Properties

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

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} -\beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{5} -\beta q^{7} + q^{9} + ( -4 + \beta ) q^{11} + 4 q^{13} + \beta q^{15} + ( -2 - \beta ) q^{17} - q^{19} -\beta q^{21} + ( 2 - 2 \beta ) q^{23} + ( 3 + \beta ) q^{25} + q^{27} + ( 2 + 2 \beta ) q^{29} + 6 q^{31} + ( -4 + \beta ) q^{33} + ( -8 - \beta ) q^{35} + 4 q^{37} + 4 q^{39} + ( 2 + 2 \beta ) q^{41} + ( 4 + \beta ) q^{43} + \beta q^{45} + ( -2 + 3 \beta ) q^{47} + ( 1 + \beta ) q^{49} + ( -2 - \beta ) q^{51} + 10 q^{53} + ( 8 - 3 \beta ) q^{55} - q^{57} + ( 4 - 2 \beta ) q^{59} + ( -6 - \beta ) q^{61} -\beta q^{63} + 4 \beta q^{65} -2 \beta q^{67} + ( 2 - 2 \beta ) q^{69} + ( 8 + 2 \beta ) q^{71} + ( 6 - \beta ) q^{73} + ( 3 + \beta ) q^{75} + ( -8 + 3 \beta ) q^{77} + 2 q^{79} + q^{81} + ( -8 - 3 \beta ) q^{85} + ( 2 + 2 \beta ) q^{87} + ( 6 - 4 \beta ) q^{89} -4 \beta q^{91} + 6 q^{93} -\beta q^{95} + ( 14 - 2 \beta ) q^{97} + ( -4 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{5} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{5} - q^{7} + 2q^{9} - 7q^{11} + 8q^{13} + q^{15} - 5q^{17} - 2q^{19} - q^{21} + 2q^{23} + 7q^{25} + 2q^{27} + 6q^{29} + 12q^{31} - 7q^{33} - 17q^{35} + 8q^{37} + 8q^{39} + 6q^{41} + 9q^{43} + q^{45} - q^{47} + 3q^{49} - 5q^{51} + 20q^{53} + 13q^{55} - 2q^{57} + 6q^{59} - 13q^{61} - q^{63} + 4q^{65} - 2q^{67} + 2q^{69} + 18q^{71} + 11q^{73} + 7q^{75} - 13q^{77} + 4q^{79} + 2q^{81} - 19q^{85} + 6q^{87} + 8q^{89} - 4q^{91} + 12q^{93} - q^{95} + 26q^{97} - 7q^{99} + O(q^{100}) \)

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
−2.37228
3.37228
0 1.00000 0 −2.37228 0 2.37228 0 1.00000 0
1.2 0 1.00000 0 3.37228 0 −3.37228 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.bt 2
4.b odd 2 1 3648.2.a.bm 2
8.b even 2 1 1824.2.a.o 2
8.d odd 2 1 1824.2.a.r yes 2
24.f even 2 1 5472.2.a.be 2
24.h odd 2 1 5472.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.o 2 8.b even 2 1
1824.2.a.r yes 2 8.d odd 2 1
3648.2.a.bm 2 4.b odd 2 1
3648.2.a.bt 2 1.a even 1 1 trivial
5472.2.a.bd 2 24.h odd 2 1
5472.2.a.be 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -8 - T + T^{2} \)
$7$ \( -8 + T + T^{2} \)
$11$ \( 4 + 7 T + T^{2} \)
$13$ \( ( -4 + T )^{2} \)
$17$ \( -2 + 5 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -32 - 2 T + T^{2} \)
$29$ \( -24 - 6 T + T^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( ( -4 + T )^{2} \)
$41$ \( -24 - 6 T + T^{2} \)
$43$ \( 12 - 9 T + T^{2} \)
$47$ \( -74 + T + T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( -24 - 6 T + T^{2} \)
$61$ \( 34 + 13 T + T^{2} \)
$67$ \( -32 + 2 T + T^{2} \)
$71$ \( 48 - 18 T + T^{2} \)
$73$ \( 22 - 11 T + T^{2} \)
$79$ \( ( -2 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( -116 - 8 T + T^{2} \)
$97$ \( 136 - 26 T + T^{2} \)
show more
show less