LMFDB - Embedded newform 751.1.b.d.750.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [751,1,Mod(750,751)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(751, 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("751.750");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 751 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	751.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:	$0.374797824487$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{15}$
Projective field:	Galois closure of 15.1.134734730815558692751.1

Embedding invariants

Embedding label			750.3
Root			$1.33826$ of defining polynomial
Character	$\chi$	$=$	751.750

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33826 q^{2} +0.790943 q^{4} +0.618034 q^{5} -0.279773 q^{8} +1.00000 q^{9} +O(q^{10})$	$q+1.33826 q^{2} +0.790943 q^{4} +0.618034 q^{5} -0.279773 q^{8} +1.00000 q^{9} +0.827091 q^{10} -1.95630 q^{13} -1.16535 q^{16} +1.33826 q^{18} -1.00000 q^{19} +0.488830 q^{20} +1.82709 q^{23} -0.618034 q^{25} -2.61803 q^{26} -1.27977 q^{32} +0.790943 q^{36} +1.82709 q^{37} -1.33826 q^{38} -0.172909 q^{40} -1.61803 q^{43} +0.618034 q^{45} +2.44512 q^{46} -0.209057 q^{47} +1.00000 q^{49} -0.827091 q^{50} -1.54732 q^{52} -1.00000 q^{53} -1.95630 q^{59} -0.209057 q^{61} -0.547318 q^{64} -1.20906 q^{65} -1.61803 q^{71} -0.279773 q^{72} +2.44512 q^{74} -0.790943 q^{76} -0.720227 q^{80} +1.00000 q^{81} -2.16535 q^{86} +1.82709 q^{89} +0.827091 q^{90} +1.44512 q^{92} -0.279773 q^{94} -0.618034 q^{95} +1.33826 q^{97} +1.33826 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 5 q^{4} - 2 q^{5} - q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + q^2 + 5 * q^4 - 2 * q^5 - q^8 + 4 * q^9	$ 4 q + q^{2} + 5 q^{4} - 2 q^{5} - q^{8} + 4 q^{9} - 3 q^{10} + q^{13} + 6 q^{16} + q^{18} - 4 q^{19} + q^{23} + 2 q^{25} - 6 q^{26} - 5 q^{32} + 5 q^{36} + q^{37} - q^{38} - 7 q^{40} - 2 q^{43} - 2 q^{45} - q^{46} + q^{47} + 4 q^{49} + 3 q^{50} - 4 q^{53} + q^{59} + q^{61} + 4 q^{64} - 3 q^{65} - 2 q^{71} - q^{72} - q^{74} - 5 q^{76} - 3 q^{80} + 4 q^{81} + 2 q^{86} + q^{89} - 3 q^{90} - 5 q^{92} - q^{94} + 2 q^{95} + q^{97} + q^{98}+O(q^{100}) $ 4 * q + q^2 + 5 * q^4 - 2 * q^5 - q^8 + 4 * q^9 - 3 * q^10 + q^13 + 6 * q^16 + q^18 - 4 * q^19 + q^23 + 2 * q^25 - 6 * q^26 - 5 * q^32 + 5 * q^36 + q^37 - q^38 - 7 * q^40 - 2 * q^43 - 2 * q^45 - q^46 + q^47 + 4 * q^49 + 3 * q^50 - 4 * q^53 + q^59 + q^61 + 4 * q^64 - 3 * q^65 - 2 * q^71 - q^72 - q^74 - 5 * q^76 - 3 * q^80 + 4 * q^81 + 2 * q^86 + q^89 - 3 * q^90 - 5 * q^92 - q^94 + 2 * q^95 + q^97 + q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/751\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$2$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0.790943	0.790943
$5$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$5$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−0.279773	−0.279773
$9$	1.00000	1.00000
$10$	0.827091	0.827091
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$13$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$14$	0	0
$15$	0	0
$16$	−1.16535	−1.16535
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.33826	1.33826
$19$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$20$	0.488830	0.488830
$21$	0	0
$22$	0	0
$23$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$23$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$24$	0	0
$25$	−0.618034	−0.618034
$26$	−2.61803	−2.61803
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.27977	−1.27977
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0.790943	0.790943
$37$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$37$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$38$	−1.33826	−1.33826
$39$	0	0
$40$	−0.172909	−0.172909
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$43$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$44$	0	0
$45$	0.618034	0.618034
$46$	2.44512	2.44512
$47$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$47$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	−0.827091	−0.827091
$51$	0	0
$52$	−1.54732	−1.54732
$53$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$53$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$59$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$60$	0	0
$61$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$61$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$62$	0	0
$63$	0	0
$64$	−0.547318	−0.547318
$65$	−1.20906	−1.20906
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$72$	−0.279773	−0.279773
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	2.44512	2.44512
$75$	0	0
$76$	−0.790943	−0.790943
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.720227	−0.720227
$81$	1.00000	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−2.16535	−2.16535
$87$	0	0
$88$	0	0
$89$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$89$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$90$	0.827091	0.827091
$91$	0	0
$92$	1.44512	1.44512
$93$	0	0
$94$	−0.279773	−0.279773
$95$	−0.618034	−0.618034
$96$	0	0
$97$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$97$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$98$	1.33826	1.33826
$99$	0	0
$100$	−0.488830	−0.488830
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0.547318	0.547318
$105$	0	0
$106$	−1.33826	−1.33826
$107$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$107$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$108$	0	0
$109$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$109$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	1.12920	1.12920
$116$	0	0
$117$	−1.95630	−1.95630
$118$	−2.61803	−2.61803
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	−0.279773	−0.279773
$123$	0	0
$124$	0	0
$125$	−1.00000	−1.00000
$126$	0	0
$127$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$127$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$128$	0.547318	0.547318
$129$	0	0
$130$	−1.61803	−1.61803
$131$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$131$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$139$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$140$	0	0
$141$	0	0
$142$	−2.16535	−2.16535
$143$	0	0
$144$	−1.16535	−1.16535
$145$	0	0
$146$	0	0
$147$	0	0
$148$	1.44512	1.44512
$149$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$149$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$150$	0	0
$151$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$151$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$152$	0.279773	0.279773
$153$	0	0
$154$	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
$160$	−0.790943	−0.790943
$161$	0	0
$162$	1.33826	1.33826
$163$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$163$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$167$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$168$	0	0
$169$	2.82709	2.82709
$170$	0	0
$171$	−1.00000	−1.00000
$172$	−1.27977	−1.27977
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	2.44512	2.44512
$179$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$179$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$180$	0.488830	0.488830
$181$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$181$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$182$	0	0
$183$	0	0
$184$	−0.511170	−0.511170
$185$	1.12920	1.12920
$186$	0	0
$187$	0	0
$188$	−0.165352	−0.165352
$189$	0	0
$190$	−0.827091	−0.827091
$191$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$192$	0	0
$193$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$194$	1.79094	1.79094
$195$	0	0
$196$	0.790943	0.790943
$197$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$197$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$198$	0	0
$199$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$199$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$200$	0.172909	0.172909
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.82709	1.82709
$208$	2.27977	2.27977
$209$	0	0
$210$	0	0
$211$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$211$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$212$	−0.790943	−0.790943
$213$	0	0
$214$	−0.279773	−0.279773
$215$	−1.00000	−1.00000
$216$	0	0
$217$	0	0
$218$	1.79094	1.79094
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−0.618034	−0.618034
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$230$	1.51117	1.51117
$231$	0	0
$232$	0	0
$233$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$233$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$234$	−2.61803	−2.61803
$235$	−0.129204	−0.129204
$236$	−1.54732	−1.54732
$237$	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			$\pi$
$242$	1.33826	1.33826
$243$	0	0
$244$	−0.165352	−0.165352
$245$	0.618034	0.618034
$246$	0	0
$247$	1.95630	1.95630
$248$	0	0
$249$	0	0
$250$	−1.33826	−1.33826
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−2.61803	−2.61803
$255$	0	0
$256$	1.27977	1.27977
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	−0.956295	−0.956295
$261$	0	0
$262$	2.44512	2.44512
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−0.618034	−0.618034
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$269$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$270$	0	0
$271$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$271$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.79094	1.79094
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−1.27977	−1.27977
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.27977	−1.27977
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	−1.20906	−1.20906
$296$	−0.511170	−0.511170
$297$	0	0
$298$	1.79094	1.79094
$299$	−3.57433	−3.57433
$300$	0	0
$301$	0	0
$302$	0.827091	0.827091
$303$	0	0
$304$	1.16535	1.16535
$305$	−0.129204	−0.129204
$306$	0	0
$307$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$307$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$311$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−0.338261	−0.338261
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.790943	0.790943
$325$	1.20906	1.20906
$326$	0.827091	0.827091
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	1.82709	1.82709
$334$	−1.33826	−1.33826
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	3.78339	3.78339
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−1.33826	−1.33826
$343$	0	0
$344$	0.452682	0.452682
$345$	0	0
$346$	0	0
$347$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$347$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−1.00000	−1.00000
$356$	1.44512	1.44512
$357$	0	0
$358$	0.827091	0.827091
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.172909	−0.172909
$361$	0	0
$362$	−2.16535	−2.16535
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$368$	−2.12920	−2.12920
$369$	0	0
$370$	1.51117	1.51117
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0.0584884	0.0584884
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−0.488830	−0.488830
$381$	0	0
$382$	−1.33826	−1.33826
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−1.33826	−1.33826
$387$	−1.61803	−1.61803
$388$	1.05849	1.05849
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−0.279773	−0.279773
$393$	0	0
$394$	−0.279773	−0.279773
$395$	0	0
$396$	0	0
$397$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$398$	1.79094	1.79094
$399$	0	0
$400$	0.720227	0.720227
$401$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$401$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.618034	0.618034
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	2.44512	2.44512
$415$	0	0
$416$	2.50361	2.50361
$417$	0	0
$418$	0	0
$419$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$419$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	2.44512	2.44512
$423$	−0.209057	−0.209057
$424$	0.279773	0.279773
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−0.165352	−0.165352
$429$	0	0
$430$	−1.33826	−1.33826
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$433$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$434$	0	0
$435$	0	0
$436$	1.05849	1.05849
$437$	−1.82709	−1.82709
$438$	0	0
$439$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$439$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	1.12920	1.12920
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−0.827091	−0.827091
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$457$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$458$	−1.33826	−1.33826
$459$	0	0
$460$	0.893136	0.893136
$461$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$461$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−2.61803	−2.61803
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−1.54732	−1.54732
$469$	0	0
$470$	−0.172909	−0.172909
$471$	0	0
$472$	0.547318	0.547318
$473$	0	0
$474$	0	0
$475$	0.618034	0.618034
$476$	0	0
$477$	−1.00000	−1.00000
$478$	0	0
$479$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$479$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$480$	0	0
$481$	−3.57433	−3.57433
$482$	0	0
$483$	0	0
$484$	0.790943	0.790943
$485$	0.827091	0.827091
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0.0584884	0.0584884
$489$	0	0
$490$	0.827091	0.827091
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	2.61803	2.61803
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$499$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$500$	−0.790943	−0.790943
$501$	0	0
$502$	0	0
$503$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$503$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−1.54732	−1.54732
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.16535	1.16535
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0.338261	0.338261
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$523$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$524$	1.44512	1.44512
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	2.33826	2.33826
$530$	−0.827091	−0.827091
$531$	−1.95630	−1.95630
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−0.129204	−0.129204
$536$	0	0
$537$	0	0
$538$	−2.16535	−2.16535
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−2.16535	−2.16535
$543$	0	0
$544$	0	0
$545$	0.827091	0.827091
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−0.209057	−0.209057
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	1.05849	1.05849
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	3.16535	3.16535
$560$	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
$567$	0	0
$568$	0.452682	0.452682
$569$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$569$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.12920	−1.12920
$576$	−0.547318	−0.547318
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.33826	1.33826
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.20906	−1.20906
$586$	0	0
$587$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$587$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$588$	0	0
$589$	0	0
$590$	−1.61803	−1.61803
$591$	0	0
$592$	−2.12920	−2.12920
$593$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$593$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$594$	0	0
$595$	0	0
$596$	1.05849	1.05849
$597$	0	0
$598$	−4.78339	−4.78339
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$601$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$602$	0	0
$603$	0	0
$604$	0.488830	0.488830
$605$	0.618034	0.618034
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.27977	1.27977
$609$	0	0
$610$	−0.172909	−0.172909
$611$	0.408977	0.408977
$612$	0	0
$613$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$614$	0.827091	0.827091
$615$	0	0
$616$	0	0
$617$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$617$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	−0.279773	−0.279773
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$631$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.20906	−1.20906
$636$	0	0
$637$	−1.95630	−1.95630
$638$	0	0
$639$	−1.61803	−1.61803
$640$	0.338261	0.338261
$641$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$641$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$642$	0	0
$643$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$643$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−0.279773	−0.279773
$649$	0	0
$650$	1.61803	1.61803
$651$	0	0
$652$	0.488830	0.488830
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	1.12920	1.12920
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	2.44512	2.44512
$667$	0	0
$668$	−0.790943	−0.790943
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$673$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$674$	0	0
$675$	0	0
$676$	2.23607	2.23607
$677$	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	0
$683$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$684$	−0.790943	−0.790943
$685$	0	0
$686$	0	0
$687$	0	0
$688$	1.88558	1.88558
$689$	1.95630	1.95630
$690$	0	0
$691$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$691$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$692$	0	0
$693$	0	0
$694$	1.79094	1.79094
$695$	0.827091	0.827091
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−1.82709	−1.82709
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−1.33826	−1.33826
$711$	0	0
$712$	−0.511170	−0.511170
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0.488830	0.488830
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−0.720227	−0.720227
$721$	0	0
$722$	0	0
$723$	0	0
$724$	−1.27977	−1.27977
$725$	0	0
$726$	0	0
$727$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$727$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−1.33826	−1.33826
$735$	0	0
$736$	−2.33826	−2.33826
$737$	0	0
$738$	0	0
$739$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$740$	0.893136	0.893136
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0.827091	0.827091
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.00000	1.00000
$751$	1.00000	1.00000
$752$	0.243625	0.243625
$753$	0	0
$754$	0	0
$755$	0.381966	0.381966
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0.172909	0.172909
$761$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$761$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$762$	0	0
$763$	0	0
$764$	−0.790943	−0.790943
$765$	0	0
$766$	0	0
$767$	3.82709	3.82709
$768$	0	0
$769$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$769$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$770$	0	0
$771$	0	0
$772$	−0.790943	−0.790943
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−2.16535	−2.16535
$775$	0	0
$776$	−0.374409	−0.374409
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.16535	−1.16535
$785$	0	0
$786$	0	0
$787$	2.00000	2.00000	1.00000			$0$
$787$	2.00000	2.00000	1.00000			$0$
$788$	−0.165352	−0.165352
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.408977	0.408977
$794$	−1.33826	−1.33826
$795$	0	0
$796$	1.05849	1.05849
$797$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$797$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$798$	0	0
$799$	0	0
$800$	0.790943	0.790943
$801$	1.82709	1.82709
$802$	−2.61803	−2.61803
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0.827091	0.827091
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.381966	0.381966
$816$	0	0
$817$	1.61803	1.61803
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$823$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$827$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$828$	1.44512	1.44512
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	1.07072	1.07072
$833$	0	0
$834$	0	0
$835$	−0.618034	−0.618034
$836$	0	0
$837$	0	0
$838$	1.79094	1.79094
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	1.44512	1.44512
$845$	1.74724	1.74724
$846$	−0.279773	−0.279773
$847$	0	0
$848$	1.16535	1.16535
$849$	0	0
$850$	0	0
$851$	3.33826	3.33826
$852$	0	0
$853$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$853$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$854$	0	0
$855$	−0.618034	−0.618034
$856$	0.0584884	0.0584884
$857$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$857$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	−0.790943	−0.790943
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	−0.279773	−0.279773
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−0.374409	−0.374409
$873$	1.33826	1.33826
$874$	−2.44512	−2.44512
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0.827091	0.827091
$879$	0	0
$880$	0	0
$881$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$881$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$882$	1.33826	1.33826
$883$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$883$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$884$	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
$890$	1.51117	1.51117
$891$	0	0
$892$	0	0
$893$	0.209057	0.209057
$894$	0	0
$895$	0.381966	0.381966
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−0.488830	−0.488830
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.00000	−1.00000
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$911$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$912$	0	0
$913$	0	0
$914$	−0.279773	−0.279773
$915$	0	0
$916$	−0.790943	−0.790943
$917$	0	0
$918$	0	0
$919$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$919$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$920$	−0.315921	−0.315921
$921$	0	0
$922$	2.44512	2.44512
$923$	3.16535	3.16535
$924$	0	0
$925$	−1.12920	−1.12920
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$930$	0	0
$931$	−1.00000	−1.00000
$932$	−1.54732	−1.54732
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0.547318	0.547318
$937$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$937$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$938$	0	0
$939$	0	0
$940$	−0.102193	−0.102193
$941$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$941$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$942$	0	0
$943$	0	0
$944$	2.27977	2.27977
$945$	0	0
$946$	0	0
$947$	2.00000	2.00000	1.00000			$0$
$947$	2.00000	2.00000	1.00000			$0$
$948$	0	0
$949$	0	0
$950$	0.827091	0.827091
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−1.33826	−1.33826
$955$	−0.618034	−0.618034
$956$	0	0
$957$	0	0
$958$	−2.61803	−2.61803
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	−4.78339	−4.78339
$963$	−0.209057	−0.209057
$964$	0	0
$965$	−0.618034	−0.618034
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.279773	−0.279773
$969$	0	0
$970$	1.10686	1.10686
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0.243625	0.243625
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0.488830	0.488830
$981$	1.33826	1.33826
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−0.129204	−0.129204
$986$	0	0
$987$	0	0
$988$	1.54732	1.54732
$989$	−2.95630	−2.95630
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.827091	0.827091
$996$	0	0
$997$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$997$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$998$	−0.279773	−0.279773
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	751.1.b.d.750.3	✓	4
751.750	odd	2	CM	751.1.b.d.750.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
751.1.b.d.750.3	✓	4	1.1	even	1	trivial
751.1.b.d.750.3	✓	4	751.750	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:	\( N \)	\(=\)	\( 751 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	751.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.33826 q^{2} +0.790943 q^{4} +0.618034 q^{5} -0.279773 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.33826 q^{2} +0.790943 q^{4} +0.618034 q^{5} -0.279773 q^{8} +1.00000 q^{9} +0.827091 q^{10} -1.95630 q^{13} -1.16535 q^{16} +1.33826 q^{18} -1.00000 q^{19} +0.488830 q^{20} +1.82709 q^{23} -0.618034 q^{25} -2.61803 q^{26} -1.27977 q^{32} +0.790943 q^{36} +1.82709 q^{37} -1.33826 q^{38} -0.172909 q^{40} -1.61803 q^{43} +0.618034 q^{45} +2.44512 q^{46} -0.209057 q^{47} +1.00000 q^{49} -0.827091 q^{50} -1.54732 q^{52} -1.00000 q^{53} -1.95630 q^{59} -0.209057 q^{61} -0.547318 q^{64} -1.20906 q^{65} -1.61803 q^{71} -0.279773 q^{72} +2.44512 q^{74} -0.790943 q^{76} -0.720227 q^{80} +1.00000 q^{81} -2.16535 q^{86} +1.82709 q^{89} +0.827091 q^{90} +1.44512 q^{92} -0.279773 q^{94} -0.618034 q^{95} +1.33826 q^{97} +1.33826 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 5 q^{4} - 2 q^{5} - q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + q^2 + 5 * q^4 - 2 * q^5 - q^8 + 4 * q^9	\( 4 q + q^{2} + 5 q^{4} - 2 q^{5} - q^{8} + 4 q^{9} - 3 q^{10} + q^{13} + 6 q^{16} + q^{18} - 4 q^{19} + q^{23} + 2 q^{25} - 6 q^{26} - 5 q^{32} + 5 q^{36} + q^{37} - q^{38} - 7 q^{40} - 2 q^{43} - 2 q^{45} - q^{46} + q^{47} + 4 q^{49} + 3 q^{50} - 4 q^{53} + q^{59} + q^{61} + 4 q^{64} - 3 q^{65} - 2 q^{71} - q^{72} - q^{74} - 5 q^{76} - 3 q^{80} + 4 q^{81} + 2 q^{86} + q^{89} - 3 q^{90} - 5 q^{92} - q^{94} + 2 q^{95} + q^{97} + q^{98}+O(q^{100}) \) 4 * q + q^2 + 5 * q^4 - 2 * q^5 - q^8 + 4 * q^9 - 3 * q^10 + q^13 + 6 * q^16 + q^18 - 4 * q^19 + q^23 + 2 * q^25 - 6 * q^26 - 5 * q^32 + 5 * q^36 + q^37 - q^38 - 7 * q^40 - 2 * q^43 - 2 * q^45 - q^46 + q^47 + 4 * q^49 + 3 * q^50 - 4 * q^53 + q^59 + q^61 + 4 * q^64 - 3 * q^65 - 2 * q^71 - q^72 - q^74 - 5 * q^76 - 3 * q^80 + 4 * q^81 + 2 * q^86 + q^89 - 3 * q^90 - 5 * q^92 - q^94 + 2 * q^95 + q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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