Properties

 Label 1078.2.a.e 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} - 2 q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 - q^8 - 2 * q^9 $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} - 2 q^{9} - q^{11} + q^{12} - q^{13} + q^{16} - 6 q^{17} + 2 q^{18} + 2 q^{19} + q^{22} - 6 q^{23} - q^{24} - 5 q^{25} + q^{26} - 5 q^{27} + 9 q^{29} - 4 q^{31} - q^{32} - q^{33} + 6 q^{34} - 2 q^{36} + 2 q^{37} - 2 q^{38} - q^{39} - 6 q^{41} - 4 q^{43} - q^{44} + 6 q^{46} - 6 q^{47} + q^{48} + 5 q^{50} - 6 q^{51} - q^{52} + 5 q^{54} + 2 q^{57} - 9 q^{58} - 3 q^{59} + 11 q^{61} + 4 q^{62} + q^{64} + q^{66} + 11 q^{67} - 6 q^{68} - 6 q^{69} + 2 q^{72} + 2 q^{73} - 2 q^{74} - 5 q^{75} + 2 q^{76} + q^{78} + 5 q^{79} + q^{81} + 6 q^{82} - 6 q^{83} + 4 q^{86} + 9 q^{87} + q^{88} - 18 q^{89} - 6 q^{92} - 4 q^{93} + 6 q^{94} - q^{96} - 13 q^{97} + 2 q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - q^8 - 2 * q^9 - q^11 + q^12 - q^13 + q^16 - 6 * q^17 + 2 * q^18 + 2 * q^19 + q^22 - 6 * q^23 - q^24 - 5 * q^25 + q^26 - 5 * q^27 + 9 * q^29 - 4 * q^31 - q^32 - q^33 + 6 * q^34 - 2 * q^36 + 2 * q^37 - 2 * q^38 - q^39 - 6 * q^41 - 4 * q^43 - q^44 + 6 * q^46 - 6 * q^47 + q^48 + 5 * q^50 - 6 * q^51 - q^52 + 5 * q^54 + 2 * q^57 - 9 * q^58 - 3 * q^59 + 11 * q^61 + 4 * q^62 + q^64 + q^66 + 11 * q^67 - 6 * q^68 - 6 * q^69 + 2 * q^72 + 2 * q^73 - 2 * q^74 - 5 * q^75 + 2 * q^76 + q^78 + 5 * q^79 + q^81 + 6 * q^82 - 6 * q^83 + 4 * q^86 + 9 * q^87 + q^88 - 18 * q^89 - 6 * q^92 - 4 * q^93 + 6 * q^94 - q^96 - 13 * q^97 + 2 * 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 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.e 1
3.b odd 2 1 9702.2.a.br 1
4.b odd 2 1 8624.2.a.k 1
7.b odd 2 1 1078.2.a.c 1
7.c even 3 2 154.2.e.c 2
7.d odd 6 2 1078.2.e.k 2
21.c even 2 1 9702.2.a.bs 1
21.h odd 6 2 1386.2.k.e 2
28.d even 2 1 8624.2.a.u 1
28.g odd 6 2 1232.2.q.d 2

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

 $$T_{3} - 1$$ T3 - 1 $$T_{5}$$ T5 $$T_{13} + 1$$ T13 + 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 1$$
$17$ $$T + 6$$
$19$ $$T - 2$$
$23$ $$T + 6$$
$29$ $$T - 9$$
$31$ $$T + 4$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T + 6$$
$53$ $$T$$
$59$ $$T + 3$$
$61$ $$T - 11$$
$67$ $$T - 11$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T - 5$$
$83$ $$T + 6$$
$89$ $$T + 18$$
$97$ $$T + 13$$