Properties

Label 720.4.a.t
Level $720$
Weight $4$
Character orbit 720.a
Self dual yes
Analytic conductor $42.481$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5 q^{5} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} - 14 q^{7} + 6 q^{11} + 68 q^{13} - 78 q^{17} - 44 q^{19} + 120 q^{23} + 25 q^{25} - 126 q^{29} + 244 q^{31} - 70 q^{35} - 304 q^{37} + 480 q^{41} - 104 q^{43} + 600 q^{47} - 147 q^{49} + 258 q^{53} + 30 q^{55} + 534 q^{59} + 362 q^{61} + 340 q^{65} + 268 q^{67} - 972 q^{71} + 470 q^{73} - 84 q^{77} - 1244 q^{79} + 396 q^{83} - 390 q^{85} + 972 q^{89} - 952 q^{91} - 220 q^{95} - 46 q^{97}+O(q^{100}) \) 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 −14.0000 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 720.4.a.t 1
3.b odd 2 1 720.4.a.e 1
4.b odd 2 1 90.4.a.e yes 1
12.b even 2 1 90.4.a.b 1
20.d odd 2 1 450.4.a.c 1
20.e even 4 2 450.4.c.f 2
36.f odd 6 2 810.4.e.a 2
36.h even 6 2 810.4.e.u 2
60.h even 2 1 450.4.a.m 1
60.l odd 4 2 450.4.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.a.b 1 12.b even 2 1
90.4.a.e yes 1 4.b odd 2 1
450.4.a.c 1 20.d odd 2 1
450.4.a.m 1 60.h even 2 1
450.4.c.f 2 20.e even 4 2
450.4.c.g 2 60.l odd 4 2
720.4.a.e 1 3.b odd 2 1
720.4.a.t 1 1.a even 1 1 trivial
810.4.e.a 2 36.f odd 6 2
810.4.e.u 2 36.h even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} + 14 \) Copy content Toggle raw display
\( T_{11} - 6 \) 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
$7$ \( T + 14 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T - 68 \) Copy content Toggle raw display
$17$ \( T + 78 \) Copy content Toggle raw display
$19$ \( T + 44 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T + 126 \) Copy content Toggle raw display
$31$ \( T - 244 \) Copy content Toggle raw display
$37$ \( T + 304 \) Copy content Toggle raw display
$41$ \( T - 480 \) Copy content Toggle raw display
$43$ \( T + 104 \) Copy content Toggle raw display
$47$ \( T - 600 \) Copy content Toggle raw display
$53$ \( T - 258 \) Copy content Toggle raw display
$59$ \( T - 534 \) Copy content Toggle raw display
$61$ \( T - 362 \) Copy content Toggle raw display
$67$ \( T - 268 \) Copy content Toggle raw display
$71$ \( T + 972 \) Copy content Toggle raw display
$73$ \( T - 470 \) Copy content Toggle raw display
$79$ \( T + 1244 \) Copy content Toggle raw display
$83$ \( T - 396 \) Copy content Toggle raw display
$89$ \( T - 972 \) Copy content Toggle raw display
$97$ \( T + 46 \) Copy content Toggle raw display
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