Newform orbit 245.3.i.a

• Label: 245.3.i.a
• Level: $245$
• Weight: $3$
• Character orbit: 245.i
• Analytic conductor: $6.676$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -35
• Inner twists: $4$

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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} + 8 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 4 q^{4} + 5 q^{5} + 8 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

19.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−0.500000

−

0.866025i

−2.00000

−

3.46410i

2.50000

−

4.33013i

0

0

0

4.00000

−

6.92820i

0

129.1

0

−0.500000

+

0.866025i

−2.00000

+

3.46410i

2.50000

+

4.33013i

0

0

0

4.00000

+

6.92820i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
35.c	odd	2	1	CM by \(\Q(\sqrt{-35}) \)
7.c	even	3	1	inner
35.i	odd	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.a		2
5.b	even	2	1		245.3.i.b		2
7.b	odd	2	1		245.3.i.b		2
7.c	even	3	1		35.3.c.b	yes	1
7.c	even	3	1	inner	245.3.i.a		2
7.d	odd	6	1		35.3.c.a	✓	1
7.d	odd	6	1		245.3.i.b		2
21.g	even	6	1		315.3.e.a		1
21.h	odd	6	1		315.3.e.b		1
28.f	even	6	1		560.3.p.b		1
28.g	odd	6	1		560.3.p.a		1
35.c	odd	2	1	CM	245.3.i.a		2
35.i	odd	6	1		35.3.c.b	yes	1
35.i	odd	6	1	inner	245.3.i.a		2
35.j	even	6	1		35.3.c.a	✓	1
35.j	even	6	1		245.3.i.b		2
35.k	even	12	2		175.3.d.e		2
35.l	odd	12	2		175.3.d.e		2
105.o	odd	6	1		315.3.e.a		1
105.p	even	6	1		315.3.e.b		1
140.p	odd	6	1		560.3.p.b		1
140.s	even	6	1		560.3.p.a		1

    

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

Hecke kernels

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

\( T_{2} \)

\( T_{3}^{2} + T_{3} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T + 1 \)
$5$	\( T^{2} - 5T + 25 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 13T + 169 \)