Properties

Label 1728.2.r.d
Level $1728$
Weight $2$
Character orbit 1728.r
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
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: 8.0.12960000.1
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + (\beta_{7} - \beta_{5}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + (\beta_{7} - \beta_{5}) q^{7} - 3 \beta_1 q^{11} - \beta_{4} q^{13} - 2 \beta_{3} q^{19} + \beta_{7} q^{23} + 10 \beta_{2} q^{25} - \beta_{6} q^{29} - \beta_{7} q^{31} + 15 \beta_{3} q^{35} + (2 \beta_{6} + 2 \beta_{4}) q^{37} + ( - 9 \beta_{2} + 9) q^{41} - 7 \beta_1 q^{43} + ( - \beta_{7} + \beta_{5}) q^{47} + (8 \beta_{2} - 8) q^{49} + (2 \beta_{6} + 2 \beta_{4}) q^{53} + 3 \beta_{5} q^{55} + ( - 3 \beta_{3} - 3 \beta_1) q^{59} - 3 \beta_{6} q^{61} + 15 \beta_{2} q^{65} + ( - 11 \beta_{3} - 11 \beta_1) q^{67} - 2 \beta_{5} q^{71} + 8 q^{73} + 3 \beta_{4} q^{77} + ( - \beta_{7} + \beta_{5}) q^{79} + 3 \beta_1 q^{83} + 12 q^{89} + 15 \beta_{3} q^{91} - 2 \beta_{7} q^{95} - \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{25} + 36 q^{41} - 32 q^{49} + 60 q^{65} + 64 q^{73} + 96 q^{89} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 13\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} + 24\nu^{4} - 56\nu^{2} + 39 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -11\nu^{6} + 24\nu^{4} - 56\nu^{2} - 15 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{7} - 32\nu^{5} + 88\nu^{3} + 25\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + \beta_{4} - 9\beta_{2} + 9 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - \beta_{5} + 6\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{6} + 2\beta_{4} - 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 10\beta_{5} + 33\beta_{3} + 33\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{6} + 4\beta_{4} - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 13\beta_{5} + 87\beta_1 ) / 6 \) Copy content Toggle raw display

Character values

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

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

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
0.535233 + 0.309017i
−0.535233 0.309017i
1.40126 + 0.809017i
−1.40126 0.809017i
0.535233 0.309017i
−0.535233 + 0.309017i
1.40126 0.809017i
−1.40126 + 0.809017i
0 0 0 −3.35410 + 1.93649i 0 −1.93649 + 3.35410i 0 0 0
289.2 0 0 0 −3.35410 + 1.93649i 0 1.93649 3.35410i 0 0 0
289.3 0 0 0 3.35410 1.93649i 0 −1.93649 + 3.35410i 0 0 0
289.4 0 0 0 3.35410 1.93649i 0 1.93649 3.35410i 0 0 0
1441.1 0 0 0 −3.35410 1.93649i 0 −1.93649 3.35410i 0 0 0
1441.2 0 0 0 −3.35410 1.93649i 0 1.93649 + 3.35410i 0 0 0
1441.3 0 0 0 3.35410 + 1.93649i 0 −1.93649 3.35410i 0 0 0
1441.4 0 0 0 3.35410 + 1.93649i 0 1.93649 + 3.35410i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.d 8
3.b odd 2 1 576.2.r.d 8
4.b odd 2 1 inner 1728.2.r.d 8
8.b even 2 1 inner 1728.2.r.d 8
8.d odd 2 1 inner 1728.2.r.d 8
9.c even 3 1 inner 1728.2.r.d 8
9.c even 3 1 5184.2.d.g 4
9.d odd 6 1 576.2.r.d 8
9.d odd 6 1 5184.2.d.h 4
12.b even 2 1 576.2.r.d 8
24.f even 2 1 576.2.r.d 8
24.h odd 2 1 576.2.r.d 8
36.f odd 6 1 inner 1728.2.r.d 8
36.f odd 6 1 5184.2.d.g 4
36.h even 6 1 576.2.r.d 8
36.h even 6 1 5184.2.d.h 4
72.j odd 6 1 576.2.r.d 8
72.j odd 6 1 5184.2.d.h 4
72.l even 6 1 576.2.r.d 8
72.l even 6 1 5184.2.d.h 4
72.n even 6 1 inner 1728.2.r.d 8
72.n even 6 1 5184.2.d.g 4
72.p odd 6 1 inner 1728.2.r.d 8
72.p odd 6 1 5184.2.d.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.d 8 3.b odd 2 1
576.2.r.d 8 9.d odd 6 1
576.2.r.d 8 12.b even 2 1
576.2.r.d 8 24.f even 2 1
576.2.r.d 8 24.h odd 2 1
576.2.r.d 8 36.h even 6 1
576.2.r.d 8 72.j odd 6 1
576.2.r.d 8 72.l even 6 1
1728.2.r.d 8 1.a even 1 1 trivial
1728.2.r.d 8 4.b odd 2 1 inner
1728.2.r.d 8 8.b even 2 1 inner
1728.2.r.d 8 8.d odd 2 1 inner
1728.2.r.d 8 9.c even 3 1 inner
1728.2.r.d 8 36.f odd 6 1 inner
1728.2.r.d 8 72.n even 6 1 inner
1728.2.r.d 8 72.p odd 6 1 inner
5184.2.d.g 4 9.c even 3 1
5184.2.d.g 4 36.f odd 6 1
5184.2.d.g 4 72.n even 6 1
5184.2.d.g 4 72.p odd 6 1
5184.2.d.h 4 9.d odd 6 1
5184.2.d.h 4 36.h even 6 1
5184.2.d.h 4 72.j odd 6 1
5184.2.d.h 4 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 15T_{5}^{2} + 225 \) 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
$5$ \( (T^{4} - 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 81)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 49 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 135 T^{2} + 18225)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 121 T^{2} + 14641)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$73$ \( (T - 8)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 15 T^{2} + 225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T - 12)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
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