LMFDB - Embedded newform 791.1.d.c.790.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [791,1,Mod(790,791)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(791, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("791.790");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 791 = 7 \cdot 113 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	791.d (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:	$0.394760424993$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.3464395697.1

Embedding invariants

Embedding label			790.3
Root			$0.765367$ of defining polynomial
Character	$\chi$	$=$	791.790

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -0.765367 q^{3} +1.00000 q^{4} +0.765367 q^{5} -1.08239 q^{6} +1.00000 q^{7} -0.414214 q^{9} +O(q^{10})$	$q+1.41421 q^{2} -0.765367 q^{3} +1.00000 q^{4} +0.765367 q^{5} -1.08239 q^{6} +1.00000 q^{7} -0.414214 q^{9} +1.08239 q^{10} -0.765367 q^{12} +1.41421 q^{14} -0.585786 q^{15} -1.00000 q^{16} +1.84776 q^{17} -0.585786 q^{18} -1.84776 q^{19} +0.765367 q^{20} -0.765367 q^{21} -0.414214 q^{25} +1.08239 q^{27} +1.00000 q^{28} -0.828427 q^{30} -1.41421 q^{32} +2.61313 q^{34} +0.765367 q^{35} -0.414214 q^{36} -2.61313 q^{38} -1.08239 q^{42} -0.317025 q^{45} -0.765367 q^{47} +0.765367 q^{48} +1.00000 q^{49} -0.585786 q^{50} -1.41421 q^{51} -2.00000 q^{53} +1.53073 q^{54} +1.41421 q^{57} -1.84776 q^{59} -0.585786 q^{60} -0.414214 q^{63} -1.00000 q^{64} +1.84776 q^{68} +1.08239 q^{70} +0.765367 q^{73} +0.317025 q^{75} -1.84776 q^{76} -0.765367 q^{80} -0.414214 q^{81} -0.765367 q^{84} +1.41421 q^{85} -1.84776 q^{89} -0.448342 q^{90} -1.08239 q^{94} -1.41421 q^{95} +1.08239 q^{96} +1.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^4 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{4} + 4 q^{7} + 4 q^{9} - 8 q^{15} - 4 q^{16} - 8 q^{18} + 4 q^{25} + 4 q^{28} + 8 q^{30} + 4 q^{36} + 4 q^{49} - 8 q^{50} - 8 q^{53} - 8 q^{60} + 4 q^{63} - 4 q^{64} + 4 q^{81}+O(q^{100}) $ 4 * q + 4 * q^4 + 4 * q^7 + 4 * q^9 - 8 * q^15 - 4 * q^16 - 8 * q^18 + 4 * q^25 + 4 * q^28 + 8 * q^30 + 4 * q^36 + 4 * q^49 - 8 * q^50 - 8 * q^53 - 8 * q^60 + 4 * q^63 - 4 * q^64 + 4 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/791\mathbb{Z}\right)^\times$.

$n$	$227$	$568$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$3$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$4$	1.00000	1.00000
$5$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$5$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$6$	−1.08239	−1.08239
$7$	1.00000	1.00000
$7$	1.00000	1.00000
$8$	0	0
$9$	−0.414214	−0.414214
$10$	1.08239	1.08239
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−0.765367	−0.765367
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	1.41421	1.41421
$15$	−0.585786	−0.585786
$16$	−1.00000	−1.00000
$17$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$17$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$18$	−0.585786	−0.585786
$19$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$19$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$20$	0.765367	0.765367
$21$	−0.765367	−0.765367
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−0.414214	−0.414214
$26$	0	0
$27$	1.08239	1.08239
$28$	1.00000	1.00000
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−0.828427	−0.828427
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.41421	−1.41421
$33$	0	0
$34$	2.61313	2.61313
$35$	0.765367	0.765367
$36$	−0.414214	−0.414214
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−2.61313	−2.61313
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−1.08239	−1.08239
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−0.317025	−0.317025
$46$	0	0
$47$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$47$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$48$	0.765367	0.765367
$49$	1.00000	1.00000
$50$	−0.585786	−0.585786
$51$	−1.41421	−1.41421
$52$	0	0
$53$	−2.00000	−2.00000	−1.00000			$\pi$
$53$	−2.00000	−2.00000	−1.00000			$\pi$
$54$	1.53073	1.53073
$55$	0	0
$56$	0	0
$57$	1.41421	1.41421
$58$	0	0
$59$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$59$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$60$	−0.585786	−0.585786
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−0.414214	−0.414214
$64$	−1.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	1.84776	1.84776
$69$	0	0
$70$	1.08239	1.08239
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$73$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$74$	0	0
$75$	0.317025	0.317025
$76$	−1.84776	−1.84776
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.765367	−0.765367
$81$	−0.414214	−0.414214
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−0.765367	−0.765367
$85$	1.41421	1.41421
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$89$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$90$	−0.448342	−0.448342
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−1.08239	−1.08239
$95$	−1.41421	−1.41421
$96$	1.08239	1.08239
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.41421	1.41421
$99$	0	0
$100$	−0.414214	−0.414214
$101$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$101$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$102$	−2.00000	−2.00000
$103$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$103$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$104$	0	0
$105$	−0.585786	−0.585786
$106$	−2.82843	−2.82843
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.08239	1.08239
$109$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$109$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−1.00000
$113$	1.00000	1.00000
$113$	1.00000	1.00000
$114$	2.00000	2.00000
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−2.61313	−2.61313
$119$	1.84776	1.84776
$120$	0	0
$121$	−1.00000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.08239	−1.08239
$126$	−0.585786	−0.585786
$127$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$127$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−1.84776	−1.84776
$134$	0	0
$135$	0.828427	0.828427
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0.765367	0.765367
$141$	0.585786	0.585786
$142$	0	0
$143$	0	0
$144$	0.414214	0.414214
$145$	0	0
$146$	1.08239	1.08239
$147$	−0.765367	−0.765367
$148$	0	0
$149$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$149$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$150$	0.448342	0.448342
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−0.765367	−0.765367
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	1.53073	1.53073
$160$	−1.08239	−1.08239
$161$	0	0
$162$	−0.585786	−0.585786
$163$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$163$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$167$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	2.00000	2.00000
$171$	0.765367	0.765367
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−0.414214	−0.414214
$176$	0	0
$177$	1.41421	1.41421
$178$	−2.61313	−2.61313
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.317025	−0.317025
$181$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$181$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−0.765367	−0.765367
$189$	1.08239	1.08239
$190$	−2.00000	−2.00000
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0.765367	0.765367
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	1.00000	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$199$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$200$	0	0
$201$	0	0
$202$	2.61313	2.61313
$203$	0	0
$204$	−1.41421	−1.41421
$205$	0	0
$206$	1.08239	1.08239
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−0.828427	−0.828427
$211$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$211$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$212$	−2.00000	−2.00000
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	2.00000	2.00000
$219$	−0.585786	−0.585786
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$223$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$224$	−1.41421	−1.41421
$225$	0.171573	0.171573
$226$	1.41421	1.41421
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	1.41421	1.41421
$229$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$229$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$233$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$234$	0	0
$235$	−0.585786	−0.585786
$236$	−1.84776	−1.84776
$237$	0	0
$238$	2.61313	2.61313
$239$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$239$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$240$	0.585786	0.585786
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.41421	−1.41421
$243$	−0.765367	−0.765367
$244$	0	0
$245$	0.765367	0.765367
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−1.53073	−1.53073
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−0.414214	−0.414214
$253$	0	0
$254$	2.00000	2.00000
$255$	−1.08239	−1.08239
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−1.53073	−1.53073
$266$	−2.61313	−2.61313
$267$	1.41421	1.41421
$268$	0	0
$269$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$269$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$270$	1.17157	1.17157
$271$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$271$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$272$	−1.84776	−1.84776
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.00000	−2.00000	−1.00000			$\pi$
$277$	−2.00000	−2.00000	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0.828427	0.828427
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	1.08239	1.08239
$286$	0	0
$287$	0	0
$288$	0.585786	0.585786
$289$	2.41421	2.41421
$290$	0	0
$291$	0	0
$292$	0.765367	0.765367
$293$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$293$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$294$	−1.08239	−1.08239
$295$	−1.41421	−1.41421
$296$	0	0
$297$	0	0
$298$	2.00000	2.00000
$299$	0	0
$300$	0.317025	0.317025
$301$	0	0
$302$	0	0
$303$	−1.41421	−1.41421
$304$	1.84776	1.84776
$305$	0	0
$306$	−1.08239	−1.08239
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−0.585786	−0.585786
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	−0.317025	−0.317025
$316$	0	0
$317$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$317$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$318$	2.16478	2.16478
$319$	0	0
$320$	−0.765367	−0.765367
$321$	0	0
$322$	0	0
$323$	−3.41421	−3.41421
$324$	−0.414214	−0.414214
$325$	0	0
$326$	−2.00000	−2.00000
$327$	−1.08239	−1.08239
$328$	0	0
$329$	−0.765367	−0.765367
$330$	0	0
$331$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$331$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$332$	0	0
$333$	0	0
$334$	−2.61313	−2.61313
$335$	0	0
$336$	0.765367	0.765367
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	1.41421	1.41421
$339$	−0.765367	−0.765367
$340$	1.41421	1.41421
$341$	0	0
$342$	1.08239	1.08239
$343$	1.00000	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$349$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$350$	−0.585786	−0.585786
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	2.00000	2.00000
$355$	0	0
$356$	−1.84776	−1.84776
$357$	−1.41421	−1.41421
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	2.41421	2.41421
$362$	2.61313	2.61313
$363$	0.765367	0.765367
$364$	0	0
$365$	0.585786	0.585786
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.00000	−2.00000
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0.828427	0.828427
$376$	0	0
$377$	0	0
$378$	1.53073	1.53073
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−1.41421	−1.41421
$381$	−1.08239	−1.08239
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$389$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$397$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$398$	2.61313	2.61313
$399$	1.41421	1.41421
$400$	0.414214	0.414214
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	1.84776	1.84776
$405$	−0.317025	−0.317025
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$409$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$410$	0	0
$411$	0	0
$412$	0.765367	0.765367
$413$	−1.84776	−1.84776
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$419$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$420$	−0.585786	−0.585786
$421$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$421$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$422$	2.00000	2.00000
$423$	0.317025	0.317025
$424$	0	0
$425$	−0.765367	−0.765367
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.08239	−1.08239
$433$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$433$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$434$	0	0
$435$	0	0
$436$	1.41421	1.41421
$437$	0	0
$438$	−0.828427	−0.828427
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−0.414214	−0.414214
$442$	0	0
$443$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$443$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$444$	0	0
$445$	−1.41421	−1.41421
$446$	1.08239	1.08239
$447$	−1.08239	−1.08239
$448$	−1.00000	−1.00000
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0.242641	0.242641
$451$	0	0
$452$	1.00000	1.00000
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	2.61313	2.61313
$459$	2.00000	2.00000
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	2.00000	2.00000
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	−0.828427	−0.828427
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.765367	0.765367
$476$	1.84776	1.84776
$477$	0.828427	0.828427
$478$	−2.00000	−2.00000
$479$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$479$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$480$	0.828427	0.828427
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−1.00000	−1.00000
$485$	0	0
$486$	−1.08239	−1.08239
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.08239	1.08239
$490$	1.08239	1.08239
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−1.08239	−1.08239
$501$	1.41421	1.41421
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	1.41421	1.41421
$506$	0	0
$507$	−0.765367	−0.765367
$508$	1.41421	1.41421
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−1.53073	−1.53073
$511$	0.765367	0.765367
$512$	1.41421	1.41421
$513$	−2.00000	−2.00000
$514$	0	0
$515$	0.585786	0.585786
$516$	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
$523$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$523$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$524$	0	0
$525$	0.317025	0.317025
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	−2.16478	−2.16478
$531$	0.765367	0.765367
$532$	−1.84776	−1.84776
$533$	0	0
$534$	2.00000	2.00000
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−1.08239	−1.08239
$539$	0	0
$540$	0.828427	0.828427
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−1.08239	−1.08239
$543$	−1.41421	−1.41421
$544$	−2.61313	−2.61313
$545$	1.08239	1.08239
$546$	0	0
$547$	−2.00000	−2.00000	−1.00000			$\pi$
$547$	−2.00000	−2.00000	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−2.82843	−2.82843
$555$	0	0
$556$	0	0
$557$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$557$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$558$	0	0
$559$	0	0
$560$	−0.765367	−0.765367
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0.585786	0.585786
$565$	0.765367	0.765367
$566$	0	0
$567$	−0.414214	−0.414214
$568$	0	0
$569$	−2.00000	−2.00000	−1.00000			$\pi$
$569$	−2.00000	−2.00000	−1.00000			$\pi$
$570$	1.53073	1.53073
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0.414214	0.414214
$577$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$577$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$578$	3.41421	3.41421
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	1.08239	1.08239
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.765367	−0.765367
$589$	0	0
$590$	−2.00000	−2.00000
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	1.41421	1.41421
$596$	1.41421	1.41421
$597$	−1.41421	−1.41421
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.765367	−0.765367
$606$	−2.00000	−2.00000
$607$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$607$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$608$	2.61313	2.61313
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−0.765367	−0.765367
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.00000	−2.00000	−1.00000			$\pi$
$617$	−2.00000	−2.00000	−1.00000			$\pi$
$618$	−0.828427	−0.828427
$619$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$619$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.84776	−1.84776
$624$	0	0
$625$	−0.414214	−0.414214
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−0.448342	−0.448342
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−1.08239	−1.08239
$634$	−2.00000	−2.00000
$635$	1.08239	1.08239
$636$	1.53073	1.53073
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$643$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$644$	0	0
$645$	0	0
$646$	−4.82843	−4.82843
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−1.41421	−1.41421
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	−1.53073	−1.53073
$655$	0	0
$656$	0	0
$657$	−0.317025	−0.317025
$658$	−1.08239	−1.08239
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$661$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$662$	−2.00000	−2.00000
$663$	0	0
$664$	0	0
$665$	−1.41421	−1.41421
$666$	0	0
$667$	0	0
$668$	−1.84776	−1.84776
$669$	−0.585786	−0.585786
$670$	0	0
$671$	0	0
$672$	1.08239	1.08239
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−0.448342	−0.448342
$676$	1.00000	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−1.08239	−1.08239
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0.765367	0.765367
$685$	0	0
$686$	1.41421	1.41421
$687$	−1.41421	−1.41421
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−1.08239	−1.08239
$699$	−1.08239	−1.08239
$700$	−0.414214	−0.414214
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0.448342	0.448342
$706$	0	0
$707$	1.84776	1.84776
$708$	1.41421	1.41421
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	−2.00000	−2.00000
$715$	0	0
$716$	0	0
$717$	1.08239	1.08239
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.317025	0.317025
$721$	0.765367	0.765367
$722$	3.41421	3.41421
$723$	0	0
$724$	1.84776	1.84776
$725$	0	0
$726$	1.08239	1.08239
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0.828427	0.828427
$731$	0	0
$732$	0	0
$733$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$733$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$734$	0	0
$735$	−0.585786	−0.585786
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$739$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$740$	0	0
$741$	0	0
$742$	−2.82843	−2.82843
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	1.08239	1.08239
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	1.17157	1.17157
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0.765367	0.765367
$753$	0	0
$754$	0	0
$755$	0	0
$756$	1.08239	1.08239
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−1.53073	−1.53073
$763$	1.41421	1.41421
$764$	0	0
$765$	−0.585786	−0.585786
$766$	0	0
$767$	0	0
$768$	−0.765367	−0.765367
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−2.00000	−2.00000
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.00000	−1.00000
$785$	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
$791$	1.00000	1.00000
$792$	0	0
$793$	0	0
$794$	−1.08239	−1.08239
$795$	1.17157	1.17157
$796$	1.84776	1.84776
$797$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$797$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$798$	2.00000	2.00000
$799$	−1.41421	−1.41421
$800$	0.585786	0.585786
$801$	0.765367	0.765367
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.585786	0.585786
$808$	0	0
$809$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$809$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$810$	−0.448342	−0.448342
$811$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$811$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$812$	0	0
$813$	0.585786	0.585786
$814$	0	0
$815$	−1.08239	−1.08239
$816$	1.41421	1.41421
$817$	0	0
$818$	−2.61313	−2.61313
$819$	0	0
$820$	0	0
$821$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$821$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$822$	0	0
$823$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$823$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$824$	0	0
$825$	0	0
$826$	−2.61313	−2.61313
$827$	2.00000	2.00000	1.00000			$0$
$827$	2.00000	2.00000	1.00000			$0$
$828$	0	0
$829$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$829$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$830$	0	0
$831$	1.53073	1.53073
$832$	0	0
$833$	1.84776	1.84776
$834$	0	0
$835$	−1.41421	−1.41421
$836$	0	0
$837$	0	0
$838$	2.61313	2.61313
$839$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$839$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	−2.00000	−2.00000
$843$	0	0
$844$	1.41421	1.41421
$845$	0.765367	0.765367
$846$	0.448342	0.448342
$847$	−1.00000	−1.00000
$848$	2.00000	2.00000
$849$	0	0
$850$	−1.08239	−1.08239
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0.585786	0.585786
$856$	0	0
$857$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$857$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$858$	0	0
$859$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$859$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$863$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$864$	−1.53073	−1.53073
$865$	0	0
$866$	−1.08239	−1.08239
$867$	−1.84776	−1.84776
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.08239	−1.08239
$876$	−0.585786	−0.585786
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−0.585786	−0.585786
$880$	0	0
$881$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$881$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$882$	−0.585786	−0.585786
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	1.08239	1.08239
$886$	2.00000	2.00000
$887$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$887$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$888$	0	0
$889$	1.41421	1.41421
$890$	−2.00000	−2.00000
$891$	0	0
$892$	0.765367	0.765367
$893$	1.41421	1.41421
$894$	−1.53073	−1.53073
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0.171573	0.171573
$901$	−3.69552	−3.69552
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.41421	1.41421
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−0.765367	−0.765367
$910$	0	0
$911$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$911$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$912$	−1.41421	−1.41421
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1.84776	1.84776
$917$	0	0
$918$	2.82843	2.82843
$919$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$919$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.317025	−0.317025
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−1.84776	−1.84776
$932$	1.41421	1.41421
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$937$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$938$	0	0
$939$	0	0
$940$	−0.585786	−0.585786
$941$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$941$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$942$	0	0
$943$	0	0
$944$	1.84776	1.84776
$945$	0.828427	0.828427
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	1.08239	1.08239
$951$	1.08239	1.08239
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	1.17157	1.17157
$955$	0	0
$956$	−1.41421	−1.41421
$957$	0	0
$958$	2.61313	2.61313
$959$	0	0
$960$	0.585786	0.585786
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$967$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$968$	0	0
$969$	2.61313	2.61313
$970$	0	0
$971$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$971$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$972$	−0.765367	−0.765367
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	1.53073	1.53073
$979$	0	0
$980$	0.765367	0.765367
$981$	−0.585786	−0.585786
$982$	0	0
$983$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$983$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.585786	0.585786
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$991$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$992$	0	0
$993$	1.08239	1.08239
$994$	0	0
$995$	1.41421	1.41421
$996$	0	0
$997$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$997$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	791.1.d.c.790.3	✓	4
7.6	odd	2	inner	791.1.d.c.790.4	yes	4
113.112	even	2	inner	791.1.d.c.790.4	yes	4
791.790	odd	2	CM	791.1.d.c.790.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
791.1.d.c.790.3	✓	4	1.1	even	1	trivial
791.1.d.c.790.3	✓	4	791.790	odd	2	CM
791.1.d.c.790.4	yes	4	7.6	odd	2	inner
791.1.d.c.790.4	yes	4	113.112	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 791 = 7 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	791.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -0.765367 q^{3} +1.00000 q^{4} +0.765367 q^{5} -1.08239 q^{6} +1.00000 q^{7} -0.414214 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -0.765367 q^{3} +1.00000 q^{4} +0.765367 q^{5} -1.08239 q^{6} +1.00000 q^{7} -0.414214 q^{9} +1.08239 q^{10} -0.765367 q^{12} +1.41421 q^{14} -0.585786 q^{15} -1.00000 q^{16} +1.84776 q^{17} -0.585786 q^{18} -1.84776 q^{19} +0.765367 q^{20} -0.765367 q^{21} -0.414214 q^{25} +1.08239 q^{27} +1.00000 q^{28} -0.828427 q^{30} -1.41421 q^{32} +2.61313 q^{34} +0.765367 q^{35} -0.414214 q^{36} -2.61313 q^{38} -1.08239 q^{42} -0.317025 q^{45} -0.765367 q^{47} +0.765367 q^{48} +1.00000 q^{49} -0.585786 q^{50} -1.41421 q^{51} -2.00000 q^{53} +1.53073 q^{54} +1.41421 q^{57} -1.84776 q^{59} -0.585786 q^{60} -0.414214 q^{63} -1.00000 q^{64} +1.84776 q^{68} +1.08239 q^{70} +0.765367 q^{73} +0.317025 q^{75} -1.84776 q^{76} -0.765367 q^{80} -0.414214 q^{81} -0.765367 q^{84} +1.41421 q^{85} -1.84776 q^{89} -0.448342 q^{90} -1.08239 q^{94} -1.41421 q^{95} +1.08239 q^{96} +1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^4 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{4} + 4 q^{7} + 4 q^{9} - 8 q^{15} - 4 q^{16} - 8 q^{18} + 4 q^{25} + 4 q^{28} + 8 q^{30} + 4 q^{36} + 4 q^{49} - 8 q^{50} - 8 q^{53} - 8 q^{60} + 4 q^{63} - 4 q^{64} + 4 q^{81}+O(q^{100}) \) 4 * q + 4 * q^4 + 4 * q^7 + 4 * q^9 - 8 * q^15 - 4 * q^16 - 8 * q^18 + 4 * q^25 + 4 * q^28 + 8 * q^30 + 4 * q^36 + 4 * q^49 - 8 * q^50 - 8 * q^53 - 8 * q^60 + 4 * q^63 - 4 * q^64 + 4 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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