Properties

Label 1728.2.i.n
Level $1728$
Weight $2$
Character orbit 1728.i
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.170772624.1
Defining polynomial: \(x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{6} q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} +O(q^{10})\) \( q -\beta_{6} q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} + \beta_{3} q^{11} + ( 2 \beta_{4} + \beta_{6} ) q^{13} + \beta_{5} q^{17} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} -\beta_{7} q^{23} + ( -4 + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{25} + ( 3 - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} + ( -2 \beta_{2} - \beta_{7} ) q^{31} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{35} -4 q^{37} + \beta_{4} q^{41} + \beta_{3} q^{43} + ( -\beta_{1} + \beta_{7} ) q^{47} + ( -5 \beta_{4} + 3 \beta_{6} ) q^{49} -4 q^{53} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -\beta_{2} + 2 \beta_{7} ) q^{59} + ( 5 - 8 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{61} + ( 7 - 8 \beta_{4} - \beta_{5} - \beta_{6} ) q^{65} + ( -\beta_{2} + 2 \beta_{7} ) q^{67} + 2 \beta_{1} q^{71} + ( 8 + \beta_{5} ) q^{73} + ( 6 \beta_{4} + 3 \beta_{6} ) q^{77} + ( \beta_{1} + 2 \beta_{3} - \beta_{7} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{83} + 8 \beta_{4} q^{85} + ( -8 - 2 \beta_{5} ) q^{89} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{91} + 4 \beta_{7} q^{95} + ( -9 + 9 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{5} + O(q^{10}) \) \( 8q + 2q^{5} + 6q^{13} - 4q^{17} - 14q^{25} + 10q^{29} - 32q^{37} + 4q^{41} - 26q^{49} - 32q^{53} + 26q^{61} + 30q^{65} + 60q^{73} + 18q^{77} + 32q^{85} - 56q^{89} - 36q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{4} + 6 \nu^{2} + 4 \nu \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} + 3 \nu^{5} - 6 \nu^{4} + 6 \nu^{3} + 4 \nu - 24 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} - 9 \nu^{5} + 6 \nu^{4} - 6 \nu^{3} + 24 \nu^{2} - 44 \nu + 48 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{7} - 5 \nu^{6} + 7 \nu^{5} - 6 \nu^{4} + 10 \nu^{3} - 24 \nu^{2} + 28 \nu - 24 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 6 \nu^{5} + 5 \nu^{4} - 6 \nu^{3} + 14 \nu^{2} - 20 \nu + 20 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} + 4 \nu^{4} - 5 \nu^{3} + 13 \nu^{2} - 18 \nu + 20 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{7} - 3 \nu^{6} + 6 \nu^{5} - 3 \nu^{4} + 6 \nu^{3} - 15 \nu^{2} + 16 \nu - 24 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{1} + 2\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1}\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{6} + \beta_{5} + 10 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_{1} - 4\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_{1}\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} - \beta_{6} - 2 \beta_{5} - 17 \beta_{4} + \beta_{2} + 32\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{7} + 9 \beta_{5} - 5 \beta_{3} + 5 \beta_{2} - \beta_{1} + 24\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{7} - 13 \beta_{6} + 13 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + \beta_{1} + 8\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−1.02187 0.977642i
0.335728 + 1.37379i
0.774115 1.18353i
1.41203 0.0786378i
−1.02187 + 0.977642i
0.335728 1.37379i
0.774115 + 1.18353i
1.41203 + 0.0786378i
0 0 0 −1.18614 + 2.05446i 0 −1.10489 1.91373i 0 0 0
577.2 0 0 0 −1.18614 + 2.05446i 0 1.10489 + 1.91373i 0 0 0
577.3 0 0 0 1.68614 2.92048i 0 −2.35143 4.07279i 0 0 0
577.4 0 0 0 1.68614 2.92048i 0 2.35143 + 4.07279i 0 0 0
1153.1 0 0 0 −1.18614 2.05446i 0 −1.10489 + 1.91373i 0 0 0
1153.2 0 0 0 −1.18614 2.05446i 0 1.10489 1.91373i 0 0 0
1153.3 0 0 0 1.68614 + 2.92048i 0 −2.35143 + 4.07279i 0 0 0
1153.4 0 0 0 1.68614 + 2.92048i 0 2.35143 4.07279i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1153.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.n 8
3.b odd 2 1 576.2.i.n 8
4.b odd 2 1 inner 1728.2.i.n 8
8.b even 2 1 864.2.i.f 8
8.d odd 2 1 864.2.i.f 8
9.c even 3 1 inner 1728.2.i.n 8
9.c even 3 1 5184.2.a.cc 4
9.d odd 6 1 576.2.i.n 8
9.d odd 6 1 5184.2.a.cf 4
12.b even 2 1 576.2.i.n 8
24.f even 2 1 288.2.i.f 8
24.h odd 2 1 288.2.i.f 8
36.f odd 6 1 inner 1728.2.i.n 8
36.f odd 6 1 5184.2.a.cc 4
36.h even 6 1 576.2.i.n 8
36.h even 6 1 5184.2.a.cf 4
72.j odd 6 1 288.2.i.f 8
72.j odd 6 1 2592.2.a.u 4
72.l even 6 1 288.2.i.f 8
72.l even 6 1 2592.2.a.u 4
72.n even 6 1 864.2.i.f 8
72.n even 6 1 2592.2.a.x 4
72.p odd 6 1 864.2.i.f 8
72.p odd 6 1 2592.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 24.f even 2 1
288.2.i.f 8 24.h odd 2 1
288.2.i.f 8 72.j odd 6 1
288.2.i.f 8 72.l even 6 1
576.2.i.n 8 3.b odd 2 1
576.2.i.n 8 9.d odd 6 1
576.2.i.n 8 12.b even 2 1
576.2.i.n 8 36.h even 6 1
864.2.i.f 8 8.b even 2 1
864.2.i.f 8 8.d odd 2 1
864.2.i.f 8 72.n even 6 1
864.2.i.f 8 72.p odd 6 1
1728.2.i.n 8 1.a even 1 1 trivial
1728.2.i.n 8 4.b odd 2 1 inner
1728.2.i.n 8 9.c even 3 1 inner
1728.2.i.n 8 36.f odd 6 1 inner
2592.2.a.u 4 72.j odd 6 1
2592.2.a.u 4 72.l even 6 1
2592.2.a.x 4 72.n even 6 1
2592.2.a.x 4 72.p odd 6 1
5184.2.a.cc 4 9.c even 3 1
5184.2.a.cc 4 36.f odd 6 1
5184.2.a.cf 4 9.d odd 6 1
5184.2.a.cf 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} - T_{5}^{3} + 9 T_{5}^{2} + 8 T_{5} + 64 \)
\( T_{7}^{8} + 27 T_{7}^{6} + 621 T_{7}^{4} + 2916 T_{7}^{2} + 11664 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} )^{2} \)
$7$ \( 11664 + 2916 T^{2} + 621 T^{4} + 27 T^{6} + T^{8} \)
$11$ \( 729 + 972 T^{2} + 1269 T^{4} + 36 T^{6} + T^{8} \)
$13$ \( ( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$17$ \( ( -8 + T + T^{2} )^{4} \)
$19$ \( ( 432 - 45 T^{2} + T^{4} )^{2} \)
$23$ \( 11664 + 2916 T^{2} + 621 T^{4} + 27 T^{6} + T^{8} \)
$29$ \( ( 4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$31$ \( 15116544 + 524880 T^{2} + 14337 T^{4} + 135 T^{6} + T^{8} \)
$37$ \( ( 4 + T )^{8} \)
$41$ \( ( 1 - T + T^{2} )^{4} \)
$43$ \( 729 + 972 T^{2} + 1269 T^{4} + 36 T^{6} + T^{8} \)
$47$ \( 11664 + 2916 T^{2} + 621 T^{4} + 27 T^{6} + T^{8} \)
$53$ \( ( 4 + T )^{8} \)
$59$ \( 60886809 + 1404540 T^{2} + 24597 T^{4} + 180 T^{6} + T^{8} \)
$61$ \( ( 1024 + 416 T + 201 T^{2} - 13 T^{3} + T^{4} )^{2} \)
$67$ \( 60886809 + 1404540 T^{2} + 24597 T^{4} + 180 T^{6} + T^{8} \)
$71$ \( ( 1728 - 108 T^{2} + T^{4} )^{2} \)
$73$ \( ( 48 - 15 T + T^{2} )^{4} \)
$79$ \( 15116544 + 524880 T^{2} + 14337 T^{4} + 135 T^{6} + T^{8} \)
$83$ \( 2985984 + 357696 T^{2} + 41121 T^{4} + 207 T^{6} + T^{8} \)
$89$ \( ( 16 + 14 T + T^{2} )^{4} \)
$97$ \( ( 81 + 9 T + T^{2} )^{4} \)
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