Properties

Label 648.2.n.n
Level 648
Weight 2
Character orbit 648.n
Analytic conductor 5.174
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 648.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{2} + \beta_{4} + \beta_{6} ) q^{4} -\beta_{3} q^{5} -\beta_{4} q^{7} + ( -\beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{2} + \beta_{4} + \beta_{6} ) q^{4} -\beta_{3} q^{5} -\beta_{4} q^{7} + ( -\beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{8} + ( 1 + \beta_{6} ) q^{10} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{11} + ( 2 - 4 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} ) q^{13} + ( \beta_{3} - \beta_{7} ) q^{14} + ( 3 \beta_{2} + \beta_{4} ) q^{16} + ( 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} - 4 \beta_{7} ) q^{17} + ( 2 + 4 \beta_{6} ) q^{19} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{20} + ( -3 + 3 \beta_{2} + 3 \beta_{4} - 3 \beta_{6} ) q^{22} + ( -2 \beta_{3} + 4 \beta_{7} ) q^{23} -4 \beta_{4} q^{25} + ( 2 \beta_{1} + 2 \beta_{3} + 8 \beta_{5} - 2 \beta_{7} ) q^{26} + ( 1 - \beta_{6} ) q^{28} + ( 6 \beta_{3} + 6 \beta_{5} ) q^{29} + ( 7 - 7 \beta_{4} ) q^{31} + ( 5 \beta_{3} + \beta_{7} ) q^{32} + ( -2 \beta_{2} + 6 \beta_{4} ) q^{34} -\beta_{5} q^{35} + ( -2 - 4 \beta_{6} ) q^{37} + ( 2 \beta_{1} + 8 \beta_{3} + 8 \beta_{5} ) q^{38} + ( -3 + \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{40} + ( 2 \beta_{3} - 4 \beta_{7} ) q^{41} + ( -8 \beta_{2} - 4 \beta_{4} ) q^{43} + ( -3 \beta_{1} - 3 \beta_{3} - 6 \beta_{5} + 3 \beta_{7} ) q^{44} + ( -6 + 2 \beta_{6} ) q^{46} + ( 6 - 6 \beta_{4} ) q^{49} + ( 4 \beta_{3} - 4 \beta_{7} ) q^{50} + ( 6 \beta_{2} + 10 \beta_{4} ) q^{52} + 9 \beta_{5} q^{53} -3 q^{55} + ( \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{56} + ( -6 + 6 \beta_{2} + 6 \beta_{4} - 6 \beta_{6} ) q^{58} + 4 \beta_{3} q^{59} + ( 7 \beta_{1} + 7 \beta_{3} - 7 \beta_{7} ) q^{62} + ( -7 - 5 \beta_{6} ) q^{64} + ( 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{65} + ( -10 \beta_{3} + 6 \beta_{7} ) q^{68} + ( -\beta_{2} - \beta_{4} ) q^{70} + ( -12 \beta_{1} - 12 \beta_{3} - 6 \beta_{5} + 12 \beta_{7} ) q^{71} + 3 q^{73} + ( -2 \beta_{1} - 8 \beta_{3} - 8 \beta_{5} ) q^{74} + ( -10 + 6 \beta_{2} + 10 \beta_{4} - 6 \beta_{6} ) q^{76} -3 \beta_{3} q^{77} -4 \beta_{4} q^{79} + ( -3 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{7} ) q^{80} + ( 6 - 2 \beta_{6} ) q^{82} + ( -7 \beta_{3} - 7 \beta_{5} ) q^{83} + ( 2 - 4 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} ) q^{85} + ( -12 \beta_{3} - 4 \beta_{7} ) q^{86} + ( -3 \beta_{2} - 9 \beta_{4} ) q^{88} + ( 8 \beta_{1} + 8 \beta_{3} + 4 \beta_{5} - 8 \beta_{7} ) q^{89} + ( -2 - 4 \beta_{6} ) q^{91} + ( -6 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} ) q^{92} + ( -2 \beta_{3} + 4 \beta_{7} ) q^{95} -7 \beta_{4} q^{97} + ( 6 \beta_{1} + 6 \beta_{3} - 6 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 6q^{4} - 4q^{7} + O(q^{10}) \) \( 8q - 6q^{4} - 4q^{7} + 4q^{10} - 2q^{16} - 6q^{22} - 16q^{25} + 12q^{28} + 28q^{31} + 28q^{34} - 10q^{40} - 56q^{46} + 24q^{49} + 28q^{52} - 24q^{55} - 12q^{58} - 36q^{64} - 2q^{70} + 24q^{73} - 28q^{76} - 16q^{79} + 56q^{82} - 30q^{88} - 28q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} + 5 \nu^{4} - 5 \nu^{2} - 12 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 5 \nu^{5} - 5 \nu^{3} - 12 \nu \)\()/40\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 36 \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/40\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 7 \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 12 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{4} - \beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 3 \beta_{2}\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 5 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{6} - 7\)
\(\nu^{7}\)\(=\)\(-10 \beta_{5} - 10 \beta_{3} - 7 \beta_{1}\)

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(1\) \(-1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.09445 0.895644i
−0.228425 1.39564i
0.228425 + 1.39564i
1.09445 + 0.895644i
−1.09445 + 0.895644i
−0.228425 + 1.39564i
0.228425 1.39564i
1.09445 0.895644i
−1.09445 0.895644i 0 0.395644 + 1.96048i −0.866025 0.500000i 0 −0.500000 0.866025i 1.32288 2.50000i 0 0.500000 + 1.32288i
109.2 −0.228425 1.39564i 0 −1.89564 + 0.637600i 0.866025 + 0.500000i 0 −0.500000 0.866025i 1.32288 + 2.50000i 0 0.500000 1.32288i
109.3 0.228425 + 1.39564i 0 −1.89564 + 0.637600i −0.866025 0.500000i 0 −0.500000 0.866025i −1.32288 2.50000i 0 0.500000 1.32288i
109.4 1.09445 + 0.895644i 0 0.395644 + 1.96048i 0.866025 + 0.500000i 0 −0.500000 0.866025i −1.32288 + 2.50000i 0 0.500000 + 1.32288i
541.1 −1.09445 + 0.895644i 0 0.395644 1.96048i −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 1.32288 + 2.50000i 0 0.500000 1.32288i
541.2 −0.228425 + 1.39564i 0 −1.89564 0.637600i 0.866025 0.500000i 0 −0.500000 + 0.866025i 1.32288 2.50000i 0 0.500000 + 1.32288i
541.3 0.228425 1.39564i 0 −1.89564 0.637600i −0.866025 + 0.500000i 0 −0.500000 + 0.866025i −1.32288 + 2.50000i 0 0.500000 + 1.32288i
541.4 1.09445 0.895644i 0 0.395644 1.96048i 0.866025 0.500000i 0 −0.500000 + 0.866025i −1.32288 2.50000i 0 0.500000 1.32288i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
24.h odd 2 1 inner
72.j odd 6 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.n.n 8
3.b odd 2 1 inner 648.2.n.n 8
4.b odd 2 1 2592.2.r.p 8
8.b even 2 1 inner 648.2.n.n 8
8.d odd 2 1 2592.2.r.p 8
9.c even 3 1 216.2.d.b 4
9.c even 3 1 inner 648.2.n.n 8
9.d odd 6 1 216.2.d.b 4
9.d odd 6 1 inner 648.2.n.n 8
12.b even 2 1 2592.2.r.p 8
24.f even 2 1 2592.2.r.p 8
24.h odd 2 1 inner 648.2.n.n 8
36.f odd 6 1 864.2.d.a 4
36.f odd 6 1 2592.2.r.p 8
36.h even 6 1 864.2.d.a 4
36.h even 6 1 2592.2.r.p 8
72.j odd 6 1 216.2.d.b 4
72.j odd 6 1 inner 648.2.n.n 8
72.l even 6 1 864.2.d.a 4
72.l even 6 1 2592.2.r.p 8
72.n even 6 1 216.2.d.b 4
72.n even 6 1 inner 648.2.n.n 8
72.p odd 6 1 864.2.d.a 4
72.p odd 6 1 2592.2.r.p 8
144.u even 12 1 6912.2.a.bd 2
144.u even 12 1 6912.2.a.bv 2
144.v odd 12 1 6912.2.a.bd 2
144.v odd 12 1 6912.2.a.bv 2
144.w odd 12 1 6912.2.a.bc 2
144.w odd 12 1 6912.2.a.bu 2
144.x even 12 1 6912.2.a.bc 2
144.x even 12 1 6912.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 9.c even 3 1
216.2.d.b 4 9.d odd 6 1
216.2.d.b 4 72.j odd 6 1
216.2.d.b 4 72.n even 6 1
648.2.n.n 8 1.a even 1 1 trivial
648.2.n.n 8 3.b odd 2 1 inner
648.2.n.n 8 8.b even 2 1 inner
648.2.n.n 8 9.c even 3 1 inner
648.2.n.n 8 9.d odd 6 1 inner
648.2.n.n 8 24.h odd 2 1 inner
648.2.n.n 8 72.j odd 6 1 inner
648.2.n.n 8 72.n even 6 1 inner
864.2.d.a 4 36.f odd 6 1
864.2.d.a 4 36.h even 6 1
864.2.d.a 4 72.l even 6 1
864.2.d.a 4 72.p odd 6 1
2592.2.r.p 8 4.b odd 2 1
2592.2.r.p 8 8.d odd 2 1
2592.2.r.p 8 12.b even 2 1
2592.2.r.p 8 24.f even 2 1
2592.2.r.p 8 36.f odd 6 1
2592.2.r.p 8 36.h even 6 1
2592.2.r.p 8 72.l even 6 1
2592.2.r.p 8 72.p odd 6 1
6912.2.a.bc 2 144.w odd 12 1
6912.2.a.bc 2 144.x even 12 1
6912.2.a.bd 2 144.u even 12 1
6912.2.a.bd 2 144.v odd 12 1
6912.2.a.bu 2 144.w odd 12 1
6912.2.a.bu 2 144.x even 12 1
6912.2.a.bv 2 144.u even 12 1
6912.2.a.bv 2 144.v odd 12 1

Hecke kernels

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

\( T_{5}^{4} - T_{5}^{2} + 1 \)
\( T_{13}^{4} - 28 T_{13}^{2} + 784 \)
\( T_{17}^{2} - 28 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \)
$3$ 1
$5$ \( ( 1 + 9 T^{2} + 56 T^{4} + 225 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 4 T + 7 T^{2} )^{4}( 1 + 5 T + 7 T^{2} )^{4} \)
$11$ \( ( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 6 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 10 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 22 T^{2} - 357 T^{4} + 18502 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{4}( 1 + 4 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 46 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 54 T^{2} + 1235 T^{4} - 90774 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 26 T^{2} - 1173 T^{4} - 48074 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 47 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 - 25 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 102 T^{2} + 6923 T^{4} + 355062 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 61 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 + 67 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 3 T + 73 T^{2} )^{8} \)
$79$ \( ( 1 - 13 T + 79 T^{2} )^{4}( 1 + 17 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 117 T^{2} + 6800 T^{4} + 806013 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 66 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4} )^{4} \)
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