# Properties

 Label 1792.2.f.a Level $1792$ Weight $2$ Character orbit 1792.f Analytic conductor $14.309$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( -3 + 2 \zeta_{12}^{2} ) q^{7} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( -3 + 2 \zeta_{12}^{2} ) q^{7} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{13} -2 \zeta_{12}^{3} q^{15} + 4 \zeta_{12}^{3} q^{17} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{19} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{21} + ( -2 + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{23} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} -4 q^{27} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{29} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{31} + ( -4 + 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{33} + ( -1 + 2 \zeta_{12} - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} -4 q^{37} + ( 4 - 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{39} + ( -4 + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{41} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{45} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{47} + ( 5 - 8 \zeta_{12}^{2} ) q^{49} + ( 4 - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{51} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{53} + 4 q^{55} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( -9 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{59} + ( 5 - 10 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{61} + ( -3 - 8 \zeta_{12} + 2 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{63} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{65} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{67} + ( 4 - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{69} + ( 2 - 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{71} + ( 4 - 8 \zeta_{12}^{2} ) q^{73} + ( -7 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{75} + ( 2 + 4 \zeta_{12} + 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{77} + ( -2 + 4 \zeta_{12}^{2} ) q^{79} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{81} + ( -9 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{83} + ( 4 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{85} + ( 12 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{87} + ( 4 - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{89} + ( -1 - 6 \zeta_{12} - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{91} + ( -12 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{93} + ( 4 - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{95} -12 \zeta_{12}^{3} q^{97} + ( 6 - 12 \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 8q^{7} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 8q^{7} + 4q^{9} - 12q^{19} + 8q^{21} + 4q^{25} - 16q^{27} - 12q^{35} - 16q^{37} + 4q^{49} + 16q^{55} - 36q^{59} - 8q^{63} - 28q^{75} + 24q^{77} + 4q^{81} - 36q^{83} + 16q^{85} + 48q^{87} - 12q^{91} - 48q^{93} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1791.1
 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 −2.73205 0 0.732051i 0 −2.00000 1.73205i 0 4.46410 0
1791.2 0 −2.73205 0 0.732051i 0 −2.00000 + 1.73205i 0 4.46410 0
1791.3 0 0.732051 0 2.73205i 0 −2.00000 1.73205i 0 −2.46410 0
1791.4 0 0.732051 0 2.73205i 0 −2.00000 + 1.73205i 0 −2.46410 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

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

## Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} - 2$$ $$T_{5}^{4} + 8 T_{5}^{2} + 4$$ $$T_{29}^{2} - 48$$ $$T_{31}^{2} - 48$$ $$T_{37} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -2 + 2 T + T^{2} )^{2}$$
$5$ $$4 + 8 T^{2} + T^{4}$$
$7$ $$( 7 + 4 T + T^{2} )^{2}$$
$11$ $$64 + 32 T^{2} + T^{4}$$
$13$ $$36 + 24 T^{2} + T^{4}$$
$17$ $$( 16 + T^{2} )^{2}$$
$19$ $$( 6 + 6 T + T^{2} )^{2}$$
$23$ $$64 + 32 T^{2} + T^{4}$$
$29$ $$( -48 + T^{2} )^{2}$$
$31$ $$( -48 + T^{2} )^{2}$$
$37$ $$( 4 + T )^{4}$$
$41$ $$1024 + 128 T^{2} + T^{4}$$
$43$ $$576 + 96 T^{2} + T^{4}$$
$47$ $$( -48 + T^{2} )^{2}$$
$53$ $$( -48 + T^{2} )^{2}$$
$59$ $$( 54 + 18 T + T^{2} )^{2}$$
$61$ $$4356 + 168 T^{2} + T^{4}$$
$67$ $$576 + 96 T^{2} + T^{4}$$
$71$ $$2704 + 152 T^{2} + T^{4}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( 12 + T^{2} )^{2}$$
$83$ $$( 78 + 18 T + T^{2} )^{2}$$
$89$ $$256 + 224 T^{2} + T^{4}$$
$97$ $$( 144 + T^{2} )^{2}$$
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