Properties

Label 1080.4.a.h
Level $1080$
Weight $4$
Character orbit 1080.a
Self dual yes
Analytic conductor $63.722$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.7220628062\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.47977.1
Defining polynomial: \( x^{3} - x^{2} - 60x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 5 q^{5} + (\beta_{2} + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + (\beta_{2} + 3) q^{7} + (\beta_{2} - \beta_1 - 6) q^{11} + (2 \beta_{2} + \beta_1 - 7) q^{13} + (\beta_{2} + \beta_1 - 28) q^{17} + (3 \beta_{2} + 7) q^{19} + (\beta_{2} - 16) q^{23} + 25 q^{25} + ( - 5 \beta_{2} - 4 \beta_1 + 12) q^{29} + ( - 3 \beta_{2} - 5 \beta_1 + 108) q^{31} + ( - 5 \beta_{2} - 15) q^{35} + ( - 3 \beta_{2} + 4 \beta_1 + 11) q^{37} + ( - 2 \beta_{2} + 6 \beta_1 - 38) q^{41} + (7 \beta_{2} - 2 \beta_1 + 94) q^{43} + (17 \beta_{2} + 4 \beta_1 - 94) q^{47} + ( - 5 \beta_{2} - 6 \beta_1 + 76) q^{49} + ( - 6 \beta_{2} + 8 \beta_1 - 74) q^{53} + ( - 5 \beta_{2} + 5 \beta_1 + 30) q^{55} + 92 q^{59} + (11 \beta_{2} + 4 \beta_1 + 101) q^{61} + ( - 10 \beta_{2} - 5 \beta_1 + 35) q^{65} + ( - 23 \beta_{2} + 18 \beta_1 + 345) q^{67} + ( - 10 \beta_{2} - 2 \beta_1 - 170) q^{71} + ( - 25 \beta_{2} - 12 \beta_1 + 149) q^{73} + ( - 20 \beta_1 + 526) q^{77} + (22 \beta_{2} - 9 \beta_1 + 259) q^{79} + (36 \beta_{2} - 2 \beta_1 - 26) q^{83} + ( - 5 \beta_{2} - 5 \beta_1 + 140) q^{85} + (20 \beta_{2} - 10 \beta_1 + 108) q^{89} + ( - 37 \beta_{2} + 2 \beta_1 + 665) q^{91} + ( - 15 \beta_{2} - 35) q^{95} + (19 \beta_{2} + 16 \beta_1 + 397) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} + 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} + 9 q^{7} - 18 q^{11} - 21 q^{13} - 84 q^{17} + 21 q^{19} - 48 q^{23} + 75 q^{25} + 36 q^{29} + 324 q^{31} - 45 q^{35} + 33 q^{37} - 114 q^{41} + 282 q^{43} - 282 q^{47} + 228 q^{49} - 222 q^{53} + 90 q^{55} + 276 q^{59} + 303 q^{61} + 105 q^{65} + 1035 q^{67} - 510 q^{71} + 447 q^{73} + 1578 q^{77} + 777 q^{79} - 78 q^{83} + 420 q^{85} + 324 q^{89} + 1995 q^{91} - 105 q^{95} + 1191 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 60x - 44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{2} - 9\nu - 118 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8\beta_{2} + 3\beta _1 + 242 ) / 6 \) 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
−0.749725
8.58548
−6.83575
0 0 0 −5.00000 0 −24.3916 0 0 0
1.2 0 0 0 −5.00000 0 9.46549 0 0 0
1.3 0 0 0 −5.00000 0 23.9261 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 1080.4.a.h 3
3.b odd 2 1 1080.4.a.n yes 3
4.b odd 2 1 2160.4.a.bf 3
12.b even 2 1 2160.4.a.bn 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.h 3 1.a even 1 1 trivial
1080.4.a.n yes 3 3.b odd 2 1
2160.4.a.bf 3 4.b odd 2 1
2160.4.a.bn 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1080))\):

\( T_{7}^{3} - 9T_{7}^{2} - 588T_{7} + 5524 \) Copy content Toggle raw display
\( T_{11}^{3} + 18T_{11}^{2} - 3081T_{11} - 76426 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 9 T^{2} - 588 T + 5524 \) Copy content Toggle raw display
$11$ \( T^{3} + 18 T^{2} - 3081 T - 76426 \) Copy content Toggle raw display
$13$ \( T^{3} + 21 T^{2} - 3681 T - 30901 \) Copy content Toggle raw display
$17$ \( T^{3} + 84 T^{2} - 33 T - 86732 \) Copy content Toggle raw display
$19$ \( T^{3} - 21 T^{2} - 5388 T + 138464 \) Copy content Toggle raw display
$23$ \( T^{3} + 48 T^{2} + 153 T - 2038 \) Copy content Toggle raw display
$29$ \( T^{3} - 36 T^{2} - 41655 T + 3034586 \) Copy content Toggle raw display
$31$ \( T^{3} - 324 T^{2} - 18813 T + 9213176 \) Copy content Toggle raw display
$37$ \( T^{3} - 33 T^{2} - 44748 T + 2851956 \) Copy content Toggle raw display
$41$ \( T^{3} + 114 T^{2} - 81144 T - 1847232 \) Copy content Toggle raw display
$43$ \( T^{3} - 282 T^{2} - 17943 T + 1113236 \) Copy content Toggle raw display
$47$ \( T^{3} + 282 T^{2} + \cdots + 11238296 \) Copy content Toggle raw display
$53$ \( T^{3} + 222 T^{2} - 164016 T + 5908896 \) Copy content Toggle raw display
$59$ \( (T - 92)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} - 303 T^{2} + \cdots + 13330220 \) Copy content Toggle raw display
$67$ \( T^{3} - 1035 T^{2} + \cdots + 849676336 \) Copy content Toggle raw display
$71$ \( T^{3} + 510 T^{2} + \cdots - 11439360 \) Copy content Toggle raw display
$73$ \( T^{3} - 447 T^{2} + \cdots + 78574428 \) Copy content Toggle raw display
$79$ \( T^{3} - 777 T^{2} + \cdots - 14000175 \) Copy content Toggle raw display
$83$ \( T^{3} + 78 T^{2} - 832644 T + 87546568 \) Copy content Toggle raw display
$89$ \( T^{3} - 324 T^{2} + \cdots - 92454912 \) Copy content Toggle raw display
$97$ \( T^{3} - 1191 T^{2} + \cdots + 31736348 \) Copy content Toggle raw display
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