Properties

Label 1728.4.d.g
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: 8.0.1731891456.1
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + \beta_{6} q^{7} - \beta_{7} q^{11} + \beta_{5} q^{13} + 5 \beta_{4} q^{17} - 7 \beta_1 q^{19} - 11 \beta_{3} q^{23} + 17 q^{25} - 26 \beta_{2} q^{29} + 10 \beta_{6} q^{31} + 9 \beta_{7} q^{35} + 11 \beta_{5} q^{37} - 16 \beta_{4} q^{41} - 92 \beta_1 q^{43} - 11 \beta_{3} q^{47} + 116 q^{49} + 2 \beta_{2} q^{53} + 12 \beta_{6} q^{55} - 7 \beta_{7} q^{59} + 21 \beta_{5} q^{61} + 9 \beta_{4} q^{65} - 353 \beta_1 q^{67} - 76 \beta_{3} q^{71} + 425 q^{73} - 51 \beta_{2} q^{77} - 61 \beta_{6} q^{79} - 24 \beta_{7} q^{83} - 60 \beta_{5} q^{85} + 3 \beta_{4} q^{89} - 459 \beta_1 q^{91} + 7 \beta_{3} q^{95} + 799 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 136 q^{25} + 928 q^{49} + 3400 q^{73} + 6392 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -9\nu^{7} + 65\nu^{5} - 377\nu^{3} + 256\nu ) / 832 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -27\nu^{6} + 195\nu^{4} - 1755\nu^{2} + 2328 ) / 260 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 75\nu^{5} - 435\nu^{3} + 1632\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -12\nu^{6} - 1782 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} + 27\nu^{4} - 147\nu^{2} + 216 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 51\nu^{7} - 507\nu^{5} + 3939\nu^{3} - 15456\nu ) / 832 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27\nu^{7} - 195\nu^{5} + 1515\nu^{3} - 768\nu ) / 160 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} - \beta_{3} + 6\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + \beta_{4} - 9\beta_{2} + 54 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 78\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18\beta_{5} - 9\beta_{4} - 49\beta_{2} - 294 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 29\beta_{7} + 58\beta_{6} + 101\beta_{3} + 606\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -65\beta_{4} - 1782 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -181\beta_{7} + 362\beta_{6} + 701\beta_{3} - 4206\beta_1 ) / 24 \) 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
1.35234 + 0.780776i
2.21837 + 1.28078i
−1.35234 0.780776i
−2.21837 1.28078i
2.21837 1.28078i
1.35234 0.780776i
−2.21837 + 1.28078i
−1.35234 + 0.780776i
0 0 0 10.3923i 0 −21.4243 0 0 0
865.2 0 0 0 10.3923i 0 −21.4243 0 0 0
865.3 0 0 0 10.3923i 0 21.4243 0 0 0
865.4 0 0 0 10.3923i 0 21.4243 0 0 0
865.5 0 0 0 10.3923i 0 −21.4243 0 0 0
865.6 0 0 0 10.3923i 0 −21.4243 0 0 0
865.7 0 0 0 10.3923i 0 21.4243 0 0 0
865.8 0 0 0 10.3923i 0 21.4243 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.g 8
3.b odd 2 1 inner 1728.4.d.g 8
4.b odd 2 1 inner 1728.4.d.g 8
8.b even 2 1 inner 1728.4.d.g 8
8.d odd 2 1 inner 1728.4.d.g 8
12.b even 2 1 inner 1728.4.d.g 8
24.f even 2 1 inner 1728.4.d.g 8
24.h odd 2 1 inner 1728.4.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.d.g 8 1.a even 1 1 trivial
1728.4.d.g 8 3.b odd 2 1 inner
1728.4.d.g 8 4.b odd 2 1 inner
1728.4.d.g 8 8.b even 2 1 inner
1728.4.d.g 8 8.d odd 2 1 inner
1728.4.d.g 8 12.b even 2 1 inner
1728.4.d.g 8 24.f even 2 1 inner
1728.4.d.g 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 \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 108 \) Copy content Toggle raw display
\( T_{7}^{2} - 459 \) Copy content Toggle raw display
\( T_{17}^{2} - 15300 \) Copy content Toggle raw display
\( T_{23}^{2} - 13068 \) 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
$5$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 459)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 612)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 459)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 15300)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 49)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 13068)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 73008)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 45900)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 55539)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 156672)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8464)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 13068)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 432)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 29988)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 202419)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 124609)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 623808)^{4} \) Copy content Toggle raw display
$73$ \( (T - 425)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 1707939)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 352512)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 5508)^{4} \) Copy content Toggle raw display
$97$ \( (T - 799)^{8} \) Copy content Toggle raw display
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