# Properties

 Label 273.2.a.e Level $273$ Weight $2$ Character orbit 273.a Self dual yes Analytic conductor $2.180$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(1,273)]

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, 0]))

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

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

chi := DirichletCharacter("273.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.a (trivial)

## 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: yes Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.17428.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b2 + 2) * q^4 + (-b2 - 1) * q^5 + b1 * q^6 + q^7 + (b3 + 2*b1) * q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9} + ( - \beta_{3} - 3 \beta_1) q^{10} + ( - \beta_{3} - \beta_1) q^{11} + (\beta_{2} + 2) q^{12} + q^{13} + \beta_1 q^{14} + ( - \beta_{2} - 1) q^{15} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{16} - 2 \beta_1 q^{17} + \beta_1 q^{18} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 8) q^{20} + q^{21} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{22} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{23} + (\beta_{3} + 2 \beta_1) q^{24} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{25} + \beta_1 q^{26} + q^{27} + (\beta_{2} + 2) q^{28} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{29} + ( - \beta_{3} - 3 \beta_1) q^{30} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{31} + (2 \beta_{2} + \beta_1 + 2) q^{32} + ( - \beta_{3} - \beta_1) q^{33} + ( - 2 \beta_{2} - 8) q^{34} + ( - \beta_{2} - 1) q^{35} + (\beta_{2} + 2) q^{36} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{37} + (2 \beta_{2} + 2) q^{38} + q^{39} + ( - \beta_{3} - 2 \beta_{2} - 7 \beta_1 - 2) q^{40} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 4) q^{41} + \beta_1 q^{42} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{43} + ( - \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 2) q^{44} + ( - \beta_{2} - 1) q^{45} + (2 \beta_{3} + 4 \beta_1 - 6) q^{46} + (\beta_{2} + 2 \beta_1 + 1) q^{47} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{48} + q^{49} + (2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 2) q^{50} - 2 \beta_1 q^{51} + (\beta_{2} + 2) q^{52} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{53} + \beta_1 q^{54} + (2 \beta_{2} + 4 \beta_1 + 2) q^{55} + (\beta_{3} + 2 \beta_1) q^{56} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{57} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{58} + (\beta_{3} + 3 \beta_1 - 6) q^{59} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 8) q^{60} + (4 \beta_1 + 2) q^{61} + ( - 2 \beta_{2} - 14) q^{62} + q^{63} + ( - \beta_{2} + 4 \beta_1) q^{64} + ( - \beta_{2} - 1) q^{65} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{66} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{67} + ( - 2 \beta_{3} - 8 \beta_1) q^{68} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{69} + ( - \beta_{3} - 3 \beta_1) q^{70} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 2) q^{71} + (\beta_{3} + 2 \beta_1) q^{72} + ( - \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 1) q^{73} + ( - 4 \beta_{2} + 6 \beta_1 - 4) q^{74} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{75} + (2 \beta_{2} + 4 \beta_1 - 2) q^{76} + ( - \beta_{3} - \beta_1) q^{77} + \beta_1 q^{78} + (\beta_{3} - \beta_{2} + 5 \beta_1 + 1) q^{79} + ( - \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 10) q^{80} + q^{81} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{82} + (\beta_{2} - 2 \beta_1 + 1) q^{83} + (\beta_{2} + 2) q^{84} + (2 \beta_{3} + 6 \beta_1) q^{85} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{86} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{87} + ( - \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 14) q^{88} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 3) q^{89} + ( - \beta_{3} - 3 \beta_1) q^{90} + q^{91} + (4 \beta_{2} - 2 \beta_1 + 10) q^{92} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{93} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 + 8) q^{94} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{95} + (2 \beta_{2} + \beta_1 + 2) q^{96} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 - 5) q^{97} + \beta_1 q^{98} + ( - \beta_{3} - \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b2 + 2) * q^4 + (-b2 - 1) * q^5 + b1 * q^6 + q^7 + (b3 + 2*b1) * q^8 + q^9 + (-b3 - 3*b1) * q^10 + (-b3 - b1) * q^11 + (b2 + 2) * q^12 + q^13 + b1 * q^14 + (-b2 - 1) * q^15 + (b3 + b2 + b1 + 2) * q^16 - 2*b1 * q^17 + b1 * q^18 + (b3 - b2 + b1 + 1) * q^19 + (-b3 - 2*b2 - b1 - 8) * q^20 + q^21 + (-b3 - 2*b2 - b1 - 2) * q^22 + (b3 + b2 - b1 + 1) * q^23 + (b3 + 2*b1) * q^24 + (b3 + b2 + b1 + 2) * q^25 + b1 * q^26 + q^27 + (b2 + 2) * q^28 + (-b3 + b2 - b1 + 1) * q^29 + (-b3 - 3*b1) * q^30 + (b3 - b2 - 3*b1 + 1) * q^31 + (2*b2 + b1 + 2) * q^32 + (-b3 - b1) * q^33 + (-2*b2 - 8) * q^34 + (-b2 - 1) * q^35 + (b2 + 2) * q^36 + (-2*b3 + 2*b2 - 2*b1 + 4) * q^37 + (2*b2 + 2) * q^38 + q^39 + (-b3 - 2*b2 - 7*b1 - 2) * q^40 + (b3 + 2*b2 + b1 - 4) * q^41 + b1 * q^42 + (-b3 - b2 - b1 + 1) * q^43 + (-b3 - 2*b2 - 5*b1 - 2) * q^44 + (-b2 - 1) * q^45 + (2*b3 + 4*b1 - 6) * q^46 + (b2 + 2*b1 + 1) * q^47 + (b3 + b2 + b1 + 2) * q^48 + q^49 + (2*b3 + 2*b2 + 5*b1 + 2) * q^50 - 2*b1 * q^51 + (b2 + 2) * q^52 + (b3 + b2 + b1 + 1) * q^53 + b1 * q^54 + (2*b2 + 4*b1 + 2) * q^55 + (b3 + 2*b1) * q^56 + (b3 - b2 + b1 + 1) * q^57 + (-2*b2 + 2*b1 - 2) * q^58 + (b3 + 3*b1 - 6) * q^59 + (-b3 - 2*b2 - b1 - 8) * q^60 + (4*b1 + 2) * q^61 + (-2*b2 - 14) * q^62 + q^63 + (-b2 + 4*b1) * q^64 + (-b2 - 1) * q^65 + (-b3 - 2*b2 - b1 - 2) * q^66 + (-2*b3 - 2*b2 + 2*b1 - 6) * q^67 + (-2*b3 - 8*b1) * q^68 + (b3 + b2 - b1 + 1) * q^69 + (-b3 - 3*b1) * q^70 + (-3*b3 + 2*b2 - 3*b1 + 2) * q^71 + (b3 + 2*b1) * q^72 + (-b3 + 3*b2 - 5*b1 - 1) * q^73 + (-4*b2 + 6*b1 - 4) * q^74 + (b3 + b2 + b1 + 2) * q^75 + (2*b2 + 4*b1 - 2) * q^76 + (-b3 - b1) * q^77 + b1 * q^78 + (b3 - b2 + 5*b1 + 1) * q^79 + (-b3 - 4*b2 - 5*b1 - 10) * q^80 + q^81 + (3*b3 + 2*b2 + b1 + 2) * q^82 + (b2 - 2*b1 + 1) * q^83 + (b2 + 2) * q^84 + (2*b3 + 6*b1) * q^85 + (-2*b3 - 2*b2 - 2*b1 - 2) * q^86 + (-b3 + b2 - b1 + 1) * q^87 + (-b3 - 2*b2 - 5*b1 - 14) * q^88 + (2*b3 - 3*b2 + 2*b1 - 3) * q^89 + (-b3 - 3*b1) * q^90 + q^91 + (4*b2 - 2*b1 + 10) * q^92 + (b3 - b2 - 3*b1 + 1) * q^93 + (b3 + 2*b2 + 3*b1 + 8) * q^94 + (b3 - 3*b2 - 3*b1 + 3) * q^95 + (2*b2 + b1 + 2) * q^96 + (-b3 - b2 + 3*b1 - 5) * q^97 + b1 * q^98 + (-b3 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 4 q^{3} + 7 q^{4} - 3 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + q^2 + 4 * q^3 + 7 * q^4 - 3 * q^5 + q^6 + 4 * q^7 + 3 * q^8 + 4 * q^9 $$4 q + q^{2} + 4 q^{3} + 7 q^{4} - 3 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + 4 q^{9} - 4 q^{10} - 2 q^{11} + 7 q^{12} + 4 q^{13} + q^{14} - 3 q^{15} + 9 q^{16} - 2 q^{17} + q^{18} + 7 q^{19} - 32 q^{20} + 4 q^{21} - 8 q^{22} + 3 q^{23} + 3 q^{24} + 9 q^{25} + q^{26} + 4 q^{27} + 7 q^{28} + q^{29} - 4 q^{30} + 3 q^{31} + 7 q^{32} - 2 q^{33} - 30 q^{34} - 3 q^{35} + 7 q^{36} + 10 q^{37} + 6 q^{38} + 4 q^{39} - 14 q^{40} - 16 q^{41} + q^{42} + 3 q^{43} - 12 q^{44} - 3 q^{45} - 18 q^{46} + 5 q^{47} + 9 q^{48} + 4 q^{49} + 13 q^{50} - 2 q^{51} + 7 q^{52} + 5 q^{53} + q^{54} + 10 q^{55} + 3 q^{56} + 7 q^{57} - 4 q^{58} - 20 q^{59} - 32 q^{60} + 12 q^{61} - 54 q^{62} + 4 q^{63} + 5 q^{64} - 3 q^{65} - 8 q^{66} - 22 q^{67} - 10 q^{68} + 3 q^{69} - 4 q^{70} + 3 q^{72} - 13 q^{73} - 6 q^{74} + 9 q^{75} - 6 q^{76} - 2 q^{77} + q^{78} + 11 q^{79} - 42 q^{80} + 4 q^{81} + 10 q^{82} + q^{83} + 7 q^{84} + 8 q^{85} - 10 q^{86} + q^{87} - 60 q^{88} - 5 q^{89} - 4 q^{90} + 4 q^{91} + 34 q^{92} + 3 q^{93} + 34 q^{94} + 13 q^{95} + 7 q^{96} - 17 q^{97} + q^{98} - 2 q^{99}+O(q^{100})$$ 4 * q + q^2 + 4 * q^3 + 7 * q^4 - 3 * q^5 + q^6 + 4 * q^7 + 3 * q^8 + 4 * q^9 - 4 * q^10 - 2 * q^11 + 7 * q^12 + 4 * q^13 + q^14 - 3 * q^15 + 9 * q^16 - 2 * q^17 + q^18 + 7 * q^19 - 32 * q^20 + 4 * q^21 - 8 * q^22 + 3 * q^23 + 3 * q^24 + 9 * q^25 + q^26 + 4 * q^27 + 7 * q^28 + q^29 - 4 * q^30 + 3 * q^31 + 7 * q^32 - 2 * q^33 - 30 * q^34 - 3 * q^35 + 7 * q^36 + 10 * q^37 + 6 * q^38 + 4 * q^39 - 14 * q^40 - 16 * q^41 + q^42 + 3 * q^43 - 12 * q^44 - 3 * q^45 - 18 * q^46 + 5 * q^47 + 9 * q^48 + 4 * q^49 + 13 * q^50 - 2 * q^51 + 7 * q^52 + 5 * q^53 + q^54 + 10 * q^55 + 3 * q^56 + 7 * q^57 - 4 * q^58 - 20 * q^59 - 32 * q^60 + 12 * q^61 - 54 * q^62 + 4 * q^63 + 5 * q^64 - 3 * q^65 - 8 * q^66 - 22 * q^67 - 10 * q^68 + 3 * q^69 - 4 * q^70 + 3 * q^72 - 13 * q^73 - 6 * q^74 + 9 * q^75 - 6 * q^76 - 2 * q^77 + q^78 + 11 * q^79 - 42 * q^80 + 4 * q^81 + 10 * q^82 + q^83 + 7 * q^84 + 8 * q^85 - 10 * q^86 + q^87 - 60 * q^88 - 5 * q^89 - 4 * q^90 + 4 * q^91 + 34 * q^92 + 3 * q^93 + 34 * q^94 + 13 * q^95 + 7 * q^96 - 17 * q^97 + q^98 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{3}$$ $$=$$ $$-\nu^{2} + 2\nu + 3$$ -v^2 + 2*v + 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 3$$ b1 + 3 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + \beta_{2} + 2\beta _1 + 1$$ 2*b3 + b2 + 2*b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.787711 1.52616 −2.10710 2.36865
−2.37951 1.00000 3.66208 −2.66208 −2.37951 1.00000 −3.95493 1.00000 6.33445
1.2 −0.670843 1.00000 −1.54997 2.54997 −0.670843 1.00000 2.38147 1.00000 −1.71063
1.3 1.43986 1.00000 0.0731828 0.926817 1.43986 1.00000 −2.77434 1.00000 1.33448
1.4 2.61050 1.00000 4.81471 −3.81471 2.61050 1.00000 7.34780 1.00000 −9.95830
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.a.e 4
3.b odd 2 1 819.2.a.k 4
4.b odd 2 1 4368.2.a.br 4
5.b even 2 1 6825.2.a.bg 4
7.b odd 2 1 1911.2.a.s 4
13.b even 2 1 3549.2.a.w 4
21.c even 2 1 5733.2.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.e 4 1.a even 1 1 trivial
819.2.a.k 4 3.b odd 2 1
1911.2.a.s 4 7.b odd 2 1
3549.2.a.w 4 13.b even 2 1
4368.2.a.br 4 4.b odd 2 1
5733.2.a.bf 4 21.c even 2 1
6825.2.a.bg 4 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} - 7T_{2}^{2} + 5T_{2} + 6$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(273))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} - 7 T^{2} + 5 T + 6$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4} + 3 T^{3} - 10 T^{2} - 20 T + 24$$
$7$ $$(T - 1)^{4}$$
$11$ $$T^{4} + 2 T^{3} - 24 T^{2} - 32 T + 96$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} + 2 T^{3} - 28 T^{2} - 40 T + 96$$
$19$ $$T^{4} - 7 T^{3} - 12 T^{2} + 48 T + 64$$
$23$ $$T^{4} - 3 T^{3} - 52 T^{2} + 256 T - 288$$
$29$ $$T^{4} - T^{3} - 30 T^{2} + 52 T + 72$$
$31$ $$T^{4} - 3 T^{3} - 128 T^{2} + \cdots + 3968$$
$37$ $$T^{4} - 10 T^{3} - 84 T^{2} + \cdots - 128$$
$41$ $$T^{4} + 16 T^{3} - 688 T - 1392$$
$43$ $$T^{4} - 3 T^{3} - 44 T^{2} + 112 T - 64$$
$47$ $$T^{4} - 5 T^{3} - 40 T^{2} + 16 T + 144$$
$53$ $$T^{4} - 5 T^{3} - 38 T^{2} + 68 T - 24$$
$59$ $$T^{4} + 20 T^{3} + 80 T^{2} + \cdots - 1536$$
$61$ $$T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 496$$
$67$ $$T^{4} + 22 T^{3} - 40 T^{2} + \cdots - 15488$$
$71$ $$T^{4} - 232 T^{2} + 304 T + 10176$$
$73$ $$T^{4} + 13 T^{3} - 166 T^{2} + \cdots - 11672$$
$79$ $$T^{4} - 11 T^{3} - 120 T^{2} + \cdots - 3456$$
$83$ $$T^{4} - T^{3} - 36 T^{2} - 80 T - 48$$
$89$ $$T^{4} + 5 T^{3} - 162 T^{2} + \cdots - 1704$$
$97$ $$T^{4} + 17 T^{3} - 14 T^{2} + \cdots - 1528$$