Properties

Label 180.4.a.e
Level $180$
Weight $4$
Character orbit 180.a
Self dual yes
Analytic conductor $10.620$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 180.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6203438010\)
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

\(f(q)\) \(=\) \( q + 5 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + 2 q^{7} + 30 q^{11} - 4 q^{13} + 90 q^{17} - 28 q^{19} + 120 q^{23} + 25 q^{25} + 210 q^{29} - 4 q^{31} + 10 q^{35} + 200 q^{37} + 240 q^{41} - 136 q^{43} - 120 q^{47} - 339 q^{49} - 30 q^{53} + 150 q^{55} - 450 q^{59} - 166 q^{61} - 20 q^{65} + 908 q^{67} - 1020 q^{71} - 250 q^{73} + 60 q^{77} - 916 q^{79} - 1140 q^{83} + 450 q^{85} - 420 q^{89} - 8 q^{91} - 140 q^{95} + 1538 q^{97}+O(q^{100}) \) 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 180.4.a.e yes 1
3.b odd 2 1 180.4.a.b 1
4.b odd 2 1 720.4.a.w 1
5.b even 2 1 900.4.a.j 1
5.c odd 4 2 900.4.d.i 2
9.c even 3 2 1620.4.i.c 2
9.d odd 6 2 1620.4.i.i 2
12.b even 2 1 720.4.a.h 1
15.d odd 2 1 900.4.a.i 1
15.e even 4 2 900.4.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.a.b 1 3.b odd 2 1
180.4.a.e yes 1 1.a even 1 1 trivial
720.4.a.h 1 12.b even 2 1
720.4.a.w 1 4.b odd 2 1
900.4.a.i 1 15.d odd 2 1
900.4.a.j 1 5.b even 2 1
900.4.d.d 2 15.e even 4 2
900.4.d.i 2 5.c odd 4 2
1620.4.i.c 2 9.c even 3 2
1620.4.i.i 2 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(180))\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} - 30 \) 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
$11$ \( T - 30 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T - 90 \) Copy content Toggle raw display
$19$ \( T + 28 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T - 210 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 200 \) Copy content Toggle raw display
$41$ \( T - 240 \) Copy content Toggle raw display
$43$ \( T + 136 \) Copy content Toggle raw display
$47$ \( T + 120 \) Copy content Toggle raw display
$53$ \( T + 30 \) Copy content Toggle raw display
$59$ \( T + 450 \) Copy content Toggle raw display
$61$ \( T + 166 \) Copy content Toggle raw display
$67$ \( T - 908 \) Copy content Toggle raw display
$71$ \( T + 1020 \) Copy content Toggle raw display
$73$ \( T + 250 \) Copy content Toggle raw display
$79$ \( T + 916 \) Copy content Toggle raw display
$83$ \( T + 1140 \) Copy content Toggle raw display
$89$ \( T + 420 \) Copy content Toggle raw display
$97$ \( T - 1538 \) Copy content Toggle raw display
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