# Properties

 Label 2760.1.co.c Level $2760$ Weight $1$ Character orbit 2760.co Analytic conductor $1.377$ Analytic rank $0$ Dimension $10$ Projective image $D_{22}$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,1,Mod(419,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 11, 11, 11, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2760.419");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2760.co (of order $$22$$, degree $$10$$, 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: $$1.37741943487$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{22}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{22} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{22} q^{2} + \zeta_{22}^{2} q^{3} + \zeta_{22}^{2} q^{4} - \zeta_{22}^{7} q^{5} + \zeta_{22}^{3} q^{6} + \zeta_{22}^{3} q^{8} + \zeta_{22}^{4} q^{9} +O(q^{10})$$ q + z * q^2 + z^2 * q^3 + z^2 * q^4 - z^7 * q^5 + z^3 * q^6 + z^3 * q^8 + z^4 * q^9 $$q + \zeta_{22} q^{2} + \zeta_{22}^{2} q^{3} + \zeta_{22}^{2} q^{4} - \zeta_{22}^{7} q^{5} + \zeta_{22}^{3} q^{6} + \zeta_{22}^{3} q^{8} + \zeta_{22}^{4} q^{9} - \zeta_{22}^{8} q^{10} + \zeta_{22}^{4} q^{12} - \zeta_{22}^{9} q^{15} + \zeta_{22}^{4} q^{16} + (\zeta_{22}^{10} - 1) q^{17} + \zeta_{22}^{5} q^{18} + (\zeta_{22}^{7} - \zeta_{22}^{5}) q^{19} - \zeta_{22}^{9} q^{20} - \zeta_{22}^{8} q^{23} + \zeta_{22}^{5} q^{24} - \zeta_{22}^{3} q^{25} + \zeta_{22}^{6} q^{27} - \zeta_{22}^{10} q^{30} + (\zeta_{22}^{10} - \zeta_{22}^{8}) q^{31} + \zeta_{22}^{5} q^{32} + ( - \zeta_{22} - 1) q^{34} + \zeta_{22}^{6} q^{36} + (\zeta_{22}^{8} - \zeta_{22}^{6}) q^{38} - \zeta_{22}^{10} q^{40} + q^{45} - \zeta_{22}^{9} q^{46} + ( - \zeta_{22}^{8} - \zeta_{22}^{3}) q^{47} + \zeta_{22}^{6} q^{48} + \zeta_{22}^{2} q^{49} - \zeta_{22}^{4} q^{50} + ( - \zeta_{22}^{2} - \zeta_{22}) q^{51} + (\zeta_{22}^{9} - \zeta_{22}^{4}) q^{53} + \zeta_{22}^{7} q^{54} + (\zeta_{22}^{9} - \zeta_{22}^{7}) q^{57} + q^{60} + (\zeta_{22}^{6} - \zeta_{22}) q^{61} + ( - \zeta_{22}^{9} - 1) q^{62} + \zeta_{22}^{6} q^{64} + ( - \zeta_{22}^{2} - \zeta_{22}) q^{68} - \zeta_{22}^{10} q^{69} + \zeta_{22}^{7} q^{72} - \zeta_{22}^{5} q^{75} + (\zeta_{22}^{9} - \zeta_{22}^{7}) q^{76} + ( - \zeta_{22}^{6} + \zeta_{22}^{3}) q^{79} + q^{80} + \zeta_{22}^{8} q^{81} + ( - \zeta_{22}^{6} + \zeta_{22}^{2}) q^{83} + (\zeta_{22}^{7} + \zeta_{22}^{6}) q^{85} + \zeta_{22} q^{90} - \zeta_{22}^{10} q^{92} + ( - \zeta_{22}^{10} - \zeta_{22}) q^{93} + ( - \zeta_{22}^{9} - \zeta_{22}^{4}) q^{94} + (\zeta_{22}^{3} - \zeta_{22}) q^{95} + \zeta_{22}^{7} q^{96} + \zeta_{22}^{3} q^{98} +O(q^{100})$$ q + z * q^2 + z^2 * q^3 + z^2 * q^4 - z^7 * q^5 + z^3 * q^6 + z^3 * q^8 + z^4 * q^9 - z^8 * q^10 + z^4 * q^12 - z^9 * q^15 + z^4 * q^16 + (z^10 - 1) * q^17 + z^5 * q^18 + (z^7 - z^5) * q^19 - z^9 * q^20 - z^8 * q^23 + z^5 * q^24 - z^3 * q^25 + z^6 * q^27 - z^10 * q^30 + (z^10 - z^8) * q^31 + z^5 * q^32 + (-z - 1) * q^34 + z^6 * q^36 + (z^8 - z^6) * q^38 - z^10 * q^40 + q^45 - z^9 * q^46 + (-z^8 - z^3) * q^47 + z^6 * q^48 + z^2 * q^49 - z^4 * q^50 + (-z^2 - z) * q^51 + (z^9 - z^4) * q^53 + z^7 * q^54 + (z^9 - z^7) * q^57 + q^60 + (z^6 - z) * q^61 + (-z^9 - 1) * q^62 + z^6 * q^64 + (-z^2 - z) * q^68 - z^10 * q^69 + z^7 * q^72 - z^5 * q^75 + (z^9 - z^7) * q^76 + (-z^6 + z^3) * q^79 + q^80 + z^8 * q^81 + (-z^6 + z^2) * q^83 + (z^7 + z^6) * q^85 + z * q^90 - z^10 * q^92 + (-z^10 - z) * q^93 + (-z^9 - z^4) * q^94 + (z^3 - z) * q^95 + z^7 * q^96 + z^3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + q^{8} - q^{9}+O(q^{10})$$ 10 * q + q^2 - q^3 - q^4 - q^5 + q^6 + q^8 - q^9 $$10 q + q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + q^{8} - q^{9} + q^{10} - q^{12} - q^{15} - q^{16} - 11 q^{17} + q^{18} - q^{20} + q^{23} + q^{24} - q^{25} - q^{27} + q^{30} + q^{32} - 11 q^{34} - q^{36} + q^{40} + 10 q^{45} - q^{46} - q^{48} - q^{49} + q^{50} + 2 q^{53} + q^{54} + 10 q^{60} - 2 q^{61} - 11 q^{62} - q^{64} + q^{69} + q^{72} - q^{75} + 2 q^{79} + 10 q^{80} - q^{81} + q^{90} + q^{92} + q^{96} + q^{98}+O(q^{100})$$ 10 * q + q^2 - q^3 - q^4 - q^5 + q^6 + q^8 - q^9 + q^10 - q^12 - q^15 - q^16 - 11 * q^17 + q^18 - q^20 + q^23 + q^24 - q^25 - q^27 + q^30 + q^32 - 11 * q^34 - q^36 + q^40 + 10 * q^45 - q^46 - q^48 - q^49 + q^50 + 2 * q^53 + q^54 + 10 * q^60 - 2 * q^61 - 11 * q^62 - q^64 + q^69 + q^72 - q^75 + 2 * q^79 + 10 * q^80 - q^81 + q^90 + q^92 + q^96 + q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$1381$$ $$1657$$ $$1841$$ $$2071$$ $$\chi(n)$$ $$\zeta_{22}^{3}$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1
 −0.415415 − 0.909632i 0.959493 + 0.281733i −0.841254 − 0.540641i −0.841254 + 0.540641i 0.959493 − 0.281733i 0.142315 − 0.989821i −0.415415 + 0.909632i 0.142315 + 0.989821i 0.654861 + 0.755750i 0.654861 − 0.755750i
−0.415415 0.909632i −0.654861 + 0.755750i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.959493 + 0.281733i 0 0.959493 + 0.281733i −0.142315 0.989821i 0.959493 0.281733i
539.1 0.959493 + 0.281733i 0.841254 + 0.540641i 0.841254 + 0.540641i 0.415415 0.909632i 0.654861 + 0.755750i 0 0.654861 + 0.755750i 0.415415 + 0.909632i 0.654861 0.755750i
659.1 −0.841254 0.540641i 0.415415 + 0.909632i 0.415415 + 0.909632i −0.654861 0.755750i 0.142315 0.989821i 0 0.142315 0.989821i −0.654861 + 0.755750i 0.142315 + 0.989821i
779.1 −0.841254 + 0.540641i 0.415415 0.909632i 0.415415 0.909632i −0.654861 + 0.755750i 0.142315 + 0.989821i 0 0.142315 + 0.989821i −0.654861 0.755750i 0.142315 0.989821i
1019.1 0.959493 0.281733i 0.841254 0.540641i 0.841254 0.540641i 0.415415 + 0.909632i 0.654861 0.755750i 0 0.654861 0.755750i 0.415415 0.909632i 0.654861 + 0.755750i
1259.1 0.142315 0.989821i −0.959493 0.281733i −0.959493 0.281733i 0.841254 0.540641i −0.415415 + 0.909632i 0 −0.415415 + 0.909632i 0.841254 + 0.540641i −0.415415 0.909632i
1739.1 −0.415415 + 0.909632i −0.654861 0.755750i −0.654861 0.755750i −0.142315 0.989821i 0.959493 0.281733i 0 0.959493 0.281733i −0.142315 + 0.989821i 0.959493 + 0.281733i
1859.1 0.142315 + 0.989821i −0.959493 + 0.281733i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.415415 0.909632i 0 −0.415415 0.909632i 0.841254 0.540641i −0.415415 + 0.909632i
2219.1 0.654861 + 0.755750i −0.142315 + 0.989821i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.841254 + 0.540641i 0 −0.841254 + 0.540641i −0.959493 0.281733i −0.841254 0.540641i
2459.1 0.654861 0.755750i −0.142315 0.989821i −0.142315 0.989821i −0.959493 0.281733i −0.841254 0.540641i 0 −0.841254 0.540641i −0.959493 + 0.281733i −0.841254 + 0.540641i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
184.j even 22 1 inner
2760.co odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.1.co.c yes 10
3.b odd 2 1 2760.1.co.b yes 10
5.b even 2 1 2760.1.co.b yes 10
8.d odd 2 1 2760.1.co.d yes 10
15.d odd 2 1 CM 2760.1.co.c yes 10
23.d odd 22 1 2760.1.co.d yes 10
24.f even 2 1 2760.1.co.a 10
40.e odd 2 1 2760.1.co.a 10
69.g even 22 1 2760.1.co.a 10
115.i odd 22 1 2760.1.co.a 10
120.m even 2 1 2760.1.co.d yes 10
184.j even 22 1 inner 2760.1.co.c yes 10
345.n even 22 1 2760.1.co.d yes 10
552.z odd 22 1 2760.1.co.b yes 10
920.bn even 22 1 2760.1.co.b yes 10
2760.co odd 22 1 inner 2760.1.co.c yes 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.1.co.a 10 24.f even 2 1
2760.1.co.a 10 40.e odd 2 1
2760.1.co.a 10 69.g even 22 1
2760.1.co.a 10 115.i odd 22 1
2760.1.co.b yes 10 3.b odd 2 1
2760.1.co.b yes 10 5.b even 2 1
2760.1.co.b yes 10 552.z odd 22 1
2760.1.co.b yes 10 920.bn even 22 1
2760.1.co.c yes 10 1.a even 1 1 trivial
2760.1.co.c yes 10 15.d odd 2 1 CM
2760.1.co.c yes 10 184.j even 22 1 inner
2760.1.co.c yes 10 2760.co odd 22 1 inner
2760.1.co.d yes 10 8.d odd 2 1
2760.1.co.d yes 10 23.d odd 22 1
2760.1.co.d yes 10 120.m even 2 1
2760.1.co.d yes 10 345.n even 22 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2760, [\chi])$$:

 $$T_{17}^{10} + 11 T_{17}^{9} + 55 T_{17}^{8} + 165 T_{17}^{7} + 330 T_{17}^{6} + 462 T_{17}^{5} + 462 T_{17}^{4} + 330 T_{17}^{3} + 165 T_{17}^{2} + 55 T_{17} + 11$$ T17^10 + 11*T17^9 + 55*T17^8 + 165*T17^7 + 330*T17^6 + 462*T17^5 + 462*T17^4 + 330*T17^3 + 165*T17^2 + 55*T17 + 11 $$T_{31}^{10} + 22T_{31}^{5} - 33T_{31}^{3} + 11T_{31}^{2} + 11T_{31} + 11$$ T31^10 + 22*T31^5 - 33*T31^3 + 11*T31^2 + 11*T31 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$3$ $$T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1$$
$5$ $$T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1$$
$7$ $$T^{10}$$
$11$ $$T^{10}$$
$13$ $$T^{10}$$
$17$ $$T^{10} + 11 T^{9} + 55 T^{8} + 165 T^{7} + \cdots + 11$$
$19$ $$T^{10} + 11 T^{6} - 11 T^{5} + 22 T^{2} + \cdots + 11$$
$23$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$29$ $$T^{10}$$
$31$ $$T^{10} + 22 T^{5} - 33 T^{3} + 11 T^{2} + \cdots + 11$$
$37$ $$T^{10}$$
$41$ $$T^{10}$$
$43$ $$T^{10}$$
$47$ $$T^{10} + 11 T^{8} + 44 T^{6} + 77 T^{4} + \cdots + 11$$
$53$ $$T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1$$
$59$ $$T^{10}$$
$61$ $$T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1$$
$67$ $$T^{10}$$
$71$ $$T^{10}$$
$73$ $$T^{10}$$
$79$ $$T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1$$
$83$ $$T^{10} + 22 T^{5} - 33 T^{3} + 11 T^{2} + \cdots + 11$$
$89$ $$T^{10}$$
$97$ $$T^{10}$$