# 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: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - \nu^{5} - 4 \nu^{4} + 3 \nu^{3} + 2 \nu^{2} - 8 \nu + 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} - \nu^{5} - 3 \nu^{4} + 5 \nu^{3} + 3 \nu^{2} - 12 \nu + 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 4 \nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 11 \nu - 4$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{7} + 7 \nu^{6} - 3 \nu^{5} - 11 \nu^{4} + 15 \nu^{3} + 11 \nu^{2} - 40 \nu + 32$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{7} + 5 \nu^{6} - 3 \nu^{5} - 7 \nu^{4} + 11 \nu^{3} + 7 \nu^{2} - 27 \nu + 22$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{7} - 20 \nu^{6} + 11 \nu^{5} + 30 \nu^{4} - 45 \nu^{3} - 28 \nu^{2} + 116 \nu - 88$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{7} - 22 \nu^{6} + 13 \nu^{5} + 32 \nu^{4} - 47 \nu^{3} - 30 \nu^{2} + 132 \nu - 104$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_{1} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{2} - 7 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} - 4 \beta_{5} + 7 \beta_{4} + \beta_{3} + 3 \beta_{2} - 3 \beta_{1} - 3$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{6} - \beta_{5} - 5 \beta_{4} - \beta_{3} + 9 \beta_{2} + 2 \beta_{1} - 2$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{7} - 12 \beta_{6} - 3 \beta_{5} - 10 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_{1} + 11$$$$)/2$$

## 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 $$T_{3}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$1 - 10 T + 106 T^{2} + 56 T^{3} + 55 T^{4} + 8 T^{5} + 10 T^{6} + 2 T^{7} + T^{8}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$121 - 726 T + 1606 T^{2} - 924 T^{3} + 75 T^{4} + 84 T^{5} - 2 T^{6} - 6 T^{7} + T^{8}$$
$11$ $$1089 - 990 T^{2} + 867 T^{4} - 30 T^{6} + T^{8}$$
$13$ $$28561 + 2704 T^{2} + 1248 T^{3} + 30 T^{4} + 96 T^{5} + 16 T^{6} + T^{8}$$
$17$ $$169 - 130 T + 334 T^{2} + 128 T^{3} + 331 T^{4} - 16 T^{5} + 22 T^{6} + 2 T^{7} + T^{8}$$
$19$ $$( 169 - 78 T - T^{2} + 6 T^{3} + T^{4} )^{2}$$
$23$ $$89401 - 43654 T + 23110 T^{2} - 5104 T^{3} + 1795 T^{4} - 352 T^{5} + 94 T^{6} - 10 T^{7} + T^{8}$$
$29$ $$1 - 40 T + 1618 T^{2} + 704 T^{3} + 643 T^{4} - 64 T^{5} + 82 T^{6} + 8 T^{7} + T^{8}$$
$31$ $$( 64 + 32 T^{2} + T^{4} )^{2}$$
$37$ $$1 + 66 T + 1402 T^{2} - 3300 T^{3} + 2367 T^{4} + 300 T^{5} - 38 T^{6} - 6 T^{7} + T^{8}$$
$41$ $$( 1 + 6 T + 11 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$43$ $$169 + 130 T + 334 T^{2} - 128 T^{3} + 331 T^{4} + 16 T^{5} + 22 T^{6} - 2 T^{7} + T^{8}$$
$47$ $$1763584 + 350464 T^{2} + 14304 T^{4} + 208 T^{6} + T^{8}$$
$53$ $$( -48 + 36 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$59$ $$9 - 108 T + 486 T^{2} - 648 T^{3} + 183 T^{4} + 216 T^{5} + 30 T^{6} - 12 T^{7} + T^{8}$$
$61$ $$1590121 + 1215604 T + 603958 T^{2} + 178096 T^{3} + 38311 T^{4} + 5296 T^{5} + 526 T^{6} + 28 T^{7} + T^{8}$$
$67$ $$7667361 - 847314 T - 284454 T^{2} + 34884 T^{3} + 9615 T^{4} - 684 T^{5} - 102 T^{6} + 6 T^{7} + T^{8}$$
$71$ $$109767529 - 2263032 T - 2268434 T^{2} + 47088 T^{3} + 37047 T^{4} - 218 T^{6} + T^{8}$$
$73$ $$2930944 + 404608 T^{2} + 16944 T^{4} + 232 T^{6} + T^{8}$$
$79$ $$( 4432 + 640 T - 132 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$83$ $$36864 + 73728 T^{2} + 7104 T^{4} + 192 T^{6} + T^{8}$$
$89$ $$78375609 + 34420464 T + 3179718 T^{2} - 816480 T^{3} + 4143 T^{4} + 5040 T^{5} - 18 T^{6} - 24 T^{7} + T^{8}$$
$97$ $$196249 + 71766 T - 16946 T^{2} - 9396 T^{3} + 2187 T^{4} + 1740 T^{5} + 358 T^{6} + 30 T^{7} + T^{8}$$