# Properties

 Label 1040.2.q.i Level $1040$ Weight $2$ Character orbit 1040.q Analytic conductor $8.304$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 −1.00000 0 −0.500000 + 0.866025i 0 1.00000 1.73205i 0
321.1 0 0.500000 0.866025i 0 −1.00000 0 −0.500000 0.866025i 0 1.00000 + 1.73205i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.q.i 2
4.b odd 2 1 260.2.i.a 2
12.b even 2 1 2340.2.q.f 2
13.c even 3 1 inner 1040.2.q.i 2
20.d odd 2 1 1300.2.i.d 2
20.e even 4 2 1300.2.bb.b 4
52.i odd 6 1 3380.2.a.i 1
52.j odd 6 1 260.2.i.a 2
52.j odd 6 1 3380.2.a.f 1
52.l even 12 2 3380.2.f.d 2
156.p even 6 1 2340.2.q.f 2
260.v odd 6 1 1300.2.i.d 2
260.bj even 12 2 1300.2.bb.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.a 2 4.b odd 2 1
260.2.i.a 2 52.j odd 6 1
1040.2.q.i 2 1.a even 1 1 trivial
1040.2.q.i 2 13.c even 3 1 inner
1300.2.i.d 2 20.d odd 2 1
1300.2.i.d 2 260.v odd 6 1
1300.2.bb.b 4 20.e even 4 2
1300.2.bb.b 4 260.bj even 12 2
2340.2.q.f 2 12.b even 2 1
2340.2.q.f 2 156.p even 6 1
3380.2.a.f 1 52.j odd 6 1
3380.2.a.i 1 52.i odd 6 1
3380.2.f.d 2 52.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{19}^{2} - 5T_{19} + 25$$ T19^2 - 5*T19 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} - 2T + 13$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} - 9T + 81$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} - 7T + 49$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$T^{2} - 9T + 81$$
$73$ $$(T - 2)^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} + 17T + 289$$