# Properties

 Label 1078.2.a.k Level $1078$ Weight $2$ Character orbit 1078.a Self dual yes Analytic conductor $8.608$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
1.00000 1.00000 1.00000 0 1.00000 0 1.00000 −2.00000 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

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.k 1
3.b odd 2 1 9702.2.a.o 1
4.b odd 2 1 8624.2.a.l 1
7.b odd 2 1 1078.2.a.i 1
7.c even 3 2 154.2.e.b 2
7.d odd 6 2 1078.2.e.d 2
21.c even 2 1 9702.2.a.l 1
21.h odd 6 2 1386.2.k.n 2
28.d even 2 1 8624.2.a.t 1
28.g odd 6 2 1232.2.q.c 2

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

 $$T_{3} - 1$$ $$T_{5}$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$1 + T$$
$13$ $$-5 + T$$
$17$ $$-6 + T$$
$19$ $$-2 + T$$
$23$ $$-6 + T$$
$29$ $$-3 + T$$
$31$ $$-8 + T$$
$37$ $$-2 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$-6 + T$$
$53$ $$12 + T$$
$59$ $$3 + T$$
$61$ $$7 + T$$
$67$ $$13 + T$$
$71$ $$12 + T$$
$73$ $$10 + T$$
$79$ $$1 + T$$
$83$ $$-6 + T$$
$89$ $$-6 + T$$
$97$ $$13 + T$$