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: \(x^{3} - x^{2} - 23 x + 6\)
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.

\(f(q)\) \(=\) \( q -5 q^{5} + ( -15 - \beta_{1} ) q^{7} +O(q^{10})\) \( q -5 q^{5} + ( -15 - \beta_{1} ) q^{7} + ( 12 - \beta_{1} + \beta_{2} ) q^{11} + ( 8 - 5 \beta_{1} - \beta_{2} ) q^{13} + ( 6 + \beta_{1} + 2 \beta_{2} ) q^{17} + ( -65 - 7 \beta_{1} + \beta_{2} ) q^{19} + ( -22 + 16 \beta_{1} + \beta_{2} ) q^{23} + 25 q^{25} + ( -61 - 21 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -70 + 15 \beta_{1} - 2 \beta_{2} ) q^{31} + ( 75 + 5 \beta_{1} ) q^{35} + ( 16 - 32 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 107 - 21 \beta_{1} - 4 \beta_{2} ) q^{41} + ( -21 - 39 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -153 + 14 \beta_{1} + \beta_{2} ) q^{47} + ( -56 + 30 \beta_{1} + \beta_{2} ) q^{49} + ( -175 - 7 \beta_{1} - 3 \beta_{2} ) q^{53} + ( -60 + 5 \beta_{1} - 5 \beta_{2} ) q^{55} + ( 34 - 41 \beta_{1} - 3 \beta_{2} ) q^{59} + ( 207 + 30 \beta_{1} + 4 \beta_{2} ) q^{61} + ( -40 + 25 \beta_{1} + 5 \beta_{2} ) q^{65} + ( -282 + 26 \beta_{1} - 6 \beta_{2} ) q^{67} + ( -190 + 17 \beta_{1} - 15 \beta_{2} ) q^{71} + ( 430 + 11 \beta_{1} + 15 \beta_{2} ) q^{73} + ( -101 - 28 \beta_{1} - 13 \beta_{2} ) q^{77} + ( -207 + 11 \beta_{1} + 3 \beta_{2} ) q^{79} + ( -414 + 60 \beta_{1} + 15 \beta_{2} ) q^{83} + ( -30 - 5 \beta_{1} - 10 \beta_{2} ) q^{85} + ( -711 + 33 \beta_{1} + 12 \beta_{2} ) q^{89} + ( 173 + 98 \beta_{1} + 19 \beta_{2} ) q^{91} + ( 325 + 35 \beta_{1} - 5 \beta_{2} ) q^{95} + ( 474 + 32 \beta_{1} + 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 44 q^{7} + O(q^{10}) \) \( 3 q - 15 q^{5} - 44 q^{7} + 38 q^{11} + 28 q^{13} + 19 q^{17} - 187 q^{19} - 81 q^{23} + 75 q^{25} - 160 q^{29} - 227 q^{31} + 220 q^{35} + 78 q^{37} + 338 q^{41} - 22 q^{43} - 472 q^{47} - 197 q^{49} - 521 q^{53} - 190 q^{55} + 140 q^{59} + 595 q^{61} - 140 q^{65} - 878 q^{67} - 602 q^{71} + 1294 q^{73} - 288 q^{77} - 629 q^{79} - 1287 q^{83} - 95 q^{85} - 2154 q^{89} + 440 q^{91} + 935 q^{95} + 1392 q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 4 \nu - 61 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 63\)\()/4\)

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{3} + 44 T_{7}^{2} + 552 T_{7} + 1800 \)
\( T_{11}^{3} - 38 T_{11}^{2} - 2612 T_{11} + 83280 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 5 + T )^{3} \)
$7$ \( 1800 + 552 T + 44 T^{2} + T^{3} \)
$11$ \( 83280 - 2612 T - 38 T^{2} + T^{3} \)
$13$ \( 100120 - 4576 T - 28 T^{2} + T^{3} \)
$17$ \( 553887 - 11477 T - 19 T^{2} + T^{3} \)
$19$ \( -525871 + 3587 T + 187 T^{2} + T^{3} \)
$23$ \( -2043981 - 23301 T + 81 T^{2} + T^{3} \)
$29$ \( -7892760 - 47768 T + 160 T^{2} + T^{3} \)
$31$ \( 246321 - 17973 T + 227 T^{2} + T^{3} \)
$37$ \( 13637080 - 99924 T - 78 T^{2} + T^{3} \)
$41$ \( 12116640 - 42812 T - 338 T^{2} + T^{3} \)
$43$ \( -18464560 - 159916 T + 22 T^{2} + T^{3} \)
$47$ \( -283200 + 54208 T + 472 T^{2} + T^{3} \)
$53$ \( -939789 + 61387 T + 521 T^{2} + T^{3} \)
$59$ \( 34131480 - 166448 T - 140 T^{2} + T^{3} \)
$61$ \( 1782607 - 2749 T - 595 T^{2} + T^{3} \)
$67$ \( -11295000 + 75948 T + 878 T^{2} + T^{3} \)
$71$ \( -280550880 - 583652 T + 602 T^{2} + T^{3} \)
$73$ \( 404091280 - 95908 T - 1294 T^{2} + T^{3} \)
$79$ \( 2010303 + 97059 T + 629 T^{2} + T^{3} \)
$83$ \( -346404411 - 365877 T + 1287 T^{2} + T^{3} \)
$89$ \( 74325600 + 1057572 T + 2154 T^{2} + T^{3} \)
$97$ \( -63595520 + 543936 T - 1392 T^{2} + T^{3} \)
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