Properties

Label 1728.3.b.i
Level $1728$
Weight $3$
Character orbit 1728.b
Analytic conductor $47.085$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.116304318664704.2
Defining polynomial: \(x^{12} - 2 x^{11} + x^{10} + 6 x^{9} - 9 x^{8} - 2 x^{7} + 18 x^{6} - 4 x^{5} - 36 x^{4} + 48 x^{3} + 16 x^{2} - 64 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q -\beta_{4} q^{5} + \beta_{2} q^{7} + ( -2 \beta_{1} + \beta_{9} ) q^{11} + ( -2 \beta_{4} - \beta_{5} ) q^{13} + ( -4 - \beta_{10} ) q^{17} + ( 4 \beta_{1} - \beta_{9} - \beta_{11} ) q^{19} + ( 3 \beta_{2} + 3 \beta_{3} + \beta_{7} ) q^{23} + ( -6 - 2 \beta_{8} - \beta_{10} ) q^{25} + ( 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{29} + ( -4 \beta_{2} + 3 \beta_{7} ) q^{31} + ( -5 \beta_{1} - 2 \beta_{9} ) q^{35} + ( -6 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{37} + ( -28 + 3 \beta_{8} + 2 \beta_{10} ) q^{41} + ( -9 \beta_{1} + 2 \beta_{9} - \beta_{11} ) q^{43} + ( 9 \beta_{3} - 4 \beta_{7} ) q^{47} + ( 4 + 2 \beta_{8} + \beta_{10} ) q^{49} + ( \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{53} + ( 5 \beta_{2} + 2 \beta_{3} ) q^{55} + ( -24 \beta_{1} + 2 \beta_{9} + 2 \beta_{11} ) q^{59} + 4 \beta_{6} q^{61} + ( -76 - 6 \beta_{8} - \beta_{10} ) q^{65} + ( -\beta_{1} + 4 \beta_{9} + \beta_{11} ) q^{67} + ( -9 \beta_{2} + 18 \beta_{3} + \beta_{7} ) q^{71} + ( 5 - 4 \beta_{8} + \beta_{10} ) q^{73} + ( -7 \beta_{4} - 4 \beta_{5} - 10 \beta_{6} ) q^{77} + ( -3 \beta_{2} - 8 \beta_{3} - 3 \beta_{7} ) q^{79} + ( -22 \beta_{1} - 5 \beta_{9} - 4 \beta_{11} ) q^{83} + ( 14 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} ) q^{85} + ( -104 - 3 \beta_{8} + \beta_{10} ) q^{89} + ( 31 \beta_{1} - 7 \beta_{9} + 2 \beta_{11} ) q^{91} + ( -9 \beta_{2} + 27 \beta_{3} + 13 \beta_{7} ) q^{95} + ( -17 + 10 \beta_{8} + 2 \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + O(q^{10}) \) \( 12 q - 48 q^{17} - 72 q^{25} - 336 q^{41} + 48 q^{49} - 912 q^{65} + 60 q^{73} - 1248 q^{89} - 204 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} + x^{10} + 6 x^{9} - 9 x^{8} - 2 x^{7} + 18 x^{6} - 4 x^{5} - 36 x^{4} + 48 x^{3} + 16 x^{2} - 64 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} - 10 \nu^{8} - 25 \nu^{7} + 30 \nu^{6} + 30 \nu^{5} - 68 \nu^{4} - 28 \nu^{3} + 112 \nu^{2} - 64 \nu - 128 \)\()/32\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} - 8 \nu^{10} + 7 \nu^{9} + 4 \nu^{8} - 35 \nu^{7} + 16 \nu^{6} + 66 \nu^{5} - 52 \nu^{4} - 96 \nu^{3} + 144 \nu^{2} - 32 \nu - 248 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - 10 \nu^{10} + 5 \nu^{9} + 14 \nu^{8} - 37 \nu^{7} - 10 \nu^{6} + 78 \nu^{5} - 20 \nu^{4} - 156 \nu^{3} + 144 \nu^{2} + 32 \nu - 256 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{11} - 30 \nu^{10} - 21 \nu^{9} + 122 \nu^{8} - 139 \nu^{7} - 158 \nu^{6} + 330 \nu^{5} + 148 \nu^{4} - 772 \nu^{3} + 208 \nu^{2} + 960 \nu - 1408 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{11} + 11 \nu^{10} - 2 \nu^{9} - 23 \nu^{8} + 38 \nu^{7} + 27 \nu^{6} - 84 \nu^{5} - 16 \nu^{4} + 180 \nu^{3} - 96 \nu^{2} - 96 \nu + 256 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{10} + \nu^{9} - 15 \nu^{8} + 19 \nu^{7} + 19 \nu^{6} - 48 \nu^{5} - 18 \nu^{4} + 104 \nu^{3} - 44 \nu^{2} - 96 \nu + 168 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{11} - 30 \nu^{10} + 9 \nu^{9} + 66 \nu^{8} - 93 \nu^{7} - 78 \nu^{6} + 222 \nu^{5} + 60 \nu^{4} - 480 \nu^{3} + 336 \nu^{2} + 288 \nu - 552 \)\()/8\)
\(\beta_{8}\)\(=\)\((\)\( 15 \nu^{11} - 12 \nu^{10} - 21 \nu^{9} + 72 \nu^{8} - 3 \nu^{7} - 132 \nu^{6} + 66 \nu^{5} + 252 \nu^{4} - 300 \nu^{3} - 48 \nu^{2} + 480 \nu - 144 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -67 \nu^{11} + 86 \nu^{10} + 73 \nu^{9} - 386 \nu^{8} + 231 \nu^{7} + 630 \nu^{6} - 770 \nu^{5} - 900 \nu^{4} + 2020 \nu^{3} - 528 \nu^{2} - 2624 \nu + 2176 \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( 18 \nu^{11} - 15 \nu^{10} - 30 \nu^{9} + 87 \nu^{8} - 6 \nu^{7} - 171 \nu^{6} + 84 \nu^{5} + 300 \nu^{4} - 324 \nu^{3} - 48 \nu^{2} + 576 \nu - 184 \)\()/4\)
\(\beta_{11}\)\(=\)\((\)\( 157 \nu^{11} - 146 \nu^{10} - 247 \nu^{9} + 806 \nu^{8} - 153 \nu^{7} - 1554 \nu^{6} + 1022 \nu^{5} + 2652 \nu^{4} - 3484 \nu^{3} - 336 \nu^{2} + 6080 \nu - 2560 \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{2} + 8\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} + 4 \beta_{3} - 2 \beta_{2} + 15 \beta_{1} + 8\)\()/48\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{10} - 3 \beta_{8} + 2 \beta_{7} - 7 \beta_{3} + 2 \beta_{2} - 32\)\()/24\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 11 \beta_{6} - 2 \beta_{5} - 16 \beta_{4} - 16 \beta_{3} + 2 \beta_{2} + 63 \beta_{1} - 40\)\()/48\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - 7 \beta_{8} - 6 \beta_{7} - 14 \beta_{6} - 2 \beta_{5} - 28 \beta_{4} - 11 \beta_{3} + 10 \beta_{2} - 36 \beta_{1} + 48\)\()/48\)
\(\nu^{6}\)\(=\)\((\)\(-\beta_{10} + 4 \beta_{8} - \beta_{7} - 6 \beta_{3} + 9 \beta_{2} + 44\)\()/12\)
\(\nu^{7}\)\(=\)\((\)\(2 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 10 \beta_{5} - 4 \beta_{4} + 13 \beta_{3} + 2 \beta_{2} - 56 \beta_{1} - 16\)\()/16\)
\(\nu^{8}\)\(=\)\((\)\(-23 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} + 12 \beta_{8} + 30 \beta_{7} + 3 \beta_{6} + 18 \beta_{5} + 48 \beta_{4} - 40 \beta_{3} + 2 \beta_{2} - 135 \beta_{1} - 8\)\()/48\)
\(\nu^{9}\)\(=\)\((\)\(-5 \beta_{10} - \beta_{8} + 22 \beta_{7} - 29 \beta_{3} - 26 \beta_{2} - 464\)\()/24\)
\(\nu^{10}\)\(=\)\((\)\(-43 \beta_{11} + 10 \beta_{10} - 82 \beta_{9} + 16 \beta_{8} - 30 \beta_{7} - 41 \beta_{6} - 14 \beta_{5} - 64 \beta_{4} - 28 \beta_{3} - 82 \beta_{2} + 237 \beta_{1} - 152\)\()/48\)
\(\nu^{11}\)\(=\)\((\)\(18 \beta_{11} - 33 \beta_{10} - 102 \beta_{9} - 5 \beta_{8} - 58 \beta_{7} + 122 \beta_{6} - 94 \beta_{5} - 20 \beta_{4} - 33 \beta_{3} + 102 \beta_{2} - 48 \beta_{1} - 432\)\()/48\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−1.41362 0.0408194i
0.742163 + 1.20382i
1.33544 0.465413i
−1.07078 + 0.923815i
0.828615 + 1.14604i
0.578188 + 1.29062i
0.578188 1.29062i
0.828615 1.14604i
−1.07078 0.923815i
1.33544 + 0.465413i
0.742163 1.20382i
−1.41362 + 0.0408194i
0 0 0 8.36964i 0 2.43910i 0 0 0
1567.2 0 0 0 8.36964i 0 2.43910i 0 0 0
1567.3 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.4 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.5 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.6 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.7 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.8 0 0 0 2.64040i 0 8.30833i 0 0 0
1567.9 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.10 0 0 0 3.99718i 0 7.74742i 0 0 0
1567.11 0 0 0 8.36964i 0 2.43910i 0 0 0
1567.12 0 0 0 8.36964i 0 2.43910i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5}^{6} + 93 T_{5}^{4} + 1719 T_{5}^{2} + 7803 \)
\( T_{7}^{6} + 135 T_{7}^{4} + 4911 T_{7}^{2} + 24649 \)
\( T_{11}^{6} - 417 T_{11}^{4} + 23211 T_{11}^{2} - 121203 \)
\( T_{17}^{3} + 12 T_{17}^{2} - 540 T_{17} - 6912 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 7803 + 1719 T^{2} + 93 T^{4} + T^{6} )^{2} \)
$7$ \( ( 24649 + 4911 T^{2} + 135 T^{4} + T^{6} )^{2} \)
$11$ \( ( -121203 + 23211 T^{2} - 417 T^{4} + T^{6} )^{2} \)
$13$ \( ( 442368 + 219024 T^{2} + 936 T^{4} + T^{6} )^{2} \)
$17$ \( ( -6912 - 540 T + 12 T^{2} + T^{3} )^{4} \)
$19$ \( ( -51121152 + 671760 T^{2} - 1656 T^{4} + T^{6} )^{2} \)
$23$ \( ( 394896384 + 1919376 T^{2} + 2520 T^{4} + T^{6} )^{2} \)
$29$ \( ( 20155392 + 384912 T^{2} + 1512 T^{4} + T^{6} )^{2} \)
$31$ \( ( 262018969 + 2857839 T^{2} + 3447 T^{4} + T^{6} )^{2} \)
$37$ \( ( 2674142208 + 6906384 T^{2} + 5256 T^{4} + T^{6} )^{2} \)
$41$ \( ( -113184 - 972 T + 84 T^{2} + T^{3} )^{4} \)
$43$ \( ( -322486272 + 3172608 T^{2} - 4320 T^{4} + T^{6} )^{2} \)
$47$ \( ( 5780865024 + 15344640 T^{2} + 7488 T^{4} + T^{6} )^{2} \)
$53$ \( ( 8741547 + 294615 T^{2} + 1101 T^{4} + T^{6} )^{2} \)
$59$ \( ( -5159780352 + 22581504 T^{2} - 11232 T^{4} + T^{6} )^{2} \)
$61$ \( ( 768 + T^{2} )^{6} \)
$67$ \( ( -936050688 + 10036224 T^{2} - 6336 T^{4} + T^{6} )^{2} \)
$71$ \( ( 4087812096 + 76911120 T^{2} + 22392 T^{4} + T^{6} )^{2} \)
$73$ \( ( -120721 - 5409 T - 15 T^{2} + T^{3} )^{4} \)
$79$ \( ( 1846936576 + 14058624 T^{2} + 8868 T^{4} + T^{6} )^{2} \)
$83$ \( ( -11178011043 + 99655371 T^{2} - 30801 T^{4} + T^{6} )^{2} \)
$89$ \( ( 680832 + 28944 T + 312 T^{2} + T^{3} )^{4} \)
$97$ \( ( -1211279 - 22365 T + 51 T^{2} + T^{3} )^{4} \)
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