# Properties

 Label 2760.1.cm.f Level $2760$ Weight $1$ Character orbit 2760.cm Analytic conductor $1.377$ Analytic rank $0$ Dimension $20$ Projective image $D_{22}$ CM discriminant -15 Inner twists $8$

# 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: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{22})$$ Coefficient field: $$\Q(\zeta_{44})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{44}^{5} q^{2} - \zeta_{44}^{21} q^{3} + \zeta_{44}^{10} q^{4} + \zeta_{44}^{13} q^{5} - \zeta_{44}^{4} q^{6} - \zeta_{44}^{15} q^{8} - \zeta_{44}^{20} q^{9} +O(q^{10})$$ q - z^5 * q^2 - z^21 * q^3 + z^10 * q^4 + z^13 * q^5 - z^4 * q^6 - z^15 * q^8 - z^20 * q^9 $$q - \zeta_{44}^{5} q^{2} - \zeta_{44}^{21} q^{3} + \zeta_{44}^{10} q^{4} + \zeta_{44}^{13} q^{5} - \zeta_{44}^{4} q^{6} - \zeta_{44}^{15} q^{8} - \zeta_{44}^{20} q^{9} - \zeta_{44}^{18} q^{10} + \zeta_{44}^{9} q^{12} + \zeta_{44}^{12} q^{15} + \zeta_{44}^{20} q^{16} + (\zeta_{44}^{17} + \zeta_{44}^{11}) q^{17} - \zeta_{44}^{3} q^{18} + ( - \zeta_{44}^{14} + \zeta_{44}^{2}) q^{19} - \zeta_{44} q^{20} + \zeta_{44}^{7} q^{23} - \zeta_{44}^{14} q^{24} - \zeta_{44}^{4} q^{25} - \zeta_{44}^{19} q^{27} - \zeta_{44}^{17} q^{30} + (\zeta_{44}^{18} + \zeta_{44}^{6}) q^{31} + \zeta_{44}^{3} q^{32} + ( - \zeta_{44}^{16} + 1) q^{34} + \zeta_{44}^{8} q^{36} + (\zeta_{44}^{19} - \zeta_{44}^{7}) q^{38} + \zeta_{44}^{6} q^{40} + \zeta_{44}^{11} q^{45} - \zeta_{44}^{12} q^{46} + (\zeta_{44}^{15} - \zeta_{44}^{7}) q^{47} + \zeta_{44}^{19} q^{48} - \zeta_{44}^{10} q^{49} + \zeta_{44}^{9} q^{50} + (\zeta_{44}^{16} + \zeta_{44}^{10}) q^{51} + ( - \zeta_{44}^{9} - \zeta_{44}) q^{53} - \zeta_{44}^{2} q^{54} + ( - \zeta_{44}^{13} + \zeta_{44}) q^{57} - q^{60} + ( - \zeta_{44}^{16} + \zeta_{44}^{8}) q^{61} + ( - \zeta_{44}^{11} + \zeta_{44}) q^{62} - \zeta_{44}^{8} q^{64} + (\zeta_{44}^{21} - \zeta_{44}^{5}) q^{68} + \zeta_{44}^{6} q^{69} - \zeta_{44}^{13} q^{72} - \zeta_{44}^{3} q^{75} + (\zeta_{44}^{12} + \zeta_{44}^{2}) q^{76} + (\zeta_{44}^{8} + \zeta_{44}^{4}) q^{79} - \zeta_{44}^{11} q^{80} - \zeta_{44}^{18} q^{81} + (\zeta_{44}^{21} - \zeta_{44}^{19}) q^{83} + ( - \zeta_{44}^{8} - \zeta_{44}^{2}) q^{85} - \zeta_{44}^{16} q^{90} + \zeta_{44}^{17} q^{92} + (\zeta_{44}^{17} + \zeta_{44}^{5}) q^{93} + ( - \zeta_{44}^{20} + \zeta_{44}^{12}) q^{94} + (\zeta_{44}^{15} + \zeta_{44}^{5}) q^{95} + \zeta_{44}^{2} q^{96} + \zeta_{44}^{15} q^{98} +O(q^{100})$$ q - z^5 * q^2 - z^21 * q^3 + z^10 * q^4 + z^13 * q^5 - z^4 * q^6 - z^15 * q^8 - z^20 * q^9 - z^18 * q^10 + z^9 * q^12 + z^12 * q^15 + z^20 * q^16 + (z^17 + z^11) * q^17 - z^3 * q^18 + (-z^14 + z^2) * q^19 - z * q^20 + z^7 * q^23 - z^14 * q^24 - z^4 * q^25 - z^19 * q^27 - z^17 * q^30 + (z^18 + z^6) * q^31 + z^3 * q^32 + (-z^16 + 1) * q^34 + z^8 * q^36 + (z^19 - z^7) * q^38 + z^6 * q^40 + z^11 * q^45 - z^12 * q^46 + (z^15 - z^7) * q^47 + z^19 * q^48 - z^10 * q^49 + z^9 * q^50 + (z^16 + z^10) * q^51 + (-z^9 - z) * q^53 - z^2 * q^54 + (-z^13 + z) * q^57 - q^60 + (-z^16 + z^8) * q^61 + (-z^11 + z) * q^62 - z^8 * q^64 + (z^21 - z^5) * q^68 + z^6 * q^69 - z^13 * q^72 - z^3 * q^75 + (z^12 + z^2) * q^76 + (z^8 + z^4) * q^79 - z^11 * q^80 - z^18 * q^81 + (z^21 - z^19) * q^83 + (-z^8 - z^2) * q^85 - z^16 * q^90 + z^17 * q^92 + (z^17 + z^5) * q^93 + (-z^20 + z^12) * q^94 + (z^15 + z^5) * q^95 + z^2 * q^96 + z^15 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 2 q^{4} + 2 q^{6} + 2 q^{9}+O(q^{10})$$ 20 * q + 2 * q^4 + 2 * q^6 + 2 * q^9 $$20 q + 2 q^{4} + 2 q^{6} + 2 q^{9} - 2 q^{10} - 2 q^{15} - 2 q^{16} - 2 q^{24} + 2 q^{25} + 4 q^{31} + 22 q^{34} - 2 q^{36} + 2 q^{40} + 2 q^{46} - 2 q^{49} - 2 q^{54} - 20 q^{60} + 2 q^{64} + 2 q^{69} - 4 q^{79} - 2 q^{81} + 2 q^{90} + 2 q^{96}+O(q^{100})$$ 20 * q + 2 * q^4 + 2 * q^6 + 2 * q^9 - 2 * q^10 - 2 * q^15 - 2 * q^16 - 2 * q^24 + 2 * q^25 + 4 * q^31 + 22 * q^34 - 2 * q^36 + 2 * q^40 + 2 * q^46 - 2 * q^49 - 2 * q^54 - 20 * q^60 + 2 * q^64 + 2 * q^69 - 4 * q^79 - 2 * q^81 + 2 * q^90 + 2 * q^96

## Character values

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

 $$n$$ $$1201$$ $$1381$$ $$1657$$ $$1841$$ $$2071$$ $$\chi(n)$$ $$\zeta_{44}^{4}$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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
 0.281733 + 0.959493i −0.281733 − 0.959493i 0.540641 − 0.841254i −0.540641 + 0.841254i −0.909632 + 0.415415i 0.909632 − 0.415415i −0.909632 − 0.415415i 0.909632 + 0.415415i 0.540641 + 0.841254i −0.540641 − 0.841254i −0.755750 + 0.654861i 0.755750 − 0.654861i 0.989821 + 0.142315i −0.989821 − 0.142315i 0.989821 − 0.142315i −0.989821 + 0.142315i 0.281733 − 0.959493i −0.281733 + 0.959493i −0.755750 − 0.654861i 0.755750 + 0.654861i
−0.989821 0.142315i 0.281733 0.959493i 0.959493 + 0.281733i −0.540641 0.841254i −0.415415 + 0.909632i 0 −0.909632 0.415415i −0.841254 0.540641i 0.415415 + 0.909632i
29.2 0.989821 + 0.142315i −0.281733 + 0.959493i 0.959493 + 0.281733i 0.540641 + 0.841254i −0.415415 + 0.909632i 0 0.909632 + 0.415415i −0.841254 0.540641i 0.415415 + 0.909632i
269.1 −0.281733 0.959493i 0.540641 + 0.841254i −0.841254 + 0.540641i 0.909632 0.415415i 0.654861 0.755750i 0 0.755750 + 0.654861i −0.415415 + 0.909632i −0.654861 0.755750i
269.2 0.281733 + 0.959493i −0.540641 0.841254i −0.841254 + 0.540641i −0.909632 + 0.415415i 0.654861 0.755750i 0 −0.755750 0.654861i −0.415415 + 0.909632i −0.654861 0.755750i
509.1 −0.540641 0.841254i −0.909632 0.415415i −0.415415 + 0.909632i −0.755750 0.654861i 0.142315 + 0.989821i 0 0.989821 0.142315i 0.654861 + 0.755750i −0.142315 + 0.989821i
509.2 0.540641 + 0.841254i 0.909632 + 0.415415i −0.415415 + 0.909632i 0.755750 + 0.654861i 0.142315 + 0.989821i 0 −0.989821 + 0.142315i 0.654861 + 0.755750i −0.142315 + 0.989821i
629.1 −0.540641 + 0.841254i −0.909632 + 0.415415i −0.415415 0.909632i −0.755750 + 0.654861i 0.142315 0.989821i 0 0.989821 + 0.142315i 0.654861 0.755750i −0.142315 0.989821i
629.2 0.540641 0.841254i 0.909632 0.415415i −0.415415 0.909632i 0.755750 0.654861i 0.142315 0.989821i 0 −0.989821 0.142315i 0.654861 0.755750i −0.142315 0.989821i
749.1 −0.281733 + 0.959493i 0.540641 0.841254i −0.841254 0.540641i 0.909632 + 0.415415i 0.654861 + 0.755750i 0 0.755750 0.654861i −0.415415 0.909632i −0.654861 + 0.755750i
749.2 0.281733 0.959493i −0.540641 + 0.841254i −0.841254 0.540641i −0.909632 0.415415i 0.654861 + 0.755750i 0 −0.755750 + 0.654861i −0.415415 0.909632i −0.654861 + 0.755750i
869.1 −0.909632 + 0.415415i −0.755750 0.654861i 0.654861 0.755750i 0.989821 + 0.142315i 0.959493 + 0.281733i 0 −0.281733 + 0.959493i 0.142315 + 0.989821i −0.959493 + 0.281733i
869.2 0.909632 0.415415i 0.755750 + 0.654861i 0.654861 0.755750i −0.989821 0.142315i 0.959493 + 0.281733i 0 0.281733 0.959493i 0.142315 + 0.989821i −0.959493 + 0.281733i
1589.1 −0.755750 0.654861i 0.989821 0.142315i 0.142315 + 0.989821i −0.281733 + 0.959493i −0.841254 0.540641i 0 0.540641 0.841254i 0.959493 0.281733i 0.841254 0.540641i
1589.2 0.755750 + 0.654861i −0.989821 + 0.142315i 0.142315 + 0.989821i 0.281733 0.959493i −0.841254 0.540641i 0 −0.540641 + 0.841254i 0.959493 0.281733i 0.841254 0.540641i
1829.1 −0.755750 + 0.654861i 0.989821 + 0.142315i 0.142315 0.989821i −0.281733 0.959493i −0.841254 + 0.540641i 0 0.540641 + 0.841254i 0.959493 + 0.281733i 0.841254 + 0.540641i
1829.2 0.755750 0.654861i −0.989821 0.142315i 0.142315 0.989821i 0.281733 + 0.959493i −0.841254 + 0.540641i 0 −0.540641 0.841254i 0.959493 + 0.281733i 0.841254 + 0.540641i
2189.1 −0.989821 + 0.142315i 0.281733 + 0.959493i 0.959493 0.281733i −0.540641 + 0.841254i −0.415415 0.909632i 0 −0.909632 + 0.415415i −0.841254 + 0.540641i 0.415415 0.909632i
2189.2 0.989821 0.142315i −0.281733 0.959493i 0.959493 0.281733i 0.540641 0.841254i −0.415415 0.909632i 0 0.909632 0.415415i −0.841254 + 0.540641i 0.415415 0.909632i
2309.1 −0.909632 0.415415i −0.755750 + 0.654861i 0.654861 + 0.755750i 0.989821 0.142315i 0.959493 0.281733i 0 −0.281733 0.959493i 0.142315 0.989821i −0.959493 0.281733i
2309.2 0.909632 + 0.415415i 0.755750 0.654861i 0.654861 + 0.755750i −0.989821 + 0.142315i 0.959493 0.281733i 0 0.281733 + 0.959493i 0.142315 0.989821i −0.959493 0.281733i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
184.p even 22 1 inner
552.r odd 22 1 inner
920.bf even 22 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.f yes 20
3.b odd 2 1 inner 2760.1.cm.f yes 20
5.b even 2 1 inner 2760.1.cm.f yes 20
8.b even 2 1 2760.1.cm.e 20
15.d odd 2 1 CM 2760.1.cm.f yes 20
23.c even 11 1 2760.1.cm.e 20
24.h odd 2 1 2760.1.cm.e 20
40.f even 2 1 2760.1.cm.e 20
69.h odd 22 1 2760.1.cm.e 20
115.j even 22 1 2760.1.cm.e 20
120.i odd 2 1 2760.1.cm.e 20
184.p even 22 1 inner 2760.1.cm.f yes 20
345.p odd 22 1 2760.1.cm.e 20
552.r odd 22 1 inner 2760.1.cm.f yes 20
920.bf even 22 1 inner 2760.1.cm.f yes 20
2760.cm odd 22 1 inner 2760.1.cm.f yes 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.1.cm.e 20 8.b even 2 1
2760.1.cm.e 20 23.c even 11 1
2760.1.cm.e 20 24.h odd 2 1
2760.1.cm.e 20 40.f even 2 1
2760.1.cm.e 20 69.h odd 22 1
2760.1.cm.e 20 115.j even 22 1
2760.1.cm.e 20 120.i odd 2 1
2760.1.cm.e 20 345.p odd 22 1
2760.1.cm.f yes 20 1.a even 1 1 trivial
2760.1.cm.f yes 20 3.b odd 2 1 inner
2760.1.cm.f yes 20 5.b even 2 1 inner
2760.1.cm.f yes 20 15.d odd 2 1 CM
2760.1.cm.f yes 20 184.p even 22 1 inner
2760.1.cm.f yes 20 552.r odd 22 1 inner
2760.1.cm.f yes 20 920.bf even 22 1 inner
2760.1.cm.f yes 20 2760.cm odd 22 1 inner

## Hecke kernels

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

 $$T_{11}$$ T11 $$T_{13}$$ T13 $$T_{19}^{10} + 11T_{19}^{6} - 11T_{19}^{5} + 22T_{19}^{2} + 33T_{19} + 11$$ T19^10 + 11*T19^6 - 11*T19^5 + 22*T19^2 + 33*T19 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} - T^{18} + \cdots + 1$$
$3$ $$T^{20} - T^{18} + \cdots + 1$$
$5$ $$T^{20} - T^{18} + \cdots + 1$$
$7$ $$T^{20}$$
$11$ $$T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20} + 11 T^{18} + \cdots + 121$$
$19$ $$(T^{10} + 11 T^{6} + \cdots + 11)^{2}$$
$23$ $$T^{20} - T^{18} + \cdots + 1$$
$29$ $$T^{20}$$
$31$ $$(T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2}$$
$37$ $$T^{20}$$
$41$ $$T^{20}$$
$43$ $$T^{20}$$
$47$ $$(T^{10} - 11 T^{8} + \cdots - 11)^{2}$$
$53$ $$T^{20} - 4 T^{18} + \cdots + 1$$
$59$ $$T^{20}$$
$61$ $$(T^{10} + 11 T^{4} + \cdots + 11)^{2}$$
$67$ $$T^{20}$$
$71$ $$T^{20}$$
$73$ $$T^{20}$$
$79$ $$(T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2}$$
$83$ $$T^{20} - 4 T^{18} + \cdots + 1$$
$89$ $$T^{20}$$
$97$ $$T^{20}$$