Properties

Label 1728.3.b.h
Level $1728$
Weight $3$
Character orbit 1728.b
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM discriminant -24
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
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: \( 2^{14}\cdot 3^{4} \)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{5} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 5 \zeta_{24}^{6} ) q^{7} +O(q^{10})\) \( q + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{5} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 5 \zeta_{24}^{6} ) q^{7} + ( -4 \zeta_{24} + 10 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 5 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{11} + ( -2 + 12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{25} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{29} + ( -30 \zeta_{24} + 30 \zeta_{24}^{3} + 30 \zeta_{24}^{5} - 19 \zeta_{24}^{6} ) q^{31} + ( -16 \zeta_{24} + 58 \zeta_{24}^{2} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} - 29 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{35} + ( -48 + 60 \zeta_{24} + 60 \zeta_{24}^{3} - 60 \zeta_{24}^{5} ) q^{49} + ( 47 + 10 \zeta_{24} - 10 \zeta_{24}^{3} - 94 \zeta_{24}^{4} + 10 \zeta_{24}^{5} + 20 \zeta_{24}^{7} ) q^{53} + ( 42 \zeta_{24} - 42 \zeta_{24}^{3} - 42 \zeta_{24}^{5} + 63 \zeta_{24}^{6} ) q^{55} + ( 48 \zeta_{24} + 48 \zeta_{24}^{3} + 48 \zeta_{24}^{5} - 96 \zeta_{24}^{7} ) q^{59} + ( 25 - 84 \zeta_{24} - 84 \zeta_{24}^{3} + 84 \zeta_{24}^{5} ) q^{73} + ( 73 + 50 \zeta_{24} - 50 \zeta_{24}^{3} - 146 \zeta_{24}^{4} + 50 \zeta_{24}^{5} + 100 \zeta_{24}^{7} ) q^{77} -58 \zeta_{24}^{6} q^{79} + ( -20 \zeta_{24} + 134 \zeta_{24}^{2} - 20 \zeta_{24}^{3} - 20 \zeta_{24}^{5} - 67 \zeta_{24}^{6} + 40 \zeta_{24}^{7} ) q^{83} + ( 95 - 24 \zeta_{24} - 24 \zeta_{24}^{3} + 24 \zeta_{24}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 16 q^{25} - 384 q^{49} + 200 q^{73} + 760 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
−0.965926 0.258819i
0.258819 0.965926i
0 0 0 6.63103i 0 13.4853i 0 0 0
1567.2 0 0 0 6.63103i 0 13.4853i 0 0 0
1567.3 0 0 0 3.16693i 0 3.48528i 0 0 0
1567.4 0 0 0 3.16693i 0 3.48528i 0 0 0
1567.5 0 0 0 3.16693i 0 3.48528i 0 0 0
1567.6 0 0 0 3.16693i 0 3.48528i 0 0 0
1567.7 0 0 0 6.63103i 0 13.4853i 0 0 0
1567.8 0 0 0 6.63103i 0 13.4853i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.b.h 8
3.b odd 2 1 inner 1728.3.b.h 8
4.b odd 2 1 inner 1728.3.b.h 8
8.b even 2 1 inner 1728.3.b.h 8
8.d odd 2 1 inner 1728.3.b.h 8
12.b even 2 1 inner 1728.3.b.h 8
24.f even 2 1 inner 1728.3.b.h 8
24.h odd 2 1 CM 1728.3.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.h 8 1.a even 1 1 trivial
1728.3.b.h 8 3.b odd 2 1 inner
1728.3.b.h 8 4.b odd 2 1 inner
1728.3.b.h 8 8.b even 2 1 inner
1728.3.b.h 8 8.d odd 2 1 inner
1728.3.b.h 8 12.b even 2 1 inner
1728.3.b.h 8 24.f even 2 1 inner
1728.3.b.h 8 24.h odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} + 54 T_{5}^{2} + 441 \)
\( T_{7}^{4} + 194 T_{7}^{2} + 2209 \)
\( T_{11}^{4} - 342 T_{11}^{2} + 441 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 441 + 54 T^{2} + T^{4} )^{2} \)
$7$ \( ( 2209 + 194 T^{2} + T^{4} )^{2} \)
$11$ \( ( 441 - 342 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( ( 864 + T^{2} )^{4} \)
$31$ \( ( 2070721 + 4322 T^{2} + T^{4} )^{2} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( ( 36324729 + 14454 T^{2} + T^{4} )^{2} \)
$59$ \( ( -13824 + T^{2} )^{4} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( -13487 - 50 T + T^{2} )^{4} \)
$79$ \( ( 3364 + T^{2} )^{4} \)
$83$ \( ( 122478489 - 31734 T^{2} + T^{4} )^{2} \)
$89$ \( T^{8} \)
$97$ \( ( 7873 - 190 T + T^{2} )^{4} \)
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