LMFDB - Embedded newform 2671.1.b.a.2670.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2671,1,Mod(2670,2671)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2671, 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("2671.2670");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2671 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2671.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:	$1.33300264874$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\Q(\zeta_{46})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 10x^{9} + 9x^{8} + 36x^{7} - 28x^{6} - 56x^{5} + 35x^{4} + 35x^{3} - 15x^{2} - 6x + 1 $ x^11 - x^10 - 10x^9 + 9x^8 + 36x^7 - 28x^6 - 56x^5 + 35x^4 + 35x^3 - 15x^2 - 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{23}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{23} - \cdots)$

Embedding invariants

Embedding label			2670.3
Root			$-1.36511$ of defining polynomial
Character	$\chi$	$=$	2671.2670

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.55142 q^{2} +1.40691 q^{4} -1.98137 q^{5} -0.631293 q^{8} +1.00000 q^{9} +O(q^{10})$	$q-1.55142 q^{2} +1.40691 q^{4} -1.98137 q^{5} -0.631293 q^{8} +1.00000 q^{9} +3.07395 q^{10} -0.427511 q^{16} -1.15336 q^{17} -1.55142 q^{18} -2.78762 q^{20} +2.92583 q^{25} +1.29454 q^{32} +1.78935 q^{34} +1.40691 q^{36} -0.669759 q^{37} +1.25083 q^{40} -1.83442 q^{43} -1.98137 q^{45} -0.136485 q^{47} +1.00000 q^{49} -4.53921 q^{50} -0.136485 q^{61} -1.58087 q^{64} +1.70884 q^{67} -1.62268 q^{68} +1.36511 q^{71} -0.631293 q^{72} +1.03908 q^{74} +0.847057 q^{80} +1.00000 q^{81} -0.669759 q^{83} +2.28524 q^{85} +2.84596 q^{86} -1.83442 q^{89} +3.07395 q^{90} +0.211746 q^{94} -1.55142 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - q^{2} + 10 q^{4} - q^{5} - 2 q^{8} + 11 q^{9}+O(q^{10}) $ 11 * q - q^2 + 10 * q^4 - q^5 - 2 * q^8 + 11 * q^9	$ 11 q - q^{2} + 10 q^{4} - q^{5} - 2 q^{8} + 11 q^{9} - 2 q^{10} + 9 q^{16} - q^{17} - q^{18} - 3 q^{20} + 10 q^{25} - 3 q^{32} - 2 q^{34} + 10 q^{36} - q^{37} - 4 q^{40} - q^{43} - q^{45} - q^{47} + 11 q^{49} - 3 q^{50} - q^{61} + 8 q^{64} - q^{67} - 3 q^{68} - q^{71} - 2 q^{72} - 2 q^{74} - 5 q^{80} + 11 q^{81} - q^{83} - 2 q^{85} - 2 q^{86} - q^{89} - 2 q^{90} - 2 q^{94} - q^{98}+O(q^{100}) $ 11 * q - q^2 + 10 * q^4 - q^5 - 2 * q^8 + 11 * q^9 - 2 * q^10 + 9 * q^16 - q^17 - q^18 - 3 * q^20 + 10 * q^25 - 3 * q^32 - 2 * q^34 + 10 * q^36 - q^37 - 4 * q^40 - q^43 - q^45 - q^47 + 11 * q^49 - 3 * q^50 - q^61 + 8 * q^64 - q^67 - 3 * q^68 - q^71 - 2 * q^72 - 2 * q^74 - 5 * q^80 + 11 * q^81 - q^83 - 2 * q^85 - 2 * q^86 - q^89 - 2 * q^90 - 2 * q^94 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2671\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

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 $

$2$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$2$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.40691	1.40691
$5$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$5$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−0.631293	−0.631293
$9$	1.00000	1.00000
$10$	3.07395	3.07395
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	−0.427511	−0.427511
$17$	−1.15336	−1.15336	−0.576680	−	0.816970i	$-0.695652\pi$
$17$	−1.15336	−1.15336	−0.576680	+	0.816970i	$0.695652\pi$
$18$	−1.55142	−1.55142
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−2.78762	−2.78762
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	2.92583	2.92583
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.29454	1.29454
$33$	0	0
$34$	1.78935	1.78935
$35$	0	0
$36$	1.40691	1.40691
$37$	−0.669759	−0.669759	−0.334880	−	0.942261i	$-0.608696\pi$
$37$	−0.669759	−0.669759	−0.334880	+	0.942261i	$0.608696\pi$
$38$	0	0
$39$	0	0
$40$	1.25083	1.25083
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−1.83442	−1.83442	−0.917211	−	0.398401i	$-0.869565\pi$
$43$	−1.83442	−1.83442	−0.917211	+	0.398401i	$0.869565\pi$
$44$	0	0
$45$	−1.98137	−1.98137
$46$	0	0
$47$	−0.136485	−0.136485	−0.0682424	−	0.997669i	$-0.521739\pi$
$47$	−0.136485	−0.136485	−0.0682424	+	0.997669i	$0.521739\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	−4.53921	−4.53921
$51$	0	0
$52$	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	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−0.136485	−0.136485	−0.0682424	−	0.997669i	$-0.521739\pi$
$61$	−0.136485	−0.136485	−0.0682424	+	0.997669i	$0.521739\pi$
$62$	0	0
$63$	0	0
$64$	−1.58087	−1.58087
$65$	0	0
$66$	0	0
$67$	1.70884	1.70884	0.854419	−	0.519584i	$-0.173913\pi$
$67$	1.70884	1.70884	0.854419	+	0.519584i	$0.173913\pi$
$68$	−1.62268	−1.62268
$69$	0	0
$70$	0	0
$71$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$71$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$72$	−0.631293	−0.631293
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	1.03908	1.03908
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.847057	0.847057
$81$	1.00000	1.00000
$82$	0	0
$83$	−0.669759	−0.669759	−0.334880	−	0.942261i	$-0.608696\pi$
$83$	−0.669759	−0.669759	−0.334880	+	0.942261i	$0.608696\pi$
$84$	0	0
$85$	2.28524	2.28524
$86$	2.84596	2.84596
$87$	0	0
$88$	0	0
$89$	−1.83442	−1.83442	−0.917211	−	0.398401i	$-0.869565\pi$
$89$	−1.83442	−1.83442	−0.917211	+	0.398401i	$0.869565\pi$
$90$	3.07395	3.07395
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0.211746	0.211746
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.55142	−1.55142
$99$	0	0
$100$	4.11639	4.11639
$101$	1.70884	1.70884	0.854419	−	0.519584i	$-0.173913\pi$
$101$	1.70884	1.70884	0.854419	+	0.519584i	$0.173913\pi$
$102$	0	0
$103$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$103$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.92583	1.92583	0.962917	−	0.269797i	$-0.0869565\pi$
$107$	1.92583	1.92583	0.962917	+	0.269797i	$0.0869565\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.70884	1.70884	0.854419	−	0.519584i	$-0.173913\pi$
$113$	1.70884	1.70884	0.854419	+	0.519584i	$0.173913\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0.211746	0.211746
$123$	0	0
$124$	0	0
$125$	−3.81579	−3.81579
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.15806	1.15806
$129$	0	0
$130$	0	0
$131$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$131$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$132$	0	0
$133$	0	0
$134$	−2.65113	−2.65113
$135$	0	0
$136$	0.728108	0.728108
$137$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$137$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	−2.11786	−2.11786
$143$	0	0
$144$	−0.427511	−0.427511
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−0.942292	−0.942292
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−0.669759	−0.669759	−0.334880	−	0.942261i	$-0.608696\pi$
$151$	−0.669759	−0.669759	−0.334880	+	0.942261i	$0.608696\pi$
$152$	0	0
$153$	−1.15336	−1.15336
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−2.56497	−2.56497
$161$	0	0
$162$	−1.55142	−1.55142
$163$	−1.15336	−1.15336	−0.576680	−	0.816970i	$-0.695652\pi$
$163$	−1.15336	−1.15336	−0.576680	+	0.816970i	$0.695652\pi$
$164$	0	0
$165$	0	0
$166$	1.03908	1.03908
$167$	0.406912	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$167$	0.406912	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	−3.54537	−3.54537
$171$	0	0
$172$	−2.58087	−2.58087
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	2.84596	2.84596
$179$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$179$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$180$	−2.78762	−2.78762
$181$	0.406912	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$181$	0.406912	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.32704	1.32704
$186$	0	0
$187$	0	0
$188$	−0.192022	−0.192022
$189$	0	0
$190$	0	0
$191$	−1.15336	−1.15336	−0.576680	−	0.816970i	$-0.695652\pi$
$191$	−1.15336	−1.15336	−0.576680	+	0.816970i	$0.695652\pi$
$192$	0	0
$193$	−0.136485	−0.136485	−0.0682424	−	0.997669i	$-0.521739\pi$
$193$	−0.136485	−0.136485	−0.0682424	+	0.997669i	$0.521739\pi$
$194$	0	0
$195$	0	0
$196$	1.40691	1.40691
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	1.92583	1.92583	0.962917	−	0.269797i	$-0.0869565\pi$
$199$	1.92583	1.92583	0.962917	+	0.269797i	$0.0869565\pi$
$200$	−1.84706	−1.84706
$201$	0	0
$202$	−2.65113	−2.65113
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−1.42751	−1.42751
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	−2.98778	−2.98778
$215$	3.63467	3.63467
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	2.92583	2.92583
$226$	−2.65113	−2.65113
$227$	0.406912	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$227$	0.406912	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0.270427	0.270427
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.70884	1.70884	0.854419	−	0.519584i	$-0.173913\pi$
$239$	1.70884	1.70884	0.854419	+	0.519584i	$0.173913\pi$
$240$	0	0
$241$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$241$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$242$	−1.55142	−1.55142
$243$	0	0
$244$	−0.192022	−0.192022
$245$	−1.98137	−1.98137
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	5.91991	5.91991
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.215765	−0.215765
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−1.42751	−1.42751
$263$	−1.83442	−1.83442	−0.917211	−	0.398401i	$-0.869565\pi$
$263$	−1.83442	−1.83442	−0.917211	+	0.398401i	$0.869565\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	2.40419	2.40419
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−1.15336	−1.15336	−0.576680	−	0.816970i	$-0.695652\pi$
$271$	−1.15336	−1.15336	−0.576680	+	0.816970i	$0.695652\pi$
$272$	0.493074	0.493074
$273$	0	0
$274$	−2.11786	−2.11786
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$283$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$284$	1.92058	1.92058
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.29454	1.29454
$289$	0.330241	0.330241
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0.422814	0.422814
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	1.03908	1.03908
$303$	0	0
$304$	0	0
$305$	0.270427	0.270427
$306$	1.78935	1.78935
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.669759	−0.669759	−0.334880	−	0.942261i	$-0.608696\pi$
$311$	−0.669759	−0.669759	−0.334880	+	0.942261i	$0.608696\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.406912	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$317$	0.406912	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$318$	0	0
$319$	0	0
$320$	3.13229	3.13229
$321$	0	0
$322$	0	0
$323$	0	0
$324$	1.40691	1.40691
$325$	0	0
$326$	1.78935	1.78935
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$331$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$332$	−0.942292	−0.942292
$333$	−0.669759	−0.669759
$334$	−0.631293	−0.631293
$335$	−3.38585	−3.38585
$336$	0	0
$337$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$337$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$338$	−1.55142	−1.55142
$339$	0	0
$340$	3.21513	3.21513
$341$	0	0
$342$	0	0
$343$	0	0
$344$	1.15806	1.15806
$345$	0	0
$346$	0	0
$347$	−0.136485	−0.136485	−0.0682424	−	0.997669i	$-0.521739\pi$
$347$	−0.136485	−0.136485	−0.0682424	+	0.997669i	$0.521739\pi$
$348$	0	0
$349$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$349$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$353$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$354$	0	0
$355$	−2.70478	−2.70478
$356$	−2.58087	−2.58087
$357$	0	0
$358$	−2.11786	−2.11786
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	1.25083	1.25083
$361$	1.00000	1.00000
$362$	−0.631293	−0.631293
$363$	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	0
$369$	0	0
$370$	−2.05880	−2.05880
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0.0861618	0.0861618
$377$	0	0
$378$	0	0
$379$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$379$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$380$	0	0
$381$	0	0
$382$	1.78935	1.78935
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0.211746	0.211746
$387$	−1.83442	−1.83442
$388$	0	0
$389$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$389$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$390$	0	0
$391$	0	0
$392$	−0.631293	−0.631293
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.83442	−1.83442	−0.917211	−	0.398401i	$-0.869565\pi$
$397$	−1.83442	−1.83442	−0.917211	+	0.398401i	$0.869565\pi$
$398$	−2.98778	−2.98778
$399$	0	0
$400$	−1.25083	−1.25083
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	2.40419	2.40419
$405$	−1.98137	−1.98137
$406$	0	0
$407$	0	0
$408$	0	0
$409$	1.70884	1.70884	0.854419	−	0.519584i	$-0.173913\pi$
$409$	1.70884	1.70884	0.854419	+	0.519584i	$0.173913\pi$
$410$	0	0
$411$	0	0
$412$	1.29454	1.29454
$413$	0	0
$414$	0	0
$415$	1.32704	1.32704
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	−0.136485	−0.136485
$424$	0	0
$425$	−3.37454	−3.37454
$426$	0	0
$427$	0	0
$428$	2.70948	2.70948
$429$	0	0
$430$	−5.63891	−5.63891
$431$	0.406912	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$431$	0.406912	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$439$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	3.63467	3.63467
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.669759	−0.669759	−0.334880	−	0.942261i	$-0.608696\pi$
$449$	−0.669759	−0.669759	−0.334880	+	0.942261i	$0.608696\pi$
$450$	−4.53921	−4.53921
$451$	0	0
$452$	2.40419	2.40419
$453$	0	0
$454$	−0.631293	−0.631293
$455$	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	0
$461$	1.92583	1.92583	0.962917	−	0.269797i	$-0.0869565\pi$
$461$	1.92583	1.92583	0.962917	+	0.269797i	$0.0869565\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	−0.419547	−0.419547
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−2.65113	−2.65113
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	−1.42751	−1.42751
$483$	0	0
$484$	1.40691	1.40691
$485$	0	0
$486$	0	0
$487$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$487$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$488$	0.0861618	0.0861618
$489$	0	0
$490$	3.07395	3.07395
$491$	1.92583	1.92583	0.962917	−	0.269797i	$-0.0869565\pi$
$491$	1.92583	1.92583	0.962917	+	0.269797i	$0.0869565\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.136485	−0.136485	−0.0682424	−	0.997669i	$-0.521739\pi$
$499$	−0.136485	−0.136485	−0.0682424	+	0.997669i	$0.521739\pi$
$500$	−5.36849	−5.36849
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−3.38585	−3.38585
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$509$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$510$	0	0
$511$	0	0
$512$	−0.823315	−0.823315
$513$	0	0
$514$	0	0
$515$	−1.82312	−1.82312
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	1.92583	1.92583	0.962917	−	0.269797i	$-0.0869565\pi$
$523$	1.92583	1.92583	0.962917	+	0.269797i	$0.0869565\pi$
$524$	1.29454	1.29454
$525$	0	0
$526$	2.84596	2.84596
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−3.81579	−3.81579
$536$	−1.07878	−1.07878
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$541$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$542$	1.78935	1.78935
$543$	0	0
$544$	−1.49307	−1.49307
$545$	0	0
$546$	0	0
$547$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$547$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$548$	1.92058	1.92058
$549$	−0.136485	−0.136485
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.406912	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$557$	0.406912	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$563$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$564$	0	0
$565$	−3.38585	−3.38585
$566$	−2.11786	−2.11786
$567$	0	0
$568$	−0.861781	−0.861781
$569$	−0.136485	−0.136485	−0.0682424	−	0.997669i	$-0.521739\pi$
$569$	−0.136485	−0.136485	−0.0682424	+	0.997669i	$0.521739\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−1.58087	−1.58087
$577$	0.406912	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$577$	0.406912	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$578$	−0.512343	−0.512343
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$587$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0.286329	0.286329
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$599$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$600$	0	0
$601$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$601$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$602$	0	0
$603$	1.70884	1.70884
$604$	−0.942292	−0.942292
$605$	−1.98137	−1.98137
$606$	0	0
$607$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$607$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$608$	0	0
$609$	0	0
$610$	−0.419547	−0.419547
$611$	0	0
$612$	−1.62268	−1.62268
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	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
$622$	1.03908	1.03908
$623$	0	0
$624$	0	0
$625$	4.63467	4.63467
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.772474	0.772474
$630$	0	0
$631$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$631$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$632$	0	0
$633$	0	0
$634$	−0.631293	−0.631293
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.36511	1.36511
$640$	−2.29454	−2.29454
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$643$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−0.631293	−0.631293
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−1.62268	−1.62268
$653$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$653$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$654$	0	0
$655$	−1.82312	−1.82312
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$659$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$660$	0	0
$661$	1.70884	1.70884	0.854419	−	0.519584i	$-0.173913\pi$
$661$	1.70884	1.70884	0.854419	+	0.519584i	$0.173913\pi$
$662$	−1.42751	−1.42751
$663$	0	0
$664$	0.422814	0.422814
$665$	0	0
$666$	1.03908	1.03908
$667$	0	0
$668$	0.572489	0.572489
$669$	0	0
$670$	5.25288	5.25288
$671$	0	0
$672$	0	0
$673$	−1.83442	−1.83442	−0.917211	−	0.398401i	$-0.869565\pi$
$673$	−1.83442	−1.83442	−0.917211	+	0.398401i	$0.869565\pi$
$674$	2.40691	2.40691
$675$	0	0
$676$	1.40691	1.40691
$677$	1.92583	1.92583	0.962917	−	0.269797i	$-0.0869565\pi$
$677$	1.92583	1.92583	0.962917	+	0.269797i	$0.0869565\pi$
$678$	0	0
$679$	0	0
$680$	−1.44265	−1.44265
$681$	0	0
$682$	0	0
$683$	1.70884	1.70884	0.854419	−	0.519584i	$-0.173913\pi$
$683$	1.70884	1.70884	0.854419	+	0.519584i	$0.173913\pi$
$684$	0	0
$685$	−2.70478	−2.70478
$686$	0	0
$687$	0	0
$688$	0.784235	0.784235
$689$	0	0
$690$	0	0
$691$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$691$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$692$	0	0
$693$	0	0
$694$	0.211746	0.211746
$695$	0	0
$696$	0	0
$697$	0	0
$698$	2.40691	2.40691
$699$	0	0
$700$	0	0
$701$	−0.136485	−0.136485	−0.0682424	−	0.997669i	$-0.521739\pi$
$701$	−0.136485	−0.136485	−0.0682424	+	0.997669i	$0.521739\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−2.11786	−2.11786
$707$	0	0
$708$	0	0
$709$	−1.15336	−1.15336	−0.576680	−	0.816970i	$-0.695652\pi$
$709$	−1.15336	−1.15336	−0.576680	+	0.816970i	$0.695652\pi$
$710$	4.19626	4.19626
$711$	0	0
$712$	1.15806	1.15806
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.92058	1.92058
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.847057	0.847057
$721$	0	0
$722$	−1.55142	−1.55142
$723$	0	0
$724$	0.572489	0.572489
$725$	0	0
$726$	0	0
$727$	1.70884	1.70884	0.854419	−	0.519584i	$-0.173913\pi$
$727$	1.70884	1.70884	0.854419	+	0.519584i	$0.173913\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	2.11575	2.11575
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$739$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$740$	1.86703	1.86703
$741$	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
$747$	−0.669759	−0.669759
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.83442	−1.83442	−0.917211	−	0.398401i	$-0.869565\pi$
$751$	−1.83442	−1.83442	−0.917211	+	0.398401i	$0.869565\pi$
$752$	0.0583487	0.0583487
$753$	0	0
$754$	0	0
$755$	1.32704	1.32704
$756$	0	0
$757$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$757$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$758$	2.40691	2.40691
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−1.62268	−1.62268
$765$	2.28524	2.28524
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$769$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$770$	0	0
$771$	0	0
$772$	−0.192022	−0.192022
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	2.84596	2.84596
$775$	0	0
$776$	0	0
$777$	0	0
$778$	2.40691	2.40691
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−0.427511	−0.427511
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	2.84596	2.84596
$795$	0	0
$796$	2.70948	2.70948
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0.157416	0.157416
$800$	3.78762	3.78762
$801$	−1.83442	−1.83442
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−1.07878	−1.07878
$809$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$809$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$810$	3.07395	3.07395
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.28524	2.28524
$816$	0	0
$817$	0	0
$818$	−2.65113	−2.65113
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$823$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$824$	−0.580871	−0.580871
$825$	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			$\pi$
$830$	−2.05880	−2.05880
$831$	0	0
$832$	0	0
$833$	−1.15336	−1.15336
$834$	0	0
$835$	−0.806244	−0.806244
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$839$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.98137	−1.98137
$846$	0.211746	0.211746
$847$	0	0
$848$	0	0
$849$	0	0
$850$	5.23534	5.23534
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−1.21576	−1.21576
$857$	−0.669759	−0.669759	−0.334880	−	0.942261i	$-0.608696\pi$
$857$	−0.669759	−0.669759	−0.334880	+	0.942261i	$0.608696\pi$
$858$	0	0
$859$	−0.669759	−0.669759	−0.334880	−	0.942261i	$-0.608696\pi$
$859$	−0.669759	−0.669759	−0.334880	+	0.942261i	$0.608696\pi$
$860$	5.11367	5.11367
$861$	0	0
$862$	−0.631293	−0.631293
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.15336	−1.15336	−0.576680	−	0.816970i	$-0.695652\pi$
$877$	−1.15336	−1.15336	−0.576680	+	0.816970i	$0.695652\pi$
$878$	2.40691	2.40691
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−1.55142	−1.55142
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.98137	−1.98137	−0.990686	−	0.136167i	$-0.956522\pi$
$887$	−1.98137	−1.98137	−0.990686	+	0.136167i	$0.956522\pi$
$888$	0	0
$889$	0	0
$890$	−5.63891	−5.63891
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2.70478	−2.70478
$896$	0	0
$897$	0	0
$898$	1.03908	1.03908
$899$	0	0
$900$	4.11639	4.11639
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−1.07878	−1.07878
$905$	−0.806244	−0.806244
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0.572489	0.572489
$909$	1.70884	1.70884
$910$	0	0
$911$	−0.136485	−0.136485	−0.0682424	−	0.997669i	$-0.521739\pi$
$911$	−0.136485	−0.136485	−0.0682424	+	0.997669i	$0.521739\pi$
$912$	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	0
$921$	0	0
$922$	−2.98778	−2.98778
$923$	0	0
$924$	0	0
$925$	−1.95960	−1.95960
$926$	0	0
$927$	0.920130	0.920130
$928$	0	0
$929$	−1.55142	−1.55142	−0.775711	−	0.631088i	$-0.782609\pi$
$929$	−1.55142	−1.55142	−0.775711	+	0.631088i	$0.782609\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.406912	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$937$	0.406912	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$938$	0	0
$939$	0	0
$940$	0.380467	0.380467
$941$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$941$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$942$	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
$949$	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	0
$955$	2.28524	2.28524
$956$	2.40419	2.40419
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	1.92583	1.92583
$964$	1.29454	1.29454
$965$	0.270427	0.270427
$966$	0	0
$967$	1.36511	1.36511	0.682553	−	0.730836i	$-0.260870\pi$
$967$	1.36511	1.36511	0.682553	+	0.730836i	$0.260870\pi$
$968$	−0.631293	−0.631293
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	2.40691	2.40691
$975$	0	0
$976$	0.0583487	0.0583487
$977$	0.920130	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$977$	0.920130	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$978$	0	0
$979$	0	0
$980$	−2.78762	−2.78762
$981$	0	0
$982$	−2.98778	−2.98778
$983$	1.92583	1.92583	0.962917	−	0.269797i	$-0.0869565\pi$
$983$	1.92583	1.92583	0.962917	+	0.269797i	$0.0869565\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.81579	−3.81579
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.211746	0.211746
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2671.1.b.a.2670.3	✓	11
2671.2670	odd	2	CM	2671.1.b.a.2670.3	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2671.1.b.a.2670.3	✓	11	1.1	even	1	trivial
2671.1.b.a.2670.3	✓	11	2671.2670	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:	\( N \)	\(=\)	\( 2671 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2671.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.55142 q^{2} +1.40691 q^{4} -1.98137 q^{5} -0.631293 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.55142 q^{2} +1.40691 q^{4} -1.98137 q^{5} -0.631293 q^{8} +1.00000 q^{9} +3.07395 q^{10} -0.427511 q^{16} -1.15336 q^{17} -1.55142 q^{18} -2.78762 q^{20} +2.92583 q^{25} +1.29454 q^{32} +1.78935 q^{34} +1.40691 q^{36} -0.669759 q^{37} +1.25083 q^{40} -1.83442 q^{43} -1.98137 q^{45} -0.136485 q^{47} +1.00000 q^{49} -4.53921 q^{50} -0.136485 q^{61} -1.58087 q^{64} +1.70884 q^{67} -1.62268 q^{68} +1.36511 q^{71} -0.631293 q^{72} +1.03908 q^{74} +0.847057 q^{80} +1.00000 q^{81} -0.669759 q^{83} +2.28524 q^{85} +2.84596 q^{86} -1.83442 q^{89} +3.07395 q^{90} +0.211746 q^{94} -1.55142 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - q^{2} + 10 q^{4} - q^{5} - 2 q^{8} + 11 q^{9}+O(q^{10}) \) 11 * q - q^2 + 10 * q^4 - q^5 - 2 * q^8 + 11 * q^9	\( 11 q - q^{2} + 10 q^{4} - q^{5} - 2 q^{8} + 11 q^{9} - 2 q^{10} + 9 q^{16} - q^{17} - q^{18} - 3 q^{20} + 10 q^{25} - 3 q^{32} - 2 q^{34} + 10 q^{36} - q^{37} - 4 q^{40} - q^{43} - q^{45} - q^{47} + 11 q^{49} - 3 q^{50} - q^{61} + 8 q^{64} - q^{67} - 3 q^{68} - q^{71} - 2 q^{72} - 2 q^{74} - 5 q^{80} + 11 q^{81} - q^{83} - 2 q^{85} - 2 q^{86} - q^{89} - 2 q^{90} - 2 q^{94} - q^{98}+O(q^{100}) \) 11 * q - q^2 + 10 * q^4 - q^5 - 2 * q^8 + 11 * q^9 - 2 * q^10 + 9 * q^16 - q^17 - q^18 - 3 * q^20 + 10 * q^25 - 3 * q^32 - 2 * q^34 + 10 * q^36 - q^37 - 4 * q^40 - q^43 - q^45 - q^47 + 11 * q^49 - 3 * q^50 - q^61 + 8 * q^64 - q^67 - 3 * q^68 - q^71 - 2 * q^72 - 2 * q^74 - 5 * q^80 + 11 * q^81 - q^83 - 2 * q^85 - 2 * q^86 - q^89 - 2 * q^90 - 2 * q^94 - q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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