Properties

Label 1728.3.b.b
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: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{5} + \beta_1 q^{7} - 10 \beta_{3} q^{11} - \beta_{2} q^{13} - 6 q^{17} + \beta_{3} q^{19} + 30 \beta_1 q^{23} + 13 q^{25} + 12 \beta_{2} q^{29} + 14 \beta_1 q^{31} + 2 \beta_{3} q^{35} - 11 \beta_{2} q^{37} + 48 q^{41} + 12 \beta_{3} q^{43} - 66 \beta_1 q^{47} + 48 q^{49} + 28 \beta_{2} q^{53} + 60 \beta_1 q^{55} + 18 \beta_{3} q^{59} - 25 \beta_{2} q^{61} - 6 q^{65} + 35 \beta_{3} q^{67} + 48 \beta_1 q^{71} + 49 q^{73} - 10 \beta_{2} q^{77} + 83 \beta_1 q^{79} - 8 \beta_{3} q^{83} + 12 \beta_{2} q^{85} - 66 q^{89} + \beta_{3} q^{91} - 6 \beta_1 q^{95} + 107 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{17} + 52 q^{25} + 192 q^{41} + 192 q^{49} - 24 q^{65} + 196 q^{73} - 264 q^{89} + 428 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) 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 3.46410i 0 1.00000i 0 0 0
1567.2 0 0 0 3.46410i 0 1.00000i 0 0 0
1567.3 0 0 0 3.46410i 0 1.00000i 0 0 0
1567.4 0 0 0 3.46410i 0 1.00000i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.b 4
3.b odd 2 1 1728.3.b.e yes 4
4.b odd 2 1 inner 1728.3.b.b 4
8.b even 2 1 inner 1728.3.b.b 4
8.d odd 2 1 inner 1728.3.b.b 4
12.b even 2 1 1728.3.b.e yes 4
24.f even 2 1 1728.3.b.e yes 4
24.h odd 2 1 1728.3.b.e yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.b 4 1.a even 1 1 trivial
1728.3.b.b 4 4.b odd 2 1 inner
1728.3.b.b 4 8.b even 2 1 inner
1728.3.b.b 4 8.d odd 2 1 inner
1728.3.b.e yes 4 3.b odd 2 1
1728.3.b.e yes 4 12.b even 2 1
1728.3.b.e yes 4 24.f even 2 1
1728.3.b.e yes 4 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 300 \) Copy content Toggle raw display
\( T_{17} + 6 \) 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
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 900)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 432)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 363)^{2} \) Copy content Toggle raw display
$41$ \( (T - 48)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4356)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2352)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 972)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 1875)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3675)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2304)^{2} \) Copy content Toggle raw display
$73$ \( (T - 49)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 6889)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T + 66)^{4} \) Copy content Toggle raw display
$97$ \( (T - 107)^{4} \) Copy content Toggle raw display
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