Properties

Label 648.2.a.h
Level $648$
Weight $2$
Character orbit 648.a
Self dual yes
Analytic conductor $5.174$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \beta ) q^{5} + 2 \beta q^{7} +O(q^{10})\) \( q + ( 2 + \beta ) q^{5} + 2 \beta q^{7} + 2 q^{11} + ( 1 - 2 \beta ) q^{13} + ( 4 - \beta ) q^{17} + ( -4 - 2 \beta ) q^{19} + ( 2 - 4 \beta ) q^{23} + ( 2 + 4 \beta ) q^{25} + ( 6 - \beta ) q^{29} + ( -4 - 4 \beta ) q^{31} + ( 6 + 4 \beta ) q^{35} + ( 3 - 2 \beta ) q^{37} + 4 \beta q^{41} + ( -8 + 2 \beta ) q^{43} + 4 \beta q^{47} + 5 q^{49} + ( -4 - 4 \beta ) q^{53} + ( 4 + 2 \beta ) q^{55} + 8 q^{59} + ( 7 + 2 \beta ) q^{61} + ( -4 - 3 \beta ) q^{65} + ( -4 + 2 \beta ) q^{67} -2 q^{71} + q^{73} + 4 \beta q^{77} + ( 4 - 2 \beta ) q^{79} + ( 4 - 4 \beta ) q^{83} + ( 5 + 2 \beta ) q^{85} -3 \beta q^{89} + ( -12 + 2 \beta ) q^{91} + ( -14 - 8 \beta ) q^{95} + ( 2 - 8 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + O(q^{10}) \) \( 2q + 4q^{5} + 4q^{11} + 2q^{13} + 8q^{17} - 8q^{19} + 4q^{23} + 4q^{25} + 12q^{29} - 8q^{31} + 12q^{35} + 6q^{37} - 16q^{43} + 10q^{49} - 8q^{53} + 8q^{55} + 16q^{59} + 14q^{61} - 8q^{65} - 8q^{67} - 4q^{71} + 2q^{73} + 8q^{79} + 8q^{83} + 10q^{85} - 24q^{91} - 28q^{95} + 4q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 0 0 0.267949 0 −3.46410 0 0 0
1.2 0 0 0 3.73205 0 3.46410 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 648.2.a.h yes 2
3.b odd 2 1 648.2.a.e 2
4.b odd 2 1 1296.2.a.q 2
8.b even 2 1 5184.2.a.bg 2
8.d odd 2 1 5184.2.a.bi 2
9.c even 3 2 648.2.i.i 4
9.d odd 6 2 648.2.i.j 4
12.b even 2 1 1296.2.a.m 2
24.f even 2 1 5184.2.a.bz 2
24.h odd 2 1 5184.2.a.cb 2
36.f odd 6 2 1296.2.i.r 4
36.h even 6 2 1296.2.i.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.2.a.e 2 3.b odd 2 1
648.2.a.h yes 2 1.a even 1 1 trivial
648.2.i.i 4 9.c even 3 2
648.2.i.j 4 9.d odd 6 2
1296.2.a.m 2 12.b even 2 1
1296.2.a.q 2 4.b odd 2 1
1296.2.i.r 4 36.f odd 6 2
1296.2.i.t 4 36.h even 6 2
5184.2.a.bg 2 8.b even 2 1
5184.2.a.bi 2 8.d odd 2 1
5184.2.a.bz 2 24.f even 2 1
5184.2.a.cb 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(648))\):

\( T_{5}^{2} - 4 T_{5} + 1 \)
\( T_{7}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - 4 T + T^{2} \)
$7$ \( -12 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -11 - 2 T + T^{2} \)
$17$ \( 13 - 8 T + T^{2} \)
$19$ \( 4 + 8 T + T^{2} \)
$23$ \( -44 - 4 T + T^{2} \)
$29$ \( 33 - 12 T + T^{2} \)
$31$ \( -32 + 8 T + T^{2} \)
$37$ \( -3 - 6 T + T^{2} \)
$41$ \( -48 + T^{2} \)
$43$ \( 52 + 16 T + T^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( -32 + 8 T + T^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( 37 - 14 T + T^{2} \)
$67$ \( 4 + 8 T + T^{2} \)
$71$ \( ( 2 + T )^{2} \)
$73$ \( ( -1 + T )^{2} \)
$79$ \( 4 - 8 T + T^{2} \)
$83$ \( -32 - 8 T + T^{2} \)
$89$ \( -27 + T^{2} \)
$97$ \( -188 - 4 T + T^{2} \)
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