Properties

Label 2160.4.a.bo
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: 3.3.985.1
Defining polynomial: \( x^{3} - x^{2} - 6x + 1 \) 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.

\(f(q)\) \(=\) \( q + 5 q^{5} + (\beta_{2} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + (\beta_{2} - 2) q^{7} + (\beta_{2} + \beta_1 - 4) q^{11} + ( - \beta_{2} - \beta_1 + 6) q^{13} + ( - 3 \beta_{2} - 2 \beta_1 - 7) q^{17} + ( - 3 \beta_{2} + 3 \beta_1 - 19) q^{19} + (2 \beta_{2} - 3 \beta_1 - 29) q^{23} + 25 q^{25} + ( - \beta_{2} + 8 \beta_1 + 46) q^{29} + (5 \beta_{2} - 8 \beta_1 - 39) q^{31} + (5 \beta_{2} - 10) q^{35} + (4 \beta_{2} + 8 \beta_1 + 50) q^{37} + ( - 5 \beta_{2} - 12 \beta_1) q^{41} + ( - 21 \beta_{2} + 4 \beta_1 - 60) q^{43} + ( - 16 \beta_{2} - \beta_1 - 228) q^{47} + ( - 16 \beta_{2} - 3 \beta_1 - 27) q^{49} + (19 \beta_{2} - \beta_1 + 29) q^{53} + (5 \beta_{2} + 5 \beta_1 - 20) q^{55} + ( - 15 \beta_{2} + 3 \beta_1 - 238) q^{59} + ( - 22 \beta_{2} - 10 \beta_1 - 171) q^{61} + ( - 5 \beta_{2} - 5 \beta_1 + 30) q^{65} + ( - 14 \beta_{2} + 12 \beta_1 + 58) q^{67} + (11 \beta_{2} - 19 \beta_1 - 256) q^{71} + (7 \beta_{2} + 15 \beta_1 - 84) q^{73} + ( - 22 \beta_{2} + 7 \beta_1 + 296) q^{77} + ( - 17 \beta_{2} - 21 \beta_1 - 69) q^{79} + (10 \beta_{2} + 17 \beta_1 - 563) q^{83} + ( - 15 \beta_{2} - 10 \beta_1 - 35) q^{85} + (33 \beta_{2} + 14 \beta_1 + 104) q^{89} + (24 \beta_{2} - 7 \beta_1 - 300) q^{91} + ( - 15 \beta_{2} + 15 \beta_1 - 95) q^{95} + (60 \beta_{2} - 4 \beta_1 - 360) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} - 6 q^{7} - 12 q^{11} + 18 q^{13} - 21 q^{17} - 57 q^{19} - 87 q^{23} + 75 q^{25} + 138 q^{29} - 117 q^{31} - 30 q^{35} + 150 q^{37} - 180 q^{43} - 684 q^{47} - 81 q^{49} + 87 q^{53} - 60 q^{55} - 714 q^{59} - 513 q^{61} + 90 q^{65} + 174 q^{67} - 768 q^{71} - 252 q^{73} + 888 q^{77} - 207 q^{79} - 1689 q^{83} - 105 q^{85} + 312 q^{89} - 900 q^{91} - 285 q^{95} - 1080 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 6\nu^{2} - 6\nu - 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 4 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 52 ) / 12 \) 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
0.162962
2.93080
−2.09376
0 0 0 5.00000 0 −26.8184 0 0 0
1.2 0 0 0 5.00000 0 7.95278 0 0 0
1.3 0 0 0 5.00000 0 12.8657 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.bo 3
3.b odd 2 1 2160.4.a.bg 3
4.b odd 2 1 1080.4.a.m yes 3
12.b even 2 1 1080.4.a.g 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.g 3 12.b even 2 1
1080.4.a.m yes 3 4.b odd 2 1
2160.4.a.bg 3 3.b odd 2 1
2160.4.a.bo 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{3} + 6T_{7}^{2} - 456T_{7} + 2744 \) Copy content Toggle raw display
\( T_{11}^{3} + 12T_{11}^{2} - 1260T_{11} - 20920 \) 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
$7$ \( T^{3} + 6 T^{2} - 456 T + 2744 \) Copy content Toggle raw display
$11$ \( T^{3} + 12 T^{2} - 1260 T - 20920 \) Copy content Toggle raw display
$13$ \( T^{3} - 18 T^{2} - 1200 T + 23384 \) Copy content Toggle raw display
$17$ \( T^{3} + 21 T^{2} - 7281 T + 47219 \) Copy content Toggle raw display
$19$ \( T^{3} + 57 T^{2} - 11985 T + 332479 \) Copy content Toggle raw display
$23$ \( T^{3} + 87 T^{2} - 7989 T - 655435 \) Copy content Toggle raw display
$29$ \( T^{3} - 138 T^{2} - 53064 T + 3137800 \) Copy content Toggle raw display
$31$ \( T^{3} + 117 T^{2} - 68385 T - 9394301 \) Copy content Toggle raw display
$37$ \( T^{3} - 150 T^{2} - 56052 T - 2749896 \) Copy content Toggle raw display
$41$ \( T^{3} - 138708 T + 17332488 \) Copy content Toggle raw display
$43$ \( T^{3} + 180 T^{2} + \cdots - 31951208 \) Copy content Toggle raw display
$47$ \( T^{3} + 684 T^{2} + \cdots - 31272640 \) Copy content Toggle raw display
$53$ \( T^{3} - 87 T^{2} - 168705 T + 28044495 \) Copy content Toggle raw display
$59$ \( T^{3} + 714 T^{2} + \cdots - 20643704 \) Copy content Toggle raw display
$61$ \( T^{3} + 513 T^{2} + \cdots - 57378877 \) Copy content Toggle raw display
$67$ \( T^{3} - 174 T^{2} + \cdots + 55705400 \) Copy content Toggle raw display
$71$ \( T^{3} + 768 T^{2} + \cdots - 166892184 \) Copy content Toggle raw display
$73$ \( T^{3} + 252 T^{2} + \cdots - 54480168 \) Copy content Toggle raw display
$79$ \( T^{3} + 207 T^{2} + \cdots + 102453849 \) Copy content Toggle raw display
$83$ \( T^{3} + 1689 T^{2} + \cdots - 51094085 \) Copy content Toggle raw display
$89$ \( T^{3} - 312 T^{2} + \cdots + 120696696 \) Copy content Toggle raw display
$97$ \( T^{3} + 1080 T^{2} + \cdots + 134280704 \) Copy content Toggle raw display
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