Properties

Label 315.2.a.a
Level 315
Weight 2
Character orbit 315.a
Self dual yes
Analytic conductor 2.515
Analytic rank 1
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} - q^{5} + q^{7} + 3q^{8} + O(q^{10}) \) \( q - q^{2} - q^{4} - q^{5} + q^{7} + 3q^{8} + q^{10} - 6q^{13} - q^{14} - q^{16} - 2q^{17} - 8q^{19} + q^{20} - 8q^{23} + q^{25} + 6q^{26} - q^{28} + 2q^{29} + 4q^{31} - 5q^{32} + 2q^{34} - q^{35} - 2q^{37} + 8q^{38} - 3q^{40} + 6q^{41} + 4q^{43} + 8q^{46} - 8q^{47} + q^{49} - q^{50} + 6q^{52} - 10q^{53} + 3q^{56} - 2q^{58} - 4q^{59} - 2q^{61} - 4q^{62} + 7q^{64} + 6q^{65} + 4q^{67} + 2q^{68} + q^{70} + 12q^{71} - 2q^{73} + 2q^{74} + 8q^{76} + 8q^{79} + q^{80} - 6q^{82} + 4q^{83} + 2q^{85} - 4q^{86} + 6q^{89} - 6q^{91} + 8q^{92} + 8q^{94} + 8q^{95} - 18q^{97} - q^{98} + O(q^{100}) \)

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
−1.00000 0 −1.00000 −1.00000 0 1.00000 3.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.a 1
3.b odd 2 1 105.2.a.a 1
4.b odd 2 1 5040.2.a.d 1
5.b even 2 1 1575.2.a.h 1
5.c odd 4 2 1575.2.d.b 2
7.b odd 2 1 2205.2.a.b 1
12.b even 2 1 1680.2.a.f 1
15.d odd 2 1 525.2.a.a 1
15.e even 4 2 525.2.d.b 2
21.c even 2 1 735.2.a.f 1
21.g even 6 2 735.2.i.b 2
21.h odd 6 2 735.2.i.a 2
24.f even 2 1 6720.2.a.bk 1
24.h odd 2 1 6720.2.a.p 1
60.h even 2 1 8400.2.a.co 1
105.g even 2 1 3675.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 3.b odd 2 1
315.2.a.a 1 1.a even 1 1 trivial
525.2.a.a 1 15.d odd 2 1
525.2.d.b 2 15.e even 4 2
735.2.a.f 1 21.c even 2 1
735.2.i.a 2 21.h odd 6 2
735.2.i.b 2 21.g even 6 2
1575.2.a.h 1 5.b even 2 1
1575.2.d.b 2 5.c odd 4 2
1680.2.a.f 1 12.b even 2 1
2205.2.a.b 1 7.b odd 2 1
3675.2.a.f 1 105.g even 2 1
5040.2.a.d 1 4.b odd 2 1
6720.2.a.p 1 24.h odd 2 1
6720.2.a.bk 1 24.f even 2 1
8400.2.a.co 1 60.h even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\).