LMFDB - Embedded newform 1231.1.b.c.1230.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1231,1,Mod(1230,1231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1231, 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("1231.1230");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1231 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1231.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:	$0.614349030551$
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:	$\Z[a_1, a_2]$
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			1230.6
Root			$-0.792160$ of defining polynomial
Character	$\chi$	$=$	1231.1230

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.792160 q^{2} -0.372483 q^{4} +1.53209 q^{5} +1.94609 q^{7} -1.08723 q^{8} +1.00000 q^{9} +1.21366 q^{10} -1.00000 q^{11} -1.67098 q^{13} +1.54161 q^{14} -0.488773 q^{16} +0.792160 q^{18} -1.98648 q^{19} -0.570677 q^{20} -0.792160 q^{22} +1.34730 q^{25} -1.32368 q^{26} -0.724886 q^{28} -0.116290 q^{29} -0.573606 q^{31} +0.700040 q^{32} +2.98158 q^{35} -0.372483 q^{36} +1.19432 q^{37} -1.57361 q^{38} -1.66573 q^{40} -1.87939 q^{41} -0.116290 q^{43} +0.372483 q^{44} +1.53209 q^{45} +2.78727 q^{49} +1.06727 q^{50} +0.622410 q^{52} -1.53209 q^{55} -2.11584 q^{56} -0.0921200 q^{58} -0.454388 q^{62} +1.94609 q^{63} +1.04332 q^{64} -2.56008 q^{65} +2.36189 q^{70} -1.08723 q^{72} +0.946090 q^{74} +0.739929 q^{76} -1.94609 q^{77} -0.748844 q^{80} +1.00000 q^{81} -1.48877 q^{82} -0.0921200 q^{86} +1.08723 q^{88} +1.21366 q^{90} -3.25187 q^{91} -3.04346 q^{95} +2.20796 q^{98} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{4} + 9 q^{9} - 9 q^{11} + 9 q^{16} + 9 q^{25} - 9 q^{28} + 9 q^{36} - 9 q^{38} - 9 q^{40} - 9 q^{44} + 9 q^{49} + 9 q^{64} - 9 q^{70} - 9 q^{74} + 9 q^{81} - 9 q^{99}+O(q^{100}) $ 9 * q + 9 * q^4 + 9 * q^9 - 9 * q^11 + 9 * q^16 + 9 * q^25 - 9 * q^28 + 9 * q^36 - 9 * q^38 - 9 * q^40 - 9 * q^44 + 9 * q^49 + 9 * q^64 - 9 * q^70 - 9 * q^74 + 9 * q^81 - 9 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1231\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$2$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−0.372483	−0.372483
$5$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$5$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$6$	0	0
$7$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$7$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$8$	−1.08723	−1.08723
$9$	1.00000	1.00000
$10$	1.21366	1.21366
$11$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$12$	0	0
$13$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$13$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$14$	1.54161	1.54161
$15$	0	0
$16$	−0.488773	−0.488773
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.792160	0.792160
$19$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$19$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$20$	−0.570677	−0.570677
$21$	0	0
$22$	−0.792160	−0.792160
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	1.34730	1.34730
$26$	−1.32368	−1.32368
$27$	0	0
$28$	−0.724886	−0.724886
$29$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$29$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$30$	0	0
$31$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$31$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$32$	0.700040	0.700040
$33$	0	0
$34$	0	0
$35$	2.98158	2.98158
$36$	−0.372483	−0.372483
$37$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$37$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$38$	−1.57361	−1.57361
$39$	0	0
$40$	−1.66573	−1.66573
$41$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$41$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$42$	0	0
$43$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$43$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$44$	0.372483	0.372483
$45$	1.53209	1.53209
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	2.78727	2.78727
$50$	1.06727	1.06727
$51$	0	0
$52$	0.622410	0.622410
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−1.53209	−1.53209
$56$	−2.11584	−2.11584
$57$	0	0
$58$	−0.0921200	−0.0921200
$59$	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			$\pi$
$62$	−0.454388	−0.454388
$63$	1.94609	1.94609
$64$	1.04332	1.04332
$65$	−2.56008	−2.56008
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	2.36189	2.36189
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.08723	−1.08723
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0.946090	0.946090
$75$	0	0
$76$	0.739929	0.739929
$77$	−1.94609	−1.94609
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.748844	−0.748844
$81$	1.00000	1.00000
$82$	−1.48877	−1.48877
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−0.0921200	−0.0921200
$87$	0	0
$88$	1.08723	1.08723
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	1.21366	1.21366
$91$	−3.25187	−3.25187
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.04346	−3.04346
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	2.20796	2.20796
$99$	−1.00000	−1.00000
$100$	−0.501845	−0.501845
$101$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$101$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1.81673	1.81673
$105$	0	0
$106$	0	0
$107$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$107$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−1.21366	−1.21366
$111$	0	0
$112$	−0.951196	−0.951196
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0.0433160	0.0433160
$117$	−1.67098	−1.67098
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0.213659	0.213659
$125$	0.532089	0.532089
$126$	1.54161	1.54161
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0.126433	0.126433
$129$	0	0
$130$	−2.02799	−2.02799
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−3.86586	−3.86586
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$137$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$138$	0	0
$139$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$139$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$140$	−1.11059	−1.11059
$141$	0	0
$142$	0	0
$143$	1.67098	1.67098
$144$	−0.488773	−0.488773
$145$	−0.178166	−0.178166
$146$	0	0
$147$	0	0
$148$	−0.444863	−0.444863
$149$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$149$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$150$	0	0
$151$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$151$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$152$	2.15975	2.15975
$153$	0	0
$154$	−1.54161	−1.54161
$155$	−0.878816	−0.878816
$156$	0	0
$157$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$157$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$158$	0	0
$159$	0	0
$160$	1.07252	1.07252
$161$	0	0
$162$	0.792160	0.792160
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0.700040	0.700040
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.79216	1.79216
$170$	0	0
$171$	−1.98648	−1.98648
$172$	0.0433160	0.0433160
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	2.62196	2.62196
$176$	0.488773	0.488773
$177$	0	0
$178$	0	0
$179$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$179$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$180$	−0.570677	−0.570677
$181$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$181$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$182$	−2.57600	−2.57600
$183$	0	0
$184$	0	0
$185$	1.82980	1.82980
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−2.41090	−2.41090
$191$	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
$196$	−1.03821	−1.03821
$197$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$197$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$198$	−0.792160	−0.792160
$199$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$199$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$200$	−1.46482	−1.46482
$201$	0	0
$202$	1.41580	1.41580
$203$	−0.226310	−0.226310
$204$	0	0
$205$	−2.87939	−2.87939
$206$	0	0
$207$	0	0
$208$	0.816728	0.816728
$209$	1.98648	1.98648
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0.275114	0.275114
$215$	−0.178166	−0.178166
$216$	0	0
$217$	−1.11629	−1.11629
$218$	0	0
$219$	0	0
$220$	0.570677	0.570677
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.36234	1.36234
$225$	1.34730	1.34730
$226$	0	0
$227$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$227$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0.126433	0.126433
$233$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$233$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$234$	−1.32368	−1.32368
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.27034	4.27034
$246$	0	0
$247$	3.31935	3.31935
$248$	0.623640	0.623640
$249$	0	0
$250$	0.421499	0.421499
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−0.724886	−0.724886
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.943161	−0.943161
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	2.32425	2.32425
$260$	0.953588	0.953588
$261$	−0.116290	−0.116290
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−3.06238	−3.06238
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$271$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$272$	0	0
$273$	0	0
$274$	−1.08723	−1.08723
$275$	−1.34730	−1.34730
$276$	0	0
$277$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$277$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$278$	−1.32368	−1.32368
$279$	−0.573606	−0.573606
$280$	−3.24165	−3.24165
$281$	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			$\pi$
$284$	0	0
$285$	0	0
$286$	1.32368	1.32368
$287$	−3.65745	−3.65745
$288$	0.700040	0.700040
$289$	1.00000	1.00000
$290$	−0.141136	−0.141136
$291$	0	0
$292$	0	0
$293$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$293$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$294$	0	0
$295$	0	0
$296$	−1.29849	−1.29849
$297$	0	0
$298$	−1.57361	−1.57361
$299$	0	0
$300$	0	0
$301$	−0.226310	−0.226310
$302$	0.946090	0.946090
$303$	0	0
$304$	0.970936	0.970936
$305$	0	0
$306$	0	0
$307$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$307$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$308$	0.724886	0.724886
$309$	0	0
$310$	−0.696163	−0.696163
$311$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$311$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0.627517	0.627517
$315$	2.98158	2.98158
$316$	0	0
$317$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$317$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$318$	0	0
$319$	0.116290	0.116290
$320$	1.59845	1.59845
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.372483	−0.372483
$325$	−2.25130	−2.25130
$326$	0	0
$327$	0	0
$328$	2.04332	2.04332
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	1.19432	1.19432
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$337$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$338$	1.41968	1.41968
$339$	0	0
$340$	0	0
$341$	0.573606	0.573606
$342$	−1.57361	−1.57361
$343$	3.47818	3.47818
$344$	0.126433	0.126433
$345$	0	0
$346$	0	0
$347$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$347$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	2.07701	2.07701
$351$	0	0
$352$	−0.700040	−0.700040
$353$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$353$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−0.454388	−0.454388
$359$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$359$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$360$	−1.66573	−1.66573
$361$	2.94609	2.94609
$362$	1.54161	1.54161
$363$	0	0
$364$	1.21127	1.21127
$365$	0	0
$366$	0	0
$367$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$367$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$368$	0	0
$369$	−1.87939	−1.87939
$370$	1.44949	1.44949
$371$	0	0
$372$	0	0
$373$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$373$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.194317	0.194317
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	1.13364	1.13364
$381$	0	0
$382$	0	0
$383$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$383$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$384$	0	0
$385$	−2.98158	−2.98158
$386$	0	0
$387$	−0.116290	−0.116290
$388$	0	0
$389$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$389$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$390$	0	0
$391$	0	0
$392$	−3.03039	−3.03039
$393$	0	0
$394$	0.275114	0.275114
$395$	0	0
$396$	0.372483	0.372483
$397$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$398$	0.627517	0.627517
$399$	0	0
$400$	−0.658522	−0.658522
$401$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$401$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$402$	0	0
$403$	0.958482	0.958482
$404$	−0.665726	−0.665726
$405$	1.53209	1.53209
$406$	−0.179274	−0.179274
$407$	−1.19432	−1.19432
$408$	0	0
$409$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$409$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$410$	−2.28093	−2.28093
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1.16975	−1.16975
$417$	0	0
$418$	1.57361	1.57361
$419$	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	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−0.129362	−0.129362
$429$	0	0
$430$	−0.141136	−0.141136
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$433$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$434$	−0.884279	−0.884279
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	1.66573	1.66573
$441$	2.78727	2.78727
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	2.03039	2.03039
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	1.06727	1.06727
$451$	1.87939	1.87939
$452$	0	0
$453$	0	0
$454$	−1.08723	−1.08723
$455$	−4.98215	−4.98215
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$464$	0.0568392	0.0568392
$465$	0	0
$466$	1.54161	1.54161
$467$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$467$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$468$	0.622410	0.622410
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.116290	0.116290
$474$	0	0
$475$	−2.67637	−2.67637
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$479$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$480$	0	0
$481$	−1.99567	−1.99567
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$487$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$488$	0	0
$489$	0	0
$490$	3.38279	3.38279
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	2.62946	2.62946
$495$	−1.53209	−1.53209
$496$	0.280363	0.280363
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−0.198194	−0.198194
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−2.11584	−2.11584
$505$	2.73825	2.73825
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−0.873567	−0.873567
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	1.84118	1.84118
$519$	0	0
$520$	2.78339	2.78339
$521$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$521$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$522$	−0.0921200	−0.0921200
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	1.43997	1.43997
$533$	3.14041	3.14041
$534$	0	0
$535$	0.532089	0.532089
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.78727	−2.78727
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−1.57361	−1.57361
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0.511227	0.511227
$549$	0	0
$550$	−1.06727	−1.06727
$551$	0.231007	0.231007
$552$	0	0
$553$	0	0
$554$	1.41580	1.41580
$555$	0	0
$556$	0.622410	0.622410
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−0.454388	−0.454388
$559$	0.194317	0.194317
$560$	−1.45732	−1.45732
$561$	0	0
$562$	0	0
$563$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$563$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.94609	1.94609
$568$	0	0
$569$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$569$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$570$	0	0
$571$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$571$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$572$	−0.622410	−0.622410
$573$	0	0
$574$	−2.89729	−2.89729
$575$	0	0
$576$	1.04332	1.04332
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0.792160	0.792160
$579$	0	0
$580$	0.0663639	0.0663639
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.56008	−2.56008
$586$	0.275114	0.275114
$587$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$587$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$588$	0	0
$589$	1.13946	1.13946
$590$	0	0
$591$	0	0
$592$	−0.583750	−0.583750
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0.739929	0.739929
$597$	0	0
$598$	0	0
$599$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$599$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.179274	−0.179274
$603$	0	0
$604$	−0.444863	−0.444863
$605$	0	0
$606$	0	0
$607$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$607$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$608$	−1.39061	−1.39061
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0.627517	0.627517
$615$	0	0
$616$	2.11584	2.11584
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$619$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$620$	0.327344	0.327344
$621$	0	0
$622$	1.21366	1.21366
$623$	0	0
$624$	0	0
$625$	−0.532089	−0.532089
$626$	0	0
$627$	0	0
$628$	−0.295066	−0.295066
$629$	0	0
$630$	2.36189	2.36189
$631$	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
$634$	1.41580	1.41580
$635$	0	0
$636$	0	0
$637$	−4.65745	−4.65745
$638$	0.0921200	0.0921200
$639$	0	0
$640$	0.193707	0.193707
$641$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$641$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$642$	0	0
$643$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$643$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$647$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$648$	−1.08723	−1.08723
$649$	0	0
$650$	−1.78339	−1.78339
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0.918593	0.918593
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$661$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.92284	−5.92284
$666$	0.946090	0.946090
$667$	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			$\pi$
$674$	−0.454388	−0.454388
$675$	0	0
$676$	−0.667549	−0.667549
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0.454388	0.454388
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0.739929	0.739929
$685$	−2.10277	−2.10277
$686$	2.75527	2.75527
$687$	0	0
$688$	0.0568392	0.0568392
$689$	0	0
$690$	0	0
$691$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$691$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$692$	0	0
$693$	−1.94609	−1.94609
$694$	−1.48877	−1.48877
$695$	−2.56008	−2.56008
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−0.976636	−0.976636
$701$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$701$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$702$	0	0
$703$	−2.37248	−2.37248
$704$	−1.04332	−1.04332
$705$	0	0
$706$	−1.57361	−1.57361
$707$	3.47818	3.47818
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	2.56008	2.56008
$716$	0.213659	0.213659
$717$	0	0
$718$	0.275114	0.275114
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−0.748844	−0.748844
$721$	0	0
$722$	2.33377	2.33377
$723$	0	0
$724$	−0.724886	−0.724886
$725$	−0.156677	−0.156677
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	3.53552	3.53552
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0.946090	0.946090
$735$	0	0
$736$	0	0
$737$	0	0
$738$	−1.48877	−1.48877
$739$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$739$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$740$	−0.681570	−0.681570
$741$	0	0
$742$	0	0
$743$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$744$	0	0
$745$	−3.04346	−3.04346
$746$	−0.454388	−0.454388
$747$	0	0
$748$	0	0
$749$	0.675870	0.675870
$750$	0	0
$751$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$751$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$752$	0	0
$753$	0	0
$754$	0.153930	0.153930
$755$	1.82980	1.82980
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	3.30893	3.30893
$761$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$761$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−1.48877	−1.48877
$767$	0	0
$768$	0	0
$769$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$769$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$770$	−2.36189	−2.36189
$771$	0	0
$772$	0	0
$773$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$773$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$774$	−0.0921200	−0.0921200
$775$	−0.772818	−0.772818
$776$	0	0
$777$	0	0
$778$	−1.48877	−1.48877
$779$	3.73336	3.73336
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.36234	−1.36234
$785$	1.21366	1.21366
$786$	0	0
$787$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$787$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$788$	−0.129362	−0.129362
$789$	0	0
$790$	0	0
$791$	0	0
$792$	1.08723	1.08723
$793$	0	0
$794$	−0.792160	−0.792160
$795$	0	0
$796$	−0.295066	−0.295066
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0.943161	0.943161
$801$	0	0
$802$	−1.32368	−1.32368
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0.759271	0.759271
$807$	0	0
$808$	−1.94316	−1.94316
$809$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$809$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$810$	1.21366	1.21366
$811$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$811$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$812$	0.0842967	0.0842967
$813$	0	0
$814$	−0.946090	−0.946090
$815$	0	0
$816$	0	0
$817$	0.231007	0.231007
$818$	−0.454388	−0.454388
$819$	−3.25187	−3.25187
$820$	1.07252	1.07252
$821$	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	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−1.74336	−1.74336
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−0.739929	−0.739929
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−0.986477	−0.986477
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.74575	2.74575
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$853$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$854$	0	0
$855$	−3.04346	−3.04346
$856$	−0.377590	−0.377590
$857$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$857$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$858$	0	0
$859$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$859$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$860$	0.0663639	0.0663639
$861$	0	0
$862$	0	0
$863$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$863$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$864$	0	0
$865$	0	0
$866$	1.41580	1.41580
$867$	0	0
$868$	0.415799	0.415799
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.03549	1.03549
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0.748844	0.748844
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	2.20796	2.20796
$883$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$883$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−1.00000
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−0.878816	−0.878816
$896$	0.246050	0.246050
$897$	0	0
$898$	0	0
$899$	0.0667045	0.0667045
$900$	−0.501845	−0.501845
$901$	0	0
$902$	1.48877	1.48877
$903$	0	0
$904$	0	0
$905$	2.98158	2.98158
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0.511227	0.511227
$909$	1.78727	1.78727
$910$	−3.94666	−3.94666
$911$	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
$919$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$919$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.60910	1.60910
$926$	−0.792160	−0.792160
$927$	0	0
$928$	−0.0814074	−0.0814074
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−5.53684	−5.53684
$932$	−0.724886	−0.724886
$933$	0	0
$934$	−1.08723	−1.08723
$935$	0	0
$936$	1.81673	1.81673
$937$	−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
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0.0921200	0.0921200
$947$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$947$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$948$	0	0
$949$	0	0
$950$	−2.12011	−2.12011
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0.627517	0.627517
$959$	−2.67098	−2.67098
$960$	0	0
$961$	−0.670976	−0.670976
$962$	−1.58089	−1.58089
$963$	0.347296	0.347296
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$967$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	−3.25187	−3.25187
$974$	−1.08723	−1.08723
$975$	0	0
$976$	0	0
$977$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$977$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$978$	0	0
$979$	0	0
$980$	−1.59063	−1.59063
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0.532089	0.532089
$986$	0	0
$987$	0	0
$988$	−1.23640	−1.23640
$989$	0	0
$990$	−1.21366	−1.21366
$991$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$991$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$992$	−0.401547	−0.401547
$993$	0	0
$994$	0	0
$995$	1.21366	1.21366
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1231.1.b.c.1230.6	✓	9
1231.1230	odd	2	CM	1231.1.b.c.1230.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1231.1.b.c.1230.6	✓	9	1.1	even	1	trivial
1231.1.b.c.1230.6	✓	9	1231.1230	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:	\( N \)	\(=\)	\( 1231 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1231.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.792160 q^{2} -0.372483 q^{4} +1.53209 q^{5} +1.94609 q^{7} -1.08723 q^{8} +1.00000 q^{9} +1.21366 q^{10} -1.00000 q^{11} -1.67098 q^{13} +1.54161 q^{14} -0.488773 q^{16} +0.792160 q^{18} -1.98648 q^{19} -0.570677 q^{20} -0.792160 q^{22} +1.34730 q^{25} -1.32368 q^{26} -0.724886 q^{28} -0.116290 q^{29} -0.573606 q^{31} +0.700040 q^{32} +2.98158 q^{35} -0.372483 q^{36} +1.19432 q^{37} -1.57361 q^{38} -1.66573 q^{40} -1.87939 q^{41} -0.116290 q^{43} +0.372483 q^{44} +1.53209 q^{45} +2.78727 q^{49} +1.06727 q^{50} +0.622410 q^{52} -1.53209 q^{55} -2.11584 q^{56} -0.0921200 q^{58} -0.454388 q^{62} +1.94609 q^{63} +1.04332 q^{64} -2.56008 q^{65} +2.36189 q^{70} -1.08723 q^{72} +0.946090 q^{74} +0.739929 q^{76} -1.94609 q^{77} -0.748844 q^{80} +1.00000 q^{81} -1.48877 q^{82} -0.0921200 q^{86} +1.08723 q^{88} +1.21366 q^{90} -3.25187 q^{91} -3.04346 q^{95} +2.20796 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{4} + 9 q^{9} - 9 q^{11} + 9 q^{16} + 9 q^{25} - 9 q^{28} + 9 q^{36} - 9 q^{38} - 9 q^{40} - 9 q^{44} + 9 q^{49} + 9 q^{64} - 9 q^{70} - 9 q^{74} + 9 q^{81} - 9 q^{99}+O(q^{100}) \) 9 * q + 9 * q^4 + 9 * q^9 - 9 * q^11 + 9 * q^16 + 9 * q^25 - 9 * q^28 + 9 * q^36 - 9 * q^38 - 9 * q^40 - 9 * q^44 + 9 * q^49 + 9 * q^64 - 9 * q^70 - 9 * q^74 + 9 * q^81 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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