Properties

Label 3120.2.a.bj
Level $3120$
Weight $2$
Character orbit 3120.a
Self dual yes
Analytic conductor $24.913$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 195)
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.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + \beta_1 q^{7} + q^{9} + \beta_1 q^{11} + q^{13} - q^{15} + (\beta_{2} + \beta_1) q^{17} + ( - \beta_{2} - 2) q^{19} + \beta_1 q^{21} + ( - \beta_{2} - \beta_1 + 2) q^{23} + q^{25} + q^{27} + 6 q^{29} + (\beta_{2} - 2) q^{31} + \beta_1 q^{33} - \beta_1 q^{35} + (\beta_{2} - \beta_1 + 4) q^{37} + q^{39} + ( - \beta_{2} - \beta_1) q^{41} + 2 \beta_{2} q^{43} - q^{45} + ( - \beta_{2} + 6) q^{47} + ( - \beta_{2} - 3 \beta_1 + 3) q^{49} + (\beta_{2} + \beta_1) q^{51} + (\beta_{2} + \beta_1 + 4) q^{53} - \beta_1 q^{55} + ( - \beta_{2} - 2) q^{57} + ( - \beta_{2} - 2 \beta_1 + 2) q^{59} + (\beta_{2} + 3 \beta_1 + 4) q^{61} + \beta_1 q^{63} - q^{65} + ( - \beta_{2} - 2 \beta_1 - 2) q^{67} + ( - \beta_{2} - \beta_1 + 2) q^{69} + (\beta_1 + 4) q^{71} + ( - 2 \beta_{2} - 2) q^{73} + q^{75} + ( - \beta_{2} - 3 \beta_1 + 10) q^{77} + (\beta_{2} - \beta_1 - 2) q^{79} + q^{81} + (\beta_{2} + 2 \beta_1 - 2) q^{83} + ( - \beta_{2} - \beta_1) q^{85} + 6 q^{87} + (\beta_{2} + \beta_1 + 4) q^{89} + \beta_1 q^{91} + (\beta_{2} - 2) q^{93} + (\beta_{2} + 2) q^{95} + (\beta_{2} + \beta_1 - 8) q^{97} + \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 6 ) / 2 \) 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
−1.81361
2.34292
0.470683
0 1.00000 0 −1.00000 0 −4.91638 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 1.19656 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 2.71982 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 3120.2.a.bj 3
3.b odd 2 1 9360.2.a.dd 3
4.b odd 2 1 195.2.a.e 3
12.b even 2 1 585.2.a.n 3
20.d odd 2 1 975.2.a.o 3
20.e even 4 2 975.2.c.i 6
28.d even 2 1 9555.2.a.bq 3
52.b odd 2 1 2535.2.a.bc 3
60.h even 2 1 2925.2.a.bh 3
60.l odd 4 2 2925.2.c.w 6
156.h even 2 1 7605.2.a.bx 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.e 3 4.b odd 2 1
585.2.a.n 3 12.b even 2 1
975.2.a.o 3 20.d odd 2 1
975.2.c.i 6 20.e even 4 2
2535.2.a.bc 3 52.b odd 2 1
2925.2.a.bh 3 60.h even 2 1
2925.2.c.w 6 60.l odd 4 2
3120.2.a.bj 3 1.a even 1 1 trivial
7605.2.a.bx 3 156.h even 2 1
9360.2.a.dd 3 3.b odd 2 1
9555.2.a.bq 3 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3120))\):

\( T_{7}^{3} + T_{7}^{2} - 16T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 16T_{11} + 16 \) Copy content Toggle raw display
\( T_{17}^{3} + T_{17}^{2} - 32T_{17} - 76 \) Copy content Toggle raw display
\( T_{19}^{3} + 6T_{19}^{2} - 16T_{19} - 64 \) Copy content Toggle raw display
\( T_{31}^{3} + 6T_{31}^{2} - 16T_{31} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 16 T + 16 \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 16 T + 16 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + T^{2} - 32 T - 76 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} - 16 T - 64 \) Copy content Toggle raw display
$23$ \( T^{3} - 7 T^{2} - 16 T + 128 \) Copy content Toggle raw display
$29$ \( (T - 6)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} - 16 T - 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 13T^{2} + 316 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 32 T + 76 \) Copy content Toggle raw display
$43$ \( T^{3} - 112T + 128 \) Copy content Toggle raw display
$47$ \( T^{3} - 18 T^{2} + 80 T - 64 \) Copy content Toggle raw display
$53$ \( T^{3} - 11 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 48 T + 128 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} - 112 T + 844 \) Copy content Toggle raw display
$67$ \( T^{3} + 4 T^{2} - 64 T - 128 \) Copy content Toggle raw display
$71$ \( T^{3} - 11 T^{2} + 24 T + 32 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} - 100 T - 344 \) Copy content Toggle raw display
$79$ \( T^{3} + 5 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} - 48 T - 128 \) Copy content Toggle raw display
$89$ \( T^{3} - 11 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$97$ \( T^{3} + 25 T^{2} + 176 T + 244 \) Copy content Toggle raw display
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