Properties

Label 2160.4.a.bk
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1257.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q - 5 q^{5} + ( - \beta_{2} - \beta_1 + 4) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + ( - \beta_{2} - \beta_1 + 4) q^{7} + ( - 5 \beta_1 + 11) q^{11} + (3 \beta_1 - 27) q^{13} + (3 \beta_{2} + \beta_1 - 5) q^{17} + (2 \beta_{2} - 3 \beta_1 + 24) q^{19} + ( - 7 \beta_{2} + 12 \beta_1 - 10) q^{23} + 25 q^{25} + ( - \beta_{2} + 23 \beta_1 + 32) q^{29} + (11 \beta_{2} + 3 \beta_1 + 31) q^{31} + (5 \beta_{2} + 5 \beta_1 - 20) q^{35} + (12 \beta_{2} + 4 \beta_1 - 142) q^{37} + ( - \beta_{2} - 13 \beta_1 + 202) q^{41} + ( - 7 \beta_{2} + 21 \beta_1 - 22) q^{43} + ( - 7 \beta_{2} + 30 \beta_1 - 201) q^{47} + (5 \beta_{2} - 42 \beta_1 + 172) q^{49} + ( - 4 \beta_{2} + 17 \beta_1 + 52) q^{53} + (25 \beta_1 - 55) q^{55} + (14 \beta_{2} - 41 \beta_1 + 87) q^{59} + (26 \beta_1 - 233) q^{61} + ( - 15 \beta_1 + 135) q^{65} + (26 \beta_{2} - 34 \beta_1 - 126) q^{67} + (22 \beta_{2} - 3 \beta_1 - 1) q^{71} + ( - 8 \beta_{2} + 91 \beta_1 - 325) q^{73} + ( - 21 \beta_{2} - 96 \beta_1 + 639) q^{77} + ( - 28 \beta_{2} - 53 \beta_1 - 394) q^{79} + (17 \beta_{2} - 68 \beta_1 - 588) q^{83} + ( - 15 \beta_{2} - 5 \beta_1 + 25) q^{85} + (35 \beta_{2} - 11 \beta_1 + 592) q^{89} + (33 \beta_{2} + 78 \beta_1 - 465) q^{91} + ( - 10 \beta_{2} + 15 \beta_1 - 120) q^{95} + (48 \beta_{2} - 12 \beta_1 - 292) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53} - 140 q^{55} + 234 q^{59} - 673 q^{61} + 390 q^{65} - 386 q^{67} + 16 q^{71} - 892 q^{73} + 1800 q^{77} - 1263 q^{79} - 1815 q^{83} + 55 q^{85} + 1800 q^{89} - 1284 q^{91} - 355 q^{95} - 840 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 8x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{2} + 2\nu + 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 8\nu - 25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 4\beta _1 + 69 ) / 12 \) 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
2.72396
1.14974
−2.87370
0 0 0 −5.00000 0 −24.0794 0 0 0
1.2 0 0 0 −5.00000 0 3.85875 0 0 0
1.3 0 0 0 −5.00000 0 30.2207 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.bk 3
3.b odd 2 1 2160.4.a.bs 3
4.b odd 2 1 1080.4.a.d 3
12.b even 2 1 1080.4.a.j yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.d 3 4.b odd 2 1
1080.4.a.j yes 3 12.b even 2 1
2160.4.a.bk 3 1.a even 1 1 trivial
2160.4.a.bs 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{3} - 10T_{7}^{2} - 704T_{7} + 2808 \) Copy content Toggle raw display
\( T_{11}^{3} - 28T_{11}^{2} - 2772T_{11} + 8424 \) 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
$7$ \( T^{3} - 10 T^{2} - 704 T + 2808 \) Copy content Toggle raw display
$11$ \( T^{3} - 28 T^{2} - 2772 T + 8424 \) Copy content Toggle raw display
$13$ \( T^{3} + 78 T^{2} + 936 T - 6696 \) Copy content Toggle raw display
$17$ \( T^{3} + 11 T^{2} - 5033 T - 120315 \) Copy content Toggle raw display
$19$ \( T^{3} - 71 T^{2} - 889 T + 58295 \) Copy content Toggle raw display
$23$ \( T^{3} + 25 T^{2} - 34325 T - 1363653 \) Copy content Toggle raw display
$29$ \( T^{3} - 118 T^{2} - 57792 T + 2593368 \) Copy content Toggle raw display
$31$ \( T^{3} - 107 T^{2} - 63129 T - 2882997 \) Copy content Toggle raw display
$37$ \( T^{3} + 410 T^{2} + \cdots - 15053832 \) Copy content Toggle raw display
$41$ \( T^{3} - 592 T^{2} + 94516 T - 4157400 \) Copy content Toggle raw display
$43$ \( T^{3} + 52 T^{2} - 63452 T - 7351224 \) Copy content Toggle raw display
$47$ \( T^{3} + 580 T^{2} + \cdots - 28223296 \) Copy content Toggle raw display
$53$ \( T^{3} - 169 T^{2} - 27113 T - 581991 \) Copy content Toggle raw display
$59$ \( T^{3} - 234 T^{2} + \cdots + 66081528 \) Copy content Toggle raw display
$61$ \( T^{3} + 673 T^{2} + 68955 T - 4428477 \) Copy content Toggle raw display
$67$ \( T^{3} + 386 T^{2} + \cdots - 51000008 \) Copy content Toggle raw display
$71$ \( T^{3} - 16 T^{2} - 244884 T - 45142200 \) Copy content Toggle raw display
$73$ \( T^{3} + 892 T^{2} + \cdots - 350087736 \) Copy content Toggle raw display
$79$ \( T^{3} + 1263 T^{2} + \cdots - 504123839 \) Copy content Toggle raw display
$83$ \( T^{3} + 1815 T^{2} + \cdots + 28157301 \) Copy content Toggle raw display
$89$ \( T^{3} - 1800 T^{2} + \cdots - 30835240 \) Copy content Toggle raw display
$97$ \( T^{3} + 840 T^{2} + \cdots - 775886336 \) Copy content Toggle raw display
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