Properties

Label 3234.2.a.bc
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.

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

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 2.00000 −1.00000 0 1.00000 1.00000 2.00000
1.2 1.00000 −1.00000 1.00000 2.00000 −1.00000 0 1.00000 1.00000 2.00000
\(n\): 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.bc 2
3.b odd 2 1 9702.2.a.ci 2
7.b odd 2 1 3234.2.a.bd yes 2
21.c even 2 1 9702.2.a.cy 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bc 2 1.a even 1 1 trivial
3234.2.a.bd yes 2 7.b odd 2 1
9702.2.a.ci 2 3.b odd 2 1
9702.2.a.cy 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5} - 2 \)
\( T_{13}^{2} - 8 T_{13} + 14 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 - 2 T + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( ( 1 + T )^{2} \)
$13$ \( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 4 T + 24 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 4 T + 42 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - 8 T + 42 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 4 T + 48 T^{2} - 124 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T + 70 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 4 T + 82 T^{2} + 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 4 T + 38 T^{2} + 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 8 T + 136 T^{2} - 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T + 142 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 110 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 12 T + 150 T^{2} - 876 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 16 T + 150 T^{2} + 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 4 T + 120 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 16 T + 224 T^{2} - 1424 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 96 T^{2} + 9409 T^{4} \)
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