# Properties

 Label 480.4.a.g Level $480$ Weight $4$ Character orbit 480.a Self dual yes Analytic conductor $28.321$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 5 q^{5} - 8 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 5 * q^5 - 8 * q^7 + 9 * q^9 $$q + 3 q^{3} - 5 q^{5} - 8 q^{7} + 9 q^{9} + 4 q^{11} - 6 q^{13} - 15 q^{15} - 2 q^{17} - 16 q^{19} - 24 q^{21} - 60 q^{23} + 25 q^{25} + 27 q^{27} - 142 q^{29} - 176 q^{31} + 12 q^{33} + 40 q^{35} - 214 q^{37} - 18 q^{39} - 278 q^{41} - 68 q^{43} - 45 q^{45} + 116 q^{47} - 279 q^{49} - 6 q^{51} - 350 q^{53} - 20 q^{55} - 48 q^{57} + 684 q^{59} - 394 q^{61} - 72 q^{63} + 30 q^{65} + 108 q^{67} - 180 q^{69} - 96 q^{71} - 398 q^{73} + 75 q^{75} - 32 q^{77} + 136 q^{79} + 81 q^{81} + 436 q^{83} + 10 q^{85} - 426 q^{87} - 750 q^{89} + 48 q^{91} - 528 q^{93} + 80 q^{95} + 82 q^{97} + 36 q^{99}+O(q^{100})$$ q + 3 * q^3 - 5 * q^5 - 8 * q^7 + 9 * q^9 + 4 * q^11 - 6 * q^13 - 15 * q^15 - 2 * q^17 - 16 * q^19 - 24 * q^21 - 60 * q^23 + 25 * q^25 + 27 * q^27 - 142 * q^29 - 176 * q^31 + 12 * q^33 + 40 * q^35 - 214 * q^37 - 18 * q^39 - 278 * q^41 - 68 * q^43 - 45 * q^45 + 116 * q^47 - 279 * q^49 - 6 * q^51 - 350 * q^53 - 20 * q^55 - 48 * q^57 + 684 * q^59 - 394 * q^61 - 72 * q^63 + 30 * q^65 + 108 * q^67 - 180 * q^69 - 96 * q^71 - 398 * q^73 + 75 * q^75 - 32 * q^77 + 136 * q^79 + 81 * q^81 + 436 * q^83 + 10 * q^85 - 426 * q^87 - 750 * q^89 + 48 * q^91 - 528 * q^93 + 80 * q^95 + 82 * q^97 + 36 * 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 −8.00000 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.g yes 1
3.b odd 2 1 1440.4.a.m 1
4.b odd 2 1 480.4.a.b 1
5.b even 2 1 2400.4.a.f 1
8.b even 2 1 960.4.a.m 1
8.d odd 2 1 960.4.a.bh 1
12.b even 2 1 1440.4.a.p 1
20.d odd 2 1 2400.4.a.q 1

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

## Hecke kernels

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

 $$T_{7} + 8$$ T7 + 8 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T + 8$$
$11$ $$T - 4$$
$13$ $$T + 6$$
$17$ $$T + 2$$
$19$ $$T + 16$$
$23$ $$T + 60$$
$29$ $$T + 142$$
$31$ $$T + 176$$
$37$ $$T + 214$$
$41$ $$T + 278$$
$43$ $$T + 68$$
$47$ $$T - 116$$
$53$ $$T + 350$$
$59$ $$T - 684$$
$61$ $$T + 394$$
$67$ $$T - 108$$
$71$ $$T + 96$$
$73$ $$T + 398$$
$79$ $$T - 136$$
$83$ $$T - 436$$
$89$ $$T + 750$$
$97$ $$T - 82$$