# Properties

 Label 240.2.f.a 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 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 $$i = \sqrt{-1}$$. 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 $$q + i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9} - 2 q^{11} + 6 i q^{13} + ( - 2 i - 1) q^{15} - 2 i q^{17} - 2 q^{21} + 4 i q^{23} + ( - 4 i + 3) q^{25} - i q^{27} + 8 q^{31} - 2 i q^{33} + ( - 4 i - 2) q^{35} - 2 i q^{37} - 6 q^{39} + 2 q^{41} + 4 i q^{43} + ( - i + 2) q^{45} - 8 i q^{47} + 3 q^{49} + 2 q^{51} + 6 i q^{53} + ( - 2 i + 4) q^{55} + 10 q^{59} + 2 q^{61} - 2 i q^{63} + ( - 12 i - 6) q^{65} - 8 i q^{67} - 4 q^{69} - 12 q^{71} - 4 i q^{73} + (3 i + 4) q^{75} - 4 i q^{77} + q^{81} + 4 i q^{83} + (4 i + 2) q^{85} + 10 q^{89} - 12 q^{91} + 8 i q^{93} + 8 i q^{97} + 2 q^{99} +O(q^{100})$$ q + i * q^3 + (i - 2) * q^5 + 2*i * q^7 - q^9 - 2 * q^11 + 6*i * q^13 + (-2*i - 1) * q^15 - 2*i * q^17 - 2 * q^21 + 4*i * q^23 + (-4*i + 3) * q^25 - i * q^27 + 8 * q^31 - 2*i * q^33 + (-4*i - 2) * q^35 - 2*i * q^37 - 6 * q^39 + 2 * q^41 + 4*i * q^43 + (-i + 2) * q^45 - 8*i * q^47 + 3 * q^49 + 2 * q^51 + 6*i * q^53 + (-2*i + 4) * q^55 + 10 * q^59 + 2 * q^61 - 2*i * q^63 + (-12*i - 6) * q^65 - 8*i * q^67 - 4 * q^69 - 12 * q^71 - 4*i * q^73 + (3*i + 4) * q^75 - 4*i * q^77 + q^81 + 4*i * q^83 + (4*i + 2) * q^85 + 10 * q^89 - 12 * q^91 + 8*i * q^93 + 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 $$2 q - 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} - 4 q^{21} + 6 q^{25} + 16 q^{31} - 4 q^{35} - 12 q^{39} + 4 q^{41} + 4 q^{45} + 6 q^{49} + 4 q^{51} + 8 q^{55} + 20 q^{59} + 4 q^{61} - 12 q^{65} - 8 q^{69} - 24 q^{71} + 8 q^{75} + 2 q^{81} + 4 q^{85} + 20 q^{89} - 24 q^{91} + 4 q^{99}+O(q^{100})$$ 2 * q - 4 * q^5 - 2 * q^9 - 4 * q^11 - 2 * q^15 - 4 * q^21 + 6 * q^25 + 16 * q^31 - 4 * q^35 - 12 * q^39 + 4 * q^41 + 4 * q^45 + 6 * q^49 + 4 * q^51 + 8 * q^55 + 20 * q^59 + 4 * q^61 - 12 * q^65 - 8 * q^69 - 24 * q^71 + 8 * q^75 + 2 * q^81 + 4 * q^85 + 20 * q^89 - 24 * q^91 + 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.a 2
3.b odd 2 1 720.2.f.f 2
4.b odd 2 1 30.2.c.a 2
5.b even 2 1 inner 240.2.f.a 2
5.c odd 4 1 1200.2.a.g 1
5.c odd 4 1 1200.2.a.m 1
8.b even 2 1 960.2.f.i 2
8.d odd 2 1 960.2.f.h 2
12.b even 2 1 90.2.c.a 2
15.d odd 2 1 720.2.f.f 2
15.e even 4 1 3600.2.a.o 1
15.e even 4 1 3600.2.a.bg 1
16.e even 4 1 3840.2.d.j 2
16.e even 4 1 3840.2.d.x 2
16.f odd 4 1 3840.2.d.g 2
16.f odd 4 1 3840.2.d.y 2
20.d odd 2 1 30.2.c.a 2
20.e even 4 1 150.2.a.a 1
20.e even 4 1 150.2.a.c 1
24.f even 2 1 2880.2.f.e 2
24.h odd 2 1 2880.2.f.c 2
28.d even 2 1 1470.2.g.g 2
28.f even 6 2 1470.2.n.a 4
28.g odd 6 2 1470.2.n.h 4
36.f odd 6 2 810.2.i.e 4
36.h even 6 2 810.2.i.b 4
40.e odd 2 1 960.2.f.h 2
40.f even 2 1 960.2.f.i 2
40.i odd 4 1 4800.2.a.m 1
40.i odd 4 1 4800.2.a.cj 1
40.k even 4 1 4800.2.a.l 1
40.k even 4 1 4800.2.a.cg 1
60.h even 2 1 90.2.c.a 2
60.l odd 4 1 450.2.a.b 1
60.l odd 4 1 450.2.a.f 1
80.k odd 4 1 3840.2.d.g 2
80.k odd 4 1 3840.2.d.y 2
80.q even 4 1 3840.2.d.j 2
80.q even 4 1 3840.2.d.x 2
120.i odd 2 1 2880.2.f.c 2
120.m even 2 1 2880.2.f.e 2
140.c even 2 1 1470.2.g.g 2
140.j odd 4 1 7350.2.a.bg 1
140.j odd 4 1 7350.2.a.cc 1
140.p odd 6 2 1470.2.n.h 4
140.s even 6 2 1470.2.n.a 4
180.n even 6 2 810.2.i.b 4
180.p odd 6 2 810.2.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 4.b odd 2 1
30.2.c.a 2 20.d odd 2 1
90.2.c.a 2 12.b even 2 1
90.2.c.a 2 60.h even 2 1
150.2.a.a 1 20.e even 4 1
150.2.a.c 1 20.e even 4 1
240.2.f.a 2 1.a even 1 1 trivial
240.2.f.a 2 5.b even 2 1 inner
450.2.a.b 1 60.l odd 4 1
450.2.a.f 1 60.l odd 4 1
720.2.f.f 2 3.b odd 2 1
720.2.f.f 2 15.d odd 2 1
810.2.i.b 4 36.h even 6 2
810.2.i.b 4 180.n even 6 2
810.2.i.e 4 36.f odd 6 2
810.2.i.e 4 180.p odd 6 2
960.2.f.h 2 8.d odd 2 1
960.2.f.h 2 40.e odd 2 1
960.2.f.i 2 8.b even 2 1
960.2.f.i 2 40.f even 2 1
1200.2.a.g 1 5.c odd 4 1
1200.2.a.m 1 5.c odd 4 1
1470.2.g.g 2 28.d even 2 1
1470.2.g.g 2 140.c even 2 1
1470.2.n.a 4 28.f even 6 2
1470.2.n.a 4 140.s even 6 2
1470.2.n.h 4 28.g odd 6 2
1470.2.n.h 4 140.p odd 6 2
2880.2.f.c 2 24.h odd 2 1
2880.2.f.c 2 120.i odd 2 1
2880.2.f.e 2 24.f even 2 1
2880.2.f.e 2 120.m even 2 1
3600.2.a.o 1 15.e even 4 1
3600.2.a.bg 1 15.e even 4 1
3840.2.d.g 2 16.f odd 4 1
3840.2.d.g 2 80.k odd 4 1
3840.2.d.j 2 16.e even 4 1
3840.2.d.j 2 80.q even 4 1
3840.2.d.x 2 16.e even 4 1
3840.2.d.x 2 80.q even 4 1
3840.2.d.y 2 16.f odd 4 1
3840.2.d.y 2 80.k odd 4 1
4800.2.a.l 1 40.k even 4 1
4800.2.a.m 1 40.i odd 4 1
4800.2.a.cg 1 40.k even 4 1
4800.2.a.cj 1 40.i odd 4 1
7350.2.a.bg 1 140.j odd 4 1
7350.2.a.cc 1 140.j 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 $$T_{13}^{2} + 36$$ T13^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 64$$