# Properties

 Label 1064.2.q.j Level $1064$ Weight $2$ Character orbit 1064.q Analytic conductor $8.496$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{11} + 2 q^{13} + 2 q^{15} + ( 7 - 7 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( 2 - 6 \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + 4 q^{27} + 4 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} + 8 \zeta_{6} q^{33} + ( 2 + \zeta_{6} ) q^{35} -10 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{39} -9 q^{43} + ( 1 - \zeta_{6} ) q^{45} + 4 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} -14 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{53} -4 q^{55} + 2 q^{57} + ( 12 - 12 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( -2 - \zeta_{6} ) q^{63} + 2 \zeta_{6} q^{65} + ( -16 + 16 \zeta_{6} ) q^{67} + 6 q^{69} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} -8 \zeta_{6} q^{75} + ( -4 + 12 \zeta_{6} ) q^{77} -4 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -9 q^{83} + 7 q^{85} + ( 8 - 8 \zeta_{6} ) q^{87} + 4 \zeta_{6} q^{89} + ( 6 - 4 \zeta_{6} ) q^{91} + 8 \zeta_{6} q^{93} + ( -1 + \zeta_{6} ) q^{95} -4 q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + q^{5} + 4 q^{7} - q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + q^{5} + 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} + 7 q^{17} + q^{19} - 2 q^{21} + 3 q^{23} + 4 q^{25} + 8 q^{27} + 8 q^{29} - 4 q^{31} + 8 q^{33} + 5 q^{35} - 10 q^{37} + 4 q^{39} - 18 q^{43} + q^{45} + 4 q^{47} + 2 q^{49} - 14 q^{51} - 6 q^{53} - 8 q^{55} + 4 q^{57} + 12 q^{59} + 2 q^{61} - 5 q^{63} + 2 q^{65} - 16 q^{67} + 12 q^{69} + 12 q^{71} - 2 q^{73} - 8 q^{75} + 4 q^{77} - 4 q^{79} + 11 q^{81} - 18 q^{83} + 14 q^{85} + 8 q^{87} + 4 q^{89} + 8 q^{91} + 8 q^{93} - q^{95} - 8 q^{97} + 8 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$533$$ $$799$$ $$913$$ $$1009$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.00000 1.73205i 0 0.500000 + 0.866025i 0 2.00000 1.73205i 0 −0.500000 0.866025i 0
457.1 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 2.00000 + 1.73205i 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.j 2
7.c even 3 1 inner 1064.2.q.j 2
7.c even 3 1 7448.2.a.c 1
7.d odd 6 1 7448.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.j 2 1.a even 1 1 trivial
1064.2.q.j 2 7.c even 3 1 inner
7448.2.a.c 1 7.c even 3 1
7448.2.a.v 1 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 $$S_{2}^{\mathrm{new}}(1064, [\chi])$$:

 $$T_{3}^{2} - 2 T_{3} + 4$$ $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{2} + 4 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$49 - 7 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$100 + 10 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 9 + T )^{2}$$
$47$ $$16 - 4 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$144 - 12 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$256 + 16 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$16 + 4 T + T^{2}$$
$83$ $$( 9 + T )^{2}$$
$89$ $$16 - 4 T + T^{2}$$
$97$ $$( 4 + T )^{2}$$