# Properties

 Label 1064.2.q.l Level $1064$ Weight $2$ Character orbit 1064.q Analytic conductor $8.496$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1064 = 2^{3} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1064.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.49608277506$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + 2 \beta_{2} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + 2 q^{13} + 2 q^{15} + ( 1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{17} + ( 1 - \beta_{2} ) q^{19} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{21} + ( 1 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{23} + 4 \beta_{2} q^{25} + 4 q^{27} -4 q^{29} + 4 \beta_{2} q^{31} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{33} + ( -1 - \beta_{1} + \beta_{3} ) q^{35} + ( -2 + 2 \beta_{2} ) q^{37} + 4 \beta_{2} q^{39} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} - q^{43} + \beta_{2} q^{45} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 4 - 3 \beta_{1} - 4 \beta_{2} ) q^{49} + ( 8 - 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{53} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + 2 q^{57} + 4 \beta_{2} q^{59} + ( -9 + \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 1 + \beta_{1} - \beta_{3} ) q^{63} + ( 2 - 2 \beta_{2} ) q^{65} + ( -2 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{69} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( -1 + 2 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{73} + ( -8 + 8 \beta_{2} ) q^{75} + ( -4 + 3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} ) q^{77} + ( 6 + 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{79} + 11 \beta_{2} q^{81} + ( 3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{85} -8 \beta_{2} q^{87} + ( -10 + 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -8 + 8 \beta_{2} ) q^{93} -\beta_{2} q^{95} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{97} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 2q^{5} - 3q^{7} - 2q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 2q^{5} - 3q^{7} - 2q^{9} + 3q^{11} + 8q^{13} + 8q^{15} - 7q^{17} + 2q^{19} + 6q^{21} + 8q^{25} + 16q^{27} - 16q^{29} + 8q^{31} - 6q^{33} - 6q^{35} - 4q^{37} + 8q^{39} + 12q^{41} - 4q^{43} + 2q^{45} - 3q^{47} + 5q^{49} + 14q^{51} - 2q^{53} + 6q^{55} + 8q^{57} + 8q^{59} - 17q^{61} + 6q^{63} + 4q^{65} + 12q^{71} - 15q^{73} - 16q^{75} - 24q^{77} + 14q^{79} + 22q^{81} + 16q^{83} - 14q^{85} - 16q^{87} - 18q^{89} - 6q^{91} - 16q^{93} - 2q^{95} + 4q^{97} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 5$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} + 4 \beta_{1} + 5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1064\mathbb{Z}\right)^\times$$.

 $$n$$ $$533$$ $$799$$ $$913$$ $$1009$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
305.1
 −1.63746 + 1.52274i 2.13746 − 0.656712i −1.63746 − 1.52274i 2.13746 + 0.656712i
0 1.00000 1.73205i 0 0.500000 + 0.866025i 0 −2.63746 + 0.209313i 0 −0.500000 0.866025i 0
305.2 0 1.00000 1.73205i 0 0.500000 + 0.866025i 0 1.13746 + 2.38876i 0 −0.500000 0.866025i 0
457.1 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 −2.63746 0.209313i 0 −0.500000 + 0.866025i 0
457.2 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 1.13746 2.38876i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1064.2.q.l 4
7.c even 3 1 inner 1064.2.q.l 4
7.c even 3 1 7448.2.a.w 2
7.d odd 6 1 7448.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.l 4 1.a even 1 1 trivial
1064.2.q.l 4 7.c even 3 1 inner
7448.2.a.w 2 7.c even 3 1
7448.2.a.be 2 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1064, [\chi])$$:

 $$T_{3}^{2} - 2 T_{3} + 4$$ $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{4} - 3 T_{11}^{3} + 21 T_{11}^{2} + 36 T_{11} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 4 - 2 T + T^{2} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$49 + 21 T + 2 T^{2} + 3 T^{3} + T^{4}$$
$11$ $$144 + 36 T + 21 T^{2} - 3 T^{3} + T^{4}$$
$13$ $$( -2 + T )^{4}$$
$17$ $$4 - 14 T + 51 T^{2} + 7 T^{3} + T^{4}$$
$19$ $$( 1 - T + T^{2} )^{2}$$
$23$ $$3249 + 57 T^{2} + T^{4}$$
$29$ $$( 4 + T )^{4}$$
$31$ $$( 16 - 4 T + T^{2} )^{2}$$
$37$ $$( 4 + 2 T + T^{2} )^{2}$$
$41$ $$( -48 - 6 T + T^{2} )^{2}$$
$43$ $$( 1 + T )^{4}$$
$47$ $$144 - 36 T + 21 T^{2} + 3 T^{3} + T^{4}$$
$53$ $$3136 - 112 T + 60 T^{2} + 2 T^{3} + T^{4}$$
$59$ $$( 16 - 4 T + T^{2} )^{2}$$
$61$ $$3364 + 986 T + 231 T^{2} + 17 T^{3} + T^{4}$$
$67$ $$T^{4}$$
$71$ $$( -48 - 6 T + T^{2} )^{2}$$
$73$ $$1764 + 630 T + 183 T^{2} + 15 T^{3} + T^{4}$$
$79$ $$64 + 112 T + 204 T^{2} - 14 T^{3} + T^{4}$$
$83$ $$( -41 - 8 T + T^{2} )^{2}$$
$89$ $$576 + 432 T + 300 T^{2} + 18 T^{3} + T^{4}$$
$97$ $$( -56 - 2 T + T^{2} )^{2}$$