Properties

Label 1728.3.b.a
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: \(x^{4} - x^{2} + 1\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 \zeta_{12}^{3} q^{5} -5 \zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + 6 \zeta_{12}^{3} q^{5} -5 \zeta_{12}^{3} q^{7} -6 q^{11} + ( 3 - 6 \zeta_{12}^{2} ) q^{13} + ( 36 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{17} + ( -30 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{19} + ( -18 + 36 \zeta_{12}^{2} ) q^{23} -11 q^{25} -36 \zeta_{12}^{3} q^{29} + 26 \zeta_{12}^{3} q^{31} + 30 q^{35} + ( -15 + 30 \zeta_{12}^{2} ) q^{37} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{43} + ( -18 + 36 \zeta_{12}^{2} ) q^{47} + 24 q^{49} + 60 \zeta_{12}^{3} q^{53} -36 \zeta_{12}^{3} q^{55} -18 q^{59} + ( -45 + 90 \zeta_{12}^{2} ) q^{61} + ( 36 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{65} + ( -90 \zeta_{12} + 45 \zeta_{12}^{3} ) q^{67} + 25 q^{73} + 30 \zeta_{12}^{3} q^{77} -31 \zeta_{12}^{3} q^{79} -120 q^{83} + ( -108 + 216 \zeta_{12}^{2} ) q^{85} + ( -180 \zeta_{12} + 90 \zeta_{12}^{3} ) q^{89} + ( -30 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{91} + ( 90 - 180 \zeta_{12}^{2} ) q^{95} -85 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 24q^{11} - 44q^{25} + 120q^{35} + 96q^{49} - 72q^{59} + 100q^{73} - 480q^{83} - 340q^{97} + O(q^{100}) \)

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

Hecke kernels

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

\( T_{5}^{2} + 36 \)
\( T_{7}^{2} + 25 \)
\( T_{11} + 6 \)
\( T_{17}^{2} - 972 \)

Hecke characteristic polynomials

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