Properties

Label 3696.2.a.bl
Level $3696$
Weight $2$
Character orbit 3696.a
Self dual yes
Analytic conductor $29.513$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{21}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + 3 q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + 3 q^{5} - q^{7} + q^{9} + q^{11} + q^{13} + 3 q^{15} + ( 3 - \beta ) q^{17} + ( 2 - \beta ) q^{19} - q^{21} + ( 1 - \beta ) q^{23} + 4 q^{25} + q^{27} + ( 1 + 2 \beta ) q^{29} + ( 1 + \beta ) q^{31} + q^{33} -3 q^{35} + q^{37} + q^{39} + ( -2 + 2 \beta ) q^{41} + ( 3 + \beta ) q^{43} + 3 q^{45} + ( -6 - \beta ) q^{47} + q^{49} + ( 3 - \beta ) q^{51} + ( -5 + \beta ) q^{53} + 3 q^{55} + ( 2 - \beta ) q^{57} + \beta q^{59} + 10 q^{61} - q^{63} + 3 q^{65} + ( -4 + \beta ) q^{67} + ( 1 - \beta ) q^{69} + ( -2 + 2 \beta ) q^{71} + 7 q^{73} + 4 q^{75} - q^{77} + ( 2 + 2 \beta ) q^{79} + q^{81} + ( 7 - \beta ) q^{83} + ( 9 - 3 \beta ) q^{85} + ( 1 + 2 \beta ) q^{87} -2 \beta q^{89} - q^{91} + ( 1 + \beta ) q^{93} + ( 6 - 3 \beta ) q^{95} + ( -7 - \beta ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} + 6 q^{43} + 6 q^{45} - 12 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 6 q^{55} + 4 q^{57} + 20 q^{61} - 2 q^{63} + 6 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} + 14 q^{73} + 8 q^{75} - 2 q^{77} + 4 q^{79} + 2 q^{81} + 14 q^{83} + 18 q^{85} + 2 q^{87} - 2 q^{91} + 2 q^{93} + 12 q^{95} - 14 q^{97} + 2 q^{99} + O(q^{100}) \)

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
2.79129
−1.79129
0 1.00000 0 3.00000 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 3.00000 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.bl 2
4.b odd 2 1 231.2.a.b 2
12.b even 2 1 693.2.a.j 2
20.d odd 2 1 5775.2.a.bn 2
28.d even 2 1 1617.2.a.o 2
44.c even 2 1 2541.2.a.z 2
84.h odd 2 1 4851.2.a.ba 2
132.d odd 2 1 7623.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 4.b odd 2 1
693.2.a.j 2 12.b even 2 1
1617.2.a.o 2 28.d even 2 1
2541.2.a.z 2 44.c even 2 1
3696.2.a.bl 2 1.a even 1 1 trivial
4851.2.a.ba 2 84.h odd 2 1
5775.2.a.bn 2 20.d odd 2 1
7623.2.a.bf 2 132.d odd 2 1

Hecke kernels

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

\( T_{5} - 3 \)
\( T_{13} - 1 \)
\( T_{17}^{2} - 6 T_{17} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -3 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -12 - 6 T + T^{2} \)
$19$ \( -17 - 4 T + T^{2} \)
$23$ \( -20 - 2 T + T^{2} \)
$29$ \( -83 - 2 T + T^{2} \)
$31$ \( -20 - 2 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -80 + 4 T + T^{2} \)
$43$ \( -12 - 6 T + T^{2} \)
$47$ \( 15 + 12 T + T^{2} \)
$53$ \( 4 + 10 T + T^{2} \)
$59$ \( -21 + T^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( -5 + 8 T + T^{2} \)
$71$ \( -80 + 4 T + T^{2} \)
$73$ \( ( -7 + T )^{2} \)
$79$ \( -80 - 4 T + T^{2} \)
$83$ \( 28 - 14 T + T^{2} \)
$89$ \( -84 + T^{2} \)
$97$ \( 28 + 14 T + T^{2} \)
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