Properties

Label 3234.2.a.bk
Level $3234$
Weight $2$
Character orbit 3234.a
Self dual yes
Analytic conductor $25.824$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.14336.1
Defining polynomial: \( x^{4} - 8x^{2} + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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} + (\beta_{3} - \beta_1) q^{13} + \beta_{2} q^{15} + q^{16} + ( - 2 \beta_{3} + \beta_{2}) q^{17} - q^{18} + (\beta_{3} - \beta_{2} + \beta_1) q^{19} + \beta_{2} q^{20} + q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{23} - q^{24} + (2 \beta_{3} + 3) q^{25} + ( - \beta_{3} + \beta_1) q^{26} + q^{27} + 2 \beta_{3} q^{29} - \beta_{2} q^{30} + ( - \beta_{3} - \beta_{2} + 4) q^{31} - q^{32} - q^{33} + (2 \beta_{3} - \beta_{2}) q^{34} + q^{36} + ( - 4 \beta_{3} + \beta_1 + 2) q^{37} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{38} + (\beta_{3} - \beta_1) q^{39} - \beta_{2} q^{40} + (2 \beta_{3} - \beta_{2}) q^{41} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{43} - q^{44} + \beta_{2} q^{45} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{46} + ( - \beta_{3} - \beta_{2} + \beta_1 + 4) q^{47} + q^{48} + ( - 2 \beta_{3} - 3) q^{50} + ( - 2 \beta_{3} + \beta_{2}) q^{51} + (\beta_{3} - \beta_1) q^{52} + (4 \beta_{3} + \beta_1 + 2) q^{53} - q^{54} - \beta_{2} q^{55} + (\beta_{3} - \beta_{2} + \beta_1) q^{57} - 2 \beta_{3} q^{58} + (2 \beta_{3} + 8) q^{59} + \beta_{2} q^{60} + ( - \beta_{3} - \beta_1 + 4) q^{61} + (\beta_{3} + \beta_{2} - 4) q^{62} + q^{64} + ( - 8 \beta_{3} + \beta_1 - 4) q^{65} + q^{66} + (2 \beta_{3} + 4 \beta_{2} - \beta_1 + 4) q^{67} + ( - 2 \beta_{3} + \beta_{2}) q^{68} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{69} + (2 \beta_{3} + 2 \beta_1) q^{71} - q^{72} + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1) q^{73} + (4 \beta_{3} - \beta_1 - 2) q^{74} + (2 \beta_{3} + 3) q^{75} + (\beta_{3} - \beta_{2} + \beta_1) q^{76} + ( - \beta_{3} + \beta_1) q^{78} + ( - 6 \beta_{3} + \beta_1 + 4) q^{79} + \beta_{2} q^{80} + q^{81} + ( - 2 \beta_{3} + \beta_{2}) q^{82} + (3 \beta_{3} + \beta_{2}) q^{83} + (2 \beta_{3} - 2 \beta_1 + 8) q^{85} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{86} + 2 \beta_{3} q^{87} + q^{88} + ( - \beta_{3} - 3 \beta_1 + 4) q^{89} - \beta_{2} q^{90} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{92} + ( - \beta_{3} - \beta_{2} + 4) q^{93} + (\beta_{3} + \beta_{2} - \beta_1 - 4) q^{94} + (6 \beta_{3} + \beta_1 - 4) q^{95} - q^{96} + ( - \beta_{3} + 2 \beta_1 - 4) q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{11} + 4 q^{12} + 4 q^{16} - 4 q^{18} + 4 q^{22} - 8 q^{23} - 4 q^{24} + 12 q^{25} + 4 q^{27} + 16 q^{31} - 4 q^{32} - 4 q^{33} + 4 q^{36} + 8 q^{37} + 8 q^{43} - 4 q^{44} + 8 q^{46} + 16 q^{47} + 4 q^{48} - 12 q^{50} + 8 q^{53} - 4 q^{54} + 32 q^{59} + 16 q^{61} - 16 q^{62} + 4 q^{64} - 16 q^{65} + 4 q^{66} + 16 q^{67} - 8 q^{69} - 4 q^{72} - 8 q^{74} + 12 q^{75} + 16 q^{79} + 4 q^{81} + 32 q^{85} - 8 q^{86} + 4 q^{88} + 16 q^{89} - 8 q^{92} + 16 q^{93} - 16 q^{94} - 16 q^{95} - 4 q^{96} - 16 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 8x^{2} + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 2\beta_1 \) 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
−2.32685
1.60804
−1.60804
2.32685
−1.00000 1.00000 1.00000 −3.29066 −1.00000 0 −1.00000 1.00000 3.29066
1.2 −1.00000 1.00000 1.00000 −2.27411 −1.00000 0 −1.00000 1.00000 2.27411
1.3 −1.00000 1.00000 1.00000 2.27411 −1.00000 0 −1.00000 1.00000 −2.27411
1.4 −1.00000 1.00000 1.00000 3.29066 −1.00000 0 −1.00000 1.00000 −3.29066
\(n\): 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.bk yes 4
3.b odd 2 1 9702.2.a.ec 4
7.b odd 2 1 3234.2.a.bj 4
21.c even 2 1 9702.2.a.eb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bj 4 7.b odd 2 1
3234.2.a.bk yes 4 1.a even 1 1 trivial
9702.2.a.eb 4 21.c even 2 1
9702.2.a.ec 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{4} - 16T_{5}^{2} + 56 \) Copy content Toggle raw display
\( T_{13}^{4} - 36T_{13}^{2} - 32T_{13} + 164 \) Copy content Toggle raw display
\( T_{17}^{4} - 32T_{17}^{2} + 32T_{17} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 16T^{2} + 56 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 36 T^{2} - 32 T + 164 \) Copy content Toggle raw display
$17$ \( T^{4} - 32 T^{2} + 32 T - 8 \) Copy content Toggle raw display
$19$ \( T^{4} - 36 T^{2} + 80 T - 4 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} - 56 T^{2} + \cdots - 1168 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 16 T^{3} + 76 T^{2} + \cdots - 100 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} - 72 T^{2} + 480 T - 400 \) Copy content Toggle raw display
$41$ \( T^{4} - 32 T^{2} - 32 T - 8 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} - 56 T^{2} + \cdots - 1168 \) Copy content Toggle raw display
$47$ \( T^{4} - 16 T^{3} + 60 T^{2} - 48 T - 4 \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} - 72 T^{2} + 224 T + 112 \) Copy content Toggle raw display
$59$ \( (T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 16 T^{3} + 60 T^{2} + \cdots - 284 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} - 144 T^{2} + \cdots + 3872 \) Copy content Toggle raw display
$71$ \( T^{4} - 144 T^{2} - 256 T + 2624 \) Copy content Toggle raw display
$73$ \( T^{4} - 176 T^{2} - 448 T + 184 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} - 80 T^{2} + \cdots - 224 \) Copy content Toggle raw display
$83$ \( T^{4} - 52 T^{2} - 48 T + 92 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} - 196 T^{2} + \cdots + 12004 \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} - 36 T^{2} + \cdots + 1988 \) Copy content Toggle raw display
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