# Properties

 Label 1040.2.da.a Level $1040$ Weight $2$ Character orbit 1040.da Analytic conductor $8.304$ Analytic rank $0$ Dimension $4$ 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.da (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 130) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −1.36603 + 2.36603i 0 1.00000i 0 2.59808 1.50000i 0 −2.23205 3.86603i 0
641.2 0 0.366025 0.633975i 0 1.00000i 0 −2.59808 + 1.50000i 0 1.23205 + 2.13397i 0
881.1 0 −1.36603 2.36603i 0 1.00000i 0 2.59808 + 1.50000i 0 −2.23205 + 3.86603i 0
881.2 0 0.366025 + 0.633975i 0 1.00000i 0 −2.59808 1.50000i 0 1.23205 2.13397i 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.da.a 4
4.b odd 2 1 130.2.l.a 4
12.b even 2 1 1170.2.bs.c 4
13.e even 6 1 inner 1040.2.da.a 4
20.d odd 2 1 650.2.m.a 4
20.e even 4 1 650.2.n.a 4
20.e even 4 1 650.2.n.b 4
52.b odd 2 1 1690.2.l.g 4
52.f even 4 1 1690.2.e.l 4
52.f even 4 1 1690.2.e.n 4
52.i odd 6 1 130.2.l.a 4
52.i odd 6 1 1690.2.d.f 4
52.j odd 6 1 1690.2.d.f 4
52.j odd 6 1 1690.2.l.g 4
52.l even 12 1 1690.2.a.j 2
52.l even 12 1 1690.2.a.m 2
52.l even 12 1 1690.2.e.l 4
52.l even 12 1 1690.2.e.n 4
156.r even 6 1 1170.2.bs.c 4
260.w odd 6 1 650.2.m.a 4
260.bc even 12 1 8450.2.a.bf 2
260.bc even 12 1 8450.2.a.bm 2
260.bg even 12 1 650.2.n.a 4
260.bg even 12 1 650.2.n.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 4.b odd 2 1
130.2.l.a 4 52.i odd 6 1
650.2.m.a 4 20.d odd 2 1
650.2.m.a 4 260.w odd 6 1
650.2.n.a 4 20.e even 4 1
650.2.n.a 4 260.bg even 12 1
650.2.n.b 4 20.e even 4 1
650.2.n.b 4 260.bg even 12 1
1040.2.da.a 4 1.a even 1 1 trivial
1040.2.da.a 4 13.e even 6 1 inner
1170.2.bs.c 4 12.b even 2 1
1170.2.bs.c 4 156.r even 6 1
1690.2.a.j 2 52.l even 12 1
1690.2.a.m 2 52.l even 12 1
1690.2.d.f 4 52.i odd 6 1
1690.2.d.f 4 52.j odd 6 1
1690.2.e.l 4 52.f even 4 1
1690.2.e.l 4 52.l even 12 1
1690.2.e.n 4 52.f even 4 1
1690.2.e.n 4 52.l even 12 1
1690.2.l.g 4 52.b odd 2 1
1690.2.l.g 4 52.j odd 6 1
8450.2.a.bf 2 260.bc even 12 1
8450.2.a.bm 2 260.bc even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 2T_{3}^{3} + 6T_{3}^{2} - 4T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$T^{4} - 9T^{2} + 81$$
$13$ $$(T^{2} - 7 T + 13)^{2}$$
$17$ $$T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324$$
$19$ $$T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9$$
$23$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$29$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$31$ $$T^{4} + 24T^{2} + 36$$
$37$ $$T^{4} - 18 T^{3} + 99 T^{2} + 162 T + 81$$
$41$ $$(T^{2} + 18 T + 108)^{2}$$
$43$ $$(T^{2} - 2 T + 4)^{2}$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$(T^{2} + 6 T - 3)^{2}$$
$59$ $$(T^{2} + 18 T + 108)^{2}$$
$61$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$67$ $$T^{4}$$
$71$ $$T^{4} - 36T^{2} + 1296$$
$73$ $$T^{4} + 168T^{2} + 4356$$
$79$ $$(T^{2} + 2 T - 26)^{2}$$
$83$ $$T^{4} + 72T^{2} + 324$$
$89$ $$T^{4} + 18 T^{3} - 9 T^{2} + \cdots + 13689$$
$97$ $$T^{4} - 42 T^{3} + 726 T^{2} + \cdots + 19044$$