# Properties

 Label 1078.2.e.u 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

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

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

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

## Character values

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

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

## 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 −0.500000 + 0.866025i −1.41421 2.44949i 0 0 −1.00000 1.50000 + 2.59808i 1.41421 2.44949i
67.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.41421 + 2.44949i 0 0 −1.00000 1.50000 + 2.59808i −1.41421 + 2.44949i
177.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.41421 + 2.44949i 0 0 −1.00000 1.50000 2.59808i 1.41421 + 2.44949i
177.2 0.500000 0.866025i 0 −0.500000 0.866025i 1.41421 2.44949i 0 0 −1.00000 1.50000 2.59808i −1.41421 2.44949i
 $$n$$: 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.u 4
7.b odd 2 1 inner 1078.2.e.u 4
7.c even 3 1 1078.2.a.o 2
7.c even 3 1 inner 1078.2.e.u 4
7.d odd 6 1 1078.2.a.o 2
7.d odd 6 1 inner 1078.2.e.u 4
21.g even 6 1 9702.2.a.dk 2
21.h odd 6 1 9702.2.a.dk 2
28.f even 6 1 8624.2.a.bo 2
28.g odd 6 1 8624.2.a.bo 2

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

 $$T_{3}$$ T3 $$T_{5}^{4} + 8T_{5}^{2} + 64$$ T5^4 + 8*T5^2 + 64 $$T_{13}^{2} - 32$$ T13^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 8T^{2} + 64$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + T + 1)^{2}$$
$13$ $$(T^{2} - 32)^{2}$$
$17$ $$T^{4} + 8T^{2} + 64$$
$19$ $$T^{4} + 72T^{2} + 5184$$
$23$ $$(T^{2} - 8 T + 64)^{2}$$
$29$ $$(T + 6)^{4}$$
$31$ $$T^{4} + 72T^{2} + 5184$$
$37$ $$(T^{2} - 6 T + 36)^{2}$$
$41$ $$(T^{2} - 72)^{2}$$
$43$ $$(T + 4)^{4}$$
$47$ $$T^{4} + 8T^{2} + 64$$
$53$ $$(T^{2} + 6 T + 36)^{2}$$
$59$ $$T^{4} + 32T^{2} + 1024$$
$61$ $$T^{4} + 32T^{2} + 1024$$
$67$ $$(T^{2} - 4 T + 16)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 72T^{2} + 5184$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 8)^{2}$$
$89$ $$T^{4} + 128 T^{2} + 16384$$
$97$ $$(T^{2} - 128)^{2}$$