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: \(x^{2} - 3\)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{8} + q^{9} -\beta q^{10} + q^{11} - q^{12} -2 q^{13} -\beta q^{15} + q^{16} -\beta q^{17} - q^{18} + ( -2 - \beta ) q^{19} + \beta q^{20} - q^{22} + 2 \beta q^{23} + q^{24} + 7 q^{25} + 2 q^{26} - q^{27} -6 q^{29} + \beta q^{30} + ( -2 - \beta ) q^{31} - q^{32} - q^{33} + \beta q^{34} + q^{36} + ( 2 - 2 \beta ) q^{37} + ( 2 + \beta ) q^{38} + 2 q^{39} -\beta q^{40} -\beta q^{41} + ( -4 - 2 \beta ) q^{43} + q^{44} + \beta q^{45} -2 \beta q^{46} + ( -6 - \beta ) q^{47} - q^{48} -7 q^{50} + \beta q^{51} -2 q^{52} + ( 6 - 2 \beta ) q^{53} + q^{54} + \beta q^{55} + ( 2 + \beta ) q^{57} + 6 q^{58} -2 \beta q^{59} -\beta q^{60} -2 q^{61} + ( 2 + \beta ) q^{62} + q^{64} -2 \beta q^{65} + q^{66} + ( 8 + 2 \beta ) q^{67} -\beta q^{68} -2 \beta q^{69} -12 q^{71} - q^{72} + ( 4 - \beta ) q^{73} + ( -2 + 2 \beta ) q^{74} -7 q^{75} + ( -2 - \beta ) q^{76} -2 q^{78} + ( -4 - 2 \beta ) q^{79} + \beta q^{80} + q^{81} + \beta q^{82} + ( 6 - 3 \beta ) q^{83} -12 q^{85} + ( 4 + 2 \beta ) q^{86} + 6 q^{87} - q^{88} + ( -6 + 2 \beta ) q^{89} -\beta q^{90} + 2 \beta q^{92} + ( 2 + \beta ) q^{93} + ( 6 + \beta ) q^{94} + ( -12 - 2 \beta ) q^{95} + q^{96} -2 q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{8} + 2q^{9} + 2q^{11} - 2q^{12} - 4q^{13} + 2q^{16} - 2q^{18} - 4q^{19} - 2q^{22} + 2q^{24} + 14q^{25} + 4q^{26} - 2q^{27} - 12q^{29} - 4q^{31} - 2q^{32} - 2q^{33} + 2q^{36} + 4q^{37} + 4q^{38} + 4q^{39} - 8q^{43} + 2q^{44} - 12q^{47} - 2q^{48} - 14q^{50} - 4q^{52} + 12q^{53} + 2q^{54} + 4q^{57} + 12q^{58} - 4q^{61} + 4q^{62} + 2q^{64} + 2q^{66} + 16q^{67} - 24q^{71} - 2q^{72} + 8q^{73} - 4q^{74} - 14q^{75} - 4q^{76} - 4q^{78} - 8q^{79} + 2q^{81} + 12q^{83} - 24q^{85} + 8q^{86} + 12q^{87} - 2q^{88} - 12q^{89} + 4q^{93} + 12q^{94} - 24q^{95} + 2q^{96} - 4q^{97} + 2q^{99} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{2} - 12 \)
\( T_{13} + 2 \)
\( T_{17}^{2} - 12 \)

Hecke characteristic polynomials

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