# Properties

 Label 799.1.e.b Level $799$ Weight $1$ Character orbit 799.e Analytic conductor $0.399$ Analytic rank $0$ Dimension $8$ Projective image $D_{20}$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{20}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{20} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{2} + (\zeta_{20}^{4} - \zeta_{20}) q^{3} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{4} + ( - \zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{4} - \zeta_{20}) q^{6} + (\zeta_{20}^{3} + \zeta_{20}^{2}) q^{7} + (\zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{8} + (\zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{9}+O(q^{10})$$ q + (z^7 + z^3) * q^2 + (z^4 - z) * q^3 + (z^6 - z^4 - 1) * q^4 + (-z^8 + z^7 - z^4 - z) * q^6 + (z^3 + z^2) * q^7 + (z^9 - z^7 - z^3 + z) * q^8 + (z^8 - z^5 + z^2) * q^9 $$q + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{2} + (\zeta_{20}^{4} - \zeta_{20}) q^{3} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{4} + ( - \zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{4} - \zeta_{20}) q^{6} + (\zeta_{20}^{3} + \zeta_{20}^{2}) q^{7} + (\zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{8} + (\zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{9} + ( - \zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20} - 1) q^{12} + (\zeta_{20}^{9} + \zeta_{20}^{6} + \zeta_{20}^{5} - 1) q^{14} + (\zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + 1) q^{16} - \zeta_{20}^{6} q^{17} + (\zeta_{20}^{9} - \zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2} - \zeta_{20}) q^{18} + (\zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3}) q^{21} + (\zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20}^{3} - \zeta_{20}^{2} + \zeta_{20} + 1) q^{24} - \zeta_{20}^{5} q^{25} + ( - \zeta_{20}^{9} + \zeta_{20}^{6} - \zeta_{20}^{3} - \zeta_{20}^{2}) q^{27} + (\zeta_{20}^{9} + \zeta_{20}^{8} - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{3} - \zeta_{20}^{2}) q^{28} + ( - \zeta_{20}^{9} + \zeta_{20}^{7} - 2 \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{32} + ( - \zeta_{20}^{9} + \zeta_{20}^{3}) q^{34} + (\zeta_{20}^{9} - \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{2} + \zeta_{20}) q^{36} + ( - \zeta_{20}^{4} + \zeta_{20}) q^{37} + (\zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}) q^{42} - q^{47} + ( - \zeta_{20}^{9} + \zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2} - \zeta_{20} + 1) q^{48} + (\zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4}) q^{49} + ( - \zeta_{20}^{8} + \zeta_{20}^{2}) q^{50} + (\zeta_{20}^{7} + 1) q^{51} + ( - \zeta_{20}^{8} - \zeta_{20}^{2}) q^{53} + (\zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}^{2} + 1) q^{54} + ( - \zeta_{20}^{9} - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2} - \zeta_{20} + 1) q^{56} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{59} + (\zeta_{20}^{8} - \zeta_{20}^{7}) q^{61} + ( - \zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20} - 1) q^{63} + ( - 2 \zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 1) q^{64} + (\zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{68} + ( - \zeta_{20}^{9} - \zeta_{20}^{6}) q^{71} + (\zeta_{20}^{8} - \zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2}) q^{72} + (\zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{4} + \zeta_{20}) q^{74} + ( - \zeta_{20}^{9} + \zeta_{20}^{6}) q^{75} + (\zeta_{20}^{4} + \zeta_{20}) q^{79} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{3} + 1) q^{81} + ( - \zeta_{20}^{9} + \zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} + \zeta_{20} + 2) q^{84} + (\zeta_{20}^{8} - \zeta_{20}^{2}) q^{89} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{94} + ( - 2 \zeta_{20}^{9} - \zeta_{20}^{8} + \zeta_{20}^{7} + 2 \zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} + \zeta_{20}^{2} + \cdots - 1) q^{96} + \cdots + (\zeta_{20}^{9} + \zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{3} - \zeta_{20}^{2} - \zeta_{20}) q^{98} +O(q^{100})$$ q + (z^7 + z^3) * q^2 + (z^4 - z) * q^3 + (z^6 - z^4 - 1) * q^4 + (-z^8 + z^7 - z^4 - z) * q^6 + (z^3 + z^2) * q^7 + (z^9 - z^7 - z^3 + z) * q^8 + (z^8 - z^5 + z^2) * q^9 + (-z^8 - z^7 + z^5 - z^4 + z - 1) * q^12 + (z^9 + z^6 + z^5 - 1) * q^14 + (z^8 - z^6 + z^4 - z^2 + 1) * q^16 - z^6 * q^17 + (z^9 - z^8 - z^5 + z^2 - z) * q^18 + (z^7 + z^6 - z^4 - z^3) * q^21 + (z^8 - z^7 + z^5 + z^4 - z^3 - z^2 + z + 1) * q^24 - z^5 * q^25 + (-z^9 + z^6 - z^3 - z^2) * q^27 + (z^9 + z^8 - z^7 - z^6 - z^3 - z^2) * q^28 + (-z^9 + z^7 - 2*z^5 + z^3 - z) * q^32 + (-z^9 + z^3) * q^34 + (z^9 - z^6 + z^5 - z^4 + z^2 + z) * q^36 + (-z^4 + z) * q^37 + (z^9 - z^7 - z^6 - z^4 - z^3 + z) * q^42 - q^47 + (-z^9 + z^8 + z^7 - z^6 - z^5 + z^4 + z^3 - z^2 - z + 1) * q^48 + (z^6 + z^5 + z^4) * q^49 + (-z^8 + z^2) * q^50 + (z^7 + 1) * q^51 + (-z^8 - z^2) * q^53 + (z^6 - z^5 - z^3 + z^2 + 1) * q^54 + (-z^9 - z^6 - z^5 + z^4 + z^3 - z^2 - z + 1) * q^56 + (z^7 + z^3) * q^59 + (z^8 - z^7) * q^61 + (-z^8 - z^7 + z^5 + z^4 - z - 1) * q^63 + (-2*z^8 + z^6 - z^4 + 2*z^2 - 1) * q^64 + (z^6 + z^2 - 1) * q^68 + (-z^9 - z^6) * q^71 + (z^8 - z^7 - z^6 + z^5 + z^4 + z^3 - z^2) * q^72 + (z^8 - z^7 + z^4 + z) * q^74 + (-z^9 + z^6) * q^75 + (z^4 + z) * q^79 + (-z^7 - z^6 + z^4 - z^3 + 1) * q^81 + (-z^9 + z^8 - z^6 + z^4 - z^2 + z + 2) * q^84 + (z^8 - z^2) * q^89 + (-z^7 - z^3) * q^94 + (-2*z^9 - z^8 + z^7 + 2*z^6 - z^5 - z^4 + z^3 + z^2 - z - 1) * q^96 + (-z^3 + z^2) * q^97 + (z^9 + z^8 + z^7 - z^3 - z^2 - z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7}+O(q^{10})$$ 8 * q - 2 * q^3 - 4 * q^4 + 4 * q^6 + 2 * q^7 $$8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7} - 4 q^{12} - 6 q^{14} - 2 q^{17} + 4 q^{18} + 4 q^{21} + 2 q^{24} - 6 q^{28} + 2 q^{37} - 8 q^{47} + 4 q^{50} + 8 q^{51} + 10 q^{54} + 2 q^{56} - 2 q^{61} - 8 q^{63} + 4 q^{64} - 4 q^{68} - 2 q^{71} - 8 q^{72} - 4 q^{74} + 2 q^{75} - 2 q^{79} - 12 q^{81} + 8 q^{84} - 4 q^{89} + 2 q^{96} + 2 q^{97} - 4 q^{98}+O(q^{100})$$ 8 * q - 2 * q^3 - 4 * q^4 + 4 * q^6 + 2 * q^7 - 4 * q^12 - 6 * q^14 - 2 * q^17 + 4 * q^18 + 4 * q^21 + 2 * q^24 - 6 * q^28 + 2 * q^37 - 8 * q^47 + 4 * q^50 + 8 * q^51 + 10 * q^54 + 2 * q^56 - 2 * q^61 - 8 * q^63 + 4 * q^64 - 4 * q^68 - 2 * q^71 - 8 * q^72 - 4 * q^74 + 2 * q^75 - 2 * q^79 - 12 * q^81 + 8 * q^84 - 4 * q^89 + 2 * q^96 + 2 * q^97 - 4 * q^98

## Character values

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

 $$n$$ $$52$$ $$377$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{20}^{5}$$

## 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
 0.587785 − 0.809017i −0.587785 − 0.809017i 0.951057 + 0.309017i −0.951057 + 0.309017i 0.951057 − 0.309017i −0.951057 − 0.309017i 0.587785 + 0.809017i −0.587785 + 0.809017i
0.618034i −1.39680 + 1.39680i 0.618034 0 0.863271 + 0.863271i −1.26007 1.26007i 1.00000i 2.90211i 0
140.2 0.618034i −0.221232 + 0.221232i 0.618034 0 0.136729 + 0.136729i 0.642040 + 0.642040i 1.00000i 0.902113i 0
140.3 1.61803i −0.642040 + 0.642040i −1.61803 0 −1.03884 1.03884i 1.39680 + 1.39680i 1.00000i 0.175571i 0
140.4 1.61803i 1.26007 1.26007i −1.61803 0 2.03884 + 2.03884i 0.221232 + 0.221232i 1.00000i 2.17557i 0
234.1 1.61803i −0.642040 0.642040i −1.61803 0 −1.03884 + 1.03884i 1.39680 1.39680i 1.00000i 0.175571i 0
234.2 1.61803i 1.26007 + 1.26007i −1.61803 0 2.03884 2.03884i 0.221232 0.221232i 1.00000i 2.17557i 0
234.3 0.618034i −1.39680 1.39680i 0.618034 0 0.863271 0.863271i −1.26007 + 1.26007i 1.00000i 2.90211i 0
234.4 0.618034i −0.221232 0.221232i 0.618034 0 0.136729 0.136729i 0.642040 0.642040i 1.00000i 0.902113i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 140.4 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.b 8
17.c even 4 1 inner 799.1.e.b 8
47.b odd 2 1 CM 799.1.e.b 8
799.e odd 4 1 inner 799.1.e.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.e.b 8 1.a even 1 1 trivial
799.1.e.b 8 17.c even 4 1 inner
799.1.e.b 8 47.b odd 2 1 CM
799.1.e.b 8 799.e odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(799, [\chi])$$.

## Hecke characteristic polynomials

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