# Properties

 Label 1078.2.a.l 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 2.00000 1.00000 2.00000 2.00000 0 1.00000 1.00000 2.00000
 $$n$$: 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.l yes 1
3.b odd 2 1 9702.2.a.e 1
4.b odd 2 1 8624.2.a.g 1
7.b odd 2 1 1078.2.a.h 1
7.c even 3 2 1078.2.e.a 2
7.d odd 6 2 1078.2.e.e 2
21.c even 2 1 9702.2.a.t 1
28.d even 2 1 8624.2.a.y 1

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

 $$T_{3} - 2$$ T3 - 2 $$T_{5} - 2$$ T5 - 2 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 2$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T + 2$$
$17$ $$T$$
$19$ $$T - 2$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T - 2$$
$41$ $$T + 8$$
$43$ $$T - 12$$
$47$ $$T + 12$$
$53$ $$T + 2$$
$59$ $$T + 10$$
$61$ $$T - 10$$
$67$ $$T + 12$$
$71$ $$T - 4$$
$73$ $$T + 12$$
$79$ $$T$$
$83$ $$T - 18$$
$89$ $$T$$
$97$ $$T - 12$$