Properties

Label 1728.2.c.g
Level $1728$
Weight $2$
Character orbit 1728.c
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 864)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} +O(q^{10})\) \( q + ( 1 + \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{11} + ( 2 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{13} + ( 2 - 4 \zeta_{24}^{4} ) q^{17} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{23} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{25} + ( 2 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{29} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} - 6 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{35} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( 4 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{41} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{43} + ( -4 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{47} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{49} + ( 5 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 10 \zeta_{24}^{4} - 3 \zeta_{24}^{5} ) q^{53} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 7 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{55} + ( -4 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{59} -4 q^{61} + ( 6 + 8 \zeta_{24} - 8 \zeta_{24}^{3} - 12 \zeta_{24}^{4} - 8 \zeta_{24}^{5} ) q^{65} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 10 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{67} + ( -2 \zeta_{24} - 12 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{71} + ( -3 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 5 + 5 \zeta_{24} - 5 \zeta_{24}^{3} - 10 \zeta_{24}^{4} - 5 \zeta_{24}^{5} ) q^{77} -2 \zeta_{24}^{6} q^{79} + ( 10 \zeta_{24} - 2 \zeta_{24}^{2} + 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{83} + ( -6 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{85} + ( 2 - 6 \zeta_{24} + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{89} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 14 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{91} + ( 6 \zeta_{24} + 8 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{95} -5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 16 q^{13} - 32 q^{61} - 24 q^{73} - 48 q^{85} - 40 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1727.1
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
0 0 0 3.14626i 0 3.44949i 0 0 0
1727.2 0 0 0 3.14626i 0 3.44949i 0 0 0
1727.3 0 0 0 0.317837i 0 1.44949i 0 0 0
1727.4 0 0 0 0.317837i 0 1.44949i 0 0 0
1727.5 0 0 0 0.317837i 0 1.44949i 0 0 0
1727.6 0 0 0 0.317837i 0 1.44949i 0 0 0
1727.7 0 0 0 3.14626i 0 3.44949i 0 0 0
1727.8 0 0 0 3.14626i 0 3.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1727.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.c.g 8
3.b odd 2 1 inner 1728.2.c.g 8
4.b odd 2 1 inner 1728.2.c.g 8
8.b even 2 1 864.2.c.a 8
8.d odd 2 1 864.2.c.a 8
12.b even 2 1 inner 1728.2.c.g 8
24.f even 2 1 864.2.c.a 8
24.h odd 2 1 864.2.c.a 8
72.j odd 6 1 2592.2.s.a 8
72.j odd 6 1 2592.2.s.h 8
72.l even 6 1 2592.2.s.a 8
72.l even 6 1 2592.2.s.h 8
72.n even 6 1 2592.2.s.a 8
72.n even 6 1 2592.2.s.h 8
72.p odd 6 1 2592.2.s.a 8
72.p odd 6 1 2592.2.s.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.a 8 8.b even 2 1
864.2.c.a 8 8.d odd 2 1
864.2.c.a 8 24.f even 2 1
864.2.c.a 8 24.h odd 2 1
1728.2.c.g 8 1.a even 1 1 trivial
1728.2.c.g 8 3.b odd 2 1 inner
1728.2.c.g 8 4.b odd 2 1 inner
1728.2.c.g 8 12.b even 2 1 inner
2592.2.s.a 8 72.j odd 6 1
2592.2.s.a 8 72.l even 6 1
2592.2.s.a 8 72.n even 6 1
2592.2.s.a 8 72.p odd 6 1
2592.2.s.h 8 72.j odd 6 1
2592.2.s.h 8 72.l even 6 1
2592.2.s.h 8 72.n even 6 1
2592.2.s.h 8 72.p odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 10 T_{5}^{2} + 1 \)
\( T_{7}^{4} + 14 T_{7}^{2} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + 10 T^{2} + T^{4} )^{2} \)
$7$ \( ( 25 + 14 T^{2} + T^{4} )^{2} \)
$11$ \( ( 25 - 22 T^{2} + T^{4} )^{2} \)
$13$ \( ( -20 - 4 T + T^{2} )^{4} \)
$17$ \( ( 12 + T^{2} )^{4} \)
$19$ \( ( 24 + T^{2} )^{4} \)
$23$ \( ( -8 + T^{2} )^{4} \)
$29$ \( ( 400 + 88 T^{2} + T^{4} )^{2} \)
$31$ \( ( 361 + 62 T^{2} + T^{4} )^{2} \)
$37$ \( ( -24 + T^{2} )^{4} \)
$41$ \( ( 1600 + 112 T^{2} + T^{4} )^{2} \)
$43$ \( ( 400 + 56 T^{2} + T^{4} )^{2} \)
$47$ \( ( 400 - 88 T^{2} + T^{4} )^{2} \)
$53$ \( ( 3249 + 186 T^{2} + T^{4} )^{2} \)
$59$ \( ( 400 - 88 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4 + T )^{8} \)
$67$ \( ( 5776 + 248 T^{2} + T^{4} )^{2} \)
$71$ \( ( 10000 - 232 T^{2} + T^{4} )^{2} \)
$73$ \( ( -15 + 6 T + T^{2} )^{4} \)
$79$ \( ( 4 + T^{2} )^{4} \)
$83$ \( ( 38809 - 406 T^{2} + T^{4} )^{2} \)
$89$ \( ( 3600 + 168 T^{2} + T^{4} )^{2} \)
$97$ \( ( 5 + T )^{8} \)
show more
show less