Properties

Label 405.2.a.f
Level $405$
Weight $2$
Character orbit 405.a
Self dual yes
Analytic conductor $3.234$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} + 2 q^{4} + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{10} + 5 q^{11} + 4 q^{13} - 4 q^{16} - 4 q^{17} - 5 q^{19} + 2 q^{20} + 10 q^{22} + 6 q^{23} + q^{25} + 8 q^{26} - 5 q^{29} - 9 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{37} - 10 q^{38} + 7 q^{41} - 2 q^{43} + 10 q^{44} + 12 q^{46} + 2 q^{47} - 7 q^{49} + 2 q^{50} + 8 q^{52} + 8 q^{53} + 5 q^{55} - 10 q^{58} - q^{59} - 2 q^{61} - 18 q^{62} - 8 q^{64} + 4 q^{65} + 6 q^{67} - 8 q^{68} + q^{71} - 8 q^{73} - 20 q^{74} - 10 q^{76} + 12 q^{79} - 4 q^{80} + 14 q^{82} + 6 q^{83} - 4 q^{85} - 4 q^{86} - 9 q^{89} + 12 q^{92} + 4 q^{94} - 5 q^{95} + 14 q^{97} - 14 q^{98}+O(q^{100}) \) 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
2.00000 0 2.00000 1.00000 0 0 0 0 2.00000
\(n\): 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.f yes 1
3.b odd 2 1 405.2.a.a 1
4.b odd 2 1 6480.2.a.r 1
5.b even 2 1 2025.2.a.a 1
5.c odd 4 2 2025.2.b.b 2
9.c even 3 2 405.2.e.a 2
9.d odd 6 2 405.2.e.g 2
12.b even 2 1 6480.2.a.f 1
15.d odd 2 1 2025.2.a.f 1
15.e even 4 2 2025.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.a 1 3.b odd 2 1
405.2.a.f yes 1 1.a even 1 1 trivial
405.2.e.a 2 9.c even 3 2
405.2.e.g 2 9.d odd 6 2
2025.2.a.a 1 5.b even 2 1
2025.2.a.f 1 15.d odd 2 1
2025.2.b.a 2 15.e even 4 2
2025.2.b.b 2 5.c odd 4 2
6480.2.a.f 1 12.b even 2 1
6480.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(405))\):

\( T_{2} - 2 \) Copy content Toggle raw display
\( T_{11} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 5 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T + 5 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 5 \) Copy content Toggle raw display
$31$ \( T + 9 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 7 \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T - 8 \) Copy content Toggle raw display
$59$ \( T + 1 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 6 \) Copy content Toggle raw display
$71$ \( T - 1 \) Copy content Toggle raw display
$73$ \( T + 8 \) Copy content Toggle raw display
$79$ \( T - 12 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T + 9 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
show more
show less