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: \(\Q(\sqrt{2}) \)
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.

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

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
1.41421
−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:

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

Hecke kernels

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

\( T_{5} \)
\( T_{13}^{2} - 2 \)
\( T_{17} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 + 24 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 4 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 8 T + 52 T^{2} - 152 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T + 18 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 26 T^{2} + 841 T^{4} \)
$31$ \( 1 - 8 T + 28 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T + 46 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 8 T + 66 T^{2} - 328 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 92 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 4 T + 78 T^{2} - 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 120 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 6 T^{2} + 4489 T^{4} \)
$71$ \( 1 - 16 T + 174 T^{2} - 1136 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 - 4 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 8 T + 142 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 16 T + 180 T^{2} + 1328 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 8 T + 192 T^{2} - 712 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 32 T^{2} + 9409 T^{4} \)
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