Properties

Label 315.2.a.b
Level $315$
Weight $2$
Character orbit 315.a
Self dual yes
Analytic conductor $2.515$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{4} + q^{5} + q^{7} + O(q^{10}) \) \( q - 2q^{4} + q^{5} + q^{7} + 3q^{11} + 5q^{13} + 4q^{16} - 3q^{17} + 2q^{19} - 2q^{20} + 6q^{23} + q^{25} - 2q^{28} - 3q^{29} - 4q^{31} + q^{35} + 2q^{37} + 12q^{41} - 10q^{43} - 6q^{44} - 9q^{47} + q^{49} - 10q^{52} - 12q^{53} + 3q^{55} + 8q^{61} - 8q^{64} + 5q^{65} - 4q^{67} + 6q^{68} + 2q^{73} - 4q^{76} + 3q^{77} - q^{79} + 4q^{80} - 12q^{83} - 3q^{85} + 12q^{89} + 5q^{91} - 12q^{92} + 2q^{95} - q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 −2.00000 1.00000 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.a.b 1
3.b odd 2 1 35.2.a.a 1
4.b odd 2 1 5040.2.a.v 1
5.b even 2 1 1575.2.a.f 1
5.c odd 4 2 1575.2.d.c 2
7.b odd 2 1 2205.2.a.e 1
12.b even 2 1 560.2.a.b 1
15.d odd 2 1 175.2.a.b 1
15.e even 4 2 175.2.b.a 2
21.c even 2 1 245.2.a.c 1
21.g even 6 2 245.2.e.b 2
21.h odd 6 2 245.2.e.a 2
24.f even 2 1 2240.2.a.u 1
24.h odd 2 1 2240.2.a.k 1
33.d even 2 1 4235.2.a.c 1
39.d odd 2 1 5915.2.a.f 1
60.h even 2 1 2800.2.a.z 1
60.l odd 4 2 2800.2.g.l 2
84.h odd 2 1 3920.2.a.ba 1
105.g even 2 1 1225.2.a.e 1
105.k odd 4 2 1225.2.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 3.b odd 2 1
175.2.a.b 1 15.d odd 2 1
175.2.b.a 2 15.e even 4 2
245.2.a.c 1 21.c even 2 1
245.2.e.a 2 21.h odd 6 2
245.2.e.b 2 21.g even 6 2
315.2.a.b 1 1.a even 1 1 trivial
560.2.a.b 1 12.b even 2 1
1225.2.a.e 1 105.g even 2 1
1225.2.b.d 2 105.k odd 4 2
1575.2.a.f 1 5.b even 2 1
1575.2.d.c 2 5.c odd 4 2
2205.2.a.e 1 7.b odd 2 1
2240.2.a.k 1 24.h odd 2 1
2240.2.a.u 1 24.f even 2 1
2800.2.a.z 1 60.h even 2 1
2800.2.g.l 2 60.l odd 4 2
3920.2.a.ba 1 84.h odd 2 1
4235.2.a.c 1 33.d even 2 1
5040.2.a.v 1 4.b odd 2 1
5915.2.a.f 1 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( -1 + T \)
$11$ \( -3 + T \)
$13$ \( -5 + T \)
$17$ \( 3 + T \)
$19$ \( -2 + T \)
$23$ \( -6 + T \)
$29$ \( 3 + T \)
$31$ \( 4 + T \)
$37$ \( -2 + T \)
$41$ \( -12 + T \)
$43$ \( 10 + T \)
$47$ \( 9 + T \)
$53$ \( 12 + T \)
$59$ \( T \)
$61$ \( -8 + T \)
$67$ \( 4 + T \)
$71$ \( T \)
$73$ \( -2 + T \)
$79$ \( 1 + T \)
$83$ \( 12 + T \)
$89$ \( -12 + T \)
$97$ \( 1 + T \)
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