# Properties

 Label 1078.2.a.o Level $1078$ Weight $2$ Character orbit 1078.a Self dual yes Analytic conductor $8.608$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta q^{5} - q^{8} -3 q^{9} +O(q^{10})$$ $$q - q^{2} + q^{4} + \beta q^{5} - q^{8} -3 q^{9} -\beta q^{10} + q^{11} -2 \beta q^{13} + q^{16} -\beta q^{17} + 3 q^{18} -3 \beta q^{19} + \beta q^{20} - q^{22} -8 q^{23} + 3 q^{25} + 2 \beta q^{26} -6 q^{29} + 3 \beta q^{31} - q^{32} + \beta q^{34} -3 q^{36} -6 q^{37} + 3 \beta q^{38} -\beta q^{40} + 3 \beta q^{41} -4 q^{43} + q^{44} -3 \beta q^{45} + 8 q^{46} + \beta q^{47} -3 q^{50} -2 \beta q^{52} + 6 q^{53} + \beta q^{55} + 6 q^{58} + 2 \beta q^{59} + 2 \beta q^{61} -3 \beta q^{62} + q^{64} -16 q^{65} -4 q^{67} -\beta q^{68} + 3 q^{72} + 3 \beta q^{73} + 6 q^{74} -3 \beta q^{76} + \beta q^{80} + 9 q^{81} -3 \beta q^{82} + \beta q^{83} -8 q^{85} + 4 q^{86} - q^{88} -4 \beta q^{89} + 3 \beta q^{90} -8 q^{92} -\beta q^{94} -24 q^{95} -4 \beta q^{97} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 6q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 6q^{9} + 2q^{11} + 2q^{16} + 6q^{18} - 2q^{22} - 16q^{23} + 6q^{25} - 12q^{29} - 2q^{32} - 6q^{36} - 12q^{37} - 8q^{43} + 2q^{44} + 16q^{46} - 6q^{50} + 12q^{53} + 12q^{58} + 2q^{64} - 32q^{65} - 8q^{67} + 6q^{72} + 12q^{74} + 18q^{81} - 16q^{85} + 8q^{86} - 2q^{88} - 16q^{92} - 48q^{95} - 6q^{99} + O(q^{100})$$

## 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 0 1.00000 −2.82843 0 0 −1.00000 −3.00000 2.82843
1.2 −1.00000 0 1.00000 2.82843 0 0 −1.00000 −3.00000 −2.82843
 $$n$$: 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.o 2
3.b odd 2 1 9702.2.a.dk 2
4.b odd 2 1 8624.2.a.bo 2
7.b odd 2 1 inner 1078.2.a.o 2
7.c even 3 2 1078.2.e.u 4
7.d odd 6 2 1078.2.e.u 4
21.c even 2 1 9702.2.a.dk 2
28.d even 2 1 8624.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.o 2 1.a even 1 1 trivial
1078.2.a.o 2 7.b odd 2 1 inner
1078.2.e.u 4 7.c even 3 2
1078.2.e.u 4 7.d odd 6 2
8624.2.a.bo 2 4.b odd 2 1
8624.2.a.bo 2 28.d even 2 1
9702.2.a.dk 2 3.b odd 2 1
9702.2.a.dk 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1078))$$:

 $$T_{3}$$ $$T_{5}^{2} - 8$$ $$T_{13}^{2} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$-8 + T^{2}$$
$19$ $$-72 + T^{2}$$
$23$ $$( 8 + T )^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$-72 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-32 + T^{2}$$
$61$ $$-32 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$-72 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$-8 + T^{2}$$
$89$ $$-128 + T^{2}$$
$97$ $$-128 + T^{2}$$