Properties

Label 3648.2.a.bo
Level $3648$
Weight $2$
Character orbit 3648.a
Self dual yes
Analytic conductor $29.129$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.1294266574\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1824)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 + \beta ) q^{5} + ( -1 + \beta ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 + \beta ) q^{5} + ( -1 + \beta ) q^{7} + q^{9} + ( 1 + \beta ) q^{11} + ( 4 - 2 \beta ) q^{13} + ( -1 - \beta ) q^{15} + ( 1 + \beta ) q^{17} - q^{19} + ( 1 - \beta ) q^{21} -2 \beta q^{23} + 3 \beta q^{25} - q^{27} + 4 q^{29} + ( -2 + 2 \beta ) q^{31} + ( -1 - \beta ) q^{33} + ( 3 + \beta ) q^{35} + ( 4 + 2 \beta ) q^{37} + ( -4 + 2 \beta ) q^{39} -4 q^{41} + ( 1 + 3 \beta ) q^{43} + ( 1 + \beta ) q^{45} + ( -7 + \beta ) q^{47} + ( -2 - \beta ) q^{49} + ( -1 - \beta ) q^{51} + ( 2 + 2 \beta ) q^{53} + ( 5 + 3 \beta ) q^{55} + q^{57} + ( 2 - 2 \beta ) q^{59} + ( 9 + \beta ) q^{61} + ( -1 + \beta ) q^{63} -4 q^{65} + ( -10 - 2 \beta ) q^{67} + 2 \beta q^{69} + ( 2 - 6 \beta ) q^{71} + ( 1 + \beta ) q^{73} -3 \beta q^{75} + ( 3 + \beta ) q^{77} + ( 2 - 2 \beta ) q^{79} + q^{81} + ( 8 - 6 \beta ) q^{83} + ( 5 + 3 \beta ) q^{85} -4 q^{87} + ( 2 + 2 \beta ) q^{89} + ( -12 + 4 \beta ) q^{91} + ( 2 - 2 \beta ) q^{93} + ( -1 - \beta ) q^{95} + ( -4 - 6 \beta ) q^{97} + ( 1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 3q^{5} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 3q^{5} - q^{7} + 2q^{9} + 3q^{11} + 6q^{13} - 3q^{15} + 3q^{17} - 2q^{19} + q^{21} - 2q^{23} + 3q^{25} - 2q^{27} + 8q^{29} - 2q^{31} - 3q^{33} + 7q^{35} + 10q^{37} - 6q^{39} - 8q^{41} + 5q^{43} + 3q^{45} - 13q^{47} - 5q^{49} - 3q^{51} + 6q^{53} + 13q^{55} + 2q^{57} + 2q^{59} + 19q^{61} - q^{63} - 8q^{65} - 22q^{67} + 2q^{69} - 2q^{71} + 3q^{73} - 3q^{75} + 7q^{77} + 2q^{79} + 2q^{81} + 10q^{83} + 13q^{85} - 8q^{87} + 6q^{89} - 20q^{91} + 2q^{93} - 3q^{95} - 14q^{97} + 3q^{99} + O(q^{100}) \)

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
−1.56155
2.56155
0 −1.00000 0 −0.561553 0 −2.56155 0 1.00000 0
1.2 0 −1.00000 0 3.56155 0 1.56155 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)

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 3648.2.a.bo 2
4.b odd 2 1 3648.2.a.bv 2
8.b even 2 1 1824.2.a.p yes 2
8.d odd 2 1 1824.2.a.n 2
24.f even 2 1 5472.2.a.bk 2
24.h odd 2 1 5472.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.n 2 8.d odd 2 1
1824.2.a.p yes 2 8.b even 2 1
3648.2.a.bo 2 1.a even 1 1 trivial
3648.2.a.bv 2 4.b odd 2 1
5472.2.a.bj 2 24.h odd 2 1
5472.2.a.bk 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\):

\( T_{5}^{2} - 3 T_{5} - 2 \)
\( T_{7}^{2} + T_{7} - 4 \)
\( T_{11}^{2} - 3 T_{11} - 2 \)
\( T_{23}^{2} + 2 T_{23} - 16 \)
\( T_{31}^{2} + 2 T_{31} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -2 - 3 T + T^{2} \)
$7$ \( -4 + T + T^{2} \)
$11$ \( -2 - 3 T + T^{2} \)
$13$ \( -8 - 6 T + T^{2} \)
$17$ \( -2 - 3 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -16 + 2 T + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( -16 + 2 T + T^{2} \)
$37$ \( 8 - 10 T + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( -32 - 5 T + T^{2} \)
$47$ \( 38 + 13 T + T^{2} \)
$53$ \( -8 - 6 T + T^{2} \)
$59$ \( -16 - 2 T + T^{2} \)
$61$ \( 86 - 19 T + T^{2} \)
$67$ \( 104 + 22 T + T^{2} \)
$71$ \( -152 + 2 T + T^{2} \)
$73$ \( -2 - 3 T + T^{2} \)
$79$ \( -16 - 2 T + T^{2} \)
$83$ \( -128 - 10 T + T^{2} \)
$89$ \( -8 - 6 T + T^{2} \)
$97$ \( -104 + 14 T + T^{2} \)
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