# Properties

 Label 1040.2.da.b Level $1040$ Weight $2$ Character orbit 1040.da Analytic conductor $8.304$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ x^8 - 4*x^7 + 5*x^6 + 2*x^5 - 11*x^4 + 4*x^3 + 20*x^2 - 32*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{2} + \beta_1) q^{3} + (\beta_{6} - \beta_1) q^{5} + (3 \beta_{6} - \beta_{5} + \beta_{4}) q^{7} + (\beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_1 - 1) q^{9}+O(q^{10})$$ q + (-b7 - 2*b6 - b4 + b2 + b1) * q^3 + (b6 - b1) * q^5 + (3*b6 - b5 + b4) * q^7 + (b7 + b5 + 2*b4 - 2*b3 - b1 - 1) * q^9 $$q + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{2} + \beta_1) q^{3} + (\beta_{6} - \beta_1) q^{5} + (3 \beta_{6} - \beta_{5} + \beta_{4}) q^{7} + (\beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_1 - 1) q^{9} + (\beta_{7} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{13} + ( - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{15} + (\beta_{7} - \beta_{6} + \beta_{4} + 3 \beta_1 - 1) q^{17} + ( - 4 \beta_{6} - \beta_{4} - 1) q^{19} + (2 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 1) q^{21} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{23}+ \cdots + ( - 2 \beta_{7} - 8 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_1) q^{99}+O(q^{100})$$ q + (-b7 - 2*b6 - b4 + b2 + b1) * q^3 + (b6 - b1) * q^5 + (3*b6 - b5 + b4) * q^7 + (b7 + b5 + 2*b4 - 2*b3 - b1 - 1) * q^9 + (b7 + b3 + b2 + b1 - 1) * q^11 + (-b7 + b6 - b5 - 2*b4 + 2*b3 + b2 + b1) * q^13 + (-b4 + b3 + b1 + 1) * q^15 + (b7 - b6 + b4 + 3*b1 - 1) * q^17 + (-4*b6 - b4 - 1) * q^19 + (2*b6 - 2*b5 - 4*b4 + 2*b3 + 1) * q^21 + (-2*b7 - 2*b6 - 2*b5 + 2*b4 + b3 + 2*b2 + 2*b1 - 1) * q^23 - q^25 + (b6 + b5 - b4 + b3 + 2*b1 + 1) * q^27 + (-b7 - 4*b6 - 2*b5 - 2*b4 + b3 + b2 + 3*b1 - 1) * q^29 + (2*b6 - 4*b4 - 2*b1 + 2) * q^31 + (-5*b6 + b5 + 3*b4 + 4) * q^33 + (-b7 + 2*b4 - b1 - 2) * q^35 + (b7 - b4 + 2*b3 + b2 + b1) * q^37 + (b7 - 4*b6 - b5 + 3*b4 + 2*b2 - b1 - 3) * q^39 + (-b4 - 2*b1 + 2) * q^41 + (-b7 + b6 - b4 - 3*b1 + 1) * q^43 + (-b7 - 2*b6 - b5 + 2*b2 + b1 - 1) * q^45 + (4*b7 + 2*b5 + 6*b4 - 2*b3 - 2*b2 - 2) * q^47 + (-5*b7 - 2*b5 + b4 + b3 + 5*b2 + b1 - 1) * q^49 + (-b5 + b4 - b3 - b2 - 2*b1 + 1) * q^51 + (-b5 + b4 - b3 + b2 - 4) * q^53 + (b7 + 2*b6 - 2*b5 + b4 + b3 - b2 - 1) * q^55 + (2*b7 + 3*b6 + 4*b5 + 6*b4 - 4*b3 - b2 - 6*b1 - 1) * q^57 + (-b7 + 2*b6 - b5 + b4 + 2*b2 + b1) * q^59 + (b7 - b5 + 7*b4 + 2*b3 + 3*b1 - 8) * q^61 + (3*b7 + b4 - 3*b3 + 3*b2 - b1 + 1) * q^63 + (b7 + 2*b6 + b4 + b3 - 2*b2 - 1) * q^65 + (b3 + 8*b1 - 1) * q^67 + (2*b5 + 5*b4 - 4*b3 - 4*b1 - 3) * q^69 + (2*b6 - 6*b5 + 3*b4 - 3) * q^71 + (-2*b7 - 2*b6 - 3*b5 + 5*b4 + 3*b3 + b2 + 4*b1 - 4) * q^73 + (b7 + 2*b6 + b4 - b2 - b1) * q^75 + (5*b6 - 2*b5 + 2*b4 - 2*b3 - b2 + 2*b1 - 5) * q^77 + (-2*b6 + 3*b5 - 3*b4 + 3*b3 - b2 + 4) * q^79 + (b7 - 8*b6 + 2*b5 + 2*b4 - b3 - b2 + 3*b1 + 1) * q^81 + (-4*b6 + 2*b5 + 6*b4 - 2*b3 + 2*b1 - 2) * q^83 + (-b5 + 2*b4 + 1) * q^85 + (4*b7 + 5*b6 + 3*b5 + 9*b4 - 6*b3 - 12*b1 - 6) * q^87 + (4*b7 + b4 - 2*b3 + 4*b2 - 2*b1) * q^89 + (-5*b7 + 2*b6 - 2*b5 + 2*b4 + 3*b3 - b2 + b1 - 4) * q^91 + (2*b7 + 2*b3 + 2*b2 - 2) * q^93 + (-b6 - 4*b4 + 2*b1 + 4) * q^95 + (b7 + 3*b6 - 2*b5 - b4 - 2*b2 - b1 - 3) * q^97 + (-2*b7 - 8*b6 + b5 + b4 - b3 + b2 + 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10})$$ 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9 $$8 q - 2 q^{3} + 6 q^{7} - 4 q^{9} + 6 q^{15} - 2 q^{17} - 12 q^{19} + 10 q^{23} - 8 q^{25} + 4 q^{27} - 8 q^{29} + 42 q^{33} - 10 q^{35} + 6 q^{37} + 12 q^{41} + 2 q^{43} + 12 q^{49} + 8 q^{51} - 24 q^{53} + 12 q^{59} - 28 q^{61} + 24 q^{63} - 8 q^{65} - 6 q^{67} - 16 q^{69} + 2 q^{75} - 36 q^{77} + 16 q^{79} + 8 q^{81} + 18 q^{85} - 22 q^{87} + 24 q^{89} - 28 q^{91} + 16 q^{95} - 30 q^{97}+O(q^{100})$$ 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9 + 6 * q^15 - 2 * q^17 - 12 * q^19 + 10 * q^23 - 8 * q^25 + 4 * q^27 - 8 * q^29 + 42 * q^33 - 10 * q^35 + 6 * q^37 + 12 * q^41 + 2 * q^43 + 12 * q^49 + 8 * q^51 - 24 * q^53 + 12 * q^59 - 28 * q^61 + 24 * q^63 - 8 * q^65 - 6 * q^67 - 16 * q^69 + 2 * q^75 - 36 * q^77 + 16 * q^79 + 8 * q^81 + 18 * q^85 - 22 * q^87 + 24 * q^89 - 28 * q^91 + 16 * q^95 - 30 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 2\nu^{2} - 8\nu + 8 ) / 8$$ (-v^7 + 2*v^6 - v^5 - 4*v^4 + 3*v^3 + 2*v^2 - 8*v + 8) / 8 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 3\nu^{6} - \nu^{5} - 3\nu^{4} + 5\nu^{3} + 3\nu^{2} - 12\nu + 12 ) / 4$$ (-v^7 + 3*v^6 - v^5 - 3*v^4 + 5*v^3 + 3*v^2 - 12*v + 12) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} + 4\nu^{4} - 2\nu^{3} - 6\nu^{2} + 11\nu - 4 ) / 2$$ (v^7 - 2*v^6 + 4*v^4 - 2*v^3 - 6*v^2 + 11*v - 4) / 2 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4$$ (-3*v^7 + 7*v^6 - 3*v^5 - 11*v^4 + 15*v^3 + 11*v^2 - 40*v + 32) / 4 $$\beta_{5}$$ $$=$$ $$( -2\nu^{7} + 5\nu^{6} - 3\nu^{5} - 7\nu^{4} + 11\nu^{3} + 7\nu^{2} - 27\nu + 22 ) / 2$$ (-2*v^7 + 5*v^6 - 3*v^5 - 7*v^4 + 11*v^3 + 7*v^2 - 27*v + 22) / 2 $$\beta_{6}$$ $$=$$ $$( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8$$ (7*v^7 - 20*v^6 + 11*v^5 + 30*v^4 - 45*v^3 - 28*v^2 + 116*v - 88) / 8 $$\beta_{7}$$ $$=$$ $$( 9\nu^{7} - 22\nu^{6} + 13\nu^{5} + 32\nu^{4} - 47\nu^{3} - 30\nu^{2} + 132\nu - 104 ) / 8$$ (9*v^7 - 22*v^6 + 13*v^5 + 32*v^4 - 47*v^3 - 30*v^2 + 132*v - 104) / 8
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{5} + \beta _1 + 1 ) / 2$$ (b7 + b5 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3\beta _1 + 3 ) / 2$$ (b7 + 2*b5 - b4 - b3 - b2 - 3*b1 + 3) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} + \beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 - 2 ) / 2$$ (2*b7 + b5 + 3*b4 + b3 - b2 - 2*b1 - 2) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{2} - 7\beta _1 - 1 ) / 2$$ (-b7 + 2*b6 + b5 + 2*b2 - 7*b1 - 1) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{7} - 4\beta_{5} + 7\beta_{4} + \beta_{3} + 3\beta_{2} - 3\beta _1 - 3 ) / 2$$ (b7 - 4*b5 + 7*b4 + b3 + 3*b2 - 3*b1 - 3) / 2 $$\nu^{6}$$ $$=$$ $$( -2\beta_{6} - \beta_{5} - 5\beta_{4} - \beta_{3} + 9\beta_{2} + 2\beta _1 - 2 ) / 2$$ (-2*b6 - b5 - 5*b4 - b3 + 9*b2 + 2*b1 - 2) / 2 $$\nu^{7}$$ $$=$$ $$( 3\beta_{7} - 12\beta_{6} - 3\beta_{5} - 10\beta_{4} - 2\beta_{3} + 2\beta_{2} - \beta _1 + 11 ) / 2$$ (3*b7 - 12*b6 - 3*b5 - 10*b4 - 2*b3 + 2*b2 - b1 + 11) / 2

## Character values

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

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{4}$$ $$1$$

## 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
 1.20036 − 0.747754i −1.27597 − 0.609843i 0.665665 + 1.24775i 1.40994 + 0.109843i 1.20036 + 0.747754i −1.27597 + 0.609843i 0.665665 − 1.24775i 1.40994 − 0.109843i
0 −1.41342 + 2.44811i 0 1.00000i 0 1.64996 0.952606i 0 −2.49551 4.32235i 0
641.2 0 −0.800098 + 1.38581i 0 1.00000i 0 0.287734 0.166123i 0 0.219687 + 0.380509i 0
641.3 0 0.0473938 0.0820885i 0 1.00000i 0 4.18016 2.41342i 0 1.49551 + 2.59030i 0
641.4 0 1.16612 2.01978i 0 1.00000i 0 −3.11786 + 1.80010i 0 −1.21969 2.11256i 0
881.1 0 −1.41342 2.44811i 0 1.00000i 0 1.64996 + 0.952606i 0 −2.49551 + 4.32235i 0
881.2 0 −0.800098 1.38581i 0 1.00000i 0 0.287734 + 0.166123i 0 0.219687 0.380509i 0
881.3 0 0.0473938 + 0.0820885i 0 1.00000i 0 4.18016 + 2.41342i 0 1.49551 2.59030i 0
881.4 0 1.16612 + 2.01978i 0 1.00000i 0 −3.11786 1.80010i 0 −1.21969 + 2.11256i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.4 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.b 8
4.b odd 2 1 65.2.m.a 8
12.b even 2 1 585.2.bu.c 8
13.e even 6 1 inner 1040.2.da.b 8
20.d odd 2 1 325.2.n.d 8
20.e even 4 1 325.2.m.b 8
20.e even 4 1 325.2.m.c 8
52.b odd 2 1 845.2.m.g 8
52.f even 4 1 845.2.e.m 8
52.f even 4 1 845.2.e.n 8
52.i odd 6 1 65.2.m.a 8
52.i odd 6 1 845.2.c.g 8
52.j odd 6 1 845.2.c.g 8
52.j odd 6 1 845.2.m.g 8
52.l even 12 1 845.2.a.l 4
52.l even 12 1 845.2.a.m 4
52.l even 12 1 845.2.e.m 8
52.l even 12 1 845.2.e.n 8
156.r even 6 1 585.2.bu.c 8
156.v odd 12 1 7605.2.a.cf 4
156.v odd 12 1 7605.2.a.cj 4
260.w odd 6 1 325.2.n.d 8
260.bc even 12 1 4225.2.a.bi 4
260.bc even 12 1 4225.2.a.bl 4
260.bg even 12 1 325.2.m.b 8
260.bg even 12 1 325.2.m.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 4.b odd 2 1
65.2.m.a 8 52.i odd 6 1
325.2.m.b 8 20.e even 4 1
325.2.m.b 8 260.bg even 12 1
325.2.m.c 8 20.e even 4 1
325.2.m.c 8 260.bg even 12 1
325.2.n.d 8 20.d odd 2 1
325.2.n.d 8 260.w odd 6 1
585.2.bu.c 8 12.b even 2 1
585.2.bu.c 8 156.r even 6 1
845.2.a.l 4 52.l even 12 1
845.2.a.m 4 52.l even 12 1
845.2.c.g 8 52.i odd 6 1
845.2.c.g 8 52.j odd 6 1
845.2.e.m 8 52.f even 4 1
845.2.e.m 8 52.l even 12 1
845.2.e.n 8 52.f even 4 1
845.2.e.n 8 52.l even 12 1
845.2.m.g 8 52.b odd 2 1
845.2.m.g 8 52.j odd 6 1
1040.2.da.b 8 1.a even 1 1 trivial
1040.2.da.b 8 13.e even 6 1 inner
4225.2.a.bi 4 260.bc even 12 1
4225.2.a.bl 4 260.bc even 12 1
7605.2.a.cf 4 156.v odd 12 1
7605.2.a.cj 4 156.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 2T_{3}^{7} + 10T_{3}^{6} + 8T_{3}^{5} + 55T_{3}^{4} + 56T_{3}^{3} + 106T_{3}^{2} - 10T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 2 T^{7} + 10 T^{6} + 8 T^{5} + \cdots + 1$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8} - 6 T^{7} - 2 T^{6} + 84 T^{5} + \cdots + 121$$
$11$ $$T^{8} - 30 T^{6} + 867 T^{4} + \cdots + 1089$$
$13$ $$T^{8} + 16 T^{6} + 96 T^{5} + \cdots + 28561$$
$17$ $$T^{8} + 2 T^{7} + 22 T^{6} - 16 T^{5} + \cdots + 169$$
$19$ $$(T^{4} + 6 T^{3} - T^{2} - 78 T + 169)^{2}$$
$23$ $$T^{8} - 10 T^{7} + 94 T^{6} + \cdots + 89401$$
$29$ $$T^{8} + 8 T^{7} + 82 T^{6} - 64 T^{5} + \cdots + 1$$
$31$ $$(T^{4} + 32 T^{2} + 64)^{2}$$
$37$ $$T^{8} - 6 T^{7} - 38 T^{6} + 300 T^{5} + \cdots + 1$$
$41$ $$(T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1)^{2}$$
$43$ $$T^{8} - 2 T^{7} + 22 T^{6} + 16 T^{5} + \cdots + 169$$
$47$ $$T^{8} + 208 T^{6} + 14304 T^{4} + \cdots + 1763584$$
$53$ $$(T^{4} + 12 T^{3} + 36 T^{2} - 48)^{2}$$
$59$ $$T^{8} - 12 T^{7} + 30 T^{6} + 216 T^{5} + \cdots + 9$$
$61$ $$T^{8} + 28 T^{7} + 526 T^{6} + \cdots + 1590121$$
$67$ $$T^{8} + 6 T^{7} - 102 T^{6} + \cdots + 7667361$$
$71$ $$T^{8} - 218 T^{6} + \cdots + 109767529$$
$73$ $$T^{8} + 232 T^{6} + 16944 T^{4} + \cdots + 2930944$$
$79$ $$(T^{4} - 8 T^{3} - 132 T^{2} + 640 T + 4432)^{2}$$
$83$ $$T^{8} + 192 T^{6} + 7104 T^{4} + \cdots + 36864$$
$89$ $$T^{8} - 24 T^{7} - 18 T^{6} + \cdots + 78375609$$
$97$ $$T^{8} + 30 T^{7} + 358 T^{6} + \cdots + 196249$$