Properties

Label 4056.2.a.w
Level $4056$
Weight $2$
Character orbit 4056.a
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) 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 \(\beta = \sqrt{13}\). 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 \( q + q^{3} + q^{5} + ( - \beta - 1) q^{7} + q^{9} + (\beta + 1) q^{11} + q^{15} + ( - \beta - 2) q^{17} + ( - \beta + 3) q^{19} + ( - \beta - 1) q^{21} + ( - \beta - 1) q^{23} - 4 q^{25} + q^{27} + 3 q^{29} + (2 \beta + 2) q^{31} + (\beta + 1) q^{33} + ( - \beta - 1) q^{35} + (\beta + 6) q^{37} + ( - \beta + 8) q^{41} + (\beta + 1) q^{43} + q^{45} + (\beta + 5) q^{47} + (2 \beta + 7) q^{49} + ( - \beta - 2) q^{51} + 3 q^{53} + (\beta + 1) q^{55} + ( - \beta + 3) q^{57} + (2 \beta - 2) q^{59} + (\beta - 4) q^{61} + ( - \beta - 1) q^{63} + ( - \beta + 7) q^{67} + ( - \beta - 1) q^{69} + (\beta + 13) q^{71} - 7 q^{73} - 4 q^{75} + ( - 2 \beta - 14) q^{77} + 12 q^{79} + q^{81} + ( - 3 \beta + 1) q^{83} + ( - \beta - 2) q^{85} + 3 q^{87} + ( - 4 \beta + 2) q^{89} + (2 \beta + 2) q^{93} + ( - \beta + 3) q^{95} + (2 \beta - 4) q^{97} + (\beta + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} - 2 q^{21} - 2 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} + 4 q^{31} + 2 q^{33} - 2 q^{35} + 12 q^{37} + 16 q^{41} + 2 q^{43} + 2 q^{45} + 10 q^{47} + 14 q^{49} - 4 q^{51} + 6 q^{53} + 2 q^{55} + 6 q^{57} - 4 q^{59} - 8 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 26 q^{71} - 14 q^{73} - 8 q^{75} - 28 q^{77} + 24 q^{79} + 2 q^{81} + 2 q^{83} - 4 q^{85} + 6 q^{87} + 4 q^{89} + 4 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 1.00000 0 1.00000 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 2.60555 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.w 2
4.b odd 2 1 8112.2.a.bn 2
13.b even 2 1 4056.2.a.v 2
13.d odd 4 2 4056.2.c.l 4
13.e even 6 2 312.2.q.d 4
39.h odd 6 2 936.2.t.e 4
52.b odd 2 1 8112.2.a.bl 2
52.i odd 6 2 624.2.q.i 4
156.r even 6 2 1872.2.t.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.d 4 13.e even 6 2
624.2.q.i 4 52.i odd 6 2
936.2.t.e 4 39.h odd 6 2
1872.2.t.q 4 156.r even 6 2
4056.2.a.v 2 13.b even 2 1
4056.2.a.w 2 1.a even 1 1 trivial
4056.2.c.l 4 13.d odd 4 2
8112.2.a.bl 2 52.b odd 2 1
8112.2.a.bn 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4056))\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 23 \) Copy content Toggle raw display
$41$ \( T^{2} - 16T + 51 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 3 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$71$ \( T^{2} - 26T + 156 \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( (T - 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 116 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 204 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
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