Properties

Label 2160.4.a.bu
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 141x^{2} + 200x + 3500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) 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 \( q - 5 q^{5} + (\beta_{2} - 3) q^{7} + (\beta_1 - 1) q^{11} + (\beta_{3} - \beta_{2} - \beta_1 + 7) q^{13} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{17} + (\beta_{3} + 3 \beta_{2} - 18) q^{19} + ( - 2 \beta_{3} - 3 \beta_{2} + 44) q^{23} + 25 q^{25} + ( - \beta_{3} - \beta_{2} + 4 \beta_1 - 51) q^{29} + ( - 2 \beta_{3} - 3 \beta_1 + 19) q^{31} + ( - 5 \beta_{2} + 15) q^{35} + ( - 3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 74) q^{37} + (\beta_{3} + 2 \beta_1 - 95) q^{41} + (2 \beta_{3} - 7 \beta_{2} + 2 \beta_1 - 48) q^{43} + ( - \beta_{3} + 5 \beta_{2} - 4 \beta_1 + 31) q^{47} + ( - 4 \beta_{3} - 7 \beta_{2} - 10 \beta_1 + 236) q^{49} + (5 \beta_{3} - 10 \beta_{2} - 8 \beta_1 + 59) q^{53} + ( - 5 \beta_1 + 5) q^{55} + (11 \beta_{3} - 4 \beta_{2} - 53) q^{59} + (4 \beta_{3} + 23 \beta_{2} + 4 \beta_1 + 203) q^{61} + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 35) q^{65} + ( - 4 \beta_{3} - \beta_{2} - 2 \beta_1 - 83) q^{67} + (3 \beta_{3} - 16 \beta_{2} + 2 \beta_1 - 273) q^{71} + (7 \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 358) q^{73} + (6 \beta_{3} - 36 \beta_{2} - 12 \beta_1 - 72) q^{77} + ( - 7 \beta_{3} - 31 \beta_{2} + 9 \beta_1 - 201) q^{79} + ( - 9 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 193) q^{83} + (5 \beta_{3} + 10 \beta_{2} + 5 \beta_1 - 30) q^{85} + (\beta_{3} + 6 \beta_{2} + 10 \beta_1 + 103) q^{89} + (3 \beta_{3} + 21 \beta_{2} + 30 \beta_1 - 756) q^{91} + ( - 5 \beta_{3} - 15 \beta_{2} + 90) q^{95} + ( - 7 \beta_{3} + 19 \beta_{2} + 16 \beta_1 + 844) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{5} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{5} - 14 q^{7} - 4 q^{11} + 30 q^{13} + 28 q^{17} - 78 q^{19} + 182 q^{23} + 100 q^{25} - 202 q^{29} + 76 q^{31} + 70 q^{35} + 302 q^{37} - 380 q^{41} - 178 q^{43} + 114 q^{47} + 958 q^{49} + 256 q^{53} + 20 q^{55} - 204 q^{59} + 766 q^{61} - 150 q^{65} - 330 q^{67} - 1060 q^{71} + 1442 q^{73} - 216 q^{77} - 742 q^{79} - 768 q^{83} - 140 q^{85} + 400 q^{89} - 3066 q^{91} + 390 q^{95} + 3338 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 141x^{2} + 200x + 3500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{3} + 3\nu^{2} - 617\nu + 250 ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 9\nu^{2} + 131\nu + 550 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 9\nu^{2} - 71\nu - 575 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 4\beta_{2} + 5 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{3} - 8\beta_{2} - 8\beta _1 + 845 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 43\beta_{3} + 178\beta_{2} + 36\beta _1 - 175 ) / 6 \) 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
−10.9094
−4.73555
9.55367
7.09129
0 0 0 −5.00000 0 −35.5942 0 0 0
1.2 0 0 0 −5.00000 0 −11.2995 0 0 0
1.3 0 0 0 −5.00000 0 2.40438 0 0 0
1.4 0 0 0 −5.00000 0 30.4893 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.bu 4
3.b odd 2 1 2160.4.a.bv 4
4.b odd 2 1 1080.4.a.o 4
12.b even 2 1 1080.4.a.p yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.o 4 4.b odd 2 1
1080.4.a.p yes 4 12.b even 2 1
2160.4.a.bu 4 1.a even 1 1 trivial
2160.4.a.bv 4 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{4} + 14T_{7}^{3} - 1067T_{7}^{2} - 9792T_{7} + 29484 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} - 3749T_{11}^{2} - 45756T_{11} + 1807596 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 14 T^{3} - 1067 T^{2} + \cdots + 29484 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} - 3749 T^{2} + \cdots + 1807596 \) Copy content Toggle raw display
$13$ \( T^{4} - 30 T^{3} - 8928 T^{2} + \cdots + 18230751 \) Copy content Toggle raw display
$17$ \( T^{4} - 28 T^{3} - 13553 T^{2} + \cdots + 4462704 \) Copy content Toggle raw display
$19$ \( T^{4} + 78 T^{3} - 11003 T^{2} + \cdots + 21407584 \) Copy content Toggle raw display
$23$ \( T^{4} - 182 T^{3} + \cdots - 146544156 \) Copy content Toggle raw display
$29$ \( T^{4} + 202 T^{3} + \cdots - 561688740 \) Copy content Toggle raw display
$31$ \( T^{4} - 76 T^{3} + \cdots + 901737216 \) Copy content Toggle raw display
$37$ \( T^{4} - 302 T^{3} - 57743 T^{2} + \cdots + 9791964 \) Copy content Toggle raw display
$41$ \( T^{4} + 380 T^{3} + 29260 T^{2} + \cdots + 1440000 \) Copy content Toggle raw display
$43$ \( T^{4} + 178 T^{3} + \cdots - 372768336 \) Copy content Toggle raw display
$47$ \( T^{4} - 114 T^{3} + \cdots + 465218416 \) Copy content Toggle raw display
$53$ \( T^{4} - 256 T^{3} + \cdots + 26357772096 \) Copy content Toggle raw display
$59$ \( T^{4} + 204 T^{3} + \cdots + 3841737984 \) Copy content Toggle raw display
$61$ \( T^{4} - 766 T^{3} + \cdots - 50264811540 \) Copy content Toggle raw display
$67$ \( T^{4} + 330 T^{3} + \cdots + 415421296 \) Copy content Toggle raw display
$71$ \( T^{4} + 1060 T^{3} + \cdots - 897110784 \) Copy content Toggle raw display
$73$ \( T^{4} - 1442 T^{3} + \cdots + 1120468716 \) Copy content Toggle raw display
$79$ \( T^{4} + 742 T^{3} + \cdots + 364066942735 \) Copy content Toggle raw display
$83$ \( T^{4} + 768 T^{3} + \cdots + 4935879504 \) Copy content Toggle raw display
$89$ \( T^{4} - 400 T^{3} + \cdots + 16886941696 \) Copy content Toggle raw display
$97$ \( T^{4} - 3338 T^{3} + \cdots - 650133282260 \) Copy content Toggle raw display
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