Properties

Label 3234.2.a.bg
Level $3234$
Weight $2$
Character orbit 3234.a
Self dual yes
Analytic conductor $25.824$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 462)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - \beta_{2} q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - \beta_{2} q^{5} - q^{6} + q^{8} + q^{9} - \beta_{2} q^{10} + q^{11} - q^{12} - 2 \beta_{2} q^{13} + \beta_{2} q^{15} + q^{16} - q^{17} + q^{18} + (\beta_{2} - \beta_1 + 1) q^{19} - \beta_{2} q^{20} + q^{22} + (\beta_1 + 3) q^{23} - q^{24} + ( - \beta_{2} - \beta_1 + 1) q^{25} - 2 \beta_{2} q^{26} - q^{27} + ( - \beta_{2} - \beta_1 + 3) q^{29} + \beta_{2} q^{30} + (2 \beta_{2} + 2 \beta_1 - 2) q^{31} + q^{32} - q^{33} - q^{34} + q^{36} + ( - \beta_{2} + \beta_1 + 3) q^{37} + (\beta_{2} - \beta_1 + 1) q^{38} + 2 \beta_{2} q^{39} - \beta_{2} q^{40} + (\beta_{2} + \beta_1 - 4) q^{41} + ( - \beta_{2} + \beta_1 + 3) q^{43} + q^{44} - \beta_{2} q^{45} + (\beta_1 + 3) q^{46} + ( - \beta_1 + 5) q^{47} - q^{48} + ( - \beta_{2} - \beta_1 + 1) q^{50} + q^{51} - 2 \beta_{2} q^{52} + 2 \beta_{2} q^{53} - q^{54} - \beta_{2} q^{55} + ( - \beta_{2} + \beta_1 - 1) q^{57} + ( - \beta_{2} - \beta_1 + 3) q^{58} + (3 \beta_{2} - \beta_1 - 3) q^{59} + \beta_{2} q^{60} + (3 \beta_{2} + 2 \beta_1 + 2) q^{61} + (2 \beta_{2} + 2 \beta_1 - 2) q^{62} + q^{64} + ( - 2 \beta_{2} - 2 \beta_1 + 12) q^{65} - q^{66} + ( - \beta_{2} - \beta_1 + 2) q^{67} - q^{68} + ( - \beta_1 - 3) q^{69} + ( - \beta_{2} - \beta_1 + 1) q^{71} + q^{72} - 2 \beta_1 q^{73} + ( - \beta_{2} + \beta_1 + 3) q^{74} + (\beta_{2} + \beta_1 - 1) q^{75} + (\beta_{2} - \beta_1 + 1) q^{76} + 2 \beta_{2} q^{78} + (3 \beta_{2} + 4) q^{79} - \beta_{2} q^{80} + q^{81} + (\beta_{2} + \beta_1 - 4) q^{82} + (3 \beta_{2} - \beta_1 + 2) q^{83} + \beta_{2} q^{85} + ( - \beta_{2} + \beta_1 + 3) q^{86} + (\beta_{2} + \beta_1 - 3) q^{87} + q^{88} + ( - 2 \beta_{2} + 12) q^{89} - \beta_{2} q^{90} + (\beta_1 + 3) q^{92} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{93} + ( - \beta_1 + 5) q^{94} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{95} - q^{96} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{11} - 3 q^{12} + 3 q^{16} - 3 q^{17} + 3 q^{18} + 3 q^{19} + 3 q^{22} + 9 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 9 q^{29} - 6 q^{31} + 3 q^{32} - 3 q^{33} - 3 q^{34} + 3 q^{36} + 9 q^{37} + 3 q^{38} - 12 q^{41} + 9 q^{43} + 3 q^{44} + 9 q^{46} + 15 q^{47} - 3 q^{48} + 3 q^{50} + 3 q^{51} - 3 q^{54} - 3 q^{57} + 9 q^{58} - 9 q^{59} + 6 q^{61} - 6 q^{62} + 3 q^{64} + 36 q^{65} - 3 q^{66} + 6 q^{67} - 3 q^{68} - 9 q^{69} + 3 q^{71} + 3 q^{72} + 9 q^{74} - 3 q^{75} + 3 q^{76} + 12 q^{79} + 3 q^{81} - 12 q^{82} + 6 q^{83} + 9 q^{86} - 9 q^{87} + 3 q^{88} + 36 q^{89} + 9 q^{92} + 6 q^{93} + 15 q^{94} - 24 q^{95} - 3 q^{96} - 9 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 8 ) / 2 \) 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
−2.14510
2.66908
−0.523976
1.00000 −1.00000 1.00000 −2.74657 −1.00000 0 1.00000 1.00000 −2.74657
1.2 1.00000 −1.00000 1.00000 −0.454904 −1.00000 0 1.00000 1.00000 −0.454904
1.3 1.00000 −1.00000 1.00000 3.20147 −1.00000 0 1.00000 1.00000 3.20147
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 3234.2.a.bg 3
3.b odd 2 1 9702.2.a.du 3
7.b odd 2 1 3234.2.a.bi 3
7.d odd 6 2 462.2.i.f 6
21.c even 2 1 9702.2.a.dt 3
21.g even 6 2 1386.2.k.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.f 6 7.d odd 6 2
1386.2.k.w 6 21.g even 6 2
3234.2.a.bg 3 1.a even 1 1 trivial
3234.2.a.bi 3 7.b odd 2 1
9702.2.a.dt 3 21.c even 2 1
9702.2.a.du 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{3} - 9T_{5} - 4 \) Copy content Toggle raw display
\( T_{13}^{3} - 36T_{13} - 32 \) Copy content Toggle raw display
\( T_{17} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 9T - 4 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 36T - 32 \) Copy content Toggle raw display
$17$ \( (T + 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 36 T - 36 \) Copy content Toggle raw display
$23$ \( T^{3} - 9 T^{2} + 3 T + 21 \) Copy content Toggle raw display
$29$ \( T^{3} - 9T^{2} + 92 \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} - 96 T - 512 \) Copy content Toggle raw display
$37$ \( T^{3} - 9 T^{2} - 12 T + 164 \) Copy content Toggle raw display
$41$ \( T^{3} + 12 T^{2} + 21 T - 82 \) Copy content Toggle raw display
$43$ \( T^{3} - 9 T^{2} - 12 T + 164 \) Copy content Toggle raw display
$47$ \( T^{3} - 15 T^{2} + 51 T + 19 \) Copy content Toggle raw display
$53$ \( T^{3} - 36T + 32 \) Copy content Toggle raw display
$59$ \( T^{3} + 9 T^{2} - 96 T - 768 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} - 129 T + 226 \) Copy content Toggle raw display
$67$ \( T^{3} - 6 T^{2} - 15 T + 84 \) Copy content Toggle raw display
$71$ \( T^{3} - 3 T^{2} - 24 T + 64 \) Copy content Toggle raw display
$73$ \( T^{3} - 96T + 192 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} - 33 T + 368 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 111 T - 188 \) Copy content Toggle raw display
$89$ \( T^{3} - 36 T^{2} + 396 T - 1328 \) Copy content Toggle raw display
$97$ \( T^{3} + 9 T^{2} - 81 T + 7 \) Copy content Toggle raw display
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