Properties

Label 648.4.i.g
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + ( 9 - 9 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + \zeta_{6} q^{5} + ( 9 - 9 \zeta_{6} ) q^{7} + ( -17 + 17 \zeta_{6} ) q^{11} + 44 \zeta_{6} q^{13} -56 q^{17} -94 q^{19} -50 \zeta_{6} q^{23} + ( 124 - 124 \zeta_{6} ) q^{25} + ( -30 + 30 \zeta_{6} ) q^{29} + 139 \zeta_{6} q^{31} + 9 q^{35} -174 q^{37} + 318 \zeta_{6} q^{41} + ( 242 - 242 \zeta_{6} ) q^{43} + ( -630 + 630 \zeta_{6} ) q^{47} + 262 \zeta_{6} q^{49} -547 q^{53} -17 q^{55} -236 \zeta_{6} q^{59} + ( -328 + 328 \zeta_{6} ) q^{61} + ( -44 + 44 \zeta_{6} ) q^{65} -614 \zeta_{6} q^{67} -296 q^{71} + 433 q^{73} + 153 \zeta_{6} q^{77} + ( 56 - 56 \zeta_{6} ) q^{79} + ( -1225 + 1225 \zeta_{6} ) q^{83} -56 \zeta_{6} q^{85} -1506 q^{89} + 396 q^{91} -94 \zeta_{6} q^{95} + ( -1391 + 1391 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} + 9q^{7} + O(q^{10}) \) \( 2q + q^{5} + 9q^{7} - 17q^{11} + 44q^{13} - 112q^{17} - 188q^{19} - 50q^{23} + 124q^{25} - 30q^{29} + 139q^{31} + 18q^{35} - 348q^{37} + 318q^{41} + 242q^{43} - 630q^{47} + 262q^{49} - 1094q^{53} - 34q^{55} - 236q^{59} - 328q^{61} - 44q^{65} - 614q^{67} - 592q^{71} + 866q^{73} + 153q^{77} + 56q^{79} - 1225q^{83} - 56q^{85} - 3012q^{89} + 792q^{91} - 94q^{95} - 1391q^{97} + O(q^{100}) \)

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 0.500000 + 0.866025i 0 4.50000 7.79423i 0 0 0
433.1 0 0 0 0.500000 0.866025i 0 4.50000 + 7.79423i 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.g 2
3.b odd 2 1 648.4.i.f 2
9.c even 3 1 216.4.a.b 1
9.c even 3 1 inner 648.4.i.g 2
9.d odd 6 1 216.4.a.c yes 1
9.d odd 6 1 648.4.i.f 2
36.f odd 6 1 432.4.a.f 1
36.h even 6 1 432.4.a.i 1
72.j odd 6 1 1728.4.a.m 1
72.l even 6 1 1728.4.a.n 1
72.n even 6 1 1728.4.a.s 1
72.p odd 6 1 1728.4.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.b 1 9.c even 3 1
216.4.a.c yes 1 9.d odd 6 1
432.4.a.f 1 36.f odd 6 1
432.4.a.i 1 36.h even 6 1
648.4.i.f 2 3.b odd 2 1
648.4.i.f 2 9.d odd 6 1
648.4.i.g 2 1.a even 1 1 trivial
648.4.i.g 2 9.c even 3 1 inner
1728.4.a.m 1 72.j odd 6 1
1728.4.a.n 1 72.l even 6 1
1728.4.a.s 1 72.n even 6 1
1728.4.a.t 1 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - T_{5} + 1 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 81 - 9 T + T^{2} \)
$11$ \( 289 + 17 T + T^{2} \)
$13$ \( 1936 - 44 T + T^{2} \)
$17$ \( ( 56 + T )^{2} \)
$19$ \( ( 94 + T )^{2} \)
$23$ \( 2500 + 50 T + T^{2} \)
$29$ \( 900 + 30 T + T^{2} \)
$31$ \( 19321 - 139 T + T^{2} \)
$37$ \( ( 174 + T )^{2} \)
$41$ \( 101124 - 318 T + T^{2} \)
$43$ \( 58564 - 242 T + T^{2} \)
$47$ \( 396900 + 630 T + T^{2} \)
$53$ \( ( 547 + T )^{2} \)
$59$ \( 55696 + 236 T + T^{2} \)
$61$ \( 107584 + 328 T + T^{2} \)
$67$ \( 376996 + 614 T + T^{2} \)
$71$ \( ( 296 + T )^{2} \)
$73$ \( ( -433 + T )^{2} \)
$79$ \( 3136 - 56 T + T^{2} \)
$83$ \( 1500625 + 1225 T + T^{2} \)
$89$ \( ( 1506 + T )^{2} \)
$97$ \( 1934881 + 1391 T + T^{2} \)
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