LMFDB - Newform orbit 825.2.bs.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(74,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("825.74");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.bs (of order $10$, degree $4$, not 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:	no
Analytic conductor:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{20}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - 2 \zeta_{20}^{6} + \zeta_{20}^{4} + \cdots + 1) q^{2}+ \cdots + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{4} + \cdots - 2) q^{9}+O(q^{10}) $ q + (-2z^6 + z^4 - z^2 + 1) q^2 + (z^7 - z^6 - z^5 - z^2 + 1) * q^3 + (-z^6 - z^2) * q^4 + (-z^6 + z^5 - z^4 - z^2 - z - 1) * q^6 + (-2z^7 - 2z) * q^7 + (z^6 - 2z^4 + 2z^2) * q^8 + (2z^7 - 2z^4 + z^2 - 2) * q^9	$ q + ( - 2 \zeta_{20}^{6} + \zeta_{20}^{4} + \cdots + 1) q^{2}+ \cdots + ( - 5 \zeta_{20}^{7} + \cdots + 2 \zeta_{20}) q^{99}+O(q^{100}) $ q + (-2z^6 + z^4 - z^2 + 1) q^2 + (z^7 - z^6 - z^5 - z^2 + 1) * q^3 + (-z^6 - z^2) * q^4 + (-z^6 + z^5 - z^4 - z^2 - z - 1) * q^6 + (-2z^7 - 2z) * q^7 + (z^6 - 2z^4 + 2z^2) * q^8 + (2z^7 - 2z^4 + z^2 - 2) * q^9 + (4z^7 - 3z^5 + 2z^3 - 2z) * q^11 + (z^6 + z^5 - z^4 - 2) * q^12 + (4z^7 - 4z^5 - 2z) q^14 + (-4z^6 + 3z^4 - 3z^2 + 4) q^16 + (-2z^6 - 2z^4 - 2z^2 - 2) q^17 + (3z^6 + 2z^5 - z^4 + 2z^3 - z^2 - 2) q^18 + (-z^6 + 2z^4 - 4z^2 - 2) * q^19 + (2z^7 - 2z^5 + 4z^4 + 2z^3 - 4z^2 - 4z + 2) * q^21 + (2z^7 + 4z^5 + z^3 - z) * q^22 + (-5z^6 + 5z^4 - 1) * q^23 + (2z^7 - 4z^5 - z^4 + 3z^3 - z + 1) q^24 + (3z^6 + z^5 - 4z^4 + 3z^3 + 4z^2 + z - 3) * q^27 + (4z^7 - 2z^5 + 2z^3 - 2z) * q^28 + (2z^7 - 5z^5 - 5z^3 + 2z) * q^29 + 5z^2 q^31 + (-5z^6 + z^4 - 6z^2 + 3) * q^32 + (3z^7 - 2z^6 + z^5 + 2z^3 + 3z^2 + z - 1) * q^33 + (6z^6 - 6z^4 - 8) * q^34 + (-2z^7 + 3z^6 + 2z^5 + z^2 + 2z - 1) * q^36 + (-z^7 - 5z^5 + 5z^3 + z) * q^37 + (8z^6 - 7z^4 + 7z^2 - 8) q^38 + (2z^5 - 2z^3 + z) * q^41 + (6z^7 - 2z^4 + 2z^3 + 6z^2 - 2z - 2) q^42 + (-5z^7 + 2z^5 + z^3 + 4z) q^43 + (-z^7 + 4z^5 + 3z) * q^44 + (2z^6 - 6z^4 + z^2 + 4) * q^46 + (2z^6 + 5z^2 - 5) * q^47 + (z^7 - 4z^6 + 2z^5 - z^4 - 2z^3 - 4z^2 - z) * q^48 + (z^6 - 5z^4 + 5z^2 - 1) * q^49 + (6z^6 + 2z^5 - 4z^4 + 2z^3 + 2z^2 - 8) q^51 + (z^4 + 6z^2 + 1) q^53 + (-5z^7 + 2z^6 + 4z^5 + 4z^4 - 3z^3 - 2z^2 + 8z + 1) q^54 + (-6z^7 + 8z^5 - 6z^3) q^56 + (-4z^7 + 4z^6 + 8z^5 + z^4 - 5z^3 + 2z^2 + 3z - 5) * q^57 + (z^7 - z^5 - 13z) q^58 + (-3z^7 - z^5 + z^3 + 3z) * q^59 + (-z^6 + 5z^4 + 5z^2 - 1) * q^61 + (-5z^6 + 5z^4 - 5z^2 + 10) q^62 + (2z^7 - 4z^6 + 6z^5 + 8z^4 - 4z^3 - 4z^2 + 2z + 4) q^63 + (-6z^4 + 5z^2 - 6) * q^64 + (-4z^7 - z^6 + 6z^5 + z^3 - 4z^2 + 6z + 5) * q^66 + (-7z^7 - z^5 - 7z^3) * q^67 + (8z^6 - 2z^4 + 4z^2 - 6) q^68 + (-6z^7 - 4z^6 + 6z^5 + z^2 - 5z - 1) * q^69 + (-6z^5 - 2z^3 - 6z) q^71 + (4z^7 - 3z^6 - 4z^5 + z^4 + 2z^3 + z^2 - 3) * q^72 + (-z^7 + z^5 + z^3 - z) * q^73 + (-7z^7 + 5z^5 - 12z^3 + 6z) * q^74 + (5z^6 - z^4 + 6z^2 - 3) * q^76 + (2z^6 + 8z^4 - 6z^2 + 8) q^77 + (-4z^6 + 3z^4 - 2z^2 + 1) q^79 + (-4z^7 - 4z^5 + z^4 + 4z^3 + 4z) * q^81 + (-z^5 + 3z^3 - z) q^82 + (-4z^6 - 6z^4 - 6z^2 - 4) q^83 + (2z^7 - 2z^6 + 2z^5 + 2z^3 + 2z^2 + 2z) * q^84 + (-9z^7 - 8z^3 + 8z) q^86 + (8z^7 + 5z^6 - 3z^5 - 9z^4 - 2z^3 + 14z^2 - 6z - 7) q^87 + (4z^7 - 8z^5 + 8z^3 - 5z) * q^88 + (3z^7 - 4z^5 + 3z^3) q^89 + (z^6 - 5z^4 + z^2) q^92 + (-5z^6 - 5z^5 + 5z^3 - 5z + 5) * q^93 + (5z^6 + 2z^4 + 2z^2 + 5) q^94 + (2z^7 + 2z^6 + 4z^5 - 4z^4 - 2z^3 - 3z^2 + z - 7) * q^96 + 17z^7 q^97 + (-3z^6 + 5z^4 - 8z^2 + 4) q^98 + (-5z^7 - 4z^6 + 2z^5 - 4z^4 + 4z^3 + 2z^2 + 2z) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 4 q^{4} - 10 q^{6} + 10 q^{8} - 10 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 - 4 * q^4 - 10 * q^6 + 10 * q^8 - 10 * q^9	$ 8 q + 4 q^{3} - 4 q^{4} - 10 q^{6} + 10 q^{8} - 10 q^{9} - 12 q^{12} + 12 q^{16} - 20 q^{17} - 10 q^{18} - 30 q^{19} - 28 q^{23} + 10 q^{24} - 2 q^{27} + 10 q^{31} - 6 q^{33} - 40 q^{34} - 20 q^{38} + 50 q^{46} - 26 q^{47} - 14 q^{48} + 14 q^{49} - 40 q^{51} + 18 q^{53} - 30 q^{57} - 10 q^{61} + 50 q^{62} - 26 q^{64} + 30 q^{66} - 20 q^{68} - 14 q^{69} - 30 q^{72} + 40 q^{77} - 10 q^{79} - 2 q^{81} - 40 q^{83} + 14 q^{92} + 30 q^{93} + 50 q^{94} - 50 q^{96} + 4 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 4 * q^4 - 10 * q^6 + 10 * q^8 - 10 * q^9 - 12 * q^12 + 12 * q^16 - 20 * q^17 - 10 * q^18 - 30 * q^19 - 28 * q^23 + 10 * q^24 - 2 * q^27 + 10 * q^31 - 6 * q^33 - 40 * q^34 - 20 * q^38 + 50 * q^46 - 26 * q^47 - 14 * q^48 + 14 * q^49 - 40 * q^51 + 18 * q^53 - 30 * q^57 - 10 * q^61 + 50 * q^62 - 26 * q^64 + 30 * q^66 - 20 * q^68 - 14 * q^69 - 30 * q^72 + 40 * q^77 - 10 * q^79 - 2 * q^81 - 40 * q^83 + 14 * q^92 + 30 * q^93 + 50 * q^94 - 50 * q^96 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$-\zeta_{20}^{4}$	$-1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

74.1

0.951057	−	0.309017i
−0.951057	+	0.309017i
−0.587785	+	0.809017i
0.587785	−	0.809017i
−0.587785	−	0.809017i
0.587785	+	0.809017i
0.951057	+	0.309017i
−0.951057	−	0.309017i

1.11803

1.53884i

−0.0877853

1.72982i

−0.500000

1.53884i

−2.76007

1.79892i

−0.726543

2.23607i

0.690983

−

0.224514i

−2.98459

−

0.303706i

74.2

1.11803

1.53884i

1.08779

1.34786i

−0.500000

1.53884i

−0.857960

3.18088i

0.726543

−

2.23607i

0.690983

−

0.224514i

−0.633446

2.93236i

149.1

−1.11803

0.363271i

−0.451057

1.67229i

−0.500000

0.363271i

−0.103198

−

2.03353i

3.07768

−

2.23607i

1.80902

−

2.48990i

−2.59310

−

1.50859i

149.2

−1.11803

0.363271i

1.45106

−

0.945746i

−0.500000

0.363271i

−1.27877

1.58450i

−3.07768

2.23607i

1.80902

−

2.48990i

1.21113

−

2.74466i

299.1

−1.11803

−

0.363271i

−0.451057

−

1.67229i

−0.500000

−

0.363271i

−0.103198

2.03353i

3.07768

2.23607i

1.80902

2.48990i

−2.59310

1.50859i

299.2

−1.11803

−

0.363271i

1.45106

0.945746i

−0.500000

−

0.363271i

−1.27877

−

1.58450i

−3.07768

−

2.23607i

1.80902

2.48990i

1.21113

2.74466i

524.1

1.11803

−

1.53884i

−0.0877853

−

1.72982i

−0.500000

−

1.53884i

−2.76007

−

1.79892i

−0.726543

−

2.23607i

0.690983

0.224514i

−2.98459

0.303706i

524.2

1.11803

−

1.53884i

1.08779

−

1.34786i

−0.500000

−

1.53884i

−0.857960

−

3.18088i

0.726543

2.23607i

0.690983

0.224514i

−0.633446

−

2.93236i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner
15.d	odd	2	1	inner
165.r	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.bs.c		8
3.b	odd	2	1		825.2.bs.b		8
5.b	even	2	1		825.2.bs.b		8
5.c	odd	4	1		825.2.bi.a	✓	8
5.c	odd	4	1		825.2.bi.c	yes	8
11.d	odd	10	1	inner	825.2.bs.c		8
15.d	odd	2	1	inner	825.2.bs.c		8
15.e	even	4	1		825.2.bi.a	✓	8
15.e	even	4	1		825.2.bi.c	yes	8
33.f	even	10	1		825.2.bs.b		8
55.h	odd	10	1		825.2.bs.b		8
55.l	even	20	1		825.2.bi.a	✓	8
55.l	even	20	1		825.2.bi.c	yes	8
165.r	even	10	1	inner	825.2.bs.c		8
165.u	odd	20	1		825.2.bi.a	✓	8
165.u	odd	20	1		825.2.bi.c	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.bi.a	✓	8	5.c	odd	4	1
825.2.bi.a	✓	8	15.e	even	4	1
825.2.bi.a	✓	8	55.l	even	20	1
825.2.bi.a	✓	8	165.u	odd	20	1
825.2.bi.c	yes	8	5.c	odd	4	1
825.2.bi.c	yes	8	15.e	even	4	1
825.2.bi.c	yes	8	55.l	even	20	1
825.2.bi.c	yes	8	165.u	odd	20	1
825.2.bs.b		8	3.b	odd	2	1
825.2.bs.b		8	5.b	even	2	1
825.2.bs.b		8	33.f	even	10	1
825.2.bs.b		8	55.h	odd	10	1
825.2.bs.c		8	1.a	even	1	1	trivial
825.2.bs.c		8	11.d	odd	10	1	inner
825.2.bs.c		8	15.d	odd	2	1	inner
825.2.bs.c		8	165.r	even	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} + 5T_{2} + 5 $ T2^4 + 5*T2 + 5 acting on $S_{2}^{\mathrm{new}}(825, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 5 T + 5)^{2} $ (T^4 + 5*T + 5)^2
$3$	$ T^{8} - 4 T^{7} + \cdots + 81 $ T^8 - 4T^7 + 13T^6 - 30T^5 + 61T^4 - 90T^3 + 117T^2 - 108*T + 81
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + 160 T^{4} + \cdots + 6400 $ T^8 + 160T^4 + 1600T^2 + 6400
$11$	$ T^{8} + 4 T^{6} + \cdots + 14641 $ T^8 + 4T^6 - 74T^4 + 484*T^2 + 14641
$13$	$ T^{8} $ T^8
$17$	$ (T^{4} + 10 T^{3} + \cdots + 80)^{2} $ (T^4 + 10T^3 + 60T^2 + 120*T + 80)^2
$19$	$ (T^{4} + 15 T^{3} + \cdots + 605)^{2} $ (T^4 + 15T^3 + 105T^2 + 385*T + 605)^2
$23$	$ (T^{2} + 7 T - 19)^{4} $ (T^2 + 7*T - 19)^4
$29$	$ T^{8} + 25 T^{6} + \cdots + 23088025 $ T^8 + 25T^6 + 6610T^4 + 576600*T^2 + 23088025
$31$	$ (T^{4} - 5 T^{3} + \cdots + 625)^{2} $ (T^4 - 5T^3 + 25T^2 - 125*T + 625)^2
$37$	$ T^{8} + 11 T^{6} + \cdots + 2825761 $ T^8 + 11T^6 + 4561T^4 - 183229*T^2 + 2825761
$41$	$ T^{8} - 5 T^{6} + \cdots + 25 $ T^8 - 5T^6 + 85T^4 + 75*T^2 + 25
$43$	$ (T^{4} - 65 T^{2} + 605)^{2} $ (T^4 - 65*T^2 + 605)^2
$47$	$ (T^{4} + 13 T^{3} + \cdots + 961)^{2} $ (T^4 + 13T^3 + 94T^2 + 372*T + 961)^2
$53$	$ (T^{4} - 9 T^{3} + \cdots + 1681)^{2} $ (T^4 - 9T^3 + 46T^2 - 164*T + 1681)^2
$59$	$ T^{8} - 41 T^{6} + \cdots + 130321 $ T^8 - 41T^6 + 681T^4 - 361*T^2 + 130321
$61$	$ (T^{4} + 5 T^{3} + \cdots + 4205)^{2} $ (T^4 + 5T^3 - 15T^2 + 435*T + 4205)^2
$67$	$ (T^{4} + 163 T^{2} + 1681)^{2} $ (T^4 + 163*T^2 + 1681)^2
$71$	$ T^{8} + 80 T^{6} + \cdots + 160000 $ T^8 + 80T^6 + 18400T^4 - 88000*T^2 + 160000
$73$	$ T^{8} - 5 T^{6} + \cdots + 25 $ T^8 - 5T^6 + 85T^4 + 75*T^2 + 25
$79$	$ (T^{4} + 5 T^{3} + 125)^{2} $ (T^4 + 5*T^3 + 125)^2
$83$	$ (T^{4} + 20 T^{3} + \cdots + 80)^{2} $ (T^4 + 20T^3 + 280T^2 + 280*T + 80)^2
$89$	$ (T^{4} + 35 T^{2} + 25)^{2} $ (T^4 + 35*T^2 + 25)^2
$97$	$ T^{8} + \cdots + 6975757441 $ T^8 - 289T^6 + 83521T^4 - 24137569*T^2 + 6975757441
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.bs (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{20}^{6} + \zeta_{20}^{4} + \cdots + 1) q^{2}+ \cdots + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{4} + \cdots - 2) q^{9}+O(q^{10}) \) q + (-2z^6 + z^4 - z^2 + 1) q^2 + (z^7 - z^6 - z^5 - z^2 + 1) * q^3 + (-z^6 - z^2) * q^4 + (-z^6 + z^5 - z^4 - z^2 - z - 1) * q^6 + (-2z^7 - 2z) * q^7 + (z^6 - 2z^4 + 2z^2) * q^8 + (2z^7 - 2z^4 + z^2 - 2) * q^9	\( q + ( - 2 \zeta_{20}^{6} + \zeta_{20}^{4} + \cdots + 1) q^{2}+ \cdots + ( - 5 \zeta_{20}^{7} + \cdots + 2 \zeta_{20}) q^{99}+O(q^{100}) \) q + (-2z^6 + z^4 - z^2 + 1) q^2 + (z^7 - z^6 - z^5 - z^2 + 1) * q^3 + (-z^6 - z^2) * q^4 + (-z^6 + z^5 - z^4 - z^2 - z - 1) * q^6 + (-2z^7 - 2z) * q^7 + (z^6 - 2z^4 + 2z^2) * q^8 + (2z^7 - 2z^4 + z^2 - 2) * q^9 + (4z^7 - 3z^5 + 2z^3 - 2z) * q^11 + (z^6 + z^5 - z^4 - 2) * q^12 + (4z^7 - 4z^5 - 2z) q^14 + (-4z^6 + 3z^4 - 3z^2 + 4) q^16 + (-2z^6 - 2z^4 - 2z^2 - 2) q^17 + (3z^6 + 2z^5 - z^4 + 2z^3 - z^2 - 2) q^18 + (-z^6 + 2z^4 - 4z^2 - 2) * q^19 + (2z^7 - 2z^5 + 4z^4 + 2z^3 - 4z^2 - 4z + 2) * q^21 + (2z^7 + 4z^5 + z^3 - z) * q^22 + (-5z^6 + 5z^4 - 1) * q^23 + (2z^7 - 4z^5 - z^4 + 3z^3 - z + 1) q^24 + (3z^6 + z^5 - 4z^4 + 3z^3 + 4z^2 + z - 3) * q^27 + (4z^7 - 2z^5 + 2z^3 - 2z) * q^28 + (2z^7 - 5z^5 - 5z^3 + 2z) * q^29 + 5z^2 q^31 + (-5z^6 + z^4 - 6z^2 + 3) * q^32 + (3z^7 - 2z^6 + z^5 + 2z^3 + 3z^2 + z - 1) * q^33 + (6z^6 - 6z^4 - 8) * q^34 + (-2z^7 + 3z^6 + 2z^5 + z^2 + 2z - 1) * q^36 + (-z^7 - 5z^5 + 5z^3 + z) * q^37 + (8z^6 - 7z^4 + 7z^2 - 8) q^38 + (2z^5 - 2z^3 + z) * q^41 + (6z^7 - 2z^4 + 2z^3 + 6z^2 - 2z - 2) q^42 + (-5z^7 + 2z^5 + z^3 + 4z) q^43 + (-z^7 + 4z^5 + 3z) * q^44 + (2z^6 - 6z^4 + z^2 + 4) * q^46 + (2z^6 + 5z^2 - 5) * q^47 + (z^7 - 4z^6 + 2z^5 - z^4 - 2z^3 - 4z^2 - z) * q^48 + (z^6 - 5z^4 + 5z^2 - 1) * q^49 + (6z^6 + 2z^5 - 4z^4 + 2z^3 + 2z^2 - 8) q^51 + (z^4 + 6z^2 + 1) q^53 + (-5z^7 + 2z^6 + 4z^5 + 4z^4 - 3z^3 - 2z^2 + 8z + 1) q^54 + (-6z^7 + 8z^5 - 6z^3) q^56 + (-4z^7 + 4z^6 + 8z^5 + z^4 - 5z^3 + 2z^2 + 3z - 5) * q^57 + (z^7 - z^5 - 13z) q^58 + (-3z^7 - z^5 + z^3 + 3z) * q^59 + (-z^6 + 5z^4 + 5z^2 - 1) * q^61 + (-5z^6 + 5z^4 - 5z^2 + 10) q^62 + (2z^7 - 4z^6 + 6z^5 + 8z^4 - 4z^3 - 4z^2 + 2z + 4) q^63 + (-6z^4 + 5z^2 - 6) * q^64 + (-4z^7 - z^6 + 6z^5 + z^3 - 4z^2 + 6z + 5) * q^66 + (-7z^7 - z^5 - 7z^3) * q^67 + (8z^6 - 2z^4 + 4z^2 - 6) q^68 + (-6z^7 - 4z^6 + 6z^5 + z^2 - 5z - 1) * q^69 + (-6z^5 - 2z^3 - 6z) q^71 + (4z^7 - 3z^6 - 4z^5 + z^4 + 2z^3 + z^2 - 3) * q^72 + (-z^7 + z^5 + z^3 - z) * q^73 + (-7z^7 + 5z^5 - 12z^3 + 6z) * q^74 + (5z^6 - z^4 + 6z^2 - 3) * q^76 + (2z^6 + 8z^4 - 6z^2 + 8) q^77 + (-4z^6 + 3z^4 - 2z^2 + 1) q^79 + (-4z^7 - 4z^5 + z^4 + 4z^3 + 4z) * q^81 + (-z^5 + 3z^3 - z) q^82 + (-4z^6 - 6z^4 - 6z^2 - 4) q^83 + (2z^7 - 2z^6 + 2z^5 + 2z^3 + 2z^2 + 2z) * q^84 + (-9z^7 - 8z^3 + 8z) q^86 + (8z^7 + 5z^6 - 3z^5 - 9z^4 - 2z^3 + 14z^2 - 6z - 7) q^87 + (4z^7 - 8z^5 + 8z^3 - 5z) * q^88 + (3z^7 - 4z^5 + 3z^3) q^89 + (z^6 - 5z^4 + z^2) q^92 + (-5z^6 - 5z^5 + 5z^3 - 5z + 5) * q^93 + (5z^6 + 2z^4 + 2z^2 + 5) q^94 + (2z^7 + 2z^6 + 4z^5 - 4z^4 - 2z^3 - 3z^2 + z - 7) * q^96 + 17z^7 q^97 + (-3z^6 + 5z^4 - 8z^2 + 4) q^98 + (-5z^7 - 4z^6 + 2z^5 - 4z^4 + 4z^3 + 2z^2 + 2z) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 4 q^{4} - 10 q^{6} + 10 q^{8} - 10 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 - 4 * q^4 - 10 * q^6 + 10 * q^8 - 10 * q^9	\( 8 q + 4 q^{3} - 4 q^{4} - 10 q^{6} + 10 q^{8} - 10 q^{9} - 12 q^{12} + 12 q^{16} - 20 q^{17} - 10 q^{18} - 30 q^{19} - 28 q^{23} + 10 q^{24} - 2 q^{27} + 10 q^{31} - 6 q^{33} - 40 q^{34} - 20 q^{38} + 50 q^{46} - 26 q^{47} - 14 q^{48} + 14 q^{49} - 40 q^{51} + 18 q^{53} - 30 q^{57} - 10 q^{61} + 50 q^{62} - 26 q^{64} + 30 q^{66} - 20 q^{68} - 14 q^{69} - 30 q^{72} + 40 q^{77} - 10 q^{79} - 2 q^{81} - 40 q^{83} + 14 q^{92} + 30 q^{93} + 50 q^{94} - 50 q^{96} + 4 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 4 * q^4 - 10 * q^6 + 10 * q^8 - 10 * q^9 - 12 * q^12 + 12 * q^16 - 20 * q^17 - 10 * q^18 - 30 * q^19 - 28 * q^23 + 10 * q^24 - 2 * q^27 + 10 * q^31 - 6 * q^33 - 40 * q^34 - 20 * q^38 + 50 * q^46 - 26 * q^47 - 14 * q^48 + 14 * q^49 - 40 * q^51 + 18 * q^53 - 30 * q^57 - 10 * q^61 + 50 * q^62 - 26 * q^64 + 30 * q^66 - 20 * q^68 - 14 * q^69 - 30 * q^72 + 40 * q^77 - 10 * q^79 - 2 * q^81 - 40 * q^83 + 14 * q^92 + 30 * q^93 + 50 * q^94 - 50 * q^96 + 4 * q^99

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	825.2.bs.c

Level	$825$
Weight	$2$
Character orbit	825.bs
Analytic conductor	$6.588$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(n\)	\(376\)	\(551\)	\(727\)
\(\chi(n)\)	\(-\zeta_{20}^{4}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 74.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 5 T + 5)^{2} \) (T^4 + 5*T + 5)^2
$3$	\( T^{8} - 4 T^{7} + \cdots + 81 \) T^8 - 4T^7 + 13T^6 - 30T^5 + 61T^4 - 90T^3 + 117T^2 - 108*T + 81
$5$	\( T^{8} \) T^8
$7$	\( T^{8} + 160 T^{4} + \cdots + 6400 \) T^8 + 160T^4 + 1600T^2 + 6400
$11$	\( T^{8} + 4 T^{6} + \cdots + 14641 \) T^8 + 4T^6 - 74T^4 + 484*T^2 + 14641
$13$	\( T^{8} \) T^8
$17$	\( (T^{4} + 10 T^{3} + \cdots + 80)^{2} \) (T^4 + 10T^3 + 60T^2 + 120*T + 80)^2
$19$	\( (T^{4} + 15 T^{3} + \cdots + 605)^{2} \) (T^4 + 15T^3 + 105T^2 + 385*T + 605)^2
$23$	\( (T^{2} + 7 T - 19)^{4} \) (T^2 + 7*T - 19)^4
$29$	\( T^{8} + 25 T^{6} + \cdots + 23088025 \) T^8 + 25T^6 + 6610T^4 + 576600*T^2 + 23088025
$31$	\( (T^{4} - 5 T^{3} + \cdots + 625)^{2} \) (T^4 - 5T^3 + 25T^2 - 125*T + 625)^2
$37$	\( T^{8} + 11 T^{6} + \cdots + 2825761 \) T^8 + 11T^6 + 4561T^4 - 183229*T^2 + 2825761
$41$	\( T^{8} - 5 T^{6} + \cdots + 25 \) T^8 - 5T^6 + 85T^4 + 75*T^2 + 25
$43$	\( (T^{4} - 65 T^{2} + 605)^{2} \) (T^4 - 65*T^2 + 605)^2
$47$	\( (T^{4} + 13 T^{3} + \cdots + 961)^{2} \) (T^4 + 13T^3 + 94T^2 + 372*T + 961)^2
$53$	\( (T^{4} - 9 T^{3} + \cdots + 1681)^{2} \) (T^4 - 9T^3 + 46T^2 - 164*T + 1681)^2
$59$	\( T^{8} - 41 T^{6} + \cdots + 130321 \) T^8 - 41T^6 + 681T^4 - 361*T^2 + 130321
$61$	\( (T^{4} + 5 T^{3} + \cdots + 4205)^{2} \) (T^4 + 5T^3 - 15T^2 + 435*T + 4205)^2
$67$	\( (T^{4} + 163 T^{2} + 1681)^{2} \) (T^4 + 163*T^2 + 1681)^2
$71$	\( T^{8} + 80 T^{6} + \cdots + 160000 \) T^8 + 80T^6 + 18400T^4 - 88000*T^2 + 160000
$73$	\( T^{8} - 5 T^{6} + \cdots + 25 \) T^8 - 5T^6 + 85T^4 + 75*T^2 + 25
$79$	\( (T^{4} + 5 T^{3} + 125)^{2} \) (T^4 + 5*T^3 + 125)^2
$83$	\( (T^{4} + 20 T^{3} + \cdots + 80)^{2} \) (T^4 + 20T^3 + 280T^2 + 280*T + 80)^2
$89$	\( (T^{4} + 35 T^{2} + 25)^{2} \) (T^4 + 35*T^2 + 25)^2
$97$	\( T^{8} + \cdots + 6975757441 \) T^8 - 289T^6 + 83521T^4 - 24137569*T^2 + 6975757441
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