# Properties

 Label 880.2.bd.e Level $880$ Weight $2$ Character orbit 880.bd Analytic conductor $7.027$ 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] = [880,2,Mod(417,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("880.417");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$880 = 2^{4} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 880.bd (of order $$4$$, 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: $$7.02683537787$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + (\beta_1 - 2) q^{5} + \beta_1 q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + (b1 - 2) * q^5 + b1 * q^9 $$q + ( - \beta_1 + 1) q^{3} + (\beta_1 - 2) q^{5} + \beta_1 q^{9} + ( - \beta_{2} - 1) q^{11} + (\beta_{3} + \beta_{2}) q^{13} + (3 \beta_1 - 1) q^{15} + (\beta_{3} - \beta_{2}) q^{17} + 2 \beta_{3} q^{19} + ( - \beta_1 + 1) q^{23} + ( - 4 \beta_1 + 3) q^{25} + (4 \beta_1 + 4) q^{27} + 2 \beta_{3} q^{29} - 2 q^{31} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{33} + (3 \beta_1 + 3) q^{37} + 2 \beta_{3} q^{39} - 2 \beta_{2} q^{41} + ( - 2 \beta_1 - 1) q^{45} + ( - 3 \beta_1 - 3) q^{47} + 7 \beta_1 q^{49} - 2 \beta_{2} q^{51} + (\beta_1 - 1) q^{53} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{55} + (2 \beta_{3} - 2 \beta_{2}) q^{57} + 6 \beta_1 q^{59} + 2 \beta_{2} q^{61} + ( - 3 \beta_{3} - \beta_{2}) q^{65} + ( - 3 \beta_1 - 3) q^{67} - 2 \beta_1 q^{69} + 8 q^{71} + ( - \beta_{3} - \beta_{2}) q^{73} + ( - 7 \beta_1 - 1) q^{75} - 2 \beta_{3} q^{79} + 5 q^{81} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{83} + ( - \beta_{3} + 3 \beta_{2}) q^{85} + (2 \beta_{3} - 2 \beta_{2}) q^{87} - 6 \beta_1 q^{89} + (2 \beta_1 - 2) q^{93} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{95} + ( - 7 \beta_1 - 7) q^{97} + (\beta_{3} - \beta_1) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + (b1 - 2) * q^5 + b1 * q^9 + (-b2 - 1) * q^11 + (b3 + b2) * q^13 + (3*b1 - 1) * q^15 + (b3 - b2) * q^17 + 2*b3 * q^19 + (-b1 + 1) * q^23 + (-4*b1 + 3) * q^25 + (4*b1 + 4) * q^27 + 2*b3 * q^29 - 2 * q^31 + (-b3 - b2 + b1 - 1) * q^33 + (3*b1 + 3) * q^37 + 2*b3 * q^39 - 2*b2 * q^41 + (-2*b1 - 1) * q^45 + (-3*b1 - 3) * q^47 + 7*b1 * q^49 - 2*b2 * q^51 + (b1 - 1) * q^53 + (b3 + 2*b2 - b1 + 2) * q^55 + (2*b3 - 2*b2) * q^57 + 6*b1 * q^59 + 2*b2 * q^61 + (-3*b3 - b2) * q^65 + (-3*b1 - 3) * q^67 - 2*b1 * q^69 + 8 * q^71 + (-b3 - b2) * q^73 + (-7*b1 - 1) * q^75 - 2*b3 * q^79 + 5 * q^81 + (-2*b3 - 2*b2) * q^83 + (-b3 + 3*b2) * q^85 + (2*b3 - 2*b2) * q^87 - 6*b1 * q^89 + (2*b1 - 2) * q^93 + (-4*b3 + 2*b2) * q^95 + (-7*b1 - 7) * q^97 + (b3 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 8 q^{5}+O(q^{10})$$ 4 * q + 4 * q^3 - 8 * q^5 $$4 q + 4 q^{3} - 8 q^{5} - 4 q^{11} - 4 q^{15} + 4 q^{23} + 12 q^{25} + 16 q^{27} - 8 q^{31} - 4 q^{33} + 12 q^{37} - 4 q^{45} - 12 q^{47} - 4 q^{53} + 8 q^{55} - 12 q^{67} + 32 q^{71} - 4 q^{75} + 20 q^{81} - 8 q^{93} - 28 q^{97}+O(q^{100})$$ 4 * q + 4 * q^3 - 8 * q^5 - 4 * q^11 - 4 * q^15 + 4 * q^23 + 12 * q^25 + 16 * q^27 - 8 * q^31 - 4 * q^33 + 12 * q^37 - 4 * q^45 - 12 * q^47 - 4 * q^53 + 8 * q^55 - 12 * q^67 + 32 * q^71 - 4 * q^75 + 20 * q^81 - 8 * q^93 - 28 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 5$$ (v^3 + 5*v) / 5 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 5\nu ) / 5$$ (-v^3 + 5*v) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$5\beta_1$$ 5*b1 $$\nu^{3}$$ $$=$$ $$( -5\beta_{3} + 5\beta_{2} ) / 2$$ (-5*b3 + 5*b2) / 2

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 1.58114 + 1.58114i −1.58114 − 1.58114i −1.58114 + 1.58114i 1.58114 − 1.58114i
0 1.00000 1.00000i 0 −2.00000 + 1.00000i 0 0 0 1.00000i 0
417.2 0 1.00000 1.00000i 0 −2.00000 + 1.00000i 0 0 0 1.00000i 0
593.1 0 1.00000 + 1.00000i 0 −2.00000 1.00000i 0 0 0 1.00000i 0
593.2 0 1.00000 + 1.00000i 0 −2.00000 1.00000i 0 0 0 1.00000i 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.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bd.e 4
4.b odd 2 1 55.2.e.a 4
5.c odd 4 1 inner 880.2.bd.e 4
11.b odd 2 1 inner 880.2.bd.e 4
12.b even 2 1 495.2.k.b 4
20.d odd 2 1 275.2.e.b 4
20.e even 4 1 55.2.e.a 4
20.e even 4 1 275.2.e.b 4
44.c even 2 1 55.2.e.a 4
44.g even 10 4 605.2.m.b 16
44.h odd 10 4 605.2.m.b 16
55.e even 4 1 inner 880.2.bd.e 4
60.l odd 4 1 495.2.k.b 4
132.d odd 2 1 495.2.k.b 4
220.g even 2 1 275.2.e.b 4
220.i odd 4 1 55.2.e.a 4
220.i odd 4 1 275.2.e.b 4
220.v even 20 4 605.2.m.b 16
220.w odd 20 4 605.2.m.b 16
660.q even 4 1 495.2.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.a 4 4.b odd 2 1
55.2.e.a 4 20.e even 4 1
55.2.e.a 4 44.c even 2 1
55.2.e.a 4 220.i odd 4 1
275.2.e.b 4 20.d odd 2 1
275.2.e.b 4 20.e even 4 1
275.2.e.b 4 220.g even 2 1
275.2.e.b 4 220.i odd 4 1
495.2.k.b 4 12.b even 2 1
495.2.k.b 4 60.l odd 4 1
495.2.k.b 4 132.d odd 2 1
495.2.k.b 4 660.q even 4 1
605.2.m.b 16 44.g even 10 4
605.2.m.b 16 44.h odd 10 4
605.2.m.b 16 220.v even 20 4
605.2.m.b 16 220.w odd 20 4
880.2.bd.e 4 1.a even 1 1 trivial
880.2.bd.e 4 5.c odd 4 1 inner
880.2.bd.e 4 11.b odd 2 1 inner
880.2.bd.e 4 55.e even 4 1 inner

## Hecke kernels

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

 $$T_{3}^{2} - 2T_{3} + 2$$ T3^2 - 2*T3 + 2 $$T_{7}$$ T7 $$T_{13}^{4} + 400$$ T13^4 + 400

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 2 T + 2)^{2}$$
$5$ $$(T^{2} + 4 T + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 2 T + 11)^{2}$$
$13$ $$T^{4} + 400$$
$17$ $$T^{4} + 400$$
$19$ $$(T^{2} - 40)^{2}$$
$23$ $$(T^{2} - 2 T + 2)^{2}$$
$29$ $$(T^{2} - 40)^{2}$$
$31$ $$(T + 2)^{4}$$
$37$ $$(T^{2} - 6 T + 18)^{2}$$
$41$ $$(T^{2} + 40)^{2}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 6 T + 18)^{2}$$
$53$ $$(T^{2} + 2 T + 2)^{2}$$
$59$ $$(T^{2} + 36)^{2}$$
$61$ $$(T^{2} + 40)^{2}$$
$67$ $$(T^{2} + 6 T + 18)^{2}$$
$71$ $$(T - 8)^{4}$$
$73$ $$T^{4} + 400$$
$79$ $$(T^{2} - 40)^{2}$$
$83$ $$T^{4} + 6400$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$(T^{2} + 14 T + 98)^{2}$$