# Properties

 Label 1078.2.a.q Level $1078$ Weight $2$ Character orbit 1078.a Self dual yes Analytic conductor $8.608$ Analytic rank $1$ 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: $$1$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} - \beta q^{5} - \beta q^{6} - q^{8} - q^{9} +O(q^{10})$$ q - q^2 + b * q^3 + q^4 - b * q^5 - b * q^6 - q^8 - q^9 $$q - q^{2} + \beta q^{3} + q^{4} - \beta q^{5} - \beta q^{6} - q^{8} - q^{9} + \beta q^{10} - q^{11} + \beta q^{12} - 2 \beta q^{13} - 2 q^{15} + q^{16} + 4 \beta q^{17} + q^{18} - 2 \beta q^{19} - \beta q^{20} + q^{22} - 2 q^{23} - \beta q^{24} - 3 q^{25} + 2 \beta q^{26} - 4 \beta q^{27} - 6 q^{29} + 2 q^{30} + \beta q^{31} - q^{32} - \beta q^{33} - 4 \beta q^{34} - q^{36} - 2 q^{37} + 2 \beta q^{38} - 4 q^{39} + \beta q^{40} - 4 \beta q^{41} + 4 q^{43} - q^{44} + \beta q^{45} + 2 q^{46} - 3 \beta q^{47} + \beta q^{48} + 3 q^{50} + 8 q^{51} - 2 \beta q^{52} - 12 q^{53} + 4 \beta q^{54} + \beta q^{55} - 4 q^{57} + 6 q^{58} - 3 \beta q^{59} - 2 q^{60} + 4 \beta q^{61} - \beta q^{62} + q^{64} + 4 q^{65} + \beta q^{66} - 2 q^{67} + 4 \beta q^{68} - 2 \beta q^{69} - 10 q^{71} + q^{72} + 10 \beta q^{73} + 2 q^{74} - 3 \beta q^{75} - 2 \beta q^{76} + 4 q^{78} - 8 q^{79} - \beta q^{80} - 5 q^{81} + 4 \beta q^{82} + 2 \beta q^{83} - 8 q^{85} - 4 q^{86} - 6 \beta q^{87} + q^{88} + 5 \beta q^{89} - \beta q^{90} - 2 q^{92} + 2 q^{93} + 3 \beta q^{94} + 4 q^{95} - \beta q^{96} + 11 \beta q^{97} + q^{99} +O(q^{100})$$ q - q^2 + b * q^3 + q^4 - b * q^5 - b * q^6 - q^8 - q^9 + b * q^10 - q^11 + b * q^12 - 2*b * q^13 - 2 * q^15 + q^16 + 4*b * q^17 + q^18 - 2*b * q^19 - b * q^20 + q^22 - 2 * q^23 - b * q^24 - 3 * q^25 + 2*b * q^26 - 4*b * q^27 - 6 * q^29 + 2 * q^30 + b * q^31 - q^32 - b * q^33 - 4*b * q^34 - q^36 - 2 * q^37 + 2*b * q^38 - 4 * q^39 + b * q^40 - 4*b * q^41 + 4 * q^43 - q^44 + b * q^45 + 2 * q^46 - 3*b * q^47 + b * q^48 + 3 * q^50 + 8 * q^51 - 2*b * q^52 - 12 * q^53 + 4*b * q^54 + b * q^55 - 4 * q^57 + 6 * q^58 - 3*b * q^59 - 2 * q^60 + 4*b * q^61 - b * q^62 + q^64 + 4 * q^65 + b * q^66 - 2 * q^67 + 4*b * q^68 - 2*b * q^69 - 10 * q^71 + q^72 + 10*b * q^73 + 2 * q^74 - 3*b * q^75 - 2*b * q^76 + 4 * q^78 - 8 * q^79 - b * q^80 - 5 * q^81 + 4*b * q^82 + 2*b * q^83 - 8 * q^85 - 4 * q^86 - 6*b * q^87 + q^88 + 5*b * q^89 - b * q^90 - 2 * q^92 + 2 * q^93 + 3*b * q^94 + 4 * q^95 - b * q^96 + 11*b * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 2 q^{11} - 4 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{22} - 4 q^{23} - 6 q^{25} - 12 q^{29} + 4 q^{30} - 2 q^{32} - 2 q^{36} - 4 q^{37} - 8 q^{39} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 6 q^{50} + 16 q^{51} - 24 q^{53} - 8 q^{57} + 12 q^{58} - 4 q^{60} + 2 q^{64} + 8 q^{65} - 4 q^{67} - 20 q^{71} + 2 q^{72} + 4 q^{74} + 8 q^{78} - 16 q^{79} - 10 q^{81} - 16 q^{85} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{93} + 8 q^{95} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 2 * q^11 - 4 * q^15 + 2 * q^16 + 2 * q^18 + 2 * q^22 - 4 * q^23 - 6 * q^25 - 12 * q^29 + 4 * q^30 - 2 * q^32 - 2 * q^36 - 4 * q^37 - 8 * q^39 + 8 * q^43 - 2 * q^44 + 4 * q^46 + 6 * q^50 + 16 * q^51 - 24 * q^53 - 8 * q^57 + 12 * q^58 - 4 * q^60 + 2 * q^64 + 8 * q^65 - 4 * q^67 - 20 * q^71 + 2 * q^72 + 4 * q^74 + 8 * q^78 - 16 * q^79 - 10 * q^81 - 16 * q^85 - 8 * q^86 + 2 * q^88 - 4 * q^92 + 4 * q^93 + 8 * q^95 + 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
 −1.41421 1.41421
−1.00000 −1.41421 1.00000 1.41421 1.41421 0 −1.00000 −1.00000 −1.41421
1.2 −1.00000 1.41421 1.00000 −1.41421 −1.41421 0 −1.00000 −1.00000 1.41421
 $$n$$: 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.q 2
3.b odd 2 1 9702.2.a.dq 2
4.b odd 2 1 8624.2.a.bw 2
7.b odd 2 1 inner 1078.2.a.q 2
7.c even 3 2 1078.2.e.s 4
7.d odd 6 2 1078.2.e.s 4
21.c even 2 1 9702.2.a.dq 2
28.d even 2 1 8624.2.a.bw 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} - 2$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 8$$
$17$ $$T^{2} - 32$$
$19$ $$T^{2} - 8$$
$23$ $$(T + 2)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 2$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 32$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 18$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} - 18$$
$61$ $$T^{2} - 32$$
$67$ $$(T + 2)^{2}$$
$71$ $$(T + 10)^{2}$$
$73$ $$T^{2} - 200$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} - 8$$
$89$ $$T^{2} - 50$$
$97$ $$T^{2} - 242$$