Properties

Label 2160.4.a.br
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.4281.1
Defining polynomial: \( x^{3} - x^{2} - 12x + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1080)
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.

\(f(q)\) \(=\) \( q + 5 q^{5} + ( - \beta_1 + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + ( - \beta_1 + 3) q^{7} + (\beta_{2} - \beta_1 + 4) q^{11} + ( - \beta_{2} - \beta_1 + 16) q^{13} + (\beta_1 + 12) q^{17} + (\beta_{2} - \beta_1 - 9) q^{19} + ( - \beta_{2} - 2 \beta_1 + 4) q^{23} + 25 q^{25} + (2 \beta_{2} + 3 \beta_1 + 9) q^{29} + ( - 2 \beta_{2} - \beta_1 - 14) q^{31} + ( - 5 \beta_1 + 15) q^{35} + (2 \beta_{2} - 8 \beta_1 + 80) q^{37} + (4 \beta_{2} - \beta_1 + 125) q^{41} + ( - 6 \beta_{2} - 11 \beta_1 + 45) q^{43} + (9 \beta_{2} - 4 \beta_1 + 23) q^{47} + ( - 3 \beta_{2} - 20 \beta_1 + 188) q^{49} + ( - \beta_{2} - 9 \beta_1 + 271) q^{53} + (5 \beta_{2} - 5 \beta_1 + 20) q^{55} + ( - 11 \beta_{2} + 19 \beta_1 - 202) q^{59} + (6 \beta_{2} + 22 \beta_1 - 91) q^{61} + ( - 5 \beta_{2} - 5 \beta_1 + 80) q^{65} + ( - 2 \beta_{2} + 18 \beta_1 - 2) q^{67} + (5 \beta_{2} + 9 \beta_1 - 414) q^{71} + ( - \beta_{2} + 3 \beta_1 - 134) q^{73} + ( - 15 \beta_{2} - 6 \beta_1 + 429) q^{77} + (17 \beta_{2} + 5 \beta_1 + 111) q^{79} + ( - 13 \beta_{2} + 26 \beta_1 - 242) q^{83} + (5 \beta_1 + 60) q^{85} + ( - 15 \beta_1 + 791) q^{89} + (9 \beta_{2} - 48 \beta_1 + 675) q^{91} + (5 \beta_{2} - 5 \beta_1 - 45) q^{95} + ( - 14 \beta_{2} - 8 \beta_1 + 190) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 8 q^{7} + 10 q^{11} + 48 q^{13} + 37 q^{17} - 29 q^{19} + 11 q^{23} + 75 q^{25} + 28 q^{29} - 41 q^{31} + 40 q^{35} + 230 q^{37} + 370 q^{41} + 130 q^{43} + 56 q^{47} + 547 q^{49} + 805 q^{53} + 50 q^{55} - 576 q^{59} - 257 q^{61} + 240 q^{65} + 14 q^{67} - 1238 q^{71} - 398 q^{73} + 1296 q^{77} + 321 q^{79} - 687 q^{83} + 185 q^{85} + 2358 q^{89} + 1968 q^{91} - 145 q^{95} + 576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 12x + 15 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{2} + 6\nu + 15 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 8\nu^{2} + 12\nu - 71 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 4\beta _1 + 11 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 2\beta _1 + 101 ) / 12 \) 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
1.29027
3.26757
−3.55784
0 0 0 5.00000 0 −16.4120 0 0 0
1.2 0 0 0 5.00000 0 −10.2514 0 0 0
1.3 0 0 0 5.00000 0 34.6634 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.br 3
3.b odd 2 1 2160.4.a.bj 3
4.b odd 2 1 1080.4.a.k yes 3
12.b even 2 1 1080.4.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.e 3 12.b even 2 1
1080.4.a.k yes 3 4.b odd 2 1
2160.4.a.bj 3 3.b odd 2 1
2160.4.a.br 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{3} - 8T_{7}^{2} - 756T_{7} - 5832 \) Copy content Toggle raw display
\( T_{11}^{3} - 10T_{11}^{2} - 2864T_{11} + 59400 \) 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
$7$ \( T^{3} - 8 T^{2} - 756 T - 5832 \) Copy content Toggle raw display
$11$ \( T^{3} - 10 T^{2} - 2864 T + 59400 \) Copy content Toggle raw display
$13$ \( T^{3} - 48 T^{2} - 2700 T + 118584 \) Copy content Toggle raw display
$17$ \( T^{3} - 37 T^{2} - 321 T + 15597 \) Copy content Toggle raw display
$19$ \( T^{3} + 29 T^{2} - 2617 T + 22675 \) Copy content Toggle raw display
$23$ \( T^{3} - 11 T^{2} - 6045 T + 44775 \) Copy content Toggle raw display
$29$ \( T^{3} - 28 T^{2} - 18068 T - 296760 \) Copy content Toggle raw display
$31$ \( T^{3} + 41 T^{2} - 10409 T + 291591 \) Copy content Toggle raw display
$37$ \( T^{3} - 230 T^{2} - 37172 T + 4001784 \) Copy content Toggle raw display
$41$ \( T^{3} - 370 T^{2} + 7512 T + 2205432 \) Copy content Toggle raw display
$43$ \( T^{3} - 130 T^{2} + \cdots + 16735752 \) Copy content Toggle raw display
$47$ \( T^{3} - 56 T^{2} - 195952 T + 7432640 \) Copy content Toggle raw display
$53$ \( T^{3} - 805 T^{2} + 148071 T - 7723035 \) Copy content Toggle raw display
$59$ \( T^{3} + 576 T^{2} + \cdots - 227290536 \) Copy content Toggle raw display
$61$ \( T^{3} + 257 T^{2} + \cdots + 37318467 \) Copy content Toggle raw display
$67$ \( T^{3} - 14 T^{2} - 251140 T + 30539000 \) Copy content Toggle raw display
$71$ \( T^{3} + 1238 T^{2} + \cdots + 9117432 \) Copy content Toggle raw display
$73$ \( T^{3} + 398 T^{2} + 44256 T + 1074888 \) Copy content Toggle raw display
$79$ \( T^{3} - 321 T^{2} + \cdots - 143418599 \) Copy content Toggle raw display
$83$ \( T^{3} + 687 T^{2} + \cdots - 435938355 \) Copy content Toggle raw display
$89$ \( T^{3} - 2358 T^{2} + \cdots - 374731256 \) Copy content Toggle raw display
$97$ \( T^{3} - 576 T^{2} + \cdots + 257454592 \) Copy content Toggle raw display
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