Newform orbit 115.3.d.a

• Label: 115.3.d.a
• Level: $115$
• Weight: $3$
• Character orbit: 115.d
• Analytic conductor: $3.134$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	115.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13352304014\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial:	\( x^{6} - 2x^{5} - 34x^{4} + 50x^{3} + 690x^{2} - 600x + 4725 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{2} + \beta_{4} q^{3} - 3 q^{4} - \beta_{3} q^{5} - \beta_{4} q^{6} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + 7 q^{8} + (\beta_{4} + \beta_1 + 8) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 2 q^{3} - 18 q^{4} - 2 q^{6} + 42 q^{8} + 48 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 34x^{4} + 50x^{3} + 690x^{2} - 600x + 4725 \) :

\(\beta_{1}\)	\(=\)	\( ( 62\nu^{5} - 5599\nu^{4} + 5812\nu^{3} + 221635\nu^{2} - 193350\nu - 3002445 ) / 249285 \)
\(\beta_{2}\)	\(=\)	\( ( 7\nu^{5} + 172\nu^{4} - 416\nu^{3} + 1167\nu^{2} + 4975\nu - 2585 ) / 16619 \)
\(\beta_{3}\)	\(=\)	\( ( 272\nu^{5} - 439\nu^{4} - 6668\nu^{3} + 7360\nu^{2} + 205185\nu - 88575 ) / 249285 \)
\(\beta_{4}\)	\(=\)	\( ( -272\nu^{5} + 439\nu^{4} + 6668\nu^{3} - 7360\nu^{2} + 44100\nu + 88575 ) / 249285 \)
\(\beta_{5}\)	\(=\)	\( ( -551\nu^{5} + 706\nu^{4} + 30371\nu^{3} - 32506\nu^{2} - 581535\nu + 262830 ) / 249285 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(-1\)

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.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

91.1

−4.73103	+	2.23607i
−4.73103	−	2.23607i
0.396209	+	2.23607i
0.396209	−	2.23607i
5.33482	+	2.23607i
5.33482	−	2.23607i

−1.00000

−4.73103

−3.00000

−

2.23607i

4.73103

7.39757i

7.00000

13.3827

2.23607i

91.2

−1.00000

−4.73103

−3.00000

2.23607i

4.73103

−

7.39757i

7.00000

13.3827

−

2.23607i

91.3

−1.00000

0.396209

−3.00000

−

2.23607i

−0.396209

8.16531i

7.00000

−8.84302

2.23607i

91.4

−1.00000

0.396209

−3.00000

2.23607i

−0.396209

−

8.16531i

7.00000

−8.84302

−

2.23607i

91.5

−1.00000

5.33482

−3.00000

−

2.23607i

−5.33482

−

13.3268i

7.00000

19.4603

2.23607i

91.6

−1.00000

5.33482

−3.00000

2.23607i

−5.33482

13.3268i

7.00000

19.4603

−

2.23607i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	115.3.d.a	✓	6
3.b	odd	2	1		1035.3.g.a		6
4.b	odd	2	1		1840.3.k.a		6
5.b	even	2	1		575.3.d.e		6
5.c	odd	4	2		575.3.c.c		12
23.b	odd	2	1	inner	115.3.d.a	✓	6
69.c	even	2	1		1035.3.g.a		6
92.b	even	2	1		1840.3.k.a		6
115.c	odd	2	1		575.3.d.e		6
115.e	even	4	2		575.3.c.c		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.3.d.a	✓	6	1.a	even	1	1	trivial
115.3.d.a	✓	6	23.b	odd	2	1	inner
575.3.c.c		12	5.c	odd	4	2
575.3.c.c		12	115.e	even	4	2
575.3.d.e		6	5.b	even	2	1
575.3.d.e		6	115.c	odd	2	1
1035.3.g.a		6	3.b	odd	2	1
1035.3.g.a		6	69.c	even	2	1
1840.3.k.a		6	4.b	odd	2	1
1840.3.k.a		6	92.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(115, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{6} \)
$3$	\( (T^{3} - T^{2} - 25 T + 10)^{2} \)
$5$	\( (T^{2} + 5)^{3} \)
$7$	T^6 + 299*T^4 + 25209*T^2 + 648000
$11$	T^6 + 731*T^4 + 157509*T^2 + 8632980