Properties

 Label 1078.2.a.p Level $1078$ Weight $2$ Character orbit 1078.a Self dual yes Analytic conductor $8.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 2 \beta q^{5} - q^{8} - 3 q^{9} +O(q^{10})$$ q - q^2 + q^4 - 2*b * q^5 - q^8 - 3 * q^9 $$q - q^{2} + q^{4} - 2 \beta q^{5} - q^{8} - 3 q^{9} + 2 \beta q^{10} + q^{11} - 3 \beta q^{13} + q^{16} + 2 \beta q^{17} + 3 q^{18} - \beta q^{19} - 2 \beta q^{20} - q^{22} + 6 q^{23} + 3 q^{25} + 3 \beta q^{26} + 8 q^{29} + \beta q^{31} - q^{32} - 2 \beta q^{34} - 3 q^{36} - 6 q^{37} + \beta q^{38} + 2 \beta q^{40} - 6 \beta q^{41} + 10 q^{43} + q^{44} + 6 \beta q^{45} - 6 q^{46} + 5 \beta q^{47} - 3 q^{50} - 3 \beta q^{52} + 6 q^{53} - 2 \beta q^{55} - 8 q^{58} + 10 \beta q^{59} + 3 \beta q^{61} - \beta q^{62} + q^{64} + 12 q^{65} - 4 q^{67} + 2 \beta q^{68} + 3 q^{72} - 6 \beta q^{73} + 6 q^{74} - \beta q^{76} - 2 \beta q^{80} + 9 q^{81} + 6 \beta q^{82} + 5 \beta q^{83} - 8 q^{85} - 10 q^{86} - q^{88} - 13 \beta q^{89} - 6 \beta q^{90} + 6 q^{92} - 5 \beta q^{94} + 4 q^{95} + \beta q^{97} - 3 q^{99} +O(q^{100})$$ q - q^2 + q^4 - 2*b * q^5 - q^8 - 3 * q^9 + 2*b * q^10 + q^11 - 3*b * q^13 + q^16 + 2*b * q^17 + 3 * q^18 - b * q^19 - 2*b * q^20 - q^22 + 6 * q^23 + 3 * q^25 + 3*b * q^26 + 8 * q^29 + b * q^31 - q^32 - 2*b * q^34 - 3 * q^36 - 6 * q^37 + b * q^38 + 2*b * q^40 - 6*b * q^41 + 10 * q^43 + q^44 + 6*b * q^45 - 6 * q^46 + 5*b * q^47 - 3 * q^50 - 3*b * q^52 + 6 * q^53 - 2*b * q^55 - 8 * q^58 + 10*b * q^59 + 3*b * q^61 - b * q^62 + q^64 + 12 * q^65 - 4 * q^67 + 2*b * q^68 + 3 * q^72 - 6*b * q^73 + 6 * q^74 - b * q^76 - 2*b * q^80 + 9 * q^81 + 6*b * q^82 + 5*b * q^83 - 8 * q^85 - 10 * q^86 - q^88 - 13*b * q^89 - 6*b * q^90 + 6 * q^92 - 5*b * q^94 + 4 * q^95 + b * q^97 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9} + 2 q^{11} + 2 q^{16} + 6 q^{18} - 2 q^{22} + 12 q^{23} + 6 q^{25} + 16 q^{29} - 2 q^{32} - 6 q^{36} - 12 q^{37} + 20 q^{43} + 2 q^{44} - 12 q^{46} - 6 q^{50} + 12 q^{53} - 16 q^{58} + 2 q^{64} + 24 q^{65} - 8 q^{67} + 6 q^{72} + 12 q^{74} + 18 q^{81} - 16 q^{85} - 20 q^{86} - 2 q^{88} + 12 q^{92} + 8 q^{95} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9 + 2 * q^11 + 2 * q^16 + 6 * q^18 - 2 * q^22 + 12 * q^23 + 6 * q^25 + 16 * q^29 - 2 * q^32 - 6 * q^36 - 12 * q^37 + 20 * q^43 + 2 * q^44 - 12 * q^46 - 6 * q^50 + 12 * q^53 - 16 * q^58 + 2 * q^64 + 24 * q^65 - 8 * q^67 + 6 * q^72 + 12 * q^74 + 18 * q^81 - 16 * q^85 - 20 * q^86 - 2 * q^88 + 12 * q^92 + 8 * q^95 - 6 * 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
 1.41421 −1.41421
−1.00000 0 1.00000 −2.82843 0 0 −1.00000 −3.00000 2.82843
1.2 −1.00000 0 1.00000 2.82843 0 0 −1.00000 −3.00000 −2.82843
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.a.p 2
3.b odd 2 1 9702.2.a.df 2
4.b odd 2 1 8624.2.a.bl 2
7.b odd 2 1 inner 1078.2.a.p 2
7.c even 3 2 1078.2.e.t 4
7.d odd 6 2 1078.2.e.t 4
21.c even 2 1 9702.2.a.df 2
28.d even 2 1 8624.2.a.bl 2

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

 $$T_{3}$$ T3 $$T_{5}^{2} - 8$$ T5^2 - 8 $$T_{13}^{2} - 18$$ T13^2 - 18

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 8$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 18$$
$17$ $$T^{2} - 8$$
$19$ $$T^{2} - 2$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2} - 2$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 72$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2} - 50$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 200$$
$61$ $$T^{2} - 18$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 50$$
$89$ $$T^{2} - 338$$
$97$ $$T^{2} - 2$$