# Properties

 Label 1792.2.b.f Level $1792$ Weight $2$ Character orbit 1792.b Analytic conductor $14.309$ Analytic rank $0$ Dimension $2$ 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + q^{7} - q^{9} -4 i q^{11} + 4 i q^{13} -2 q^{17} + 6 i q^{19} + 2 i q^{21} -8 q^{23} + 5 q^{25} + 4 i q^{27} -2 i q^{29} -4 q^{31} + 8 q^{33} + 10 i q^{37} -8 q^{39} + 10 q^{41} + 4 i q^{43} + 4 q^{47} + q^{49} -4 i q^{51} -2 i q^{53} -12 q^{57} + 10 i q^{59} + 8 i q^{61} - q^{63} + 8 i q^{67} -16 i q^{69} + 6 q^{73} + 10 i q^{75} -4 i q^{77} -16 q^{79} -11 q^{81} -2 i q^{83} + 4 q^{87} -18 q^{89} + 4 i q^{91} -8 i q^{93} -2 q^{97} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{7} - 2q^{9} - 4q^{17} - 16q^{23} + 10q^{25} - 8q^{31} + 16q^{33} - 16q^{39} + 20q^{41} + 8q^{47} + 2q^{49} - 24q^{57} - 2q^{63} + 12q^{73} - 32q^{79} - 22q^{81} + 8q^{87} - 36q^{89} - 4q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
897.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 1.00000 0 −1.00000 0
897.2 0 2.00000i 0 0 0 1.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.b.f 2
4.b odd 2 1 1792.2.b.b 2
8.b even 2 1 inner 1792.2.b.f 2
8.d odd 2 1 1792.2.b.b 2
16.e even 4 1 224.2.a.a 1
16.e even 4 1 448.2.a.f 1
16.f odd 4 1 224.2.a.b yes 1
16.f odd 4 1 448.2.a.b 1
48.i odd 4 1 2016.2.a.e 1
48.i odd 4 1 4032.2.a.p 1
48.k even 4 1 2016.2.a.g 1
48.k even 4 1 4032.2.a.z 1
80.k odd 4 1 5600.2.a.c 1
80.q even 4 1 5600.2.a.t 1
112.j even 4 1 1568.2.a.b 1
112.j even 4 1 3136.2.a.y 1
112.l odd 4 1 1568.2.a.h 1
112.l odd 4 1 3136.2.a.f 1
112.u odd 12 2 1568.2.i.b 2
112.v even 12 2 1568.2.i.j 2
112.w even 12 2 1568.2.i.k 2
112.x odd 12 2 1568.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 16.e even 4 1
224.2.a.b yes 1 16.f odd 4 1
448.2.a.b 1 16.f odd 4 1
448.2.a.f 1 16.e even 4 1
1568.2.a.b 1 112.j even 4 1
1568.2.a.h 1 112.l odd 4 1
1568.2.i.b 2 112.u odd 12 2
1568.2.i.c 2 112.x odd 12 2
1568.2.i.j 2 112.v even 12 2
1568.2.i.k 2 112.w even 12 2
1792.2.b.b 2 4.b odd 2 1
1792.2.b.b 2 8.d odd 2 1
1792.2.b.f 2 1.a even 1 1 trivial
1792.2.b.f 2 8.b even 2 1 inner
2016.2.a.e 1 48.i odd 4 1
2016.2.a.g 1 48.k even 4 1
3136.2.a.f 1 112.l odd 4 1
3136.2.a.y 1 112.j even 4 1
4032.2.a.p 1 48.i odd 4 1
4032.2.a.z 1 48.k even 4 1
5600.2.a.c 1 80.k odd 4 1
5600.2.a.t 1 80.q even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{5}$$ $$T_{11}^{2} + 16$$ $$T_{23} + 8$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$( 8 + T )^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$( -4 + T )^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$100 + T^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$4 + T^{2}$$
$89$ $$( 18 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$
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