Properties

Label 648.2.a.f
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q - \beta q^{5} + ( - \beta + 2) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{5} + ( - \beta + 2) q^{7} + q^{11} + (\beta + 2) q^{13} + (\beta - 3) q^{17} + (\beta + 3) q^{19} + (\beta + 2) q^{23} + (\beta + 3) q^{25} + (\beta - 2) q^{29} + ( - \beta + 4) q^{31} + ( - \beta + 8) q^{35} + ( - 2 \beta + 4) q^{37} + (2 \beta - 7) q^{41} + (2 \beta + 3) q^{43} + ( - \beta + 2) q^{47} + ( - 3 \beta + 5) q^{49} + ( - 2 \beta - 4) q^{53} - \beta q^{55} + 7 q^{59} - \beta q^{61} + ( - 3 \beta - 8) q^{65} + (2 \beta + 1) q^{67} - 4 q^{71} + (3 \beta - 5) q^{73} + ( - \beta + 2) q^{77} + (\beta - 4) q^{79} + (\beta + 12) q^{83} + (2 \beta - 8) q^{85} + 6 q^{89} + ( - \beta - 4) q^{91} + ( - 4 \beta - 8) q^{95} + ( - 2 \beta - 3) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 3 q^{7} + 2 q^{11} + 5 q^{13} - 5 q^{17} + 7 q^{19} + 5 q^{23} + 7 q^{25} - 3 q^{29} + 7 q^{31} + 15 q^{35} + 6 q^{37} - 12 q^{41} + 8 q^{43} + 3 q^{47} + 7 q^{49} - 10 q^{53} - q^{55} + 14 q^{59} - q^{61} - 19 q^{65} + 4 q^{67} - 8 q^{71} - 7 q^{73} + 3 q^{77} - 7 q^{79} + 25 q^{83} - 14 q^{85} + 12 q^{89} - 9 q^{91} - 20 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
3.37228
−2.37228
0 0 0 −3.37228 0 −1.37228 0 0 0
1.2 0 0 0 2.37228 0 4.37228 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.f 2
3.b odd 2 1 648.2.a.g 2
4.b odd 2 1 1296.2.a.n 2
8.b even 2 1 5184.2.a.bt 2
8.d odd 2 1 5184.2.a.bs 2
9.c even 3 2 72.2.i.b 4
9.d odd 6 2 216.2.i.b 4
12.b even 2 1 1296.2.a.p 2
24.f even 2 1 5184.2.a.bo 2
24.h odd 2 1 5184.2.a.bp 2
36.f odd 6 2 144.2.i.d 4
36.h even 6 2 432.2.i.d 4
72.j odd 6 2 1728.2.i.i 4
72.l even 6 2 1728.2.i.j 4
72.n even 6 2 576.2.i.j 4
72.p odd 6 2 576.2.i.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.b 4 9.c even 3 2
144.2.i.d 4 36.f odd 6 2
216.2.i.b 4 9.d odd 6 2
432.2.i.d 4 36.h even 6 2
576.2.i.j 4 72.n even 6 2
576.2.i.l 4 72.p odd 6 2
648.2.a.f 2 1.a even 1 1 trivial
648.2.a.g 2 3.b odd 2 1
1296.2.a.n 2 4.b odd 2 1
1296.2.a.p 2 12.b even 2 1
1728.2.i.i 4 72.j odd 6 2
1728.2.i.j 4 72.l even 6 2
5184.2.a.bo 2 24.f even 2 1
5184.2.a.bp 2 24.h odd 2 1
5184.2.a.bs 2 8.d odd 2 1
5184.2.a.bt 2 8.b even 2 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$31$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 3 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 17 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T - 8 \) Copy content Toggle raw display
$59$ \( (T - 7)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 29 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T - 62 \) Copy content Toggle raw display
$79$ \( T^{2} + 7T + 4 \) Copy content Toggle raw display
$83$ \( T^{2} - 25T + 148 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 17 \) Copy content Toggle raw display
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