Properties

Label 2160.4.a.e
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + 4 q^{7} - 42 q^{11} + 20 q^{13} + 93 q^{17} - 59 q^{19} - 9 q^{23} + 25 q^{25} + 120 q^{29} - 47 q^{31} - 20 q^{35} - 262 q^{37} + 126 q^{41} + 178 q^{43} - 144 q^{47} - 327 q^{49} + 741 q^{53} + 210 q^{55} + 444 q^{59} + 221 q^{61} - 100 q^{65} + 538 q^{67} - 690 q^{71} - 1126 q^{73} - 168 q^{77} - 665 q^{79} - 75 q^{83} - 465 q^{85} - 1086 q^{89} + 80 q^{91} + 295 q^{95} + 1544 q^{97}+O(q^{100}) \) 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 4.00000 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 2160.4.a.e 1
3.b odd 2 1 2160.4.a.o 1
4.b odd 2 1 270.4.a.i yes 1
12.b even 2 1 270.4.a.e 1
20.d odd 2 1 1350.4.a.g 1
20.e even 4 2 1350.4.c.q 2
36.f odd 6 2 810.4.e.h 2
36.h even 6 2 810.4.e.q 2
60.h even 2 1 1350.4.a.u 1
60.l odd 4 2 1350.4.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.e 1 12.b even 2 1
270.4.a.i yes 1 4.b odd 2 1
810.4.e.h 2 36.f odd 6 2
810.4.e.q 2 36.h even 6 2
1350.4.a.g 1 20.d odd 2 1
1350.4.a.u 1 60.h even 2 1
1350.4.c.d 2 60.l odd 4 2
1350.4.c.q 2 20.e even 4 2
2160.4.a.e 1 1.a even 1 1 trivial
2160.4.a.o 1 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} + 42 \) 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
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 42 \) Copy content Toggle raw display
$13$ \( T - 20 \) Copy content Toggle raw display
$17$ \( T - 93 \) Copy content Toggle raw display
$19$ \( T + 59 \) Copy content Toggle raw display
$23$ \( T + 9 \) Copy content Toggle raw display
$29$ \( T - 120 \) Copy content Toggle raw display
$31$ \( T + 47 \) Copy content Toggle raw display
$37$ \( T + 262 \) Copy content Toggle raw display
$41$ \( T - 126 \) Copy content Toggle raw display
$43$ \( T - 178 \) Copy content Toggle raw display
$47$ \( T + 144 \) Copy content Toggle raw display
$53$ \( T - 741 \) Copy content Toggle raw display
$59$ \( T - 444 \) Copy content Toggle raw display
$61$ \( T - 221 \) Copy content Toggle raw display
$67$ \( T - 538 \) Copy content Toggle raw display
$71$ \( T + 690 \) Copy content Toggle raw display
$73$ \( T + 1126 \) Copy content Toggle raw display
$79$ \( T + 665 \) Copy content Toggle raw display
$83$ \( T + 75 \) Copy content Toggle raw display
$89$ \( T + 1086 \) Copy content Toggle raw display
$97$ \( T - 1544 \) Copy content Toggle raw display
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