# 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 $2$

# 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 Defining polynomial: $$x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4$$ 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: 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.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 $$S_{3}^{\mathrm{new}}(1792, [\chi])$$.