Properties

Label 2160.4.a.be
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: 3.3.5637.1
Defining polynomial: \( x^{3} - x^{2} - 23x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 135)
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.

\(f(q)\) \(=\) \( q - 5 q^{5} + ( - \beta_1 - 15) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + ( - \beta_1 - 15) q^{7} + (\beta_{2} - \beta_1 + 12) q^{11} + ( - \beta_{2} - 5 \beta_1 + 8) q^{13} + (2 \beta_{2} + \beta_1 + 6) q^{17} + (\beta_{2} - 7 \beta_1 - 65) q^{19} + (\beta_{2} + 16 \beta_1 - 22) q^{23} + 25 q^{25} + (2 \beta_{2} - 21 \beta_1 - 61) q^{29} + ( - 2 \beta_{2} + 15 \beta_1 - 70) q^{31} + (5 \beta_1 + 75) q^{35} + ( - 2 \beta_{2} - 32 \beta_1 + 16) q^{37} + ( - 4 \beta_{2} - 21 \beta_1 + 107) q^{41} + (2 \beta_{2} - 39 \beta_1 - 21) q^{43} + (\beta_{2} + 14 \beta_1 - 153) q^{47} + (\beta_{2} + 30 \beta_1 - 56) q^{49} + ( - 3 \beta_{2} - 7 \beta_1 - 175) q^{53} + ( - 5 \beta_{2} + 5 \beta_1 - 60) q^{55} + ( - 3 \beta_{2} - 41 \beta_1 + 34) q^{59} + (4 \beta_{2} + 30 \beta_1 + 207) q^{61} + (5 \beta_{2} + 25 \beta_1 - 40) q^{65} + ( - 6 \beta_{2} + 26 \beta_1 - 282) q^{67} + ( - 15 \beta_{2} + 17 \beta_1 - 190) q^{71} + (15 \beta_{2} + 11 \beta_1 + 430) q^{73} + ( - 13 \beta_{2} - 28 \beta_1 - 101) q^{77} + (3 \beta_{2} + 11 \beta_1 - 207) q^{79} + (15 \beta_{2} + 60 \beta_1 - 414) q^{83} + ( - 10 \beta_{2} - 5 \beta_1 - 30) q^{85} + (12 \beta_{2} + 33 \beta_1 - 711) q^{89} + (19 \beta_{2} + 98 \beta_1 + 173) q^{91} + ( - 5 \beta_{2} + 35 \beta_1 + 325) q^{95} + (2 \beta_{2} + 32 \beta_1 + 474) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} - 44 q^{7} + 38 q^{11} + 28 q^{13} + 19 q^{17} - 187 q^{19} - 81 q^{23} + 75 q^{25} - 160 q^{29} - 227 q^{31} + 220 q^{35} + 78 q^{37} + 338 q^{41} - 22 q^{43} - 472 q^{47} - 197 q^{49} - 521 q^{53} - 190 q^{55} + 140 q^{59} + 595 q^{61} - 140 q^{65} - 878 q^{67} - 602 q^{71} + 1294 q^{73} - 288 q^{77} - 629 q^{79} - 1287 q^{83} - 95 q^{85} - 2154 q^{89} + 440 q^{91} + 935 q^{95} + 1392 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 23x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 4\nu - 61 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 63 ) / 4 \) 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
5.20067
0.258712
−4.45938
0 0 0 −5.00000 0 −24.4013 0 0 0
1.2 0 0 0 −5.00000 0 −14.5174 0 0 0
1.3 0 0 0 −5.00000 0 −5.08123 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.be 3
3.b odd 2 1 2160.4.a.bm 3
4.b odd 2 1 135.4.a.g yes 3
12.b even 2 1 135.4.a.f 3
20.d odd 2 1 675.4.a.q 3
20.e even 4 2 675.4.b.k 6
36.f odd 6 2 405.4.e.r 6
36.h even 6 2 405.4.e.t 6
60.h even 2 1 675.4.a.r 3
60.l odd 4 2 675.4.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 12.b even 2 1
135.4.a.g yes 3 4.b odd 2 1
405.4.e.r 6 36.f odd 6 2
405.4.e.t 6 36.h even 6 2
675.4.a.q 3 20.d odd 2 1
675.4.a.r 3 60.h even 2 1
675.4.b.k 6 20.e even 4 2
675.4.b.l 6 60.l odd 4 2
2160.4.a.be 3 1.a even 1 1 trivial
2160.4.a.bm 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{3} + 44T_{7}^{2} + 552T_{7} + 1800 \) Copy content Toggle raw display
\( T_{11}^{3} - 38T_{11}^{2} - 2612T_{11} + 83280 \) 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
$7$ \( T^{3} + 44 T^{2} + 552 T + 1800 \) Copy content Toggle raw display
$11$ \( T^{3} - 38 T^{2} - 2612 T + 83280 \) Copy content Toggle raw display
$13$ \( T^{3} - 28 T^{2} - 4576 T + 100120 \) Copy content Toggle raw display
$17$ \( T^{3} - 19 T^{2} - 11477 T + 553887 \) Copy content Toggle raw display
$19$ \( T^{3} + 187 T^{2} + 3587 T - 525871 \) Copy content Toggle raw display
$23$ \( T^{3} + 81 T^{2} - 23301 T - 2043981 \) Copy content Toggle raw display
$29$ \( T^{3} + 160 T^{2} - 47768 T - 7892760 \) Copy content Toggle raw display
$31$ \( T^{3} + 227 T^{2} - 17973 T + 246321 \) Copy content Toggle raw display
$37$ \( T^{3} - 78 T^{2} - 99924 T + 13637080 \) Copy content Toggle raw display
$41$ \( T^{3} - 338 T^{2} + \cdots + 12116640 \) Copy content Toggle raw display
$43$ \( T^{3} + 22 T^{2} - 159916 T - 18464560 \) Copy content Toggle raw display
$47$ \( T^{3} + 472 T^{2} + 54208 T - 283200 \) Copy content Toggle raw display
$53$ \( T^{3} + 521 T^{2} + 61387 T - 939789 \) Copy content Toggle raw display
$59$ \( T^{3} - 140 T^{2} + \cdots + 34131480 \) Copy content Toggle raw display
$61$ \( T^{3} - 595 T^{2} - 2749 T + 1782607 \) Copy content Toggle raw display
$67$ \( T^{3} + 878 T^{2} + \cdots - 11295000 \) Copy content Toggle raw display
$71$ \( T^{3} + 602 T^{2} + \cdots - 280550880 \) Copy content Toggle raw display
$73$ \( T^{3} - 1294 T^{2} + \cdots + 404091280 \) Copy content Toggle raw display
$79$ \( T^{3} + 629 T^{2} + 97059 T + 2010303 \) Copy content Toggle raw display
$83$ \( T^{3} + 1287 T^{2} + \cdots - 346404411 \) Copy content Toggle raw display
$89$ \( T^{3} + 2154 T^{2} + \cdots + 74325600 \) Copy content Toggle raw display
$97$ \( T^{3} - 1392 T^{2} + \cdots - 63595520 \) Copy content Toggle raw display
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