Properties

Label 2160.4.a.bq
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: 3.3.1772.1
Defining polynomial: \( x^{3} - x^{2} - 12x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + 5 q^{5} + (\beta_{2} + 3 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + (\beta_{2} + 3 \beta_1) q^{7} + ( - \beta_{2} + 5 \beta_1 - 3) q^{11} + ( - \beta_{2} + 2 \beta_1 + 2) q^{13} + ( - 2 \beta_{2} - 5 \beta_1 + 54) q^{17} + ( - 8 \beta_{2} + \beta_1 + 19) q^{19} + ( - 5 \beta_{2} + 5 \beta_1 - 95) q^{23} + 25 q^{25} + (6 \beta_{2} + 25 \beta_1 + 28) q^{29} + ( - 9 \beta_{2} - 17 \beta_1 + 47) q^{31} + (5 \beta_{2} + 15 \beta_1) q^{35} + (20 \beta_{2} + 25 \beta_1 - 143) q^{37} + (9 \beta_{2} + 10 \beta_1 + 187) q^{41} + (3 \beta_{2} + 29 \beta_1 + 255) q^{43} + (32 \beta_{2} + 5 \beta_1 + 36) q^{47} + (3 \beta_{2} - 41 \beta_1 + 205) q^{49} + (17 \beta_{2} + 30 \beta_1 - 149) q^{53} + ( - 5 \beta_{2} + 25 \beta_1 - 15) q^{55} + ( - 17 \beta_{2} - 121) q^{59} + ( - 3 \beta_{2} + 31 \beta_1 - 60) q^{61} + ( - 5 \beta_{2} + 10 \beta_1 + 10) q^{65} + (31 \beta_{2} + 3 \beta_1 - 12) q^{67} + (53 \beta_{2} + 30 \beta_1 - 41) q^{71} + ( - 14 \beta_{2} + 3 \beta_1 - 323) q^{73} + ( - 46 \beta_{2} - 80 \beta_1 + 692) q^{77} + ( - 23 \beta_{2} - 54 \beta_1 - 312) q^{79} + ( - 21 \beta_{2} - 60 \beta_1 - 63) q^{83} + ( - 10 \beta_{2} - 25 \beta_1 + 270) q^{85} + (15 \beta_{2} + 60 \beta_1 + 315) q^{89} + ( - 26 \beta_{2} - 23 \beta_1 + 227) q^{91} + ( - 40 \beta_{2} + 5 \beta_1 + 95) q^{95} + (28 \beta_{2} - \beta_1 + 231) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 4 q^{7} - 5 q^{11} + 7 q^{13} + 155 q^{17} + 50 q^{19} - 285 q^{23} + 75 q^{25} + 115 q^{29} + 115 q^{31} + 20 q^{35} - 384 q^{37} + 580 q^{41} + 797 q^{43} + 145 q^{47} + 577 q^{49} - 400 q^{53} - 25 q^{55} - 380 q^{59} - 152 q^{61} + 35 q^{65} - 2 q^{67} - 40 q^{71} - 980 q^{73} + 1950 q^{77} - 1013 q^{79} - 270 q^{83} + 775 q^{85} + 1020 q^{89} + 632 q^{91} + 250 q^{95} + 720 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 12x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{2} + 2\nu + 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 2\nu - 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - \beta _1 + 25 ) / 3 \) 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
−3.32803
0.654334
3.67370
0 0 0 5.00000 0 −30.7000 0 0 0
1.2 0 0 0 5.00000 0 11.8065 0 0 0
1.3 0 0 0 5.00000 0 22.8935 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.bq 3
3.b odd 2 1 2160.4.a.bi 3
4.b odd 2 1 135.4.a.h yes 3
12.b even 2 1 135.4.a.e 3
20.d odd 2 1 675.4.a.p 3
20.e even 4 2 675.4.b.n 6
36.f odd 6 2 405.4.e.q 6
36.h even 6 2 405.4.e.v 6
60.h even 2 1 675.4.a.s 3
60.l odd 4 2 675.4.b.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.e 3 12.b even 2 1
135.4.a.h yes 3 4.b odd 2 1
405.4.e.q 6 36.f odd 6 2
405.4.e.v 6 36.h even 6 2
675.4.a.p 3 20.d odd 2 1
675.4.a.s 3 60.h even 2 1
675.4.b.m 6 60.l odd 4 2
675.4.b.n 6 20.e even 4 2
2160.4.a.bi 3 3.b odd 2 1
2160.4.a.bq 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{3} - 4T_{7}^{2} - 795T_{7} + 8298 \) Copy content Toggle raw display
\( T_{11}^{3} + 5T_{11}^{2} - 2888T_{11} - 31260 \) 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
$7$ \( T^{3} - 4 T^{2} - 795 T + 8298 \) Copy content Toggle raw display
$11$ \( T^{3} + 5 T^{2} - 2888 T - 31260 \) Copy content Toggle raw display
$13$ \( T^{3} - 7 T^{2} - 769 T - 6425 \) Copy content Toggle raw display
$17$ \( T^{3} - 155 T^{2} + 5608 T - 41760 \) Copy content Toggle raw display
$19$ \( T^{3} - 50 T^{2} - 16663 T + 368012 \) Copy content Toggle raw display
$23$ \( T^{3} + 285 T^{2} + 16200 T - 553500 \) Copy content Toggle raw display
$29$ \( T^{3} - 115 T^{2} - 47408 T + 6440340 \) Copy content Toggle raw display
$31$ \( T^{3} - 115 T^{2} - 29232 T + 938304 \) Copy content Toggle raw display
$37$ \( T^{3} + 384 T^{2} + \cdots - 22667198 \) Copy content Toggle raw display
$41$ \( T^{3} - 580 T^{2} + 89812 T - 3917280 \) Copy content Toggle raw display
$43$ \( T^{3} - 797 T^{2} + 142520 T + 5357936 \) Copy content Toggle raw display
$47$ \( T^{3} - 145 T^{2} + \cdots - 14388240 \) Copy content Toggle raw display
$53$ \( T^{3} + 400 T^{2} + \cdots - 12658320 \) Copy content Toggle raw display
$59$ \( T^{3} + 380 T^{2} - 27392 T - 5205120 \) Copy content Toggle raw display
$61$ \( T^{3} + 152 T^{2} - 87475 T - 5069066 \) Copy content Toggle raw display
$67$ \( T^{3} + 2 T^{2} - 243999 T - 20769300 \) Copy content Toggle raw display
$71$ \( T^{3} + 40 T^{2} + \cdots - 216071280 \) Copy content Toggle raw display
$73$ \( T^{3} + 980 T^{2} + \cdots + 16447954 \) Copy content Toggle raw display
$79$ \( T^{3} + 1013 T^{2} + \cdots - 90596925 \) Copy content Toggle raw display
$83$ \( T^{3} + 270 T^{2} + \cdots - 84539160 \) Copy content Toggle raw display
$89$ \( T^{3} - 1020 T^{2} + \cdots + 125064000 \) Copy content Toggle raw display
$97$ \( T^{3} - 720 T^{2} + \cdots + 27430558 \) Copy content Toggle raw display
show more
show less