Properties

Label 3234.2.a.x
Level $3234$
Weight $2$
Character orbit 3234.a
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) 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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{8} + q^{9} - \beta q^{10} + q^{11} - q^{12} - 2 q^{13} - \beta q^{15} + q^{16} - \beta q^{17} - q^{18} + ( - \beta - 2) q^{19} + \beta q^{20} - q^{22} + 2 \beta q^{23} + q^{24} + 7 q^{25} + 2 q^{26} - q^{27} - 6 q^{29} + \beta q^{30} + ( - \beta - 2) q^{31} - q^{32} - q^{33} + \beta q^{34} + q^{36} + ( - 2 \beta + 2) q^{37} + (\beta + 2) q^{38} + 2 q^{39} - \beta q^{40} - \beta q^{41} + ( - 2 \beta - 4) q^{43} + q^{44} + \beta q^{45} - 2 \beta q^{46} + ( - \beta - 6) q^{47} - q^{48} - 7 q^{50} + \beta q^{51} - 2 q^{52} + ( - 2 \beta + 6) q^{53} + q^{54} + \beta q^{55} + (\beta + 2) q^{57} + 6 q^{58} - 2 \beta q^{59} - \beta q^{60} - 2 q^{61} + (\beta + 2) q^{62} + q^{64} - 2 \beta q^{65} + q^{66} + (2 \beta + 8) q^{67} - \beta q^{68} - 2 \beta q^{69} - 12 q^{71} - q^{72} + ( - \beta + 4) q^{73} + (2 \beta - 2) q^{74} - 7 q^{75} + ( - \beta - 2) q^{76} - 2 q^{78} + ( - 2 \beta - 4) q^{79} + \beta q^{80} + q^{81} + \beta q^{82} + ( - 3 \beta + 6) q^{83} - 12 q^{85} + (2 \beta + 4) q^{86} + 6 q^{87} - q^{88} + (2 \beta - 6) q^{89} - \beta q^{90} + 2 \beta q^{92} + (\beta + 2) q^{93} + (\beta + 6) q^{94} + ( - 2 \beta - 12) q^{95} + q^{96} - 2 q^{97} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{18} - 4 q^{19} - 2 q^{22} + 2 q^{24} + 14 q^{25} + 4 q^{26} - 2 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{32} - 2 q^{33} + 2 q^{36} + 4 q^{37} + 4 q^{38} + 4 q^{39} - 8 q^{43} + 2 q^{44} - 12 q^{47} - 2 q^{48} - 14 q^{50} - 4 q^{52} + 12 q^{53} + 2 q^{54} + 4 q^{57} + 12 q^{58} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 2 q^{66} + 16 q^{67} - 24 q^{71} - 2 q^{72} + 8 q^{73} - 4 q^{74} - 14 q^{75} - 4 q^{76} - 4 q^{78} - 8 q^{79} + 2 q^{81} + 12 q^{83} - 24 q^{85} + 8 q^{86} + 12 q^{87} - 2 q^{88} - 12 q^{89} + 4 q^{93} + 12 q^{94} - 24 q^{95} + 2 q^{96} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
−1.00000 −1.00000 1.00000 −3.46410 1.00000 0 −1.00000 1.00000 3.46410
1.2 −1.00000 −1.00000 1.00000 3.46410 1.00000 0 −1.00000 1.00000 −3.46410
\(n\): 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.x 2
3.b odd 2 1 9702.2.a.dd 2
7.b odd 2 1 462.2.a.h 2
21.c even 2 1 1386.2.a.p 2
28.d even 2 1 3696.2.a.bc 2
77.b even 2 1 5082.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.h 2 7.b odd 2 1
1386.2.a.p 2 21.c even 2 1
3234.2.a.x 2 1.a even 1 1 trivial
3696.2.a.bc 2 28.d even 2 1
5082.2.a.bu 2 77.b even 2 1
9702.2.a.dd 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 48 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$41$ \( T^{2} - 12 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} - 48 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 72 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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