Properties

Label 1728.2.i.l
Level $1728$
Weight $2$
Character orbit 1728.i
Analytic conductor $13.798$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( 1 - \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{7} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{11} + ( -3 + 3 \zeta_{12}^{2} ) q^{13} -4 q^{17} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{19} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{23} + 4 \zeta_{12}^{2} q^{25} -\zeta_{12}^{2} q^{29} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{31} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{35} + 8 q^{37} + ( 5 - 5 \zeta_{12}^{2} ) q^{41} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{43} + ( -7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{47} + ( 4 - 4 \zeta_{12}^{2} ) q^{49} -8 q^{53} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{55} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{59} -7 \zeta_{12}^{2} q^{61} + 3 \zeta_{12}^{2} q^{65} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{67} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{71} -12 q^{73} + ( 3 - 3 \zeta_{12}^{2} ) q^{77} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{79} + ( -5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{83} + ( -4 + 4 \zeta_{12}^{2} ) q^{85} + 4 q^{89} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{91} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{95} + 3 \zeta_{12}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + O(q^{10}) \) \( 4q + 2q^{5} - 6q^{13} - 16q^{17} + 8q^{25} - 2q^{29} + 32q^{37} + 10q^{41} + 8q^{49} - 32q^{53} - 14q^{61} + 6q^{65} - 48q^{73} + 6q^{77} - 8q^{85} + 16q^{89} + 6q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 0.500000 0.866025i 0 −0.866025 1.50000i 0 0 0
577.2 0 0 0 0.500000 0.866025i 0 0.866025 + 1.50000i 0 0 0
1153.1 0 0 0 0.500000 + 0.866025i 0 −0.866025 + 1.50000i 0 0 0
1153.2 0 0 0 0.500000 + 0.866025i 0 0.866025 1.50000i 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
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.l 4
3.b odd 2 1 576.2.i.k 4
4.b odd 2 1 inner 1728.2.i.l 4
8.b even 2 1 864.2.i.d 4
8.d odd 2 1 864.2.i.d 4
9.c even 3 1 inner 1728.2.i.l 4
9.c even 3 1 5184.2.a.bl 2
9.d odd 6 1 576.2.i.k 4
9.d odd 6 1 5184.2.a.bx 2
12.b even 2 1 576.2.i.k 4
24.f even 2 1 288.2.i.d 4
24.h odd 2 1 288.2.i.d 4
36.f odd 6 1 inner 1728.2.i.l 4
36.f odd 6 1 5184.2.a.bl 2
36.h even 6 1 576.2.i.k 4
36.h even 6 1 5184.2.a.bx 2
72.j odd 6 1 288.2.i.d 4
72.j odd 6 1 2592.2.a.l 2
72.l even 6 1 288.2.i.d 4
72.l even 6 1 2592.2.a.l 2
72.n even 6 1 864.2.i.d 4
72.n even 6 1 2592.2.a.p 2
72.p odd 6 1 864.2.i.d 4
72.p odd 6 1 2592.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.d 4 24.f even 2 1
288.2.i.d 4 24.h odd 2 1
288.2.i.d 4 72.j odd 6 1
288.2.i.d 4 72.l even 6 1
576.2.i.k 4 3.b odd 2 1
576.2.i.k 4 9.d odd 6 1
576.2.i.k 4 12.b even 2 1
576.2.i.k 4 36.h even 6 1
864.2.i.d 4 8.b even 2 1
864.2.i.d 4 8.d odd 2 1
864.2.i.d 4 72.n even 6 1
864.2.i.d 4 72.p odd 6 1
1728.2.i.l 4 1.a even 1 1 trivial
1728.2.i.l 4 4.b odd 2 1 inner
1728.2.i.l 4 9.c even 3 1 inner
1728.2.i.l 4 36.f odd 6 1 inner
2592.2.a.l 2 72.j odd 6 1
2592.2.a.l 2 72.l even 6 1
2592.2.a.p 2 72.n even 6 1
2592.2.a.p 2 72.p odd 6 1
5184.2.a.bl 2 9.c even 3 1
5184.2.a.bl 2 36.f odd 6 1
5184.2.a.bx 2 9.d odd 6 1
5184.2.a.bx 2 36.h even 6 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{7}^{4} + 3 T_{7}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 9 + 3 T^{2} + T^{4} \)
$11$ \( 9 + 3 T^{2} + T^{4} \)
$13$ \( ( 9 + 3 T + T^{2} )^{2} \)
$17$ \( ( 4 + T )^{4} \)
$19$ \( ( -48 + T^{2} )^{2} \)
$23$ \( 5625 + 75 T^{2} + T^{4} \)
$29$ \( ( 1 + T + T^{2} )^{2} \)
$31$ \( 729 + 27 T^{2} + T^{4} \)
$37$ \( ( -8 + T )^{4} \)
$41$ \( ( 25 - 5 T + T^{2} )^{2} \)
$43$ \( 5625 + 75 T^{2} + T^{4} \)
$47$ \( 21609 + 147 T^{2} + T^{4} \)
$53$ \( ( 8 + T )^{4} \)
$59$ \( 9 + 3 T^{2} + T^{4} \)
$61$ \( ( 49 + 7 T + T^{2} )^{2} \)
$67$ \( 5625 + 75 T^{2} + T^{4} \)
$71$ \( ( -12 + T^{2} )^{2} \)
$73$ \( ( 12 + T )^{4} \)
$79$ \( 729 + 27 T^{2} + T^{4} \)
$83$ \( 5625 + 75 T^{2} + T^{4} \)
$89$ \( ( -4 + T )^{4} \)
$97$ \( ( 9 - 3 T + T^{2} )^{2} \)
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