Properties

Label 648.4.i.l
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + 16 \zeta_{6} q^{5} + ( 12 - 12 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 16 \zeta_{6} q^{5} + ( 12 - 12 \zeta_{6} ) q^{7} + ( 64 - 64 \zeta_{6} ) q^{11} -58 \zeta_{6} q^{13} -32 q^{17} -136 q^{19} -128 \zeta_{6} q^{23} + ( -131 + 131 \zeta_{6} ) q^{25} + ( -144 + 144 \zeta_{6} ) q^{29} -20 \zeta_{6} q^{31} + 192 q^{35} -18 q^{37} -288 \zeta_{6} q^{41} + ( 200 - 200 \zeta_{6} ) q^{43} + ( 384 - 384 \zeta_{6} ) q^{47} + 199 \zeta_{6} q^{49} -496 q^{53} + 1024 q^{55} -128 \zeta_{6} q^{59} + ( 458 - 458 \zeta_{6} ) q^{61} + ( 928 - 928 \zeta_{6} ) q^{65} + 496 \zeta_{6} q^{67} -512 q^{71} -602 q^{73} -768 \zeta_{6} q^{77} + ( -1108 + 1108 \zeta_{6} ) q^{79} + ( 704 - 704 \zeta_{6} ) q^{83} -512 \zeta_{6} q^{85} + 960 q^{89} -696 q^{91} -2176 \zeta_{6} q^{95} + ( -206 + 206 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 16q^{5} + 12q^{7} + O(q^{10}) \) \( 2q + 16q^{5} + 12q^{7} + 64q^{11} - 58q^{13} - 64q^{17} - 272q^{19} - 128q^{23} - 131q^{25} - 144q^{29} - 20q^{31} + 384q^{35} - 36q^{37} - 288q^{41} + 200q^{43} + 384q^{47} + 199q^{49} - 992q^{53} + 2048q^{55} - 128q^{59} + 458q^{61} + 928q^{65} + 496q^{67} - 1024q^{71} - 1204q^{73} - 768q^{77} - 1108q^{79} + 704q^{83} - 512q^{85} + 1920q^{89} - 1392q^{91} - 2176q^{95} - 206q^{97} + O(q^{100}) \)

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 8.00000 + 13.8564i 0 6.00000 10.3923i 0 0 0
433.1 0 0 0 8.00000 13.8564i 0 6.00000 + 10.3923i 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.l 2
3.b odd 2 1 648.4.i.a 2
9.c even 3 1 72.4.a.a 1
9.c even 3 1 inner 648.4.i.l 2
9.d odd 6 1 72.4.a.d yes 1
9.d odd 6 1 648.4.i.a 2
36.f odd 6 1 144.4.a.a 1
36.h even 6 1 144.4.a.f 1
45.h odd 6 1 1800.4.a.ba 1
45.j even 6 1 1800.4.a.z 1
45.k odd 12 2 1800.4.f.b 2
45.l even 12 2 1800.4.f.x 2
72.j odd 6 1 576.4.a.c 1
72.l even 6 1 576.4.a.d 1
72.n even 6 1 576.4.a.w 1
72.p odd 6 1 576.4.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 9.c even 3 1
72.4.a.d yes 1 9.d odd 6 1
144.4.a.a 1 36.f odd 6 1
144.4.a.f 1 36.h even 6 1
576.4.a.c 1 72.j odd 6 1
576.4.a.d 1 72.l even 6 1
576.4.a.w 1 72.n even 6 1
576.4.a.x 1 72.p odd 6 1
648.4.i.a 2 3.b odd 2 1
648.4.i.a 2 9.d odd 6 1
648.4.i.l 2 1.a even 1 1 trivial
648.4.i.l 2 9.c even 3 1 inner
1800.4.a.z 1 45.j even 6 1
1800.4.a.ba 1 45.h odd 6 1
1800.4.f.b 2 45.k odd 12 2
1800.4.f.x 2 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 16 T_{5} + 256 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 256 - 16 T + T^{2} \)
$7$ \( 144 - 12 T + T^{2} \)
$11$ \( 4096 - 64 T + T^{2} \)
$13$ \( 3364 + 58 T + T^{2} \)
$17$ \( ( 32 + T )^{2} \)
$19$ \( ( 136 + T )^{2} \)
$23$ \( 16384 + 128 T + T^{2} \)
$29$ \( 20736 + 144 T + T^{2} \)
$31$ \( 400 + 20 T + T^{2} \)
$37$ \( ( 18 + T )^{2} \)
$41$ \( 82944 + 288 T + T^{2} \)
$43$ \( 40000 - 200 T + T^{2} \)
$47$ \( 147456 - 384 T + T^{2} \)
$53$ \( ( 496 + T )^{2} \)
$59$ \( 16384 + 128 T + T^{2} \)
$61$ \( 209764 - 458 T + T^{2} \)
$67$ \( 246016 - 496 T + T^{2} \)
$71$ \( ( 512 + T )^{2} \)
$73$ \( ( 602 + T )^{2} \)
$79$ \( 1227664 + 1108 T + T^{2} \)
$83$ \( 495616 - 704 T + T^{2} \)
$89$ \( ( -960 + T )^{2} \)
$97$ \( 42436 + 206 T + T^{2} \)
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