Properties

 Label 270.2.c.c Level $270$ Weight $2$ Character orbit 270.c Analytic conductor $2.156$ Analytic rank $0$ Dimension $4$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,2,Mod(109,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("270.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$270 = 2 \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 270.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: $$2.15596085457$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 9x^{2} + 25$$ x^4 - 9*x^2 + 25 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - q^{4} + \beta_1 q^{5} + (2 \beta_{3} + 1) q^{7} + \beta_{2} q^{8}+O(q^{10})$$ q - b2 * q^2 - q^4 + b1 * q^5 + (2*b3 + 1) * q^7 + b2 * q^8 $$q - \beta_{2} q^{2} - q^{4} + \beta_1 q^{5} + (2 \beta_{3} + 1) q^{7} + \beta_{2} q^{8} - \beta_{3} q^{10} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - \beta_{2} + 2 \beta_1) q^{14} + q^{16} - 4 \beta_{2} q^{17} - 6 q^{19} - \beta_1 q^{20} + ( - 2 \beta_{3} - 1) q^{22} - 2 \beta_{2} q^{23} + (\beta_{3} + 5) q^{25} + ( - 2 \beta_{3} - 1) q^{28} + 7 q^{31} - \beta_{2} q^{32} - 4 q^{34} + (10 \beta_{2} - \beta_1) q^{35} + ( - 4 \beta_{3} - 2) q^{37} + 6 \beta_{2} q^{38} + \beta_{3} q^{40} + (2 \beta_{2} - 4 \beta_1) q^{41} + (4 \beta_{3} + 2) q^{43} + (\beta_{2} - 2 \beta_1) q^{44} - 2 q^{46} - 2 \beta_{2} q^{47} - 12 q^{49} + ( - 5 \beta_{2} + \beta_1) q^{50} - 3 \beta_{2} q^{53} + (\beta_{3} + 10) q^{55} + (\beta_{2} - 2 \beta_1) q^{56} + (2 \beta_{2} - 4 \beta_1) q^{59} - 4 q^{61} - 7 \beta_{2} q^{62} - q^{64} + (4 \beta_{3} + 2) q^{67} + 4 \beta_{2} q^{68} + (\beta_{3} + 10) q^{70} + ( - 2 \beta_{3} - 1) q^{73} + (2 \beta_{2} - 4 \beta_1) q^{74} + 6 q^{76} + 19 \beta_{2} q^{77} + \beta_1 q^{80} + (4 \beta_{3} + 2) q^{82} - 5 \beta_{2} q^{83} - 4 \beta_{3} q^{85} + ( - 2 \beta_{2} + 4 \beta_1) q^{86} + (2 \beta_{3} + 1) q^{88} + ( - 2 \beta_{2} + 4 \beta_1) q^{89} + 2 \beta_{2} q^{92} - 2 q^{94} - 6 \beta_1 q^{95} + ( - 2 \beta_{3} - 1) q^{97} + 12 \beta_{2} q^{98}+O(q^{100})$$ q - b2 * q^2 - q^4 + b1 * q^5 + (2*b3 + 1) * q^7 + b2 * q^8 - b3 * q^10 + (-b2 + 2*b1) * q^11 + (-b2 + 2*b1) * q^14 + q^16 - 4*b2 * q^17 - 6 * q^19 - b1 * q^20 + (-2*b3 - 1) * q^22 - 2*b2 * q^23 + (b3 + 5) * q^25 + (-2*b3 - 1) * q^28 + 7 * q^31 - b2 * q^32 - 4 * q^34 + (10*b2 - b1) * q^35 + (-4*b3 - 2) * q^37 + 6*b2 * q^38 + b3 * q^40 + (2*b2 - 4*b1) * q^41 + (4*b3 + 2) * q^43 + (b2 - 2*b1) * q^44 - 2 * q^46 - 2*b2 * q^47 - 12 * q^49 + (-5*b2 + b1) * q^50 - 3*b2 * q^53 + (b3 + 10) * q^55 + (b2 - 2*b1) * q^56 + (2*b2 - 4*b1) * q^59 - 4 * q^61 - 7*b2 * q^62 - q^64 + (4*b3 + 2) * q^67 + 4*b2 * q^68 + (b3 + 10) * q^70 + (-2*b3 - 1) * q^73 + (2*b2 - 4*b1) * q^74 + 6 * q^76 + 19*b2 * q^77 + b1 * q^80 + (4*b3 + 2) * q^82 - 5*b2 * q^83 - 4*b3 * q^85 + (-2*b2 + 4*b1) * q^86 + (2*b3 + 1) * q^88 + (-2*b2 + 4*b1) * q^89 + 2*b2 * q^92 - 2 * q^94 - 6*b1 * q^95 + (-2*b3 - 1) * q^97 + 12*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 2 q^{10} + 4 q^{16} - 24 q^{19} + 18 q^{25} + 28 q^{31} - 16 q^{34} - 2 q^{40} - 8 q^{46} - 48 q^{49} + 38 q^{55} - 16 q^{61} - 4 q^{64} + 38 q^{70} + 24 q^{76} + 8 q^{85} - 8 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 + 2 * q^10 + 4 * q^16 - 24 * q^19 + 18 * q^25 + 28 * q^31 - 16 * q^34 - 2 * q^40 - 8 * q^46 - 48 * q^49 + 38 * q^55 - 16 * q^61 - 4 * q^64 + 38 * q^70 + 24 * q^76 + 8 * q^85 - 8 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 4\nu ) / 5$$ (v^3 - 4*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5$$ b3 + 5 $$\nu^{3}$$ $$=$$ $$5\beta_{2} + 4\beta_1$$ 5*b2 + 4*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$217$$ $$\chi(n)$$ $$1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −2.17945 + 0.500000i 2.17945 + 0.500000i −2.17945 − 0.500000i 2.17945 − 0.500000i
1.00000i 0 −1.00000 −2.17945 + 0.500000i 0 4.35890i 1.00000i 0 0.500000 + 2.17945i
109.2 1.00000i 0 −1.00000 2.17945 + 0.500000i 0 4.35890i 1.00000i 0 0.500000 2.17945i
109.3 1.00000i 0 −1.00000 −2.17945 0.500000i 0 4.35890i 1.00000i 0 0.500000 2.17945i
109.4 1.00000i 0 −1.00000 2.17945 0.500000i 0 4.35890i 1.00000i 0 0.500000 + 2.17945i
 $$n$$: 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
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.c.c 4
3.b odd 2 1 inner 270.2.c.c 4
4.b odd 2 1 2160.2.f.m 4
5.b even 2 1 inner 270.2.c.c 4
5.c odd 4 1 1350.2.a.w 2
5.c odd 4 1 1350.2.a.x 2
9.c even 3 2 810.2.i.h 8
9.d odd 6 2 810.2.i.h 8
12.b even 2 1 2160.2.f.m 4
15.d odd 2 1 inner 270.2.c.c 4
15.e even 4 1 1350.2.a.w 2
15.e even 4 1 1350.2.a.x 2
20.d odd 2 1 2160.2.f.m 4
45.h odd 6 2 810.2.i.h 8
45.j even 6 2 810.2.i.h 8
60.h even 2 1 2160.2.f.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.c 4 1.a even 1 1 trivial
270.2.c.c 4 3.b odd 2 1 inner
270.2.c.c 4 5.b even 2 1 inner
270.2.c.c 4 15.d odd 2 1 inner
810.2.i.h 8 9.c even 3 2
810.2.i.h 8 9.d odd 6 2
810.2.i.h 8 45.h odd 6 2
810.2.i.h 8 45.j even 6 2
1350.2.a.w 2 5.c odd 4 1
1350.2.a.w 2 15.e even 4 1
1350.2.a.x 2 5.c odd 4 1
1350.2.a.x 2 15.e even 4 1
2160.2.f.m 4 4.b odd 2 1
2160.2.f.m 4 12.b even 2 1
2160.2.f.m 4 20.d odd 2 1
2160.2.f.m 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(270, [\chi])$$:

 $$T_{7}^{2} + 19$$ T7^2 + 19 $$T_{11}^{2} - 19$$ T11^2 - 19

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 9T^{2} + 25$$
$7$ $$(T^{2} + 19)^{2}$$
$11$ $$(T^{2} - 19)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 16)^{2}$$
$19$ $$(T + 6)^{4}$$
$23$ $$(T^{2} + 4)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T - 7)^{4}$$
$37$ $$(T^{2} + 76)^{2}$$
$41$ $$(T^{2} - 76)^{2}$$
$43$ $$(T^{2} + 76)^{2}$$
$47$ $$(T^{2} + 4)^{2}$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} - 76)^{2}$$
$61$ $$(T + 4)^{4}$$
$67$ $$(T^{2} + 76)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 19)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 25)^{2}$$
$89$ $$(T^{2} - 76)^{2}$$
$97$ $$(T^{2} + 19)^{2}$$