# Properties

 Label 240.3.c.c Level $240$ Weight $3$ Character orbit 240.c Analytic conductor $6.540$ 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] = [240,3,Mod(209,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("240.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 240.c (of order $$2$$, degree $$1$$, not 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: $$6.53952634465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 16x^{2} + 81$$ x^4 + 16*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2}) q^{5} + ( - \beta_{2} - 2 \beta_1) q^{7} + (\beta_{3} - 8) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b3 + 2*b2) * q^5 + (-b2 - 2*b1) * q^7 + (b3 - 8) * q^9 $$q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2}) q^{5} + ( - \beta_{2} - 2 \beta_1) q^{7} + (\beta_{3} - 8) q^{9} + 4 \beta_{3} q^{11} + ( - 2 \beta_{3} + 9 \beta_{2} + \cdots - 2) q^{15}+ \cdots + ( - 32 \beta_{3} - 68) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b3 + 2*b2) * q^5 + (-b2 - 2*b1) * q^7 + (b3 - 8) * q^9 + 4*b3 * q^11 + (-2*b3 + 9*b2 + b1 - 2) * q^15 + 8*b2 * q^17 - 12 * q^19 + (-b3 + 17) * q^21 + 17*b2 * q^23 + (-4*b2 - 8*b1 - 9) * q^25 + (9*b2 - 7*b1) * q^27 + 32 * q^31 + (36*b2 + 4*b1) * q^33 + (4*b3 - 17*b2) * q^35 + (4*b2 + 8*b1) * q^37 + 14*b3 * q^41 + (7*b2 + 14*b1) * q^43 + (-8*b3 - 18*b2 - 4*b1 - 17) * q^45 - 25*b2 * q^47 + 15 * q^49 + (-8*b3 - 8) * q^51 - 48*b2 * q^53 + (-8*b2 - 16*b1 - 68) * q^55 - 12*b1 * q^57 + 4*b3 * q^59 - 16 * q^61 + (-9*b2 + 16*b1) * q^63 + (b2 + 2*b1) * q^67 + (-17*b3 - 17) * q^69 + (20*b2 + 40*b1) * q^73 + (-4*b3 - 9*b1 + 68) * q^75 - 68*b2 * q^77 + 72 * q^79 + (-16*b3 + 47) * q^81 - 31*b2 * q^83 + (-8*b2 - 16*b1 + 32) * q^85 + 16*b3 * q^89 + 32*b1 * q^93 + (-12*b3 - 24*b2) * q^95 + (-28*b2 - 56*b1) * q^97 + (-32*b3 - 68) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 32 q^{9}+O(q^{10})$$ 4 * q - 32 * q^9 $$4 q - 32 q^{9} - 8 q^{15} - 48 q^{19} + 68 q^{21} - 36 q^{25} + 128 q^{31} - 68 q^{45} + 60 q^{49} - 32 q^{51} - 272 q^{55} - 64 q^{61} - 68 q^{69} + 272 q^{75} + 288 q^{79} + 188 q^{81} + 128 q^{85} - 272 q^{99}+O(q^{100})$$ 4 * q - 32 * q^9 - 8 * q^15 - 48 * q^19 + 68 * q^21 - 36 * q^25 + 128 * q^31 - 68 * q^45 + 60 * q^49 - 32 * q^51 - 272 * q^55 - 64 * q^61 - 68 * q^69 + 272 * q^75 + 288 * q^79 + 188 * q^81 + 128 * q^85 - 272 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 16x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 7\nu ) / 9$$ (v^3 + 7*v) / 9 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 8$$ v^2 + 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 8$$ b3 - 8 $$\nu^{3}$$ $$=$$ $$9\beta_{2} - 7\beta_1$$ 9*b2 - 7*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 −0.707107 − 2.91548i −0.707107 + 2.91548i 0.707107 − 2.91548i 0.707107 + 2.91548i
0 −0.707107 2.91548i 0 2.82843 + 4.12311i 0 5.83095i 0 −8.00000 + 4.12311i 0
209.2 0 −0.707107 + 2.91548i 0 2.82843 4.12311i 0 5.83095i 0 −8.00000 4.12311i 0
209.3 0 0.707107 2.91548i 0 −2.82843 4.12311i 0 5.83095i 0 −8.00000 4.12311i 0
209.4 0 0.707107 + 2.91548i 0 −2.82843 + 4.12311i 0 5.83095i 0 −8.00000 + 4.12311i 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 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 240.3.c.c 4
3.b odd 2 1 inner 240.3.c.c 4
4.b odd 2 1 30.3.b.a 4
5.b even 2 1 inner 240.3.c.c 4
5.c odd 4 2 1200.3.l.t 4
8.b even 2 1 960.3.c.e 4
8.d odd 2 1 960.3.c.f 4
12.b even 2 1 30.3.b.a 4
15.d odd 2 1 inner 240.3.c.c 4
15.e even 4 2 1200.3.l.t 4
20.d odd 2 1 30.3.b.a 4
20.e even 4 2 150.3.d.d 4
24.f even 2 1 960.3.c.f 4
24.h odd 2 1 960.3.c.e 4
36.f odd 6 2 810.3.j.c 8
36.h even 6 2 810.3.j.c 8
40.e odd 2 1 960.3.c.f 4
40.f even 2 1 960.3.c.e 4
60.h even 2 1 30.3.b.a 4
60.l odd 4 2 150.3.d.d 4
120.i odd 2 1 960.3.c.e 4
120.m even 2 1 960.3.c.f 4
180.n even 6 2 810.3.j.c 8
180.p odd 6 2 810.3.j.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.b.a 4 4.b odd 2 1
30.3.b.a 4 12.b even 2 1
30.3.b.a 4 20.d odd 2 1
30.3.b.a 4 60.h even 2 1
150.3.d.d 4 20.e even 4 2
150.3.d.d 4 60.l odd 4 2
240.3.c.c 4 1.a even 1 1 trivial
240.3.c.c 4 3.b odd 2 1 inner
240.3.c.c 4 5.b even 2 1 inner
240.3.c.c 4 15.d odd 2 1 inner
810.3.j.c 8 36.f odd 6 2
810.3.j.c 8 36.h even 6 2
810.3.j.c 8 180.n even 6 2
810.3.j.c 8 180.p odd 6 2
960.3.c.e 4 8.b even 2 1
960.3.c.e 4 24.h odd 2 1
960.3.c.e 4 40.f even 2 1
960.3.c.e 4 120.i odd 2 1
960.3.c.f 4 8.d odd 2 1
960.3.c.f 4 24.f even 2 1
960.3.c.f 4 40.e odd 2 1
960.3.c.f 4 120.m even 2 1
1200.3.l.t 4 5.c odd 4 2
1200.3.l.t 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(240, [\chi])$$:

 $$T_{7}^{2} + 34$$ T7^2 + 34 $$T_{17}^{2} - 128$$ T17^2 - 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 16T^{2} + 81$$
$5$ $$T^{4} + 18T^{2} + 625$$
$7$ $$(T^{2} + 34)^{2}$$
$11$ $$(T^{2} + 272)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 128)^{2}$$
$19$ $$(T + 12)^{4}$$
$23$ $$(T^{2} - 578)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T - 32)^{4}$$
$37$ $$(T^{2} + 544)^{2}$$
$41$ $$(T^{2} + 3332)^{2}$$
$43$ $$(T^{2} + 1666)^{2}$$
$47$ $$(T^{2} - 1250)^{2}$$
$53$ $$(T^{2} - 4608)^{2}$$
$59$ $$(T^{2} + 272)^{2}$$
$61$ $$(T + 16)^{4}$$
$67$ $$(T^{2} + 34)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 13600)^{2}$$
$79$ $$(T - 72)^{4}$$
$83$ $$(T^{2} - 1922)^{2}$$
$89$ $$(T^{2} + 4352)^{2}$$
$97$ $$(T^{2} + 26656)^{2}$$