Properties

Label 3234.2.a.bl
Level $3234$
Weight $2$
Character orbit 3234.a
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 2 \nu^{2} + 4 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 3\)

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
2.68554
−1.74912
0.334904
−1.27133
1.00000 −1.00000 1.00000 −3.79793 −1.00000 0 1.00000 1.00000 −3.79793
1.2 1.00000 −1.00000 1.00000 −2.47363 −1.00000 0 1.00000 1.00000 −2.47363
1.3 1.00000 −1.00000 1.00000 0.473626 −1.00000 0 1.00000 1.00000 0.473626
1.4 1.00000 −1.00000 1.00000 1.79793 −1.00000 0 1.00000 1.00000 1.79793
\(n\): 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.bl 4
3.b odd 2 1 9702.2.a.ea 4
7.b odd 2 1 3234.2.a.bm yes 4
21.c even 2 1 9702.2.a.dz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bl 4 1.a even 1 1 trivial
3234.2.a.bm yes 4 7.b odd 2 1
9702.2.a.dz 4 21.c even 2 1
9702.2.a.ea 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{4} + 4 T_{5}^{3} - 4 T_{5}^{2} - 16 T_{5} + 8 \)
\( T_{13}^{4} + 8 T_{13}^{3} - 4 T_{13}^{2} - 80 T_{13} - 28 \)
\( T_{17}^{4} + 4 T_{17}^{3} - 52 T_{17}^{2} - 112 T_{17} + 776 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 8 - 16 T - 4 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( -28 - 80 T - 4 T^{2} + 8 T^{3} + T^{4} \)
$17$ \( 776 - 112 T - 52 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( -28 + 24 T + 40 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 16 - 32 T + 8 T^{3} + T^{4} \)
$29$ \( 64 - 64 T - 32 T^{2} + 8 T^{3} + T^{4} \)
$31$ \( 964 - 328 T - 88 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( 16 + 96 T - 32 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( -4984 - 1552 T - 84 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( 16 - 96 T - 32 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( 4 + 8 T - 24 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 3856 - 992 T - 160 T^{2} + 8 T^{3} + T^{4} \)
$59$ \( -448 - 512 T - 144 T^{2} + T^{4} \)
$61$ \( 164 - 624 T + 92 T^{2} + 24 T^{3} + T^{4} \)
$67$ \( 32 - 32 T - 40 T^{2} + 8 T^{3} + T^{4} \)
$71$ \( 12352 + 960 T - 224 T^{2} - 8 T^{3} + T^{4} \)
$73$ \( 712 - 80 T - 68 T^{2} + 4 T^{3} + T^{4} \)
$79$ \( 32 - 992 T - 136 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( 1988 - 136 T - 104 T^{2} + 4 T^{3} + T^{4} \)
$89$ \( -284 + 16 T + 124 T^{2} + 24 T^{3} + T^{4} \)
$97$ \( -1148 - 1280 T - 260 T^{2} + T^{4} \)
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