Properties

Label 4056.2.a.z
Level $4056$
Weight $2$
Character orbit 4056.a
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
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.

\(f(q)\) \(=\) \( q - q^{3} - \beta_{2} q^{5} + ( - \beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_{2} q^{5} + ( - \beta_1 + 1) q^{7} + q^{9} + (\beta_{2} - \beta_1) q^{11} + \beta_{2} q^{15} - \beta_1 q^{17} + (\beta_{2} + \beta_1 + 2) q^{19} + (\beta_1 - 1) q^{21} + (\beta_{2} - \beta_1) q^{23} + (\beta_{2} - 2 \beta_1 + 5) q^{25} - q^{27} + ( - \beta_{2} - 4) q^{29} + ( - \beta_{2} + 1) q^{31} + ( - \beta_{2} + \beta_1) q^{33} + ( - 3 \beta_{2} - \beta_1 + 4) q^{35} + (\beta_1 + 2) q^{37} + ( - 2 \beta_{2} + \beta_1) q^{41} + ( - \beta_1 - 7) q^{43} - \beta_{2} q^{45} + (\beta_{2} - \beta_1 - 4) q^{47} + ( - 3 \beta_{2} + 8) q^{49} + \beta_1 q^{51} + ( - 3 \beta_{2} - 2) q^{53} + ( - 3 \beta_{2} + \beta_1 - 6) q^{55} + ( - \beta_{2} - \beta_1 - 2) q^{57} + (2 \beta_1 + 2) q^{59} + (\beta_{2} + \beta_1 + 1) q^{61} + ( - \beta_1 + 1) q^{63} + ( - \beta_1 + 9) q^{67} + ( - \beta_{2} + \beta_1) q^{69} + (\beta_{2} - \beta_1 + 4) q^{71} + (2 \beta_{2} - 3) q^{73} + ( - \beta_{2} + 2 \beta_1 - 5) q^{75} + (2 \beta_1 + 10) q^{77} + (3 \beta_{2} - 2 \beta_1 + 3) q^{79} + q^{81} + (3 \beta_{2} - \beta_1 - 6) q^{83} + ( - 2 \beta_{2} - \beta_1 + 4) q^{85} + (\beta_{2} + 4) q^{87} + (2 \beta_{2} + 8) q^{89} + (\beta_{2} - 1) q^{93} + ( - \beta_{2} + 3 \beta_1 - 14) q^{95} + ( - \beta_{2} + 7) q^{97} + (\beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 6 q^{19} - 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} + 12 q^{35} + 6 q^{37} - 21 q^{43} - 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} - 6 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 27 q^{67} + 12 q^{71} - 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} - 18 q^{83} + 12 q^{85} + 12 q^{87} + 24 q^{89} - 3 q^{93} - 42 q^{95} + 21 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 8 ) / 2 \) 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
−2.36147
2.52892
−0.167449
0 −1.00000 0 −3.93800 0 1.78493 0 1.00000 0
1.2 0 −1.00000 0 0.133492 0 −3.92434 0 1.00000 0
1.3 0 −1.00000 0 3.80451 0 5.13941 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4056.2.a.z 3
4.b odd 2 1 8112.2.a.ck 3
13.b even 2 1 4056.2.a.y 3
13.c even 3 2 312.2.q.e 6
13.d odd 4 2 4056.2.c.m 6
39.i odd 6 2 936.2.t.h 6
52.b odd 2 1 8112.2.a.cl 3
52.j odd 6 2 624.2.q.j 6
156.p even 6 2 1872.2.t.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.e 6 13.c even 3 2
624.2.q.j 6 52.j odd 6 2
936.2.t.h 6 39.i odd 6 2
1872.2.t.u 6 156.p even 6 2
4056.2.a.y 3 13.b even 2 1
4056.2.a.z 3 1.a even 1 1 trivial
4056.2.c.m 6 13.d odd 4 2
8112.2.a.ck 3 4.b odd 2 1
8112.2.a.cl 3 52.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4056))\):

\( T_{5}^{3} - 15T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 18T_{7} + 36 \) 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
$5$ \( T^{3} - 15T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$11$ \( T^{3} - 24T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 21T + 16 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 36 T + 208 \) Copy content Toggle raw display
$23$ \( T^{3} - 24T + 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 12 T^{2} + 33 T + 6 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 12 T + 16 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 9 T + 18 \) Copy content Toggle raw display
$41$ \( T^{3} - 57T + 156 \) Copy content Toggle raw display
$43$ \( T^{3} + 21 T^{2} + 126 T + 212 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} + 24 T - 24 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 123 T - 208 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} - 72 T + 32 \) Copy content Toggle raw display
$61$ \( T^{3} - 3 T^{2} - 45 T + 167 \) Copy content Toggle raw display
$67$ \( T^{3} - 27 T^{2} + 222 T - 524 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} + 24 T + 40 \) Copy content Toggle raw display
$73$ \( T^{3} + 9 T^{2} - 33 T - 169 \) Copy content Toggle raw display
$79$ \( T^{3} - 9 T^{2} - 120 T - 16 \) Copy content Toggle raw display
$83$ \( T^{3} + 18 T^{2} - 12 T - 992 \) Copy content Toggle raw display
$89$ \( T^{3} - 24 T^{2} + 132 T - 48 \) Copy content Toggle raw display
$97$ \( T^{3} - 21 T^{2} + 132 T - 236 \) Copy content Toggle raw display
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