−1.05026
$797$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$797$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$798$	0	0
$799$	−3.86586	−3.86586
$800$	−0.590046	−0.590046
$801$	−1.00000	−1.00000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0.421499	0.421499
$807$	0	0
$808$	0	0
$809$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$809$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$810$	−0.199211	−0.199211
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.13946	1.13946
$816$	0	0
$817$	0	0
$818$	0.620711	0.620711
$819$	0	0
$820$	0.602439	0.602439
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0.102263	0.102263
$829$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$829$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$830$	0	0
$831$	0	0
$832$	−0.275114	−0.275114
$833$	1.94609	1.94609
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−0.0403870	−0.0403870
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0.675870	0.675870
$843$	0	0
$844$	0	0
$845$	0.213659	0.213659
$846$	−0.689896	−0.689896
$847$	0	0
$848$	1.16655	1.16655
$849$	0	0
$850$	−0.453492	−0.453492
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0.602439	0.602439
$861$	0	0
$862$	0.275114	0.275114
$863$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$863$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$864$	0	0
$865$	0.787265	0.787265
$866$	0.532089	0.532089
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	1.29658	1.29658
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.580324	−0.580324
$879$	0	0
$880$	0.513851	0.513851
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.347296	0.347296
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−1.35567	−1.35567
$885$	0	0
$886$	−0.652704	−0.652704
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0.199211	0.199211
$891$	−1.37248	−1.37248
$892$	−0.305407	−0.305407
$893$	0	0
$894$	0	0
$895$	0.573606	0.573606
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0.590046	0.590046
$901$	3.47818	3.47818
$902$	−0.569281	−0.569281
$903$	0	0
$904$	1.22668	1.22668
$905$	0	0
$906$	0	0
$907$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$907$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$908$	−1.34730	−1.34730
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	−0.0435383	−0.0435383
$921$	0	0
$922$	−0.580324	−0.580324
$923$	−1.32368	−1.32368
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$929$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$930$	0	0
$931$	0	0
$932$	0.102263	0.102263
$933$	0	0
$934$	−0.199211	−0.199211
$935$	1.53209	1.53209
$936$	−0.517045	−0.517045
$937$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$937$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$938$	0	0
$939$	0	0
$940$	−1.00202	−1.00202
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−0.138887	−0.138887
$944$	0	0
$945$	0	0
$946$	−0.569281	−0.569281
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0.620711	0.620711
$955$	0	0
$956$	−1.71136	−1.71136
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.34730	1.34730
$962$	0	0
$963$	0	0
$964$	−0.305407	−0.305407
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.576801	−0.576801
$969$	0	0
$970$	0	0
$971$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$971$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$972$	0	0
$973$	0	0
$974$	−0.199211	−0.199211
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	1.37248	1.37248
$980$	0.504421	0.504421
$981$	−1.98648	−1.98648
$982$	0.275114	0.275114
$983$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$983$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.138887	−0.138887
$990$	0.273414	0.273414
$991$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$991$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$992$	1.34730	1.34730
$993$	0	0
$994$	0	0
$995$	−0.685068	−0.685068
$996$	0	0
$997$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$997$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$998$	0.675870	0.675870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1759.1.b.c.1758.5	✓	9
1759.1758	odd	2	CM	1759.1.b.c.1758.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1759.1.b.c.1758.5	✓	9	1.1	even	1	trivial
1759.1.b.c.1758.5	✓	9	1759.1758	odd	2	CM

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:	\( N \)	\(=\)	\( 1759 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1759.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.347296 q^{2} -0.879385 q^{4} -0.573606 q^{5} -0.652704 q^{8} +1.00000 q^{9} -0.199211 q^{10} -1.37248 q^{11} +0.792160 q^{13} +0.652704 q^{16} +1.94609 q^{17} +0.347296 q^{18} +0.504421 q^{20} -0.476658 q^{22} -0.116290 q^{23} -0.670976 q^{25} +0.275114 q^{26} +1.53209 q^{31} +0.879385 q^{32} +0.675870 q^{34} -0.879385 q^{36} +0.374395 q^{40} +1.19432 q^{41} +1.19432 q^{43} +1.20694 q^{44} -0.573606 q^{45} -0.0403870 q^{46} -1.98648 q^{47} +1.00000 q^{49} -0.233027 q^{50} -0.696613 q^{52} +1.78727 q^{53} +0.787265 q^{55} +0.532089 q^{62} -0.347296 q^{64} -0.454388 q^{65} -1.71136 q^{68} -1.67098 q^{71} -0.652704 q^{72} -0.374395 q^{80} +1.00000 q^{81} +0.414782 q^{82} -1.11629 q^{85} +0.414782 q^{86} +0.895825 q^{88} -1.00000 q^{89} -0.199211 q^{90} +0.102263 q^{92} -0.689896 q^{94} +0.347296 q^{98} -1.37248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{4} - 9 q^{8} + 9 q^{9} + 9 q^{16} + 9 q^{25} - 9 q^{32} + 9 q^{36} + 9 q^{49} - 9 q^{55} - 9 q^{62} - 9 q^{72} + 9 q^{81} - 9 q^{85} - 9 q^{89}+O(q^{100}) \) 9 * q + 9 * q^4 - 9 * q^8 + 9 * q^9 + 9 * q^16 + 9 * q^25 - 9 * q^32 + 9 * q^36 + 9 * q^49 - 9 * q^55 - 9 * q^62 - 9 * q^72 + 9 * q^81 - 9 * q^85 - 9 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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