# Properties

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

# Learn more

## 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 112) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12}^{3} q^{3} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} - q^{9} +O(q^{10})$$ $$q + 2 \zeta_{12}^{3} q^{3} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} - q^{9} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{13} + ( -4 + 8 \zeta_{12}^{2} ) q^{15} + 2 \zeta_{12}^{3} q^{19} + ( -4 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{21} + ( 2 - 4 \zeta_{12}^{2} ) q^{23} + 7 q^{25} + 4 \zeta_{12}^{3} q^{27} + 6 \zeta_{12}^{3} q^{29} -8 q^{31} + ( 4 - 8 \zeta_{12}^{2} ) q^{33} + ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{35} + 2 \zeta_{12}^{3} q^{37} + ( -4 + 8 \zeta_{12}^{2} ) q^{39} + ( 4 - 8 \zeta_{12}^{2} ) q^{41} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{43} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{45} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} -6 \zeta_{12}^{3} q^{53} -12 q^{55} -4 q^{57} -6 \zeta_{12}^{3} q^{59} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{61} + ( -1 - 2 \zeta_{12}^{2} ) q^{63} + 12 q^{65} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{69} + ( -2 + 4 \zeta_{12}^{2} ) q^{71} + ( -4 + 8 \zeta_{12}^{2} ) q^{73} + 14 \zeta_{12}^{3} q^{75} + ( -8 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} + ( 2 - 4 \zeta_{12}^{2} ) q^{79} -11 q^{81} -6 \zeta_{12}^{3} q^{83} -12 q^{87} + ( 4 - 8 \zeta_{12}^{2} ) q^{89} + ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91} -16 \zeta_{12}^{3} q^{93} + ( -4 + 8 \zeta_{12}^{2} ) q^{95} + ( 8 - 16 \zeta_{12}^{2} ) q^{97} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 8q^{7} - 4q^{9} + 28q^{25} - 32q^{31} + 4q^{49} - 48q^{55} - 16q^{57} - 8q^{63} + 48q^{65} - 44q^{81} - 48q^{87} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
895.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 2.00000i 0 −3.46410 0 2.00000 + 1.73205i 0 −1.00000 0
895.2 0 2.00000i 0 3.46410 0 2.00000 1.73205i 0 −1.00000 0
895.3 0 2.00000i 0 −3.46410 0 2.00000 1.73205i 0 −1.00000 0
895.4 0 2.00000i 0 3.46410 0 2.00000 + 1.73205i 0 −1.00000 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
8.b even 2 1 inner
28.d even 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.c 4
4.b odd 2 1 1792.2.e.a 4
7.b odd 2 1 1792.2.e.a 4
8.b even 2 1 inner 1792.2.e.c 4
8.d odd 2 1 1792.2.e.a 4
16.e even 4 1 112.2.f.b yes 2
16.e even 4 1 448.2.f.a 2
16.f odd 4 1 112.2.f.a 2
16.f odd 4 1 448.2.f.c 2
28.d even 2 1 inner 1792.2.e.c 4
48.i odd 4 1 1008.2.b.b 2
48.i odd 4 1 4032.2.b.b 2
48.k even 4 1 1008.2.b.g 2
48.k even 4 1 4032.2.b.h 2
56.e even 2 1 inner 1792.2.e.c 4
56.h odd 2 1 1792.2.e.a 4
80.i odd 4 1 2800.2.e.b 4
80.j even 4 1 2800.2.e.c 4
80.k odd 4 1 2800.2.k.e 2
80.q even 4 1 2800.2.k.b 2
80.s even 4 1 2800.2.e.c 4
80.t odd 4 1 2800.2.e.b 4
112.j even 4 1 112.2.f.b yes 2
112.j even 4 1 448.2.f.a 2
112.l odd 4 1 112.2.f.a 2
112.l odd 4 1 448.2.f.c 2
112.u odd 12 1 784.2.p.e 2
112.u odd 12 1 784.2.p.f 2
112.v even 12 1 784.2.p.a 2
112.v even 12 1 784.2.p.b 2
112.w even 12 1 784.2.p.a 2
112.w even 12 1 784.2.p.b 2
112.x odd 12 1 784.2.p.e 2
112.x odd 12 1 784.2.p.f 2
336.v odd 4 1 1008.2.b.b 2
336.v odd 4 1 4032.2.b.b 2
336.y even 4 1 1008.2.b.g 2
336.y even 4 1 4032.2.b.h 2
560.r even 4 1 2800.2.e.c 4
560.u odd 4 1 2800.2.e.b 4
560.be even 4 1 2800.2.k.b 2
560.bf odd 4 1 2800.2.k.e 2
560.bm odd 4 1 2800.2.e.b 4
560.bn even 4 1 2800.2.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a 2 16.f odd 4 1
112.2.f.a 2 112.l odd 4 1
112.2.f.b yes 2 16.e even 4 1
112.2.f.b yes 2 112.j even 4 1
448.2.f.a 2 16.e even 4 1
448.2.f.a 2 112.j even 4 1
448.2.f.c 2 16.f odd 4 1
448.2.f.c 2 112.l odd 4 1
784.2.p.a 2 112.v even 12 1
784.2.p.a 2 112.w even 12 1
784.2.p.b 2 112.v even 12 1
784.2.p.b 2 112.w even 12 1
784.2.p.e 2 112.u odd 12 1
784.2.p.e 2 112.x odd 12 1
784.2.p.f 2 112.u odd 12 1
784.2.p.f 2 112.x odd 12 1
1008.2.b.b 2 48.i odd 4 1
1008.2.b.b 2 336.v odd 4 1
1008.2.b.g 2 48.k even 4 1
1008.2.b.g 2 336.y even 4 1
1792.2.e.a 4 4.b odd 2 1
1792.2.e.a 4 7.b odd 2 1
1792.2.e.a 4 8.d odd 2 1
1792.2.e.a 4 56.h odd 2 1
1792.2.e.c 4 1.a even 1 1 trivial
1792.2.e.c 4 8.b even 2 1 inner
1792.2.e.c 4 28.d even 2 1 inner
1792.2.e.c 4 56.e even 2 1 inner
2800.2.e.b 4 80.i odd 4 1
2800.2.e.b 4 80.t odd 4 1
2800.2.e.b 4 560.u odd 4 1
2800.2.e.b 4 560.bm odd 4 1
2800.2.e.c 4 80.j even 4 1
2800.2.e.c 4 80.s even 4 1
2800.2.e.c 4 560.r even 4 1
2800.2.e.c 4 560.bn even 4 1
2800.2.k.b 2 80.q even 4 1
2800.2.k.b 2 560.be even 4 1
2800.2.k.e 2 80.k odd 4 1
2800.2.k.e 2 560.bf odd 4 1
4032.2.b.b 2 48.i odd 4 1
4032.2.b.b 2 336.v odd 4 1
4032.2.b.h 2 48.k even 4 1
4032.2.b.h 2 336.y even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{11}^{2} - 12$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 4 + T^{2} )^{2}$$
$5$ $$( -12 + T^{2} )^{2}$$
$7$ $$( 7 - 4 T + T^{2} )^{2}$$
$11$ $$( -12 + T^{2} )^{2}$$
$13$ $$( -12 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$( 12 + T^{2} )^{2}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$( 8 + T )^{4}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 48 + T^{2} )^{2}$$
$43$ $$( -108 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$( -12 + T^{2} )^{2}$$
$67$ $$( -12 + T^{2} )^{2}$$
$71$ $$( 12 + T^{2} )^{2}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( 12 + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 48 + T^{2} )^{2}$$
$97$ $$( 192 + T^{2} )^{2}$$
show more
show less