Properties

Label 1728.3.e.u
Level $1728$
Weight $3$
Character orbit 1728.e
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + \beta_{4} q^{7} + \beta_{6} q^{11} + ( - \beta_{5} - 6) q^{13} + (2 \beta_{2} + \beta_1) q^{17} + (\beta_{7} + 2 \beta_{4}) q^{19} + ( - \beta_{6} + \beta_{3}) q^{23} + ( - \beta_{5} - 4) q^{25} + ( - 2 \beta_{2} + 5 \beta_1) q^{29} + ( - \beta_{7} + 5 \beta_{4}) q^{31} + ( - 2 \beta_{6} + 3 \beta_{3}) q^{35} + (3 \beta_{5} + 20) q^{37} + 7 \beta_1 q^{41} + (2 \beta_{7} - 6 \beta_{4}) q^{43} + (2 \beta_{6} + 4 \beta_{3}) q^{47} + (3 \beta_{5} - 4) q^{49} + (9 \beta_{2} - 5 \beta_1) q^{53} + ( - 2 \beta_{7} + 11 \beta_{4}) q^{55} + ( - 2 \beta_{6} + 4 \beta_{3}) q^{59} + 4 q^{61} + ( - 10 \beta_{2} - 25 \beta_1) q^{65} + (4 \beta_{7} + 2 \beta_{4}) q^{67} + (\beta_{6} + 5 \beta_{3}) q^{71} + ( - \beta_{5} + 21) q^{73} + ( - 15 \beta_{2} - 24 \beta_1) q^{77} + ( - 5 \beta_{7} - 8 \beta_{4}) q^{79} + (\beta_{6} + 10 \beta_{3}) q^{83} + ( - 3 \beta_{5} - 62) q^{85} + (14 \beta_{2} + 12 \beta_1) q^{89} + (\beta_{7} - 20 \beta_{4}) q^{91} + (\beta_{6} + 11 \beta_{3}) q^{95} + 27 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{13} - 32 q^{25} + 160 q^{37} - 32 q^{49} + 32 q^{61} + 168 q^{73} - 496 q^{85} + 216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{5} + 6\zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 9\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\zeta_{24}^{6} + 3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 6\zeta_{24}^{2} - 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 12\zeta_{24}^{7} - 6\zeta_{24}^{5} - 6\zeta_{24}^{3} - 6\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 12\zeta_{24}^{7} - 3\zeta_{24}^{6} + 6\zeta_{24}^{5} - 6\zeta_{24}^{3} + 6\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -6\zeta_{24}^{6} - 12\zeta_{24}^{5} + 12\zeta_{24}^{3} + 12\zeta_{24}^{2} + 12\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 3\beta_{6} - 3\beta_{5} - 2\beta_{4} + \beta_{3} + 9\beta_1 ) / 72 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{7} + 4\beta_{4} + 2\beta_{3} ) / 36 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} - 2\beta_{4} - 9\beta_1 ) / 36 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( 2\beta_{2} - \beta _1 + 6 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - 3\beta_{5} + 2\beta_{4} + \beta_{3} - 9\beta_1 ) / 72 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + 3\beta_{5} - 2\beta_{4} + \beta_{3} - 9\beta_1 ) / 72 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
−0.258819 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
0.965926 0.258819i
0.258819 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
0 0 0 6.61037i 0 −9.43879 0 0 0
1025.2 0 0 0 6.61037i 0 9.43879 0 0 0
1025.3 0 0 0 3.78194i 0 −0.953512 0 0 0
1025.4 0 0 0 3.78194i 0 0.953512 0 0 0
1025.5 0 0 0 3.78194i 0 −0.953512 0 0 0
1025.6 0 0 0 3.78194i 0 0.953512 0 0 0
1025.7 0 0 0 6.61037i 0 −9.43879 0 0 0
1025.8 0 0 0 6.61037i 0 9.43879 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.8
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 1728.3.e.u 8
3.b odd 2 1 inner 1728.3.e.u 8
4.b odd 2 1 inner 1728.3.e.u 8
8.b even 2 1 864.3.e.f 8
8.d odd 2 1 864.3.e.f 8
12.b even 2 1 inner 1728.3.e.u 8
24.f even 2 1 864.3.e.f 8
24.h odd 2 1 864.3.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.e.f 8 8.b even 2 1
864.3.e.f 8 8.d odd 2 1
864.3.e.f 8 24.f even 2 1
864.3.e.f 8 24.h odd 2 1
1728.3.e.u 8 1.a even 1 1 trivial
1728.3.e.u 8 3.b odd 2 1 inner
1728.3.e.u 8 4.b odd 2 1 inner
1728.3.e.u 8 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 58T_{5}^{2} + 625 \) Copy content Toggle raw display
\( T_{7}^{4} - 90T_{7}^{2} + 81 \) 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
$5$ \( (T^{4} + 58 T^{2} + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 90 T^{2} + 81)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 450 T^{2} + 42849)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12 T - 180)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 280 T^{2} + 5776)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 1008 T^{2} + 129600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 720 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 472 T^{2} + 400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 3402 T^{2} + 1476225)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 40 T - 1544)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 392)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 7272 T^{2} + 11696400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3528 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4378 T^{2} + 4774225)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 5256 T^{2} + 810000)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 12456 T^{2} + 685584)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 3960 T^{2} + 2396304)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 42 T + 225)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 22680 T^{2} + 37896336)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 15570 T^{2} + 54066609)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 16360 T^{2} + 5779216)^{2} \) Copy content Toggle raw display
$97$ \( (T - 27)^{8} \) Copy content Toggle raw display
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