Properties

Label 405.2.a.b
Level $405$
Weight $2$
Character orbit 405.a
Self dual yes
Analytic conductor $3.234$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + q^{5} - 3 q^{7} + 3 q^{8} - q^{10} + 2 q^{11} - 2 q^{13} + 3 q^{14} - q^{16} - 4 q^{17} - 8 q^{19} - q^{20} - 2 q^{22} - 3 q^{23} + q^{25} + 2 q^{26} + 3 q^{28} + q^{29} - 5 q^{32} + 4 q^{34} - 3 q^{35} - 4 q^{37} + 8 q^{38} + 3 q^{40} - 5 q^{41} - 8 q^{43} - 2 q^{44} + 3 q^{46} - 7 q^{47} + 2 q^{49} - q^{50} + 2 q^{52} + 2 q^{53} + 2 q^{55} - 9 q^{56} - q^{58} + 14 q^{59} + 7 q^{61} + 7 q^{64} - 2 q^{65} - 3 q^{67} + 4 q^{68} + 3 q^{70} - 2 q^{71} + 4 q^{73} + 4 q^{74} + 8 q^{76} - 6 q^{77} - 6 q^{79} - q^{80} + 5 q^{82} - 9 q^{83} - 4 q^{85} + 8 q^{86} + 6 q^{88} + 15 q^{89} + 6 q^{91} + 3 q^{92} + 7 q^{94} - 8 q^{95} + 2 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

Atkin-Lehner signs

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

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 405.2.a.b 1
3.b odd 2 1 405.2.a.e 1
4.b odd 2 1 6480.2.a.x 1
5.b even 2 1 2025.2.a.e 1
5.c odd 4 2 2025.2.b.d 2
9.c even 3 2 135.2.e.a 2
9.d odd 6 2 45.2.e.a 2
12.b even 2 1 6480.2.a.k 1
15.d odd 2 1 2025.2.a.b 1
15.e even 4 2 2025.2.b.c 2
36.f odd 6 2 2160.2.q.a 2
36.h even 6 2 720.2.q.d 2
45.h odd 6 2 225.2.e.a 2
45.j even 6 2 675.2.e.a 2
45.k odd 12 4 675.2.k.a 4
45.l even 12 4 225.2.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.a 2 9.d odd 6 2
135.2.e.a 2 9.c even 3 2
225.2.e.a 2 45.h odd 6 2
225.2.k.a 4 45.l even 12 4
405.2.a.b 1 1.a even 1 1 trivial
405.2.a.e 1 3.b odd 2 1
675.2.e.a 2 45.j even 6 2
675.2.k.a 4 45.k odd 12 4
720.2.q.d 2 36.h even 6 2
2025.2.a.b 1 15.d odd 2 1
2025.2.a.e 1 5.b even 2 1
2025.2.b.c 2 15.e even 4 2
2025.2.b.d 2 5.c odd 4 2
2160.2.q.a 2 36.f odd 6 2
6480.2.a.k 1 12.b even 2 1
6480.2.a.x 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(405))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 3 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 4 \) Copy content Toggle raw display
$41$ \( T + 5 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T + 7 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T - 14 \) Copy content Toggle raw display
$61$ \( T - 7 \) Copy content Toggle raw display
$67$ \( T + 3 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T + 6 \) Copy content Toggle raw display
$83$ \( T + 9 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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