Properties

Label 1728.3.b.g
Level $1728$
Weight $3$
Character orbit 1728.b
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} - 7 \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} - 7 \beta_1 q^{7} - \beta_{6} q^{11} + 11 \beta_{2} q^{13} - \beta_{5} q^{17} + 5 \beta_{4} q^{19} + \beta_{7} q^{23} - 47 q^{25} + 6 \beta_{3} q^{29} + 10 \beta_1 q^{31} - 7 \beta_{6} q^{35} - 11 \beta_{2} q^{37} - 4 \beta_{5} q^{41} - 24 \beta_{4} q^{43} + 5 \beta_{7} q^{47} - 2 \beta_{3} q^{53} + 72 \beta_1 q^{55} - 3 \beta_{6} q^{59} + 11 \beta_{2} q^{61} + 11 \beta_{5} q^{65} + 67 \beta_{4} q^{67} - 8 \beta_{7} q^{71} - 71 q^{73} - 7 \beta_{3} q^{77} + 151 \beta_1 q^{79} + 16 \beta_{6} q^{83} + 72 \beta_{2} q^{85} - 11 \beta_{5} q^{89} + 77 \beta_{4} q^{91} - 5 \beta_{7} q^{95} - 25 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 376 q^{25} - 568 q^{73} - 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -6\zeta_{24}^{5} - 6\zeta_{24}^{3} + 6\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -12\zeta_{24}^{7} + 6\zeta_{24}^{5} + 6\zeta_{24}^{3} + 6\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -6\zeta_{24}^{5} + 6\zeta_{24}^{3} + 6\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 12\zeta_{24}^{7} + 6\zeta_{24}^{5} - 6\zeta_{24}^{3} + 6\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - \beta_{3} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} ) / 24 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−0.258819 0.965926i
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 0 0 8.48528i 0 7.00000i 0 0 0
1567.2 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.3 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.4 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.5 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.6 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.7 0 0 0 8.48528i 0 7.00000i 0 0 0
1567.8 0 0 0 8.48528i 0 7.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.b.g 8
3.b odd 2 1 inner 1728.3.b.g 8
4.b odd 2 1 inner 1728.3.b.g 8
8.b even 2 1 inner 1728.3.b.g 8
8.d odd 2 1 inner 1728.3.b.g 8
12.b even 2 1 inner 1728.3.b.g 8
24.f even 2 1 inner 1728.3.b.g 8
24.h odd 2 1 inner 1728.3.b.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.g 8 1.a even 1 1 trivial
1728.3.b.g 8 3.b odd 2 1 inner
1728.3.b.g 8 4.b odd 2 1 inner
1728.3.b.g 8 8.b even 2 1 inner
1728.3.b.g 8 8.d odd 2 1 inner
1728.3.b.g 8 12.b even 2 1 inner
1728.3.b.g 8 24.f even 2 1 inner
1728.3.b.g 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 72 \) Copy content Toggle raw display
\( T_{7}^{2} + 49 \) Copy content Toggle raw display
\( T_{11}^{2} - 72 \) Copy content Toggle raw display
\( T_{17}^{2} - 216 \) 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
$5$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 72)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 363)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 216)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 75)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 216)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2592)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 363)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3456)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 1728)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 5400)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 648)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 363)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 13467)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 13824)^{4} \) Copy content Toggle raw display
$73$ \( (T + 71)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 22801)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 18432)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 26136)^{4} \) Copy content Toggle raw display
$97$ \( (T + 25)^{8} \) Copy content Toggle raw display
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