Properties

Label 3696.2.a.be
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: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} - q^{7} + q^{9} - q^{11} + ( 1 - 4 \beta ) q^{13} - q^{15} + ( 4 - 2 \beta ) q^{17} + ( 3 - 6 \beta ) q^{19} + q^{21} + ( -2 + 6 \beta ) q^{23} -4 q^{25} - q^{27} + 5 q^{29} + ( 4 - 2 \beta ) q^{31} + q^{33} - q^{35} -7 q^{37} + ( -1 + 4 \beta ) q^{39} + 4 \beta q^{41} + ( -2 + 6 \beta ) q^{43} + q^{45} + ( 1 + 2 \beta ) q^{47} + q^{49} + ( -4 + 2 \beta ) q^{51} + ( -6 + 10 \beta ) q^{53} - q^{55} + ( -3 + 6 \beta ) q^{57} + ( 5 - 10 \beta ) q^{59} + 2 q^{61} - q^{63} + ( 1 - 4 \beta ) q^{65} + ( 11 + 2 \beta ) q^{67} + ( 2 - 6 \beta ) q^{69} -4 \beta q^{71} + ( 7 + 4 \beta ) q^{73} + 4 q^{75} + q^{77} + ( 12 - 4 \beta ) q^{79} + q^{81} + ( -8 - 2 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} -5 q^{87} + ( -2 + 4 \beta ) q^{89} + ( -1 + 4 \beta ) q^{91} + ( -4 + 2 \beta ) q^{93} + ( 3 - 6 \beta ) q^{95} + ( 6 - 6 \beta ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} - 2q^{11} - 2q^{13} - 2q^{15} + 6q^{17} + 2q^{21} + 2q^{23} - 8q^{25} - 2q^{27} + 10q^{29} + 6q^{31} + 2q^{33} - 2q^{35} - 14q^{37} + 2q^{39} + 4q^{41} + 2q^{43} + 2q^{45} + 4q^{47} + 2q^{49} - 6q^{51} - 2q^{53} - 2q^{55} + 4q^{61} - 2q^{63} - 2q^{65} + 24q^{67} - 2q^{69} - 4q^{71} + 18q^{73} + 8q^{75} + 2q^{77} + 20q^{79} + 2q^{81} - 18q^{83} + 6q^{85} - 10q^{87} + 2q^{91} - 6q^{93} + 6q^{97} - 2q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 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.be 2
4.b odd 2 1 231.2.a.c 2
12.b even 2 1 693.2.a.f 2
20.d odd 2 1 5775.2.a.be 2
28.d even 2 1 1617.2.a.p 2
44.c even 2 1 2541.2.a.t 2
84.h odd 2 1 4851.2.a.w 2
132.d odd 2 1 7623.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 4.b odd 2 1
693.2.a.f 2 12.b even 2 1
1617.2.a.p 2 28.d even 2 1
2541.2.a.t 2 44.c even 2 1
3696.2.a.be 2 1.a even 1 1 trivial
4851.2.a.w 2 84.h odd 2 1
5775.2.a.be 2 20.d odd 2 1
7623.2.a.bm 2 132.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\):

\( T_{5} - 1 \)
\( T_{13}^{2} + 2 T_{13} - 19 \)
\( T_{17}^{2} - 6 T_{17} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 - T + 5 T^{2} )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 7 T^{2} + 361 T^{4} \)
$23$ \( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 5 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 6 T + 66 T^{2} - 186 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 7 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 2 T + 42 T^{2} - 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 93 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 2 T - 18 T^{2} + 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 7 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 24 T + 273 T^{2} - 1608 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 126 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 18 T + 207 T^{2} - 1314 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 20 T + 238 T^{2} - 1580 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 18 T + 242 T^{2} + 1494 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 158 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 6 T + 158 T^{2} - 582 T^{3} + 9409 T^{4} \)
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