Properties

Label 3648.2.a.bs
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.70156
3.70156
0 1.00000 0 −2.70156 0 −4.70156 0 1.00000 0
1.2 0 1.00000 0 3.70156 0 1.70156 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.bs 2
4.b odd 2 1 3648.2.a.bn 2
8.b even 2 1 456.2.a.e 2
8.d odd 2 1 912.2.a.o 2
24.f even 2 1 2736.2.a.bb 2
24.h odd 2 1 1368.2.a.l 2
152.g odd 2 1 8664.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.e 2 8.b even 2 1
912.2.a.o 2 8.d odd 2 1
1368.2.a.l 2 24.h odd 2 1
2736.2.a.bb 2 24.f even 2 1
3648.2.a.bn 2 4.b odd 2 1
3648.2.a.bs 2 1.a even 1 1 trivial
8664.2.a.v 2 152.g odd 2 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} - 10 \)
\( T_{7}^{2} + 3 T_{7} - 8 \)
\( T_{11}^{2} + 3 T_{11} - 8 \)
\( T_{23} - 4 \)
\( T_{31}^{2} - 6 T_{31} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -10 - T + T^{2} \)
$7$ \( -8 + 3 T + T^{2} \)
$11$ \( -8 + 3 T + T^{2} \)
$13$ \( ( 6 + T )^{2} \)
$17$ \( -10 - T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( -32 - 6 T + T^{2} \)
$37$ \( -32 + 6 T + T^{2} \)
$41$ \( -32 - 6 T + T^{2} \)
$43$ \( -92 + T + T^{2} \)
$47$ \( -4 - 5 T + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( 10 + 9 T + T^{2} \)
$67$ \( ( -12 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 2 - 7 T + T^{2} \)
$79$ \( -160 - 4 T + T^{2} \)
$83$ \( -160 + 4 T + T^{2} \)
$89$ \( -164 + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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