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 $1$

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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{3} + 3 q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 3 q^{5} - q^{7} + q^{9} + q^{11} + q^{13} + 3 q^{15} + ( - \beta + 3) q^{17} + ( - \beta + 2) q^{19} - q^{21} + ( - \beta + 1) q^{23} + 4 q^{25} + q^{27} + (2 \beta + 1) q^{29} + (\beta + 1) q^{31} + q^{33} - 3 q^{35} + q^{37} + q^{39} + (2 \beta - 2) q^{41} + (\beta + 3) q^{43} + 3 q^{45} + ( - \beta - 6) q^{47} + q^{49} + ( - \beta + 3) q^{51} + (\beta - 5) q^{53} + 3 q^{55} + ( - \beta + 2) q^{57} + \beta q^{59} + 10 q^{61} - q^{63} + 3 q^{65} + (\beta - 4) q^{67} + ( - \beta + 1) q^{69} + (2 \beta - 2) q^{71} + 7 q^{73} + 4 q^{75} - q^{77} + (2 \beta + 2) q^{79} + q^{81} + ( - \beta + 7) q^{83} + ( - 3 \beta + 9) q^{85} + (2 \beta + 1) q^{87} - 2 \beta q^{89} - q^{91} + (\beta + 1) q^{93} + ( - 3 \beta + 6) q^{95} + ( - \beta - 7) q^{97} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} + 6 q^{43} + 6 q^{45} - 12 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 6 q^{55} + 4 q^{57} + 20 q^{61} - 2 q^{63} + 6 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} + 14 q^{73} + 8 q^{75} - 2 q^{77} + 4 q^{79} + 2 q^{81} + 14 q^{83} + 18 q^{85} + 2 q^{87} - 2 q^{91} + 2 q^{93} + 12 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) \) 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
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

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