# Properties

 Label 1078.2.a.a Level $1078$ Weight $2$ Character orbit 1078.a Self dual yes Analytic conductor $8.608$ Analytic rank $1$ 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: $$1$$ 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

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

## 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 −3.00000 1.00000 −4.00000 3.00000 0 −1.00000 6.00000 4.00000
 $$n$$: 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.a 1
3.b odd 2 1 9702.2.a.cg 1
4.b odd 2 1 8624.2.a.bd 1
7.b odd 2 1 1078.2.a.f 1
7.c even 3 2 154.2.e.d 2
7.d odd 6 2 1078.2.e.g 2
21.c even 2 1 9702.2.a.bb 1
21.h odd 6 2 1386.2.k.a 2
28.d even 2 1 8624.2.a.d 1
28.g odd 6 2 1232.2.q.a 2

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

 $$T_{3} + 3$$ $$T_{5} + 4$$ $$T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$3 + T$$
$5$ $$4 + T$$
$7$ $$T$$
$11$ $$1 + T$$
$13$ $$1 + T$$
$17$ $$-2 + T$$
$19$ $$-6 + T$$
$23$ $$2 + T$$
$29$ $$-1 + T$$
$31$ $$-4 + T$$
$37$ $$2 + T$$
$41$ $$2 + T$$
$43$ $$-4 + T$$
$47$ $$-2 + T$$
$53$ $$12 + T$$
$59$ $$-9 + T$$
$61$ $$5 + T$$
$67$ $$9 + T$$
$71$ $$-4 + T$$
$73$ $$2 + T$$
$79$ $$15 + T$$
$83$ $$6 + T$$
$89$ $$-6 + T$$
$97$ $$5 + T$$