Properties

Label 405.4.a.a
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 5 q^{2} + 17 q^{4} + 5 q^{5} + 9 q^{7} - 45 q^{8} + O(q^{10}) \) \( q - 5 q^{2} + 17 q^{4} + 5 q^{5} + 9 q^{7} - 45 q^{8} - 25 q^{10} - 8 q^{11} + 43 q^{13} - 45 q^{14} + 89 q^{16} - 122 q^{17} - 59 q^{19} + 85 q^{20} + 40 q^{22} - 213 q^{23} + 25 q^{25} - 215 q^{26} + 153 q^{28} + 224 q^{29} - 36 q^{31} - 85 q^{32} + 610 q^{34} + 45 q^{35} + 206 q^{37} + 295 q^{38} - 225 q^{40} + 413 q^{41} - 392 q^{43} - 136 q^{44} + 1065 q^{46} - 311 q^{47} - 262 q^{49} - 125 q^{50} + 731 q^{52} - 377 q^{53} - 40 q^{55} - 405 q^{56} - 1120 q^{58} + 337 q^{59} + 40 q^{61} + 180 q^{62} - 287 q^{64} + 215 q^{65} + 348 q^{67} - 2074 q^{68} - 225 q^{70} + 62 q^{71} - 1214 q^{73} - 1030 q^{74} - 1003 q^{76} - 72 q^{77} - 294 q^{79} + 445 q^{80} - 2065 q^{82} + 534 q^{83} - 610 q^{85} + 1960 q^{86} + 360 q^{88} - 810 q^{89} + 387 q^{91} - 3621 q^{92} + 1555 q^{94} - 295 q^{95} - 928 q^{97} + 1310 q^{98} + O(q^{100}) \)

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
−5.00000 0 17.0000 5.00000 0 9.00000 −45.0000 0 −25.0000
\(n\): 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 405.4.a.a 1
3.b odd 2 1 405.4.a.b yes 1
5.b even 2 1 2025.4.a.f 1
9.c even 3 2 405.4.e.m 2
9.d odd 6 2 405.4.e.a 2
15.d odd 2 1 2025.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.a 1 1.a even 1 1 trivial
405.4.a.b yes 1 3.b odd 2 1
405.4.e.a 2 9.d odd 6 2
405.4.e.m 2 9.c even 3 2
2025.4.a.a 1 15.d odd 2 1
2025.4.a.f 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5 + T \)
$3$ \( T \)
$5$ \( -5 + T \)
$7$ \( -9 + T \)
$11$ \( 8 + T \)
$13$ \( -43 + T \)
$17$ \( 122 + T \)
$19$ \( 59 + T \)
$23$ \( 213 + T \)
$29$ \( -224 + T \)
$31$ \( 36 + T \)
$37$ \( -206 + T \)
$41$ \( -413 + T \)
$43$ \( 392 + T \)
$47$ \( 311 + T \)
$53$ \( 377 + T \)
$59$ \( -337 + T \)
$61$ \( -40 + T \)
$67$ \( -348 + T \)
$71$ \( -62 + T \)
$73$ \( 1214 + T \)
$79$ \( 294 + T \)
$83$ \( -534 + T \)
$89$ \( 810 + T \)
$97$ \( 928 + T \)
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