# Properties

 Label 480.2.f.c Level $480$ Weight $2$ Character orbit 480.f Analytic conductor $3.833$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(289,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, 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("480.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$480 = 2^{5} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 480.f (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: $$3.83281929702$$ 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: 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + (2 i + 1) q^{5} + 4 i q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + (2*i + 1) * q^5 + 4*i * q^7 - q^9 $$q + i q^{3} + (2 i + 1) q^{5} + 4 i q^{7} - q^{9} - 4 i q^{13} + (i - 2) q^{15} - 8 q^{19} - 4 q^{21} + 4 i q^{23} + (4 i - 3) q^{25} - i q^{27} + 6 q^{29} + 8 q^{31} + (4 i - 8) q^{35} - 4 i q^{37} + 4 q^{39} + 6 q^{41} + 4 i q^{43} + ( - 2 i - 1) q^{45} + 4 i q^{47} - 9 q^{49} + 12 i q^{53} - 8 i q^{57} - 6 q^{61} - 4 i q^{63} + ( - 4 i + 8) q^{65} - 12 i q^{67} - 4 q^{69} + 16 q^{71} + ( - 3 i - 4) q^{75} - 8 q^{79} + q^{81} - 12 i q^{83} + 6 i q^{87} + 10 q^{89} + 16 q^{91} + 8 i q^{93} + ( - 16 i - 8) q^{95} + 8 i q^{97} +O(q^{100})$$ q + i * q^3 + (2*i + 1) * q^5 + 4*i * q^7 - q^9 - 4*i * q^13 + (i - 2) * q^15 - 8 * q^19 - 4 * q^21 + 4*i * q^23 + (4*i - 3) * q^25 - i * q^27 + 6 * q^29 + 8 * q^31 + (4*i - 8) * q^35 - 4*i * q^37 + 4 * q^39 + 6 * q^41 + 4*i * q^43 + (-2*i - 1) * q^45 + 4*i * q^47 - 9 * q^49 + 12*i * q^53 - 8*i * q^57 - 6 * q^61 - 4*i * q^63 + (-4*i + 8) * q^65 - 12*i * q^67 - 4 * q^69 + 16 * q^71 + (-3*i - 4) * q^75 - 8 * q^79 + q^81 - 12*i * q^83 + 6*i * q^87 + 10 * q^89 + 16 * q^91 + 8*i * q^93 + (-16*i - 8) * q^95 + 8*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^9 $$2 q + 2 q^{5} - 2 q^{9} - 4 q^{15} - 16 q^{19} - 8 q^{21} - 6 q^{25} + 12 q^{29} + 16 q^{31} - 16 q^{35} + 8 q^{39} + 12 q^{41} - 2 q^{45} - 18 q^{49} - 12 q^{61} + 16 q^{65} - 8 q^{69} + 32 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} + 20 q^{89} + 32 q^{91} - 16 q^{95}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^9 - 4 * q^15 - 16 * q^19 - 8 * q^21 - 6 * q^25 + 12 * q^29 + 16 * q^31 - 16 * q^35 + 8 * q^39 + 12 * q^41 - 2 * q^45 - 18 * q^49 - 12 * q^61 + 16 * q^65 - 8 * q^69 + 32 * q^71 - 8 * q^75 - 16 * q^79 + 2 * q^81 + 20 * q^89 + 32 * q^91 - 16 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$421$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 − 1.00000i 1.00000i
0 1.00000i 0 1.00000 2.00000i 0 4.00000i 0 −1.00000 0
289.2 0 1.00000i 0 1.00000 + 2.00000i 0 4.00000i 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 480.2.f.c 2
3.b odd 2 1 1440.2.f.b 2
4.b odd 2 1 480.2.f.d yes 2
5.b even 2 1 inner 480.2.f.c 2
5.c odd 4 1 2400.2.a.p 1
5.c odd 4 1 2400.2.a.s 1
8.b even 2 1 960.2.f.d 2
8.d odd 2 1 960.2.f.e 2
12.b even 2 1 1440.2.f.d 2
15.d odd 2 1 1440.2.f.b 2
15.e even 4 1 7200.2.a.a 1
15.e even 4 1 7200.2.a.ca 1
16.e even 4 1 3840.2.d.p 2
16.e even 4 1 3840.2.d.q 2
16.f odd 4 1 3840.2.d.a 2
16.f odd 4 1 3840.2.d.bf 2
20.d odd 2 1 480.2.f.d yes 2
20.e even 4 1 2400.2.a.o 1
20.e even 4 1 2400.2.a.t 1
24.f even 2 1 2880.2.f.m 2
24.h odd 2 1 2880.2.f.o 2
40.e odd 2 1 960.2.f.e 2
40.f even 2 1 960.2.f.d 2
40.i odd 4 1 4800.2.a.e 1
40.i odd 4 1 4800.2.a.cp 1
40.k even 4 1 4800.2.a.c 1
40.k even 4 1 4800.2.a.cr 1
60.h even 2 1 1440.2.f.d 2
60.l odd 4 1 7200.2.a.c 1
60.l odd 4 1 7200.2.a.by 1
80.k odd 4 1 3840.2.d.a 2
80.k odd 4 1 3840.2.d.bf 2
80.q even 4 1 3840.2.d.p 2
80.q even 4 1 3840.2.d.q 2
120.i odd 2 1 2880.2.f.o 2
120.m even 2 1 2880.2.f.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.c 2 1.a even 1 1 trivial
480.2.f.c 2 5.b even 2 1 inner
480.2.f.d yes 2 4.b odd 2 1
480.2.f.d yes 2 20.d odd 2 1
960.2.f.d 2 8.b even 2 1
960.2.f.d 2 40.f even 2 1
960.2.f.e 2 8.d odd 2 1
960.2.f.e 2 40.e odd 2 1
1440.2.f.b 2 3.b odd 2 1
1440.2.f.b 2 15.d odd 2 1
1440.2.f.d 2 12.b even 2 1
1440.2.f.d 2 60.h even 2 1
2400.2.a.o 1 20.e even 4 1
2400.2.a.p 1 5.c odd 4 1
2400.2.a.s 1 5.c odd 4 1
2400.2.a.t 1 20.e even 4 1
2880.2.f.m 2 24.f even 2 1
2880.2.f.m 2 120.m even 2 1
2880.2.f.o 2 24.h odd 2 1
2880.2.f.o 2 120.i odd 2 1
3840.2.d.a 2 16.f odd 4 1
3840.2.d.a 2 80.k odd 4 1
3840.2.d.p 2 16.e even 4 1
3840.2.d.p 2 80.q even 4 1
3840.2.d.q 2 16.e even 4 1
3840.2.d.q 2 80.q even 4 1
3840.2.d.bf 2 16.f odd 4 1
3840.2.d.bf 2 80.k odd 4 1
4800.2.a.c 1 40.k even 4 1
4800.2.a.e 1 40.i odd 4 1
4800.2.a.cp 1 40.i odd 4 1
4800.2.a.cr 1 40.k even 4 1
7200.2.a.a 1 15.e even 4 1
7200.2.a.c 1 60.l odd 4 1
7200.2.a.by 1 60.l odd 4 1
7200.2.a.ca 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(480, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11}$$ T11 $$T_{19} + 8$$ T19 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2}$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 144$$
$59$ $$T^{2}$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 16)^{2}$$
$73$ $$T^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 64$$
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