# Properties

 Label 1792.2.b.c 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 14) 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.

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

## 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.c 2
4.b odd 2 1 1792.2.b.g 2
8.b even 2 1 inner 1792.2.b.c 2
8.d odd 2 1 1792.2.b.g 2
16.e even 4 1 14.2.a.a 1
16.e even 4 1 448.2.a.g 1
16.f odd 4 1 112.2.a.c 1
16.f odd 4 1 448.2.a.a 1
48.i odd 4 1 126.2.a.b 1
48.i odd 4 1 4032.2.a.w 1
48.k even 4 1 1008.2.a.h 1
48.k even 4 1 4032.2.a.r 1
80.i odd 4 1 350.2.c.d 2
80.j even 4 1 2800.2.g.h 2
80.k odd 4 1 2800.2.a.g 1
80.q even 4 1 350.2.a.f 1
80.s even 4 1 2800.2.g.h 2
80.t odd 4 1 350.2.c.d 2
112.j even 4 1 784.2.a.b 1
112.j even 4 1 3136.2.a.z 1
112.l odd 4 1 98.2.a.a 1
112.l odd 4 1 3136.2.a.e 1
112.u odd 12 2 784.2.i.c 2
112.v even 12 2 784.2.i.i 2
112.w even 12 2 98.2.c.b 2
112.x odd 12 2 98.2.c.a 2
144.w odd 12 2 1134.2.f.f 2
144.x even 12 2 1134.2.f.l 2
176.l odd 4 1 1694.2.a.e 1
208.m odd 4 1 2366.2.d.b 2
208.p even 4 1 2366.2.a.j 1
208.r odd 4 1 2366.2.d.b 2
240.bb even 4 1 3150.2.g.j 2
240.bf even 4 1 3150.2.g.j 2
240.bm odd 4 1 3150.2.a.i 1
272.r even 4 1 4046.2.a.f 1
304.j odd 4 1 5054.2.a.c 1
336.v odd 4 1 7056.2.a.bd 1
336.y even 4 1 882.2.a.i 1
336.bo even 12 2 882.2.g.d 2
336.bt odd 12 2 882.2.g.c 2
368.k odd 4 1 7406.2.a.a 1
560.r even 4 1 2450.2.c.c 2
560.bf odd 4 1 2450.2.a.t 1
560.bn even 4 1 2450.2.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 16.e even 4 1
98.2.a.a 1 112.l odd 4 1
98.2.c.a 2 112.x odd 12 2
98.2.c.b 2 112.w even 12 2
112.2.a.c 1 16.f odd 4 1
126.2.a.b 1 48.i odd 4 1
350.2.a.f 1 80.q even 4 1
350.2.c.d 2 80.i odd 4 1
350.2.c.d 2 80.t odd 4 1
448.2.a.a 1 16.f odd 4 1
448.2.a.g 1 16.e even 4 1
784.2.a.b 1 112.j even 4 1
784.2.i.c 2 112.u odd 12 2
784.2.i.i 2 112.v even 12 2
882.2.a.i 1 336.y even 4 1
882.2.g.c 2 336.bt odd 12 2
882.2.g.d 2 336.bo even 12 2
1008.2.a.h 1 48.k even 4 1
1134.2.f.f 2 144.w odd 12 2
1134.2.f.l 2 144.x even 12 2
1694.2.a.e 1 176.l odd 4 1
1792.2.b.c 2 1.a even 1 1 trivial
1792.2.b.c 2 8.b even 2 1 inner
1792.2.b.g 2 4.b odd 2 1
1792.2.b.g 2 8.d odd 2 1
2366.2.a.j 1 208.p even 4 1
2366.2.d.b 2 208.m odd 4 1
2366.2.d.b 2 208.r odd 4 1
2450.2.a.t 1 560.bf odd 4 1
2450.2.c.c 2 560.r even 4 1
2450.2.c.c 2 560.bn even 4 1
2800.2.a.g 1 80.k odd 4 1
2800.2.g.h 2 80.j even 4 1
2800.2.g.h 2 80.s even 4 1
3136.2.a.e 1 112.l odd 4 1
3136.2.a.z 1 112.j even 4 1
3150.2.a.i 1 240.bm odd 4 1
3150.2.g.j 2 240.bb even 4 1
3150.2.g.j 2 240.bf even 4 1
4032.2.a.r 1 48.k even 4 1
4032.2.a.w 1 48.i odd 4 1
4046.2.a.f 1 272.r even 4 1
5054.2.a.c 1 304.j odd 4 1
7056.2.a.bd 1 336.v odd 4 1
7406.2.a.a 1 368.k odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$( 12 + T )^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( 10 + T )^{2}$$
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