LMFDB - Embedded newform 1175.1.d.e.751.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1175,1,Mod(751,1175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1175.751");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1175 = 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1175.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:	$0.586401389844$
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 15.1.4947491410771484375.1

Embedding invariants

Embedding label			751.4
Root			$1.82709$ of defining polynomial
Character	$\chi$	$=$	1175.751

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82709 q^{2} -1.95630 q^{3} +2.33826 q^{4} -3.57433 q^{6} -0.209057 q^{7} +2.44512 q^{8} +2.82709 q^{9} +O(q^{10})$	$q+1.82709 q^{2} -1.95630 q^{3} +2.33826 q^{4} -3.57433 q^{6} -0.209057 q^{7} +2.44512 q^{8} +2.82709 q^{9} -4.57433 q^{12} -0.381966 q^{14} +2.12920 q^{16} +1.33826 q^{17} +5.16535 q^{18} +0.408977 q^{21} -4.78339 q^{24} -3.57433 q^{27} -0.488830 q^{28} +1.44512 q^{32} +2.44512 q^{34} +6.61048 q^{36} +0.618034 q^{37} +0.747238 q^{42} +1.00000 q^{47} -4.16535 q^{48} -0.956295 q^{49} -2.61803 q^{51} -1.61803 q^{53} -6.53062 q^{54} -0.511170 q^{56} -0.209057 q^{59} -1.61803 q^{61} -0.591023 q^{63} +0.511170 q^{64} +3.12920 q^{68} -1.95630 q^{71} +6.91259 q^{72} +1.12920 q^{74} +0.618034 q^{79} +4.16535 q^{81} -1.00000 q^{83} +0.956295 q^{84} -1.61803 q^{89} +1.82709 q^{94} -2.82709 q^{96} -1.61803 q^{97} -1.74724 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{3} + 5 q^{4} - q^{6} + q^{7} - q^{8} + 5 q^{9}+O(q^{10}) $ 4 * q + q^2 + q^3 + 5 * q^4 - q^6 + q^7 - q^8 + 5 * q^9	$ 4 q + q^{2} + q^{3} + 5 q^{4} - q^{6} + q^{7} - q^{8} + 5 q^{9} - 5 q^{12} - 6 q^{14} + 6 q^{16} + q^{17} + 10 q^{18} - q^{21} - 4 q^{24} - q^{27} - 5 q^{32} - q^{34} + 5 q^{36} - 2 q^{37} - 4 q^{42} + 4 q^{47} - 6 q^{48} + 5 q^{49} - 6 q^{51} - 2 q^{53} - 4 q^{54} - 4 q^{56} + q^{59} - 2 q^{61} - 5 q^{63} + 4 q^{64} + 10 q^{68} + q^{71} + 10 q^{72} + 2 q^{74} - 2 q^{79} + 6 q^{81} - 4 q^{83} - 5 q^{84} - 2 q^{89} + q^{94} - 5 q^{96} - 2 q^{97}+O(q^{100}) $ 4 * q + q^2 + q^3 + 5 * q^4 - q^6 + q^7 - q^8 + 5 * q^9 - 5 * q^12 - 6 * q^14 + 6 * q^16 + q^17 + 10 * q^18 - q^21 - 4 * q^24 - q^27 - 5 * q^32 - q^34 + 5 * q^36 - 2 * q^37 - 4 * q^42 + 4 * q^47 - 6 * q^48 + 5 * q^49 - 6 * q^51 - 2 * q^53 - 4 * q^54 - 4 * q^56 + q^59 - 2 * q^61 - 5 * q^63 + 4 * q^64 + 10 * q^68 + q^71 + 10 * q^72 + 2 * q^74 - 2 * q^79 + 6 * q^81 - 4 * q^83 - 5 * q^84 - 2 * q^89 + q^94 - 5 * q^96 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1175\mathbb{Z}\right)^\times$.

$n$	$377$	$851$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$2$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$3$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$3$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$4$	2.33826	2.33826
$5$	0	0
$5$	0	0
$6$	−3.57433	−3.57433
$7$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$7$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$8$	2.44512	2.44512
$9$	2.82709	2.82709
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−4.57433	−4.57433
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−0.381966	−0.381966
$15$	0	0
$16$	2.12920	2.12920
$17$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$17$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$18$	5.16535	5.16535
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0.408977	0.408977
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−4.78339	−4.78339
$25$	0	0
$26$	0	0
$27$	−3.57433	−3.57433
$28$	−0.488830	−0.488830
$29$	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			$\pi$
$32$	1.44512	1.44512
$33$	0	0
$34$	2.44512	2.44512
$35$	0	0
$36$	6.61048	6.61048
$37$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$37$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0.747238	0.747238
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.00000	1.00000
$47$	1.00000	1.00000
$48$	−4.16535	−4.16535
$49$	−0.956295	−0.956295
$50$	0	0
$51$	−2.61803	−2.61803
$52$	0	0
$53$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$53$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$54$	−6.53062	−6.53062
$55$	0	0
$56$	−0.511170	−0.511170
$57$	0	0
$58$	0	0
$59$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$59$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$60$	0	0
$61$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$61$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$62$	0	0
$63$	−0.591023	−0.591023
$64$	0.511170	0.511170
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	3.12920	3.12920
$69$	0	0
$70$	0	0
$71$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$71$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$72$	6.91259	6.91259
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	1.12920	1.12920
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$79$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$80$	0	0
$81$	4.16535	4.16535
$82$	0	0
$83$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$83$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$84$	0.956295	0.956295
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$89$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	1.82709	1.82709
$95$	0	0
$96$	−2.82709	−2.82709
$97$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$97$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$98$	−1.74724	−1.74724
$99$	0	0
$100$	0	0
$101$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$101$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$102$	−4.78339	−4.78339
$103$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$103$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$104$	0	0
$105$	0	0
$106$	−2.95630	−2.95630
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−8.35772	−8.35772
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−1.20906	−1.20906
$112$	−0.445125	−0.445125
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−0.381966	−0.381966
$119$	−0.279773	−0.279773
$120$	0	0
$121$	1.00000	1.00000
$122$	−2.95630	−2.95630
$123$	0	0
$124$	0	0
$125$	0	0
$126$	−1.07985	−1.07985
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.511170	−0.511170
$129$	0	0
$130$	0	0
$131$	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
$136$	3.27222	3.27222
$137$	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	0
$141$	−1.95630	−1.95630
$142$	−3.57433	−3.57433
$143$	0	0
$144$	6.01945	6.01945
$145$	0	0
$146$	0	0
$147$	1.87080	1.87080
$148$	1.44512	1.44512
$149$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$149$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	3.78339	3.78339
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$157$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$158$	1.12920	1.12920
$159$	3.16535	3.16535
$160$	0	0
$161$	0	0
$162$	7.61048	7.61048
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−1.82709	−1.82709
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	1.00000	1.00000
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$173$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.408977	0.408977
$178$	−2.95630	−2.95630
$179$	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
$183$	3.16535	3.16535
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	2.33826	2.33826
$189$	0.747238	0.747238
$190$	0	0
$191$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$192$	−1.00000	−1.00000
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	−2.95630	−2.95630
$195$	0	0
$196$	−2.23607	−2.23607
$197$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	2.44512	2.44512
$203$	0	0
$204$	−6.12165	−6.12165
$205$	0	0
$206$	2.44512	2.44512
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−3.78339	−3.78339
$213$	3.82709	3.82709
$214$	0	0
$215$	0	0
$216$	−8.73968	−8.73968
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−2.20906	−2.20906
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−0.302113	−0.302113
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	−0.488830	−0.488830
$237$	−1.20906	−1.20906
$238$	−0.511170	−0.511170
$239$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$239$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$240$	0	0
$241$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$241$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$242$	1.82709	1.82709
$243$	−4.57433	−4.57433
$244$	−3.78339	−3.78339
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.95630	1.95630
$250$	0	0
$251$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$251$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$252$	−1.38197	−1.38197
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−1.44512	−1.44512
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	−0.129204	−0.129204
$260$	0	0
$261$	0	0
$262$	3.33826	3.33826
$263$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$263$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.16535	3.16535
$268$	0	0
$269$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$270$	0	0
$271$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$271$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$272$	2.84943	2.84943
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$277$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−3.57433	−3.57433
$283$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$283$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$284$	−4.57433	−4.57433
$285$	0	0
$286$	0	0
$287$	0	0
$288$	4.08550	4.08550
$289$	0.790943	0.790943
$290$	0	0
$291$	3.16535	3.16535
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	3.41811	3.41811
$295$	0	0
$296$	1.51117	1.51117
$297$	0	0
$298$	−3.57433	−3.57433
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.61803	−2.61803
$304$	0	0
$305$	0	0
$306$	6.91259	6.91259
$307$	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
$309$	−2.61803	−2.61803
$310$	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			$\pi$
$314$	3.33826	3.33826
$315$	0	0
$316$	1.44512	1.44512
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	5.78339	5.78339
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	9.73968	9.73968
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.209057	−0.209057
$330$	0	0
$331$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$331$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$332$	−2.33826	−2.33826
$333$	1.74724	1.74724
$334$	0	0
$335$	0	0
$336$	0.870796	0.870796
$337$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$337$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$338$	1.82709	1.82709
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0.408977	0.408977
$344$	0	0
$345$	0	0
$346$	−0.381966	−0.381966
$347$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$347$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$353$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$354$	0.747238	0.747238
$355$	0	0
$356$	−3.78339	−3.78339
$357$	0.547318	0.547318
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	−1.95630	−1.95630
$364$	0	0
$365$	0	0
$366$	5.78339	5.78339
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.338261	0.338261
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	2.44512	2.44512
$377$	0	0
$378$	1.36527	1.36527
$379$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$379$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$380$	0	0
$381$	0	0
$382$	−1.82709	−1.82709
$383$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$383$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$384$	1.00000	1.00000
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−3.78339	−3.78339
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−2.33826	−2.33826
$393$	−3.57433	−3.57433
$394$	−1.82709	−1.82709
$395$	0	0
$396$	0	0
$397$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$397$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$401$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$402$	0	0
$403$	0	0
$404$	3.12920	3.12920
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−6.40142	−6.40142
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	3.12920	3.12920
$413$	0.0437048	0.0437048
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	2.82709	2.82709
$424$	−3.95630	−3.95630
$425$	0	0
$426$	6.99244	6.99244
$427$	0.338261	0.338261
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$431$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$432$	−7.61048	−7.61048
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$440$	0	0
$441$	−2.70353	−2.70353
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−2.82709	−2.82709
$445$	0	0
$446$	0	0
$447$	3.82709	3.82709
$448$	−0.106864	−0.106864
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$457$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$458$	0	0
$459$	−4.78339	−4.78339
$460$	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	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.57433	−3.57433
$472$	−0.511170	−0.511170
$473$	0	0
$474$	−2.20906	−2.20906
$475$	0	0
$476$	−0.654182	−0.654182
$477$	−4.57433	−4.57433
$478$	1.12920	1.12920
$479$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$479$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$480$	0	0
$481$	0	0
$482$	1.12920	1.12920
$483$	0	0
$484$	2.33826	2.33826
$485$	0	0
$486$	−8.35772	−8.35772
$487$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$487$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$488$	−3.95630	−3.95630
$489$	0	0
$490$	0	0
$491$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$491$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.408977	0.408977
$498$	3.57433	3.57433
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	1.12920	1.12920
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−1.44512	−1.44512
$505$	0	0
$506$	0	0
$507$	−1.95630	−1.95630
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−2.12920	−2.12920
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−0.236068	−0.236068
$519$	0.408977	0.408977
$520$	0	0
$521$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$521$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$522$	0	0
$523$	2.00000	2.00000	1.00000			$0$
$523$	2.00000	2.00000	1.00000			$0$
$524$	4.27222	4.27222
$525$	0	0
$526$	−2.95630	−2.95630
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	−0.591023	−0.591023
$532$	0	0
$533$	0	0
$534$	5.78339	5.78339
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−1.82709	−1.82709
$539$	0	0
$540$	0	0
$541$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$541$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$542$	−3.57433	−3.57433
$543$	0	0
$544$	1.93395	1.93395
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−4.57433	−4.57433
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−0.129204	−0.129204
$554$	2.44512	2.44512
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−4.57433	−4.57433
$565$	0	0
$566$	3.33826	3.33826
$567$	−0.870796	−0.870796
$568$	−4.78339	−4.78339
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$571$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$572$	0	0
$573$	1.95630	1.95630
$574$	0	0
$575$	0	0
$576$	1.44512	1.44512
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.44512	1.44512
$579$	0	0
$580$	0	0
$581$	0.209057	0.209057
$582$	5.78339	5.78339
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	4.37441	4.37441
$589$	0	0
$590$	0	0
$591$	1.95630	1.95630
$592$	1.31592	1.31592
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	−4.57433	−4.57433
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$601$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	−4.78339	−4.78339
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	8.84655	8.84655
$613$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$614$	1.12920	1.12920
$615$	0	0
$616$	0	0
$617$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$617$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$618$	−4.78339	−4.78339
$619$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.338261	0.338261
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	4.27222	4.27222
$629$	0.827091	0.827091
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	1.51117	1.51117
$633$	0	0
$634$	0	0
$635$	0	0
$636$	7.40142	7.40142
$637$	0	0
$638$	0	0
$639$	−5.53062	−5.53062
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$643$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$647$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$648$	10.1848	10.1848
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$653$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−0.381966	−0.381966
$659$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$659$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$660$	0	0
$661$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$661$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$662$	−0.381966	−0.381966
$663$	0	0
$664$	−2.44512	−2.44512
$665$	0	0
$666$	3.19236	3.19236
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0.591023	0.591023
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	3.33826	3.33826
$675$	0	0
$676$	2.33826	2.33826
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0.338261	0.338261
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$683$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$684$	0	0
$685$	0	0
$686$	0.747238	0.747238
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−0.488830	−0.488830
$693$	0	0
$694$	1.12920	1.12920
$695$	0	0
$696$	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
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−0.381966	−0.381966
$707$	−0.279773	−0.279773
$708$	0.956295	0.956295
$709$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$709$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$710$	0	0
$711$	1.74724	1.74724
$712$	−3.95630	−3.95630
$713$	0	0
$714$	1.00000	1.00000
$715$	0	0
$716$	0	0
$717$	−1.20906	−1.20906
$718$	0	0
$719$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$719$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$720$	0	0
$721$	−0.279773	−0.279773
$722$	1.82709	1.82709
$723$	−1.20906	−1.20906
$724$	0	0
$725$	0	0
$726$	−3.57433	−3.57433
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	4.78339	4.78339
$730$	0	0
$731$	0	0
$732$	7.40142	7.40142
$733$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$733$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$739$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$740$	0	0
$741$	0	0
$742$	0.618034	0.618034
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.82709	−2.82709
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	2.12920	2.12920
$753$	−1.20906	−1.20906
$754$	0	0
$755$	0	0
$756$	1.74724	1.74724
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−2.95630	−2.95630
$759$	0	0
$760$	0	0
$761$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$762$	0	0
$763$	0	0
$764$	−2.33826	−2.33826
$765$	0	0
$766$	1.12920	1.12920
$767$	0	0
$768$	2.82709	2.82709
$769$	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	0
$773$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$773$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$774$	0	0
$775$	0	0
$776$	−3.95630	−3.95630
$777$	0.252762	0.252762
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−2.03615	−2.03615
$785$	0	0
$786$	−6.53062	−6.53062
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−2.33826	−2.33826
$789$	3.16535	3.16535
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−3.57433	−3.57433
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	1.33826	1.33826
$800$	0	0
$801$	−4.57433	−4.57433
$802$	1.12920	1.12920
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.95630	1.95630
$808$	3.27222	3.27222
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$811$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$812$	0	0
$813$	3.82709	3.82709
$814$	0	0
$815$	0	0
$816$	−5.57433	−5.57433
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	2.00000	2.00000	1.00000			$0$
$823$	2.00000	2.00000	1.00000			$0$
$824$	3.27222	3.27222
$825$	0	0
$826$	0.0798526	0.0798526
$827$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$827$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	−2.61803	−2.61803
$832$	0	0
$833$	−1.27977	−1.27977
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	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
$844$	0	0
$845$	0	0
$846$	5.16535	5.16535
$847$	−0.209057	−0.209057
$848$	−3.44512	−3.44512
$849$	−3.57433	−3.57433
$850$	0	0
$851$	0	0
$852$	8.94874	8.94874
$853$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$853$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$854$	0.618034	0.618034
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	3.33826	3.33826
$863$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$863$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$864$	−5.16535	−5.16535
$865$	0	0
$866$	0	0
$867$	−1.54732	−1.54732
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−4.57433	−4.57433
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−1.82709	−1.82709
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−4.93960	−4.93960
$883$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$883$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−2.95630	−2.95630
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	6.99244	6.99244
$895$	0	0
$896$	0.106864	0.106864
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−2.16535	−2.16535
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$908$	0	0
$909$	3.78339	3.78339
$910$	0	0
$911$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$911$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$912$	0	0
$913$	0	0
$914$	−3.57433	−3.57433
$915$	0	0
$916$	0	0
$917$	−0.381966	−0.381966
$918$	−8.73968	−8.73968
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1.20906	−1.20906
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.78339	3.78339
$928$	0	0
$929$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$929$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$941$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$942$	−6.53062	−6.53062
$943$	0	0
$944$	−0.445125	−0.445125
$945$	0	0
$946$	0	0
$947$	2.00000	2.00000	1.00000			$0$
$947$	2.00000	2.00000	1.00000			$0$
$948$	−2.82709	−2.82709
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−0.684079	−0.684079
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−8.35772	−8.35772
$955$	0	0
$956$	1.44512	1.44512
$957$	0	0
$958$	3.33826	3.33826
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	1.44512	1.44512
$965$	0	0
$966$	0	0
$967$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$967$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$968$	2.44512	2.44512
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−10.6960	−10.6960
$973$	0	0
$974$	−1.82709	−1.82709
$975$	0	0
$976$	−3.44512	−3.44512
$977$	2.00000	2.00000	1.00000			$0$
$977$	2.00000	2.00000	1.00000			$0$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−2.95630	−2.95630
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.408977	0.408977
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$991$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$992$	0	0
$993$	0.408977	0.408977
$994$	0.747238	0.747238
$995$	0	0
$996$	4.57433	4.57433
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	−2.20906	−2.20906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1175.1.d.e.751.4	yes	4
5.2	odd	4		1175.1.b.c.1174.7		8
5.3	odd	4		1175.1.b.c.1174.2		8
5.4	even	2		1175.1.d.d.751.1	✓	4
47.46	odd	2	CM	1175.1.d.e.751.4	yes	4
235.93	even	4		1175.1.b.c.1174.2		8
235.187	even	4		1175.1.b.c.1174.7		8
235.234	odd	2		1175.1.d.d.751.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1175.1.b.c.1174.2		8	5.3	odd	4
1175.1.b.c.1174.2		8	235.93	even	4
1175.1.b.c.1174.7		8	5.2	odd	4
1175.1.b.c.1174.7		8	235.187	even	4
1175.1.d.d.751.1	✓	4	5.4	even	2
1175.1.d.d.751.1	✓	4	235.234	odd	2
1175.1.d.e.751.4	yes	4	1.1	even	1	trivial
1175.1.d.e.751.4	yes	4	47.46	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:	\( N \)	\(=\)	\( 1175 = 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1175.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.82709 q^{2} -1.95630 q^{3} +2.33826 q^{4} -3.57433 q^{6} -0.209057 q^{7} +2.44512 q^{8} +2.82709 q^{9} +O(q^{10})\)	\(q+1.82709 q^{2} -1.95630 q^{3} +2.33826 q^{4} -3.57433 q^{6} -0.209057 q^{7} +2.44512 q^{8} +2.82709 q^{9} -4.57433 q^{12} -0.381966 q^{14} +2.12920 q^{16} +1.33826 q^{17} +5.16535 q^{18} +0.408977 q^{21} -4.78339 q^{24} -3.57433 q^{27} -0.488830 q^{28} +1.44512 q^{32} +2.44512 q^{34} +6.61048 q^{36} +0.618034 q^{37} +0.747238 q^{42} +1.00000 q^{47} -4.16535 q^{48} -0.956295 q^{49} -2.61803 q^{51} -1.61803 q^{53} -6.53062 q^{54} -0.511170 q^{56} -0.209057 q^{59} -1.61803 q^{61} -0.591023 q^{63} +0.511170 q^{64} +3.12920 q^{68} -1.95630 q^{71} +6.91259 q^{72} +1.12920 q^{74} +0.618034 q^{79} +4.16535 q^{81} -1.00000 q^{83} +0.956295 q^{84} -1.61803 q^{89} +1.82709 q^{94} -2.82709 q^{96} -1.61803 q^{97} -1.74724 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{3} + 5 q^{4} - q^{6} + q^{7} - q^{8} + 5 q^{9}+O(q^{10}) \) 4 * q + q^2 + q^3 + 5 * q^4 - q^6 + q^7 - q^8 + 5 * q^9	\( 4 q + q^{2} + q^{3} + 5 q^{4} - q^{6} + q^{7} - q^{8} + 5 q^{9} - 5 q^{12} - 6 q^{14} + 6 q^{16} + q^{17} + 10 q^{18} - q^{21} - 4 q^{24} - q^{27} - 5 q^{32} - q^{34} + 5 q^{36} - 2 q^{37} - 4 q^{42} + 4 q^{47} - 6 q^{48} + 5 q^{49} - 6 q^{51} - 2 q^{53} - 4 q^{54} - 4 q^{56} + q^{59} - 2 q^{61} - 5 q^{63} + 4 q^{64} + 10 q^{68} + q^{71} + 10 q^{72} + 2 q^{74} - 2 q^{79} + 6 q^{81} - 4 q^{83} - 5 q^{84} - 2 q^{89} + q^{94} - 5 q^{96} - 2 q^{97}+O(q^{100}) \) 4 * q + q^2 + q^3 + 5 * q^4 - q^6 + q^7 - q^8 + 5 * q^9 - 5 * q^12 - 6 * q^14 + 6 * q^16 + q^17 + 10 * q^18 - q^21 - 4 * q^24 - q^27 - 5 * q^32 - q^34 + 5 * q^36 - 2 * q^37 - 4 * q^42 + 4 * q^47 - 6 * q^48 + 5 * q^49 - 6 * q^51 - 2 * q^53 - 4 * q^54 - 4 * q^56 + q^59 - 2 * q^61 - 5 * q^63 + 4 * q^64 + 10 * q^68 + q^71 + 10 * q^72 + 2 * q^74 - 2 * q^79 + 6 * q^81 - 4 * q^83 - 5 * q^84 - 2 * q^89 + q^94 - 5 * q^96 - 2 * q^97

Introduction

L-functions

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Database

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