# Properties

 Label 480.4.a.e Level $480$ Weight $4$ Character orbit 480.a Self dual yes Analytic conductor $28.321$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(1,480)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("480.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$480 = 2^{5} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 480.a (trivial)

## 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: yes Analytic conductor: $$28.3209168028$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 5 q^{5} + 12 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 5 * q^5 + 12 * q^7 + 9 * q^9 $$q - 3 q^{3} + 5 q^{5} + 12 q^{7} + 9 q^{9} + 24 q^{11} + 38 q^{13} - 15 q^{15} - 6 q^{17} - 104 q^{19} - 36 q^{21} - 100 q^{23} + 25 q^{25} - 27 q^{27} + 230 q^{29} + 56 q^{31} - 72 q^{33} + 60 q^{35} + 190 q^{37} - 114 q^{39} + 202 q^{41} + 148 q^{43} + 45 q^{45} - 124 q^{47} - 199 q^{49} + 18 q^{51} + 206 q^{53} + 120 q^{55} + 312 q^{57} + 128 q^{59} + 190 q^{61} + 108 q^{63} + 190 q^{65} + 204 q^{67} + 300 q^{69} + 440 q^{71} + 1210 q^{73} - 75 q^{75} + 288 q^{77} - 816 q^{79} + 81 q^{81} + 1412 q^{83} - 30 q^{85} - 690 q^{87} - 214 q^{89} + 456 q^{91} - 168 q^{93} - 520 q^{95} + 1202 q^{97} + 216 q^{99}+O(q^{100})$$ q - 3 * q^3 + 5 * q^5 + 12 * q^7 + 9 * q^9 + 24 * q^11 + 38 * q^13 - 15 * q^15 - 6 * q^17 - 104 * q^19 - 36 * q^21 - 100 * q^23 + 25 * q^25 - 27 * q^27 + 230 * q^29 + 56 * q^31 - 72 * q^33 + 60 * q^35 + 190 * q^37 - 114 * q^39 + 202 * q^41 + 148 * q^43 + 45 * q^45 - 124 * q^47 - 199 * q^49 + 18 * q^51 + 206 * q^53 + 120 * q^55 + 312 * q^57 + 128 * q^59 + 190 * q^61 + 108 * q^63 + 190 * q^65 + 204 * q^67 + 300 * q^69 + 440 * q^71 + 1210 * q^73 - 75 * q^75 + 288 * q^77 - 816 * q^79 + 81 * q^81 + 1412 * q^83 - 30 * q^85 - 690 * q^87 - 214 * q^89 + 456 * q^91 - 168 * q^93 - 520 * q^95 + 1202 * q^97 + 216 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −3.00000 0 5.00000 0 12.0000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.4.a.e 1
3.b odd 2 1 1440.4.a.g 1
4.b odd 2 1 480.4.a.j yes 1
5.b even 2 1 2400.4.a.p 1
8.b even 2 1 960.4.a.y 1
8.d odd 2 1 960.4.a.d 1
12.b even 2 1 1440.4.a.d 1
20.d odd 2 1 2400.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.a.e 1 1.a even 1 1 trivial
480.4.a.j yes 1 4.b odd 2 1
960.4.a.d 1 8.d odd 2 1
960.4.a.y 1 8.b even 2 1
1440.4.a.d 1 12.b even 2 1
1440.4.a.g 1 3.b odd 2 1
2400.4.a.g 1 20.d odd 2 1
2400.4.a.p 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(480))$$:

 $$T_{7} - 12$$ T7 - 12 $$T_{11} - 24$$ T11 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 5$$
$7$ $$T - 12$$
$11$ $$T - 24$$
$13$ $$T - 38$$
$17$ $$T + 6$$
$19$ $$T + 104$$
$23$ $$T + 100$$
$29$ $$T - 230$$
$31$ $$T - 56$$
$37$ $$T - 190$$
$41$ $$T - 202$$
$43$ $$T - 148$$
$47$ $$T + 124$$
$53$ $$T - 206$$
$59$ $$T - 128$$
$61$ $$T - 190$$
$67$ $$T - 204$$
$71$ $$T - 440$$
$73$ $$T - 1210$$
$79$ $$T + 816$$
$83$ $$T - 1412$$
$89$ $$T + 214$$
$97$ $$T - 1202$$