# Properties

 Label 1078.2.e.k Level $1078$ Weight $2$ Character orbit 1078.e 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(67,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("1078.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.e (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + q^{6} - q^{8} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + q^6 - q^8 + 2*z * q^9 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + q^{6} - q^{8} + 2 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} + \zeta_{6} q^{12} + q^{13} - \zeta_{6} q^{16} + (6 \zeta_{6} - 6) q^{17} + (2 \zeta_{6} - 2) q^{18} + 2 \zeta_{6} q^{19} + q^{22} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} + ( - 5 \zeta_{6} + 5) q^{25} + \zeta_{6} q^{26} + 5 q^{27} + 9 q^{29} + (4 \zeta_{6} - 4) q^{31} + ( - \zeta_{6} + 1) q^{32} - \zeta_{6} q^{33} - 6 q^{34} - 2 q^{36} - 2 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{38} + ( - \zeta_{6} + 1) q^{39} + 6 q^{41} - 4 q^{43} + \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{46} - 6 \zeta_{6} q^{47} - q^{48} + 5 q^{50} + 6 \zeta_{6} q^{51} + (\zeta_{6} - 1) q^{52} + 5 \zeta_{6} q^{54} + 2 q^{57} + 9 \zeta_{6} q^{58} + (3 \zeta_{6} - 3) q^{59} + 11 \zeta_{6} q^{61} - 4 q^{62} + q^{64} + ( - \zeta_{6} + 1) q^{66} + (11 \zeta_{6} - 11) q^{67} - 6 \zeta_{6} q^{68} + 6 q^{69} - 2 \zeta_{6} q^{72} + ( - 2 \zeta_{6} + 2) q^{73} + ( - 2 \zeta_{6} + 2) q^{74} - 5 \zeta_{6} q^{75} - 2 q^{76} + q^{78} - 5 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 6 \zeta_{6} q^{82} + 6 q^{83} - 4 \zeta_{6} q^{86} + ( - 9 \zeta_{6} + 9) q^{87} + (\zeta_{6} - 1) q^{88} - 18 \zeta_{6} q^{89} - 6 q^{92} + 4 \zeta_{6} q^{93} + ( - 6 \zeta_{6} + 6) q^{94} - \zeta_{6} q^{96} + 13 q^{97} + 2 q^{99} +O(q^{100})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + q^6 - q^8 + 2*z * q^9 + (-z + 1) * q^11 + z * q^12 + q^13 - z * q^16 + (6*z - 6) * q^17 + (2*z - 2) * q^18 + 2*z * q^19 + q^22 + 6*z * q^23 + (z - 1) * q^24 + (-5*z + 5) * q^25 + z * q^26 + 5 * q^27 + 9 * q^29 + (4*z - 4) * q^31 + (-z + 1) * q^32 - z * q^33 - 6 * q^34 - 2 * q^36 - 2*z * q^37 + (2*z - 2) * q^38 + (-z + 1) * q^39 + 6 * q^41 - 4 * q^43 + z * q^44 + (6*z - 6) * q^46 - 6*z * q^47 - q^48 + 5 * q^50 + 6*z * q^51 + (z - 1) * q^52 + 5*z * q^54 + 2 * q^57 + 9*z * q^58 + (3*z - 3) * q^59 + 11*z * q^61 - 4 * q^62 + q^64 + (-z + 1) * q^66 + (11*z - 11) * q^67 - 6*z * q^68 + 6 * q^69 - 2*z * q^72 + (-2*z + 2) * q^73 + (-2*z + 2) * q^74 - 5*z * q^75 - 2 * q^76 + q^78 - 5*z * q^79 + (z - 1) * q^81 + 6*z * q^82 + 6 * q^83 - 4*z * q^86 + (-9*z + 9) * q^87 + (z - 1) * q^88 - 18*z * q^89 - 6 * q^92 + 4*z * q^93 + (-6*z + 6) * q^94 - z * q^96 + 13 * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} - q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 - q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q + q^{2} + q^{3} - q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + q^{11} + q^{12} + 2 q^{13} - q^{16} - 6 q^{17} - 2 q^{18} + 2 q^{19} + 2 q^{22} + 6 q^{23} - q^{24} + 5 q^{25} + q^{26} + 10 q^{27} + 18 q^{29} - 4 q^{31} + q^{32} - q^{33} - 12 q^{34} - 4 q^{36} - 2 q^{37} - 2 q^{38} + q^{39} + 12 q^{41} - 8 q^{43} + q^{44} - 6 q^{46} - 6 q^{47} - 2 q^{48} + 10 q^{50} + 6 q^{51} - q^{52} + 5 q^{54} + 4 q^{57} + 9 q^{58} - 3 q^{59} + 11 q^{61} - 8 q^{62} + 2 q^{64} + q^{66} - 11 q^{67} - 6 q^{68} + 12 q^{69} - 2 q^{72} + 2 q^{73} + 2 q^{74} - 5 q^{75} - 4 q^{76} + 2 q^{78} - 5 q^{79} - q^{81} + 6 q^{82} + 12 q^{83} - 4 q^{86} + 9 q^{87} - q^{88} - 18 q^{89} - 12 q^{92} + 4 q^{93} + 6 q^{94} - q^{96} + 26 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 - q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + q^11 + q^12 + 2 * q^13 - q^16 - 6 * q^17 - 2 * q^18 + 2 * q^19 + 2 * q^22 + 6 * q^23 - q^24 + 5 * q^25 + q^26 + 10 * q^27 + 18 * q^29 - 4 * q^31 + q^32 - q^33 - 12 * q^34 - 4 * q^36 - 2 * q^37 - 2 * q^38 + q^39 + 12 * q^41 - 8 * q^43 + q^44 - 6 * q^46 - 6 * q^47 - 2 * q^48 + 10 * q^50 + 6 * q^51 - q^52 + 5 * q^54 + 4 * q^57 + 9 * q^58 - 3 * q^59 + 11 * q^61 - 8 * q^62 + 2 * q^64 + q^66 - 11 * q^67 - 6 * q^68 + 12 * q^69 - 2 * q^72 + 2 * q^73 + 2 * q^74 - 5 * q^75 - 4 * q^76 + 2 * q^78 - 5 * q^79 - q^81 + 6 * q^82 + 12 * q^83 - 4 * q^86 + 9 * q^87 - q^88 - 18 * q^89 - 12 * q^92 + 4 * q^93 + 6 * q^94 - q^96 + 26 * q^97 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 1.00000 0 −1.00000 1.00000 + 1.73205i 0
177.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 1.00000 0 −1.00000 1.00000 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.k 2
7.b odd 2 1 154.2.e.c 2
7.c even 3 1 1078.2.a.c 1
7.c even 3 1 inner 1078.2.e.k 2
7.d odd 6 1 154.2.e.c 2
7.d odd 6 1 1078.2.a.e 1
21.c even 2 1 1386.2.k.e 2
21.g even 6 1 1386.2.k.e 2
21.g even 6 1 9702.2.a.br 1
21.h odd 6 1 9702.2.a.bs 1
28.d even 2 1 1232.2.q.d 2
28.f even 6 1 1232.2.q.d 2
28.f even 6 1 8624.2.a.k 1
28.g odd 6 1 8624.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.c 2 7.b odd 2 1
154.2.e.c 2 7.d odd 6 1
1078.2.a.c 1 7.c even 3 1
1078.2.a.e 1 7.d odd 6 1
1078.2.e.k 2 1.a even 1 1 trivial
1078.2.e.k 2 7.c even 3 1 inner
1232.2.q.d 2 28.d even 2 1
1232.2.q.d 2 28.f even 6 1
1386.2.k.e 2 21.c even 2 1
1386.2.k.e 2 21.g even 6 1
8624.2.a.k 1 28.f even 6 1
8624.2.a.u 1 28.g odd 6 1
9702.2.a.br 1 21.g even 6 1
9702.2.a.bs 1 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1078, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{5}$$ T5 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} - 2T + 4$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} - 11T + 121$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 18T + 324$$
$97$ $$(T - 13)^{2}$$