# Properties

 Label 1078.2.e.f 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (\zeta_{6} - 1) q^{4} + 2 \zeta_{6} q^{5} - 3 q^{6} + q^{8} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (-3*z + 3) * q^3 + (z - 1) * q^4 + 2*z * q^5 - 3 * q^6 + q^8 - 6*z * q^9 $$q - \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (\zeta_{6} - 1) q^{4} + 2 \zeta_{6} q^{5} - 3 q^{6} + q^{8} - 6 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{10} + ( - \zeta_{6} + 1) q^{11} + 3 \zeta_{6} q^{12} + 7 q^{13} + 6 q^{15} - \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{17} + (6 \zeta_{6} - 6) q^{18} - 2 q^{20} - q^{22} + 8 \zeta_{6} q^{23} + ( - 3 \zeta_{6} + 3) q^{24} + ( - \zeta_{6} + 1) q^{25} - 7 \zeta_{6} q^{26} - 9 q^{27} - 5 q^{29} - 6 \zeta_{6} q^{30} + ( - 4 \zeta_{6} + 4) q^{31} + (\zeta_{6} - 1) q^{32} - 3 \zeta_{6} q^{33} - 2 q^{34} + 6 q^{36} - 4 \zeta_{6} q^{37} + ( - 21 \zeta_{6} + 21) q^{39} + 2 \zeta_{6} q^{40} - 4 q^{41} - 8 q^{43} + \zeta_{6} q^{44} + ( - 12 \zeta_{6} + 12) q^{45} + ( - 8 \zeta_{6} + 8) q^{46} + 2 \zeta_{6} q^{47} - 3 q^{48} - q^{50} - 6 \zeta_{6} q^{51} + (7 \zeta_{6} - 7) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} + 9 \zeta_{6} q^{54} + 2 q^{55} + 5 \zeta_{6} q^{58} + ( - 3 \zeta_{6} + 3) q^{59} + (6 \zeta_{6} - 6) q^{60} + \zeta_{6} q^{61} - 4 q^{62} + q^{64} + 14 \zeta_{6} q^{65} + (3 \zeta_{6} - 3) q^{66} + (9 \zeta_{6} - 9) q^{67} + 2 \zeta_{6} q^{68} + 24 q^{69} - 2 q^{71} - 6 \zeta_{6} q^{72} + ( - 4 \zeta_{6} + 4) q^{73} + (4 \zeta_{6} - 4) q^{74} - 3 \zeta_{6} q^{75} - 21 q^{78} - 9 \zeta_{6} q^{79} + ( - 2 \zeta_{6} + 2) q^{80} + (9 \zeta_{6} - 9) q^{81} + 4 \zeta_{6} q^{82} - 6 q^{83} + 4 q^{85} + 8 \zeta_{6} q^{86} + (15 \zeta_{6} - 15) q^{87} + ( - \zeta_{6} + 1) q^{88} + 6 \zeta_{6} q^{89} - 12 q^{90} - 8 q^{92} - 12 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{94} + 3 \zeta_{6} q^{96} - 7 q^{97} - 6 q^{99} +O(q^{100})$$ q - z * q^2 + (-3*z + 3) * q^3 + (z - 1) * q^4 + 2*z * q^5 - 3 * q^6 + q^8 - 6*z * q^9 + (-2*z + 2) * q^10 + (-z + 1) * q^11 + 3*z * q^12 + 7 * q^13 + 6 * q^15 - z * q^16 + (-2*z + 2) * q^17 + (6*z - 6) * q^18 - 2 * q^20 - q^22 + 8*z * q^23 + (-3*z + 3) * q^24 + (-z + 1) * q^25 - 7*z * q^26 - 9 * q^27 - 5 * q^29 - 6*z * q^30 + (-4*z + 4) * q^31 + (z - 1) * q^32 - 3*z * q^33 - 2 * q^34 + 6 * q^36 - 4*z * q^37 + (-21*z + 21) * q^39 + 2*z * q^40 - 4 * q^41 - 8 * q^43 + z * q^44 + (-12*z + 12) * q^45 + (-8*z + 8) * q^46 + 2*z * q^47 - 3 * q^48 - q^50 - 6*z * q^51 + (7*z - 7) * q^52 + (-6*z + 6) * q^53 + 9*z * q^54 + 2 * q^55 + 5*z * q^58 + (-3*z + 3) * q^59 + (6*z - 6) * q^60 + z * q^61 - 4 * q^62 + q^64 + 14*z * q^65 + (3*z - 3) * q^66 + (9*z - 9) * q^67 + 2*z * q^68 + 24 * q^69 - 2 * q^71 - 6*z * q^72 + (-4*z + 4) * q^73 + (4*z - 4) * q^74 - 3*z * q^75 - 21 * q^78 - 9*z * q^79 + (-2*z + 2) * q^80 + (9*z - 9) * q^81 + 4*z * q^82 - 6 * q^83 + 4 * q^85 + 8*z * q^86 + (15*z - 15) * q^87 + (-z + 1) * q^88 + 6*z * q^89 - 12 * q^90 - 8 * q^92 - 12*z * q^93 + (-2*z + 2) * q^94 + 3*z * q^96 - 7 * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{3} - q^{4} + 2 q^{5} - 6 q^{6} + 2 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q - q^2 + 3 * q^3 - q^4 + 2 * q^5 - 6 * q^6 + 2 * q^8 - 6 * q^9 $$2 q - q^{2} + 3 q^{3} - q^{4} + 2 q^{5} - 6 q^{6} + 2 q^{8} - 6 q^{9} + 2 q^{10} + q^{11} + 3 q^{12} + 14 q^{13} + 12 q^{15} - q^{16} + 2 q^{17} - 6 q^{18} - 4 q^{20} - 2 q^{22} + 8 q^{23} + 3 q^{24} + q^{25} - 7 q^{26} - 18 q^{27} - 10 q^{29} - 6 q^{30} + 4 q^{31} - q^{32} - 3 q^{33} - 4 q^{34} + 12 q^{36} - 4 q^{37} + 21 q^{39} + 2 q^{40} - 8 q^{41} - 16 q^{43} + q^{44} + 12 q^{45} + 8 q^{46} + 2 q^{47} - 6 q^{48} - 2 q^{50} - 6 q^{51} - 7 q^{52} + 6 q^{53} + 9 q^{54} + 4 q^{55} + 5 q^{58} + 3 q^{59} - 6 q^{60} + q^{61} - 8 q^{62} + 2 q^{64} + 14 q^{65} - 3 q^{66} - 9 q^{67} + 2 q^{68} + 48 q^{69} - 4 q^{71} - 6 q^{72} + 4 q^{73} - 4 q^{74} - 3 q^{75} - 42 q^{78} - 9 q^{79} + 2 q^{80} - 9 q^{81} + 4 q^{82} - 12 q^{83} + 8 q^{85} + 8 q^{86} - 15 q^{87} + q^{88} + 6 q^{89} - 24 q^{90} - 16 q^{92} - 12 q^{93} + 2 q^{94} + 3 q^{96} - 14 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - q^2 + 3 * q^3 - q^4 + 2 * q^5 - 6 * q^6 + 2 * q^8 - 6 * q^9 + 2 * q^10 + q^11 + 3 * q^12 + 14 * q^13 + 12 * q^15 - q^16 + 2 * q^17 - 6 * q^18 - 4 * q^20 - 2 * q^22 + 8 * q^23 + 3 * q^24 + q^25 - 7 * q^26 - 18 * q^27 - 10 * q^29 - 6 * q^30 + 4 * q^31 - q^32 - 3 * q^33 - 4 * q^34 + 12 * q^36 - 4 * q^37 + 21 * q^39 + 2 * q^40 - 8 * q^41 - 16 * q^43 + q^44 + 12 * q^45 + 8 * q^46 + 2 * q^47 - 6 * q^48 - 2 * q^50 - 6 * q^51 - 7 * q^52 + 6 * q^53 + 9 * q^54 + 4 * q^55 + 5 * q^58 + 3 * q^59 - 6 * q^60 + q^61 - 8 * q^62 + 2 * q^64 + 14 * q^65 - 3 * q^66 - 9 * q^67 + 2 * q^68 + 48 * q^69 - 4 * q^71 - 6 * q^72 + 4 * q^73 - 4 * q^74 - 3 * q^75 - 42 * q^78 - 9 * q^79 + 2 * q^80 - 9 * q^81 + 4 * q^82 - 12 * q^83 + 8 * q^85 + 8 * q^86 - 15 * q^87 + q^88 + 6 * q^89 - 24 * q^90 - 16 * q^92 - 12 * q^93 + 2 * q^94 + 3 * q^96 - 14 * q^97 - 12 * 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 1.50000 2.59808i −0.500000 + 0.866025i 1.00000 + 1.73205i −3.00000 0 1.00000 −3.00000 5.19615i 1.00000 1.73205i
177.1 −0.500000 + 0.866025i 1.50000 + 2.59808i −0.500000 0.866025i 1.00000 1.73205i −3.00000 0 1.00000 −3.00000 + 5.19615i 1.00000 + 1.73205i
 $$n$$: 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.f 2
7.b odd 2 1 154.2.e.a 2
7.c even 3 1 1078.2.a.g 1
7.c even 3 1 inner 1078.2.e.f 2
7.d odd 6 1 154.2.e.a 2
7.d odd 6 1 1078.2.a.m 1
21.c even 2 1 1386.2.k.o 2
21.g even 6 1 1386.2.k.o 2
21.g even 6 1 9702.2.a.i 1
21.h odd 6 1 9702.2.a.y 1
28.d even 2 1 1232.2.q.e 2
28.f even 6 1 1232.2.q.e 2
28.f even 6 1 8624.2.a.b 1
28.g odd 6 1 8624.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 7.b odd 2 1
154.2.e.a 2 7.d odd 6 1
1078.2.a.g 1 7.c even 3 1
1078.2.a.m 1 7.d odd 6 1
1078.2.e.f 2 1.a even 1 1 trivial
1078.2.e.f 2 7.c even 3 1 inner
1232.2.q.e 2 28.d even 2 1
1232.2.q.e 2 28.f even 6 1
1386.2.k.o 2 21.c even 2 1
1386.2.k.o 2 21.g even 6 1
8624.2.a.b 1 28.f even 6 1
8624.2.a.be 1 28.g odd 6 1
9702.2.a.i 1 21.g even 6 1
9702.2.a.y 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 $$S_{2}^{\mathrm{new}}(1078, [\chi])$$:

 $$T_{3}^{2} - 3T_{3} + 9$$ T3^2 - 3*T3 + 9 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{13} - 7$$ T13 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 7)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$(T + 5)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$(T + 4)^{2}$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 9T + 81$$
$71$ $$(T + 2)^{2}$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} + 9T + 81$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T + 7)^{2}$$