# Properties

 Label 1078.2.e.b Level $1078$ Weight $2$ Character orbit 1078.e Analytic conductor $8.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - 2 \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (z - 1) * q^4 - 2*z * q^5 + q^8 + 3*z * q^9 $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - 2 \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + ( - \zeta_{6} + 1) q^{11} + 2 q^{13} - \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + ( - 3 \zeta_{6} + 3) q^{18} + 2 q^{20} - q^{22} + 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 2 \zeta_{6} q^{26} - 2 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + (\zeta_{6} - 1) q^{32} + 2 q^{34} - 3 q^{36} + 2 \zeta_{6} q^{37} - 2 \zeta_{6} q^{40} + 10 q^{41} + 4 q^{43} + \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{45} + ( - 8 \zeta_{6} + 8) q^{46} - 8 \zeta_{6} q^{47} - q^{50} + (2 \zeta_{6} - 2) q^{52} + (6 \zeta_{6} - 6) q^{53} - 2 q^{55} + 2 \zeta_{6} q^{58} - 10 \zeta_{6} q^{61} - 8 q^{62} + q^{64} - 4 \zeta_{6} q^{65} + ( - 12 \zeta_{6} + 12) q^{67} - 2 \zeta_{6} q^{68} + 16 q^{71} + 3 \zeta_{6} q^{72} + ( - 14 \zeta_{6} + 14) q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + (2 \zeta_{6} - 2) q^{80} + (9 \zeta_{6} - 9) q^{81} - 10 \zeta_{6} q^{82} + 4 q^{85} - 4 \zeta_{6} q^{86} + ( - \zeta_{6} + 1) q^{88} + 6 \zeta_{6} q^{89} - 6 q^{90} - 8 q^{92} + (8 \zeta_{6} - 8) q^{94} + 10 q^{97} + 3 q^{99} +O(q^{100})$$ q - z * q^2 + (z - 1) * q^4 - 2*z * q^5 + q^8 + 3*z * q^9 + (2*z - 2) * q^10 + (-z + 1) * q^11 + 2 * q^13 - z * q^16 + (2*z - 2) * q^17 + (-3*z + 3) * q^18 + 2 * q^20 - q^22 + 8*z * q^23 + (-z + 1) * q^25 - 2*z * q^26 - 2 * q^29 + (-8*z + 8) * q^31 + (z - 1) * q^32 + 2 * q^34 - 3 * q^36 + 2*z * q^37 - 2*z * q^40 + 10 * q^41 + 4 * q^43 + z * q^44 + (-6*z + 6) * q^45 + (-8*z + 8) * q^46 - 8*z * q^47 - q^50 + (2*z - 2) * q^52 + (6*z - 6) * q^53 - 2 * q^55 + 2*z * q^58 - 10*z * q^61 - 8 * q^62 + q^64 - 4*z * q^65 + (-12*z + 12) * q^67 - 2*z * q^68 + 16 * q^71 + 3*z * q^72 + (-14*z + 14) * q^73 + (-2*z + 2) * q^74 + (2*z - 2) * q^80 + (9*z - 9) * q^81 - 10*z * q^82 + 4 * q^85 - 4*z * q^86 + (-z + 1) * q^88 + 6*z * q^89 - 6 * q^90 - 8 * q^92 + (8*z - 8) * q^94 + 10 * q^97 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 - 2 * q^5 + 2 * q^8 + 3 * q^9 $$2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} + 3 q^{9} - 2 q^{10} + q^{11} + 4 q^{13} - q^{16} - 2 q^{17} + 3 q^{18} + 4 q^{20} - 2 q^{22} + 8 q^{23} + q^{25} - 2 q^{26} - 4 q^{29} + 8 q^{31} - q^{32} + 4 q^{34} - 6 q^{36} + 2 q^{37} - 2 q^{40} + 20 q^{41} + 8 q^{43} + q^{44} + 6 q^{45} + 8 q^{46} - 8 q^{47} - 2 q^{50} - 2 q^{52} - 6 q^{53} - 4 q^{55} + 2 q^{58} - 10 q^{61} - 16 q^{62} + 2 q^{64} - 4 q^{65} + 12 q^{67} - 2 q^{68} + 32 q^{71} + 3 q^{72} + 14 q^{73} + 2 q^{74} - 2 q^{80} - 9 q^{81} - 10 q^{82} + 8 q^{85} - 4 q^{86} + q^{88} + 6 q^{89} - 12 q^{90} - 16 q^{92} - 8 q^{94} + 20 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^4 - 2 * q^5 + 2 * q^8 + 3 * q^9 - 2 * q^10 + q^11 + 4 * q^13 - q^16 - 2 * q^17 + 3 * q^18 + 4 * q^20 - 2 * q^22 + 8 * q^23 + q^25 - 2 * q^26 - 4 * q^29 + 8 * q^31 - q^32 + 4 * q^34 - 6 * q^36 + 2 * q^37 - 2 * q^40 + 20 * q^41 + 8 * q^43 + q^44 + 6 * q^45 + 8 * q^46 - 8 * q^47 - 2 * q^50 - 2 * q^52 - 6 * q^53 - 4 * q^55 + 2 * q^58 - 10 * q^61 - 16 * q^62 + 2 * q^64 - 4 * q^65 + 12 * q^67 - 2 * q^68 + 32 * q^71 + 3 * q^72 + 14 * q^73 + 2 * q^74 - 2 * q^80 - 9 * q^81 - 10 * q^82 + 8 * q^85 - 4 * q^86 + q^88 + 6 * q^89 - 12 * q^90 - 16 * q^92 - 8 * q^94 + 20 * q^97 + 6 * 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 −0.500000 + 0.866025i −1.00000 1.73205i 0 0 1.00000 1.50000 + 2.59808i −1.00000 + 1.73205i
177.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 + 1.73205i 0 0 1.00000 1.50000 2.59808i −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.b 2
7.b odd 2 1 1078.2.e.c 2
7.c even 3 1 154.2.a.c 1
7.c even 3 1 inner 1078.2.e.b 2
7.d odd 6 1 1078.2.a.j 1
7.d odd 6 1 1078.2.e.c 2
21.g even 6 1 9702.2.a.v 1
21.h odd 6 1 1386.2.a.b 1
28.f even 6 1 8624.2.a.o 1
28.g odd 6 1 1232.2.a.h 1
35.j even 6 1 3850.2.a.f 1
35.l odd 12 2 3850.2.c.l 2
56.k odd 6 1 4928.2.a.o 1
56.p even 6 1 4928.2.a.n 1
77.h odd 6 1 1694.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.c 1 7.c even 3 1
1078.2.a.j 1 7.d odd 6 1
1078.2.e.b 2 1.a even 1 1 trivial
1078.2.e.b 2 7.c even 3 1 inner
1078.2.e.c 2 7.b odd 2 1
1078.2.e.c 2 7.d odd 6 1
1232.2.a.h 1 28.g odd 6 1
1386.2.a.b 1 21.h odd 6 1
1694.2.a.c 1 77.h odd 6 1
3850.2.a.f 1 35.j even 6 1
3850.2.c.l 2 35.l odd 12 2
4928.2.a.n 1 56.p even 6 1
4928.2.a.o 1 56.k odd 6 1
8624.2.a.o 1 28.f even 6 1
9702.2.a.v 1 21.g even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$T^{2} - 2T + 4$$
$41$ $$(T - 10)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} - 12T + 144$$
$71$ $$(T - 16)^{2}$$
$73$ $$T^{2} - 14T + 196$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T - 10)^{2}$$
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