Properties

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + q^{11} - q^{12} - \beta q^{13} + q^{16} + 4 q^{17} - q^{18} + ( - \beta + 4) q^{19} - q^{22} + ( - 4 \beta + 2) q^{23} + q^{24} - 5 q^{25} + \beta q^{26} - q^{27} + 4 \beta q^{29} + (5 \beta + 4) q^{31} - q^{32} - q^{33} - 4 q^{34} + q^{36} + ( - 4 \beta - 2) q^{37} + (\beta - 4) q^{38} + \beta q^{39} + ( - 4 \beta + 4) q^{41} - 10 q^{43} + q^{44} + (4 \beta - 2) q^{46} + \beta q^{47} - q^{48} + 5 q^{50} - 4 q^{51} - \beta q^{52} + (4 \beta + 2) q^{53} + q^{54} + (\beta - 4) q^{57} - 4 \beta q^{58} + ( - 2 \beta + 4) q^{59} + \beta q^{61} + ( - 5 \beta - 4) q^{62} + q^{64} + q^{66} + 8 \beta q^{67} + 4 q^{68} + (4 \beta - 2) q^{69} + ( - 4 \beta + 8) q^{71} - q^{72} + 4 q^{73} + (4 \beta + 2) q^{74} + 5 q^{75} + ( - \beta + 4) q^{76} - \beta q^{78} + ( - 4 \beta - 4) q^{79} + q^{81} + (4 \beta - 4) q^{82} + (5 \beta - 8) q^{83} + 10 q^{86} - 4 \beta q^{87} - q^{88} + (\beta + 4) q^{89} + ( - 4 \beta + 2) q^{92} + ( - 5 \beta - 4) q^{93} - \beta q^{94} + q^{96} - 9 \beta q^{97} + q^{99} +O(q^{100}) \) 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 \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 8 q^{19} - 2 q^{22} + 4 q^{23} + 2 q^{24} - 10 q^{25} - 2 q^{27} + 8 q^{31} - 2 q^{32} - 2 q^{33} - 8 q^{34} + 2 q^{36} - 4 q^{37} - 8 q^{38} + 8 q^{41} - 20 q^{43} + 2 q^{44} - 4 q^{46} - 2 q^{48} + 10 q^{50} - 8 q^{51} + 4 q^{53} + 2 q^{54} - 8 q^{57} + 8 q^{59} - 8 q^{62} + 2 q^{64} + 2 q^{66} + 8 q^{68} - 4 q^{69} + 16 q^{71} - 2 q^{72} + 8 q^{73} + 4 q^{74} + 10 q^{75} + 8 q^{76} - 8 q^{79} + 2 q^{81} - 8 q^{82} - 16 q^{83} + 20 q^{86} - 2 q^{88} + 8 q^{89} + 4 q^{92} - 8 q^{93} + 2 q^{96} + 2 q^{99}+O(q^{100}) \) 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
1.41421
−1.41421
−1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 1.00000 0
\(n\): 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.y 2
3.b odd 2 1 9702.2.a.de 2
7.b odd 2 1 3234.2.a.z yes 2
21.c even 2 1 9702.2.a.dl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.y 2 1.a even 1 1 trivial
3234.2.a.z yes 2 7.b odd 2 1
9702.2.a.de 2 3.b odd 2 1
9702.2.a.dl 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13}^{2} - 2 \) Copy content Toggle raw display
\( T_{17} - 4 \) 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
$5$ \( T^{2} \) 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
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$29$ \( T^{2} - 32 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T - 34 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} - 2 \) Copy content Toggle raw display
$67$ \( T^{2} - 128 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 16T + 14 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$97$ \( T^{2} - 162 \) Copy content Toggle raw display
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