# Properties

 Label 1078.2.a.x Level $1078$ Weight $2$ Character orbit 1078.a Self dual yes Analytic conductor $8.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$1$$ Twist minimal: no (minimal twist has level 154) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + ( 2 - \beta ) q^{5} + ( 1 + \beta ) q^{6} + q^{8} + 2 \beta q^{9} +O(q^{10})$$ $$q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + ( 2 - \beta ) q^{5} + ( 1 + \beta ) q^{6} + q^{8} + 2 \beta q^{9} + ( 2 - \beta ) q^{10} + q^{11} + ( 1 + \beta ) q^{12} + ( 1 + 2 \beta ) q^{13} + \beta q^{15} + q^{16} + ( 2 - 4 \beta ) q^{17} + 2 \beta q^{18} + ( 2 - \beta ) q^{19} + ( 2 - \beta ) q^{20} + q^{22} + ( -2 - 3 \beta ) q^{23} + ( 1 + \beta ) q^{24} + ( 1 - 4 \beta ) q^{25} + ( 1 + 2 \beta ) q^{26} + ( 1 - \beta ) q^{27} + ( -3 + 4 \beta ) q^{29} + \beta q^{30} + 4 q^{31} + q^{32} + ( 1 + \beta ) q^{33} + ( 2 - 4 \beta ) q^{34} + 2 \beta q^{36} + ( -8 - \beta ) q^{37} + ( 2 - \beta ) q^{38} + ( 5 + 3 \beta ) q^{39} + ( 2 - \beta ) q^{40} + ( 4 + \beta ) q^{41} -4 \beta q^{43} + q^{44} + ( -4 + 4 \beta ) q^{45} + ( -2 - 3 \beta ) q^{46} + ( 2 + 6 \beta ) q^{47} + ( 1 + \beta ) q^{48} + ( 1 - 4 \beta ) q^{50} + ( -6 - 2 \beta ) q^{51} + ( 1 + 2 \beta ) q^{52} + ( -2 + 7 \beta ) q^{53} + ( 1 - \beta ) q^{54} + ( 2 - \beta ) q^{55} + \beta q^{57} + ( -3 + 4 \beta ) q^{58} + ( 7 - \beta ) q^{59} + \beta q^{60} + ( -9 - 2 \beta ) q^{61} + 4 q^{62} + q^{64} + ( -2 + 3 \beta ) q^{65} + ( 1 + \beta ) q^{66} + ( 7 - 3 \beta ) q^{67} + ( 2 - 4 \beta ) q^{68} + ( -8 - 5 \beta ) q^{69} + ( -4 - 5 \beta ) q^{71} + 2 \beta q^{72} + ( 8 + \beta ) q^{73} + ( -8 - \beta ) q^{74} + ( -7 - 3 \beta ) q^{75} + ( 2 - \beta ) q^{76} + ( 5 + 3 \beta ) q^{78} + ( -9 - 3 \beta ) q^{79} + ( 2 - \beta ) q^{80} + ( -1 - 6 \beta ) q^{81} + ( 4 + \beta ) q^{82} + ( -2 + 10 \beta ) q^{83} + ( 12 - 10 \beta ) q^{85} -4 \beta q^{86} + ( 5 + \beta ) q^{87} + q^{88} + ( -4 - 6 \beta ) q^{89} + ( -4 + 4 \beta ) q^{90} + ( -2 - 3 \beta ) q^{92} + ( 4 + 4 \beta ) q^{93} + ( 2 + 6 \beta ) q^{94} + ( 6 - 4 \beta ) q^{95} + ( 1 + \beta ) q^{96} + ( 1 + 2 \beta ) q^{97} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 2 q^{27} - 6 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{33} + 4 q^{34} - 16 q^{37} + 4 q^{38} + 10 q^{39} + 4 q^{40} + 8 q^{41} + 2 q^{44} - 8 q^{45} - 4 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{58} + 14 q^{59} - 18 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} + 2 q^{66} + 14 q^{67} + 4 q^{68} - 16 q^{69} - 8 q^{71} + 16 q^{73} - 16 q^{74} - 14 q^{75} + 4 q^{76} + 10 q^{78} - 18 q^{79} + 4 q^{80} - 2 q^{81} + 8 q^{82} - 4 q^{83} + 24 q^{85} + 10 q^{87} + 2 q^{88} - 8 q^{89} - 8 q^{90} - 4 q^{92} + 8 q^{93} + 4 q^{94} + 12 q^{95} + 2 q^{96} + 2 q^{97} + O(q^{100})$$

## 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 −0.414214 1.00000 3.41421 −0.414214 0 1.00000 −2.82843 3.41421
1.2 1.00000 2.41421 1.00000 0.585786 2.41421 0 1.00000 2.82843 0.585786
 $$n$$: 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

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 1078.2.a.x 2
3.b odd 2 1 9702.2.a.ch 2
4.b odd 2 1 8624.2.a.bh 2
7.b odd 2 1 1078.2.a.t 2
7.c even 3 2 1078.2.e.m 4
7.d odd 6 2 154.2.e.e 4
21.c even 2 1 9702.2.a.cx 2
21.g even 6 2 1386.2.k.t 4
28.d even 2 1 8624.2.a.cc 2
28.f even 6 2 1232.2.q.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 7.d odd 6 2
1078.2.a.t 2 7.b odd 2 1
1078.2.a.x 2 1.a even 1 1 trivial
1078.2.e.m 4 7.c even 3 2
1232.2.q.f 4 28.f even 6 2
1386.2.k.t 4 21.g even 6 2
8624.2.a.bh 2 4.b odd 2 1
8624.2.a.cc 2 28.d even 2 1
9702.2.a.ch 2 3.b odd 2 1
9702.2.a.cx 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1078))$$:

 $$T_{3}^{2} - 2 T_{3} - 1$$ $$T_{5}^{2} - 4 T_{5} + 2$$ $$T_{13}^{2} - 2 T_{13} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-1 - 2 T + T^{2}$$
$5$ $$2 - 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-7 - 2 T + T^{2}$$
$17$ $$-28 - 4 T + T^{2}$$
$19$ $$2 - 4 T + T^{2}$$
$23$ $$-14 + 4 T + T^{2}$$
$29$ $$-23 + 6 T + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$62 + 16 T + T^{2}$$
$41$ $$14 - 8 T + T^{2}$$
$43$ $$-32 + T^{2}$$
$47$ $$-68 - 4 T + T^{2}$$
$53$ $$-94 + 4 T + T^{2}$$
$59$ $$47 - 14 T + T^{2}$$
$61$ $$73 + 18 T + T^{2}$$
$67$ $$31 - 14 T + T^{2}$$
$71$ $$-34 + 8 T + T^{2}$$
$73$ $$62 - 16 T + T^{2}$$
$79$ $$63 + 18 T + T^{2}$$
$83$ $$-196 + 4 T + T^{2}$$
$89$ $$-56 + 8 T + T^{2}$$
$97$ $$-7 - 2 T + T^{2}$$