Properties

Label 1792.3.g.d
Level 1792
Weight 3
Character orbit 1792.g
Analytic conductor 48.828
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1792.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.8284633734\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1997017344.2
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 224)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( \beta_{5} - \beta_{6} ) q^{5} + \beta_{5} q^{7} + ( 5 - 2 \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( \beta_{5} - \beta_{6} ) q^{5} + \beta_{5} q^{7} + ( 5 - 2 \beta_{1} - 2 \beta_{2} ) q^{9} + ( -4 + \beta_{2} - \beta_{7} ) q^{11} + ( -2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} + ( -\beta_{3} + \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{15} + ( -2 + 3 \beta_{2} - \beta_{7} ) q^{17} + ( -11 + 3 \beta_{1} ) q^{19} + ( -\beta_{3} - \beta_{5} + \beta_{6} ) q^{21} + ( 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{23} + ( -13 + 6 \beta_{1} - 4 \beta_{2} - 2 \beta_{7} ) q^{25} + ( -22 + 6 \beta_{1} + 6 \beta_{2} + 2 \beta_{7} ) q^{27} + ( 5 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{29} + ( 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{31} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{33} + ( -7 + 3 \beta_{1} - \beta_{7} ) q^{35} + ( 6 \beta_{3} - 11 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} ) q^{37} + ( -3 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{39} + ( -18 - 12 \beta_{1} - 3 \beta_{2} + \beta_{7} ) q^{41} + ( 28 - 5 \beta_{2} - 3 \beta_{7} ) q^{43} + ( 8 \beta_{3} - 16 \beta_{4} + 15 \beta_{5} + \beta_{6} ) q^{45} + ( 3 \beta_{3} + 5 \beta_{4} - 8 \beta_{6} ) q^{47} -7 q^{49} + ( -2 - 14 \beta_{1} + 2 \beta_{2} - 2 \beta_{7} ) q^{51} + ( 6 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{53} + ( -24 \beta_{4} - 20 \beta_{5} + 4 \beta_{6} ) q^{55} + ( 50 - 14 \beta_{1} - 6 \beta_{2} ) q^{57} + ( -29 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{7} ) q^{59} + ( 12 \beta_{4} + 19 \beta_{5} - 3 \beta_{6} ) q^{61} + ( 2 \beta_{3} - 14 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} ) q^{63} + ( -38 + 6 \beta_{1} - 2 \beta_{2} ) q^{65} + ( 46 + 6 \beta_{1} + 9 \beta_{2} - \beta_{7} ) q^{67} + ( -4 \beta_{3} + 36 \beta_{4} - 16 \beta_{5} ) q^{69} + ( 8 \beta_{4} - 4 \beta_{5} - 12 \beta_{6} ) q^{71} + ( -34 + 10 \beta_{2} - 2 \beta_{7} ) q^{73} + ( 83 - 3 \beta_{1} + 2 \beta_{2} + 6 \beta_{7} ) q^{75} + ( 3 \beta_{3} + 7 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{77} + ( -6 \beta_{3} - 30 \beta_{4} + 8 \beta_{5} ) q^{79} + ( 63 - 34 \beta_{1} - 10 \beta_{2} - 8 \beta_{7} ) q^{81} + ( 53 + 3 \beta_{1} + 14 \beta_{2} + 2 \beta_{7} ) q^{83} + ( -6 \beta_{3} - 10 \beta_{4} - 18 \beta_{5} - 6 \beta_{6} ) q^{85} + ( 3 \beta_{3} - 3 \beta_{4} - 12 \beta_{5} + 4 \beta_{6} ) q^{87} + ( 10 + 12 \beta_{1} + 6 \beta_{2} - 2 \beta_{7} ) q^{89} + ( -7 + 3 \beta_{1} + 2 \beta_{2} - \beta_{7} ) q^{91} + ( -2 \beta_{3} + 38 \beta_{4} - 8 \beta_{6} ) q^{93} + ( -3 \beta_{3} + 3 \beta_{4} - 26 \beta_{5} + 14 \beta_{6} ) q^{95} + ( 66 + 12 \beta_{1} - 3 \beta_{2} + 5 \beta_{7} ) q^{97} + ( -68 - 8 \beta_{1} + 3 \beta_{2} + 5 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} + 40q^{9} + O(q^{10}) \) \( 8q - 8q^{3} + 40q^{9} - 32q^{11} - 16q^{17} - 88q^{19} - 104q^{25} - 176q^{27} - 56q^{35} - 144q^{41} + 224q^{43} - 56q^{49} - 16q^{51} + 400q^{57} - 232q^{59} - 304q^{65} + 368q^{67} - 272q^{73} + 664q^{75} + 504q^{81} + 424q^{83} + 80q^{89} - 56q^{91} + 528q^{97} - 544q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 11 \nu^{4} + 24 \nu^{2} + 8 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} - 11 \nu^{4} - 16 \nu^{2} + 20 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 13 \nu^{5} + 42 \nu^{3} + 48 \nu \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 13 \nu^{5} + 42 \nu^{3} + 32 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 14 \nu^{5} + 51 \nu^{3} + 42 \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 16 \nu^{5} + 77 \nu^{3} + 110 \nu \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{6} - 41 \nu^{4} - 136 \nu^{2} - 68 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + \beta_{3}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{6} - 3 \beta_{5} + 5 \beta_{4} - 3 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{7} - 8 \beta_{2} - 11 \beta_{1} + 45\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-9 \beta_{6} + 35 \beta_{5} - 48 \beta_{4} + 22 \beta_{3}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(11 \beta_{7} + 64 \beta_{2} + 105 \beta_{1} - 343\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(75 \beta_{6} - 329 \beta_{5} + 438 \beta_{4} - 176 \beta_{3}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
2.92812i
2.92812i
1.27733i
1.27733i
0.277334i
0.277334i
1.92812i
1.92812i
0 −5.85623 0 5.78167i 0 2.64575i 0 25.2955 0
127.2 0 −5.85623 0 5.78167i 0 2.64575i 0 25.2955 0
127.3 0 −2.55467 0 9.86836i 0 2.64575i 0 −2.47367 0
127.4 0 −2.55467 0 9.86836i 0 2.64575i 0 −2.47367 0
127.5 0 0.554669 0 4.57685i 0 2.64575i 0 −8.69234 0
127.6 0 0.554669 0 4.57685i 0 2.64575i 0 −8.69234 0
127.7 0 3.85623 0 0.490168i 0 2.64575i 0 5.87054 0
127.8 0 3.85623 0 0.490168i 0 2.64575i 0 5.87054 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1792.3.g.d 8
4.b odd 2 1 1792.3.g.f 8
8.b even 2 1 1792.3.g.f 8
8.d odd 2 1 inner 1792.3.g.d 8
16.e even 4 1 224.3.d.b 8
16.e even 4 1 448.3.d.e 8
16.f odd 4 1 224.3.d.b 8
16.f odd 4 1 448.3.d.e 8
48.i odd 4 1 2016.3.m.c 8
48.k even 4 1 2016.3.m.c 8
112.j even 4 1 1568.3.d.n 8
112.l odd 4 1 1568.3.d.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.d.b 8 16.e even 4 1
224.3.d.b 8 16.f odd 4 1
448.3.d.e 8 16.e even 4 1
448.3.d.e 8 16.f odd 4 1
1568.3.d.n 8 112.j even 4 1
1568.3.d.n 8 112.l odd 4 1
1792.3.g.d 8 1.a even 1 1 trivial
1792.3.g.d 8 8.d odd 2 1 inner
1792.3.g.f 8 4.b odd 2 1
1792.3.g.f 8 8.b even 2 1
2016.3.m.c 8 48.i odd 4 1
2016.3.m.c 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4 T_{3}^{3} - 20 T_{3}^{2} - 48 T_{3} + 32 \) acting on \(S_{3}^{\mathrm{new}}(1792, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 4 T + 16 T^{2} + 60 T^{3} + 158 T^{4} + 540 T^{5} + 1296 T^{6} + 2916 T^{7} + 6561 T^{8} )^{2} \)
$5$ \( 1 - 48 T^{2} + 732 T^{4} + 16432 T^{6} - 1001466 T^{8} + 10270000 T^{10} + 285937500 T^{12} - 11718750000 T^{14} + 152587890625 T^{16} \)
$7$ \( ( 1 + 7 T^{2} )^{4} \)
$11$ \( ( 1 + 16 T + 276 T^{2} + 1840 T^{3} + 27782 T^{4} + 222640 T^{5} + 4040916 T^{6} + 28344976 T^{7} + 214358881 T^{8} )^{2} \)
$13$ \( 1 - 1136 T^{2} + 596060 T^{4} - 187756944 T^{6} + 38751742342 T^{8} - 5362526077584 T^{10} + 486224453559260 T^{12} - 26466624699138416 T^{14} + 665416609183179841 T^{16} \)
$17$ \( ( 1 + 8 T + 428 T^{2} - 1416 T^{3} + 75814 T^{4} - 409224 T^{5} + 35746988 T^{6} + 193100552 T^{7} + 6975757441 T^{8} )^{2} \)
$19$ \( ( 1 + 44 T + 1936 T^{2} + 47828 T^{3} + 1128094 T^{4} + 17265908 T^{5} + 252301456 T^{6} + 2070018764 T^{7} + 16983563041 T^{8} )^{2} \)
$23$ \( 1 - 1928 T^{2} + 2366876 T^{4} - 1917103032 T^{6} + 1186521008582 T^{8} - 536484029577912 T^{10} + 185352391597952156 T^{12} - 42251395904935178888 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( 1 - 5720 T^{2} + 14827292 T^{4} - 22985018856 T^{6} + 23490964042630 T^{8} - 16256867121490536 T^{10} + 7417299636925331612 T^{12} - \)\(20\!\cdots\!20\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} \)
$31$ \( 1 - 3624 T^{2} + 5953500 T^{4} - 5905968152 T^{6} + 5172694104774 T^{8} - 5454285613703192 T^{10} + 5077686791404993500 T^{12} - \)\(28\!\cdots\!64\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( 1 + 360 T^{2} + 3032604 T^{4} + 2655359320 T^{6} + 6705554738694 T^{8} + 4976570878530520 T^{10} + 10651959241878640284 T^{12} + \)\(23\!\cdots\!60\)\( T^{14} + \)\(12\!\cdots\!41\)\( T^{16} \)
$41$ \( ( 1 + 72 T + 4364 T^{2} + 180408 T^{3} + 8880422 T^{4} + 303265848 T^{5} + 12331621004 T^{6} + 342007505352 T^{7} + 7984925229121 T^{8} )^{2} \)
$43$ \( ( 1 - 112 T + 8468 T^{2} - 425040 T^{3} + 19578758 T^{4} - 785898960 T^{5} + 28950406868 T^{6} - 707992661488 T^{7} + 11688200277601 T^{8} )^{2} \)
$47$ \( 1 - 6696 T^{2} + 34041564 T^{4} - 108976927256 T^{6} + 286725887177670 T^{8} - 531772641369485336 T^{10} + \)\(81\!\cdots\!04\)\( T^{12} - \)\(77\!\cdots\!36\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 - 11640 T^{2} + 63508956 T^{4} - 235884348872 T^{6} + 714590361466758 T^{8} - 1861240972971887432 T^{10} + \)\(39\!\cdots\!16\)\( T^{12} - \)\(57\!\cdots\!40\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} \)
$59$ \( ( 1 + 116 T + 17424 T^{2} + 1222412 T^{3} + 96865118 T^{4} + 4255216172 T^{5} + 211132898064 T^{6} + 4892941902356 T^{7} + 146830437604321 T^{8} )^{2} \)
$61$ \( 1 - 16240 T^{2} + 130126300 T^{4} - 693430250128 T^{6} + 2856274871188870 T^{8} - 9601124987862517648 T^{10} + \)\(24\!\cdots\!00\)\( T^{12} - \)\(43\!\cdots\!40\)\( T^{14} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( ( 1 - 184 T + 25028 T^{2} - 2353896 T^{3} + 180133414 T^{4} - 10566639144 T^{5} + 504342256388 T^{6} - 16644342319096 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( 1 - 21000 T^{2} + 247457180 T^{4} - 1991232124728 T^{6} + 11615880585488070 T^{8} - 50600555550540147768 T^{10} + \)\(15\!\cdots\!80\)\( T^{12} - \)\(34\!\cdots\!00\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} \)
$73$ \( ( 1 + 136 T + 21660 T^{2} + 1811512 T^{3} + 170528390 T^{4} + 9653547448 T^{5} + 615105900060 T^{6} + 20581454775304 T^{7} + 806460091894081 T^{8} )^{2} \)
$79$ \( 1 - 26248 T^{2} + 289540636 T^{4} - 1851543245752 T^{6} + 10296312398061382 T^{8} - 72117759397043305912 T^{10} + \)\(43\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!68\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \)
$83$ \( ( 1 - 212 T + 32112 T^{2} - 3651372 T^{3} + 342998878 T^{4} - 25154301708 T^{5} + 1523981603952 T^{6} - 69311359154228 T^{7} + 2252292232139041 T^{8} )^{2} \)
$89$ \( ( 1 - 40 T + 25916 T^{2} - 837912 T^{3} + 285914566 T^{4} - 6637100952 T^{5} + 1626027917756 T^{6} - 19879251638440 T^{7} + 3936588805702081 T^{8} )^{2} \)
$97$ \( ( 1 - 264 T + 52364 T^{2} - 7260024 T^{3} + 808802534 T^{4} - 68309565816 T^{5} + 4635747270284 T^{6} - 219904609301256 T^{7} + 7837433594376961 T^{8} )^{2} \)
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