# Properties

 Label 273.2.p.b Level $273$ Weight $2$ Character orbit 273.p Analytic conductor $2.180$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(34,273)]

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([0, 2, 1]))

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

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

chi := DirichletCharacter("273.34");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.p (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: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_{2} - 1) q^{7} - q^{9}+O(q^{10})$$ q - b2 * q^3 + 2*b2 * q^4 + (-b2 + b1 - 1) * q^5 + (-b3 - b2 - 1) * q^7 - q^9 $$q - \beta_{2} q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_{2} - 1) q^{7} - q^{9} + (\beta_{2} + 2 \beta_1 + 1) q^{11} + 2 q^{12} + ( - 2 \beta_{2} + \beta_1 + 2) q^{13} + ( - \beta_{3} + \beta_{2} - 1) q^{15} - 4 q^{16} + ( - \beta_{3} + \beta_1 - 2) q^{17} + (2 \beta_{2} - \beta_1 + 2) q^{19} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{20} + (\beta_{2} - \beta_1 - 1) q^{21} + (\beta_{3} + \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{25} + \beta_{2} q^{27} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{28} + (\beta_{3} - \beta_1 + 5) q^{29} + ( - 4 \beta_{2} - \beta_1 - 4) q^{31} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{33} + (2 \beta_{2} - 2 \beta_1 + 5) q^{35} - 2 \beta_{2} q^{36} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{37} + ( - \beta_{3} - 2 \beta_{2} - 2) q^{39} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{41} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{43} + (4 \beta_{3} + 2 \beta_{2} - 2) q^{44} + (\beta_{2} - \beta_1 + 1) q^{45} + (\beta_{3} + 5 \beta_{2} - 5) q^{47} + 4 \beta_{2} q^{48} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{49} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{51} + (2 \beta_{3} + 4 \beta_{2} + 4) q^{52} + (\beta_{3} - \beta_1 - 1) q^{53} + ( - \beta_{3} + 8 \beta_{2} - \beta_1) q^{55} + (\beta_{3} - 2 \beta_{2} + 2) q^{57} + (4 \beta_{3} + 2 \beta_{2} - 2) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{60} + ( - \beta_{3} - \beta_1) q^{61} + (\beta_{3} + \beta_{2} + 1) q^{63} - 8 \beta_{2} q^{64} + ( - 3 \beta_{3} + 5 \beta_{2} + \beta_1 - 4) q^{65} + 4 \beta_{3} q^{67} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{68} + ( - \beta_{3} + \beta_1 + 1) q^{69} + ( - 6 \beta_{2} + 6) q^{71} + ( - \beta_{3} - 4 \beta_{2} + 4) q^{73} + (2 \beta_{3} - 2 \beta_1 + 2) q^{75} + ( - 2 \beta_{3} + 4 \beta_{2} - 4) q^{76} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 10) q^{77} + (2 \beta_{3} - 2 \beta_1 - 5) q^{79} + (4 \beta_{2} - 4 \beta_1 + 4) q^{80} + q^{81} + ( - 5 \beta_{2} - \beta_1 - 5) q^{83} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{84} + (7 \beta_{2} - 4 \beta_1 + 7) q^{85} + (\beta_{3} - 5 \beta_{2} + \beta_1) q^{87} + ( - \beta_{3} + 7 \beta_{2} - 7) q^{89} + ( - 3 \beta_{3} - 3 \beta_1 + 1) q^{91} + (2 \beta_{3} - 2 \beta_1 - 2) q^{92} + (\beta_{3} + 4 \beta_{2} - 4) q^{93} + (3 \beta_{3} - 9 \beta_{2} + 3 \beta_1) q^{95} + ( - 2 \beta_{2} + \beta_1 - 2) q^{97} + ( - \beta_{2} - 2 \beta_1 - 1) q^{99}+O(q^{100})$$ q - b2 * q^3 + 2*b2 * q^4 + (-b2 + b1 - 1) * q^5 + (-b3 - b2 - 1) * q^7 - q^9 + (b2 + 2*b1 + 1) * q^11 + 2 * q^12 + (-2*b2 + b1 + 2) * q^13 + (-b3 + b2 - 1) * q^15 - 4 * q^16 + (-b3 + b1 - 2) * q^17 + (2*b2 - b1 + 2) * q^19 + (2*b3 - 2*b2 + 2) * q^20 + (b2 - b1 - 1) * q^21 + (b3 + b2 + b1) * q^23 + (-2*b3 + 2*b2 - 2*b1) * q^25 + b2 * q^27 + (-2*b2 + 2*b1 + 2) * q^28 + (b3 - b1 + 5) * q^29 + (-4*b2 - b1 - 4) * q^31 + (-2*b3 - b2 + 1) * q^33 + (2*b2 - 2*b1 + 5) * q^35 - 2*b2 * q^36 + (-2*b2 + 4*b1 - 2) * q^37 + (-b3 - 2*b2 - 2) * q^39 + (-4*b2 - 2*b1 - 4) * q^41 + (-2*b3 + b2 - 2*b1) * q^43 + (4*b3 + 2*b2 - 2) * q^44 + (b2 - b1 + 1) * q^45 + (b3 + 5*b2 - 5) * q^47 + 4*b2 * q^48 + (2*b3 - 3*b2 - 2*b1) * q^49 + (-b3 + 2*b2 - b1) * q^51 + (2*b3 + 4*b2 + 4) * q^52 + (b3 - b1 - 1) * q^53 + (-b3 + 8*b2 - b1) * q^55 + (b3 - 2*b2 + 2) * q^57 + (4*b3 + 2*b2 - 2) * q^59 + (-2*b2 + 2*b1 - 2) * q^60 + (-b3 - b1) * q^61 + (b3 + b2 + 1) * q^63 - 8*b2 * q^64 + (-3*b3 + 5*b2 + b1 - 4) * q^65 + 4*b3 * q^67 + (2*b3 - 4*b2 + 2*b1) * q^68 + (-b3 + b1 + 1) * q^69 + (-6*b2 + 6) * q^71 + (-b3 - 4*b2 + 4) * q^73 + (2*b3 - 2*b1 + 2) * q^75 + (-2*b3 + 4*b2 - 4) * q^76 + (-3*b3 - 2*b2 - b1 + 10) * q^77 + (2*b3 - 2*b1 - 5) * q^79 + (4*b2 - 4*b1 + 4) * q^80 + q^81 + (-5*b2 - b1 - 5) * q^83 + (-2*b3 - 2*b2 - 2) * q^84 + (7*b2 - 4*b1 + 7) * q^85 + (b3 - 5*b2 + b1) * q^87 + (-b3 + 7*b2 - 7) * q^89 + (-3*b3 - 3*b1 + 1) * q^91 + (2*b3 - 2*b1 - 2) * q^92 + (b3 + 4*b2 - 4) * q^93 + (3*b3 - 9*b2 + 3*b1) * q^95 + (-2*b2 + b1 - 2) * q^97 + (-b2 - 2*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^5 - 4 * q^7 - 4 * q^9 $$4 q - 4 q^{5} - 4 q^{7} - 4 q^{9} + 4 q^{11} + 8 q^{12} + 8 q^{13} - 4 q^{15} - 16 q^{16} - 8 q^{17} + 8 q^{19} + 8 q^{20} - 4 q^{21} + 8 q^{28} + 20 q^{29} - 16 q^{31} + 4 q^{33} + 20 q^{35} - 8 q^{37} - 8 q^{39} - 16 q^{41} - 8 q^{44} + 4 q^{45} - 20 q^{47} + 16 q^{52} - 4 q^{53} + 8 q^{57} - 8 q^{59} - 8 q^{60} + 4 q^{63} - 16 q^{65} + 4 q^{69} + 24 q^{71} + 16 q^{73} + 8 q^{75} - 16 q^{76} + 40 q^{77} - 20 q^{79} + 16 q^{80} + 4 q^{81} - 20 q^{83} - 8 q^{84} + 28 q^{85} - 28 q^{89} + 4 q^{91} - 8 q^{92} - 16 q^{93} - 8 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^5 - 4 * q^7 - 4 * q^9 + 4 * q^11 + 8 * q^12 + 8 * q^13 - 4 * q^15 - 16 * q^16 - 8 * q^17 + 8 * q^19 + 8 * q^20 - 4 * q^21 + 8 * q^28 + 20 * q^29 - 16 * q^31 + 4 * q^33 + 20 * q^35 - 8 * q^37 - 8 * q^39 - 16 * q^41 - 8 * q^44 + 4 * q^45 - 20 * q^47 + 16 * q^52 - 4 * q^53 + 8 * q^57 - 8 * q^59 - 8 * q^60 + 4 * q^63 - 16 * q^65 + 4 * q^69 + 24 * q^71 + 16 * q^73 + 8 * q^75 - 16 * q^76 + 40 * q^77 - 20 * q^79 + 16 * q^80 + 4 * q^81 - 20 * q^83 - 8 * q^84 + 28 * q^85 - 28 * q^89 + 4 * q^91 - 8 * q^92 - 16 * q^93 - 8 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 −1.58114 + 1.58114i 1.58114 − 1.58114i −1.58114 − 1.58114i 1.58114 + 1.58114i
0 1.00000i 2.00000i −2.58114 + 2.58114i 0 −2.58114 0.581139i 0 −1.00000 0
34.2 0 1.00000i 2.00000i 0.581139 0.581139i 0 0.581139 + 2.58114i 0 −1.00000 0
265.1 0 1.00000i 2.00000i −2.58114 2.58114i 0 −2.58114 + 0.581139i 0 −1.00000 0
265.2 0 1.00000i 2.00000i 0.581139 + 0.581139i 0 0.581139 2.58114i 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
91.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.p.b 4
3.b odd 2 1 819.2.y.c 4
7.b odd 2 1 273.2.p.c yes 4
13.d odd 4 1 273.2.p.c yes 4
21.c even 2 1 819.2.y.b 4
39.f even 4 1 819.2.y.b 4
91.i even 4 1 inner 273.2.p.b 4
273.o odd 4 1 819.2.y.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.p.b 4 1.a even 1 1 trivial
273.2.p.b 4 91.i even 4 1 inner
273.2.p.c yes 4 7.b odd 2 1
273.2.p.c yes 4 13.d odd 4 1
819.2.y.b 4 21.c even 2 1
819.2.y.b 4 39.f even 4 1
819.2.y.c 4 3.b odd 2 1
819.2.y.c 4 273.o odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}^{4} + 4T_{5}^{3} + 8T_{5}^{2} - 12T_{5} + 9$$ T5^4 + 4*T5^3 + 8*T5^2 - 12*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4} + 4 T^{3} + 8 T^{2} - 12 T + 9$$
$7$ $$T^{4} + 4 T^{3} + 8 T^{2} + 28 T + 49$$
$11$ $$T^{4} - 4 T^{3} + 8 T^{2} + 72 T + 324$$
$13$ $$T^{4} - 8 T^{3} + 32 T^{2} - 104 T + 169$$
$17$ $$(T^{2} + 4 T - 6)^{2}$$
$19$ $$T^{4} - 8 T^{3} + 32 T^{2} - 24 T + 9$$
$23$ $$T^{4} + 22T^{2} + 81$$
$29$ $$(T^{2} - 10 T + 15)^{2}$$
$31$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 729$$
$37$ $$T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 5184$$
$41$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 144$$
$43$ $$T^{4} + 82T^{2} + 1521$$
$47$ $$T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 2025$$
$53$ $$(T^{2} + 2 T - 9)^{2}$$
$59$ $$T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 5184$$
$61$ $$(T^{2} + 10)^{2}$$
$67$ $$T^{4} + 6400$$
$71$ $$(T^{2} - 12 T + 72)^{2}$$
$73$ $$T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 729$$
$79$ $$(T^{2} + 10 T - 15)^{2}$$
$83$ $$T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 2025$$
$89$ $$T^{4} + 28 T^{3} + 392 T^{2} + \cdots + 8649$$
$97$ $$T^{4} + 8 T^{3} + 32 T^{2} + 24 T + 9$$
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