# Properties

 Label 240.2.f.c Level $240$ Weight $2$ Character orbit 240.f Analytic conductor $1.916$ 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] = [240,2,Mod(49,240)]

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, 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("240.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.f (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: $$1.91640964851$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + (i + 2) q^{5} + 2 i q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + (i + 2) * q^5 + 2*i * q^7 - q^9 $$q + i q^{3} + (i + 2) q^{5} + 2 i q^{7} - q^{9} - 2 q^{11} - 2 i q^{13} + (2 i - 1) q^{15} + 6 i q^{17} + 8 q^{19} - 2 q^{21} - 4 i q^{23} + (4 i + 3) q^{25} - i q^{27} - 8 q^{29} - 2 i q^{33} + (4 i - 2) q^{35} - 10 i q^{37} + 2 q^{39} + 2 q^{41} - 12 i q^{43} + ( - i - 2) q^{45} + 3 q^{49} - 6 q^{51} - 10 i q^{53} + ( - 2 i - 4) q^{55} + 8 i q^{57} - 6 q^{59} + 2 q^{61} - 2 i q^{63} + ( - 4 i + 2) q^{65} + 8 i q^{67} + 4 q^{69} + 4 q^{71} - 4 i q^{73} + (3 i - 4) q^{75} - 4 i q^{77} - 8 q^{79} + q^{81} + 4 i q^{83} + (12 i - 6) q^{85} - 8 i q^{87} - 6 q^{89} + 4 q^{91} + (8 i + 16) q^{95} + 8 i q^{97} + 2 q^{99} +O(q^{100})$$ q + i * q^3 + (i + 2) * q^5 + 2*i * q^7 - q^9 - 2 * q^11 - 2*i * q^13 + (2*i - 1) * q^15 + 6*i * q^17 + 8 * q^19 - 2 * q^21 - 4*i * q^23 + (4*i + 3) * q^25 - i * q^27 - 8 * q^29 - 2*i * q^33 + (4*i - 2) * q^35 - 10*i * q^37 + 2 * q^39 + 2 * q^41 - 12*i * q^43 + (-i - 2) * q^45 + 3 * q^49 - 6 * q^51 - 10*i * q^53 + (-2*i - 4) * q^55 + 8*i * q^57 - 6 * q^59 + 2 * q^61 - 2*i * q^63 + (-4*i + 2) * q^65 + 8*i * q^67 + 4 * q^69 + 4 * q^71 - 4*i * q^73 + (3*i - 4) * q^75 - 4*i * q^77 - 8 * q^79 + q^81 + 4*i * q^83 + (12*i - 6) * q^85 - 8*i * q^87 - 6 * q^89 + 4 * q^91 + (8*i + 16) * q^95 + 8*i * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 - 2 * q^9 $$2 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} + 16 q^{19} - 4 q^{21} + 6 q^{25} - 16 q^{29} - 4 q^{35} + 4 q^{39} + 4 q^{41} - 4 q^{45} + 6 q^{49} - 12 q^{51} - 8 q^{55} - 12 q^{59} + 4 q^{61} + 4 q^{65} + 8 q^{69} + 8 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{85} - 12 q^{89} + 8 q^{91} + 32 q^{95} + 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^11 - 2 * q^15 + 16 * q^19 - 4 * q^21 + 6 * q^25 - 16 * q^29 - 4 * q^35 + 4 * q^39 + 4 * q^41 - 4 * q^45 + 6 * q^49 - 12 * q^51 - 8 * q^55 - 12 * q^59 + 4 * q^61 + 4 * q^65 + 8 * q^69 + 8 * q^71 - 8 * q^75 - 16 * q^79 + 2 * q^81 - 12 * q^85 - 12 * q^89 + 8 * q^91 + 32 * q^95 + 4 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 1.00000i 0 2.00000 1.00000i 0 2.00000i 0 −1.00000 0
49.2 0 1.00000i 0 2.00000 + 1.00000i 0 2.00000i 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.f.c 2
3.b odd 2 1 720.2.f.b 2
4.b odd 2 1 120.2.f.a 2
5.b even 2 1 inner 240.2.f.c 2
5.c odd 4 1 1200.2.a.h 1
5.c odd 4 1 1200.2.a.l 1
8.b even 2 1 960.2.f.b 2
8.d odd 2 1 960.2.f.a 2
12.b even 2 1 360.2.f.a 2
15.d odd 2 1 720.2.f.b 2
15.e even 4 1 3600.2.a.n 1
15.e even 4 1 3600.2.a.bi 1
16.e even 4 1 3840.2.d.m 2
16.e even 4 1 3840.2.d.v 2
16.f odd 4 1 3840.2.d.d 2
16.f odd 4 1 3840.2.d.ba 2
20.d odd 2 1 120.2.f.a 2
20.e even 4 1 600.2.a.d 1
20.e even 4 1 600.2.a.g 1
24.f even 2 1 2880.2.f.t 2
24.h odd 2 1 2880.2.f.r 2
40.e odd 2 1 960.2.f.a 2
40.f even 2 1 960.2.f.b 2
40.i odd 4 1 4800.2.a.n 1
40.i odd 4 1 4800.2.a.ci 1
40.k even 4 1 4800.2.a.k 1
40.k even 4 1 4800.2.a.ch 1
60.h even 2 1 360.2.f.a 2
60.l odd 4 1 1800.2.a.g 1
60.l odd 4 1 1800.2.a.q 1
80.k odd 4 1 3840.2.d.d 2
80.k odd 4 1 3840.2.d.ba 2
80.q even 4 1 3840.2.d.m 2
80.q even 4 1 3840.2.d.v 2
120.i odd 2 1 2880.2.f.r 2
120.m even 2 1 2880.2.f.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.f.a 2 4.b odd 2 1
120.2.f.a 2 20.d odd 2 1
240.2.f.c 2 1.a even 1 1 trivial
240.2.f.c 2 5.b even 2 1 inner
360.2.f.a 2 12.b even 2 1
360.2.f.a 2 60.h even 2 1
600.2.a.d 1 20.e even 4 1
600.2.a.g 1 20.e even 4 1
720.2.f.b 2 3.b odd 2 1
720.2.f.b 2 15.d odd 2 1
960.2.f.a 2 8.d odd 2 1
960.2.f.a 2 40.e odd 2 1
960.2.f.b 2 8.b even 2 1
960.2.f.b 2 40.f even 2 1
1200.2.a.h 1 5.c odd 4 1
1200.2.a.l 1 5.c odd 4 1
1800.2.a.g 1 60.l odd 4 1
1800.2.a.q 1 60.l odd 4 1
2880.2.f.r 2 24.h odd 2 1
2880.2.f.r 2 120.i odd 2 1
2880.2.f.t 2 24.f even 2 1
2880.2.f.t 2 120.m even 2 1
3600.2.a.n 1 15.e even 4 1
3600.2.a.bi 1 15.e even 4 1
3840.2.d.d 2 16.f odd 4 1
3840.2.d.d 2 80.k odd 4 1
3840.2.d.m 2 16.e even 4 1
3840.2.d.m 2 80.q even 4 1
3840.2.d.v 2 16.e even 4 1
3840.2.d.v 2 80.q even 4 1
3840.2.d.ba 2 16.f odd 4 1
3840.2.d.ba 2 80.k odd 4 1
4800.2.a.k 1 40.k even 4 1
4800.2.a.n 1 40.i odd 4 1
4800.2.a.ch 1 40.k even 4 1
4800.2.a.ci 1 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{13}^{2} + 4$$ T13^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 8)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 144$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 100$$
$59$ $$(T + 6)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 64$$