Properties

Label 3234.2.a.bb
Level $3234$
Weight $2$
Character orbit 3234.a
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + q^{8} + q^{9} + ( - \beta - 1) q^{10} - q^{11} - q^{12} + (\beta + 1) q^{15} + q^{16} + (\beta - 3) q^{17} + q^{18} + (3 \beta - 1) q^{19} + ( - \beta - 1) q^{20} - q^{22} + (2 \beta + 2) q^{23} - q^{24} + (2 \beta + 1) q^{25} - q^{27} - 2 \beta q^{29} + (\beta + 1) q^{30} + ( - \beta - 5) q^{31} + q^{32} + q^{33} + (\beta - 3) q^{34} + q^{36} + (4 \beta - 2) q^{37} + (3 \beta - 1) q^{38} + ( - \beta - 1) q^{40} + (\beta - 3) q^{41} + ( - 2 \beta + 2) q^{43} - q^{44} + ( - \beta - 1) q^{45} + (2 \beta + 2) q^{46} + ( - 3 \beta - 3) q^{47} - q^{48} + (2 \beta + 1) q^{50} + ( - \beta + 3) q^{51} - 6 q^{53} - q^{54} + (\beta + 1) q^{55} + ( - 3 \beta + 1) q^{57} - 2 \beta q^{58} + ( - 2 \beta - 6) q^{59} + (\beta + 1) q^{60} + ( - \beta - 5) q^{62} + q^{64} + q^{66} + ( - 6 \beta + 2) q^{67} + (\beta - 3) q^{68} + ( - 2 \beta - 2) q^{69} + (2 \beta + 2) q^{71} + q^{72} + (3 \beta - 9) q^{73} + (4 \beta - 2) q^{74} + ( - 2 \beta - 1) q^{75} + (3 \beta - 1) q^{76} + ( - 6 \beta - 2) q^{79} + ( - \beta - 1) q^{80} + q^{81} + (\beta - 3) q^{82} + ( - \beta - 1) q^{83} + (2 \beta - 2) q^{85} + ( - 2 \beta + 2) q^{86} + 2 \beta q^{87} - q^{88} + (2 \beta - 2) q^{89} + ( - \beta - 1) q^{90} + (2 \beta + 2) q^{92} + (\beta + 5) q^{93} + ( - 3 \beta - 3) q^{94} + ( - 2 \beta - 14) q^{95} - q^{96} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 2 q^{30} - 10 q^{31} + 2 q^{32} + 2 q^{33} - 6 q^{34} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{40} - 6 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{45} + 4 q^{46} - 6 q^{47} - 2 q^{48} + 2 q^{50} + 6 q^{51} - 12 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{57} - 12 q^{59} + 2 q^{60} - 10 q^{62} + 2 q^{64} + 2 q^{66} + 4 q^{67} - 6 q^{68} - 4 q^{69} + 4 q^{71} + 2 q^{72} - 18 q^{73} - 4 q^{74} - 2 q^{75} - 2 q^{76} - 4 q^{79} - 2 q^{80} + 2 q^{81} - 6 q^{82} - 2 q^{83} - 4 q^{85} + 4 q^{86} - 2 q^{88} - 4 q^{89} - 2 q^{90} + 4 q^{92} + 10 q^{93} - 6 q^{94} - 28 q^{95} - 2 q^{96} - 2 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
1.00000 −1.00000 1.00000 −3.23607 −1.00000 0 1.00000 1.00000 −3.23607
1.2 1.00000 −1.00000 1.00000 1.23607 −1.00000 0 1.00000 1.00000 1.23607
\(n\): 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.bb 2
3.b odd 2 1 9702.2.a.cw 2
7.b odd 2 1 3234.2.a.be yes 2
21.c even 2 1 9702.2.a.ck 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bb 2 1.a even 1 1 trivial
3234.2.a.be yes 2 7.b odd 2 1
9702.2.a.ck 2 21.c even 2 1
9702.2.a.cw 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 4 \) 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
$5$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 176 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 18T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 176 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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