Properties

Label 648.4.i.i
Level $648$
Weight $4$
Character orbit 648.i
Analytic conductor $38.233$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.2332376837\)
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 4 \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} + ( 28 - 28 \zeta_{6} ) q^{11} + 11 \zeta_{6} q^{13} -44 q^{17} + 29 q^{19} + 172 \zeta_{6} q^{23} + ( 109 - 109 \zeta_{6} ) q^{25} + ( 192 - 192 \zeta_{6} ) q^{29} -116 \zeta_{6} q^{31} -12 q^{35} -69 q^{37} + 384 \zeta_{6} q^{41} + ( -328 + 328 \zeta_{6} ) q^{43} + ( 156 - 156 \zeta_{6} ) q^{47} + 334 \zeta_{6} q^{49} + 392 q^{53} + 112 q^{55} + 412 \zeta_{6} q^{59} + ( 425 - 425 \zeta_{6} ) q^{61} + ( -44 + 44 \zeta_{6} ) q^{65} -257 \zeta_{6} q^{67} + 1000 q^{71} -359 q^{73} + 84 \zeta_{6} q^{77} + ( -877 + 877 \zeta_{6} ) q^{79} + ( -328 + 328 \zeta_{6} ) q^{83} -176 \zeta_{6} q^{85} + 1572 q^{89} -33 q^{91} + 116 \zeta_{6} q^{95} + ( 1483 - 1483 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 3q^{7} + O(q^{10}) \) \( 2q + 4q^{5} - 3q^{7} + 28q^{11} + 11q^{13} - 88q^{17} + 58q^{19} + 172q^{23} + 109q^{25} + 192q^{29} - 116q^{31} - 24q^{35} - 138q^{37} + 384q^{41} - 328q^{43} + 156q^{47} + 334q^{49} + 784q^{53} + 224q^{55} + 412q^{59} + 425q^{61} - 44q^{65} - 257q^{67} + 2000q^{71} - 718q^{73} + 84q^{77} - 877q^{79} - 328q^{83} - 176q^{85} + 3144q^{89} - 66q^{91} + 116q^{95} + 1483q^{97} + O(q^{100}) \)

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 648.4.i.i 2
3.b odd 2 1 648.4.i.d 2
9.c even 3 1 216.4.a.a 1
9.c even 3 1 inner 648.4.i.i 2
9.d odd 6 1 216.4.a.d yes 1
9.d odd 6 1 648.4.i.d 2
36.f odd 6 1 432.4.a.d 1
36.h even 6 1 432.4.a.k 1
72.j odd 6 1 1728.4.a.j 1
72.l even 6 1 1728.4.a.i 1
72.n even 6 1 1728.4.a.x 1
72.p odd 6 1 1728.4.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.a 1 9.c even 3 1
216.4.a.d yes 1 9.d odd 6 1
432.4.a.d 1 36.f odd 6 1
432.4.a.k 1 36.h even 6 1
648.4.i.d 2 3.b odd 2 1
648.4.i.d 2 9.d odd 6 1
648.4.i.i 2 1.a even 1 1 trivial
648.4.i.i 2 9.c even 3 1 inner
1728.4.a.i 1 72.l even 6 1
1728.4.a.j 1 72.j odd 6 1
1728.4.a.w 1 72.p odd 6 1
1728.4.a.x 1 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4 T_{5} + 16 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 - 4 T + T^{2} \)
$7$ \( 9 + 3 T + T^{2} \)
$11$ \( 784 - 28 T + T^{2} \)
$13$ \( 121 - 11 T + T^{2} \)
$17$ \( ( 44 + T )^{2} \)
$19$ \( ( -29 + T )^{2} \)
$23$ \( 29584 - 172 T + T^{2} \)
$29$ \( 36864 - 192 T + T^{2} \)
$31$ \( 13456 + 116 T + T^{2} \)
$37$ \( ( 69 + T )^{2} \)
$41$ \( 147456 - 384 T + T^{2} \)
$43$ \( 107584 + 328 T + T^{2} \)
$47$ \( 24336 - 156 T + T^{2} \)
$53$ \( ( -392 + T )^{2} \)
$59$ \( 169744 - 412 T + T^{2} \)
$61$ \( 180625 - 425 T + T^{2} \)
$67$ \( 66049 + 257 T + T^{2} \)
$71$ \( ( -1000 + T )^{2} \)
$73$ \( ( 359 + T )^{2} \)
$79$ \( 769129 + 877 T + T^{2} \)
$83$ \( 107584 + 328 T + T^{2} \)
$89$ \( ( -1572 + T )^{2} \)
$97$ \( 2199289 - 1483 T + T^{2} \)
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