# Properties

 Label 880.2.b.h Level $880$ Weight $2$ Character orbit 880.b Analytic conductor $7.027$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 55) 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_{3} - 1) q^{5} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} + \beta_1 - 1) q^{9}+O(q^{10})$$ q - b2 * q^3 + (b3 - 1) * q^5 + (b3 + b2 - b1) * q^7 + (b3 + b1 - 1) * q^9 $$q - \beta_{2} q^{3} + (\beta_{3} - 1) q^{5} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} + \beta_1 - 1) q^{9} + q^{11} + (2 \beta_{2} - \beta_1) q^{15} + 2 \beta_{2} q^{17} + 4 q^{19} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{21} + \beta_{2} q^{23} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{25} + (\beta_{3} - \beta_1) q^{27} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{29} + ( - \beta_{3} - \beta_1) q^{31} - \beta_{2} q^{33} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 - 4) q^{35} + (\beta_{3} + 4 \beta_{2} - \beta_1) q^{37} + (2 \beta_{3} + 2 \beta_1 + 2) q^{41} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{43} + ( - \beta_{2} - 2 \beta_1 + 5) q^{45} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{47} - 5 q^{49} + ( - 2 \beta_{3} - 2 \beta_1 + 8) q^{51} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{53} + (\beta_{3} - 1) q^{55} - 4 \beta_{2} q^{57} + ( - \beta_{3} - \beta_1 - 4) q^{59} + ( - 2 \beta_{3} - 2 \beta_1 + 6) q^{61} + (\beta_{3} - 5 \beta_{2} - \beta_1) q^{63} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{67} + ( - \beta_{3} - \beta_1 + 4) q^{69} + (3 \beta_{3} + 3 \beta_1) q^{71} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{73} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 4) q^{75} + (\beta_{3} + \beta_{2} - \beta_1) q^{77} + ( - 2 \beta_{3} - 2 \beta_1 + 8) q^{79} + (2 \beta_{3} + 2 \beta_1 - 3) q^{81} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{83} + ( - 4 \beta_{2} + 2 \beta_1) q^{85} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{87} + ( - \beta_{3} - \beta_1 + 2) q^{89} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{93} + (4 \beta_{3} - 4) q^{95} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{97} + (\beta_{3} + \beta_1 - 1) q^{99}+O(q^{100})$$ q - b2 * q^3 + (b3 - 1) * q^5 + (b3 + b2 - b1) * q^7 + (b3 + b1 - 1) * q^9 + q^11 + (2*b2 - b1) * q^15 + 2*b2 * q^17 + 4 * q^19 + (-2*b3 - 2*b1 + 4) * q^21 + b2 * q^23 + (-b3 - b2 - 2*b1 + 1) * q^25 + (b3 - b1) * q^27 + (-2*b3 - 2*b1 - 2) * q^29 + (-b3 - b1) * q^31 - b2 * q^33 + (-b3 - 3*b2 - b1 - 4) * q^35 + (b3 + 4*b2 - b1) * q^37 + (2*b3 + 2*b1 + 2) * q^41 + (-b3 - b2 + b1) * q^43 + (-b2 - 2*b1 + 5) * q^45 + (-b3 + 3*b2 + b1) * q^47 - 5 * q^49 + (-2*b3 - 2*b1 + 8) * q^51 + (2*b3 - 2*b2 - 2*b1) * q^53 + (b3 - 1) * q^55 - 4*b2 * q^57 + (-b3 - b1 - 4) * q^59 + (-2*b3 - 2*b1 + 6) * q^61 + (b3 - 5*b2 - b1) * q^63 + (2*b3 - b2 - 2*b1) * q^67 + (-b3 - b1 + 4) * q^69 + (3*b3 + 3*b1) * q^71 + (-2*b3 - 2*b2 + 2*b1) * q^73 + (-b3 - 4*b2 + 2*b1 - 4) * q^75 + (b3 + b2 - b1) * q^77 + (-2*b3 - 2*b1 + 8) * q^79 + (2*b3 + 2*b1 - 3) * q^81 + (-b3 + 3*b2 + b1) * q^83 + (-4*b2 + 2*b1) * q^85 + (-2*b3 - 2*b2 + 2*b1) * q^87 + (-b3 - b1 + 2) * q^89 + (-b3 - 2*b2 + b1) * q^93 + (4*b3 - 4) * q^95 + (-b3 + 2*b2 + b1) * q^97 + (b3 + b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{5} - 2 q^{9}+O(q^{10})$$ 4 * q - 3 * q^5 - 2 * q^9 $$4 q - 3 q^{5} - 2 q^{9} + 4 q^{11} - q^{15} + 16 q^{19} + 12 q^{21} + q^{25} - 12 q^{29} - 2 q^{31} - 18 q^{35} + 12 q^{41} + 18 q^{45} - 20 q^{49} + 28 q^{51} - 3 q^{55} - 18 q^{59} + 20 q^{61} + 14 q^{69} + 6 q^{71} - 15 q^{75} + 28 q^{79} - 8 q^{81} + 2 q^{85} + 6 q^{89} - 12 q^{95} - 2 q^{99}+O(q^{100})$$ 4 * q - 3 * q^5 - 2 * q^9 + 4 * q^11 - q^15 + 16 * q^19 + 12 * q^21 + q^25 - 12 * q^29 - 2 * q^31 - 18 * q^35 + 12 * q^41 + 18 * q^45 - 20 * q^49 + 28 * q^51 - 3 * q^55 - 18 * q^59 + 20 * q^61 + 14 * q^69 + 6 * q^71 - 15 * q^75 + 28 * q^79 - 8 * q^81 + 2 * q^85 + 6 * q^89 - 12 * q^95 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + \nu - 6 ) / 3$$ (v^3 + 2*v^2 + v - 6) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} + \nu - 3 ) / 3$$ (v^3 - v^2 + v - 3) / 3 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} - \nu^{2} + 4\nu + 9 ) / 3$$ (-2*v^3 - v^2 + 4*v + 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b3 + b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{2} + \beta _1 + 1$$ -b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} + 3\beta_{2} + \beta _1 + 8 ) / 2$$ (-b3 + 3*b2 + b1 + 8) / 2

## 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
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
0 2.52434i 0 −2.18614 + 0.469882i 0 3.46410i 0 −3.37228 0
529.2 0 0.792287i 0 0.686141 2.12819i 0 3.46410i 0 2.37228 0
529.3 0 0.792287i 0 0.686141 + 2.12819i 0 3.46410i 0 2.37228 0
529.4 0 2.52434i 0 −2.18614 0.469882i 0 3.46410i 0 −3.37228 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.h 4
4.b odd 2 1 55.2.b.a 4
5.b even 2 1 inner 880.2.b.h 4
5.c odd 4 2 4400.2.a.cc 4
12.b even 2 1 495.2.c.a 4
20.d odd 2 1 55.2.b.a 4
20.e even 4 2 275.2.a.h 4
44.c even 2 1 605.2.b.c 4
44.g even 10 4 605.2.j.j 16
44.h odd 10 4 605.2.j.i 16
60.h even 2 1 495.2.c.a 4
60.l odd 4 2 2475.2.a.bi 4
220.g even 2 1 605.2.b.c 4
220.i odd 4 2 3025.2.a.ba 4
220.n odd 10 4 605.2.j.i 16
220.o even 10 4 605.2.j.j 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 4.b odd 2 1
55.2.b.a 4 20.d odd 2 1
275.2.a.h 4 20.e even 4 2
495.2.c.a 4 12.b even 2 1
495.2.c.a 4 60.h even 2 1
605.2.b.c 4 44.c even 2 1
605.2.b.c 4 220.g even 2 1
605.2.j.i 16 44.h odd 10 4
605.2.j.i 16 220.n odd 10 4
605.2.j.j 16 44.g even 10 4
605.2.j.j 16 220.o even 10 4
880.2.b.h 4 1.a even 1 1 trivial
880.2.b.h 4 5.b even 2 1 inner
2475.2.a.bi 4 60.l odd 4 2
3025.2.a.ba 4 220.i odd 4 2
4400.2.a.cc 4 5.c odd 4 2

## Hecke kernels

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

 $$T_{3}^{4} + 7T_{3}^{2} + 4$$ T3^4 + 7*T3^2 + 4 $$T_{7}^{2} + 12$$ T7^2 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 7T^{2} + 4$$
$5$ $$T^{4} + 3 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 12)^{2}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 28T^{2} + 64$$
$19$ $$(T - 4)^{4}$$
$23$ $$T^{4} + 7T^{2} + 4$$
$29$ $$(T^{2} + 6 T - 24)^{2}$$
$31$ $$(T^{2} + T - 8)^{2}$$
$37$ $$T^{4} + 123T^{2} + 144$$
$41$ $$(T^{2} - 6 T - 24)^{2}$$
$43$ $$(T^{2} + 12)^{2}$$
$47$ $$(T^{2} + 44)^{2}$$
$53$ $$T^{4} + 112T^{2} + 1024$$
$59$ $$(T^{2} + 9 T + 12)^{2}$$
$61$ $$(T^{2} - 10 T - 8)^{2}$$
$67$ $$T^{4} + 87T^{2} + 36$$
$71$ $$(T^{2} - 3 T - 72)^{2}$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T^{2} - 14 T + 16)^{2}$$
$83$ $$(T^{2} + 44)^{2}$$
$89$ $$(T^{2} - 3 T - 6)^{2}$$
$97$ $$T^{4} + 51T^{2} + 576$$