Properties

Label 3696.2.a.t
Level 3696
Weight 2
Character orbit 3696.a
Self dual yes
Analytic conductor 29.513
Analytic rank 0
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{5} - q^{7} + q^{9} + q^{11} + 6q^{13} - 2q^{15} + 2q^{17} - 4q^{19} - q^{21} - q^{25} + q^{27} - 2q^{29} - 8q^{31} + q^{33} + 2q^{35} + 6q^{37} + 6q^{39} + 10q^{41} + 4q^{43} - 2q^{45} + 8q^{47} + q^{49} + 2q^{51} + 6q^{53} - 2q^{55} - 4q^{57} - 4q^{59} - 10q^{61} - q^{63} - 12q^{65} + 12q^{67} + 2q^{73} - q^{75} - q^{77} - 16q^{79} + q^{81} - 4q^{83} - 4q^{85} - 2q^{87} + 18q^{89} - 6q^{91} - 8q^{93} + 8q^{95} + 2q^{97} + q^{99} + O(q^{100}) \)

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:

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.t 1
4.b odd 2 1 231.2.a.a 1
12.b even 2 1 693.2.a.d 1
20.d odd 2 1 5775.2.a.t 1
28.d even 2 1 1617.2.a.e 1
44.c even 2 1 2541.2.a.h 1
84.h odd 2 1 4851.2.a.p 1
132.d odd 2 1 7623.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.a 1 4.b odd 2 1
693.2.a.d 1 12.b even 2 1
1617.2.a.e 1 28.d even 2 1
2541.2.a.h 1 44.c even 2 1
3696.2.a.t 1 1.a even 1 1 trivial
4851.2.a.p 1 84.h odd 2 1
5775.2.a.t 1 20.d odd 2 1
7623.2.a.f 1 132.d odd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - T \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 - T \)
$13$ \( 1 - 6 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 - 10 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 - 12 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 + 16 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 18 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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