# 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

# Learn more about

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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