Properties

Label 2160.4.a.bt
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: 3.3.697.1
Defining polynomial: \( x^{3} - 7x - 5 \) 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.

\(f(q)\) \(=\) \( q + 5 q^{5} + (\beta_{2} + 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + (\beta_{2} + 8) q^{7} + ( - 3 \beta_{2} + \beta_1 - 2) q^{11} + ( - 3 \beta_{2} - \beta_1 + 16) q^{13} + ( - \beta_{2} + 4 \beta_1 - 9) q^{17} + ( - 3 \beta_{2} - 3 \beta_1 + 65) q^{19} + ( - 10 \beta_{2} + 3 \beta_1 + 9) q^{23} + 25 q^{25} + ( - 3 \beta_{2} + 2 \beta_1 + 20) q^{29} + (\beta_{2} - 2 \beta_1 + 93) q^{31} + (5 \beta_{2} + 40) q^{35} + (8 \beta_{2} + 2 \beta_1 - 46) q^{37} + (\beta_{2} - 12 \beta_1 + 22) q^{41} + (11 \beta_{2} + 10 \beta_1 - 74) q^{43} + (24 \beta_{2} - 7 \beta_1 + 88) q^{47} + (8 \beta_{2} - 3 \beta_1 - 79) q^{49} + (5 \beta_{2} - \beta_1 - 169) q^{53} + ( - 15 \beta_{2} + 5 \beta_1 - 10) q^{55} + ( - 15 \beta_{2} - 3 \beta_1 + 320) q^{59} + (42 \beta_{2} + 2 \beta_1 + 181) q^{61} + ( - 15 \beta_{2} - 5 \beta_1 + 80) q^{65} + (46 \beta_{2} + 6 \beta_1 + 362) q^{67} + (3 \beta_{2} + 5 \beta_1 + 606) q^{71} + ( - 23 \beta_{2} - 9 \beta_1 + 454) q^{73} + ( - 14 \beta_{2} + 25 \beta_1 - 592) q^{77} + ( - 17 \beta_{2} - 15 \beta_1 + 43) q^{79} + (2 \beta_{2} - 13 \beta_1 + 523) q^{83} + ( - 5 \beta_{2} + 20 \beta_1 - 45) q^{85} + (55 \beta_{2} - 16 \beta_1 - 590) q^{89} + (28 \beta_{2} - 7 \beta_1 - 496) q^{91} + ( - 15 \beta_{2} - 15 \beta_1 + 325) q^{95} + ( - 56 \beta_{2} + 26 \beta_1 - 112) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 24 q^{7} - 6 q^{11} + 48 q^{13} - 27 q^{17} + 195 q^{19} + 27 q^{23} + 75 q^{25} + 60 q^{29} + 279 q^{31} + 120 q^{35} - 138 q^{37} + 66 q^{41} - 222 q^{43} + 264 q^{47} - 237 q^{49} - 507 q^{53} - 30 q^{55} + 960 q^{59} + 543 q^{61} + 240 q^{65} + 1086 q^{67} + 1818 q^{71} + 1362 q^{73} - 1776 q^{77} + 129 q^{79} + 1569 q^{83} - 135 q^{85} - 1770 q^{89} - 1488 q^{91} + 975 q^{95} - 336 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 7x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 6\nu^{2} - 6\nu - 28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 56 ) / 12 \) 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
−0.782816
2.94883
−2.16601
0 0 0 5.00000 0 −11.6263 0 0 0
1.2 0 0 0 5.00000 0 14.4806 0 0 0
1.3 0 0 0 5.00000 0 21.1457 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.bt 3
3.b odd 2 1 2160.4.a.bl 3
4.b odd 2 1 1080.4.a.i yes 3
12.b even 2 1 1080.4.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.c 3 12.b even 2 1
1080.4.a.i yes 3 4.b odd 2 1
2160.4.a.bl 3 3.b odd 2 1
2160.4.a.bt 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{3} - 24T_{7}^{2} - 108T_{7} + 3560 \) Copy content Toggle raw display
\( T_{11}^{3} + 6T_{11}^{2} - 3480T_{11} + 44648 \) 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
$7$ \( T^{3} - 24 T^{2} - 108 T + 3560 \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} - 3480 T + 44648 \) Copy content Toggle raw display
$13$ \( T^{3} - 48 T^{2} - 3156 T + 8360 \) Copy content Toggle raw display
$17$ \( T^{3} + 27 T^{2} - 15897 T - 428443 \) Copy content Toggle raw display
$19$ \( T^{3} - 195 T^{2} + 255 T + 954091 \) Copy content Toggle raw display
$23$ \( T^{3} - 27 T^{2} - 36669 T + 1787159 \) Copy content Toggle raw display
$29$ \( T^{3} - 60 T^{2} - 5100 T + 306136 \) Copy content Toggle raw display
$31$ \( T^{3} - 279 T^{2} + 21759 T - 418385 \) Copy content Toggle raw display
$37$ \( T^{3} + 138 T^{2} - 18036 T + 122040 \) Copy content Toggle raw display
$41$ \( T^{3} - 66 T^{2} - 143136 T + 15824376 \) Copy content Toggle raw display
$43$ \( T^{3} + 222 T^{2} + \cdots - 25516184 \) Copy content Toggle raw display
$47$ \( T^{3} - 264 T^{2} - 186864 T - 774208 \) Copy content Toggle raw display
$53$ \( T^{3} + 507 T^{2} + 77535 T + 3425517 \) Copy content Toggle raw display
$59$ \( T^{3} - 960 T^{2} + \cdots - 15048440 \) Copy content Toggle raw display
$61$ \( T^{3} - 543 T^{2} + \cdots + 236291075 \) Copy content Toggle raw display
$67$ \( T^{3} - 1086 T^{2} + \cdots + 422390264 \) Copy content Toggle raw display
$71$ \( T^{3} - 1818 T^{2} + \cdots - 206864712 \) Copy content Toggle raw display
$73$ \( T^{3} - 1362 T^{2} + \cdots + 5163048 \) Copy content Toggle raw display
$79$ \( T^{3} - 129 T^{2} + \cdots + 64805697 \) Copy content Toggle raw display
$83$ \( T^{3} - 1569 T^{2} + \cdots - 40821251 \) Copy content Toggle raw display
$89$ \( T^{3} + 1770 T^{2} + \cdots - 666862200 \) Copy content Toggle raw display
$97$ \( T^{3} + 336 T^{2} + \cdots + 503900672 \) Copy content Toggle raw display
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