Properties

Label 1728.4.d.f
Level $1728$
Weight $4$
Character orbit 1728.d
Analytic conductor $101.955$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.592240896.1
Defining polynomial: \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} + 7 \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{5} + 7 \beta_1 q^{7} - \beta_{7} q^{11} - 9 \beta_{2} q^{13} - 3 \beta_{4} q^{17} + 49 \beta_{3} q^{19} + 5 \beta_{5} q^{23} - 31 q^{25} + 2 \beta_{6} q^{29} + 14 \beta_1 q^{31} - 7 \beta_{7} q^{35} - 59 \beta_{2} q^{37} - 16 \beta_{4} q^{41} + 260 \beta_{3} q^{43} + 29 \beta_{5} q^{47} - 196 q^{49} + 46 \beta_{6} q^{53} - 156 \beta_1 q^{55} - 15 \beta_{7} q^{59} - 101 \beta_{2} q^{61} + 9 \beta_{4} q^{65} - 241 \beta_{3} q^{67} - 20 \beta_{5} q^{71} + 353 q^{73} - 21 \beta_{6} q^{77} - 3 \beta_1 q^{79} - 48 \beta_{7} q^{83} - 468 \beta_{2} q^{85} - 37 \beta_{4} q^{89} + 189 \beta_{3} q^{91} - 49 \beta_{5} q^{95} + 1111 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 248 q^{25} - 1568 q^{49} + 2824 q^{73} + 8888 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 16\nu^{5} + 76\nu^{3} - 207\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{6} - 40\nu^{4} + 280\nu^{2} - 261 ) / 180 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{7} - 40\nu^{5} + 190\nu^{3} - 81\nu ) / 270 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 231 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{7} + 100\nu^{5} - 610\nu^{3} + 1719\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{6} + 28\nu^{4} - 124\nu^{2} + 126 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 40\nu^{5} + 244\nu^{3} - 81\nu ) / 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{5} - 6\beta_{3} + 6\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} - \beta_{4} + 42\beta_{2} + 42 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 15\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21\beta_{6} + 7\beta_{4} + 186\beta_{2} - 186 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{7} - 57\beta_{5} - 366\beta_{3} - 366\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10\beta_{4} - 231 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -97\beta_{7} - 291\beta_{5} + 2022\beta_{3} - 2022\beta_1 ) / 24 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
−1.99426 1.15139i
1.12824 0.651388i
1.99426 + 1.15139i
−1.12824 + 0.651388i
1.12824 + 0.651388i
−1.99426 + 1.15139i
−1.12824 0.651388i
1.99426 1.15139i
0 0 0 12.4900i 0 −12.1244 0 0 0
865.2 0 0 0 12.4900i 0 −12.1244 0 0 0
865.3 0 0 0 12.4900i 0 12.1244 0 0 0
865.4 0 0 0 12.4900i 0 12.1244 0 0 0
865.5 0 0 0 12.4900i 0 −12.1244 0 0 0
865.6 0 0 0 12.4900i 0 −12.1244 0 0 0
865.7 0 0 0 12.4900i 0 12.1244 0 0 0
865.8 0 0 0 12.4900i 0 12.1244 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 865.8
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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.d.f 8
3.b odd 2 1 inner 1728.4.d.f 8
4.b odd 2 1 inner 1728.4.d.f 8
8.b even 2 1 inner 1728.4.d.f 8
8.d odd 2 1 inner 1728.4.d.f 8
12.b even 2 1 inner 1728.4.d.f 8
24.f even 2 1 inner 1728.4.d.f 8
24.h odd 2 1 inner 1728.4.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.d.f 8 1.a even 1 1 trivial
1728.4.d.f 8 3.b odd 2 1 inner
1728.4.d.f 8 4.b odd 2 1 inner
1728.4.d.f 8 8.b even 2 1 inner
1728.4.d.f 8 8.d odd 2 1 inner
1728.4.d.f 8 12.b even 2 1 inner
1728.4.d.f 8 24.f even 2 1 inner
1728.4.d.f 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 156 \) Copy content Toggle raw display
\( T_{7}^{2} - 147 \) Copy content Toggle raw display
\( T_{17}^{2} - 4212 \) Copy content Toggle raw display
\( T_{23}^{2} - 3900 \) 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
$5$ \( (T^{2} + 156)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 147)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 468)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 243)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4212)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2401)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3900)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 624)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 588)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10443)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 119808)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 67600)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 131196)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 330096)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 105300)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 30603)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 58081)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 62400)^{4} \) Copy content Toggle raw display
$73$ \( (T - 353)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1078272)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 640692)^{4} \) Copy content Toggle raw display
$97$ \( (T - 1111)^{8} \) Copy content Toggle raw display
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