Properties

Label 3696.2.a.bp
Level $3696$
Weight $2$
Character orbit 3696.a
Self dual yes
Analytic conductor $29.513$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -\beta_{2} q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta_{2} q^{5} + q^{7} + q^{9} - q^{11} -\beta_{2} q^{13} -\beta_{2} q^{15} + ( -\beta_{1} + \beta_{2} ) q^{17} + ( -4 + \beta_{1} ) q^{19} + q^{21} + ( 2 + \beta_{1} - \beta_{2} ) q^{23} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( 4 + \beta_{2} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} ) q^{31} - q^{33} -\beta_{2} q^{35} + ( 2 \beta_{1} - \beta_{2} ) q^{37} -\beta_{2} q^{39} + ( 2 + 2 \beta_{1} ) q^{41} + ( -2 + \beta_{1} - \beta_{2} ) q^{43} -\beta_{2} q^{45} + ( 8 - \beta_{1} ) q^{47} + q^{49} + ( -\beta_{1} + \beta_{2} ) q^{51} + ( -\beta_{1} - \beta_{2} ) q^{53} + \beta_{2} q^{55} + ( -4 + \beta_{1} ) q^{57} + ( 8 + \beta_{1} + 2 \beta_{2} ) q^{59} + 6 q^{61} + q^{63} + ( 10 - 2 \beta_{1} + \beta_{2} ) q^{65} + ( -4 - \beta_{1} ) q^{67} + ( 2 + \beta_{1} - \beta_{2} ) q^{69} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 8 + \beta_{2} ) q^{73} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{75} - q^{77} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( 6 - \beta_{1} - \beta_{2} ) q^{83} + ( -6 + \beta_{1} - 3 \beta_{2} ) q^{85} + ( 4 + \beta_{2} ) q^{87} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} -\beta_{2} q^{91} + ( 2 - \beta_{1} - \beta_{2} ) q^{93} + ( -4 + \beta_{1} + 6 \beta_{2} ) q^{95} + ( 8 + \beta_{1} + \beta_{2} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 3q^{7} + 3q^{9} - 3q^{11} - 12q^{19} + 3q^{21} + 6q^{23} + 15q^{25} + 3q^{27} + 12q^{29} + 6q^{31} - 3q^{33} + 6q^{41} - 6q^{43} + 24q^{47} + 3q^{49} - 12q^{57} + 24q^{59} + 18q^{61} + 3q^{63} + 30q^{65} - 12q^{67} + 6q^{69} - 12q^{71} + 24q^{73} + 15q^{75} - 3q^{77} - 12q^{79} + 3q^{81} + 18q^{83} - 18q^{85} + 12q^{87} + 18q^{89} + 6q^{93} - 12q^{95} + 24q^{97} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 4 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 8\)\()/2\)

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
−2.36147
2.52892
−0.167449
0 1.00000 0 −3.93800 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 0.133492 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 3.80451 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.bp 3
4.b odd 2 1 231.2.a.d 3
12.b even 2 1 693.2.a.m 3
20.d odd 2 1 5775.2.a.bw 3
28.d even 2 1 1617.2.a.s 3
44.c even 2 1 2541.2.a.bi 3
84.h odd 2 1 4851.2.a.bp 3
132.d odd 2 1 7623.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.d 3 4.b odd 2 1
693.2.a.m 3 12.b even 2 1
1617.2.a.s 3 28.d even 2 1
2541.2.a.bi 3 44.c even 2 1
3696.2.a.bp 3 1.a even 1 1 trivial
4851.2.a.bp 3 84.h odd 2 1
5775.2.a.bw 3 20.d odd 2 1
7623.2.a.cb 3 132.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\):

\( T_{5}^{3} - 15 T_{5} + 2 \)
\( T_{13}^{3} - 15 T_{13} + 2 \)
\( T_{17}^{3} - 24 T_{17} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( 2 - 15 T + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( 2 - 15 T + T^{3} \)
$17$ \( 8 - 24 T + T^{3} \)
$19$ \( -36 + 27 T + 12 T^{2} + T^{3} \)
$23$ \( 32 - 12 T - 6 T^{2} + T^{3} \)
$29$ \( -6 + 33 T - 12 T^{2} + T^{3} \)
$31$ \( -32 - 36 T - 6 T^{2} + T^{3} \)
$37$ \( -246 - 75 T + T^{3} \)
$41$ \( 32 - 72 T - 6 T^{2} + T^{3} \)
$43$ \( -48 - 12 T + 6 T^{2} + T^{3} \)
$47$ \( -328 + 171 T - 24 T^{2} + T^{3} \)
$53$ \( -120 - 48 T + T^{3} \)
$59$ \( 716 + 87 T - 24 T^{2} + T^{3} \)
$61$ \( ( -6 + T )^{3} \)
$67$ \( -4 + 27 T + 12 T^{2} + T^{3} \)
$71$ \( -384 - 48 T + 12 T^{2} + T^{3} \)
$73$ \( -394 + 177 T - 24 T^{2} + T^{3} \)
$79$ \( -256 - 48 T + 12 T^{2} + T^{3} \)
$83$ \( -48 + 60 T - 18 T^{2} + T^{3} \)
$89$ \( 1896 - 84 T - 18 T^{2} + T^{3} \)
$97$ \( -8 + 144 T - 24 T^{2} + T^{3} \)
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