Properties

Label 1728.2.d.h
Level $1728$
Weight $2$
Character orbit 1728.d
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + ( 1 - 2 \zeta_{12}^{2} ) q^{13} + \zeta_{12}^{3} q^{19} + 5 q^{25} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{31} + ( 7 - 14 \zeta_{12}^{2} ) q^{37} + 8 \zeta_{12}^{3} q^{43} -4 q^{49} + ( 5 - 10 \zeta_{12}^{2} ) q^{61} + 11 \zeta_{12}^{3} q^{67} + 17 q^{73} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{79} -3 \zeta_{12}^{3} q^{91} -5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 20q^{25} - 16q^{49} + 68q^{73} - 20q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 0 0 −1.73205 0 0 0
865.2 0 0 0 0 0 −1.73205 0 0 0
865.3 0 0 0 0 0 1.73205 0 0 0
865.4 0 0 0 0 0 1.73205 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
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 1728.2.d.h 4
3.b odd 2 1 CM 1728.2.d.h 4
4.b odd 2 1 inner 1728.2.d.h 4
8.b even 2 1 inner 1728.2.d.h 4
8.d odd 2 1 inner 1728.2.d.h 4
12.b even 2 1 inner 1728.2.d.h 4
16.e even 4 1 6912.2.a.bl 2
16.e even 4 1 6912.2.a.bm 2
16.f odd 4 1 6912.2.a.bl 2
16.f odd 4 1 6912.2.a.bm 2
24.f even 2 1 inner 1728.2.d.h 4
24.h odd 2 1 inner 1728.2.d.h 4
48.i odd 4 1 6912.2.a.bl 2
48.i odd 4 1 6912.2.a.bm 2
48.k even 4 1 6912.2.a.bl 2
48.k even 4 1 6912.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.2.d.h 4 1.a even 1 1 trivial
1728.2.d.h 4 3.b odd 2 1 CM
1728.2.d.h 4 4.b odd 2 1 inner
1728.2.d.h 4 8.b even 2 1 inner
1728.2.d.h 4 8.d odd 2 1 inner
1728.2.d.h 4 12.b even 2 1 inner
1728.2.d.h 4 24.f even 2 1 inner
1728.2.d.h 4 24.h odd 2 1 inner
6912.2.a.bl 2 16.e even 4 1
6912.2.a.bl 2 16.f odd 4 1
6912.2.a.bl 2 48.i odd 4 1
6912.2.a.bl 2 48.k even 4 1
6912.2.a.bm 2 16.e even 4 1
6912.2.a.bm 2 16.f odd 4 1
6912.2.a.bm 2 48.i odd 4 1
6912.2.a.bm 2 48.k even 4 1

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{2} - 3 \)
\( T_{17} \)
\( T_{41} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -3 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -108 + T^{2} )^{2} \)
$37$ \( ( 147 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 64 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 75 + T^{2} )^{2} \)
$67$ \( ( 121 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( -17 + T )^{4} \)
$79$ \( ( -147 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 5 + T )^{4} \)
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