# Properties

 Label 1078.2.e.s Level $1078$ Weight $2$ Character orbit 1078.e Analytic conductor $8.608$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} + \beta_1) q^{5} - \beta_{3} q^{6} - q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q - b2 * q^2 + b1 * q^3 + (-b2 - 1) * q^4 + (b3 + b1) * q^5 - b3 * q^6 - q^8 - b2 * q^9 $$q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} + \beta_1) q^{5} - \beta_{3} q^{6} - q^{8} - \beta_{2} q^{9} + \beta_1 q^{10} + (\beta_{2} + 1) q^{11} + ( - \beta_{3} - \beta_1) q^{12} - 2 \beta_{3} q^{13} - 2 q^{15} + \beta_{2} q^{16} + 4 \beta_1 q^{17} + ( - \beta_{2} - 1) q^{18} + (2 \beta_{3} + 2 \beta_1) q^{19} - \beta_{3} q^{20} + q^{22} - 2 \beta_{2} q^{23} - \beta_1 q^{24} + (3 \beta_{2} + 3) q^{25} + ( - 2 \beta_{3} - 2 \beta_1) q^{26} - 4 \beta_{3} q^{27} - 6 q^{29} + 2 \beta_{2} q^{30} + \beta_1 q^{31} + (\beta_{2} + 1) q^{32} + (\beta_{3} + \beta_1) q^{33} - 4 \beta_{3} q^{34} - q^{36} - 2 \beta_{2} q^{37} + 2 \beta_1 q^{38} + (4 \beta_{2} + 4) q^{39} + ( - \beta_{3} - \beta_1) q^{40} - 4 \beta_{3} q^{41} + 4 q^{43} - \beta_{2} q^{44} + \beta_1 q^{45} + ( - 2 \beta_{2} - 2) q^{46} + (3 \beta_{3} + 3 \beta_1) q^{47} + \beta_{3} q^{48} + 3 q^{50} + 8 \beta_{2} q^{51} - 2 \beta_1 q^{52} + (12 \beta_{2} + 12) q^{53} + ( - 4 \beta_{3} - 4 \beta_1) q^{54} + \beta_{3} q^{55} - 4 q^{57} + 6 \beta_{2} q^{58} - 3 \beta_1 q^{59} + (2 \beta_{2} + 2) q^{60} + ( - 4 \beta_{3} - 4 \beta_1) q^{61} - \beta_{3} q^{62} + q^{64} + 4 \beta_{2} q^{65} + \beta_1 q^{66} + (2 \beta_{2} + 2) q^{67} + ( - 4 \beta_{3} - 4 \beta_1) q^{68} - 2 \beta_{3} q^{69} - 10 q^{71} + \beta_{2} q^{72} + 10 \beta_1 q^{73} + ( - 2 \beta_{2} - 2) q^{74} + (3 \beta_{3} + 3 \beta_1) q^{75} - 2 \beta_{3} q^{76} + 4 q^{78} - 8 \beta_{2} q^{79} - \beta_1 q^{80} + (5 \beta_{2} + 5) q^{81} + ( - 4 \beta_{3} - 4 \beta_1) q^{82} + 2 \beta_{3} q^{83} - 8 q^{85} - 4 \beta_{2} q^{86} - 6 \beta_1 q^{87} + ( - \beta_{2} - 1) q^{88} + ( - 5 \beta_{3} - 5 \beta_1) q^{89} - \beta_{3} q^{90} - 2 q^{92} + 2 \beta_{2} q^{93} + 3 \beta_1 q^{94} + ( - 4 \beta_{2} - 4) q^{95} + (\beta_{3} + \beta_1) q^{96} + 11 \beta_{3} q^{97} + q^{99}+O(q^{100})$$ q - b2 * q^2 + b1 * q^3 + (-b2 - 1) * q^4 + (b3 + b1) * q^5 - b3 * q^6 - q^8 - b2 * q^9 + b1 * q^10 + (b2 + 1) * q^11 + (-b3 - b1) * q^12 - 2*b3 * q^13 - 2 * q^15 + b2 * q^16 + 4*b1 * q^17 + (-b2 - 1) * q^18 + (2*b3 + 2*b1) * q^19 - b3 * q^20 + q^22 - 2*b2 * q^23 - b1 * q^24 + (3*b2 + 3) * q^25 + (-2*b3 - 2*b1) * q^26 - 4*b3 * q^27 - 6 * q^29 + 2*b2 * q^30 + b1 * q^31 + (b2 + 1) * q^32 + (b3 + b1) * q^33 - 4*b3 * q^34 - q^36 - 2*b2 * q^37 + 2*b1 * q^38 + (4*b2 + 4) * q^39 + (-b3 - b1) * q^40 - 4*b3 * q^41 + 4 * q^43 - b2 * q^44 + b1 * q^45 + (-2*b2 - 2) * q^46 + (3*b3 + 3*b1) * q^47 + b3 * q^48 + 3 * q^50 + 8*b2 * q^51 - 2*b1 * q^52 + (12*b2 + 12) * q^53 + (-4*b3 - 4*b1) * q^54 + b3 * q^55 - 4 * q^57 + 6*b2 * q^58 - 3*b1 * q^59 + (2*b2 + 2) * q^60 + (-4*b3 - 4*b1) * q^61 - b3 * q^62 + q^64 + 4*b2 * q^65 + b1 * q^66 + (2*b2 + 2) * q^67 + (-4*b3 - 4*b1) * q^68 - 2*b3 * q^69 - 10 * q^71 + b2 * q^72 + 10*b1 * q^73 + (-2*b2 - 2) * q^74 + (3*b3 + 3*b1) * q^75 - 2*b3 * q^76 + 4 * q^78 - 8*b2 * q^79 - b1 * q^80 + (5*b2 + 5) * q^81 + (-4*b3 - 4*b1) * q^82 + 2*b3 * q^83 - 8 * q^85 - 4*b2 * q^86 - 6*b1 * q^87 + (-b2 - 1) * q^88 + (-5*b3 - 5*b1) * q^89 - b3 * q^90 - 2 * q^92 + 2*b2 * q^93 + 3*b1 * q^94 + (-4*b2 - 4) * q^95 + (b3 + b1) * q^96 + 11*b3 * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^8 + 2 * q^9 $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} + 2 q^{9} + 2 q^{11} - 8 q^{15} - 2 q^{16} - 2 q^{18} + 4 q^{22} + 4 q^{23} + 6 q^{25} - 24 q^{29} - 4 q^{30} + 2 q^{32} - 4 q^{36} + 4 q^{37} + 8 q^{39} + 16 q^{43} + 2 q^{44} - 4 q^{46} + 12 q^{50} - 16 q^{51} + 24 q^{53} - 16 q^{57} - 12 q^{58} + 4 q^{60} + 4 q^{64} - 8 q^{65} + 4 q^{67} - 40 q^{71} - 2 q^{72} - 4 q^{74} + 16 q^{78} + 16 q^{79} + 10 q^{81} - 32 q^{85} + 8 q^{86} - 2 q^{88} - 8 q^{92} - 4 q^{93} - 8 q^{95} + 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^8 + 2 * q^9 + 2 * q^11 - 8 * q^15 - 2 * q^16 - 2 * q^18 + 4 * q^22 + 4 * q^23 + 6 * q^25 - 24 * q^29 - 4 * q^30 + 2 * q^32 - 4 * q^36 + 4 * q^37 + 8 * q^39 + 16 * q^43 + 2 * q^44 - 4 * q^46 + 12 * q^50 - 16 * q^51 + 24 * q^53 - 16 * q^57 - 12 * q^58 + 4 * q^60 + 4 * q^64 - 8 * q^65 + 4 * q^67 - 40 * q^71 - 2 * q^72 - 4 * q^74 + 16 * q^78 + 16 * q^79 + 10 * q^81 - 32 * q^85 + 8 * q^86 - 2 * q^88 - 8 * q^92 - 4 * q^93 - 8 * q^95 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0.500000 + 0.866025i −0.707107 + 1.22474i −0.500000 + 0.866025i 0.707107 + 1.22474i −1.41421 0 −1.00000 0.500000 + 0.866025i −0.707107 + 1.22474i
67.2 0.500000 + 0.866025i 0.707107 1.22474i −0.500000 + 0.866025i −0.707107 1.22474i 1.41421 0 −1.00000 0.500000 + 0.866025i 0.707107 1.22474i
177.1 0.500000 0.866025i −0.707107 1.22474i −0.500000 0.866025i 0.707107 1.22474i −1.41421 0 −1.00000 0.500000 0.866025i −0.707107 1.22474i
177.2 0.500000 0.866025i 0.707107 + 1.22474i −0.500000 0.866025i −0.707107 + 1.22474i 1.41421 0 −1.00000 0.500000 0.866025i 0.707107 + 1.22474i
 $$n$$: 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.s 4
7.b odd 2 1 inner 1078.2.e.s 4
7.c even 3 1 1078.2.a.q 2
7.c even 3 1 inner 1078.2.e.s 4
7.d odd 6 1 1078.2.a.q 2
7.d odd 6 1 inner 1078.2.e.s 4
21.g even 6 1 9702.2.a.dq 2
21.h odd 6 1 9702.2.a.dq 2
28.f even 6 1 8624.2.a.bw 2
28.g odd 6 1 8624.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.q 2 7.c even 3 1
1078.2.a.q 2 7.d odd 6 1
1078.2.e.s 4 1.a even 1 1 trivial
1078.2.e.s 4 7.b odd 2 1 inner
1078.2.e.s 4 7.c even 3 1 inner
1078.2.e.s 4 7.d odd 6 1 inner
8624.2.a.bw 2 28.f even 6 1
8624.2.a.bw 2 28.g odd 6 1
9702.2.a.dq 2 21.g even 6 1
9702.2.a.dq 2 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 $$S_{2}^{\mathrm{new}}(1078, [\chi])$$:

 $$T_{3}^{4} + 2T_{3}^{2} + 4$$ T3^4 + 2*T3^2 + 4 $$T_{5}^{4} + 2T_{5}^{2} + 4$$ T5^4 + 2*T5^2 + 4 $$T_{13}^{2} - 8$$ T13^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} + 2T^{2} + 4$$
$5$ $$T^{4} + 2T^{2} + 4$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$(T^{2} - 8)^{2}$$
$17$ $$T^{4} + 32T^{2} + 1024$$
$19$ $$T^{4} + 8T^{2} + 64$$
$23$ $$(T^{2} - 2 T + 4)^{2}$$
$29$ $$(T + 6)^{4}$$
$31$ $$T^{4} + 2T^{2} + 4$$
$37$ $$(T^{2} - 2 T + 4)^{2}$$
$41$ $$(T^{2} - 32)^{2}$$
$43$ $$(T - 4)^{4}$$
$47$ $$T^{4} + 18T^{2} + 324$$
$53$ $$(T^{2} - 12 T + 144)^{2}$$
$59$ $$T^{4} + 18T^{2} + 324$$
$61$ $$T^{4} + 32T^{2} + 1024$$
$67$ $$(T^{2} - 2 T + 4)^{2}$$
$71$ $$(T + 10)^{4}$$
$73$ $$T^{4} + 200 T^{2} + 40000$$
$79$ $$(T^{2} - 8 T + 64)^{2}$$
$83$ $$(T^{2} - 8)^{2}$$
$89$ $$T^{4} + 50T^{2} + 2500$$
$97$ $$(T^{2} - 242)^{2}$$