Properties

 Label 1078.2.e.h 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - 2 \zeta_{6} q^{5} - 2 q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (2*z - 2) * q^3 + (z - 1) * q^4 - 2*z * q^5 - 2 * q^6 - q^8 - z * q^9 $$q + \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - 2 \zeta_{6} q^{5} - 2 q^{6} - q^{8} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{10} + (\zeta_{6} - 1) q^{11} - 2 \zeta_{6} q^{12} - 4 q^{13} + 4 q^{15} - \zeta_{6} q^{16} + ( - \zeta_{6} + 1) q^{18} - 4 \zeta_{6} q^{19} + 2 q^{20} - q^{22} - 4 \zeta_{6} q^{23} + ( - 2 \zeta_{6} + 2) q^{24} + ( - \zeta_{6} + 1) q^{25} - 4 \zeta_{6} q^{26} - 4 q^{27} + 2 q^{29} + 4 \zeta_{6} q^{30} + ( - 10 \zeta_{6} + 10) q^{31} + ( - \zeta_{6} + 1) q^{32} - 2 \zeta_{6} q^{33} + q^{36} + 6 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + ( - 8 \zeta_{6} + 8) q^{39} + 2 \zeta_{6} q^{40} - 4 q^{43} - \zeta_{6} q^{44} + (2 \zeta_{6} - 2) q^{45} + ( - 4 \zeta_{6} + 4) q^{46} - 10 \zeta_{6} q^{47} + 2 q^{48} + q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + ( - 14 \zeta_{6} + 14) q^{53} - 4 \zeta_{6} q^{54} + 2 q^{55} + 8 q^{57} + 2 \zeta_{6} q^{58} + (10 \zeta_{6} - 10) q^{59} + (4 \zeta_{6} - 4) q^{60} + 8 \zeta_{6} q^{61} + 10 q^{62} + q^{64} + 8 \zeta_{6} q^{65} + ( - 2 \zeta_{6} + 2) q^{66} + (8 \zeta_{6} - 8) q^{67} + 8 q^{69} - 4 q^{71} + \zeta_{6} q^{72} + (4 \zeta_{6} - 4) q^{73} + (6 \zeta_{6} - 6) q^{74} + 2 \zeta_{6} q^{75} + 4 q^{76} + 8 q^{78} - 16 \zeta_{6} q^{79} + (2 \zeta_{6} - 2) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 4 q^{83} - 4 \zeta_{6} q^{86} + (4 \zeta_{6} - 4) q^{87} + ( - \zeta_{6} + 1) q^{88} - 10 \zeta_{6} q^{89} - 2 q^{90} + 4 q^{92} + 20 \zeta_{6} q^{93} + ( - 10 \zeta_{6} + 10) q^{94} + (8 \zeta_{6} - 8) q^{95} + 2 \zeta_{6} q^{96} + 6 q^{97} + q^{99} +O(q^{100})$$ q + z * q^2 + (2*z - 2) * q^3 + (z - 1) * q^4 - 2*z * q^5 - 2 * q^6 - q^8 - z * q^9 + (-2*z + 2) * q^10 + (z - 1) * q^11 - 2*z * q^12 - 4 * q^13 + 4 * q^15 - z * q^16 + (-z + 1) * q^18 - 4*z * q^19 + 2 * q^20 - q^22 - 4*z * q^23 + (-2*z + 2) * q^24 + (-z + 1) * q^25 - 4*z * q^26 - 4 * q^27 + 2 * q^29 + 4*z * q^30 + (-10*z + 10) * q^31 + (-z + 1) * q^32 - 2*z * q^33 + q^36 + 6*z * q^37 + (-4*z + 4) * q^38 + (-8*z + 8) * q^39 + 2*z * q^40 - 4 * q^43 - z * q^44 + (2*z - 2) * q^45 + (-4*z + 4) * q^46 - 10*z * q^47 + 2 * q^48 + q^50 + (-4*z + 4) * q^52 + (-14*z + 14) * q^53 - 4*z * q^54 + 2 * q^55 + 8 * q^57 + 2*z * q^58 + (10*z - 10) * q^59 + (4*z - 4) * q^60 + 8*z * q^61 + 10 * q^62 + q^64 + 8*z * q^65 + (-2*z + 2) * q^66 + (8*z - 8) * q^67 + 8 * q^69 - 4 * q^71 + z * q^72 + (4*z - 4) * q^73 + (6*z - 6) * q^74 + 2*z * q^75 + 4 * q^76 + 8 * q^78 - 16*z * q^79 + (2*z - 2) * q^80 + (-11*z + 11) * q^81 + 4 * q^83 - 4*z * q^86 + (4*z - 4) * q^87 + (-z + 1) * q^88 - 10*z * q^89 - 2 * q^90 + 4 * q^92 + 20*z * q^93 + (-10*z + 10) * q^94 + (8*z - 8) * q^95 + 2*z * q^96 + 6 * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 2 * q^8 - q^9 $$2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{8} - q^{9} + 2 q^{10} - q^{11} - 2 q^{12} - 8 q^{13} + 8 q^{15} - q^{16} + q^{18} - 4 q^{19} + 4 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{24} + q^{25} - 4 q^{26} - 8 q^{27} + 4 q^{29} + 4 q^{30} + 10 q^{31} + q^{32} - 2 q^{33} + 2 q^{36} + 6 q^{37} + 4 q^{38} + 8 q^{39} + 2 q^{40} - 8 q^{43} - q^{44} - 2 q^{45} + 4 q^{46} - 10 q^{47} + 4 q^{48} + 2 q^{50} + 4 q^{52} + 14 q^{53} - 4 q^{54} + 4 q^{55} + 16 q^{57} + 2 q^{58} - 10 q^{59} - 4 q^{60} + 8 q^{61} + 20 q^{62} + 2 q^{64} + 8 q^{65} + 2 q^{66} - 8 q^{67} + 16 q^{69} - 8 q^{71} + q^{72} - 4 q^{73} - 6 q^{74} + 2 q^{75} + 8 q^{76} + 16 q^{78} - 16 q^{79} - 2 q^{80} + 11 q^{81} + 8 q^{83} - 4 q^{86} - 4 q^{87} + q^{88} - 10 q^{89} - 4 q^{90} + 8 q^{92} + 20 q^{93} + 10 q^{94} - 8 q^{95} + 2 q^{96} + 12 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 2 * q^8 - q^9 + 2 * q^10 - q^11 - 2 * q^12 - 8 * q^13 + 8 * q^15 - q^16 + q^18 - 4 * q^19 + 4 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^24 + q^25 - 4 * q^26 - 8 * q^27 + 4 * q^29 + 4 * q^30 + 10 * q^31 + q^32 - 2 * q^33 + 2 * q^36 + 6 * q^37 + 4 * q^38 + 8 * q^39 + 2 * q^40 - 8 * q^43 - q^44 - 2 * q^45 + 4 * q^46 - 10 * q^47 + 4 * q^48 + 2 * q^50 + 4 * q^52 + 14 * q^53 - 4 * q^54 + 4 * q^55 + 16 * q^57 + 2 * q^58 - 10 * q^59 - 4 * q^60 + 8 * q^61 + 20 * q^62 + 2 * q^64 + 8 * q^65 + 2 * q^66 - 8 * q^67 + 16 * q^69 - 8 * q^71 + q^72 - 4 * q^73 - 6 * q^74 + 2 * q^75 + 8 * q^76 + 16 * q^78 - 16 * q^79 - 2 * q^80 + 11 * q^81 + 8 * q^83 - 4 * q^86 - 4 * q^87 + q^88 - 10 * q^89 - 4 * q^90 + 8 * q^92 + 20 * q^93 + 10 * q^94 - 8 * q^95 + 2 * q^96 + 12 * q^97 + 2 * 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 −1.00000 + 1.73205i −0.500000 + 0.866025i −1.00000 1.73205i −2.00000 0 −1.00000 −0.500000 0.866025i 1.00000 1.73205i
177.1 0.500000 0.866025i −1.00000 1.73205i −0.500000 0.866025i −1.00000 + 1.73205i −2.00000 0 −1.00000 −0.500000 + 0.866025i 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.h 2
7.b odd 2 1 1078.2.e.l 2
7.c even 3 1 154.2.a.b 1
7.c even 3 1 inner 1078.2.e.h 2
7.d odd 6 1 1078.2.a.b 1
7.d odd 6 1 1078.2.e.l 2
21.g even 6 1 9702.2.a.bz 1
21.h odd 6 1 1386.2.a.f 1
28.f even 6 1 8624.2.a.z 1
28.g odd 6 1 1232.2.a.c 1
35.j even 6 1 3850.2.a.o 1
35.l odd 12 2 3850.2.c.d 2
56.k odd 6 1 4928.2.a.bf 1
56.p even 6 1 4928.2.a.d 1
77.h odd 6 1 1694.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 7.c even 3 1
1078.2.a.b 1 7.d odd 6 1
1078.2.e.h 2 1.a even 1 1 trivial
1078.2.e.h 2 7.c even 3 1 inner
1078.2.e.l 2 7.b odd 2 1
1078.2.e.l 2 7.d odd 6 1
1232.2.a.c 1 28.g odd 6 1
1386.2.a.f 1 21.h odd 6 1
1694.2.a.i 1 77.h odd 6 1
3850.2.a.o 1 35.j even 6 1
3850.2.c.d 2 35.l odd 12 2
4928.2.a.d 1 56.p even 6 1
4928.2.a.bf 1 56.k odd 6 1
8624.2.a.z 1 28.f even 6 1
9702.2.a.bz 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 $$S_{2}^{\mathrm{new}}(1078, [\chi])$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + T + 1$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 10T + 100$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$T^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 10T + 100$$
$53$ $$T^{2} - 14T + 196$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} - 8T + 64$$
$67$ $$T^{2} + 8T + 64$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} + 4T + 16$$
$79$ $$T^{2} + 16T + 256$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 10T + 100$$
$97$ $$(T - 6)^{2}$$