Properties

Label 1728.3.g.m
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: 8.0.56070144.2
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{5} + (\beta_{6} + \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{5} + (\beta_{6} + \beta_{2}) q^{7} + ( - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2}) q^{11} + (\beta_{3} - \beta_1 - 1) q^{13} + ( - \beta_{3} - 2 \beta_1 - 3) q^{17} + (4 \beta_{7} - \beta_{4} + \beta_{2}) q^{19} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + \beta_{2}) q^{23} + (\beta_{5} + \beta_1 + 3) q^{25} + ( - 2 \beta_{5} - 2 \beta_1 - 16) q^{29} + (4 \beta_{7} + 3 \beta_{2}) q^{31} + (3 \beta_{7} + 3 \beta_{6} + 5 \beta_{4} + 5 \beta_{2}) q^{35} + (\beta_{5} - 7 \beta_{3} - 3) q^{37} + (2 \beta_{5} - 2 \beta_1 - 20) q^{41} + ( - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + \beta_{2}) q^{43} + ( - 7 \beta_{7} + 3 \beta_{6} - 7 \beta_{4} - 5 \beta_{2}) q^{47} + ( - 10 \beta_{3} - 6 \beta_1 - 18) q^{49} + ( - 2 \beta_{5} + 6 \beta_{3} + 2 \beta_1 - 6) q^{53} + ( - 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 9 \beta_{2}) q^{55} + (3 \beta_{7} - \beta_{6} + 9 \beta_{4} - 9 \beta_{2}) q^{59} + (4 \beta_{5} + \beta_{3} + 3 \beta_1 + 17) q^{61} + (2 \beta_{5} - 7 \beta_{3} - 4 \beta_1 + 35) q^{65} + ( - 8 \beta_{7} - \beta_{4} + 2 \beta_{2}) q^{67} + (14 \beta_{7} - 2 \beta_{6} - 14 \beta_{4} + 2 \beta_{2}) q^{71} + (4 \beta_{5} + 6 \beta_{3} - 2 \beta_1 + 9) q^{73} + ( - 2 \beta_{5} + 9 \beta_{3} + 6 \beta_1 + 65) q^{77} + ( - 4 \beta_{7} - 5 \beta_{6} + 4 \beta_{4} - 4 \beta_{2}) q^{79} + ( - 18 \beta_{7} + 2 \beta_{6} + 14 \beta_{4} + 8 \beta_{2}) q^{83} + (\beta_{5} - 12 \beta_{3} - 11 \beta_1 - 12) q^{85} + (4 \beta_{5} - 9 \beta_{3} + 2 \beta_1 + 21) q^{89} + ( - 4 \beta_{7} - 4 \beta_{6} + 5 \beta_{4} - 17 \beta_{2}) q^{91} + ( - \beta_{7} + 5 \beta_{6} - 9 \beta_{4} + 27 \beta_{2}) q^{95} + (5 \beta_{5} - 10 \beta_{3} - \beta_1 + 13) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} - 8 q^{13} - 24 q^{17} + 24 q^{25} - 128 q^{29} - 24 q^{37} - 160 q^{41} - 144 q^{49} - 48 q^{53} + 136 q^{61} + 280 q^{65} + 72 q^{73} + 520 q^{77} - 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu^{2} - 6\nu + 15 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\nu^{7} - 84\nu^{6} + 268\nu^{5} - 460\nu^{4} + 468\nu^{3} - 284\nu^{2} - 164\nu + 116 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{6} - 6\nu^{5} + 24\nu^{4} - 38\nu^{3} + 62\nu^{2} - 44\nu + 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -48\nu^{7} + 168\nu^{6} - 684\nu^{5} + 1290\nu^{4} - 2268\nu^{3} + 2196\nu^{2} - 1596\nu + 471 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -8\nu^{6} + 24\nu^{5} - 84\nu^{4} + 128\nu^{3} - 170\nu^{2} + 110\nu - 43 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -100\nu^{7} + 350\nu^{6} - 1314\nu^{5} + 2410\nu^{4} - 3874\nu^{3} + 3576\nu^{2} - 2622\nu + 787 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -112\nu^{7} + 392\nu^{6} - 1448\nu^{5} + 2640\nu^{4} - 4108\nu^{3} + 3718\nu^{2} - 2318\nu + 618 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + \beta_{2} + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + \beta_{2} + 2\beta _1 - 24 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 4\beta_{6} + 4\beta_{4} - 8\beta_{2} + 3\beta _1 - 39 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{7} + 10\beta_{6} + \beta_{5} + 8\beta_{4} + 4\beta_{3} - 17\beta_{2} - 7\beta _1 + 54 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 38\beta_{7} - 14\beta_{6} + 5\beta_{5} - 38\beta_{4} + 20\beta_{3} + 43\beta_{2} - 45\beta _1 + 402 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 196\beta_{7} - 94\beta_{6} - 9\beta_{5} - 154\beta_{4} - 24\beta_{3} + 215\beta_{2} + 23\beta _1 - 120 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -4\beta_{7} - 20\beta_{6} - 49\beta_{5} + 36\beta_{4} - 154\beta_{3} + 7\beta_{2} + 245\beta _1 - 1920 ) / 24 \) 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
0.500000 + 1.19293i
0.500000 1.19293i
0.500000 + 1.56488i
0.500000 1.56488i
0.500000 + 2.19293i
0.500000 2.19293i
0.500000 + 0.564882i
0.500000 0.564882i
0 0 0 −6.59655 0 4.56106i 0 0 0
703.2 0 0 0 −6.59655 0 4.56106i 0 0 0
703.3 0 0 0 −0.956810 0 6.34610i 0 0 0
703.4 0 0 0 −0.956810 0 6.34610i 0 0 0
703.5 0 0 0 5.13244 0 4.02516i 0 0 0
703.6 0 0 0 5.13244 0 4.02516i 0 0 0
703.7 0 0 0 6.42091 0 13.8102i 0 0 0
703.8 0 0 0 6.42091 0 13.8102i 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.m 8
3.b odd 2 1 1728.3.g.j 8
4.b odd 2 1 inner 1728.3.g.m 8
8.b even 2 1 864.3.g.b 8
8.d odd 2 1 864.3.g.b 8
12.b even 2 1 1728.3.g.j 8
24.f even 2 1 864.3.g.d yes 8
24.h odd 2 1 864.3.g.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.g.b 8 8.b even 2 1
864.3.g.b 8 8.d odd 2 1
864.3.g.d yes 8 24.f even 2 1
864.3.g.d yes 8 24.h odd 2 1
1728.3.g.j 8 3.b odd 2 1
1728.3.g.j 8 12.b even 2 1
1728.3.g.m 8 1.a even 1 1 trivial
1728.3.g.m 8 4.b 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 \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} - 4T_{5}^{3} - 48T_{5}^{2} + 176T_{5} + 208 \) Copy content Toggle raw display
\( T_{7}^{8} + 268T_{7}^{6} + 16566T_{7}^{4} + 362092T_{7}^{2} + 2588881 \) 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
$5$ \( (T^{4} - 4 T^{3} - 48 T^{2} + 176 T + 208)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 268 T^{6} + 16566 T^{4} + \cdots + 2588881 \) Copy content Toggle raw display
$11$ \( T^{8} + 400 T^{6} + 43296 T^{4} + \cdots + 891136 \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} - 282 T^{2} - 2300 T - 3167)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{3} - 720 T^{2} - 3024 T + 5328)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 1548 T^{6} + \cdots + 13809305169 \) Copy content Toggle raw display
$23$ \( T^{8} + 1360 T^{6} + \cdots + 522762496 \) Copy content Toggle raw display
$29$ \( (T^{4} + 64 T^{3} - 768 T^{2} + \cdots - 966656)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 1984 T^{6} + \cdots + 7929856 \) Copy content Toggle raw display
$37$ \( (T^{4} + 12 T^{3} - 2826 T^{2} + \cdots - 661167)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 80 T^{3} - 768 T^{2} + \cdots - 929792)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 2368 T^{6} + \cdots + 8446345216 \) Copy content Toggle raw display
$47$ \( T^{8} + 14800 T^{6} + \cdots + 80138017536256 \) Copy content Toggle raw display
$53$ \( (T^{4} + 24 T^{3} - 4032 T^{2} + \cdots + 3624192)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 16528 T^{6} + \cdots + 1539862591744 \) Copy content Toggle raw display
$61$ \( (T^{4} - 68 T^{3} - 6618 T^{2} + \cdots - 11656223)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 8620 T^{6} + \cdots + 701937325489 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 121012672331776 \) Copy content Toggle raw display
$73$ \( (T^{4} - 36 T^{3} - 12186 T^{2} + \cdots + 1325601)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 8620 T^{6} + \cdots + 40981548721 \) Copy content Toggle raw display
$83$ \( T^{8} + 46912 T^{6} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{4} - 84 T^{3} - 8784 T^{2} + \cdots - 146736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 52 T^{3} - 15114 T^{2} + \cdots + 32388241)^{2} \) Copy content Toggle raw display
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