Properties

Label 1792.3.d.g
Level 1792
Weight 3
Character orbit 1792.d
Analytic conductor 48.828
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(48.8284633734\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{6} q^{5} -\beta_{1} q^{7} + ( 3 - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + \beta_{6} q^{5} -\beta_{1} q^{7} + ( 3 - \beta_{4} ) q^{9} + ( -4 \beta_{2} + 2 \beta_{3} ) q^{11} + \beta_{6} q^{13} + 2 \beta_{1} q^{15} + ( 18 + \beta_{4} ) q^{17} + ( 10 \beta_{2} - 9 \beta_{3} ) q^{19} + \beta_{5} q^{21} + ( -2 \beta_{1} - 2 \beta_{7} ) q^{23} + ( 17 - 7 \beta_{4} ) q^{25} + ( 2 \beta_{2} + 6 \beta_{3} ) q^{27} + ( 3 \beta_{5} - 3 \beta_{6} ) q^{29} + ( -12 \beta_{1} - \beta_{7} ) q^{31} + ( -4 - 2 \beta_{4} ) q^{33} -7 \beta_{2} q^{35} + ( -7 \beta_{5} + 3 \beta_{6} ) q^{37} + 2 \beta_{1} q^{39} + ( 10 + 3 \beta_{4} ) q^{41} + ( 6 \beta_{2} + 4 \beta_{3} ) q^{43} + ( -2 \beta_{5} + 9 \beta_{6} ) q^{45} + ( -8 \beta_{1} - \beta_{7} ) q^{47} -7 q^{49} + ( -2 \beta_{2} + 24 \beta_{3} ) q^{51} + ( 11 \beta_{5} - \beta_{6} ) q^{53} + ( -20 \beta_{1} + 4 \beta_{7} ) q^{55} + ( 34 + 9 \beta_{4} ) q^{57} + ( -6 \beta_{2} - 17 \beta_{3} ) q^{59} + ( 4 \beta_{5} + \beta_{6} ) q^{61} + ( -3 \beta_{1} + \beta_{7} ) q^{63} + ( 42 - 7 \beta_{4} ) q^{65} + ( -22 \beta_{2} - 6 \beta_{3} ) q^{67} + ( 14 \beta_{5} - 4 \beta_{6} ) q^{69} + ( -16 \beta_{1} - 4 \beta_{7} ) q^{71} + ( -58 - 2 \beta_{4} ) q^{73} + ( 14 \beta_{2} - 25 \beta_{3} ) q^{75} + ( 2 \beta_{5} - 4 \beta_{6} ) q^{77} + ( -8 \beta_{1} + 4 \beta_{7} ) q^{79} + ( -13 - 15 \beta_{4} ) q^{81} + ( 12 \beta_{2} - \beta_{3} ) q^{83} + ( 2 \beta_{5} + 12 \beta_{6} ) q^{85} + ( 12 \beta_{1} + 3 \beta_{7} ) q^{87} + ( -78 + 6 \beta_{4} ) q^{89} -7 \beta_{2} q^{91} + ( 18 \beta_{5} - 2 \beta_{6} ) q^{93} + ( 42 \beta_{1} - 10 \beta_{7} ) q^{95} + ( -34 + 23 \beta_{4} ) q^{97} + ( -32 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} + 144q^{17} + 136q^{25} - 32q^{33} + 80q^{41} - 56q^{49} + 272q^{57} + 336q^{65} - 464q^{73} - 104q^{81} - 624q^{89} - 272q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{4} + 1 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 4 \nu^{6} - 4 \nu^{5} - 7 \nu^{3} + 20 \nu^{2} + 4 \nu \)\()/24\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} + 4 \nu^{5} + 7 \nu^{3} + 20 \nu^{2} - 4 \nu \)\()/24\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} - 12 \nu^{6} + 4 \nu^{5} - 13 \nu^{3} + 36 \nu^{2} + 44 \nu \)\()/24\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} - 12 \nu^{6} - 4 \nu^{5} + 13 \nu^{3} + 36 \nu^{2} - 44 \nu \)\()/24\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2}\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{5} + 3 \beta_{3} + 3 \beta_{2}\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + 10 \beta_{3} - 10 \beta_{2}\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 11 \beta_{4} + 22 \beta_{3} - 22 \beta_{2}\)\()/16\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{6} - 5 \beta_{5} + 9 \beta_{3} + 9 \beta_{2}\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(7 \beta_{7} + 14 \beta_{6} - 14 \beta_{5} - 13 \beta_{4} - 26 \beta_{3} + 26 \beta_{2}\)\()/16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1023.1
−0.581861 + 1.28897i
1.28897 0.581861i
0.581861 1.28897i
−1.28897 + 0.581861i
0.581861 + 1.28897i
−1.28897 0.581861i
−0.581861 1.28897i
1.28897 + 0.581861i
0 3.41421i 0 −1.54985 0 2.64575i 0 −2.65685 0
1023.2 0 3.41421i 0 1.54985 0 2.64575i 0 −2.65685 0
1023.3 0 0.585786i 0 −9.03316 0 2.64575i 0 8.65685 0
1023.4 0 0.585786i 0 9.03316 0 2.64575i 0 8.65685 0
1023.5 0 0.585786i 0 −9.03316 0 2.64575i 0 8.65685 0
1023.6 0 0.585786i 0 9.03316 0 2.64575i 0 8.65685 0
1023.7 0 3.41421i 0 −1.54985 0 2.64575i 0 −2.65685 0
1023.8 0 3.41421i 0 1.54985 0 2.64575i 0 −2.65685 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1023.8
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 1792.3.d.g 8
4.b odd 2 1 inner 1792.3.d.g 8
8.b even 2 1 inner 1792.3.d.g 8
8.d odd 2 1 inner 1792.3.d.g 8
16.e even 4 1 56.3.g.a 4
16.e even 4 1 224.3.g.a 4
16.f odd 4 1 56.3.g.a 4
16.f odd 4 1 224.3.g.a 4
48.i odd 4 1 504.3.g.a 4
48.i odd 4 1 2016.3.g.a 4
48.k even 4 1 504.3.g.a 4
48.k even 4 1 2016.3.g.a 4
112.j even 4 1 392.3.g.h 4
112.j even 4 1 1568.3.g.h 4
112.l odd 4 1 392.3.g.h 4
112.l odd 4 1 1568.3.g.h 4
112.u odd 12 2 392.3.k.i 8
112.v even 12 2 392.3.k.j 8
112.w even 12 2 392.3.k.i 8
112.x odd 12 2 392.3.k.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.a 4 16.e even 4 1
56.3.g.a 4 16.f odd 4 1
224.3.g.a 4 16.e even 4 1
224.3.g.a 4 16.f odd 4 1
392.3.g.h 4 112.j even 4 1
392.3.g.h 4 112.l odd 4 1
392.3.k.i 8 112.u odd 12 2
392.3.k.i 8 112.w even 12 2
392.3.k.j 8 112.v even 12 2
392.3.k.j 8 112.x odd 12 2
504.3.g.a 4 48.i odd 4 1
504.3.g.a 4 48.k even 4 1
1568.3.g.h 4 112.j even 4 1
1568.3.g.h 4 112.l odd 4 1
1792.3.d.g 8 1.a even 1 1 trivial
1792.3.d.g 8 4.b odd 2 1 inner
1792.3.d.g 8 8.b even 2 1 inner
1792.3.d.g 8 8.d odd 2 1 inner
2016.3.g.a 4 48.i odd 4 1
2016.3.g.a 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{4} + 12 T_{3}^{2} + 4 \)
\( T_{5}^{4} - 84 T_{5}^{2} + 196 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 24 T^{2} + 274 T^{4} - 1944 T^{6} + 6561 T^{8} )^{2} \)
$5$ \( ( 1 + 16 T^{2} - 254 T^{4} + 10000 T^{6} + 390625 T^{8} )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{4} \)
$11$ \( ( 1 - 308 T^{2} + 48390 T^{4} - 4509428 T^{6} + 214358881 T^{8} )^{2} \)
$13$ \( ( 1 + 592 T^{2} + 143170 T^{4} + 16908112 T^{6} + 815730721 T^{8} )^{2} \)
$17$ \( ( 1 - 36 T + 870 T^{2} - 10404 T^{3} + 83521 T^{4} )^{4} \)
$19$ \( ( 1 + 8 T^{2} + 249106 T^{4} + 1042568 T^{6} + 16983563041 T^{8} )^{2} \)
$23$ \( ( 1 - 268 T^{2} + 477286 T^{4} - 74997388 T^{6} + 78310985281 T^{8} )^{2} \)
$29$ \( ( 1 + 1178 T^{2} + 707281 T^{4} )^{4} \)
$31$ \( ( 1 - 1380 T^{2} + 1419974 T^{4} - 1274458980 T^{6} + 852891037441 T^{8} )^{2} \)
$37$ \( ( 1 + 1780 T^{2} + 2031622 T^{4} + 3336006580 T^{6} + 3512479453921 T^{8} )^{2} \)
$41$ \( ( 1 - 20 T + 3174 T^{2} - 33620 T^{3} + 2825761 T^{4} )^{4} \)
$43$ \( ( 1 - 6580 T^{2} + 17648902 T^{4} - 22495710580 T^{6} + 11688200277601 T^{8} )^{2} \)
$47$ \( ( 1 - 7492 T^{2} + 23390470 T^{4} - 36558570052 T^{6} + 23811286661761 T^{8} )^{2} \)
$53$ \( ( 1 + 1604 T^{2} - 6155034 T^{4} + 12656331524 T^{6} + 62259690411361 T^{8} )^{2} \)
$59$ \( ( 1 - 9208 T^{2} + 43383250 T^{4} - 111576660088 T^{6} + 146830437604321 T^{8} )^{2} \)
$61$ \( ( 1 + 13232 T^{2} + 71110338 T^{4} + 183208168112 T^{6} + 191707312997281 T^{8} )^{2} \)
$67$ \( ( 1 - 10660 T^{2} + 62288614 T^{4} - 214810949860 T^{6} + 406067677556641 T^{8} )^{2} \)
$71$ \( ( 1 - 9412 T^{2} + 47279686 T^{4} - 239174741572 T^{6} + 645753531245761 T^{8} )^{2} \)
$73$ \( ( 1 + 116 T + 13894 T^{2} + 618164 T^{3} + 28398241 T^{4} )^{4} \)
$79$ \( ( 1 - 16900 T^{2} + 142880134 T^{4} - 658256368900 T^{6} + 1517108809906561 T^{8} )^{2} \)
$83$ \( ( 1 - 25912 T^{2} + 262120210 T^{4} - 1229740013752 T^{6} + 2252292232139041 T^{8} )^{2} \)
$89$ \( ( 1 + 156 T + 20774 T^{2} + 1235676 T^{3} + 62742241 T^{4} )^{4} \)
$97$ \( ( 1 + 68 T + 3046 T^{2} + 639812 T^{3} + 88529281 T^{4} )^{4} \)
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