# Properties

 Label 880.2.b.a Level $880$ Weight $2$ Character orbit 880.b Analytic conductor $7.027$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(529,880)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("880.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$880 = 2^{4} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 880.b (of order $$2$$, degree $$1$$, 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: $$7.02683537787$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( - i - 2) q^{5} + 3 i q^{7} + 2 q^{9}+O(q^{10})$$ q + i * q^3 + (-i - 2) * q^5 + 3*i * q^7 + 2 * q^9 $$q + i q^{3} + ( - i - 2) q^{5} + 3 i q^{7} + 2 q^{9} - q^{11} + 4 i q^{13} + ( - 2 i + 1) q^{15} - 3 i q^{17} - 5 q^{19} - 3 q^{21} - 4 i q^{23} + (4 i + 3) q^{25} + 5 i q^{27} - 5 q^{29} - 7 q^{31} - i q^{33} + ( - 6 i + 3) q^{35} + 7 i q^{37} - 4 q^{39} - 8 q^{41} + 6 i q^{43} + ( - 2 i - 4) q^{45} + 8 i q^{47} - 2 q^{49} + 3 q^{51} + 9 i q^{53} + (i + 2) q^{55} - 5 i q^{57} - 13 q^{61} + 6 i q^{63} + ( - 8 i + 4) q^{65} - 12 i q^{67} + 4 q^{69} + 3 q^{71} - 6 i q^{73} + (3 i - 4) q^{75} - 3 i q^{77} + q^{81} - 4 i q^{83} + (6 i - 3) q^{85} - 5 i q^{87} + 15 q^{89} - 12 q^{91} - 7 i q^{93} + (5 i + 10) q^{95} + 12 i q^{97} - 2 q^{99} +O(q^{100})$$ q + i * q^3 + (-i - 2) * q^5 + 3*i * q^7 + 2 * q^9 - q^11 + 4*i * q^13 + (-2*i + 1) * q^15 - 3*i * q^17 - 5 * q^19 - 3 * q^21 - 4*i * q^23 + (4*i + 3) * q^25 + 5*i * q^27 - 5 * q^29 - 7 * q^31 - i * q^33 + (-6*i + 3) * q^35 + 7*i * q^37 - 4 * q^39 - 8 * q^41 + 6*i * q^43 + (-2*i - 4) * q^45 + 8*i * q^47 - 2 * q^49 + 3 * q^51 + 9*i * q^53 + (i + 2) * q^55 - 5*i * q^57 - 13 * q^61 + 6*i * q^63 + (-8*i + 4) * q^65 - 12*i * q^67 + 4 * q^69 + 3 * q^71 - 6*i * q^73 + (3*i - 4) * q^75 - 3*i * q^77 + q^81 - 4*i * q^83 + (6*i - 3) * q^85 - 5*i * q^87 + 15 * q^89 - 12 * q^91 - 7*i * q^93 + (5*i + 10) * q^95 + 12*i * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} + 4 q^{9}+O(q^{10})$$ 2 * q - 4 * q^5 + 4 * q^9 $$2 q - 4 q^{5} + 4 q^{9} - 2 q^{11} + 2 q^{15} - 10 q^{19} - 6 q^{21} + 6 q^{25} - 10 q^{29} - 14 q^{31} + 6 q^{35} - 8 q^{39} - 16 q^{41} - 8 q^{45} - 4 q^{49} + 6 q^{51} + 4 q^{55} - 26 q^{61} + 8 q^{65} + 8 q^{69} + 6 q^{71} - 8 q^{75} + 2 q^{81} - 6 q^{85} + 30 q^{89} - 24 q^{91} + 20 q^{95} - 4 q^{99}+O(q^{100})$$ 2 * q - 4 * q^5 + 4 * q^9 - 2 * q^11 + 2 * q^15 - 10 * q^19 - 6 * q^21 + 6 * q^25 - 10 * q^29 - 14 * q^31 + 6 * q^35 - 8 * q^39 - 16 * q^41 - 8 * q^45 - 4 * q^49 + 6 * q^51 + 4 * q^55 - 26 * q^61 + 8 * q^65 + 8 * q^69 + 6 * q^71 - 8 * q^75 + 2 * q^81 - 6 * q^85 + 30 * q^89 - 24 * q^91 + 20 * q^95 - 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$177$$ $$321$$ $$661$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 − 1.00000i 1.00000i
0 1.00000i 0 −2.00000 + 1.00000i 0 3.00000i 0 2.00000 0
529.2 0 1.00000i 0 −2.00000 1.00000i 0 3.00000i 0 2.00000 0
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.b.a 2
4.b odd 2 1 110.2.b.a 2
5.b even 2 1 inner 880.2.b.a 2
5.c odd 4 1 4400.2.a.k 1
5.c odd 4 1 4400.2.a.s 1
12.b even 2 1 990.2.c.d 2
20.d odd 2 1 110.2.b.a 2
20.e even 4 1 550.2.a.e 1
20.e even 4 1 550.2.a.j 1
44.c even 2 1 1210.2.b.a 2
60.h even 2 1 990.2.c.d 2
60.l odd 4 1 4950.2.a.q 1
60.l odd 4 1 4950.2.a.ba 1
220.g even 2 1 1210.2.b.a 2
220.i odd 4 1 6050.2.a.f 1
220.i odd 4 1 6050.2.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.a 2 4.b odd 2 1
110.2.b.a 2 20.d odd 2 1
550.2.a.e 1 20.e even 4 1
550.2.a.j 1 20.e even 4 1
880.2.b.a 2 1.a even 1 1 trivial
880.2.b.a 2 5.b even 2 1 inner
990.2.c.d 2 12.b even 2 1
990.2.c.d 2 60.h even 2 1
1210.2.b.a 2 44.c even 2 1
1210.2.b.a 2 220.g even 2 1
4400.2.a.k 1 5.c odd 4 1
4400.2.a.s 1 5.c odd 4 1
4950.2.a.q 1 60.l odd 4 1
4950.2.a.ba 1 60.l odd 4 1
6050.2.a.f 1 220.i odd 4 1
6050.2.a.bk 1 220.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(880, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 9$$ T7^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 5)^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 81$$
$59$ $$T^{2}$$
$61$ $$(T + 13)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 144$$