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Properties

Label 1728.2.r.c
Level $1728$
Weight $2$
Character orbit 1728.r
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{11} + (\beta_{7} + \beta_{5} - 3) q^{17} + ( - 3 \beta_{6} - 2 \beta_{4}) q^{19} + (5 \beta_{2} - 5) q^{25} + (2 \beta_{7} + 3 \beta_{2}) q^{41} + (3 \beta_{3} - 4 \beta_1) q^{43} + 7 \beta_{2} q^{49} + ( - 5 \beta_{6} - \beta_{4} - 5 \beta_{3} + 4 \beta_1) q^{59} + ( - 3 \beta_{6} - 5 \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{67} + (3 \beta_{7} + 3 \beta_{5} - 1) q^{73} - 9 \beta_1 q^{83} - 18 q^{89} + (3 \beta_{5} - 5 \beta_{2} + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{17} - 20 q^{25} + 12 q^{41} + 28 q^{49} - 8 q^{73} - 144 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} - 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{7} + 4\zeta_{24}^{5} + 4\zeta_{24}^{3} - 2\zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} + 4\beta_{3} - 2\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} - 4\beta_{6} + \beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_1 ) / 12 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.965926 + 0.258819i
−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 0 0 0 0 0 0 0 0
289.2 0 0 0 0 0 0 0 0 0
289.3 0 0 0 0 0 0 0 0 0
289.4 0 0 0 0 0 0 0 0 0
1441.1 0 0 0 0 0 0 0 0 0
1441.2 0 0 0 0 0 0 0 0 0
1441.3 0 0 0 0 0 0 0 0 0
1441.4 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1441.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p 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.r.c 8
3.b odd 2 1 576.2.r.c ✓ 8
4.b odd 2 1 inner 1728.2.r.c 8
8.b even 2 1 inner 1728.2.r.c 8
8.d odd 2 1 CM 1728.2.r.c 8
9.c even 3 1 inner 1728.2.r.c 8
9.c even 3 1 5184.2.d.e 4
9.d odd 6 1 576.2.r.c ✓ 8
9.d odd 6 1 5184.2.d.l 4
12.b even 2 1 576.2.r.c ✓ 8
24.f even 2 1 576.2.r.c ✓ 8
24.h odd 2 1 576.2.r.c ✓ 8
36.f odd 6 1 inner 1728.2.r.c 8
36.f odd 6 1 5184.2.d.e 4
36.h even 6 1 576.2.r.c ✓ 8
36.h even 6 1 5184.2.d.l 4
72.j odd 6 1 576.2.r.c ✓ 8
72.j odd 6 1 5184.2.d.l 4
72.l even 6 1 576.2.r.c ✓ 8
72.l even 6 1 5184.2.d.l 4
72.n even 6 1 inner 1728.2.r.c 8
72.n even 6 1 5184.2.d.e 4
72.p odd 6 1 inner 1728.2.r.c 8
72.p odd 6 1 5184.2.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.c ✓ 8 3.b odd 2 1
576.2.r.c ✓ 8 9.d odd 6 1
576.2.r.c ✓ 8 12.b even 2 1
576.2.r.c ✓ 8 24.f even 2 1
576.2.r.c ✓ 8 24.h odd 2 1
576.2.r.c ✓ 8 36.h even 6 1
576.2.r.c ✓ 8 72.j odd 6 1
576.2.r.c ✓ 8 72.l even 6 1
1728.2.r.c 8 1.a even 1 1 trivial
1728.2.r.c 8 4.b odd 2 1 inner
1728.2.r.c 8 8.b even 2 1 inner
1728.2.r.c 8 8.d odd 2 1 CM
1728.2.r.c 8 9.c even 3 1 inner
1728.2.r.c 8 36.f odd 6 1 inner
1728.2.r.c 8 72.n even 6 1 inner
1728.2.r.c 8 72.p odd 6 1 inner
5184.2.d.e 4 9.c even 3 1
5184.2.d.e 4 36.f odd 6 1
5184.2.d.e 4 72.n even 6 1
5184.2.d.e 4 72.p odd 6 1
5184.2.d.l 4 9.d odd 6 1
5184.2.d.l 4 36.h even 6 1
5184.2.d.l 4 72.j odd 6 1
5184.2.d.l 4 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 30 T^{6} + 891 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 15)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 110 T^{2} + 2809)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + 123 T^{2} + 522 T + 7569)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 158 T^{6} + 24123 T^{4} + \cdots + 707281 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 318 T^{6} + \cdots + 395254161 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - 206 T^{6} + 42411 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 215)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 324 T^{2} + 104976)^{2} \) Copy content Toggle raw display
$89$ \( (T + 18)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481)^{2} \) Copy content Toggle raw display
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