Properties

 Label 273.1.o.a Level $273$ Weight $1$ Character orbit 273.o Analytic conductor $0.136$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ 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(83,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

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

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

chi := DirichletCharacter("273.83");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 273.o (of order $$4$$, 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: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.968877.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.8719893.1

$q$-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.

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

Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$-1$$ $$i$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
83.1
 − 1.00000i 1.00000i
0 1.00000i 1.00000i 0 0 1.00000i 0 −1.00000 0
125.1 0 1.00000i 1.00000i 0 0 1.00000i 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.i even 4 1 inner
273.o odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.1.o.a 2
3.b odd 2 1 CM 273.1.o.a 2
7.b odd 2 1 273.1.o.b yes 2
7.c even 3 2 1911.1.cc.b 4
7.d odd 6 2 1911.1.cc.a 4
13.b even 2 1 3549.1.o.b 2
13.c even 3 2 3549.1.ca.b 4
13.d odd 4 1 273.1.o.b yes 2
13.d odd 4 1 3549.1.o.a 2
13.e even 6 2 3549.1.ca.c 4
13.f odd 12 2 3549.1.ca.a 4
13.f odd 12 2 3549.1.ca.d 4
21.c even 2 1 273.1.o.b yes 2
21.g even 6 2 1911.1.cc.a 4
21.h odd 6 2 1911.1.cc.b 4
39.d odd 2 1 3549.1.o.b 2
39.f even 4 1 273.1.o.b yes 2
39.f even 4 1 3549.1.o.a 2
39.h odd 6 2 3549.1.ca.c 4
39.i odd 6 2 3549.1.ca.b 4
39.k even 12 2 3549.1.ca.a 4
39.k even 12 2 3549.1.ca.d 4
91.b odd 2 1 3549.1.o.a 2
91.i even 4 1 inner 273.1.o.a 2
91.i even 4 1 3549.1.o.b 2
91.n odd 6 2 3549.1.ca.a 4
91.t odd 6 2 3549.1.ca.d 4
91.z odd 12 2 1911.1.cc.a 4
91.bb even 12 2 1911.1.cc.b 4
91.bc even 12 2 3549.1.ca.b 4
91.bc even 12 2 3549.1.ca.c 4
273.g even 2 1 3549.1.o.a 2
273.o odd 4 1 inner 273.1.o.a 2
273.o odd 4 1 3549.1.o.b 2
273.u even 6 2 3549.1.ca.d 4
273.bn even 6 2 3549.1.ca.a 4
273.ca odd 12 2 3549.1.ca.b 4
273.ca odd 12 2 3549.1.ca.c 4
273.cb odd 12 2 1911.1.cc.b 4
273.cd even 12 2 1911.1.cc.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.o.a 2 1.a even 1 1 trivial
273.1.o.a 2 3.b odd 2 1 CM
273.1.o.a 2 91.i even 4 1 inner
273.1.o.a 2 273.o odd 4 1 inner
273.1.o.b yes 2 7.b odd 2 1
273.1.o.b yes 2 13.d odd 4 1
273.1.o.b yes 2 21.c even 2 1
273.1.o.b yes 2 39.f even 4 1
1911.1.cc.a 4 7.d odd 6 2
1911.1.cc.a 4 21.g even 6 2
1911.1.cc.a 4 91.z odd 12 2
1911.1.cc.a 4 273.cd even 12 2
1911.1.cc.b 4 7.c even 3 2
1911.1.cc.b 4 21.h odd 6 2
1911.1.cc.b 4 91.bb even 12 2
1911.1.cc.b 4 273.cb odd 12 2
3549.1.o.a 2 13.d odd 4 1
3549.1.o.a 2 39.f even 4 1
3549.1.o.a 2 91.b odd 2 1
3549.1.o.a 2 273.g even 2 1
3549.1.o.b 2 13.b even 2 1
3549.1.o.b 2 39.d odd 2 1
3549.1.o.b 2 91.i even 4 1
3549.1.o.b 2 273.o odd 4 1
3549.1.ca.a 4 13.f odd 12 2
3549.1.ca.a 4 39.k even 12 2
3549.1.ca.a 4 91.n odd 6 2
3549.1.ca.a 4 273.bn even 6 2
3549.1.ca.b 4 13.c even 3 2
3549.1.ca.b 4 39.i odd 6 2
3549.1.ca.b 4 91.bc even 12 2
3549.1.ca.b 4 273.ca odd 12 2
3549.1.ca.c 4 13.e even 6 2
3549.1.ca.c 4 39.h odd 6 2
3549.1.ca.c 4 91.bc even 12 2
3549.1.ca.c 4 273.ca odd 12 2
3549.1.ca.d 4 13.f odd 12 2
3549.1.ca.d 4 39.k even 12 2
3549.1.ca.d 4 91.t odd 6 2
3549.1.ca.d 4 273.u even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{2} - 2T_{19} + 2$$ acting on $$S_{1}^{\mathrm{new}}(273, [\chi])$$.

Hecke characteristic polynomials

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