Properties

Label 360.4.a.d
Level $360$
Weight $4$
Character orbit 360.a
Self dual yes
Analytic conductor $21.241$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.2406876021\)
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

\(f(q)\) \(=\) \( q - 5 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + 2 q^{7} + 34 q^{11} - 68 q^{13} + 38 q^{17} + 4 q^{19} - 152 q^{23} + 25 q^{25} + 46 q^{29} - 260 q^{31} - 10 q^{35} - 312 q^{37} - 48 q^{41} - 200 q^{43} - 104 q^{47} - 339 q^{49} + 414 q^{53} - 170 q^{55} + 2 q^{59} - 38 q^{61} + 340 q^{65} - 244 q^{67} - 708 q^{71} - 378 q^{73} + 68 q^{77} - 852 q^{79} - 844 q^{83} - 190 q^{85} + 1380 q^{89} - 136 q^{91} - 20 q^{95} + 514 q^{97}+O(q^{100}) \) 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
0 0 0 −5.00000 0 2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 360.4.a.d 1
3.b odd 2 1 360.4.a.k yes 1
4.b odd 2 1 720.4.a.g 1
5.b even 2 1 1800.4.a.r 1
5.c odd 4 2 1800.4.f.t 2
12.b even 2 1 720.4.a.x 1
15.d odd 2 1 1800.4.a.q 1
15.e even 4 2 1800.4.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.d 1 1.a even 1 1 trivial
360.4.a.k yes 1 3.b odd 2 1
720.4.a.g 1 4.b odd 2 1
720.4.a.x 1 12.b even 2 1
1800.4.a.q 1 15.d odd 2 1
1800.4.a.r 1 5.b even 2 1
1800.4.f.f 2 15.e even 4 2
1800.4.f.t 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(360))\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} - 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 34 \) Copy content Toggle raw display
$13$ \( T + 68 \) Copy content Toggle raw display
$17$ \( T - 38 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 152 \) Copy content Toggle raw display
$29$ \( T - 46 \) Copy content Toggle raw display
$31$ \( T + 260 \) Copy content Toggle raw display
$37$ \( T + 312 \) Copy content Toggle raw display
$41$ \( T + 48 \) Copy content Toggle raw display
$43$ \( T + 200 \) Copy content Toggle raw display
$47$ \( T + 104 \) Copy content Toggle raw display
$53$ \( T - 414 \) Copy content Toggle raw display
$59$ \( T - 2 \) Copy content Toggle raw display
$61$ \( T + 38 \) Copy content Toggle raw display
$67$ \( T + 244 \) Copy content Toggle raw display
$71$ \( T + 708 \) Copy content Toggle raw display
$73$ \( T + 378 \) Copy content Toggle raw display
$79$ \( T + 852 \) Copy content Toggle raw display
$83$ \( T + 844 \) Copy content Toggle raw display
$89$ \( T - 1380 \) Copy content Toggle raw display
$97$ \( T - 514 \) Copy content Toggle raw display
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