Properties

 Label 273.2.c.a Level $273$ Weight $2$ Character orbit 273.c Analytic conductor $2.180$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(64,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("273.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.c (of order $$2$$, degree $$1$$, 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: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + q^{3} - 2 q^{4} + 3 i q^{5} + 2 i q^{6} - i q^{7} + q^{9} +O(q^{10})$$ q + 2*i * q^2 + q^3 - 2 * q^4 + 3*i * q^5 + 2*i * q^6 - i * q^7 + q^9 $$q + 2 i q^{2} + q^{3} - 2 q^{4} + 3 i q^{5} + 2 i q^{6} - i q^{7} + q^{9} - 6 q^{10} - 2 q^{12} + ( - 3 i + 2) q^{13} + 2 q^{14} + 3 i q^{15} - 4 q^{16} - 2 q^{17} + 2 i q^{18} + i q^{19} - 6 i q^{20} - i q^{21} - q^{23} - 4 q^{25} + (4 i + 6) q^{26} + q^{27} + 2 i q^{28} + 5 q^{29} - 6 q^{30} + 5 i q^{31} - 8 i q^{32} - 4 i q^{34} + 3 q^{35} - 2 q^{36} - 8 i q^{37} - 2 q^{38} + ( - 3 i + 2) q^{39} - 10 i q^{41} + 2 q^{42} + 9 q^{43} + 3 i q^{45} - 2 i q^{46} + 7 i q^{47} - 4 q^{48} - q^{49} - 8 i q^{50} - 2 q^{51} + (6 i - 4) q^{52} + 9 q^{53} + 2 i q^{54} + i q^{57} + 10 i q^{58} - 4 i q^{59} - 6 i q^{60} - 8 q^{61} - 10 q^{62} - i q^{63} + 8 q^{64} + (6 i + 9) q^{65} + 2 i q^{67} + 4 q^{68} - q^{69} + 6 i q^{70} + 9 i q^{73} + 16 q^{74} - 4 q^{75} - 2 i q^{76} + (4 i + 6) q^{78} + 15 q^{79} - 12 i q^{80} + q^{81} + 20 q^{82} + 9 i q^{83} + 2 i q^{84} - 6 i q^{85} + 18 i q^{86} + 5 q^{87} - 9 i q^{89} - 6 q^{90} + ( - 2 i - 3) q^{91} + 2 q^{92} + 5 i q^{93} - 14 q^{94} - 3 q^{95} - 8 i q^{96} - 13 i q^{97} - 2 i q^{98} +O(q^{100})$$ q + 2*i * q^2 + q^3 - 2 * q^4 + 3*i * q^5 + 2*i * q^6 - i * q^7 + q^9 - 6 * q^10 - 2 * q^12 + (-3*i + 2) * q^13 + 2 * q^14 + 3*i * q^15 - 4 * q^16 - 2 * q^17 + 2*i * q^18 + i * q^19 - 6*i * q^20 - i * q^21 - q^23 - 4 * q^25 + (4*i + 6) * q^26 + q^27 + 2*i * q^28 + 5 * q^29 - 6 * q^30 + 5*i * q^31 - 8*i * q^32 - 4*i * q^34 + 3 * q^35 - 2 * q^36 - 8*i * q^37 - 2 * q^38 + (-3*i + 2) * q^39 - 10*i * q^41 + 2 * q^42 + 9 * q^43 + 3*i * q^45 - 2*i * q^46 + 7*i * q^47 - 4 * q^48 - q^49 - 8*i * q^50 - 2 * q^51 + (6*i - 4) * q^52 + 9 * q^53 + 2*i * q^54 + i * q^57 + 10*i * q^58 - 4*i * q^59 - 6*i * q^60 - 8 * q^61 - 10 * q^62 - i * q^63 + 8 * q^64 + (6*i + 9) * q^65 + 2*i * q^67 + 4 * q^68 - q^69 + 6*i * q^70 + 9*i * q^73 + 16 * q^74 - 4 * q^75 - 2*i * q^76 + (4*i + 6) * q^78 + 15 * q^79 - 12*i * q^80 + q^81 + 20 * q^82 + 9*i * q^83 + 2*i * q^84 - 6*i * q^85 + 18*i * q^86 + 5 * q^87 - 9*i * q^89 - 6 * q^90 + (-2*i - 3) * q^91 + 2 * q^92 + 5*i * q^93 - 14 * q^94 - 3 * q^95 - 8*i * q^96 - 13*i * q^97 - 2*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9 $$2 q + 2 q^{3} - 4 q^{4} + 2 q^{9} - 12 q^{10} - 4 q^{12} + 4 q^{13} + 4 q^{14} - 8 q^{16} - 4 q^{17} - 2 q^{23} - 8 q^{25} + 12 q^{26} + 2 q^{27} + 10 q^{29} - 12 q^{30} + 6 q^{35} - 4 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{42} + 18 q^{43} - 8 q^{48} - 2 q^{49} - 4 q^{51} - 8 q^{52} + 18 q^{53} - 16 q^{61} - 20 q^{62} + 16 q^{64} + 18 q^{65} + 8 q^{68} - 2 q^{69} + 32 q^{74} - 8 q^{75} + 12 q^{78} + 30 q^{79} + 2 q^{81} + 40 q^{82} + 10 q^{87} - 12 q^{90} - 6 q^{91} + 4 q^{92} - 28 q^{94} - 6 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9 - 12 * q^10 - 4 * q^12 + 4 * q^13 + 4 * q^14 - 8 * q^16 - 4 * q^17 - 2 * q^23 - 8 * q^25 + 12 * q^26 + 2 * q^27 + 10 * q^29 - 12 * q^30 + 6 * q^35 - 4 * q^36 - 4 * q^38 + 4 * q^39 + 4 * q^42 + 18 * q^43 - 8 * q^48 - 2 * q^49 - 4 * q^51 - 8 * q^52 + 18 * q^53 - 16 * q^61 - 20 * q^62 + 16 * q^64 + 18 * q^65 + 8 * q^68 - 2 * q^69 + 32 * q^74 - 8 * q^75 + 12 * q^78 + 30 * q^79 + 2 * q^81 + 40 * q^82 + 10 * q^87 - 12 * q^90 - 6 * q^91 + 4 * q^92 - 28 * q^94 - 6 * q^95

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 − 1.00000i 1.00000i
2.00000i 1.00000 −2.00000 3.00000i 2.00000i 1.00000i 0 1.00000 −6.00000
64.2 2.00000i 1.00000 −2.00000 3.00000i 2.00000i 1.00000i 0 1.00000 −6.00000
 $$n$$: 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
13.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.c.a 2
3.b odd 2 1 819.2.c.a 2
4.b odd 2 1 4368.2.h.e 2
7.b odd 2 1 1911.2.c.a 2
13.b even 2 1 inner 273.2.c.a 2
13.d odd 4 1 3549.2.a.a 1
13.d odd 4 1 3549.2.a.e 1
39.d odd 2 1 819.2.c.a 2
52.b odd 2 1 4368.2.h.e 2
91.b odd 2 1 1911.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.a 2 1.a even 1 1 trivial
273.2.c.a 2 13.b even 2 1 inner
819.2.c.a 2 3.b odd 2 1
819.2.c.a 2 39.d odd 2 1
1911.2.c.a 2 7.b odd 2 1
1911.2.c.a 2 91.b odd 2 1
3549.2.a.a 1 13.d odd 4 1
3549.2.a.e 1 13.d odd 4 1
4368.2.h.e 2 4.b odd 2 1
4368.2.h.e 2 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 9$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T + 13$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$(T + 1)^{2}$$
$29$ $$(T - 5)^{2}$$
$31$ $$T^{2} + 25$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 100$$
$43$ $$(T - 9)^{2}$$
$47$ $$T^{2} + 49$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 81$$
$79$ $$(T - 15)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$T^{2} + 81$$
$97$ $$T^{2} + 169$$