# Properties

 Label 1064.2.q.k 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( 1 + \beta_{2} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{7} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( 1 + \beta_{2} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{7} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{9} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{11} + ( -2 - \beta_{3} ) q^{13} + ( -2 - \beta_{3} ) q^{15} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 1 + \beta_{2} ) q^{19} + ( -4 + 4 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{21} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{23} -4 \beta_{2} q^{25} + ( 8 + 8 \beta_{3} ) q^{27} + ( -8 - \beta_{3} ) q^{29} + ( \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{31} + ( 4 - 5 \beta_{1} + 4 \beta_{2} ) q^{33} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{35} + ( 8 + \beta_{1} + 8 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{39} -4 \beta_{3} q^{41} + ( 1 - 3 \beta_{3} ) q^{43} + ( 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{45} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 12 - 8 \beta_{1} + 12 \beta_{2} ) q^{51} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{53} + ( -1 + 3 \beta_{3} ) q^{55} + ( -2 - \beta_{3} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 5 + 5 \beta_{2} ) q^{61} + ( 16 + \beta_{1} + 5 \beta_{2} + 7 \beta_{3} ) q^{63} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 8 + 5 \beta_{3} ) q^{69} + ( -12 + 3 \beta_{3} ) q^{71} -7 \beta_{2} q^{73} + ( 8 - 4 \beta_{1} + 8 \beta_{2} ) q^{75} + ( 5 + 2 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{77} + ( 6 + 7 \beta_{1} + 6 \beta_{2} ) q^{79} + ( -12 \beta_{1} + 23 \beta_{2} - 12 \beta_{3} ) q^{81} + ( 5 + 7 \beta_{3} ) q^{83} + ( 4 + 2 \beta_{3} ) q^{85} + ( 10 \beta_{1} - 18 \beta_{2} + 10 \beta_{3} ) q^{87} + ( 2 + 11 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{91} + ( -10 + 4 \beta_{1} - 10 \beta_{2} ) q^{93} + \beta_{2} q^{95} -4 \beta_{3} q^{97} + ( -21 - 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} - 6 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} - 6 q^{9} - 2 q^{11} - 8 q^{13} - 8 q^{15} + 8 q^{17} + 2 q^{19} - 8 q^{21} - 6 q^{23} + 8 q^{25} + 32 q^{27} - 32 q^{29} - 12 q^{31} + 8 q^{33} - 2 q^{35} + 16 q^{37} + 12 q^{39} + 4 q^{43} + 6 q^{45} - 6 q^{47} + 10 q^{49} + 24 q^{51} + 8 q^{53} - 4 q^{55} - 8 q^{57} - 4 q^{59} + 10 q^{61} + 54 q^{63} - 4 q^{65} - 12 q^{67} + 32 q^{69} - 48 q^{71} + 14 q^{73} + 16 q^{75} - 4 q^{77} + 12 q^{79} - 46 q^{81} + 20 q^{83} + 16 q^{85} + 36 q^{87} + 4 q^{89} + 8 q^{91} - 20 q^{93} - 2 q^{95} - 84 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −1.70711 + 2.95680i 0 0.500000 + 0.866025i 0 −1.62132 + 2.09077i 0 −4.32843 7.49706i 0
305.2 0 −0.292893 + 0.507306i 0 0.500000 + 0.866025i 0 2.62132 0.358719i 0 1.32843 + 2.30090i 0
457.1 0 −1.70711 2.95680i 0 0.500000 0.866025i 0 −1.62132 2.09077i 0 −4.32843 + 7.49706i 0
457.2 0 −0.292893 0.507306i 0 0.500000 0.866025i 0 2.62132 + 0.358719i 0 1.32843 2.30090i 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.k 4
7.c even 3 1 inner 1064.2.q.k 4
7.c even 3 1 7448.2.a.bd 2
7.d odd 6 1 7448.2.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.k 4 1.a even 1 1 trivial
1064.2.q.k 4 7.c even 3 1 inner
7448.2.a.x 2 7.d odd 6 1
7448.2.a.bd 2 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 $$S_{2}^{\mathrm{new}}(1064, [\chi])$$:

 $$T_{3}^{4} + 4 T_{3}^{3} + 14 T_{3}^{2} + 8 T_{3} + 4$$ $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{4} + 2 T_{11}^{3} + 21 T_{11}^{2} - 34 T_{11} + 289$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$289 - 34 T + 21 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$( 2 + 4 T + T^{2} )^{2}$$
$17$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$( 1 - T + T^{2} )^{2}$$
$23$ $$49 + 42 T + 29 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$( 62 + 16 T + T^{2} )^{2}$$
$31$ $$1156 + 408 T + 110 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4}$$
$41$ $$( -32 + T^{2} )^{2}$$
$43$ $$( -17 - 2 T + T^{2} )^{2}$$
$47$ $$81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4}$$
$53$ $$196 - 112 T + 50 T^{2} - 8 T^{3} + T^{4}$$
$59$ $$16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$61$ $$( 25 - 5 T + T^{2} )^{2}$$
$67$ $$784 + 336 T + 116 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$( 126 + 24 T + T^{2} )^{2}$$
$73$ $$( 49 - 7 T + T^{2} )^{2}$$
$79$ $$3844 + 744 T + 206 T^{2} - 12 T^{3} + T^{4}$$
$83$ $$( -73 - 10 T + T^{2} )^{2}$$
$89$ $$56644 + 952 T + 254 T^{2} - 4 T^{3} + T^{4}$$
$97$ $$( -32 + T^{2} )^{2}$$