# Properties

 Label 799.1.e.a Level $799$ Weight $1$ Character orbit 799.e Analytic conductor $0.399$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -47 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [799,1,Mod(140,799)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(799, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3, 2]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("799.140");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$799 = 17 \cdot 47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 799.e (of order $$4$$, degree $$2$$, 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: $$0.398752945094$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ 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: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.230911.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{2} + ( - i + 1) q^{3} - 3 q^{4} + ( - 2 i - 2) q^{6} + ( - i - 1) q^{7} + 4 i q^{8} - i q^{9} +O(q^{10})$$ q - z * q^2 + (-z + 1) * q^3 - 3 * q^4 + (-2*z - 2) * q^6 + (-z - 1) * q^7 + 4*z * q^8 - z * q^9 $$q - i q^{2} + ( - i + 1) q^{3} - 3 q^{4} + ( - 2 i - 2) q^{6} + ( - i - 1) q^{7} + 4 i q^{8} - i q^{9} + (3 i - 3) q^{12} + (2 i - 2) q^{14} + 5 q^{16} + q^{17} - 2 q^{18} - q^{21} + (4 i + 4) q^{24} - i q^{25} + q^{27} + (3 i + 3) q^{28} - 6 i q^{32} - 2 i q^{34} + 3 i q^{36} + (i - 1) q^{37} + (4 i - 2) q^{42} - q^{47} + ( - 5 i + 5) q^{48} + i q^{49} - 2 q^{50} + ( - i + 1) q^{51} + ( - 4 i + 4) q^{56} - i q^{59} + (i + 1) q^{61} + (i - 1) q^{63} - 7 q^{64} - 3 q^{68} + ( - i + 1) q^{71} + 4 q^{72} + (2 i + 2) q^{74} + ( - i - 1) q^{75} + (i + 1) q^{79} + q^{81} + 3 q^{84} + q^{89} + 2 i q^{94} + ( - 6 i - 6) q^{96} + (i - 1) q^{97} + 2 q^{98} +O(q^{100})$$ q - z * q^2 + (-z + 1) * q^3 - 3 * q^4 + (-2*z - 2) * q^6 + (-z - 1) * q^7 + 4*z * q^8 - z * q^9 + (3*z - 3) * q^12 + (2*z - 2) * q^14 + 5 * q^16 + q^17 - 2 * q^18 - q^21 + (4*z + 4) * q^24 - z * q^25 + q^27 + (3*z + 3) * q^28 - 6*z * q^32 - 2*z * q^34 + 3*z * q^36 + (z - 1) * q^37 + (4*z - 2) * q^42 - q^47 + (-5*z + 5) * q^48 + z * q^49 - 2 * q^50 + (-z + 1) * q^51 + (-4*z + 4) * q^56 - z * q^59 + (z + 1) * q^61 + (z - 1) * q^63 - 7 * q^64 - 3 * q^68 + (-z + 1) * q^71 + 4 * q^72 + (2*z + 2) * q^74 + (-z - 1) * q^75 + (z + 1) * q^79 + q^81 + 3 * q^84 + q^89 + 2*z * q^94 + (-6*z - 6) * q^96 + (z - 1) * q^97 + 2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 6 q^{4} - 4 q^{6} - 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 - 6 * q^4 - 4 * q^6 - 2 * q^7 $$2 q + 2 q^{3} - 6 q^{4} - 4 q^{6} - 2 q^{7} - 6 q^{12} - 4 q^{14} + 10 q^{16} + 2 q^{17} - 4 q^{18} - 4 q^{21} + 8 q^{24} + 6 q^{28} - 2 q^{37} - 2 q^{47} + 10 q^{48} - 4 q^{50} + 2 q^{51} + 8 q^{56} + 2 q^{61} - 2 q^{63} - 14 q^{64} - 6 q^{68} + 2 q^{71} + 8 q^{72} + 4 q^{74} - 2 q^{75} + 2 q^{79} + 2 q^{81} + 12 q^{84} + 4 q^{89} - 12 q^{96} - 2 q^{97} + 4 q^{98}+O(q^{100})$$ 2 * q + 2 * q^3 - 6 * q^4 - 4 * q^6 - 2 * q^7 - 6 * q^12 - 4 * q^14 + 10 * q^16 + 2 * q^17 - 4 * q^18 - 4 * q^21 + 8 * q^24 + 6 * q^28 - 2 * q^37 - 2 * q^47 + 10 * q^48 - 4 * q^50 + 2 * q^51 + 8 * q^56 + 2 * q^61 - 2 * q^63 - 14 * q^64 - 6 * q^68 + 2 * q^71 + 8 * q^72 + 4 * q^74 - 2 * q^75 + 2 * q^79 + 2 * q^81 + 12 * q^84 + 4 * q^89 - 12 * q^96 - 2 * q^97 + 4 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/799\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$377$$ $$\chi(n)$$ $$-1$$ $$-i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
140.1
 1.00000i − 1.00000i
2.00000i 1.00000 1.00000i −3.00000 0 −2.00000 2.00000i −1.00000 1.00000i 4.00000i 1.00000i 0
234.1 2.00000i 1.00000 + 1.00000i −3.00000 0 −2.00000 + 2.00000i −1.00000 + 1.00000i 4.00000i 1.00000i 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
47.b odd 2 1 CM by $$\Q(\sqrt{-47})$$
17.c even 4 1 inner
799.e odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.1.e.a 2
17.c even 4 1 inner 799.1.e.a 2
47.b odd 2 1 CM 799.1.e.a 2
799.e odd 4 1 inner 799.1.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.e.a 2 1.a even 1 1 trivial
799.1.e.a 2 17.c even 4 1 inner
799.1.e.a 2 47.b odd 2 1 CM
799.1.e.a 2 799.e odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{1}^{\mathrm{new}}(799, [\chi])$$.

## Hecke characteristic polynomials

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