Properties

Label 1728.3.g.k
Level $1728$
Weight $3$
Character orbit 1728.g
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{5} + \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{5} + \beta_{7} q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{11} + (\beta_{4} + 2) q^{13} + (\beta_{4} + 6) q^{17} + (\beta_{7} - 3 \beta_{5} + \beta_{3}) q^{19} + ( - 3 \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{3}) q^{23} + ( - 6 \beta_{2} - \beta_1 + 6) q^{25} + (\beta_1 + 4) q^{29} + ( - 2 \beta_{7} + \beta_{6} + 5 \beta_{5} + \beta_{3}) q^{31} + ( - 4 \beta_{6} - \beta_{5}) q^{35} + (\beta_{4} - 4 \beta_{2} + 2 \beta_1 + 12) q^{37} + ( - \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 16) q^{41} + ( - 2 \beta_{7} - 4 \beta_{5} - 4 \beta_{3}) q^{43} + (4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3}) q^{47} + ( - 2 \beta_{2} - 3 \beta_1) q^{49} + (4 \beta_{4} + \beta_{2} + \beta_1 - 21) q^{53} + ( - 3 \beta_{7} - 6 \beta_{6} + 12 \beta_{5} - 4 \beta_{3}) q^{55} + ( - 2 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} - 6 \beta_{3}) q^{59} + (4 \beta_{4} + 12 \beta_{2} + 2 \beta_1 - 4) q^{61} + (3 \beta_{4} + 8 \beta_{2} + 2 \beta_1 - 14) q^{65} + ( - 2 \beta_{7} + 2 \beta_{6} - 12 \beta_{5} + 4 \beta_{3}) q^{67} + (3 \beta_{7} + 14 \beta_{6} - 3 \beta_{5} + \beta_{3}) q^{71} + ( - 6 \beta_{4} - 6 \beta_{2} - \beta_1 + 3) q^{73} + (2 \beta_{4} + 5 \beta_{2} + 4 \beta_1 + 55) q^{77} + (3 \beta_{7} + 13 \beta_{6} + 9 \beta_{5} + 5 \beta_{3}) q^{79} + (13 \beta_{7} + 3 \beta_{6} - 10 \beta_{5} + 3 \beta_{3}) q^{83} + (3 \beta_{4} + 12 \beta_{2} + 2 \beta_1 - 18) q^{85} + (4 \beta_{4} + 10 \beta_{2} + 4 \beta_1 + 78) q^{89} + (\beta_{7} - 12 \beta_{6} - 13 \beta_{5} - \beta_{3}) q^{91} + ( - \beta_{7} - 22 \beta_{6} + 11 \beta_{5} - 11 \beta_{3}) q^{95} + ( - 6 \beta_{4} + 4 \beta_{2} - 2 \beta_1 - 17) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} + 16 q^{13} + 48 q^{17} + 48 q^{25} + 32 q^{29} + 96 q^{37} - 128 q^{41} - 168 q^{53} - 32 q^{61} - 112 q^{65} + 24 q^{73} + 440 q^{77} - 144 q^{85} + 624 q^{89} - 136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{7} + 3\nu^{6} - \nu^{5} - 7\nu^{4} + 9\nu^{3} + 11\nu^{2} - 32\nu + 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{7} + 30\nu^{6} - 11\nu^{5} - 44\nu^{4} + 57\nu^{3} + 46\nu^{2} - 160\nu + 112 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{7} - 34\nu^{6} + 27\nu^{5} + 40\nu^{4} - 81\nu^{3} - 18\nu^{2} + 208\nu - 192 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 17\nu^{5} - 20\nu^{4} + 51\nu^{3} + 10\nu^{2} - 100\nu + 100 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} - 21\nu^{6} + 9\nu^{5} + 33\nu^{4} - 45\nu^{3} - 33\nu^{2} + 120\nu - 90 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\nu^{7} - 11\nu^{6} + 6\nu^{5} + 17\nu^{4} - 24\nu^{3} - 15\nu^{2} + 62\nu - 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -25\nu^{7} + 64\nu^{6} - 33\nu^{5} - 94\nu^{4} + 127\nu^{3} + 104\nu^{2} - 360\nu + 268 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} + 2\beta_{3} - \beta _1 + 12 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} + \beta_{5} + 2\beta_{4} + 3\beta_{3} + \beta _1 + 18 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} - 5\beta_{5} + 4\beta_{4} + 3\beta_{3} + 2\beta_{2} + \beta _1 - 6 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 17\beta_{6} - 3\beta_{5} + 4\beta_{4} - \beta_{3} + 8\beta_{2} + \beta _1 - 6 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{7} + 8\beta_{6} - 7\beta_{5} + \beta_{4} - \beta_{3} + 18\beta_{2} + 2\beta _1 + 42 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{6} + 6\beta_{5} + \beta_{4} + 14\beta_{2} - 2\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6\beta_{7} - 17\beta_{6} + 30\beta_{5} + 3\beta_{4} + 4\beta_{3} + 28\beta_{2} + 7\beta _1 + 96 ) / 24 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
1.20036 0.747754i
1.20036 + 0.747754i
−1.27597 + 0.609843i
−1.27597 0.609843i
1.40994 + 0.109843i
1.40994 0.109843i
0.665665 1.24775i
0.665665 + 1.24775i
0 0 0 −9.64469 0 1.12019i 0 0 0
703.2 0 0 0 −9.64469 0 1.12019i 0 0 0
703.3 0 0 0 −1.76102 0 12.0363i 0 0 0
703.4 0 0 0 −1.76102 0 12.0363i 0 0 0
703.5 0 0 0 3.22512 0 6.57221i 0 0 0
703.6 0 0 0 3.22512 0 6.57221i 0 0 0
703.7 0 0 0 4.18059 0 2.58429i 0 0 0
703.8 0 0 0 4.18059 0 2.58429i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.g.k 8
3.b odd 2 1 1728.3.g.n 8
4.b odd 2 1 inner 1728.3.g.k 8
8.b even 2 1 864.3.g.c yes 8
8.d odd 2 1 864.3.g.c yes 8
12.b even 2 1 1728.3.g.n 8
24.f even 2 1 864.3.g.a 8
24.h odd 2 1 864.3.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.g.a 8 24.f even 2 1
864.3.g.a 8 24.h odd 2 1
864.3.g.c yes 8 8.b even 2 1
864.3.g.c yes 8 8.d odd 2 1
1728.3.g.k 8 1.a even 1 1 trivial
1728.3.g.k 8 4.b odd 2 1 inner
1728.3.g.n 8 3.b odd 2 1
1728.3.g.n 8 12.b even 2 1

Hecke kernels

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

\( T_{5}^{4} + 4T_{5}^{3} - 54T_{5}^{2} + 28T_{5} + 229 \) Copy content Toggle raw display
\( T_{7}^{8} + 196T_{7}^{6} + 7758T_{7}^{4} + 51220T_{7}^{2} + 52441 \) 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
$5$ \( (T^{4} + 4 T^{3} - 54 T^{2} + 28 T + 229)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 196 T^{6} + 7758 T^{4} + \cdots + 52441 \) Copy content Toggle raw display
$11$ \( T^{8} + 604 T^{6} + \cdots + 269517889 \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} - 288 T^{2} + 640 T + 15616)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 24 T^{3} - 96 T^{2} + 2304 T + 9216)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 1200 T^{6} + 360288 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$23$ \( T^{8} + 2128 T^{6} + \cdots + 8962787584 \) Copy content Toggle raw display
$29$ \( (T^{4} - 16 T^{3} - 648 T^{2} - 3520 T - 2288)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 3220 T^{6} + \cdots + 116920969 \) Copy content Toggle raw display
$37$ \( (T^{4} - 48 T^{3} - 3480 T^{2} + \cdots - 354096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 64 T^{3} - 2376 T^{2} + \cdots - 3217904)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 7696 T^{6} + \cdots + 947180525824 \) Copy content Toggle raw display
$47$ \( T^{8} + 7216 T^{6} + \cdots + 1218851328256 \) Copy content Toggle raw display
$53$ \( (T^{4} + 84 T^{3} - 1950 T^{2} + \cdots - 1618803)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 157628225880064 \) Copy content Toggle raw display
$61$ \( (T^{4} + 16 T^{3} - 11136 T^{2} + \cdots + 3739648)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 23824 T^{6} + \cdots + 54344260884736 \) Copy content Toggle raw display
$71$ \( T^{8} + 15616 T^{6} + \cdots + 55832696848384 \) Copy content Toggle raw display
$73$ \( (T^{4} - 12 T^{3} - 10626 T^{2} + \cdots - 7709463)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 29440 T^{6} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 122448233690689 \) Copy content Toggle raw display
$89$ \( (T^{4} - 312 T^{3} + 21288 T^{2} + \cdots - 3170736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 68 T^{3} - 12090 T^{2} + \cdots - 19173311)^{2} \) Copy content Toggle raw display
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