Properties

Label 1728.1.o.a
Level $1728$
Weight $1$
Character orbit 1728.o
Analytic conductor $0.862$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1728.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.5184.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} +O(q^{10})\) \( q -\zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} -\zeta_{12}^{5} q^{11} -\zeta_{12}^{4} q^{13} -\zeta_{12} q^{23} -\zeta_{12}^{2} q^{29} + \zeta_{12} q^{31} + \zeta_{12}^{3} q^{35} -\zeta_{12}^{4} q^{41} -\zeta_{12}^{5} q^{43} -\zeta_{12}^{5} q^{47} -\zeta_{12}^{3} q^{55} -\zeta_{12} q^{59} + \zeta_{12}^{2} q^{61} -\zeta_{12}^{2} q^{65} -\zeta_{12} q^{67} + 2 \zeta_{12}^{3} q^{71} + \zeta_{12}^{4} q^{77} -\zeta_{12}^{5} q^{79} + \zeta_{12}^{5} q^{83} + \zeta_{12}^{3} q^{91} -\zeta_{12}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + O(q^{10}) \) \( 4 q + 2 q^{5} + 2 q^{13} - 2 q^{29} + 2 q^{41} + 2 q^{61} - 2 q^{65} - 2 q^{77} - 2 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
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 0.500000i 0 0 0
127.2 0 0 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0 0 0
1279.1 0 0 0 0.500000 0.866025i 0 −0.866025 + 0.500000i 0 0 0
1279.2 0 0 0 0.500000 0.866025i 0 0.866025 0.500000i 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 1728.1.o.a 4
3.b odd 2 1 576.1.o.a 4
4.b odd 2 1 inner 1728.1.o.a 4
8.b even 2 1 864.1.o.a 4
8.d odd 2 1 864.1.o.a 4
9.c even 3 1 inner 1728.1.o.a 4
9.d odd 6 1 576.1.o.a 4
12.b even 2 1 576.1.o.a 4
24.f even 2 1 288.1.o.a 4
24.h odd 2 1 288.1.o.a 4
36.f odd 6 1 inner 1728.1.o.a 4
36.h even 6 1 576.1.o.a 4
48.i odd 4 1 2304.1.t.a 4
48.i odd 4 1 2304.1.t.b 4
48.k even 4 1 2304.1.t.a 4
48.k even 4 1 2304.1.t.b 4
72.j odd 6 1 288.1.o.a 4
72.j odd 6 1 2592.1.g.a 2
72.l even 6 1 288.1.o.a 4
72.l even 6 1 2592.1.g.a 2
72.n even 6 1 864.1.o.a 4
72.n even 6 1 2592.1.g.b 2
72.p odd 6 1 864.1.o.a 4
72.p odd 6 1 2592.1.g.b 2
144.u even 12 1 2304.1.t.a 4
144.u even 12 1 2304.1.t.b 4
144.w odd 12 1 2304.1.t.a 4
144.w odd 12 1 2304.1.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.1.o.a 4 24.f even 2 1
288.1.o.a 4 24.h odd 2 1
288.1.o.a 4 72.j odd 6 1
288.1.o.a 4 72.l even 6 1
576.1.o.a 4 3.b odd 2 1
576.1.o.a 4 9.d odd 6 1
576.1.o.a 4 12.b even 2 1
576.1.o.a 4 36.h even 6 1
864.1.o.a 4 8.b even 2 1
864.1.o.a 4 8.d odd 2 1
864.1.o.a 4 72.n even 6 1
864.1.o.a 4 72.p odd 6 1
1728.1.o.a 4 1.a even 1 1 trivial
1728.1.o.a 4 4.b odd 2 1 inner
1728.1.o.a 4 9.c even 3 1 inner
1728.1.o.a 4 36.f odd 6 1 inner
2304.1.t.a 4 48.i odd 4 1
2304.1.t.a 4 48.k even 4 1
2304.1.t.a 4 144.u even 12 1
2304.1.t.a 4 144.w odd 12 1
2304.1.t.b 4 48.i odd 4 1
2304.1.t.b 4 48.k even 4 1
2304.1.t.b 4 144.u even 12 1
2304.1.t.b 4 144.w odd 12 1
2592.1.g.a 2 72.j odd 6 1
2592.1.g.a 2 72.l even 6 1
2592.1.g.b 2 72.n even 6 1
2592.1.g.b 2 72.p odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1728, [\chi])\).

Hecke characteristic polynomials

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