Properties

Label 1728.1.j.a
Level $1728$
Weight $1$
Character orbit 1728.j
Analytic conductor $0.862$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
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.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 432)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.55296.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{8} q^{5} + \zeta_{8}^{2} q^{7} +O(q^{10})\) \( q -\zeta_{8} q^{5} + \zeta_{8}^{2} q^{7} -\zeta_{8} q^{11} + ( -1 - \zeta_{8}^{2} ) q^{13} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} - q^{31} -\zeta_{8}^{3} q^{35} + ( 1 - \zeta_{8}^{2} ) q^{43} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{47} + \zeta_{8} q^{53} + \zeta_{8}^{2} q^{55} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{65} + ( -1 - \zeta_{8}^{2} ) q^{67} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{71} + \zeta_{8}^{2} q^{73} -\zeta_{8}^{3} q^{77} + \zeta_{8}^{3} q^{83} + ( -1 + \zeta_{8}^{2} ) q^{85} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{89} + ( 1 - \zeta_{8}^{2} ) q^{91} + q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{13} - 4q^{31} + 4q^{43} - 4q^{67} - 4q^{85} + 4q^{91} + 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 0 0 −0.707107 0.707107i 0 1.00000i 0 0 0
593.2 0 0 0 0.707107 + 0.707107i 0 1.00000i 0 0 0
1457.1 0 0 0 −0.707107 + 0.707107i 0 1.00000i 0 0 0
1457.2 0 0 0 0.707107 0.707107i 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
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.1.j.a 4
3.b odd 2 1 inner 1728.1.j.a 4
4.b odd 2 1 432.1.j.a 4
8.b even 2 1 3456.1.j.a 4
8.d odd 2 1 3456.1.j.b 4
12.b even 2 1 432.1.j.a 4
16.e even 4 1 inner 1728.1.j.a 4
16.e even 4 1 3456.1.j.a 4
16.f odd 4 1 432.1.j.a 4
16.f odd 4 1 3456.1.j.b 4
24.f even 2 1 3456.1.j.b 4
24.h odd 2 1 3456.1.j.a 4
36.f odd 6 2 1296.1.x.a 8
36.h even 6 2 1296.1.x.a 8
48.i odd 4 1 inner 1728.1.j.a 4
48.i odd 4 1 3456.1.j.a 4
48.k even 4 1 432.1.j.a 4
48.k even 4 1 3456.1.j.b 4
144.u even 12 2 1296.1.x.a 8
144.v odd 12 2 1296.1.x.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.1.j.a 4 4.b odd 2 1
432.1.j.a 4 12.b even 2 1
432.1.j.a 4 16.f odd 4 1
432.1.j.a 4 48.k even 4 1
1296.1.x.a 8 36.f odd 6 2
1296.1.x.a 8 36.h even 6 2
1296.1.x.a 8 144.u even 12 2
1296.1.x.a 8 144.v odd 12 2
1728.1.j.a 4 1.a even 1 1 trivial
1728.1.j.a 4 3.b odd 2 1 inner
1728.1.j.a 4 16.e even 4 1 inner
1728.1.j.a 4 48.i odd 4 1 inner
3456.1.j.a 4 8.b even 2 1
3456.1.j.a 4 16.e even 4 1
3456.1.j.a 4 24.h odd 2 1
3456.1.j.a 4 48.i odd 4 1
3456.1.j.b 4 8.d odd 2 1
3456.1.j.b 4 16.f odd 4 1
3456.1.j.b 4 24.f even 2 1
3456.1.j.b 4 48.k even 4 1

Hecke kernels

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

Hecke characteristic polynomials

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