Properties

 Label 1064.2.q.a 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} + 4 q^{13} + 2 q^{15} + ( -2 + 2 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} + ( 6 - 4 \zeta_{6} ) q^{21} + 7 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} -4 q^{27} + 2 q^{29} + ( -6 + 6 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{33} + ( -1 + 3 \zeta_{6} ) q^{35} + 10 \zeta_{6} q^{37} + ( -8 + 8 \zeta_{6} ) q^{39} + 8 q^{41} + 7 q^{43} + ( -1 + \zeta_{6} ) q^{45} -9 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{53} -3 q^{55} + 2 q^{57} + ( -14 + 14 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} + ( -1 + 3 \zeta_{6} ) q^{63} -4 \zeta_{6} q^{65} + ( -14 + 14 \zeta_{6} ) q^{67} -14 q^{69} + 8 q^{71} + ( -1 + \zeta_{6} ) q^{73} + 8 \zeta_{6} q^{75} + ( -9 + 6 \zeta_{6} ) q^{77} -10 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} + 17 q^{83} + 2 q^{85} + ( -4 + 4 \zeta_{6} ) q^{87} + ( -8 - 4 \zeta_{6} ) q^{91} -12 \zeta_{6} q^{93} + ( -1 + \zeta_{6} ) q^{95} + 12 q^{97} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - q^{5} - 5 q^{7} - q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} - q^{5} - 5 q^{7} - q^{9} + 3 q^{11} + 8 q^{13} + 4 q^{15} - 2 q^{17} - q^{19} + 8 q^{21} + 7 q^{23} + 4 q^{25} - 8 q^{27} + 4 q^{29} - 6 q^{31} + 6 q^{33} + q^{35} + 10 q^{37} - 8 q^{39} + 16 q^{41} + 14 q^{43} - q^{45} - 9 q^{47} + 11 q^{49} - 4 q^{51} - 6 q^{53} - 6 q^{55} + 4 q^{57} - 14 q^{59} - 5 q^{61} + q^{63} - 4 q^{65} - 14 q^{67} - 28 q^{69} + 16 q^{71} - q^{73} + 8 q^{75} - 12 q^{77} - 10 q^{79} + 11 q^{81} + 34 q^{83} + 4 q^{85} - 4 q^{87} - 20 q^{91} - 12 q^{93} - q^{95} + 24 q^{97} - 6 q^{99} + O(q^{100})$$

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 −2.50000 0.866025i 0 −0.500000 0.866025i 0
457.1 0 −1.00000 1.73205i 0 −0.500000 + 0.866025i 0 −2.50000 + 0.866025i 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.a 2
7.c even 3 1 inner 1064.2.q.a 2
7.c even 3 1 7448.2.a.t 1
7.d odd 6 1 7448.2.a.a 1

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

Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} + 4$$ $$T_{5}^{2} + T_{5} + 1$$ $$T_{11}^{2} - 3 T_{11} + 9$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + 2 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$49 - 7 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$36 + 6 T + T^{2}$$
$37$ $$100 - 10 T + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$( -7 + T )^{2}$$
$47$ $$81 + 9 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$196 + 14 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$196 + 14 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$1 + T + T^{2}$$
$79$ $$100 + 10 T + T^{2}$$
$83$ $$( -17 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$( -12 + T )^{2}$$