# Properties

 Label 273.1.ch.a Level $273$ Weight $1$ Character orbit 273.ch Analytic conductor $0.136$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,1,Mod(80,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 2, 1]))

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

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

chi := DirichletCharacter("273.80");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 273.ch (of order $$12$$, 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: $$0.136244748449$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{3} q^{3} - \zeta_{12}^{5} q^{4} + \zeta_{12}^{4} q^{7} - q^{9} +O(q^{10})$$ q - z^3 * q^3 - z^5 * q^4 + z^4 * q^7 - q^9 $$q - \zeta_{12}^{3} q^{3} - \zeta_{12}^{5} q^{4} + \zeta_{12}^{4} q^{7} - q^{9} - \zeta_{12}^{2} q^{12} + \zeta_{12}^{5} q^{13} - \zeta_{12}^{4} q^{16} + (\zeta_{12}^{2} - \zeta_{12}) q^{19} + \zeta_{12} q^{21} + \zeta_{12} q^{25} + \zeta_{12}^{3} q^{27} + \zeta_{12}^{3} q^{28} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{31} + \zeta_{12}^{5} q^{36} + (\zeta_{12}^{3} + \zeta_{12}^{2}) q^{37} + \zeta_{12}^{2} q^{39} + ( - \zeta_{12}^{2} - 1) q^{43} - \zeta_{12} q^{48} - \zeta_{12}^{2} q^{49} + \zeta_{12}^{4} q^{52} + ( - \zeta_{12}^{5} + \zeta_{12}^{4}) q^{57} + \zeta_{12}^{3} q^{61} - \zeta_{12}^{4} q^{63} - \zeta_{12}^{3} q^{64} + ( - \zeta_{12}^{3} - 1) q^{67} + ( - \zeta_{12}^{5} + 1) q^{73} - \zeta_{12}^{4} q^{75} + (\zeta_{12} - 1) q^{76} + q^{81} + q^{84} - \zeta_{12}^{3} q^{91} + (\zeta_{12}^{4} - \zeta_{12}) q^{93} + ( - \zeta_{12}^{3} - \zeta_{12}^{2}) q^{97} +O(q^{100})$$ q - z^3 * q^3 - z^5 * q^4 + z^4 * q^7 - q^9 - z^2 * q^12 + z^5 * q^13 - z^4 * q^16 + (z^2 - z) * q^19 + z * q^21 + z * q^25 + z^3 * q^27 + z^3 * q^28 + (-z^4 - z) * q^31 + z^5 * q^36 + (z^3 + z^2) * q^37 + z^2 * q^39 + (-z^2 - 1) * q^43 - z * q^48 - z^2 * q^49 + z^4 * q^52 + (-z^5 + z^4) * q^57 + z^3 * q^61 - z^4 * q^63 - z^3 * q^64 + (-z^3 - 1) * q^67 + (-z^5 + 1) * q^73 - z^4 * q^75 + (z - 1) * q^76 + q^81 + q^84 - z^3 * q^91 + (z^4 - z) * q^93 + (-z^3 - z^2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^7 - 4 * q^9 $$4 q - 2 q^{7} - 4 q^{9} - 2 q^{12} + 2 q^{16} + 2 q^{19} + 2 q^{31} + 2 q^{37} + 2 q^{39} - 6 q^{43} - 2 q^{49} - 2 q^{52} - 2 q^{57} + 2 q^{63} - 4 q^{67} + 4 q^{73} + 2 q^{75} - 4 q^{76} + 4 q^{81} + 4 q^{84} - 2 q^{93} - 2 q^{97}+O(q^{100})$$ 4 * q - 2 * q^7 - 4 * q^9 - 2 * q^12 + 2 * q^16 + 2 * q^19 + 2 * q^31 + 2 * q^37 + 2 * q^39 - 6 * q^43 - 2 * q^49 - 2 * q^52 - 2 * q^57 + 2 * q^63 - 4 * q^67 + 4 * q^73 + 2 * q^75 - 4 * q^76 + 4 * q^81 + 4 * q^84 - 2 * q^93 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 1.00000i −0.866025 + 0.500000i 0 0 −0.500000 + 0.866025i 0 −1.00000 0
110.1 0 1.00000i 0.866025 + 0.500000i 0 0 −0.500000 0.866025i 0 −1.00000 0
206.1 0 1.00000i 0.866025 0.500000i 0 0 −0.500000 + 0.866025i 0 −1.00000 0
215.1 0 1.00000i −0.866025 0.500000i 0 0 −0.500000 0.866025i 0 −1.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
91.w even 12 1 inner
273.ch odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.1.ch.a yes 4
3.b odd 2 1 CM 273.1.ch.a yes 4
7.b odd 2 1 1911.1.ci.a 4
7.c even 3 1 1911.1.bt.a 4
7.c even 3 1 1911.1.cb.b 4
7.d odd 6 1 273.1.bs.a 4
7.d odd 6 1 1911.1.cb.a 4
13.b even 2 1 3549.1.ch.c 4
13.c even 3 1 3549.1.bs.a 4
13.c even 3 1 3549.1.cb.d 4
13.d odd 4 1 3549.1.ch.a 4
13.d odd 4 1 3549.1.ch.b 4
13.e even 6 1 3549.1.bs.c 4
13.e even 6 1 3549.1.cb.a 4
13.f odd 12 1 273.1.bs.a 4
13.f odd 12 1 3549.1.bs.b 4
13.f odd 12 1 3549.1.cb.b 4
13.f odd 12 1 3549.1.cb.c 4
21.c even 2 1 1911.1.ci.a 4
21.g even 6 1 273.1.bs.a 4
21.g even 6 1 1911.1.cb.a 4
21.h odd 6 1 1911.1.bt.a 4
21.h odd 6 1 1911.1.cb.b 4
39.d odd 2 1 3549.1.ch.c 4
39.f even 4 1 3549.1.ch.a 4
39.f even 4 1 3549.1.ch.b 4
39.h odd 6 1 3549.1.bs.c 4
39.h odd 6 1 3549.1.cb.a 4
39.i odd 6 1 3549.1.bs.a 4
39.i odd 6 1 3549.1.cb.d 4
39.k even 12 1 273.1.bs.a 4
39.k even 12 1 3549.1.bs.b 4
39.k even 12 1 3549.1.cb.b 4
39.k even 12 1 3549.1.cb.c 4
91.l odd 6 1 3549.1.cb.b 4
91.m odd 6 1 3549.1.ch.a 4
91.p odd 6 1 3549.1.ch.b 4
91.s odd 6 1 3549.1.bs.b 4
91.v odd 6 1 3549.1.cb.c 4
91.w even 12 1 inner 273.1.ch.a yes 4
91.w even 12 1 3549.1.ch.c 4
91.x odd 12 1 1911.1.cb.a 4
91.ba even 12 1 1911.1.cb.b 4
91.ba even 12 1 3549.1.cb.a 4
91.ba even 12 1 3549.1.cb.d 4
91.bb even 12 1 3549.1.bs.a 4
91.bb even 12 1 3549.1.bs.c 4
91.bc even 12 1 1911.1.bt.a 4
91.bd odd 12 1 1911.1.ci.a 4
273.r even 6 1 3549.1.cb.c 4
273.y even 6 1 3549.1.ch.b 4
273.ba even 6 1 3549.1.bs.b 4
273.bf even 6 1 3549.1.ch.a 4
273.br even 6 1 3549.1.cb.b 4
273.bs odd 12 1 1911.1.cb.b 4
273.bs odd 12 1 3549.1.cb.a 4
273.bs odd 12 1 3549.1.cb.d 4
273.bv even 12 1 1911.1.cb.a 4
273.bw even 12 1 1911.1.ci.a 4
273.ca odd 12 1 1911.1.bt.a 4
273.cb odd 12 1 3549.1.bs.a 4
273.cb odd 12 1 3549.1.bs.c 4
273.ch odd 12 1 inner 273.1.ch.a yes 4
273.ch odd 12 1 3549.1.ch.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.bs.a 4 7.d odd 6 1
273.1.bs.a 4 13.f odd 12 1
273.1.bs.a 4 21.g even 6 1
273.1.bs.a 4 39.k even 12 1
273.1.ch.a yes 4 1.a even 1 1 trivial
273.1.ch.a yes 4 3.b odd 2 1 CM
273.1.ch.a yes 4 91.w even 12 1 inner
273.1.ch.a yes 4 273.ch odd 12 1 inner
1911.1.bt.a 4 7.c even 3 1
1911.1.bt.a 4 21.h odd 6 1
1911.1.bt.a 4 91.bc even 12 1
1911.1.bt.a 4 273.ca odd 12 1
1911.1.cb.a 4 7.d odd 6 1
1911.1.cb.a 4 21.g even 6 1
1911.1.cb.a 4 91.x odd 12 1
1911.1.cb.a 4 273.bv even 12 1
1911.1.cb.b 4 7.c even 3 1
1911.1.cb.b 4 21.h odd 6 1
1911.1.cb.b 4 91.ba even 12 1
1911.1.cb.b 4 273.bs odd 12 1
1911.1.ci.a 4 7.b odd 2 1
1911.1.ci.a 4 21.c even 2 1
1911.1.ci.a 4 91.bd odd 12 1
1911.1.ci.a 4 273.bw even 12 1
3549.1.bs.a 4 13.c even 3 1
3549.1.bs.a 4 39.i odd 6 1
3549.1.bs.a 4 91.bb even 12 1
3549.1.bs.a 4 273.cb odd 12 1
3549.1.bs.b 4 13.f odd 12 1
3549.1.bs.b 4 39.k even 12 1
3549.1.bs.b 4 91.s odd 6 1
3549.1.bs.b 4 273.ba even 6 1
3549.1.bs.c 4 13.e even 6 1
3549.1.bs.c 4 39.h odd 6 1
3549.1.bs.c 4 91.bb even 12 1
3549.1.bs.c 4 273.cb odd 12 1
3549.1.cb.a 4 13.e even 6 1
3549.1.cb.a 4 39.h odd 6 1
3549.1.cb.a 4 91.ba even 12 1
3549.1.cb.a 4 273.bs odd 12 1
3549.1.cb.b 4 13.f odd 12 1
3549.1.cb.b 4 39.k even 12 1
3549.1.cb.b 4 91.l odd 6 1
3549.1.cb.b 4 273.br even 6 1
3549.1.cb.c 4 13.f odd 12 1
3549.1.cb.c 4 39.k even 12 1
3549.1.cb.c 4 91.v odd 6 1
3549.1.cb.c 4 273.r even 6 1
3549.1.cb.d 4 13.c even 3 1
3549.1.cb.d 4 39.i odd 6 1
3549.1.cb.d 4 91.ba even 12 1
3549.1.cb.d 4 273.bs odd 12 1
3549.1.ch.a 4 13.d odd 4 1
3549.1.ch.a 4 39.f even 4 1
3549.1.ch.a 4 91.m odd 6 1
3549.1.ch.a 4 273.bf even 6 1
3549.1.ch.b 4 13.d odd 4 1
3549.1.ch.b 4 39.f even 4 1
3549.1.ch.b 4 91.p odd 6 1
3549.1.ch.b 4 273.y even 6 1
3549.1.ch.c 4 13.b even 2 1
3549.1.ch.c 4 39.d odd 2 1
3549.1.ch.c 4 91.w even 12 1
3549.1.ch.c 4 273.ch odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

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