Properties

Label 3696.2.a.s
Level $3696$
Weight $2$
Character orbit 3696.a
Self dual yes
Analytic conductor $29.513$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.5127085871\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2 q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2 q^{5} - q^{7} + q^{9} - q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{17} + 4 q^{19} - q^{21} + 4 q^{23} - q^{25} + q^{27} + 2 q^{29} + 4 q^{31} - q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} - 6 q^{41} - 2 q^{45} + 8 q^{47} + q^{49} - 6 q^{51} - 14 q^{53} + 2 q^{55} + 4 q^{57} - 12 q^{59} - 14 q^{61} - q^{63} - 4 q^{65} - 4 q^{67} + 4 q^{69} - 12 q^{71} + 6 q^{73} - q^{75} + q^{77} + q^{81} + 12 q^{85} + 2 q^{87} - 6 q^{89} - 2 q^{91} + 4 q^{93} - 8 q^{95} - 14 q^{97} - q^{99} + O(q^{100}) \)

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
0
0 1.00000 0 −2.00000 0 −1.00000 0 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 3696.2.a.s 1
4.b odd 2 1 462.2.a.a 1
12.b even 2 1 1386.2.a.k 1
28.d even 2 1 3234.2.a.n 1
44.c even 2 1 5082.2.a.q 1
84.h odd 2 1 9702.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.a 1 4.b odd 2 1
1386.2.a.k 1 12.b even 2 1
3234.2.a.n 1 28.d even 2 1
3696.2.a.s 1 1.a even 1 1 trivial
5082.2.a.q 1 44.c even 2 1
9702.2.a.bf 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\):

\( T_{5} + 2 \)
\( T_{13} - 2 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( 2 + T \)
$7$ \( 1 + T \)
$11$ \( 1 + T \)
$13$ \( -2 + T \)
$17$ \( 6 + T \)
$19$ \( -4 + T \)
$23$ \( -4 + T \)
$29$ \( -2 + T \)
$31$ \( -4 + T \)
$37$ \( 2 + T \)
$41$ \( 6 + T \)
$43$ \( T \)
$47$ \( -8 + T \)
$53$ \( 14 + T \)
$59$ \( 12 + T \)
$61$ \( 14 + T \)
$67$ \( 4 + T \)
$71$ \( 12 + T \)
$73$ \( -6 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( 6 + T \)
$97$ \( 14 + T \)
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