Properties

Label 1728.2.r.f
Level $1728$
Weight $2$
Character orbit 1728.r
Analytic conductor $13.798$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 2) q^{5} + (\beta_{11} + \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 2) q^{5} + (\beta_{11} + \beta_{2}) q^{7} + ( - \beta_{11} - \beta_{9} - \beta_{5}) q^{11} - \beta_{10} q^{13} + \beta_1 q^{17} - \beta_{7} q^{19} + ( - 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2}) q^{23} + ( - 2 \beta_{4} - 2) q^{25} + ( - \beta_{6} + \beta_{4} + \beta_1 - 1) q^{29} + (2 \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2}) q^{31} + (2 \beta_{11} + \beta_{9} + \beta_{5} + 2 \beta_{2}) q^{35} + ( - \beta_{10} + \beta_{8} + 2 \beta_{4} + 2) q^{37} + (\beta_{6} + 3 \beta_{4}) q^{41} + (\beta_{11} + 2 \beta_{9} + 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{43} + (\beta_{11} - 2 \beta_{9} + \beta_{7} + 2 \beta_{5} + \beta_{3} - \beta_{2}) q^{47} + ( - \beta_{10} + 2 \beta_{8} + 5 \beta_{4} + 2) q^{49} + ( - 2 \beta_{6} - 4 \beta_{4} - \beta_1 - 2) q^{53} + ( - \beta_{11} - \beta_{9} - 2 \beta_{5} + \beta_{2}) q^{55} + ( - \beta_{11} + \beta_{9} - \beta_{5} + 3 \beta_{3} + \beta_{2}) q^{59} + ( - \beta_{6} - \beta_{4} + \beta_1 + 1) q^{61} + ( - 2 \beta_{10} + \beta_{8} + 1) q^{65} + ( - 2 \beta_{11} + 2 \beta_{9} - \beta_{5} + 2 \beta_{3} + \beta_{2}) q^{67} + (\beta_{11} - 2 \beta_{9} + 2 \beta_{7} - \beta_{5} - 4 \beta_{3} - \beta_{2}) q^{71} + (\beta_{10} + \beta_{8}) q^{73} + ( - 3 \beta_{10} - \beta_{6} + 6 \beta_{4} - 2 \beta_1 + 12) q^{77} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3}) q^{79} + (\beta_{11} + 2 \beta_{9} + 3 \beta_{7} + 2 \beta_{5} - 3 \beta_{3} + \beta_{2}) q^{83} + (\beta_{6} + 2 \beta_1) q^{85} + (\beta_{10} + \beta_{8} + 4) q^{89} + (3 \beta_{11} + 3 \beta_{9} + 3 \beta_{2}) q^{91} + ( - 2 \beta_{7} + \beta_{3}) q^{95} + (3 \beta_{6} - \beta_{4} + 3 \beta_1 - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{5} - 6 q^{13} - 12 q^{25} - 18 q^{29} - 18 q^{41} - 24 q^{49} + 18 q^{61} - 6 q^{65} + 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + 5062 \nu^{5} - 27262 \nu^{4} - 51252 \nu^{3} - 44964 \nu^{2} - 26082 \nu - 96192 ) / 51972 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 839 \nu^{11} + 7708 \nu^{10} - 10784 \nu^{9} + 8976 \nu^{8} - 22648 \nu^{7} - 131336 \nu^{6} - 31862 \nu^{5} + 209952 \nu^{4} + 921960 \nu^{3} + 1654212 \nu^{2} + \cdots + 275832 ) / 103944 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + 11380 \nu^{5} + 57708 \nu^{4} + 98118 \nu^{3} - 132080 \nu^{2} + 49284 \nu + 15984 ) / 51972 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 5172 ) / 1704 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2793 \nu^{11} - 3602 \nu^{10} + 4065 \nu^{9} - 2808 \nu^{8} - 44568 \nu^{7} - 7530 \nu^{6} + 17238 \nu^{5} + 219348 \nu^{4} + 562962 \nu^{3} + 348980 \nu^{2} + \cdots + 133704 ) / 34648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6049 \nu^{11} - 4740 \nu^{10} + 3454 \nu^{9} - 2632 \nu^{8} - 93803 \nu^{7} - 72588 \nu^{6} + 65542 \nu^{5} + 530594 \nu^{4} + 1432764 \nu^{3} + 1177668 \nu^{2} + \cdots + 268416 ) / 51972 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + 12648 \nu^{5} + 151968 \nu^{4} + 420176 \nu^{3} + 400856 \nu^{2} + 282372 \nu + 105120 ) / 12993 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15676 \nu^{11} + 10866 \nu^{10} - 681 \nu^{9} - 11340 \nu^{8} + 267718 \nu^{7} + 188112 \nu^{6} - 234336 \nu^{5} - 1340808 \nu^{4} - 3664246 \nu^{3} + \cdots - 1037028 ) / 103944 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19630 \nu^{11} + 7150 \nu^{10} - 3611 \nu^{9} + 1848 \nu^{8} + 317744 \nu^{7} + 348166 \nu^{6} - 103124 \nu^{5} - 1841148 \nu^{4} - 5346606 \nu^{3} + \cdots - 1369008 ) / 103944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 284 \nu^{11} - 222 \nu^{10} + 159 \nu^{9} - 192 \nu^{8} - 4304 \nu^{7} - 3408 \nu^{6} + 2664 \nu^{5} + 26820 \nu^{4} + 65498 \nu^{3} + 53952 \nu^{2} + 29640 \nu + 4860 ) / 1464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6916 \nu^{11} + 3388 \nu^{10} - 2205 \nu^{9} + 1476 \nu^{8} + 110866 \nu^{7} + 108978 \nu^{6} - 40656 \nu^{5} - 637476 \nu^{4} - 1820914 \nu^{3} - 1805344 \nu^{2} + \cdots - 462528 ) / 34648 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{11} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{11} + 2\beta_{9} + 2\beta_{5} - 9\beta_{3} - 2\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{11} + 8 \beta_{9} + 3 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 12 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} + 4 \beta _1 + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{10} - 2\beta_{8} + 9\beta_{6} + 30\beta_{4} + 9\beta _1 + 28 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 22 \beta_{11} + 3 \beta_{10} - 22 \beta_{9} - 42 \beta_{7} + 19 \beta_{6} - 38 \beta_{5} + 39 \beta_{4} + 21 \beta_{3} - 38 \beta_{2} + 38 \beta _1 + 78 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -56\beta_{11} - 16\beta_{9} - 81\beta_{7} - 40\beta_{5} - 56\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 94 \beta_{11} + 35 \beta_{9} - 12 \beta_{8} - 60 \beta_{7} + 47 \beta_{6} - 35 \beta_{5} + 108 \beta_{4} - 60 \beta_{3} - 59 \beta_{2} - 47 \beta _1 - 120 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 35\beta_{10} - 70\beta_{8} + 249\beta_{6} + 669\beta_{4} - 70 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 164 \beta_{11} + 72 \beta_{10} - 472 \beta_{9} - 72 \beta_{8} - 321 \beta_{7} + 472 \beta_{6} - 308 \beta_{5} + 1140 \beta_{4} + 642 \beta_{3} - 164 \beta_{2} + 236 \beta _1 + 498 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -328\beta_{11} - 1442\beta_{9} - 1908\beta_{7} - 1442\beta_{5} + 1908\beta_{3} - 1114\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1586 \beta_{11} - 393 \beta_{10} - 1586 \beta_{9} - 3342 \beta_{7} - 1193 \beta_{6} - 2386 \beta_{5} - 2949 \beta_{4} + 1671 \beta_{3} - 2386 \beta_{2} - 2386 \beta _1 - 5898 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.180407 0.673288i
2.17840 0.583700i
−0.403293 1.50511i
−1.50511 + 0.403293i
0.583700 + 2.17840i
−0.673288 + 0.180407i
−0.180407 + 0.673288i
2.17840 + 0.583700i
−0.403293 + 1.50511i
−1.50511 0.403293i
0.583700 2.17840i
−0.673288 0.180407i
0 0 0 1.50000 0.866025i 0 −2.17731 + 3.77121i 0 0 0
289.2 0 0 0 1.50000 0.866025i 0 −1.80664 + 3.12920i 0 0 0
289.3 0 0 0 1.50000 0.866025i 0 −0.495361 + 0.857990i 0 0 0
289.4 0 0 0 1.50000 0.866025i 0 0.495361 0.857990i 0 0 0
289.5 0 0 0 1.50000 0.866025i 0 1.80664 3.12920i 0 0 0
289.6 0 0 0 1.50000 0.866025i 0 2.17731 3.77121i 0 0 0
1441.1 0 0 0 1.50000 + 0.866025i 0 −2.17731 3.77121i 0 0 0
1441.2 0 0 0 1.50000 + 0.866025i 0 −1.80664 3.12920i 0 0 0
1441.3 0 0 0 1.50000 + 0.866025i 0 −0.495361 0.857990i 0 0 0
1441.4 0 0 0 1.50000 + 0.866025i 0 0.495361 + 0.857990i 0 0 0
1441.5 0 0 0 1.50000 + 0.866025i 0 1.80664 + 3.12920i 0 0 0
1441.6 0 0 0 1.50000 + 0.866025i 0 2.17731 + 3.77121i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1441.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.r.f 12
3.b odd 2 1 576.2.r.e 12
4.b odd 2 1 inner 1728.2.r.f 12
8.b even 2 1 1728.2.r.e 12
8.d odd 2 1 1728.2.r.e 12
9.c even 3 1 1728.2.r.e 12
9.c even 3 1 5184.2.d.r 12
9.d odd 6 1 576.2.r.f yes 12
9.d odd 6 1 5184.2.d.q 12
12.b even 2 1 576.2.r.e 12
24.f even 2 1 576.2.r.f yes 12
24.h odd 2 1 576.2.r.f yes 12
36.f odd 6 1 1728.2.r.e 12
36.f odd 6 1 5184.2.d.r 12
36.h even 6 1 576.2.r.f yes 12
36.h even 6 1 5184.2.d.q 12
72.j odd 6 1 576.2.r.e 12
72.j odd 6 1 5184.2.d.q 12
72.l even 6 1 576.2.r.e 12
72.l even 6 1 5184.2.d.q 12
72.n even 6 1 inner 1728.2.r.f 12
72.n even 6 1 5184.2.d.r 12
72.p odd 6 1 inner 1728.2.r.f 12
72.p odd 6 1 5184.2.d.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.e 12 3.b odd 2 1
576.2.r.e 12 12.b even 2 1
576.2.r.e 12 72.j odd 6 1
576.2.r.e 12 72.l even 6 1
576.2.r.f yes 12 9.d odd 6 1
576.2.r.f yes 12 24.f even 2 1
576.2.r.f yes 12 24.h odd 2 1
576.2.r.f yes 12 36.h even 6 1
1728.2.r.e 12 8.b even 2 1
1728.2.r.e 12 8.d odd 2 1
1728.2.r.e 12 9.c even 3 1
1728.2.r.e 12 36.f odd 6 1
1728.2.r.f 12 1.a even 1 1 trivial
1728.2.r.f 12 4.b odd 2 1 inner
1728.2.r.f 12 72.n even 6 1 inner
1728.2.r.f 12 72.p odd 6 1 inner
5184.2.d.q 12 9.d odd 6 1
5184.2.d.q 12 36.h even 6 1
5184.2.d.q 12 72.j odd 6 1
5184.2.d.q 12 72.l even 6 1
5184.2.d.r 12 9.c even 3 1
5184.2.d.r 12 36.f odd 6 1
5184.2.d.r 12 72.n even 6 1
5184.2.d.r 12 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 3)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 33 T^{10} + 810 T^{8} + \cdots + 59049 \) Copy content Toggle raw display
$11$ \( T^{12} - 39 T^{10} + 1350 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( (T^{6} + 3 T^{5} - 24 T^{4} - 81 T^{3} + \cdots + 243)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 24 T - 36)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 117 T^{10} + \cdots + 2492305929 \) Copy content Toggle raw display
$29$ \( (T^{6} + 9 T^{5} - 36 T^{4} - 567 T^{3} + \cdots + 93987)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 117 T^{10} + \cdots + 95004009 \) Copy content Toggle raw display
$37$ \( (T^{6} + 60 T^{4} + 1008 T^{2} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 9 T^{5} + 78 T^{4} + 45 T^{3} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 123 T^{10} + \cdots + 47458321 \) Copy content Toggle raw display
$47$ \( T^{12} + 297 T^{10} + \cdots + 514609673769 \) Copy content Toggle raw display
$53$ \( (T^{6} + 180 T^{4} + 1728 T^{2} + \cdots + 432)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 147 T^{10} + 20898 T^{8} + \cdots + 531441 \) Copy content Toggle raw display
$61$ \( (T^{6} - 9 T^{5} - 36 T^{4} + 567 T^{3} + \cdots + 4563)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 231 T^{10} + \cdots + 352275361 \) Copy content Toggle raw display
$71$ \( (T^{6} - 288 T^{4} + 19872 T^{2} + \cdots - 124848)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 84 T - 164)^{4} \) Copy content Toggle raw display
$79$ \( T^{12} + 93 T^{10} + 7758 T^{8} + \cdots + 4782969 \) Copy content Toggle raw display
$83$ \( T^{12} - 171 T^{10} + \cdots + 43046721 \) Copy content Toggle raw display
$89$ \( (T^{3} - 12 T^{2} - 36 T + 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + 3 T^{5} + 222 T^{4} + \cdots + 1408969)^{2} \) Copy content Toggle raw display
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