Newform orbit 315.2.t.a

• Label: 315.2.t.a
• Level: $315$
• Weight: $2$
• Character orbit: 315.t
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 1) q^{2} + (\zeta_{6} - 2) q^{3} - q^{4} + ( - \zeta_{6} + 1) q^{5} + 3 \zeta_{6} q^{6} + (2 \zeta_{6} + 1) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 2 q^{4} + q^{5} + 3 q^{6} + 4 q^{7} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(1\)	\(\zeta_{6}\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

101.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−

1.73205i

−1.50000

+

0.866025i

−1.00000

0.500000

−

0.866025i

1.50000

+

2.59808i

2.00000

+

1.73205i

−

1.73205i

1.50000

−

2.59808i

−1.50000

−

0.866025i

131.1

1.73205i

−1.50000

−

0.866025i

−1.00000

0.500000

+

0.866025i

1.50000

−

2.59808i

2.00000

−

1.73205i

1.73205i

1.50000

+

2.59808i

−1.50000

+

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.t.a	✓	2
3.b	odd	2	1		945.2.t.a		2
7.d	odd	6	1		315.2.be.a	yes	2
9.c	even	3	1		945.2.be.a		2
9.d	odd	6	1		315.2.be.a	yes	2
21.g	even	6	1		945.2.be.a		2
63.i	even	6	1	inner	315.2.t.a	✓	2
63.t	odd	6	1		945.2.t.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.t.a	✓	2	1.a	even	1	1	trivial
315.2.t.a	✓	2	63.i	even	6	1	inner
315.2.be.a	yes	2	7.d	odd	6	1
315.2.be.a	yes	2	9.d	odd	6	1
945.2.t.a		2	3.b	odd	2	1
945.2.t.a		2	63.t	odd	6	1
945.2.be.a		2	9.c	even	3	1
945.2.be.a		2	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3 \)
$3$	\( T^{2} + 3T + 3 \)
$5$	\( T^{2} - T + 1 \)
$7$	\( T^{2} - 4T + 7 \)
$11$	\( T^{2} + 6T + 12 \)