Properties

Label 1728.3.b.f
Level $1728$
Weight $3$
Character orbit 1728.b
Analytic conductor $47.085$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 \beta_1 q^{5} + 5 \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 \beta_1 q^{5} + 5 \beta_1 q^{7} + 6 q^{11} + \beta_{2} q^{13} - 6 \beta_{3} q^{17} - 5 \beta_{3} q^{19} + 6 \beta_{2} q^{23} - 11 q^{25} - 36 \beta_1 q^{29} - 26 \beta_1 q^{31} - 30 q^{35} - 5 \beta_{2} q^{37} - 4 \beta_{3} q^{43} + 6 \beta_{2} q^{47} + 24 q^{49} + 60 \beta_1 q^{53} + 36 \beta_1 q^{55} + 18 q^{59} - 15 \beta_{2} q^{61} - 6 \beta_{3} q^{65} - 15 \beta_{3} q^{67} + 25 q^{73} + 30 \beta_1 q^{77} + 31 \beta_1 q^{79} + 120 q^{83} - 36 \beta_{2} q^{85} + 30 \beta_{3} q^{89} - 5 \beta_{3} q^{91} - 30 \beta_{2} q^{95} - 85 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{11} - 44 q^{25} - 120 q^{35} + 96 q^{49} + 72 q^{59} + 100 q^{73} + 480 q^{83} - 340 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 6\zeta_{12}^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\zeta_{12}^{3} + 6\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 3 ) / 6 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) 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
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 0 0 6.00000i 0 5.00000i 0 0 0
1567.2 0 0 0 6.00000i 0 5.00000i 0 0 0
1567.3 0 0 0 6.00000i 0 5.00000i 0 0 0
1567.4 0 0 0 6.00000i 0 5.00000i 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
8.d odd 2 1 inner
12.b 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.f yes 4
3.b odd 2 1 1728.3.b.a 4
4.b odd 2 1 1728.3.b.a 4
8.b even 2 1 1728.3.b.a 4
8.d odd 2 1 inner 1728.3.b.f yes 4
12.b even 2 1 inner 1728.3.b.f yes 4
24.f even 2 1 1728.3.b.a 4
24.h odd 2 1 inner 1728.3.b.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.a 4 3.b odd 2 1
1728.3.b.a 4 4.b odd 2 1
1728.3.b.a 4 8.b even 2 1
1728.3.b.a 4 24.f even 2 1
1728.3.b.f yes 4 1.a even 1 1 trivial
1728.3.b.f yes 4 8.d odd 2 1 inner
1728.3.b.f yes 4 12.b even 2 1 inner
1728.3.b.f yes 4 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 \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{17}^{2} - 972 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$11$ \( (T - 6)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 972)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 675)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 972)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1296)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 675)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 972)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 3600)^{2} \) Copy content Toggle raw display
$59$ \( (T - 18)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6075)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 6075)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 25)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 961)^{2} \) Copy content Toggle raw display
$83$ \( (T - 120)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 24300)^{2} \) Copy content Toggle raw display
$97$ \( (T + 85)^{4} \) Copy content Toggle raw display
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