Properties

Label 3234.2.a.ba
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: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} - q^{8} + q^{9} - \beta q^{10} + q^{11} + q^{12} + 4 q^{13} + \beta q^{15} + q^{16} + 3 q^{17} - q^{18} + 2 \beta q^{19} + \beta q^{20} - q^{22} + \beta q^{23} - q^{24} + 2 q^{25} - 4 q^{26} + q^{27} + 2 q^{29} - \beta q^{30} + 4 q^{31} - q^{32} + q^{33} - 3 q^{34} + q^{36} + ( - 2 \beta - 4) q^{37} - 2 \beta q^{38} + 4 q^{39} - \beta q^{40} - 9 q^{41} + ( - 2 \beta + 4) q^{43} + q^{44} + \beta q^{45} - \beta q^{46} + ( - 3 \beta - 4) q^{47} + q^{48} - 2 q^{50} + 3 q^{51} + 4 q^{52} + 4 q^{53} - q^{54} + \beta q^{55} + 2 \beta q^{57} - 2 q^{58} + (4 \beta + 4) q^{59} + \beta q^{60} + (3 \beta - 4) q^{61} - 4 q^{62} + q^{64} + 4 \beta q^{65} - q^{66} + ( - 4 \beta - 3) q^{67} + 3 q^{68} + \beta q^{69} + ( - 2 \beta - 8) q^{71} - q^{72} + ( - 2 \beta + 10) q^{73} + (2 \beta + 4) q^{74} + 2 q^{75} + 2 \beta q^{76} - 4 q^{78} + ( - \beta + 8) q^{79} + \beta q^{80} + q^{81} + 9 q^{82} + (4 \beta - 5) q^{83} + 3 \beta q^{85} + (2 \beta - 4) q^{86} + 2 q^{87} - q^{88} + ( - 2 \beta - 8) q^{89} - \beta q^{90} + \beta q^{92} + 4 q^{93} + (3 \beta + 4) q^{94} + 14 q^{95} - q^{96} + ( - 4 \beta + 1) q^{97} + q^{99} +O(q^{100}) \) 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 \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} + 8 q^{13} + 2 q^{16} + 6 q^{17} - 2 q^{18} - 2 q^{22} - 2 q^{24} + 4 q^{25} - 8 q^{26} + 2 q^{27} + 4 q^{29} + 8 q^{31} - 2 q^{32} + 2 q^{33} - 6 q^{34} + 2 q^{36} - 8 q^{37} + 8 q^{39} - 18 q^{41} + 8 q^{43} + 2 q^{44} - 8 q^{47} + 2 q^{48} - 4 q^{50} + 6 q^{51} + 8 q^{52} + 8 q^{53} - 2 q^{54} - 4 q^{58} + 8 q^{59} - 8 q^{61} - 8 q^{62} + 2 q^{64} - 2 q^{66} - 6 q^{67} + 6 q^{68} - 16 q^{71} - 2 q^{72} + 20 q^{73} + 8 q^{74} + 4 q^{75} - 8 q^{78} + 16 q^{79} + 2 q^{81} + 18 q^{82} - 10 q^{83} - 8 q^{86} + 4 q^{87} - 2 q^{88} - 16 q^{89} + 8 q^{93} + 8 q^{94} + 28 q^{95} - 2 q^{96} + 2 q^{97} + 2 q^{99}+O(q^{100}) \) 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
−2.64575
2.64575
−1.00000 1.00000 1.00000 −2.64575 −1.00000 0 −1.00000 1.00000 2.64575
1.2 −1.00000 1.00000 1.00000 2.64575 −1.00000 0 −1.00000 1.00000 −2.64575
\(n\): 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.ba 2
3.b odd 2 1 9702.2.a.dm 2
7.b odd 2 1 3234.2.a.w 2
7.d odd 6 2 462.2.i.e 4
21.c even 2 1 9702.2.a.db 2
21.g even 6 2 1386.2.k.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.e 4 7.d odd 6 2
1386.2.k.r 4 21.g even 6 2
3234.2.a.w 2 7.b odd 2 1
3234.2.a.ba 2 1.a even 1 1 trivial
9702.2.a.db 2 21.c even 2 1
9702.2.a.dm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{2} - 7 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} - 3 \) 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
$5$ \( T^{2} - 7 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 28 \) Copy content Toggle raw display
$23$ \( T^{2} - 7 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 12 \) Copy content Toggle raw display
$41$ \( (T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 12 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T - 47 \) Copy content Toggle raw display
$53$ \( (T - 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 96 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 47 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T - 103 \) Copy content Toggle raw display
$71$ \( T^{2} + 16T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 20T + 72 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 57 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T - 87 \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 111 \) Copy content Toggle raw display
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