# Properties

 Label 273.1.s.b Level $273$ Weight $1$ Character orbit 273.s Analytic conductor $0.136$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,1,Mod(74,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4, 4]))

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

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

chi := DirichletCharacter("273.74");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 273.s (of order $$6$$, degree $$2$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.74529.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{3} q^{2} - \zeta_{12}^{5} q^{3} + \zeta_{12}^{5} q^{5} - \zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} +O(q^{10})$$ q - z^3 * q^2 - z^5 * q^3 + z^5 * q^5 - z^2 * q^6 + z^4 * q^7 - z^3 * q^8 - z^4 * q^9 $$q - \zeta_{12}^{3} q^{2} - \zeta_{12}^{5} q^{3} + \zeta_{12}^{5} q^{5} - \zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} + \zeta_{12}^{2} q^{10} - q^{13} + \zeta_{12} q^{14} + \zeta_{12}^{4} q^{15} - q^{16} + \zeta_{12}^{3} q^{17} - \zeta_{12} q^{18} + \zeta_{12}^{3} q^{21} + \zeta_{12}^{3} q^{23} - \zeta_{12}^{2} q^{24} + \zeta_{12}^{3} q^{26} - \zeta_{12}^{3} q^{27} - \zeta_{12} q^{29} + \zeta_{12} q^{30} - \zeta_{12}^{4} q^{31} + q^{34} - \zeta_{12}^{3} q^{35} + q^{37} + \zeta_{12}^{5} q^{39} + \zeta_{12}^{2} q^{40} + \zeta_{12} q^{41} + q^{42} - \zeta_{12}^{2} q^{43} + \zeta_{12}^{3} q^{45} + q^{46} + \zeta_{12}^{5} q^{47} + \zeta_{12}^{5} q^{48} - \zeta_{12}^{2} q^{49} + \zeta_{12}^{2} q^{51} + \zeta_{12} q^{53} - q^{54} + \zeta_{12} q^{56} + \zeta_{12}^{4} q^{58} - \zeta_{12}^{3} q^{59} - \zeta_{12} q^{62} + \zeta_{12}^{2} q^{63} - q^{64} - \zeta_{12}^{5} q^{65} + \zeta_{12}^{2} q^{69} - q^{70} - \zeta_{12}^{5} q^{71} - \zeta_{12} q^{72} + \zeta_{12}^{4} q^{73} - \zeta_{12}^{3} q^{74} + \zeta_{12}^{2} q^{78} - \zeta_{12}^{2} q^{79} - \zeta_{12}^{5} q^{80} - \zeta_{12}^{2} q^{81} - \zeta_{12}^{4} q^{82} - \zeta_{12}^{2} q^{85} + \zeta_{12}^{5} q^{86} - q^{87} - \zeta_{12}^{3} q^{89} + q^{90} - \zeta_{12}^{4} q^{91} - \zeta_{12}^{3} q^{93} + \zeta_{12}^{2} q^{94} + \zeta_{12}^{2} q^{97} + \zeta_{12}^{5} q^{98} +O(q^{100})$$ q - z^3 * q^2 - z^5 * q^3 + z^5 * q^5 - z^2 * q^6 + z^4 * q^7 - z^3 * q^8 - z^4 * q^9 + z^2 * q^10 - q^13 + z * q^14 + z^4 * q^15 - q^16 + z^3 * q^17 - z * q^18 + z^3 * q^21 + z^3 * q^23 - z^2 * q^24 + z^3 * q^26 - z^3 * q^27 - z * q^29 + z * q^30 - z^4 * q^31 + q^34 - z^3 * q^35 + q^37 + z^5 * q^39 + z^2 * q^40 + z * q^41 + q^42 - z^2 * q^43 + z^3 * q^45 + q^46 + z^5 * q^47 + z^5 * q^48 - z^2 * q^49 + z^2 * q^51 + z * q^53 - q^54 + z * q^56 + z^4 * q^58 - z^3 * q^59 - z * q^62 + z^2 * q^63 - q^64 - z^5 * q^65 + z^2 * q^69 - q^70 - z^5 * q^71 - z * q^72 + z^4 * q^73 - z^3 * q^74 + z^2 * q^78 - z^2 * q^79 - z^5 * q^80 - z^2 * q^81 - z^4 * q^82 - z^2 * q^85 + z^5 * q^86 - q^87 - z^3 * q^89 + q^90 - z^4 * q^91 - z^3 * q^93 + z^2 * q^94 + z^2 * q^97 + z^5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^6 - 2 * q^7 + 2 * q^9 $$4 q - 2 q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{15} - 4 q^{16} - 2 q^{24} + 2 q^{31} + 4 q^{34} + 4 q^{37} + 2 q^{40} + 4 q^{42} - 2 q^{43} + 4 q^{46} - 2 q^{49} + 2 q^{51} - 4 q^{54} - 2 q^{58} + 2 q^{63} - 4 q^{64} + 2 q^{69} - 4 q^{70} - 2 q^{73} + 2 q^{78} - 2 q^{79} - 2 q^{81} + 2 q^{82} - 2 q^{85} - 4 q^{87} + 4 q^{90} + 2 q^{91} + 2 q^{94} + 2 q^{97}+O(q^{100})$$ 4 * q - 2 * q^6 - 2 * q^7 + 2 * q^9 + 2 * q^10 - 4 * q^13 - 2 * q^15 - 4 * q^16 - 2 * q^24 + 2 * q^31 + 4 * q^34 + 4 * q^37 + 2 * q^40 + 4 * q^42 - 2 * q^43 + 4 * q^46 - 2 * q^49 + 2 * q^51 - 4 * q^54 - 2 * q^58 + 2 * q^63 - 4 * q^64 + 2 * q^69 - 4 * q^70 - 2 * q^73 + 2 * q^78 - 2 * q^79 - 2 * q^81 + 2 * q^82 - 2 * q^85 - 4 * q^87 + 4 * q^90 + 2 * q^91 + 2 * q^94 + 2 * q^97

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.00000i 0.866025 0.500000i 0 −0.866025 + 0.500000i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i 0.500000 0.866025i 0.500000 + 0.866025i
74.2 1.00000i −0.866025 + 0.500000i 0 0.866025 0.500000i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i 0.500000 0.866025i 0.500000 + 0.866025i
107.1 1.00000i −0.866025 0.500000i 0 0.866025 + 0.500000i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i 0.500000 + 0.866025i 0.500000 0.866025i
107.2 1.00000i 0.866025 + 0.500000i 0 −0.866025 0.500000i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i 0.500000 + 0.866025i 0.500000 0.866025i
 $$n$$: 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 inner
91.h even 3 1 inner
273.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.1.s.b 4
3.b odd 2 1 inner 273.1.s.b 4
7.b odd 2 1 1911.1.s.b 4
7.c even 3 1 273.1.bm.b yes 4
7.c even 3 1 1911.1.be.c 4
7.d odd 6 1 1911.1.be.d 4
7.d odd 6 1 1911.1.bm.b 4
13.b even 2 1 3549.1.s.b 4
13.c even 3 1 273.1.bm.b yes 4
13.c even 3 1 3549.1.bk.d 4
13.d odd 4 1 3549.1.bp.b 4
13.d odd 4 1 3549.1.bp.d 4
13.e even 6 1 3549.1.bk.c 4
13.e even 6 1 3549.1.bm.c 4
13.f odd 12 1 3549.1.w.c 4
13.f odd 12 1 3549.1.w.e 4
13.f odd 12 1 3549.1.x.b 4
13.f odd 12 1 3549.1.x.d 4
21.c even 2 1 1911.1.s.b 4
21.g even 6 1 1911.1.be.d 4
21.g even 6 1 1911.1.bm.b 4
21.h odd 6 1 273.1.bm.b yes 4
21.h odd 6 1 1911.1.be.c 4
39.d odd 2 1 3549.1.s.b 4
39.f even 4 1 3549.1.bp.b 4
39.f even 4 1 3549.1.bp.d 4
39.h odd 6 1 3549.1.bk.c 4
39.h odd 6 1 3549.1.bm.c 4
39.i odd 6 1 273.1.bm.b yes 4
39.i odd 6 1 3549.1.bk.d 4
39.k even 12 1 3549.1.w.c 4
39.k even 12 1 3549.1.w.e 4
39.k even 12 1 3549.1.x.b 4
39.k even 12 1 3549.1.x.d 4
91.g even 3 1 1911.1.be.c 4
91.g even 3 1 3549.1.bk.d 4
91.h even 3 1 inner 273.1.s.b 4
91.k even 6 1 3549.1.s.b 4
91.m odd 6 1 1911.1.be.d 4
91.n odd 6 1 1911.1.bm.b 4
91.r even 6 1 3549.1.bm.c 4
91.u even 6 1 3549.1.bk.c 4
91.v odd 6 1 1911.1.s.b 4
91.x odd 12 1 3549.1.bp.b 4
91.x odd 12 1 3549.1.bp.d 4
91.z odd 12 1 3549.1.x.b 4
91.z odd 12 1 3549.1.x.d 4
91.bd odd 12 1 3549.1.w.c 4
91.bd odd 12 1 3549.1.w.e 4
273.r even 6 1 1911.1.s.b 4
273.s odd 6 1 inner 273.1.s.b 4
273.w odd 6 1 3549.1.bm.c 4
273.x odd 6 1 3549.1.bk.c 4
273.bf even 6 1 1911.1.be.d 4
273.bm odd 6 1 1911.1.be.c 4
273.bm odd 6 1 3549.1.bk.d 4
273.bn even 6 1 1911.1.bm.b 4
273.bp odd 6 1 3549.1.s.b 4
273.bv even 12 1 3549.1.bp.b 4
273.bv even 12 1 3549.1.bp.d 4
273.bw even 12 1 3549.1.w.c 4
273.bw even 12 1 3549.1.w.e 4
273.cd even 12 1 3549.1.x.b 4
273.cd even 12 1 3549.1.x.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.s.b 4 1.a even 1 1 trivial
273.1.s.b 4 3.b odd 2 1 inner
273.1.s.b 4 91.h even 3 1 inner
273.1.s.b 4 273.s odd 6 1 inner
273.1.bm.b yes 4 7.c even 3 1
273.1.bm.b yes 4 13.c even 3 1
273.1.bm.b yes 4 21.h odd 6 1
273.1.bm.b yes 4 39.i odd 6 1
1911.1.s.b 4 7.b odd 2 1
1911.1.s.b 4 21.c even 2 1
1911.1.s.b 4 91.v odd 6 1
1911.1.s.b 4 273.r even 6 1
1911.1.be.c 4 7.c even 3 1
1911.1.be.c 4 21.h odd 6 1
1911.1.be.c 4 91.g even 3 1
1911.1.be.c 4 273.bm odd 6 1
1911.1.be.d 4 7.d odd 6 1
1911.1.be.d 4 21.g even 6 1
1911.1.be.d 4 91.m odd 6 1
1911.1.be.d 4 273.bf even 6 1
1911.1.bm.b 4 7.d odd 6 1
1911.1.bm.b 4 21.g even 6 1
1911.1.bm.b 4 91.n odd 6 1
1911.1.bm.b 4 273.bn even 6 1
3549.1.s.b 4 13.b even 2 1
3549.1.s.b 4 39.d odd 2 1
3549.1.s.b 4 91.k even 6 1
3549.1.s.b 4 273.bp odd 6 1
3549.1.w.c 4 13.f odd 12 1
3549.1.w.c 4 39.k even 12 1
3549.1.w.c 4 91.bd odd 12 1
3549.1.w.c 4 273.bw even 12 1
3549.1.w.e 4 13.f odd 12 1
3549.1.w.e 4 39.k even 12 1
3549.1.w.e 4 91.bd odd 12 1
3549.1.w.e 4 273.bw even 12 1
3549.1.x.b 4 13.f odd 12 1
3549.1.x.b 4 39.k even 12 1
3549.1.x.b 4 91.z odd 12 1
3549.1.x.b 4 273.cd even 12 1
3549.1.x.d 4 13.f odd 12 1
3549.1.x.d 4 39.k even 12 1
3549.1.x.d 4 91.z odd 12 1
3549.1.x.d 4 273.cd even 12 1
3549.1.bk.c 4 13.e even 6 1
3549.1.bk.c 4 39.h odd 6 1
3549.1.bk.c 4 91.u even 6 1
3549.1.bk.c 4 273.x odd 6 1
3549.1.bk.d 4 13.c even 3 1
3549.1.bk.d 4 39.i odd 6 1
3549.1.bk.d 4 91.g even 3 1
3549.1.bk.d 4 273.bm odd 6 1
3549.1.bm.c 4 13.e even 6 1
3549.1.bm.c 4 39.h odd 6 1
3549.1.bm.c 4 91.r even 6 1
3549.1.bm.c 4 273.w odd 6 1
3549.1.bp.b 4 13.d odd 4 1
3549.1.bp.b 4 39.f even 4 1
3549.1.bp.b 4 91.x odd 12 1
3549.1.bp.b 4 273.bv even 12 1
3549.1.bp.d 4 13.d odd 4 1
3549.1.bp.d 4 39.f even 4 1
3549.1.bp.d 4 91.x odd 12 1
3549.1.bp.d 4 273.bv even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

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