# Properties

 Label 440.2.y.a Level $440$ Weight $2$ Character orbit 440.y Analytic conductor $3.513$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(81,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("440.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$440 = 2^{3} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 440.y (of order $$5$$, degree $$4$$, 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: $$3.51341768894$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.13140625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1$$ x^8 - 3*x^7 + 5*x^6 - 3*x^5 + 4*x^4 + 3*x^3 + 5*x^2 + 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_1) q^{3} + \beta_{7} q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1) q^{9}+O(q^{10})$$ q + (-b5 + b1) * q^3 + b7 * q^5 + (-b7 + b6 + b5 - 3*b4 - b2 - 1) * q^7 + (-b6 - b5 + b4 + b3 + b1 + 1) * q^9 $$q + ( - \beta_{5} + \beta_1) q^{3} + \beta_{7} q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \cdots + 7) q^{99}+O(q^{100})$$ q + (-b5 + b1) * q^3 + b7 * q^5 + (-b7 + b6 + b5 - 3*b4 - b2 - 1) * q^7 + (-b6 - b5 + b4 + b3 + b1 + 1) * q^9 + (-b7 + b6 - 2*b5 + b4 - b1 + 1) * q^11 + (b6 + b5 - 2*b4 - b3 + 2*b1 - 2) * q^13 + (-b5 + b2) * q^15 + (b7 - 2*b4 + 2*b3 - 3*b2) * q^17 + (-4*b7 + 2*b6 - b5 - 4*b4 + b3 - 2*b2 + b1 - 1) * q^19 + (-b5 + b2 + b1 - 1) * q^21 + (-b7 - b5 + b3 - 3) * q^23 + (-b7 - b4 + b3 - 1) * q^25 + (-b7 + 3*b6 + b4 - b3 - b2 - 3*b1) * q^27 + (-3*b7 + 3*b5 - 2*b4 - 3*b2 - 3) * q^29 + (-3*b6 - 3*b5 - 2*b4 + 7*b3 + 2*b1 - 2) * q^31 + (b7 - 2*b6 - 2*b5 - b4 - 2*b3 + b2 + 2*b1 + 3) * q^33 + (-b6 - b5 + b4 + 2*b3 + 1) * q^35 + (4*b7 + b6 + 4*b5 - b4 - 4*b2 + 4) * q^37 + (b7 + 2*b4 - 2*b3 - b2) * q^39 + (-b7 - 5*b6 - 5*b5 - b4 + 3*b3 + 5*b2 + 5*b1 - 3) * q^41 + (-2*b7 + 2*b5 + 2*b3 - 3*b2 - 3*b1 - 1) * q^43 + (b7 - b5 - b3 + b2 + b1 - 1) * q^45 + (b7 + 3*b6 + 5*b5 + b4 - 3*b2 - 5*b1) * q^47 + (5*b7 - 4*b6 + 6*b4 - 6*b3 + 5*b2 + 4*b1) * q^49 + (-3*b7 - b6 - 3*b5 + 3*b2 - 3) * q^51 + (2*b6 + 2*b5 + 6*b4 - 3*b1 + 6) * q^53 + (2*b7 + b5 + b4 - 2*b3 + 2*b2 - b1 + 1) * q^55 + (-2*b6 - 2*b5 + b4 - 4*b3 + 3*b1 + 1) * q^57 + (-b7 + 7*b6 + b5 - 7*b4 - b2 - 1) * q^59 + (6*b7 + 7*b4 - 7*b3 + 6*b2) * q^61 + (-8*b7 + 3*b6 - 8*b4 + 4*b3 - 3*b2 - 4) * q^63 + (-2*b7 - 2*b5 + 2*b3 - b2 - b1 + 1) * q^65 + (6*b7 + 6*b5 - 6*b3 - b2 - b1 + 4) * q^67 + (3*b5 - b3 - 3*b1 + 1) * q^69 + (-6*b7 + 3*b6 - 4*b4 + 4*b3 + 5*b2 - 3*b1) * q^71 + (6*b7 - 7*b6 + b5 + 6*b4 - b2 + 6) * q^73 + (b6 + b5 - b1) * q^75 + (-6*b7 + 6*b6 + 3*b5 - 7*b4 - 2*b3 - 2*b2 - 2*b1 - 4) * q^77 + (b6 + b5 + 7*b4 - b3 - b1 + 7) * q^79 + (5*b7 - 2*b6 - 4*b5 + 3*b4 + 4*b2 + 5) * q^81 + (3*b7 + b4 - b3 + b2) * q^83 + (-b7 - 3*b6 - 3*b5 - b4 + 3*b3 + 3*b2 + 3*b1 - 3) * q^85 + (-3*b7 + 7*b5 + 3*b3 - 4*b2 - 4*b1 - 6) * q^87 + (2*b7 + 6*b5 - 2*b3 - 6*b2 - 6*b1 + 1) * q^89 + (15*b7 - 8*b6 - 4*b5 + 15*b4 - 5*b3 + 8*b2 + 4*b1 + 5) * q^91 + (-3*b7 + 4*b6 + 2*b4 - 2*b3 + 3*b2 - 4*b1) * q^93 + (3*b7 - 2*b6 - 3*b5 + 4*b4 + 3*b2 + 3) * q^95 + (-2*b4 - b3 + 6*b1 - 2) * q^97 + (2*b7 - 3*b6 - 2*b5 + 5*b4 - 2*b3 - b2 - b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{3} - 2 q^{5} + q^{7} + 7 q^{9}+O(q^{10})$$ 8 * q + q^3 - 2 * q^5 + q^7 + 7 * q^9 $$8 q + q^{3} - 2 q^{5} + q^{7} + 7 q^{9} + 3 q^{11} - 4 q^{13} + q^{15} - 3 q^{17} + 9 q^{19} - 4 q^{21} - 22 q^{23} - 2 q^{25} - 8 q^{27} - 17 q^{29} - 4 q^{31} + 21 q^{33} + 6 q^{35} + 24 q^{37} - 13 q^{39} - 4 q^{41} - 14 q^{43} - 8 q^{45} - 12 q^{47} - 15 q^{49} - 17 q^{51} + 35 q^{53} + 3 q^{55} - q^{57} + 21 q^{59} - 22 q^{61} + 5 q^{63} + 6 q^{65} + 14 q^{67} + 3 q^{69} + 40 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 41 q^{79} + 24 q^{81} - 7 q^{83} - 8 q^{85} - 46 q^{87} - 24 q^{89} - 18 q^{91} + 3 q^{93} + 9 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100})$$ 8 * q + q^3 - 2 * q^5 + q^7 + 7 * q^9 + 3 * q^11 - 4 * q^13 + q^15 - 3 * q^17 + 9 * q^19 - 4 * q^21 - 22 * q^23 - 2 * q^25 - 8 * q^27 - 17 * q^29 - 4 * q^31 + 21 * q^33 + 6 * q^35 + 24 * q^37 - 13 * q^39 - 4 * q^41 - 14 * q^43 - 8 * q^45 - 12 * q^47 - 15 * q^49 - 17 * q^51 + 35 * q^53 + 3 * q^55 - q^57 + 21 * q^59 - 22 * q^61 + 5 * q^63 + 6 * q^65 + 14 * q^67 + 3 * q^69 + 40 * q^71 + 9 * q^73 + q^75 - 4 * q^77 + 41 * q^79 + 24 * q^81 - 7 * q^83 - 8 * q^85 - 46 * q^87 - 24 * q^89 - 18 * q^91 + 3 * q^93 + 9 * q^95 + 4 * q^97 + 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8$$ (-v^7 + 2*v^6 - 3*v^5 - 4*v^3 - 7*v^2 - 12*v - 7) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8$$ (v^7 - 7*v^5 + 20*v^4 - 16*v^3 + 19*v^2 + 6*v + 9) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8$$ (-v^7 + 4*v^6 - 9*v^5 + 12*v^4 - 16*v^3 + 13*v^2 - 10*v - 1) / 8 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8$$ (-3*v^7 + 10*v^6 - 17*v^5 + 8*v^4 - 4*v^3 - 13*v^2 - 8*v - 5) / 8 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8$$ (3*v^7 - 12*v^6 + 23*v^5 - 20*v^4 + 16*v^3 + v^2 + 6*v - 1) / 8 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8$$ (-5*v^7 + 18*v^6 - 35*v^5 + 32*v^4 - 28*v^3 - 11*v^2 - 12*v - 7) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}$$ b6 + b5 + b4 - b2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1$$ b7 + 3*b6 + 2*b5 + b4 - 3*b2 - 2*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1$$ 3*b7 + 4*b6 + b4 - b3 - 5*b2 - 4*b1 $$\nu^{5}$$ $$=$$ $$4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1$$ 4*b7 - 6*b5 - 4*b3 - 6*b2 - 6*b1 - 1 $$\nu^{6}$$ $$=$$ $$-16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6$$ -16*b6 - 16*b5 - 6*b4 - 6*b3 - 7*b1 - 6 $$\nu^{7}$$ $$=$$ $$-16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16$$ -16*b7 - 51*b6 - 29*b5 - 23*b4 + 29*b2 - 16

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$177$$ $$221$$ $$321$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 1.69513 + 1.23158i −0.386111 − 0.280526i 1.69513 − 1.23158i −0.386111 + 0.280526i −0.227943 + 0.701538i 0.418926 − 1.28932i −0.227943 − 0.701538i 0.418926 + 1.28932i
0 −0.400166 + 1.23158i 0 −0.809017 + 0.587785i 0 −0.0703870 0.216629i 0 1.07039 + 0.777682i 0
81.2 0 0.0911485 0.280526i 0 −0.809017 + 0.587785i 0 −1.35666 4.17538i 0 2.35666 + 1.71222i 0
201.1 0 −0.400166 1.23158i 0 −0.809017 0.587785i 0 −0.0703870 + 0.216629i 0 1.07039 0.777682i 0
201.2 0 0.0911485 + 0.280526i 0 −0.809017 0.587785i 0 −1.35666 + 4.17538i 0 2.35666 1.71222i 0
361.1 0 −0.965584 + 0.701538i 0 0.309017 + 0.951057i 0 1.48685 + 1.08026i 0 −0.486854 + 1.49838i 0
361.2 0 1.77460 1.28932i 0 0.309017 + 0.951057i 0 0.440197 + 0.319822i 0 0.559803 1.72290i 0
401.1 0 −0.965584 0.701538i 0 0.309017 0.951057i 0 1.48685 1.08026i 0 −0.486854 1.49838i 0
401.2 0 1.77460 + 1.28932i 0 0.309017 0.951057i 0 0.440197 0.319822i 0 0.559803 + 1.72290i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 81.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.y.a 8
4.b odd 2 1 880.2.bo.d 8
11.c even 5 1 inner 440.2.y.a 8
11.c even 5 1 4840.2.a.y 4
11.d odd 10 1 4840.2.a.z 4
44.g even 10 1 9680.2.a.ct 4
44.h odd 10 1 880.2.bo.d 8
44.h odd 10 1 9680.2.a.cu 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.a 8 1.a even 1 1 trivial
440.2.y.a 8 11.c even 5 1 inner
880.2.bo.d 8 4.b odd 2 1
880.2.bo.d 8 44.h odd 10 1
4840.2.a.y 4 11.c even 5 1
4840.2.a.z 4 11.d odd 10 1
9680.2.a.ct 4 44.g even 10 1
9680.2.a.cu 4 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - T_{3}^{7} + T_{3}^{5} + 9T_{3}^{4} + 11T_{3}^{3} + 10T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(440, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{7} + T^{5} + \cdots + 1$$
$5$ $$(T^{4} + T^{3} + T^{2} + \cdots + 1)^{2}$$
$7$ $$T^{8} - T^{7} + 15 T^{6} + \cdots + 1$$
$11$ $$T^{8} - 3 T^{7} + \cdots + 14641$$
$13$ $$T^{8} + 4 T^{7} + \cdots + 1681$$
$17$ $$T^{8} + 3 T^{7} + \cdots + 121$$
$19$ $$T^{8} - 9 T^{7} + \cdots + 10201$$
$23$ $$(T^{4} + 11 T^{3} + \cdots + 31)^{2}$$
$29$ $$T^{8} + 17 T^{7} + \cdots + 1$$
$31$ $$T^{8} + 4 T^{7} + \cdots + 143641$$
$37$ $$T^{8} - 24 T^{7} + \cdots + 1$$
$41$ $$T^{8} + 4 T^{7} + \cdots + 292681$$
$43$ $$(T^{4} + 7 T^{3} - 17 T^{2} + \cdots + 61)^{2}$$
$47$ $$T^{8} + 12 T^{7} + \cdots + 22201$$
$53$ $$T^{8} - 35 T^{7} + \cdots + 1852321$$
$59$ $$T^{8} - 21 T^{7} + \cdots + 2825761$$
$61$ $$T^{8} + 22 T^{7} + \cdots + 657721$$
$67$ $$(T^{4} - 7 T^{3} + \cdots + 431)^{2}$$
$71$ $$T^{8} - 40 T^{7} + \cdots + 97436641$$
$73$ $$T^{8} - 9 T^{7} + \cdots + 14070001$$
$79$ $$T^{8} - 41 T^{7} + \cdots + 5669161$$
$83$ $$T^{8} + 7 T^{7} + \cdots + 121$$
$89$ $$(T^{4} + 12 T^{3} + \cdots - 479)^{2}$$
$97$ $$T^{8} - 4 T^{7} + \cdots + 4748041$$