# Properties

 Label 1792.2.e.h Level $1792$ Weight $2$ Character orbit 1792.e Analytic conductor $14.309$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 896) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( \beta_{5} + \beta_{6} + \beta_{7} ) q^{5} + ( \beta_{4} + \beta_{7} ) q^{7} -\beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( \beta_{5} + \beta_{6} + \beta_{7} ) q^{5} + ( \beta_{4} + \beta_{7} ) q^{7} -\beta_{2} q^{9} + ( -1 + \beta_{2} ) q^{11} + \beta_{6} q^{13} -\beta_{4} q^{15} + ( -\beta_{1} + \beta_{3} ) q^{17} + ( \beta_{1} + 2 \beta_{3} ) q^{19} + ( -\beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{21} + ( \beta_{4} - \beta_{5} + \beta_{7} ) q^{23} + ( 2 - 3 \beta_{2} ) q^{25} + ( -\beta_{1} + \beta_{3} ) q^{27} + ( \beta_{5} - \beta_{7} ) q^{29} + ( 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{31} + ( \beta_{1} + \beta_{3} ) q^{33} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( -4 \beta_{4} + \beta_{5} - \beta_{7} ) q^{37} + ( \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{39} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{41} + ( 5 + 3 \beta_{2} ) q^{43} + ( 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{45} + ( -\beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{47} + ( -3 + \beta_{1} - 3 \beta_{3} ) q^{49} -4 q^{51} + ( -2 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} ) q^{53} + ( -3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{55} + ( -5 - 3 \beta_{2} ) q^{57} + 3 \beta_{1} q^{59} + ( -\beta_{5} - 7 \beta_{6} - \beta_{7} ) q^{61} + ( 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{63} + ( 1 - \beta_{2} ) q^{65} + ( 3 - 3 \beta_{2} ) q^{67} + ( \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{69} + ( 5 \beta_{5} - 5 \beta_{7} ) q^{71} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{73} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{75} + ( -\beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{77} + ( -2 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{79} + ( -4 - 3 \beta_{2} ) q^{81} + ( -4 \beta_{1} - 3 \beta_{3} ) q^{83} + ( -4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{85} + 2 \beta_{6} q^{87} + ( \beta_{1} - 7 \beta_{3} ) q^{89} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{91} + ( -4 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{93} + ( \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{95} + ( \beta_{1} - \beta_{3} ) q^{97} + ( -5 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{11} + 16q^{25} + 24q^{35} + 40q^{43} - 24q^{49} - 32q^{51} - 40q^{57} + 8q^{65} + 24q^{67} - 32q^{81} - 8q^{91} - 40q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 5 \nu^{3} + 2 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{4} + 7$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{7} + \nu^{5} + 21 \nu^{3} + 8 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{6} - 10 \nu^{2}$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{7} + \nu^{6} + \nu^{5} - 13 \nu^{3} + 8 \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} - \nu^{5} + 21 \nu^{3} - 8 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{7} - \nu^{6} + \nu^{5} - 13 \nu^{3} - 8 \nu^{2} + 5 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} + \beta_{4}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + \beta_{3} - 3 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{2} - 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{7} + 5 \beta_{6} + 4 \beta_{5} - \beta_{3} + 4 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{7} - 5 \beta_{5} - 8 \beta_{4}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-21 \beta_{7} - 26 \beta_{6} - 21 \beta_{5} - 5 \beta_{3} + 21 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
895.1
 −0.437016 − 0.437016i 0.437016 − 0.437016i 1.14412 − 1.14412i −1.14412 − 1.14412i 1.14412 + 1.14412i −1.14412 + 1.14412i −0.437016 + 0.437016i 0.437016 + 0.437016i
0 2.28825i 0 −0.540182 0 −1.41421 2.23607i 0 −2.23607 0
895.2 0 2.28825i 0 0.540182 0 1.41421 + 2.23607i 0 −2.23607 0
895.3 0 0.874032i 0 −3.70246 0 −1.41421 2.23607i 0 2.23607 0
895.4 0 0.874032i 0 3.70246 0 1.41421 + 2.23607i 0 2.23607 0
895.5 0 0.874032i 0 −3.70246 0 −1.41421 + 2.23607i 0 2.23607 0
895.6 0 0.874032i 0 3.70246 0 1.41421 2.23607i 0 2.23607 0
895.7 0 2.28825i 0 −0.540182 0 −1.41421 + 2.23607i 0 −2.23607 0
895.8 0 2.28825i 0 0.540182 0 1.41421 2.23607i 0 −2.23607 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 895.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.e.h 8
4.b odd 2 1 1792.2.e.i 8
7.b odd 2 1 inner 1792.2.e.h 8
8.b even 2 1 1792.2.e.i 8
8.d odd 2 1 inner 1792.2.e.h 8
16.e even 4 1 896.2.f.a 8
16.e even 4 1 896.2.f.b yes 8
16.f odd 4 1 896.2.f.a 8
16.f odd 4 1 896.2.f.b yes 8
28.d even 2 1 1792.2.e.i 8
56.e even 2 1 inner 1792.2.e.h 8
56.h odd 2 1 1792.2.e.i 8
112.j even 4 1 896.2.f.a 8
112.j even 4 1 896.2.f.b yes 8
112.l odd 4 1 896.2.f.a 8
112.l odd 4 1 896.2.f.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.f.a 8 16.e even 4 1
896.2.f.a 8 16.f odd 4 1
896.2.f.a 8 112.j even 4 1
896.2.f.a 8 112.l odd 4 1
896.2.f.b yes 8 16.e even 4 1
896.2.f.b yes 8 16.f odd 4 1
896.2.f.b yes 8 112.j even 4 1
896.2.f.b yes 8 112.l odd 4 1
1792.2.e.h 8 1.a even 1 1 trivial
1792.2.e.h 8 7.b odd 2 1 inner
1792.2.e.h 8 8.d odd 2 1 inner
1792.2.e.h 8 56.e even 2 1 inner
1792.2.e.i 8 4.b odd 2 1
1792.2.e.i 8 8.b even 2 1
1792.2.e.i 8 28.d even 2 1
1792.2.e.i 8 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1792, [\chi])$$:

 $$T_{3}^{4} + 6 T_{3}^{2} + 4$$ $$T_{11}^{2} + 2 T_{11} - 4$$ $$T_{31}^{4} - 120 T_{31}^{2} + 1600$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 4 + 6 T^{2} + T^{4} )^{2}$$
$5$ $$( 4 - 14 T^{2} + T^{4} )^{2}$$
$7$ $$( 49 + 6 T^{2} + T^{4} )^{2}$$
$11$ $$( -4 + 2 T + T^{2} )^{4}$$
$13$ $$( 4 - 6 T^{2} + T^{4} )^{2}$$
$17$ $$( 64 + 24 T^{2} + T^{4} )^{2}$$
$19$ $$( 100 + 30 T^{2} + T^{4} )^{2}$$
$23$ $$( 16 + 12 T^{2} + T^{4} )^{2}$$
$29$ $$( 4 + T^{2} )^{4}$$
$31$ $$( 1600 - 120 T^{2} + T^{4} )^{2}$$
$37$ $$( 5776 + 168 T^{2} + T^{4} )^{2}$$
$41$ $$( 1600 + 120 T^{2} + T^{4} )^{2}$$
$43$ $$( -20 - 10 T + T^{2} )^{4}$$
$47$ $$( 7744 - 184 T^{2} + T^{4} )^{2}$$
$53$ $$( 16 + 72 T^{2} + T^{4} )^{2}$$
$59$ $$( 324 + 126 T^{2} + T^{4} )^{2}$$
$61$ $$( 13924 - 254 T^{2} + T^{4} )^{2}$$
$67$ $$( -36 - 6 T + T^{2} )^{4}$$
$71$ $$( 100 + T^{2} )^{4}$$
$73$ $$( 200 + T^{2} )^{4}$$
$79$ $$( 1936 + 168 T^{2} + T^{4} )^{2}$$
$83$ $$( 12100 + 230 T^{2} + T^{4} )^{2}$$
$89$ $$( 23104 + 336 T^{2} + T^{4} )^{2}$$
$97$ $$( 64 + 24 T^{2} + T^{4} )^{2}$$
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