Properties

Label 648.2.i.a
Level $648$
Weight $2$
Character orbit 648.i
Analytic conductor $5.174$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -4 \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{7} + ( -4 + 4 \zeta_{6} ) q^{11} -\zeta_{6} q^{13} -4 q^{17} - q^{19} -4 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{31} -12 q^{35} -9 q^{37} + ( 8 - 8 \zeta_{6} ) q^{43} + ( 12 - 12 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} -8 q^{53} + 16 q^{55} -4 \zeta_{6} q^{59} + ( 5 - 5 \zeta_{6} ) q^{61} + ( -4 + 4 \zeta_{6} ) q^{65} -11 \zeta_{6} q^{67} + 8 q^{71} + q^{73} + 12 \zeta_{6} q^{77} + ( 5 - 5 \zeta_{6} ) q^{79} + ( -8 + 8 \zeta_{6} ) q^{83} + 16 \zeta_{6} q^{85} + 12 q^{89} -3 q^{91} + 4 \zeta_{6} q^{95} + ( -5 + 5 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + 3q^{7} + O(q^{10}) \) \( 2q - 4q^{5} + 3q^{7} - 4q^{11} - q^{13} - 8q^{17} - 2q^{19} - 4q^{23} - 11q^{25} + 4q^{31} - 24q^{35} - 18q^{37} + 8q^{43} + 12q^{47} - 2q^{49} - 16q^{53} + 32q^{55} - 4q^{59} + 5q^{61} - 4q^{65} - 11q^{67} + 16q^{71} + 2q^{73} + 12q^{77} + 5q^{79} - 8q^{83} + 16q^{85} + 24q^{89} - 6q^{91} + 4q^{95} - 5q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −2.00000 3.46410i 0 1.50000 2.59808i 0 0 0
433.1 0 0 0 −2.00000 + 3.46410i 0 1.50000 + 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.i.a 2
3.b odd 2 1 648.2.i.h 2
4.b odd 2 1 1296.2.i.a 2
9.c even 3 1 216.2.a.d yes 1
9.c even 3 1 inner 648.2.i.a 2
9.d odd 6 1 216.2.a.a 1
9.d odd 6 1 648.2.i.h 2
12.b even 2 1 1296.2.i.q 2
36.f odd 6 1 432.2.a.h 1
36.f odd 6 1 1296.2.i.a 2
36.h even 6 1 432.2.a.a 1
36.h even 6 1 1296.2.i.q 2
45.h odd 6 1 5400.2.a.bn 1
45.j even 6 1 5400.2.a.bp 1
45.k odd 12 2 5400.2.f.v 2
45.l even 12 2 5400.2.f.e 2
72.j odd 6 1 1728.2.a.ba 1
72.l even 6 1 1728.2.a.bb 1
72.n even 6 1 1728.2.a.a 1
72.p odd 6 1 1728.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.a 1 9.d odd 6 1
216.2.a.d yes 1 9.c even 3 1
432.2.a.a 1 36.h even 6 1
432.2.a.h 1 36.f odd 6 1
648.2.i.a 2 1.a even 1 1 trivial
648.2.i.a 2 9.c even 3 1 inner
648.2.i.h 2 3.b odd 2 1
648.2.i.h 2 9.d odd 6 1
1296.2.i.a 2 4.b odd 2 1
1296.2.i.a 2 36.f odd 6 1
1296.2.i.q 2 12.b even 2 1
1296.2.i.q 2 36.h even 6 1
1728.2.a.a 1 72.n even 6 1
1728.2.a.b 1 72.p odd 6 1
1728.2.a.ba 1 72.j odd 6 1
1728.2.a.bb 1 72.l even 6 1
5400.2.a.bn 1 45.h odd 6 1
5400.2.a.bp 1 45.j even 6 1
5400.2.f.e 2 45.l even 12 2
5400.2.f.v 2 45.k odd 12 2

Hecke kernels

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

\( T_{5}^{2} + 4 T_{5} + 16 \)
\( T_{7}^{2} - 3 T_{7} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 + 4 T + T^{2} \)
$7$ \( 9 - 3 T + T^{2} \)
$11$ \( 16 + 4 T + T^{2} \)
$13$ \( 1 + T + T^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( ( 9 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 - 8 T + T^{2} \)
$47$ \( 144 - 12 T + T^{2} \)
$53$ \( ( 8 + T )^{2} \)
$59$ \( 16 + 4 T + T^{2} \)
$61$ \( 25 - 5 T + T^{2} \)
$67$ \( 121 + 11 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( -1 + T )^{2} \)
$79$ \( 25 - 5 T + T^{2} \)
$83$ \( 64 + 8 T + T^{2} \)
$89$ \( ( -12 + T )^{2} \)
$97$ \( 25 + 5 T + T^{2} \)
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