Properties

 Label 799.1.c.d Level $799$ Weight $1$ Character orbit 799.c Analytic conductor $0.399$ Analytic rank $0$ Dimension $4$ Projective image $D_{10}$ 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(798,799)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(799, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("799.798");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$799 = 17 \cdot 47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 799.c (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.2.6928449225617.1

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{2} + (\zeta_{10}^{4} + \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10} + 1) q^{4} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{6} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{7} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{8} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{9}+O(q^{10})$$ q + (z^3 - z^2) * q^2 + (z^4 + z) * q^3 + (z^4 - z + 1) * q^4 + (z^4 - z^3 - z^2 + z) * q^6 + (z^3 + z^2) * q^7 + (-z^4 + z^3 - z^2 + z) * q^8 + (-z^3 + z^2 - 1) * q^9 $$q + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{2} + (\zeta_{10}^{4} + \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10} + 1) q^{4} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{6} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{7} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{8} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{9} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{12} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{14} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{16} - \zeta_{10} q^{17} + ( - \zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10} - 2) q^{18} + (\zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10}) q^{21} + (\zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{24} - q^{25} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{27} + ( - \zeta_{10}^{4} - \zeta_{10}^{2} - \zeta_{10}) q^{28} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{32} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{34} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{36} + (\zeta_{10}^{4} + \zeta_{10}) q^{37} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{42} + q^{47} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{48} + (\zeta_{10}^{4} - \zeta_{10} - 1) q^{49} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{50} + ( - \zeta_{10}^{2} + 1) q^{51} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{53} + ( - 2 \zeta_{10}^{4} + \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{54} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{56} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{59} + ( - \zeta_{10}^{3} - \zeta_{10}^{2}) q^{61} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{63} + (\zeta_{10}^{4} - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} + 1) q^{64} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{68} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{71} + ( - \zeta_{10}^{4} - \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{72} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{74} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{75} + (\zeta_{10}^{4} + \zeta_{10}) q^{79} + (\zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} + 1) q^{81} + q^{83} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{84} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{89} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{94} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{96} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{97} + ( - \zeta_{10}^{4} - \zeta_{10}^{2} + \zeta_{10}) q^{98} +O(q^{100})$$ q + (z^3 - z^2) * q^2 + (z^4 + z) * q^3 + (z^4 - z + 1) * q^4 + (z^4 - z^3 - z^2 + z) * q^6 + (z^3 + z^2) * q^7 + (-z^4 + z^3 - z^2 + z) * q^8 + (-z^3 + z^2 - 1) * q^9 + (z^4 - z^3 - z^2 + z) * q^12 + (-z^4 - z) * q^14 + (z^4 - z^3 + z^2 - z + 1) * q^16 - z * q^17 + (-z^4 - z^3 + z^2 + z - 2) * q^18 + (z^4 + z^3 - z^2 - z) * q^21 + (z^4 + z^3 - z^2 + z) * q^24 - q^25 + (-z^4 + z^3 + z^2 - z) * q^27 + (-z^4 - z^2 - z) * q^28 + (-z^4 + z^3 - z^2 + z - 2) * q^32 + (-z^4 + z^3) * q^34 + (-2*z^3 + z^2 - z - 1) * q^36 + (z^4 + z) * q^37 + (z^3 - z^2 + 2) * q^42 + q^47 + (-z^3 + z^2 - z) * q^48 + (z^4 - z - 1) * q^49 + (-z^3 + z^2) * q^50 + (-z^2 + 1) * q^51 + (-z^3 + z^2) * q^53 + (-2*z^4 + z^3 + z^2 - 2*z) * q^54 + (z^3 + z^2) * q^56 + (z^3 - z^2) * q^59 + (-z^3 - z^2) * q^61 + (z^4 - z^3 - z^2 + z) * q^63 + (z^4 - 2*z^3 + 2*z^2 - z + 1) * q^64 + (z^2 - z + 1) * q^68 + (-z^4 - z) * q^71 + (-z^4 - z^2 + 2*z - 2) * q^72 + (z^4 - z^3 - z^2 + z) * q^74 + (-z^4 - z) * q^75 + (z^4 + z) * q^79 + (z^4 + z^3 - z^2 - z + 1) * q^81 + q^83 + (z^3 - z^2 + 2) * q^84 + (-z^3 + z^2) * q^89 + (z^3 - z^2) * q^94 + (-z^4 + z^3 - z^2 + z) * q^96 + (z^3 + z^2) * q^97 + (-z^4 - z^2 + z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{4} + 4 q^{8} - 6 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 2 * q^4 + 4 * q^8 - 6 * q^9 $$4 q + 2 q^{2} + 2 q^{4} + 4 q^{8} - 6 q^{9} - q^{17} - 8 q^{18} - 4 q^{25} - 4 q^{32} + 2 q^{34} - 8 q^{36} + 10 q^{42} + 4 q^{47} - 6 q^{49} - 2 q^{50} + 5 q^{51} - 2 q^{53} + 2 q^{59} - 2 q^{64} + 2 q^{68} - 6 q^{72} + 4 q^{81} + 8 q^{83} + 10 q^{84} - 2 q^{89} + 2 q^{94} + 2 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 2 * q^4 + 4 * q^8 - 6 * q^9 - q^17 - 8 * q^18 - 4 * q^25 - 4 * q^32 + 2 * q^34 - 8 * q^36 + 10 * q^42 + 4 * q^47 - 6 * q^49 - 2 * q^50 + 5 * q^51 - 2 * q^53 + 2 * q^59 - 2 * q^64 + 2 * q^68 - 6 * q^72 + 4 * q^81 + 8 * q^83 + 10 * q^84 - 2 * q^89 + 2 * q^94 + 2 * q^98

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
798.1
 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i
−0.618034 1.17557i −0.618034 0 0.726543i 1.90211i 1.00000 −0.381966 0
798.2 −0.618034 1.17557i −0.618034 0 0.726543i 1.90211i 1.00000 −0.381966 0
798.3 1.61803 1.90211i 1.61803 0 3.07768i 1.17557i 1.00000 −2.61803 0
798.4 1.61803 1.90211i 1.61803 0 3.07768i 1.17557i 1.00000 −2.61803 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.b even 2 1 inner
799.c odd 2 1 inner

Twists

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

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

Hecke kernels

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

Hecke characteristic polynomials

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