Properties

Label 1728.3.h.h
Level $1728$
Weight $3$
Character orbit 1728.h
Analytic conductor $47.085$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} - 5 \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} - 5 \beta_{2} q^{7} - \beta_{4} q^{11} + 3 \beta_1 q^{13} + \beta_{7} q^{17} + 23 \beta_{3} q^{19} - \beta_{6} q^{23} + 35 q^{25} - 4 \beta_{5} q^{29} - 4 \beta_{2} q^{31} - 5 \beta_{4} q^{35} - 17 \beta_1 q^{37} + 6 \beta_{7} q^{41} - 38 \beta_{3} q^{43} - 7 \beta_{6} q^{47} + 26 q^{49} + 10 \beta_{5} q^{53} - 60 \beta_{2} q^{55} + 7 \beta_{4} q^{59} - 35 \beta_1 q^{61} - 3 \beta_{7} q^{65} - 107 \beta_{3} q^{67} - 2 \beta_{6} q^{71} - 97 q^{73} - 15 \beta_{5} q^{77} + 39 \beta_{2} q^{79} - 60 \beta_1 q^{85} - 13 \beta_{7} q^{89} - 45 \beta_{3} q^{91} + 23 \beta_{6} q^{95} + 109 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 280 q^{25} + 208 q^{49} - 776 q^{73} + 872 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 10\nu^{3} + 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 20\nu^{3} - \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} - 27 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\nu^{6} + 12\nu^{4} - 28\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 9\nu^{7} - 24\nu^{5} + 66\nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{5} - 6\beta_{3} - 6\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} + \beta_{4} + 18\beta _1 + 18 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 12\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{6} - \beta_{4} + 14\beta _1 - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 15\beta_{5} - 66\beta_{3} + 66\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{4} - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 39\beta_{5} + 174\beta_{3} + 174\beta_{2} ) / 24 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.535233 0.309017i
0.535233 + 0.309017i
1.40126 0.809017i
1.40126 + 0.809017i
−1.40126 + 0.809017i
−1.40126 0.809017i
−0.535233 + 0.309017i
−0.535233 0.309017i
0 0 0 −7.74597 0 −8.66025 0 0 0
161.2 0 0 0 −7.74597 0 −8.66025 0 0 0
161.3 0 0 0 −7.74597 0 8.66025 0 0 0
161.4 0 0 0 −7.74597 0 8.66025 0 0 0
161.5 0 0 0 7.74597 0 −8.66025 0 0 0
161.6 0 0 0 7.74597 0 −8.66025 0 0 0
161.7 0 0 0 7.74597 0 8.66025 0 0 0
161.8 0 0 0 7.74597 0 8.66025 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
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 1728.3.h.h 8
3.b odd 2 1 inner 1728.3.h.h 8
4.b odd 2 1 inner 1728.3.h.h 8
8.b even 2 1 inner 1728.3.h.h 8
8.d odd 2 1 inner 1728.3.h.h 8
12.b even 2 1 inner 1728.3.h.h 8
24.f even 2 1 inner 1728.3.h.h 8
24.h odd 2 1 inner 1728.3.h.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.h.h 8 1.a even 1 1 trivial
1728.3.h.h 8 3.b odd 2 1 inner
1728.3.h.h 8 4.b odd 2 1 inner
1728.3.h.h 8 8.b even 2 1 inner
1728.3.h.h 8 8.d odd 2 1 inner
1728.3.h.h 8 12.b even 2 1 inner
1728.3.h.h 8 24.f even 2 1 inner
1728.3.h.h 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 \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} - 60 \) Copy content Toggle raw display
\( T_{7}^{2} - 75 \) Copy content Toggle raw display
\( T_{11}^{2} - 180 \) 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
$5$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 75)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 180)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 180)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 529)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 960)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 867)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 6480)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1444)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 2940)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 6000)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8820)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3675)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 11449)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 240)^{4} \) Copy content Toggle raw display
$73$ \( (T + 97)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4563)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 30420)^{4} \) Copy content Toggle raw display
$97$ \( (T - 109)^{8} \) Copy content Toggle raw display
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