# Properties

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

# Related objects

## 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
 − 1.92812i 1.92812i 0.277334i − 0.277334i 1.27733i − 1.27733i − 2.92812i 2.92812i
0 −3.85623 0 0.490168i 0 2.64575i 0 5.87054 0
127.2 0 −3.85623 0 0.490168i 0 2.64575i 0 5.87054 0
127.3 0 −0.554669 0 4.57685i 0 2.64575i 0 −8.69234 0
127.4 0 −0.554669 0 4.57685i 0 2.64575i 0 −8.69234 0
127.5 0 2.55467 0 9.86836i 0 2.64575i 0 −2.47367 0
127.6 0 2.55467 0 9.86836i 0 2.64575i 0 −2.47367 0
127.7 0 5.85623 0 5.78167i 0 2.64575i 0 25.2955 0
127.8 0 5.85623 0 5.78167i 0 2.64575i 0 25.2955 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.f 8
4.b odd 2 1 1792.3.g.d 8
8.b even 2 1 1792.3.g.d 8
8.d odd 2 1 inner 1792.3.g.f 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 4.b odd 2 1
1792.3.g.d 8 8.b even 2 1
1792.3.g.f 8 1.a even 1 1 trivial
1792.3.g.f 8 8.d odd 2 1 inner
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
$3$ $$( 1 - 4 T + 16 T^{2} - 60 T^{3} + 158 T^{4} - 540 T^{5} + 1296 T^{6} - 2916 T^{7} + 6561 T^{8} )^{2}$$
$5$ $$1 - 48 T^{2} + 732 T^{4} + 16432 T^{6} - 1001466 T^{8} + 10270000 T^{10} + 285937500 T^{12} - 11718750000 T^{14} + 152587890625 T^{16}$$
$7$ $$( 1 + 7 T^{2} )^{4}$$
$11$ $$( 1 - 16 T + 276 T^{2} - 1840 T^{3} + 27782 T^{4} - 222640 T^{5} + 4040916 T^{6} - 28344976 T^{7} + 214358881 T^{8} )^{2}$$
$13$ $$1 - 1136 T^{2} + 596060 T^{4} - 187756944 T^{6} + 38751742342 T^{8} - 5362526077584 T^{10} + 486224453559260 T^{12} - 26466624699138416 T^{14} + 665416609183179841 T^{16}$$
$17$ $$( 1 + 8 T + 428 T^{2} - 1416 T^{3} + 75814 T^{4} - 409224 T^{5} + 35746988 T^{6} + 193100552 T^{7} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 44 T + 1936 T^{2} - 47828 T^{3} + 1128094 T^{4} - 17265908 T^{5} + 252301456 T^{6} - 2070018764 T^{7} + 16983563041 T^{8} )^{2}$$
$23$ $$1 - 1928 T^{2} + 2366876 T^{4} - 1917103032 T^{6} + 1186521008582 T^{8} - 536484029577912 T^{10} + 185352391597952156 T^{12} - 42251395904935178888 T^{14} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$1 - 5720 T^{2} + 14827292 T^{4} - 22985018856 T^{6} + 23490964042630 T^{8} - 16256867121490536 T^{10} + 7417299636925331612 T^{12} -$$$$20\!\cdots\!20$$$$T^{14} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$1 - 3624 T^{2} + 5953500 T^{4} - 5905968152 T^{6} + 5172694104774 T^{8} - 5454285613703192 T^{10} + 5077686791404993500 T^{12} -$$$$28\!\cdots\!64$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 + 360 T^{2} + 3032604 T^{4} + 2655359320 T^{6} + 6705554738694 T^{8} + 4976570878530520 T^{10} + 10651959241878640284 T^{12} +$$$$23\!\cdots\!60$$$$T^{14} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$( 1 + 72 T + 4364 T^{2} + 180408 T^{3} + 8880422 T^{4} + 303265848 T^{5} + 12331621004 T^{6} + 342007505352 T^{7} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 + 112 T + 8468 T^{2} + 425040 T^{3} + 19578758 T^{4} + 785898960 T^{5} + 28950406868 T^{6} + 707992661488 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 - 6696 T^{2} + 34041564 T^{4} - 108976927256 T^{6} + 286725887177670 T^{8} - 531772641369485336 T^{10} +$$$$81\!\cdots\!04$$$$T^{12} -$$$$77\!\cdots\!36$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 - 11640 T^{2} + 63508956 T^{4} - 235884348872 T^{6} + 714590361466758 T^{8} - 1861240972971887432 T^{10} +$$$$39\!\cdots\!16$$$$T^{12} -$$$$57\!\cdots\!40$$$$T^{14} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$( 1 - 116 T + 17424 T^{2} - 1222412 T^{3} + 96865118 T^{4} - 4255216172 T^{5} + 211132898064 T^{6} - 4892941902356 T^{7} + 146830437604321 T^{8} )^{2}$$
$61$ $$1 - 16240 T^{2} + 130126300 T^{4} - 693430250128 T^{6} + 2856274871188870 T^{8} - 9601124987862517648 T^{10} +$$$$24\!\cdots\!00$$$$T^{12} -$$$$43\!\cdots\!40$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$( 1 + 184 T + 25028 T^{2} + 2353896 T^{3} + 180133414 T^{4} + 10566639144 T^{5} + 504342256388 T^{6} + 16644342319096 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$1 - 21000 T^{2} + 247457180 T^{4} - 1991232124728 T^{6} + 11615880585488070 T^{8} - 50600555550540147768 T^{10} +$$$$15\!\cdots\!80$$$$T^{12} -$$$$34\!\cdots\!00$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16}$$
$73$ $$( 1 + 136 T + 21660 T^{2} + 1811512 T^{3} + 170528390 T^{4} + 9653547448 T^{5} + 615105900060 T^{6} + 20581454775304 T^{7} + 806460091894081 T^{8} )^{2}$$
$79$ $$1 - 26248 T^{2} + 289540636 T^{4} - 1851543245752 T^{6} + 10296312398061382 T^{8} - 72117759397043305912 T^{10} +$$$$43\!\cdots\!96$$$$T^{12} -$$$$15\!\cdots\!68$$$$T^{14} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$( 1 + 212 T + 32112 T^{2} + 3651372 T^{3} + 342998878 T^{4} + 25154301708 T^{5} + 1523981603952 T^{6} + 69311359154228 T^{7} + 2252292232139041 T^{8} )^{2}$$
$89$ $$( 1 - 40 T + 25916 T^{2} - 837912 T^{3} + 285914566 T^{4} - 6637100952 T^{5} + 1626027917756 T^{6} - 19879251638440 T^{7} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 264 T + 52364 T^{2} - 7260024 T^{3} + 808802534 T^{4} - 68309565816 T^{5} + 4635747270284 T^{6} - 219904609301256 T^{7} + 7837433594376961 T^{8} )^{2}$$