Newform orbit 245.3.i.c

• Label: 245.3.i.c
• Level: $245$
• Weight: $3$
• Character orbit: 245.i
• Analytic conductor: $6.676$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	245.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.67576647683\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.3317760000.2
Defining polynomial:	\( x^{8} - 25x^{4} + 625 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{4} \)
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{3} - \beta_1) q^{2} - \beta_{7} q^{3} + ( - 5 \beta_{2} + 5) q^{4} - \beta_{6} q^{5} + (\beta_{7} + 2 \beta_{5} - \beta_{4}) q^{6} + \beta_{3} q^{8} - \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{4} - 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 25x^{4} + 625 \) :

\(\beta_{1}\)	\(=\)	\( ( 3\nu^{2} ) / 5 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{4} ) / 25 \)
\(\beta_{3}\)	\(=\)	\( ( 3\nu^{6} ) / 125 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} + 125\nu ) / 125 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} + 10\nu^{3} - 25\nu ) / 25 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{7} + 250\nu ) / 125 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} + 5\nu^{5} - 25\nu^{3} ) / 125 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(\beta_{2}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

19.1

−2.15988	+	0.578737i
2.15988	−	0.578737i
0.578737	+	2.15988i
−0.578737	−	2.15988i
−2.15988	−	0.578737i
2.15988	+	0.578737i
0.578737	−	2.15988i
−0.578737	+	2.15988i

−2.59808

−

1.50000i

−1.58114

−

2.73861i

2.50000

+

4.33013i

4.89849

+

1.00240i

9.48683i

0

−

3.00000i

−0.500000

+

0.866025i

−11.2230

−

9.95205i

19.2

−2.59808

−

1.50000i

1.58114

+

2.73861i

2.50000

+

4.33013i

−4.89849

−

1.00240i

−

9.48683i

0

−

3.00000i

−0.500000

+

0.866025i

11.2230

+

9.95205i

19.3

2.59808

+

1.50000i

−1.58114

−

2.73861i

2.50000

+

4.33013i

−3.31735

−

3.74101i

−

9.48683i

0

3.00000i

−0.500000

+

0.866025i

−3.00721

−

14.6955i

19.4

2.59808

+

1.50000i

1.58114

+

2.73861i

2.50000

+

4.33013i

3.31735

+

3.74101i

9.48683i

0

3.00000i

−0.500000

+

0.866025i

3.00721

+

14.6955i

129.1

−2.59808

+

1.50000i

−1.58114

+

2.73861i

2.50000

−

4.33013i

4.89849

−

1.00240i

−

9.48683i

0

3.00000i

−0.500000

−

0.866025i

−11.2230

+

9.95205i

129.2

−2.59808

+

1.50000i

1.58114

−

2.73861i

2.50000

−

4.33013i

−4.89849

+

1.00240i

9.48683i

0

3.00000i

−0.500000

−

0.866025i

11.2230

−

9.95205i

129.3

2.59808

−

1.50000i

−1.58114

+

2.73861i

2.50000

−

4.33013i

−3.31735

+

3.74101i

9.48683i

0

−

3.00000i

−0.500000

−

0.866025i

−3.00721

+

14.6955i

129.4

2.59808

−

1.50000i

1.58114

−

2.73861i

2.50000

−

4.33013i

3.31735

−

3.74101i

−

9.48683i

0

−

3.00000i

−0.500000

−

0.866025i

3.00721

−

14.6955i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
35.c	odd	2	1	inner
35.i	odd	6	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	245.3.i.c		8
5.b	even	2	1	inner	245.3.i.c		8
7.b	odd	2	1	inner	245.3.i.c		8
7.c	even	3	1		35.3.c.c	✓	4
7.c	even	3	1	inner	245.3.i.c		8
7.d	odd	6	1		35.3.c.c	✓	4
7.d	odd	6	1	inner	245.3.i.c		8
21.g	even	6	1		315.3.e.c		4
21.h	odd	6	1		315.3.e.c		4
28.f	even	6	1		560.3.p.f		4
28.g	odd	6	1		560.3.p.f		4
35.c	odd	2	1	inner	245.3.i.c		8
35.i	odd	6	1		35.3.c.c	✓	4
35.i	odd	6	1	inner	245.3.i.c		8
35.j	even	6	1		35.3.c.c	✓	4
35.j	even	6	1	inner	245.3.i.c		8
35.k	even	12	1		175.3.d.b		2
35.k	even	12	1		175.3.d.h		2
35.l	odd	12	1		175.3.d.b		2
35.l	odd	12	1		175.3.d.h		2
105.o	odd	6	1		315.3.e.c		4
105.p	even	6	1		315.3.e.c		4
140.p	odd	6	1		560.3.p.f		4
140.s	even	6	1		560.3.p.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.3.c.c	✓	4	7.c	even	3	1
35.3.c.c	✓	4	7.d	odd	6	1
35.3.c.c	✓	4	35.i	odd	6	1
35.3.c.c	✓	4	35.j	even	6	1
175.3.d.b		2	35.k	even	12	1
175.3.d.b		2	35.l	odd	12	1
175.3.d.h		2	35.k	even	12	1
175.3.d.h		2	35.l	odd	12	1
245.3.i.c		8	1.a	even	1	1	trivial
245.3.i.c		8	5.b	even	2	1	inner
245.3.i.c		8	7.b	odd	2	1	inner
245.3.i.c		8	7.c	even	3	1	inner
245.3.i.c		8	7.d	odd	6	1	inner
245.3.i.c		8	35.c	odd	2	1	inner
245.3.i.c		8	35.i	odd	6	1	inner
245.3.i.c		8	35.j	even	6	1	inner
315.3.e.c		4	21.g	even	6	1
315.3.e.c		4	21.h	odd	6	1
315.3.e.c		4	105.o	odd	6	1
315.3.e.c		4	105.p	even	6	1
560.3.p.f		4	28.f	even	6	1
560.3.p.f		4	28.g	odd	6	1
560.3.p.f		4	140.p	odd	6	1
560.3.p.f		4	140.s	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} - 9T_{2}^{2} + 81 \)

\( T_{3}^{4} + 10T_{3}^{2} + 100 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - 9 T^{2} + 81)^{2} \)
$3$	\( (T^{4} + 10 T^{2} + 100)^{2} \)
$5$	T^8 - 40*T^6 + 975*T^4 - 25000*T^2 + 390625
$7$	\( T^{8} \)
$11$	\( (T^{2} + 14 T + 196)^{4} \)