# Properties

 Label 2760.1.cm.b Level $2760$ Weight $1$ Character orbit 2760.cm Analytic conductor $1.377$ Analytic rank $0$ Dimension $10$ Projective image $D_{11}$ CM discriminant -120 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,1,Mod(29,2760)]

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([0, 11, 11, 11, 18]))

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

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

chi := DirichletCharacter("2760.29");

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.cm (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: $$D_{11}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{11} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{22}^{8} q^{2} + \zeta_{22}^{9} q^{3} - \zeta_{22}^{5} q^{4} - \zeta_{22}^{4} q^{5} - \zeta_{22}^{6} q^{6} + \zeta_{22}^{2} q^{8} - \zeta_{22}^{7} q^{9} +O(q^{10})$$ q + z^8 * q^2 + z^9 * q^3 - z^5 * q^4 - z^4 * q^5 - z^6 * q^6 + z^2 * q^8 - z^7 * q^9 $$q + \zeta_{22}^{8} q^{2} + \zeta_{22}^{9} q^{3} - \zeta_{22}^{5} q^{4} - \zeta_{22}^{4} q^{5} - \zeta_{22}^{6} q^{6} + \zeta_{22}^{2} q^{8} - \zeta_{22}^{7} q^{9} + \zeta_{22} q^{10} + ( - \zeta_{22}^{10} + \zeta_{22}^{7}) q^{11} + \zeta_{22}^{3} q^{12} + ( - \zeta_{22}^{2} - 1) q^{13} + \zeta_{22}^{2} q^{15} + \zeta_{22}^{10} q^{16} + (\zeta_{22}^{8} + \zeta_{22}^{4}) q^{17} + \zeta_{22}^{4} q^{18} + \zeta_{22}^{9} q^{20} + (\zeta_{22}^{7} - \zeta_{22}^{4}) q^{22} - \zeta_{22}^{7} q^{23} - q^{24} + \zeta_{22}^{8} q^{25} + ( - \zeta_{22}^{10} - \zeta_{22}^{8}) q^{26} + \zeta_{22}^{5} q^{27} + (\zeta_{22}^{9} + \zeta_{22}^{3}) q^{29} + \zeta_{22}^{10} q^{30} + (\zeta_{22}^{8} - \zeta_{22}^{7}) q^{31} - \zeta_{22}^{7} q^{32} + (\zeta_{22}^{8} - \zeta_{22}^{5}) q^{33} + ( - \zeta_{22}^{5} - \zeta_{22}) q^{34} - \zeta_{22} q^{36} + ( - \zeta_{22}^{10} - \zeta_{22}^{4}) q^{37} + ( - \zeta_{22}^{9} + 1) q^{39} - \zeta_{22}^{6} q^{40} + ( - \zeta_{22}^{10} - \zeta_{22}^{8}) q^{43} + ( - \zeta_{22}^{4} + \zeta_{22}) q^{44} - q^{45} + \zeta_{22}^{4} q^{46} + ( - \zeta_{22}^{9} + \zeta_{22}^{2}) q^{47} - \zeta_{22}^{8} q^{48} - \zeta_{22}^{9} q^{49} - \zeta_{22}^{5} q^{50} + ( - \zeta_{22}^{6} - \zeta_{22}^{2}) q^{51} + (\zeta_{22}^{7} + \zeta_{22}^{5}) q^{52} - \zeta_{22}^{2} q^{54} + ( - \zeta_{22}^{3} + 1) q^{55} + ( - \zeta_{22}^{6} - 1) q^{58} + ( - \zeta_{22}^{8} + \zeta_{22}^{5}) q^{59} - \zeta_{22}^{7} q^{60} + ( - \zeta_{22}^{5} + \zeta_{22}^{4}) q^{62} + \zeta_{22}^{4} q^{64} + (\zeta_{22}^{6} + \zeta_{22}^{4}) q^{65} + ( - \zeta_{22}^{5} + \zeta_{22}^{2}) q^{66} + (\zeta_{22}^{3} - \zeta_{22}^{2}) q^{67} + ( - \zeta_{22}^{9} + \zeta_{22}^{2}) q^{68} + \zeta_{22}^{5} q^{69} - \zeta_{22}^{9} q^{72} + (\zeta_{22}^{7} + \zeta_{22}) q^{74} - \zeta_{22}^{6} q^{75} + (\zeta_{22}^{8} + \zeta_{22}^{6}) q^{78} + ( - \zeta_{22}^{7} + \zeta_{22}^{6}) q^{79} + \zeta_{22}^{3} q^{80} - \zeta_{22}^{3} q^{81} + ( - \zeta_{22}^{8} + \zeta_{22}) q^{85} + (\zeta_{22}^{7} + \zeta_{22}^{5}) q^{86} + ( - \zeta_{22}^{7} - \zeta_{22}) q^{87} + (\zeta_{22}^{9} + \zeta_{22}) q^{88} - \zeta_{22}^{8} q^{90} - \zeta_{22} q^{92} + ( - \zeta_{22}^{6} + \zeta_{22}^{5}) q^{93} + (\zeta_{22}^{10} + \zeta_{22}^{6}) q^{94} + \zeta_{22}^{5} q^{96} + \zeta_{22}^{6} q^{98} + ( - \zeta_{22}^{6} + \zeta_{22}^{3}) q^{99} +O(q^{100})$$ q + z^8 * q^2 + z^9 * q^3 - z^5 * q^4 - z^4 * q^5 - z^6 * q^6 + z^2 * q^8 - z^7 * q^9 + z * q^10 + (-z^10 + z^7) * q^11 + z^3 * q^12 + (-z^2 - 1) * q^13 + z^2 * q^15 + z^10 * q^16 + (z^8 + z^4) * q^17 + z^4 * q^18 + z^9 * q^20 + (z^7 - z^4) * q^22 - z^7 * q^23 - q^24 + z^8 * q^25 + (-z^10 - z^8) * q^26 + z^5 * q^27 + (z^9 + z^3) * q^29 + z^10 * q^30 + (z^8 - z^7) * q^31 - z^7 * q^32 + (z^8 - z^5) * q^33 + (-z^5 - z) * q^34 - z * q^36 + (-z^10 - z^4) * q^37 + (-z^9 + 1) * q^39 - z^6 * q^40 + (-z^10 - z^8) * q^43 + (-z^4 + z) * q^44 - q^45 + z^4 * q^46 + (-z^9 + z^2) * q^47 - z^8 * q^48 - z^9 * q^49 - z^5 * q^50 + (-z^6 - z^2) * q^51 + (z^7 + z^5) * q^52 - z^2 * q^54 + (-z^3 + 1) * q^55 + (-z^6 - 1) * q^58 + (-z^8 + z^5) * q^59 - z^7 * q^60 + (-z^5 + z^4) * q^62 + z^4 * q^64 + (z^6 + z^4) * q^65 + (-z^5 + z^2) * q^66 + (z^3 - z^2) * q^67 + (-z^9 + z^2) * q^68 + z^5 * q^69 - z^9 * q^72 + (z^7 + z) * q^74 - z^6 * q^75 + (z^8 + z^6) * q^78 + (-z^7 + z^6) * q^79 + z^3 * q^80 - z^3 * q^81 + (-z^8 + z) * q^85 + (z^7 + z^5) * q^86 + (-z^7 - z) * q^87 + (z^9 + z) * q^88 - z^8 * q^90 - z * q^92 + (-z^6 + z^5) * q^93 + (z^10 + z^6) * q^94 + z^5 * q^96 + z^6 * q^98 + (-z^6 + z^3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - q^{8} - q^{9}+O(q^{10})$$ 10 * q - q^2 + q^3 - q^4 + q^5 + q^6 - q^8 - q^9 $$10 q - q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - q^{8} - q^{9} + q^{10} + 2 q^{11} + q^{12} - 9 q^{13} - q^{15} - q^{16} - 2 q^{17} - q^{18} + q^{20} + 2 q^{22} - q^{23} - 10 q^{24} - q^{25} + 2 q^{26} + q^{27} + 2 q^{29} - q^{30} - 2 q^{31} - q^{32} - 2 q^{33} - 2 q^{34} - q^{36} + 2 q^{37} + 9 q^{39} + q^{40} + 2 q^{43} + 2 q^{44} - 10 q^{45} - q^{46} - 2 q^{47} + q^{48} - q^{49} - q^{50} + 2 q^{51} + 2 q^{52} + q^{54} + 9 q^{55} - 9 q^{58} + 2 q^{59} - q^{60} - 2 q^{62} - q^{64} - 2 q^{65} - 2 q^{66} + 2 q^{67} - 2 q^{68} + q^{69} - q^{72} + 2 q^{74} + q^{75} - 2 q^{78} - 2 q^{79} + q^{80} - q^{81} + 2 q^{85} + 2 q^{86} - 2 q^{87} + 2 q^{88} + q^{90} - q^{92} + 2 q^{93} - 2 q^{94} + q^{96} - q^{98} + 2 q^{99}+O(q^{100})$$ 10 * q - q^2 + q^3 - q^4 + q^5 + q^6 - q^8 - q^9 + q^10 + 2 * q^11 + q^12 - 9 * q^13 - q^15 - q^16 - 2 * q^17 - q^18 + q^20 + 2 * q^22 - q^23 - 10 * q^24 - q^25 + 2 * q^26 + q^27 + 2 * q^29 - q^30 - 2 * q^31 - q^32 - 2 * q^33 - 2 * q^34 - q^36 + 2 * q^37 + 9 * q^39 + q^40 + 2 * q^43 + 2 * q^44 - 10 * q^45 - q^46 - 2 * q^47 + q^48 - q^49 - q^50 + 2 * q^51 + 2 * q^52 + q^54 + 9 * q^55 - 9 * q^58 + 2 * q^59 - q^60 - 2 * q^62 - q^64 - 2 * q^65 - 2 * q^66 + 2 * q^67 - 2 * q^68 + q^69 - q^72 + 2 * q^74 + q^75 - 2 * q^78 - 2 * q^79 + q^80 - q^81 + 2 * q^85 + 2 * q^86 - 2 * q^87 + 2 * q^88 + q^90 - q^92 + 2 * q^93 - 2 * q^94 + q^96 - q^98 + 2 * q^99

## Character values

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

 $$n$$ $$1201$$ $$1381$$ $$1657$$ $$1841$$ $$2071$$ $$\chi(n)$$ $$\zeta_{22}^{8}$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 0.142315 + 0.989821i 0.959493 + 0.281733i −0.841254 − 0.540641i −0.841254 + 0.540641i 0.959493 − 0.281733i −0.415415 + 0.909632i 0.654861 + 0.755750i 0.654861 − 0.755750i 0.142315 − 0.989821i −0.415415 − 0.909632i
0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i −0.841254 + 0.540641i 0.654861 0.755750i 0 −0.959493 + 0.281733i 0.841254 + 0.540641i 0.142315 + 0.989821i
269.1 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −0.415415 0.909632i 0.142315 0.989821i 0 0.841254 + 0.540641i 0.415415 0.909632i 0.959493 + 0.281733i
509.1 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.654861 0.755750i 0.959493 + 0.281733i 0 0.415415 + 0.909632i −0.654861 0.755750i −0.841254 0.540641i
629.1 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.654861 + 0.755750i 0.959493 0.281733i 0 0.415415 0.909632i −0.654861 + 0.755750i −0.841254 + 0.540641i
749.1 −0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i 0.142315 + 0.989821i 0 0.841254 0.540641i 0.415415 + 0.909632i 0.959493 0.281733i
869.1 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i 0.142315 0.989821i −0.841254 + 0.540641i 0 −0.654861 0.755750i −0.142315 0.989821i −0.415415 + 0.909632i
1589.1 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.415415 + 0.909632i 0 −0.142315 + 0.989821i −0.959493 + 0.281733i 0.654861 + 0.755750i
1829.1 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i 0.959493 0.281733i −0.415415 0.909632i 0 −0.142315 0.989821i −0.959493 0.281733i 0.654861 0.755750i
2189.1 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −0.841254 0.540641i 0.654861 + 0.755750i 0 −0.959493 0.281733i 0.841254 0.540641i 0.142315 0.989821i
2309.1 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i 0.142315 + 0.989821i −0.841254 0.540641i 0 −0.654861 + 0.755750i −0.142315 + 0.989821i −0.415415 0.909632i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
120.i odd 2 1 CM by $$\Q(\sqrt{-30})$$
23.c even 11 1 inner
2760.cm 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.cm.b yes 10
3.b odd 2 1 2760.1.cm.d yes 10
5.b even 2 1 2760.1.cm.c yes 10
8.b even 2 1 2760.1.cm.a 10
15.d odd 2 1 2760.1.cm.a 10
23.c even 11 1 inner 2760.1.cm.b yes 10
24.h odd 2 1 2760.1.cm.c yes 10
40.f even 2 1 2760.1.cm.d yes 10
69.h odd 22 1 2760.1.cm.d yes 10
115.j even 22 1 2760.1.cm.c yes 10
120.i odd 2 1 CM 2760.1.cm.b yes 10
184.p even 22 1 2760.1.cm.a 10
345.p odd 22 1 2760.1.cm.a 10
552.r odd 22 1 2760.1.cm.c yes 10
920.bf even 22 1 2760.1.cm.d yes 10
2760.cm odd 22 1 inner 2760.1.cm.b yes 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.1.cm.a 10 8.b even 2 1
2760.1.cm.a 10 15.d odd 2 1
2760.1.cm.a 10 184.p even 22 1
2760.1.cm.a 10 345.p odd 22 1
2760.1.cm.b yes 10 1.a even 1 1 trivial
2760.1.cm.b yes 10 23.c even 11 1 inner
2760.1.cm.b yes 10 120.i odd 2 1 CM
2760.1.cm.b yes 10 2760.cm odd 22 1 inner
2760.1.cm.c yes 10 5.b even 2 1
2760.1.cm.c yes 10 24.h odd 2 1
2760.1.cm.c yes 10 115.j even 22 1
2760.1.cm.c yes 10 552.r odd 22 1
2760.1.cm.d yes 10 3.b odd 2 1
2760.1.cm.d yes 10 40.f even 2 1
2760.1.cm.d yes 10 69.h odd 22 1
2760.1.cm.d yes 10 920.bf even 22 1

## Hecke kernels

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

 $$T_{11}^{10} - 2 T_{11}^{9} + 4 T_{11}^{8} - 8 T_{11}^{7} + 16 T_{11}^{6} - 10 T_{11}^{5} + 20 T_{11}^{4} - 7 T_{11}^{3} + 3 T_{11}^{2} + 5 T_{11} + 1$$ T11^10 - 2*T11^9 + 4*T11^8 - 8*T11^7 + 16*T11^6 - 10*T11^5 + 20*T11^4 - 7*T11^3 + 3*T11^2 + 5*T11 + 1 $$T_{13}^{10} + 9 T_{13}^{9} + 37 T_{13}^{8} + 91 T_{13}^{7} + 148 T_{13}^{6} + 166 T_{13}^{5} + 130 T_{13}^{4} + 70 T_{13}^{3} + 25 T_{13}^{2} + 5 T_{13} + 1$$ T13^10 + 9*T13^9 + 37*T13^8 + 91*T13^7 + 148*T13^6 + 166*T13^5 + 130*T13^4 + 70*T13^3 + 25*T13^2 + 5*T13 + 1 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1$$
$3$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$5$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$7$ $$T^{10}$$
$11$ $$T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1$$
$13$ $$T^{10} + 9 T^{9} + 37 T^{8} + 91 T^{7} + \cdots + 1$$
$17$ $$T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1$$
$19$ $$T^{10}$$
$23$ $$T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1$$
$29$ $$T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1$$
$31$ $$T^{10} + 2 T^{9} + 4 T^{8} - 3 T^{7} + \cdots + 1$$
$37$ $$T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1$$
$41$ $$T^{10}$$
$43$ $$T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1$$
$47$ $$(T^{5} + T^{4} - 4 T^{3} - 3 T^{2} + 3 T + 1)^{2}$$
$53$ $$T^{10}$$
$59$ $$T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1$$
$61$ $$T^{10}$$
$67$ $$T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1$$
$71$ $$T^{10}$$
$73$ $$T^{10}$$
$79$ $$T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1$$
$83$ $$T^{10}$$
$89$ $$T^{10}$$
$97$ $$T^{10}$$