Properties

Label 648.4.i.e
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 8)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{5} + ( -24 + 24 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 \zeta_{6} q^{5} + ( -24 + 24 \zeta_{6} ) q^{7} + ( -44 + 44 \zeta_{6} ) q^{11} -22 \zeta_{6} q^{13} -50 q^{17} + 44 q^{19} -56 \zeta_{6} q^{23} + ( 121 - 121 \zeta_{6} ) q^{25} + ( 198 - 198 \zeta_{6} ) q^{29} + 160 \zeta_{6} q^{31} + 48 q^{35} -162 q^{37} -198 \zeta_{6} q^{41} + ( -52 + 52 \zeta_{6} ) q^{43} + ( 528 - 528 \zeta_{6} ) q^{47} -233 \zeta_{6} q^{49} + 242 q^{53} + 88 q^{55} -668 \zeta_{6} q^{59} + ( -550 + 550 \zeta_{6} ) q^{61} + ( -44 + 44 \zeta_{6} ) q^{65} -188 \zeta_{6} q^{67} -728 q^{71} + 154 q^{73} -1056 \zeta_{6} q^{77} + ( 656 - 656 \zeta_{6} ) q^{79} + ( 236 - 236 \zeta_{6} ) q^{83} + 100 \zeta_{6} q^{85} -714 q^{89} + 528 q^{91} -88 \zeta_{6} q^{95} + ( 478 - 478 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 24q^{7} + O(q^{10}) \) \( 2q - 2q^{5} - 24q^{7} - 44q^{11} - 22q^{13} - 100q^{17} + 88q^{19} - 56q^{23} + 121q^{25} + 198q^{29} + 160q^{31} + 96q^{35} - 324q^{37} - 198q^{41} - 52q^{43} + 528q^{47} - 233q^{49} + 484q^{53} + 176q^{55} - 668q^{59} - 550q^{61} - 44q^{65} - 188q^{67} - 1456q^{71} + 308q^{73} - 1056q^{77} + 656q^{79} + 236q^{83} + 100q^{85} - 1428q^{89} + 1056q^{91} - 88q^{95} + 478q^{97} + O(q^{100}) \)

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 −1.00000 1.73205i 0 −12.0000 + 20.7846i 0 0 0
433.1 0 0 0 −1.00000 + 1.73205i 0 −12.0000 20.7846i 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.e 2
3.b odd 2 1 648.4.i.h 2
9.c even 3 1 72.4.a.c 1
9.c even 3 1 inner 648.4.i.e 2
9.d odd 6 1 8.4.a.a 1
9.d odd 6 1 648.4.i.h 2
36.f odd 6 1 144.4.a.e 1
36.h even 6 1 16.4.a.a 1
45.h odd 6 1 200.4.a.g 1
45.j even 6 1 1800.4.a.d 1
45.k odd 12 2 1800.4.f.u 2
45.l even 12 2 200.4.c.e 2
63.i even 6 1 392.4.i.b 2
63.j odd 6 1 392.4.i.g 2
63.n odd 6 1 392.4.i.g 2
63.o even 6 1 392.4.a.e 1
63.s even 6 1 392.4.i.b 2
72.j odd 6 1 64.4.a.d 1
72.l even 6 1 64.4.a.b 1
72.n even 6 1 576.4.a.k 1
72.p odd 6 1 576.4.a.j 1
99.g even 6 1 968.4.a.a 1
117.n odd 6 1 1352.4.a.a 1
144.u even 12 2 256.4.b.g 2
144.w odd 12 2 256.4.b.a 2
153.i odd 6 1 2312.4.a.a 1
180.n even 6 1 400.4.a.g 1
180.v odd 12 2 400.4.c.i 2
252.s odd 6 1 784.4.a.e 1
360.bd even 6 1 1600.4.a.bm 1
360.bh odd 6 1 1600.4.a.o 1
396.o odd 6 1 1936.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 9.d odd 6 1
16.4.a.a 1 36.h even 6 1
64.4.a.b 1 72.l even 6 1
64.4.a.d 1 72.j odd 6 1
72.4.a.c 1 9.c even 3 1
144.4.a.e 1 36.f odd 6 1
200.4.a.g 1 45.h odd 6 1
200.4.c.e 2 45.l even 12 2
256.4.b.a 2 144.w odd 12 2
256.4.b.g 2 144.u even 12 2
392.4.a.e 1 63.o even 6 1
392.4.i.b 2 63.i even 6 1
392.4.i.b 2 63.s even 6 1
392.4.i.g 2 63.j odd 6 1
392.4.i.g 2 63.n odd 6 1
400.4.a.g 1 180.n even 6 1
400.4.c.i 2 180.v odd 12 2
576.4.a.j 1 72.p odd 6 1
576.4.a.k 1 72.n even 6 1
648.4.i.e 2 1.a even 1 1 trivial
648.4.i.e 2 9.c even 3 1 inner
648.4.i.h 2 3.b odd 2 1
648.4.i.h 2 9.d odd 6 1
784.4.a.e 1 252.s odd 6 1
968.4.a.a 1 99.g even 6 1
1352.4.a.a 1 117.n odd 6 1
1600.4.a.o 1 360.bh odd 6 1
1600.4.a.bm 1 360.bd even 6 1
1800.4.a.d 1 45.j even 6 1
1800.4.f.u 2 45.k odd 12 2
1936.4.a.l 1 396.o odd 6 1
2312.4.a.a 1 153.i odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T - 121 T^{2} + 250 T^{3} + 15625 T^{4} \)
$7$ \( 1 + 24 T + 233 T^{2} + 8232 T^{3} + 117649 T^{4} \)
$11$ \( 1 + 44 T + 605 T^{2} + 58564 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 22 T - 1713 T^{2} + 48334 T^{3} + 4826809 T^{4} \)
$17$ \( ( 1 + 50 T + 4913 T^{2} )^{2} \)
$19$ \( ( 1 - 44 T + 6859 T^{2} )^{2} \)
$23$ \( 1 + 56 T - 9031 T^{2} + 681352 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 198 T + 14815 T^{2} - 4829022 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 160 T - 4191 T^{2} - 4766560 T^{3} + 887503681 T^{4} \)
$37$ \( ( 1 + 162 T + 50653 T^{2} )^{2} \)
$41$ \( 1 + 198 T - 29717 T^{2} + 13646358 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 52 T - 76803 T^{2} + 4134364 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 528 T + 174961 T^{2} - 54818544 T^{3} + 10779215329 T^{4} \)
$53$ \( ( 1 - 242 T + 148877 T^{2} )^{2} \)
$59$ \( 1 + 668 T + 240845 T^{2} + 137193172 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 550 T + 75519 T^{2} + 124839550 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 188 T - 265419 T^{2} + 56543444 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 728 T + 357911 T^{2} )^{2} \)
$73$ \( ( 1 - 154 T + 389017 T^{2} )^{2} \)
$79$ \( 1 - 656 T - 62703 T^{2} - 323433584 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 236 T - 516091 T^{2} - 134941732 T^{3} + 326940373369 T^{4} \)
$89$ \( ( 1 + 714 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 478 T - 684189 T^{2} - 436257694 T^{3} + 832972004929 T^{4} \)
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