# Properties

 Label 1078.2.a.j 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

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: $$1$$ 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

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 2 q^{5} + q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 + q^4 - 2 * q^5 + q^8 - 3 * q^9 $$q + q^{2} + q^{4} - 2 q^{5} + q^{8} - 3 q^{9} - 2 q^{10} - q^{11} - 2 q^{13} + q^{16} - 2 q^{17} - 3 q^{18} - 2 q^{20} - q^{22} - 8 q^{23} - q^{25} - 2 q^{26} - 2 q^{29} + 8 q^{31} + q^{32} - 2 q^{34} - 3 q^{36} - 2 q^{37} - 2 q^{40} - 10 q^{41} + 4 q^{43} - q^{44} + 6 q^{45} - 8 q^{46} - 8 q^{47} - q^{50} - 2 q^{52} + 6 q^{53} + 2 q^{55} - 2 q^{58} - 10 q^{61} + 8 q^{62} + q^{64} + 4 q^{65} - 12 q^{67} - 2 q^{68} + 16 q^{71} - 3 q^{72} + 14 q^{73} - 2 q^{74} - 2 q^{80} + 9 q^{81} - 10 q^{82} + 4 q^{85} + 4 q^{86} - q^{88} + 6 q^{89} + 6 q^{90} - 8 q^{92} - 8 q^{94} - 10 q^{97} + 3 q^{99}+O(q^{100})$$ q + q^2 + q^4 - 2 * q^5 + q^8 - 3 * q^9 - 2 * q^10 - q^11 - 2 * q^13 + q^16 - 2 * q^17 - 3 * q^18 - 2 * q^20 - q^22 - 8 * q^23 - q^25 - 2 * q^26 - 2 * q^29 + 8 * q^31 + q^32 - 2 * q^34 - 3 * q^36 - 2 * q^37 - 2 * q^40 - 10 * q^41 + 4 * q^43 - q^44 + 6 * q^45 - 8 * q^46 - 8 * q^47 - q^50 - 2 * q^52 + 6 * q^53 + 2 * q^55 - 2 * q^58 - 10 * q^61 + 8 * q^62 + q^64 + 4 * q^65 - 12 * q^67 - 2 * q^68 + 16 * q^71 - 3 * q^72 + 14 * q^73 - 2 * q^74 - 2 * q^80 + 9 * q^81 - 10 * q^82 + 4 * q^85 + 4 * q^86 - q^88 + 6 * q^89 + 6 * q^90 - 8 * q^92 - 8 * q^94 - 10 * q^97 + 3 * 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 0 1.00000 −2.00000 0 0 1.00000 −3.00000 −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.j 1
3.b odd 2 1 9702.2.a.v 1
4.b odd 2 1 8624.2.a.o 1
7.b odd 2 1 154.2.a.c 1
7.c even 3 2 1078.2.e.c 2
7.d odd 6 2 1078.2.e.b 2
21.c even 2 1 1386.2.a.b 1
28.d even 2 1 1232.2.a.h 1
35.c odd 2 1 3850.2.a.f 1
35.f even 4 2 3850.2.c.l 2
56.e even 2 1 4928.2.a.o 1
56.h odd 2 1 4928.2.a.n 1
77.b even 2 1 1694.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.c 1 7.b odd 2 1
1078.2.a.j 1 1.a even 1 1 trivial
1078.2.e.b 2 7.d odd 6 2
1078.2.e.c 2 7.c even 3 2
1232.2.a.h 1 28.d even 2 1
1386.2.a.b 1 21.c even 2 1
1694.2.a.c 1 77.b even 2 1
3850.2.a.f 1 35.c odd 2 1
3850.2.c.l 2 35.f even 4 2
4928.2.a.n 1 56.h odd 2 1
4928.2.a.o 1 56.e even 2 1
8624.2.a.o 1 4.b odd 2 1
9702.2.a.v 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1078))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T + 8$$
$29$ $$T + 2$$
$31$ $$T - 8$$
$37$ $$T + 2$$
$41$ $$T + 10$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T + 10$$
$67$ $$T + 12$$
$71$ $$T - 16$$
$73$ $$T - 14$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 6$$
$97$ $$T + 10$$