# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(897,1792)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 2 i q^{5} + q^{7} - q^{9} +O(q^{10})$$ q + i * q^3 - 2*i * q^5 + q^7 - q^9 $$q + i q^{3} - 2 i q^{5} + q^{7} - q^{9} + 8 q^{15} - 2 q^{17} - i q^{19} + i q^{21} + 8 q^{23} - 11 q^{25} + 2 i q^{27} - i q^{29} - 4 q^{31} - 2 i q^{35} - 3 i q^{37} + 2 q^{41} - 4 i q^{43} + 2 i q^{45} + 4 q^{47} + q^{49} - 2 i q^{51} - 5 i q^{53} + 4 q^{57} - 3 i q^{59} - 2 i q^{61} - q^{63} - 6 i q^{67} + 8 i q^{69} + 14 q^{73} - 11 i q^{75} + 8 q^{79} - 11 q^{81} + 3 i q^{83} + 4 i q^{85} + 4 q^{87} - 10 q^{89} - 4 i q^{93} - 8 q^{95} - 2 q^{97} +O(q^{100})$$ q + i * q^3 - 2*i * q^5 + q^7 - q^9 + 8 * q^15 - 2 * q^17 - i * q^19 + i * q^21 + 8 * q^23 - 11 * q^25 + 2*i * q^27 - i * q^29 - 4 * q^31 - 2*i * q^35 - 3*i * q^37 + 2 * q^41 - 4*i * q^43 + 2*i * q^45 + 4 * q^47 + q^49 - 2*i * q^51 - 5*i * q^53 + 4 * q^57 - 3*i * q^59 - 2*i * q^61 - q^63 - 6*i * q^67 + 8*i * q^69 + 14 * q^73 - 11*i * q^75 + 8 * q^79 - 11 * q^81 + 3*i * q^83 + 4*i * q^85 + 4 * q^87 - 10 * q^89 - 4*i * q^93 - 8 * q^95 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^7 - 2 * q^9 $$2 q + 2 q^{7} - 2 q^{9} + 16 q^{15} - 4 q^{17} + 16 q^{23} - 22 q^{25} - 8 q^{31} + 4 q^{41} + 8 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} + 28 q^{73} + 16 q^{79} - 22 q^{81} + 8 q^{87} - 20 q^{89} - 16 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^7 - 2 * q^9 + 16 * q^15 - 4 * q^17 + 16 * q^23 - 22 * q^25 - 8 * q^31 + 4 * q^41 + 8 * q^47 + 2 * q^49 + 8 * q^57 - 2 * q^63 + 28 * q^73 + 16 * q^79 - 22 * q^81 + 8 * q^87 - 20 * q^89 - 16 * q^95 - 4 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 4.00000i 0 1.00000 0 −1.00000 0
897.2 0 2.00000i 0 4.00000i 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.h 2
4.b odd 2 1 1792.2.b.a 2
8.b even 2 1 inner 1792.2.b.h 2
8.d odd 2 1 1792.2.b.a 2
16.e even 4 1 112.2.a.a 1
16.e even 4 1 448.2.a.h 1
16.f odd 4 1 56.2.a.b 1
16.f odd 4 1 448.2.a.c 1
48.i odd 4 1 1008.2.a.m 1
48.i odd 4 1 4032.2.a.a 1
48.k even 4 1 504.2.a.h 1
48.k even 4 1 4032.2.a.d 1
80.i odd 4 1 2800.2.g.g 2
80.j even 4 1 1400.2.g.b 2
80.k odd 4 1 1400.2.a.a 1
80.q even 4 1 2800.2.a.bd 1
80.s even 4 1 1400.2.g.b 2
80.t odd 4 1 2800.2.g.g 2
112.j even 4 1 392.2.a.b 1
112.j even 4 1 3136.2.a.w 1
112.l odd 4 1 784.2.a.i 1
112.l odd 4 1 3136.2.a.c 1
112.u odd 12 2 392.2.i.a 2
112.v even 12 2 392.2.i.e 2
112.w even 12 2 784.2.i.j 2
112.x odd 12 2 784.2.i.b 2
176.i even 4 1 6776.2.a.h 1
208.o odd 4 1 9464.2.a.h 1
336.v odd 4 1 3528.2.a.b 1
336.y even 4 1 7056.2.a.c 1
336.br odd 12 2 3528.2.s.ba 2
336.bu even 12 2 3528.2.s.a 2
560.be even 4 1 9800.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 16.f odd 4 1
112.2.a.a 1 16.e even 4 1
392.2.a.b 1 112.j even 4 1
392.2.i.a 2 112.u odd 12 2
392.2.i.e 2 112.v even 12 2
448.2.a.c 1 16.f odd 4 1
448.2.a.h 1 16.e even 4 1
504.2.a.h 1 48.k even 4 1
784.2.a.i 1 112.l odd 4 1
784.2.i.b 2 112.x odd 12 2
784.2.i.j 2 112.w even 12 2
1008.2.a.m 1 48.i odd 4 1
1400.2.a.a 1 80.k odd 4 1
1400.2.g.b 2 80.j even 4 1
1400.2.g.b 2 80.s even 4 1
1792.2.b.a 2 4.b odd 2 1
1792.2.b.a 2 8.d odd 2 1
1792.2.b.h 2 1.a even 1 1 trivial
1792.2.b.h 2 8.b even 2 1 inner
2800.2.a.bd 1 80.q even 4 1
2800.2.g.g 2 80.i odd 4 1
2800.2.g.g 2 80.t odd 4 1
3136.2.a.c 1 112.l odd 4 1
3136.2.a.w 1 112.j even 4 1
3528.2.a.b 1 336.v odd 4 1
3528.2.s.a 2 336.bu even 12 2
3528.2.s.ba 2 336.br odd 12 2
4032.2.a.a 1 48.i odd 4 1
4032.2.a.d 1 48.k even 4 1
6776.2.a.h 1 176.i even 4 1
7056.2.a.c 1 336.y even 4 1
9464.2.a.h 1 208.o odd 4 1
9800.2.a.bj 1 560.be even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{5}^{2} + 16$$ T5^2 + 16 $$T_{11}$$ T11 $$T_{23} - 8$$ T23 - 8 $$T_{31} + 4$$ T31 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 16$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$(T - 8)^{2}$$
$29$ $$T^{2} + 4$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} + 100$$
$59$ $$T^{2} + 36$$
$61$ $$T^{2} + 16$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2}$$
$73$ $$(T - 14)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 10)^{2}$$
$97$ $$(T + 2)^{2}$$