# Properties

 Label 1078.2.a.r Level $1078$ Weight $2$ Character orbit 1078.a Self dual yes Analytic conductor $8.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} + 5 q^{9} +O(q^{10})$$ $$q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} + 5 q^{9} - q^{11} + \beta q^{12} -\beta q^{13} + q^{16} + \beta q^{17} -5 q^{18} + 2 \beta q^{19} + q^{22} + 8 q^{23} -\beta q^{24} -5 q^{25} + \beta q^{26} + 2 \beta q^{27} + 2 q^{29} + 3 \beta q^{31} - q^{32} -\beta q^{33} -\beta q^{34} + 5 q^{36} + 2 q^{37} -2 \beta q^{38} -8 q^{39} + \beta q^{41} -4 q^{43} - q^{44} -8 q^{46} + \beta q^{47} + \beta q^{48} + 5 q^{50} + 8 q^{51} -\beta q^{52} + 14 q^{53} -2 \beta q^{54} + 16 q^{57} -2 q^{58} -3 \beta q^{59} -3 \beta q^{61} -3 \beta q^{62} + q^{64} + \beta q^{66} + 4 q^{67} + \beta q^{68} + 8 \beta q^{69} -5 q^{72} -5 \beta q^{73} -2 q^{74} -5 \beta q^{75} + 2 \beta q^{76} + 8 q^{78} -16 q^{79} + q^{81} -\beta q^{82} -6 \beta q^{83} + 4 q^{86} + 2 \beta q^{87} + q^{88} -4 \beta q^{89} + 8 q^{92} + 24 q^{93} -\beta q^{94} -\beta q^{96} -6 \beta q^{97} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} + 10q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} + 10q^{9} - 2q^{11} + 2q^{16} - 10q^{18} + 2q^{22} + 16q^{23} - 10q^{25} + 4q^{29} - 2q^{32} + 10q^{36} + 4q^{37} - 16q^{39} - 8q^{43} - 2q^{44} - 16q^{46} + 10q^{50} + 16q^{51} + 28q^{53} + 32q^{57} - 4q^{58} + 2q^{64} + 8q^{67} - 10q^{72} - 4q^{74} + 16q^{78} - 32q^{79} + 2q^{81} + 8q^{86} + 2q^{88} + 16q^{92} + 48q^{93} - 10q^{99} + O(q^{100})$$

## 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
−1.00000 −2.82843 1.00000 0 2.82843 0 −1.00000 5.00000 0
1.2 −1.00000 2.82843 1.00000 0 −2.82843 0 −1.00000 5.00000 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
$$2$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.a.r 2
3.b odd 2 1 9702.2.a.dn 2
4.b odd 2 1 8624.2.a.by 2
7.b odd 2 1 inner 1078.2.a.r 2
7.c even 3 2 1078.2.e.r 4
7.d odd 6 2 1078.2.e.r 4
21.c even 2 1 9702.2.a.dn 2
28.d even 2 1 8624.2.a.by 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.r 2 1.a even 1 1 trivial
1078.2.a.r 2 7.b odd 2 1 inner
1078.2.e.r 4 7.c even 3 2
1078.2.e.r 4 7.d odd 6 2
8624.2.a.by 2 4.b odd 2 1
8624.2.a.by 2 28.d even 2 1
9702.2.a.dn 2 3.b odd 2 1
9702.2.a.dn 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1078))$$:

 $$T_{3}^{2} - 8$$ $$T_{5}$$ $$T_{13}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$-8 + T^{2}$$
$19$ $$-32 + T^{2}$$
$23$ $$( -8 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$-8 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -14 + T )^{2}$$
$59$ $$-72 + T^{2}$$
$61$ $$-72 + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$-200 + T^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$-288 + T^{2}$$
$89$ $$-128 + T^{2}$$
$97$ $$-288 + T^{2}$$