Properties

Label 3696.2.a.bo
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: 3.3.229.1
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + q^{7} + q^{9} + q^{11} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 2 + \beta_{1} - 3 \beta_{2} ) q^{17} + ( 3 + \beta_{2} ) q^{19} - q^{21} + ( -4 - \beta_{1} - \beta_{2} ) q^{23} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{25} - q^{27} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{31} - q^{33} + ( 1 - \beta_{1} ) q^{35} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{37} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{39} + ( 4 - 2 \beta_{2} ) q^{41} + ( 4 - 3 \beta_{1} + \beta_{2} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{51} + ( -\beta_{1} + \beta_{2} ) q^{53} + ( 1 - \beta_{1} ) q^{55} + ( -3 - \beta_{2} ) q^{57} + ( -1 - 3 \beta_{2} ) q^{59} -2 q^{61} + q^{63} + ( 7 + 3 \beta_{1} + 4 \beta_{2} ) q^{65} + ( 1 + 2 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 4 + \beta_{1} + \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -7 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{75} + q^{77} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( -2 + 3 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -6 - 5 \beta_{1} - 7 \beta_{2} ) q^{85} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{87} + ( 10 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{91} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{93} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{95} + ( 3 \beta_{1} + \beta_{2} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 4q^{5} + 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 4q^{5} + 3q^{7} + 3q^{9} + 3q^{11} - 4q^{13} - 4q^{15} + 8q^{17} + 8q^{19} - 3q^{21} - 10q^{23} + 15q^{25} - 3q^{27} - 4q^{29} + 2q^{31} - 3q^{33} + 4q^{35} + 4q^{39} + 14q^{41} + 14q^{43} + 4q^{45} + 3q^{49} - 8q^{51} + 4q^{55} - 8q^{57} - 6q^{61} + 3q^{63} + 14q^{65} + 4q^{67} + 10q^{69} + 12q^{71} - 20q^{73} - 15q^{75} + 3q^{77} - 12q^{79} + 3q^{81} - 6q^{83} - 6q^{85} + 4q^{87} + 26q^{89} - 4q^{91} - 2q^{93} + 8q^{95} - 4q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
2.11491
−1.86081
−0.254102
0 −1.00000 0 −2.58774 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 2.39821 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 4.18953 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.bo 3
4.b odd 2 1 231.2.a.e 3
12.b even 2 1 693.2.a.l 3
20.d odd 2 1 5775.2.a.bp 3
28.d even 2 1 1617.2.a.t 3
44.c even 2 1 2541.2.a.bg 3
84.h odd 2 1 4851.2.a.bi 3
132.d odd 2 1 7623.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 4.b odd 2 1
693.2.a.l 3 12.b even 2 1
1617.2.a.t 3 28.d even 2 1
2541.2.a.bg 3 44.c even 2 1
3696.2.a.bo 3 1.a even 1 1 trivial
4851.2.a.bi 3 84.h odd 2 1
5775.2.a.bp 3 20.d odd 2 1
7623.2.a.cd 3 132.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\):

\( T_{5}^{3} - 4 T_{5}^{2} - 7 T_{5} + 26 \)
\( T_{13}^{3} + 4 T_{13}^{2} - 27 T_{13} - 94 \)
\( T_{17}^{3} - 8 T_{17}^{2} - 40 T_{17} + 328 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 1 - 4 T + 8 T^{2} - 14 T^{3} + 40 T^{4} - 100 T^{5} + 125 T^{6} \)
$7$ \( ( 1 - T )^{3} \)
$11$ \( ( 1 - T )^{3} \)
$13$ \( 1 + 4 T + 12 T^{2} + 10 T^{3} + 156 T^{4} + 676 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 8 T + 11 T^{2} + 56 T^{3} + 187 T^{4} - 2312 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 8 T + 72 T^{2} - 308 T^{3} + 1368 T^{4} - 2888 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 10 T + 81 T^{2} + 396 T^{3} + 1863 T^{4} + 5290 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 4 T + 60 T^{2} + 138 T^{3} + 1740 T^{4} + 3364 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 2 T + 17 T^{2} + 132 T^{3} + 527 T^{4} - 1922 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 68 T^{2} + 106 T^{3} + 2516 T^{4} + 50653 T^{6} \)
$41$ \( 1 - 14 T + 163 T^{2} - 1116 T^{3} + 6683 T^{4} - 23534 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 14 T + 85 T^{2} - 356 T^{3} + 3655 T^{4} - 25886 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 80 T^{2} - 32 T^{3} + 3760 T^{4} + 103823 T^{6} \)
$53$ \( 1 + 143 T^{2} + 8 T^{3} + 7579 T^{4} + 148877 T^{6} \)
$59$ \( 1 + 120 T^{2} + 52 T^{3} + 7080 T^{4} + 205379 T^{6} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{3} \)
$67$ \( 1 - 4 T + 116 T^{2} - 300 T^{3} + 7772 T^{4} - 17956 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 12 T + 197 T^{2} - 1448 T^{3} + 13987 T^{4} - 60492 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 20 T + 320 T^{2} + 3054 T^{3} + 23360 T^{4} + 106580 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 12 T + 221 T^{2} + 1640 T^{3} + 17459 T^{4} + 74892 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 6 T + 117 T^{2} + 500 T^{3} + 9711 T^{4} + 41334 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 26 T + 407 T^{2} - 4300 T^{3} + 36223 T^{4} - 205946 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 4 T + 171 T^{2} + 544 T^{3} + 16587 T^{4} + 37636 T^{5} + 912673 T^{6} \)
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