Properties

 Label 799.1.c.c Level $799$ Weight $1$ Character orbit 799.c Self dual yes Analytic conductor $0.399$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -799 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$799 = 17 \cdot 47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 799.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: yes Analytic conductor: $$0.398752945094$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.8671400783.1

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + q^{4} -\beta_{1} q^{5} + q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + q^{4} -\beta_{1} q^{5} + q^{9} + ( \beta_{1} + \beta_{3} ) q^{10} -\beta_{3} q^{11} - q^{16} - q^{17} -\beta_{2} q^{18} -\beta_{1} q^{20} + ( \beta_{1} - \beta_{3} ) q^{22} + \beta_{1} q^{23} + ( 1 + \beta_{2} ) q^{25} + \beta_{3} q^{29} + \beta_{3} q^{31} + \beta_{2} q^{32} + \beta_{2} q^{34} + q^{36} + \beta_{1} q^{41} -\beta_{3} q^{44} -\beta_{1} q^{45} + ( -\beta_{1} - \beta_{3} ) q^{46} - q^{47} + q^{49} + ( -2 - \beta_{2} ) q^{50} + \beta_{2} q^{55} + ( -\beta_{1} + \beta_{3} ) q^{58} + \beta_{2} q^{59} + ( -\beta_{1} + \beta_{3} ) q^{62} - q^{64} - q^{68} -\beta_{3} q^{73} + \beta_{1} q^{80} + q^{81} + ( -\beta_{1} - \beta_{3} ) q^{82} + \beta_{1} q^{85} + ( \beta_{1} + \beta_{3} ) q^{90} + \beta_{1} q^{92} + \beta_{2} q^{94} -\beta_{2} q^{98} -\beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{4} + 4q^{9} - 4q^{16} - 4q^{17} + 4q^{25} + 4q^{36} - 4q^{47} + 4q^{49} - 8q^{50} - 4q^{64} - 4q^{68} + 4q^{81} + O(q^{100})$$

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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
798.1
 1.84776 −1.84776 0.765367 −0.765367
−1.41421 0 1.00000 −1.84776 0 0 0 1.00000 2.61313
798.2 −1.41421 0 1.00000 1.84776 0 0 0 1.00000 −2.61313
798.3 1.41421 0 1.00000 −0.765367 0 0 0 1.00000 −1.08239
798.4 1.41421 0 1.00000 0.765367 0 0 0 1.00000 1.08239
 $$n$$: 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
799.c odd 2 1 CM by $$\Q(\sqrt{-799})$$
17.b even 2 1 inner
47.b 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.c 4
17.b even 2 1 inner 799.1.c.c 4
47.b odd 2 1 inner 799.1.c.c 4
799.c odd 2 1 CM 799.1.c.c 4

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

Hecke kernels

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

Hecke characteristic polynomials

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