LMFDB - Embedded newform 383.1.b.a.382.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [383,1,Mod(382,383)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(383, 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("383.382");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 383 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	383.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:	$0.191141899838$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{34})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 7x^{6} + 6x^{5} + 15x^{4} - 10x^{3} - 10x^{2} + 4x + 1 $ x^8 - x^7 - 7x^6 + 6x^5 + 15x^4 - 10x^3 - 10x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{17}$
Projective field:	Galois closure of 17.1.463009808974713123841.1

Embedding invariants

Embedding label			382.2
Root			$-0.184537$ of defining polynomial
Character	$\chi$	$=$	383.382

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70043 q^{2} +1.86494 q^{3} +1.89148 q^{4} -3.17122 q^{6} -0.547326 q^{7} -1.51590 q^{8} +2.47802 q^{9} +3.52750 q^{12} +0.930692 q^{14} +0.686207 q^{16} -1.96595 q^{17} -4.21371 q^{18} +0.184537 q^{19} -1.02073 q^{21} -1.20527 q^{23} -2.82706 q^{24} +1.00000 q^{25} +2.75642 q^{27} -1.03525 q^{28} +0.891477 q^{29} -1.96595 q^{31} +0.349047 q^{32} +3.34296 q^{34} +4.68711 q^{36} -0.313793 q^{38} +1.73569 q^{42} +1.47802 q^{43} +2.04948 q^{46} +1.27974 q^{48} -0.700434 q^{49} -1.70043 q^{50} -3.66638 q^{51} -4.68711 q^{54} +0.829690 q^{56} +0.344151 q^{57} -1.51590 q^{58} +3.34296 q^{62} -1.35628 q^{63} -1.27974 q^{64} -1.20527 q^{67} -3.71854 q^{68} -2.24776 q^{69} +1.47802 q^{71} -3.75642 q^{72} -1.20527 q^{73} +1.86494 q^{75} +0.349047 q^{76} +2.66255 q^{81} -1.93069 q^{84} -2.51327 q^{86} +1.66255 q^{87} -2.27974 q^{92} -3.66638 q^{93} +0.650953 q^{96} +1.19104 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - q^{3} + 7 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 7 q^{9} - 3 q^{12} - 2 q^{14} + 6 q^{16} - q^{17} - 3 q^{18} - q^{19} - 2 q^{21} - q^{23} - 4 q^{24} + 8 q^{25} - 2 q^{27} - 3 q^{28} - q^{29}+ \cdots + 14 q^{98}+O(q^{100}) $ 8 * q - q^2 - q^3 + 7 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 7 * q^9 - 3 * q^12 - 2 * q^14 + 6 * q^16 - q^17 - 3 * q^18 - q^19 - 2 * q^21 - q^23 - 4 * q^24 + 8 * q^25 - 2 * q^27 - 3 * q^28 - q^29 - q^31 - 3 * q^32 - 2 * q^34 + 4 * q^36 - 2 * q^38 - 4 * q^42 - q^43 - 2 * q^46 - 5 * q^48 + 7 * q^49 - q^50 - 2 * q^51 - 4 * q^54 - 4 * q^56 - 2 * q^57 - 2 * q^58 - 2 * q^62 - 3 * q^63 + 5 * q^64 - q^67 - 3 * q^68 - 2 * q^69 - q^71 - 6 * q^72 - q^73 - q^75 - 3 * q^76 + 6 * q^81 - 6 * q^84 - 2 * q^86 - 2 * q^87 - 3 * q^92 - 2 * q^93 + 11 * q^96 + 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/383\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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 $

$2$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$2$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$3$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$3$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$4$	1.89148	1.89148
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−3.17122	−3.17122
$7$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$7$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$8$	−1.51590	−1.51590
$9$	2.47802	2.47802
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	3.52750	3.52750
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0.930692	0.930692
$15$	0	0
$16$	0.686207	0.686207
$17$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$17$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$18$	−4.21371	−4.21371
$19$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$19$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$20$	0	0
$21$	−1.02073	−1.02073
$22$	0	0
$23$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$23$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$24$	−2.82706	−2.82706
$25$	1.00000	1.00000
$26$	0	0
$27$	2.75642	2.75642
$28$	−1.03525	−1.03525
$29$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$29$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$30$	0	0
$31$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$31$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$32$	0.349047	0.349047
$33$	0	0
$34$	3.34296	3.34296
$35$	0	0
$36$	4.68711	4.68711
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−0.313793	−0.313793
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	1.73569	1.73569
$43$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$43$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$44$	0	0
$45$	0	0
$46$	2.04948	2.04948
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.27974	1.27974
$49$	−0.700434	−0.700434
$50$	−1.70043	−1.70043
$51$	−3.66638	−3.66638
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−4.68711	−4.68711
$55$	0	0
$56$	0.829690	0.829690
$57$	0.344151	0.344151
$58$	−1.51590	−1.51590
$59$	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			$\pi$
$62$	3.34296	3.34296
$63$	−1.35628	−1.35628
$64$	−1.27974	−1.27974
$65$	0	0
$66$	0	0
$67$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$67$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$68$	−3.71854	−3.71854
$69$	−2.24776	−2.24776
$70$	0	0
$71$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$71$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$72$	−3.75642	−3.75642
$73$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$73$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$74$	0	0
$75$	1.86494	1.86494
$76$	0.349047	0.349047
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	2.66255	2.66255
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−1.93069	−1.93069
$85$	0	0
$86$	−2.51327	−2.51327
$87$	1.66255	1.66255
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−2.27974	−2.27974
$93$	−3.66638	−3.66638
$94$	0	0
$95$	0	0
$96$	0.650953	0.650953
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.19104	1.19104
$99$	0	0
$100$	1.89148	1.89148
$101$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$101$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$102$	6.23444	6.23444
$103$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$103$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	5.21371	5.21371
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−0.375579	−0.375579
$113$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$113$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$114$	−0.585206	−0.585206
$115$	0	0
$116$	1.68621	1.68621
$117$	0	0
$118$	0	0
$119$	1.07601	1.07601
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	0	0
$124$	−3.71854	−3.71854
$125$	0	0
$126$	2.30627	2.30627
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.82706	1.82706
$129$	2.75642	2.75642
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−0.101002	−0.101002
$134$	2.04948	2.04948
$135$	0	0
$136$	2.98017	2.98017
$137$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$137$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$138$	3.82217	3.82217
$139$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$139$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$140$	0	0
$141$	0	0
$142$	−2.51327	−2.51327
$143$	0	0
$144$	1.70043	1.70043
$145$	0	0
$146$	2.04948	2.04948
$147$	−1.30627	−1.30627
$148$	0	0
$149$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$149$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$150$	−3.17122	−3.17122
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−0.279739	−0.279739
$153$	−4.87165	−4.87165
$154$	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
$160$	0	0
$161$	0.659675	0.659675
$162$	−4.52750	−4.52750
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	1.54733	1.54733
$169$	1.00000	1.00000
$170$	0	0
$171$	0.457285	0.457285
$172$	2.79564	2.79564
$173$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$173$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$174$	−2.82706	−2.82706
$175$	−0.547326	−0.547326
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	1.82706	1.82706
$185$	0	0
$186$	6.23444	6.23444
$187$	0	0
$188$	0	0
$189$	−1.50866	−1.50866
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−2.38664	−2.38664
$193$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$193$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$194$	0	0
$195$	0	0
$196$	−1.32486	−1.32486
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−1.51590	−1.51590
$201$	−2.24776	−2.24776
$202$	0.930692	0.930692
$203$	−0.487928	−0.487928
$204$	−6.93487	−6.93487
$205$	0	0
$206$	2.89148	2.89148
$207$	−2.98668	−2.98668
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	2.75642	2.75642
$214$	0	0
$215$	0	0
$216$	−4.17845	−4.17845
$217$	1.07601	1.07601
$218$	0	0
$219$	−2.24776	−2.24776
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$223$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$224$	−0.191042	−0.191042
$225$	2.47802	2.47802
$226$	−0.313793	−0.313793
$227$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$227$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$228$	0.650953	0.650953
$229$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$229$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$230$	0	0
$231$	0	0
$232$	−1.35139	−1.35139
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−1.82969	−1.82969
$239$	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			$\pi$
$242$	−1.70043	−1.70043
$243$	2.20910	2.20910
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	2.98017	2.98017
$249$	0	0
$250$	0	0
$251$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$251$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$252$	−2.56538	−2.56538
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−1.82706	−1.82706
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−4.68711	−4.68711
$259$	0	0
$260$	0	0
$261$	2.20910	2.20910
$262$	0	0
$263$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$263$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$264$	0	0
$265$	0	0
$266$	0.171747	0.171747
$267$	0	0
$268$	−2.27974	−2.27974
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−1.34905	−1.34905
$273$	0	0
$274$	0.930692	0.930692
$275$	0	0
$276$	−4.25159	−4.25159
$277$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$277$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$278$	2.89148	2.89148
$279$	−4.87165	−4.87165
$280$	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			$\pi$
$284$	2.79564	2.79564
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.864944	0.864944
$289$	2.86494	2.86494
$290$	0	0
$291$	0	0
$292$	−2.27974	−2.27974
$293$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$293$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$294$	2.22123	2.22123
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−3.17122	−3.17122
$299$	0	0
$300$	3.52750	3.52750
$301$	−0.808958	−0.808958
$302$	0	0
$303$	−1.02073	−1.02073
$304$	0.126630	0.126630
$305$	0	0
$306$	8.28392	8.28392
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−3.17122	−3.17122
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$313$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$317$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−1.12173	−1.12173
$323$	−0.362789	−0.362789
$324$	5.03616	5.03616
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$331$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−0.700434	−0.700434
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.70043	−1.70043
$339$	0.344151	0.344151
$340$	0	0
$341$	0	0
$342$	−0.777584	−0.777584
$343$	0.930692	0.930692
$344$	−2.24052	−2.24052
$345$	0	0
$346$	−2.51327	−2.51327
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	3.14468	3.14468
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0.930692	0.930692
$351$	0	0
$352$	0	0
$353$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$353$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.00671	2.00671
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−0.965946	−0.965946
$362$	0	0
$363$	1.86494	1.86494
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−0.827065	−0.827065
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−6.93487	−6.93487
$373$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$373$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	2.56538	2.56538
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.00000	1.00000
$383$	1.00000	1.00000
$384$	3.40737	3.40737
$385$	0	0
$386$	−3.17122	−3.17122
$387$	3.66255	3.66255
$388$	0	0
$389$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$389$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$390$	0	0
$391$	2.36949	2.36949
$392$	1.06179	1.06179
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$397$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$398$	0	0
$399$	−0.188363	−0.188363
$400$	0.686207	0.686207
$401$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$401$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$402$	3.82217	3.82217
$403$	0	0
$404$	−1.03525	−1.03525
$405$	0	0
$406$	0.829690	0.829690
$407$	0	0
$408$	5.55786	5.55786
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−1.02073	−1.02073
$412$	−3.21633	−3.21633
$413$	0	0
$414$	5.07865	5.07865
$415$	0	0
$416$	0	0
$417$	−3.17122	−3.17122
$418$	0	0
$419$	2.00000	2.00000	1.00000			$0$
$419$	2.00000	2.00000	1.00000			$0$
$420$	0	0
$421$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$421$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.96595	−1.96595
$426$	−4.68711	−4.68711
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$431$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$432$	1.89148	1.89148
$433$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$433$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$434$	−1.82969	−1.82969
$435$	0	0
$436$	0	0
$437$	−0.222416	−0.222416
$438$	3.82217	3.82217
$439$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$439$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$440$	0	0
$441$	−1.73569	−1.73569
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	−1.51590	−1.51590
$447$	3.47802	3.47802
$448$	0.700434	0.700434
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−4.21371	−4.21371
$451$	0	0
$452$	0.349047	0.349047
$453$	0	0
$454$	−1.51590	−1.51590
$455$	0	0
$456$	−0.521697	−0.521697
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−2.51327	−2.51327
$459$	−5.41898	−5.41898
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0.611738	0.611738
$465$	0	0
$466$	0	0
$467$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$467$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$468$	0	0
$469$	0.659675	0.659675
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.184537	0.184537
$476$	2.03525	2.03525
$477$	0	0
$478$	0	0
$479$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$479$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.23026	1.23026
$484$	1.89148	1.89148
$485$	0	0
$486$	−3.75642	−3.75642
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$491$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$492$	0	0
$493$	−1.75260	−1.75260
$494$	0	0
$495$	0	0
$496$	−1.34905	−1.34905
$497$	−0.808958	−0.808958
$498$	0	0
$499$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$499$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$500$	0	0
$501$	0	0
$502$	0.930692	0.930692
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	2.05599	2.05599
$505$	0	0
$506$	0	0
$507$	1.86494	1.86494
$508$	0	0
$509$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$509$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$510$	0	0
$511$	0.659675	0.659675
$512$	1.27974	1.27974
$513$	0.508661	0.508661
$514$	0	0
$515$	0	0
$516$	5.21371	5.21371
$517$	0	0
$518$	0	0
$519$	2.75642	2.75642
$520$	0	0
$521$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$521$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$522$	−3.75642	−3.75642
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	−1.02073	−1.02073
$526$	3.34296	3.34296
$527$	3.86494	3.86494
$528$	0	0
$529$	0.452674	0.452674
$530$	0	0
$531$	0	0
$532$	−0.191042	−0.191042
$533$	0	0
$534$	0	0
$535$	0	0
$536$	1.82706	1.82706
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−0.686207	−0.686207
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−1.03525	−1.03525
$549$	0	0
$550$	0	0
$551$	0.164510	0.164510
$552$	3.40737	3.40737
$553$	0	0
$554$	−2.51327	−2.51327
$555$	0	0
$556$	−3.21633	−3.21633
$557$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$557$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$558$	8.28392	8.28392
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.45729	−1.45729
$568$	−2.24052	−2.24052
$569$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$569$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.20527	−1.20527
$576$	−3.17122	−3.17122
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−4.87165	−4.87165
$579$	3.47802	3.47802
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	1.82706	1.82706
$585$	0	0
$586$	−0.313793	−0.313793
$587$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$587$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$588$	−2.47078	−2.47078
$589$	−0.362789	−0.362789
$590$	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
$596$	3.52750	3.52750
$597$	0	0
$598$	0	0
$599$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$599$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$600$	−2.82706	−2.82706
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	1.37558	1.37558
$603$	−2.98668	−2.98668
$604$	0	0
$605$	0	0
$606$	1.73569	1.73569
$607$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$607$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$608$	0.0644120	0.0644120
$609$	−0.909959	−0.909959
$610$	0	0
$611$	0	0
$612$	−9.21461	−9.21461
$613$	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			$\pi$
$618$	5.39244	5.39244
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−3.32223	−3.32223
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	−0.313793	−0.313793
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$631$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$632$	0	0
$633$	0	0
$634$	2.04948	2.04948
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.66255	3.66255
$640$	0	0
$641$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$641$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$642$	0	0
$643$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$643$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$644$	1.24776	1.24776
$645$	0	0
$646$	0.616899	0.616899
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−4.03616	−4.03616
$649$	0	0
$650$	0	0
$651$	2.00671	2.00671
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.98668	−2.98668
$658$	0	0
$659$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$659$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$660$	0	0
$661$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$661$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$662$	−1.51590	−1.51590
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.07447	−1.07447
$668$	0	0
$669$	1.66255	1.66255
$670$	0	0
$671$	0	0
$672$	−0.356284	−0.356284
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	2.75642	2.75642
$676$	1.89148	1.89148
$677$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$677$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$678$	−0.585206	−0.585206
$679$	0	0
$680$	0	0
$681$	1.66255	1.66255
$682$	0	0
$683$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$683$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$684$	0.864944	0.864944
$685$	0	0
$686$	−1.58258	−1.58258
$687$	2.75642	2.75642
$688$	1.01423	1.01423
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	2.79564	2.79564
$693$	0	0
$694$	0	0
$695$	0	0
$696$	−2.52026	−2.52026
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−1.03525	−1.03525
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−3.17122	−3.17122
$707$	0.299566	0.299566
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.36949	2.36949
$714$	−3.41227	−3.41227
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$719$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$720$	0	0
$721$	0.930692	0.930692
$722$	1.64253	1.64253
$723$	0	0
$724$	0	0
$725$	0.891477	0.891477
$726$	−3.17122	−3.17122
$727$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$727$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$728$	0	0
$729$	1.45729	1.45729
$730$	0	0
$731$	−2.90570	−2.90570
$732$	0	0
$733$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$733$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$734$	0	0
$735$	0	0
$736$	−0.420696	−0.420696
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	5.55786	5.55786
$745$	0	0
$746$	2.89148	2.89148
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$751$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$752$	0	0
$753$	−1.02073	−1.02073
$754$	0	0
$755$	0	0
$756$	−2.85360	−2.85360
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$761$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−1.70043	−1.70043
$767$	0	0
$768$	−3.40737	−3.40737
$769$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$769$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$770$	0	0
$771$	0	0
$772$	3.52750	3.52750
$773$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$773$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$774$	−6.22793	−6.22793
$775$	−1.96595	−1.96595
$776$	0	0
$777$	0	0
$778$	−3.17122	−3.17122
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−4.02917	−4.02917
$783$	2.45729	2.45729
$784$	−0.480643	−0.480643
$785$	0	0
$786$	0	0
$787$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$787$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$788$	0	0
$789$	−3.66638	−3.66638
$790$	0	0
$791$	−0.101002	−0.101002
$792$	0	0
$793$	0	0
$794$	−0.313793	−0.313793
$795$	0	0
$796$	0	0
$797$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$797$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$798$	0.320298	0.320298
$799$	0	0
$800$	0.349047	0.349047
$801$	0	0
$802$	2.04948	2.04948
$803$	0	0
$804$	−4.25159	−4.25159
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0.829690	0.829690
$809$	−1.70043	−1.70043	−0.850217	−	0.526432i	$-0.823529\pi$
$809$	−1.70043	−1.70043	−0.850217	+	0.526432i	$0.823529\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	−0.922905	−0.922905
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−2.51590	−2.51590
$817$	0.272749	0.272749
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$821$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$822$	1.73569	1.73569
$823$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$823$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$824$	2.57768	2.57768
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−5.64923	−5.64923
$829$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$829$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$830$	0	0
$831$	2.75642	2.75642
$832$	0	0
$833$	1.37702	1.37702
$834$	5.39244	5.39244
$835$	0	0
$836$	0	0
$837$	−5.41898	−5.41898
$838$	−3.40087	−3.40087
$839$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$839$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$840$	0	0
$841$	−0.205269	−0.205269
$842$	3.34296	3.34296
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.547326	−0.547326
$848$	0	0
$849$	0	0
$850$	3.34296	3.34296
$851$	0	0
$852$	5.21371	5.21371
$853$	0.891477	0.891477	0.445738	−	0.895163i	$-0.352941\pi$
$853$	0.891477	0.891477	0.445738	+	0.895163i	$0.352941\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.20527	−1.20527	−0.602635	−	0.798017i	$-0.705882\pi$
$859$	−1.20527	−1.20527	−0.602635	+	0.798017i	$0.705882\pi$
$860$	0	0
$861$	0	0
$862$	2.89148	2.89148
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.962120	0.962120
$865$	0	0
$866$	−1.51590	−1.51590
$867$	5.34296	5.34296
$868$	2.03525	2.03525
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0.378205	0.378205
$875$	0	0
$876$	−4.25159	−4.25159
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	2.04948	2.04948
$879$	0.344151	0.344151
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	2.95142	2.95142
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.547326	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$887$	−0.547326	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1.68621	1.68621
$893$	0	0
$894$	−5.91414	−5.91414
$895$	0	0
$896$	−1.00000	−1.00000
$897$	0	0
$898$	0	0
$899$	−1.75260	−1.75260
$900$	4.68711	4.68711
$901$	0	0
$902$	0	0
$903$	−1.50866	−1.50866
$904$	−0.279739	−0.279739
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1.68621	1.68621
$909$	−1.35628	−1.35628
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.236159	0.236159
$913$	0	0
$914$	0	0
$915$	0	0
$916$	2.79564	2.79564
$917$	0	0
$918$	9.21461	9.21461
$919$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$919$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.21371	−4.21371
$928$	0.311167	0.311167
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−0.129256	−0.129256
$932$	0	0
$933$	0	0
$934$	2.89148	2.89148
$935$	0	0
$936$	0	0
$937$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$937$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$938$	−1.12173	−1.12173
$939$	0.344151	0.344151
$940$	0	0
$941$	−1.96595	−1.96595	−0.982973	−	0.183750i	$-0.941176\pi$
$941$	−1.96595	−1.96595	−0.982973	+	0.183750i	$0.941176\pi$
$942$	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
$950$	−0.313793	−0.313793
$951$	−2.24776	−2.24776
$952$	−1.63113	−1.63113
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−0.313793	−0.313793
$959$	0.299566	0.299566
$960$	0	0
$961$	2.86494	2.86494
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	−2.09197	−2.09197
$967$	1.86494	1.86494	0.932472	−	0.361242i	$-0.117647\pi$
$967$	1.86494	1.86494	0.932472	+	0.361242i	$0.117647\pi$
$968$	−1.51590	−1.51590
$969$	−0.676582	−0.676582
$970$	0	0
$971$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$971$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$972$	4.17845	4.17845
$973$	0.930692	0.930692
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	3.34296	3.34296
$983$	1.47802	1.47802	0.739009	−	0.673696i	$-0.235294\pi$
$983$	1.47802	1.47802	0.739009	+	0.673696i	$0.235294\pi$
$984$	0	0
$985$	0	0
$986$	2.98017	2.98017
$987$	0	0
$988$	0	0
$989$	−1.78141	−1.78141
$990$	0	0
$991$	0.184537	0.184537	0.0922684	−	0.995734i	$-0.470588\pi$
$991$	0.184537	0.184537	0.0922684	+	0.995734i	$0.470588\pi$
$992$	−0.686207	−0.686207
$993$	1.66255	1.66255
$994$	1.37558	1.37558
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	3.34296	3.34296
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	383.1.b.a.382.2	✓	8
3.2	odd	2		3447.1.d.a.1531.7		8
383.382	odd	2	CM	383.1.b.a.382.2	✓	8
1149.1148	even	2		3447.1.d.a.1531.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
383.1.b.a.382.2	✓	8	1.1	even	1	trivial
383.1.b.a.382.2	✓	8	383.382	odd	2	CM
3447.1.d.a.1531.7		8	3.2	odd	2
3447.1.d.a.1531.7		8	1149.1148	even	2

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:	\( N \)	\(=\)	\( 383 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	383.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.70043 q^{2} +1.86494 q^{3} +1.89148 q^{4} -3.17122 q^{6} -0.547326 q^{7} -1.51590 q^{8} +2.47802 q^{9} +3.52750 q^{12} +0.930692 q^{14} +0.686207 q^{16} -1.96595 q^{17} -4.21371 q^{18} +0.184537 q^{19} -1.02073 q^{21} -1.20527 q^{23} -2.82706 q^{24} +1.00000 q^{25} +2.75642 q^{27} -1.03525 q^{28} +0.891477 q^{29} -1.96595 q^{31} +0.349047 q^{32} +3.34296 q^{34} +4.68711 q^{36} -0.313793 q^{38} +1.73569 q^{42} +1.47802 q^{43} +2.04948 q^{46} +1.27974 q^{48} -0.700434 q^{49} -1.70043 q^{50} -3.66638 q^{51} -4.68711 q^{54} +0.829690 q^{56} +0.344151 q^{57} -1.51590 q^{58} +3.34296 q^{62} -1.35628 q^{63} -1.27974 q^{64} -1.20527 q^{67} -3.71854 q^{68} -2.24776 q^{69} +1.47802 q^{71} -3.75642 q^{72} -1.20527 q^{73} +1.86494 q^{75} +0.349047 q^{76} +2.66255 q^{81} -1.93069 q^{84} -2.51327 q^{86} +1.66255 q^{87} -2.27974 q^{92} -3.66638 q^{93} +0.650953 q^{96} +1.19104 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - q^{3} + 7 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 7 q^{9} - 3 q^{12} - 2 q^{14} + 6 q^{16} - q^{17} - 3 q^{18} - q^{19} - 2 q^{21} - q^{23} - 4 q^{24} + 8 q^{25} - 2 q^{27} - 3 q^{28} - q^{29}+ \cdots + 14 q^{98}+O(q^{100}) \) 8 * q - q^2 - q^3 + 7 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 7 * q^9 - 3 * q^12 - 2 * q^14 + 6 * q^16 - q^17 - 3 * q^18 - q^19 - 2 * q^21 - q^23 - 4 * q^24 + 8 * q^25 - 2 * q^27 - 3 * q^28 - q^29 - q^31 - 3 * q^32 - 2 * q^34 + 4 * q^36 - 2 * q^38 - 4 * q^42 - q^43 - 2 * q^46 - 5 * q^48 + 7 * q^49 - q^50 - 2 * q^51 - 4 * q^54 - 4 * q^56 - 2 * q^57 - 2 * q^58 - 2 * q^62 - 3 * q^63 + 5 * q^64 - q^67 - 3 * q^68 - 2 * q^69 - q^71 - 6 * q^72 - q^73 - q^75 - 3 * q^76 + 6 * q^81 - 6 * q^84 - 2 * q^86 - 2 * q^87 - 3 * q^92 - 2 * q^93 + 11 * q^96 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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