# Properties

 Label 1040.2.a.j Level $1040$ Weight $2$ Character orbit 1040.a Self dual yes Analytic conductor $8.304$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.30444181021$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 −1.41421 0 1.00000 0 0.828427 0 −1.00000 0
1.2 0 1.41421 0 1.00000 0 −4.82843 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.a.j 2
3.b odd 2 1 9360.2.a.cd 2
4.b odd 2 1 65.2.a.b 2
5.b even 2 1 5200.2.a.bu 2
8.b even 2 1 4160.2.a.z 2
8.d odd 2 1 4160.2.a.bf 2
12.b even 2 1 585.2.a.m 2
20.d odd 2 1 325.2.a.i 2
20.e even 4 2 325.2.b.f 4
28.d even 2 1 3185.2.a.j 2
44.c even 2 1 7865.2.a.j 2
52.b odd 2 1 845.2.a.g 2
52.f even 4 2 845.2.c.b 4
52.i odd 6 2 845.2.e.c 4
52.j odd 6 2 845.2.e.h 4
52.l even 12 4 845.2.m.f 8
60.h even 2 1 2925.2.a.u 2
60.l odd 4 2 2925.2.c.r 4
156.h even 2 1 7605.2.a.x 2
260.g odd 2 1 4225.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 4.b odd 2 1
325.2.a.i 2 20.d odd 2 1
325.2.b.f 4 20.e even 4 2
585.2.a.m 2 12.b even 2 1
845.2.a.g 2 52.b odd 2 1
845.2.c.b 4 52.f even 4 2
845.2.e.c 4 52.i odd 6 2
845.2.e.h 4 52.j odd 6 2
845.2.m.f 8 52.l even 12 4
1040.2.a.j 2 1.a even 1 1 trivial
2925.2.a.u 2 60.h even 2 1
2925.2.c.r 4 60.l odd 4 2
3185.2.a.j 2 28.d even 2 1
4160.2.a.z 2 8.b even 2 1
4160.2.a.bf 2 8.d odd 2 1
4225.2.a.r 2 260.g odd 2 1
5200.2.a.bu 2 5.b even 2 1
7605.2.a.x 2 156.h even 2 1
7865.2.a.j 2 44.c even 2 1
9360.2.a.cd 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1040))$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{7}^{2} + 4T_{7} - 4$$ T7^2 + 4*T7 - 4 $$T_{11}^{2} + 4T_{11} + 2$$ T11^2 + 4*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 4T - 4$$
$11$ $$T^{2} + 4T + 2$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$T^{2} + 4T + 2$$
$23$ $$T^{2} - 2$$
$29$ $$T^{2} - 32$$
$31$ $$T^{2} + 12T + 18$$
$37$ $$T^{2} - 72$$
$41$ $$T^{2} + 12T + 28$$
$43$ $$T^{2} - 8T - 34$$
$47$ $$T^{2} - 4T - 4$$
$53$ $$T^{2} + 12T - 36$$
$59$ $$T^{2} + 12T + 18$$
$61$ $$(T + 8)^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$T^{2} + 4T - 94$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} - 72$$
$83$ $$T^{2} - 12T + 28$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 4T - 28$$