Properties

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

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q + q^{3} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{7} + q^{9} + q^{11} - 2 q^{13} - 4 q^{17} - 6 q^{19} + q^{21} + 4 q^{23} - 5 q^{25} + q^{27} - 10 q^{29} - 6 q^{31} + q^{33} - 6 q^{37} - 2 q^{39} - 12 q^{41} + 8 q^{43} - 2 q^{47} + q^{49} - 4 q^{51} + 6 q^{53} - 6 q^{57} + 8 q^{59} + 6 q^{61} + q^{63} + 4 q^{67} + 4 q^{69} - 12 q^{73} - 5 q^{75} + q^{77} + q^{81} - 14 q^{83} - 10 q^{87} + 10 q^{89} - 2 q^{91} - 6 q^{93} + 10 q^{97} + q^{99}+O(q^{100}) \) 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
0
0 1.00000 0 0 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.y 1
4.b odd 2 1 462.2.a.b 1
12.b even 2 1 1386.2.a.i 1
28.d even 2 1 3234.2.a.k 1
44.c even 2 1 5082.2.a.s 1
84.h odd 2 1 9702.2.a.bt 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.b 1 4.b odd 2 1
1386.2.a.i 1 12.b even 2 1
3234.2.a.k 1 28.d even 2 1
3696.2.a.y 1 1.a even 1 1 trivial
5082.2.a.s 1 44.c even 2 1
9702.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 10 \) Copy content Toggle raw display
$31$ \( T + 6 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T + 12 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T + 2 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 12 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 14 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
show more
show less