Newform orbit 245.4.e.e

• Label: 245.4.e.e
• Level: $245$
• Weight: $4$
• Character orbit: 245.e
• Analytic conductor: $14.455$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.4554679514\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{2} + ( - 8 \zeta_{6} + 8) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} + 5 \zeta_{6} q^{5} - 8 q^{6} - 15 q^{8} - 37 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 8 q^{3} + 7 q^{4} + 5 q^{5} - 16 q^{6} - 30 q^{8} - 37 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

116.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

4.00000

−

6.92820i

3.50000

−

6.06218i

2.50000

+

4.33013i

−8.00000

0

−15.0000

−18.5000

−

32.0429i

2.50000

−

4.33013i

226.1

−0.500000

+

0.866025i

4.00000

+

6.92820i

3.50000

+

6.06218i

2.50000

−

4.33013i

−8.00000

0

−15.0000

−18.5000

+

32.0429i

2.50000

+

4.33013i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	245.4.e.e		2
7.b	odd	2	1		245.4.e.b		2
7.c	even	3	1		35.4.a.a	✓	1
7.c	even	3	1	inner	245.4.e.e		2
7.d	odd	6	1		245.4.a.d		1
7.d	odd	6	1		245.4.e.b		2
21.g	even	6	1		2205.4.a.i		1
21.h	odd	6	1		315.4.a.c		1
28.g	odd	6	1		560.4.a.p		1
35.i	odd	6	1		1225.4.a.e		1
35.j	even	6	1		175.4.a.a		1
35.l	odd	12	2		175.4.b.a		2
56.k	odd	6	1		2240.4.a.b		1
56.p	even	6	1		2240.4.a.bk		1
105.o	odd	6	1		1575.4.a.g		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.a.a	✓	1	7.c	even	3	1
175.4.a.a		1	35.j	even	6	1
175.4.b.a		2	35.l	odd	12	2
245.4.a.d		1	7.d	odd	6	1
245.4.e.b		2	7.b	odd	2	1
245.4.e.b		2	7.d	odd	6	1
245.4.e.e		2	1.a	even	1	1	trivial
245.4.e.e		2	7.c	even	3	1	inner
315.4.a.c		1	21.h	odd	6	1
560.4.a.p		1	28.g	odd	6	1
1225.4.a.e		1	35.i	odd	6	1
1575.4.a.g		1	105.o	odd	6	1
2205.4.a.i		1	21.g	even	6	1
2240.4.a.b		1	56.k	odd	6	1
2240.4.a.bk		1	56.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \)

\( T_{3}^{2} - 8T_{3} + 64 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} - 8T + 64 \)
$5$	\( T^{2} - 5T + 25 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 12T + 144 \)