# Properties

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

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## 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: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -3 + 3 \zeta_{6} ) q^{3} + q^{5} + 3 \zeta_{6} q^{7} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -3 + 3 \zeta_{6} ) q^{3} + q^{5} + 3 \zeta_{6} q^{7} -6 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} + ( -1 + 4 \zeta_{6} ) q^{13} + ( -3 + 3 \zeta_{6} ) q^{15} + 7 \zeta_{6} q^{17} + \zeta_{6} q^{19} -9 q^{21} + ( -7 + 7 \zeta_{6} ) q^{23} + q^{25} + 9 q^{27} + ( 5 - 5 \zeta_{6} ) q^{29} + 4 q^{31} + 9 \zeta_{6} q^{33} + 3 \zeta_{6} q^{35} + ( 3 - 3 \zeta_{6} ) q^{37} + ( -9 - 3 \zeta_{6} ) q^{39} + ( -7 + 7 \zeta_{6} ) q^{41} -9 \zeta_{6} q^{43} -6 \zeta_{6} q^{45} -8 q^{47} + ( -2 + 2 \zeta_{6} ) q^{49} -21 q^{51} -6 q^{53} + ( 3 - 3 \zeta_{6} ) q^{55} -3 q^{57} + 5 \zeta_{6} q^{59} + 5 \zeta_{6} q^{61} + ( 18 - 18 \zeta_{6} ) q^{63} + ( -1 + 4 \zeta_{6} ) q^{65} + ( 13 - 13 \zeta_{6} ) q^{67} -21 \zeta_{6} q^{69} -3 \zeta_{6} q^{71} -14 q^{73} + ( -3 + 3 \zeta_{6} ) q^{75} + 9 q^{77} + 8 q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -12 q^{83} + 7 \zeta_{6} q^{85} + 15 \zeta_{6} q^{87} + ( -7 + 7 \zeta_{6} ) q^{89} + ( -12 + 9 \zeta_{6} ) q^{91} + ( -12 + 12 \zeta_{6} ) q^{93} + \zeta_{6} q^{95} + 11 \zeta_{6} q^{97} -18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 2 q^{5} + 3 q^{7} - 6 q^{9} + O(q^{10})$$ $$2 q - 3 q^{3} + 2 q^{5} + 3 q^{7} - 6 q^{9} + 3 q^{11} + 2 q^{13} - 3 q^{15} + 7 q^{17} + q^{19} - 18 q^{21} - 7 q^{23} + 2 q^{25} + 18 q^{27} + 5 q^{29} + 8 q^{31} + 9 q^{33} + 3 q^{35} + 3 q^{37} - 21 q^{39} - 7 q^{41} - 9 q^{43} - 6 q^{45} - 16 q^{47} - 2 q^{49} - 42 q^{51} - 12 q^{53} + 3 q^{55} - 6 q^{57} + 5 q^{59} + 5 q^{61} + 18 q^{63} + 2 q^{65} + 13 q^{67} - 21 q^{69} - 3 q^{71} - 28 q^{73} - 3 q^{75} + 18 q^{77} + 16 q^{79} - 9 q^{81} - 24 q^{83} + 7 q^{85} + 15 q^{87} - 7 q^{89} - 15 q^{91} - 12 q^{93} + q^{95} + 11 q^{97} - 36 q^{99} + O(q^{100})$$

## 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 −1.50000 2.59808i 0 1.00000 0 1.50000 2.59808i 0 −3.00000 + 5.19615i 0
321.1 0 −1.50000 + 2.59808i 0 1.00000 0 1.50000 + 2.59808i 0 −3.00000 5.19615i 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.b 2
4.b odd 2 1 260.2.i.d 2
12.b even 2 1 2340.2.q.a 2
13.c even 3 1 inner 1040.2.q.b 2
20.d odd 2 1 1300.2.i.a 2
20.e even 4 2 1300.2.bb.e 4
52.i odd 6 1 3380.2.a.a 1
52.j odd 6 1 260.2.i.d 2
52.j odd 6 1 3380.2.a.b 1
52.l even 12 2 3380.2.f.a 2
156.p even 6 1 2340.2.q.a 2
260.v odd 6 1 1300.2.i.a 2
260.bj even 12 2 1300.2.bb.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.d 2 4.b odd 2 1
260.2.i.d 2 52.j odd 6 1
1040.2.q.b 2 1.a even 1 1 trivial
1040.2.q.b 2 13.c even 3 1 inner
1300.2.i.a 2 20.d odd 2 1
1300.2.i.a 2 260.v odd 6 1
1300.2.bb.e 4 20.e even 4 2
1300.2.bb.e 4 260.bj even 12 2
2340.2.q.a 2 12.b even 2 1
2340.2.q.a 2 156.p even 6 1
3380.2.a.a 1 52.i odd 6 1
3380.2.a.b 1 52.j odd 6 1
3380.2.f.a 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 $$S_{2}^{\mathrm{new}}(1040, [\chi])$$:

 $$T_{3}^{2} + 3 T_{3} + 9$$ $$T_{7}^{2} - 3 T_{7} + 9$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{19}^{2} - T_{19} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$9 - 3 T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$13 - 2 T + T^{2}$$
$17$ $$49 - 7 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$49 + 7 T + T^{2}$$
$29$ $$25 - 5 T + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$9 - 3 T + T^{2}$$
$41$ $$49 + 7 T + T^{2}$$
$43$ $$81 + 9 T + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$25 - 5 T + T^{2}$$
$61$ $$25 - 5 T + T^{2}$$
$67$ $$169 - 13 T + T^{2}$$
$71$ $$9 + 3 T + T^{2}$$
$73$ $$( 14 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$49 + 7 T + T^{2}$$
$97$ $$121 - 11 T + T^{2}$$
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