Properties

Label 1728.2.f.g
Level $1728$
Weight $2$
Character orbit 1728.f
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + \beta_{3} q^{7} - \beta_{6} q^{11} + ( - \beta_{5} + 4) q^{25} + ( - \beta_{7} - 2 \beta_1) q^{29} + ( - 3 \beta_{4} - \beta_{3}) q^{31} + (2 \beta_{6} - 3 \beta_{2}) q^{35} + (\beta_{5} - 12) q^{49} + ( - \beta_{7} + 3 \beta_1) q^{53} + (4 \beta_{4} - \beta_{3}) q^{55} - 2 \beta_{2} q^{59} + (\beta_{5} + 7) q^{73} + ( - 2 \beta_{7} - 3 \beta_1) q^{77} + 5 \beta_{4} q^{79} + (\beta_{6} - 2 \beta_{2}) q^{83} + (2 \beta_{5} - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{25} - 96 q^{49} + 56 q^{73} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 6\zeta_{24}^{4} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} - 3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -6\zeta_{24}^{5} + 6\zeta_{24}^{3} + 6\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{4} - 2\zeta_{24}^{3} + 2\zeta_{24} - 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -4\zeta_{24}^{7} - 4\zeta_{24}^{6} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 8\zeta_{24}^{2} + 2\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} - 4\beta_1 ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{7} - 3\beta_{4} + 2\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - 2\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 3 ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + 3\beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} - 4\beta_1 ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} + 4\beta_1 ) / 24 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
−0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.965926 0.258819i
0.258819 0.965926i
0.258819 + 0.965926i
0 0 0 −4.18154 0 5.24264i 0 0 0
863.2 0 0 0 −4.18154 0 5.24264i 0 0 0
863.3 0 0 0 −0.717439 0 3.24264i 0 0 0
863.4 0 0 0 −0.717439 0 3.24264i 0 0 0
863.5 0 0 0 0.717439 0 3.24264i 0 0 0
863.6 0 0 0 0.717439 0 3.24264i 0 0 0
863.7 0 0 0 4.18154 0 5.24264i 0 0 0
863.8 0 0 0 4.18154 0 5.24264i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.f.g 8
3.b odd 2 1 inner 1728.2.f.g 8
4.b odd 2 1 inner 1728.2.f.g 8
8.b even 2 1 inner 1728.2.f.g 8
8.d odd 2 1 inner 1728.2.f.g 8
12.b even 2 1 inner 1728.2.f.g 8
24.f even 2 1 inner 1728.2.f.g 8
24.h odd 2 1 CM 1728.2.f.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.2.f.g 8 1.a even 1 1 trivial
1728.2.f.g 8 3.b odd 2 1 inner
1728.2.f.g 8 4.b odd 2 1 inner
1728.2.f.g 8 8.b even 2 1 inner
1728.2.f.g 8 8.d odd 2 1 inner
1728.2.f.g 8 12.b even 2 1 inner
1728.2.f.g 8 24.f even 2 1 inner
1728.2.f.g 8 24.h odd 2 1 CM

Hecke kernels

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

\( T_{5}^{4} - 18T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 38T_{7}^{2} + 289 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 18 T^{2} + 9)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 38 T^{2} + 289)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 54 T^{2} + 441)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 86 T^{2} + 49)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} - 306 T^{2} + 21609)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 14 T - 23)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 198 T^{2} + 2601)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 287)^{4} \) Copy content Toggle raw display
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