# Properties

 Label 120.4.a.b Level $120$ Weight $4$ Character orbit 120.a Self dual yes Analytic conductor $7.080$ 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] = [120,4,Mod(1,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, 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("120.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 120.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: $$7.08022920069$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 5 q^{5} + 20 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 5 * q^5 + 20 * q^7 + 9 * q^9 $$q - 3 q^{3} - 5 q^{5} + 20 q^{7} + 9 q^{9} - 56 q^{11} - 86 q^{13} + 15 q^{15} - 106 q^{17} + 4 q^{19} - 60 q^{21} + 136 q^{23} + 25 q^{25} - 27 q^{27} - 206 q^{29} - 152 q^{31} + 168 q^{33} - 100 q^{35} + 282 q^{37} + 258 q^{39} - 246 q^{41} + 412 q^{43} - 45 q^{45} + 40 q^{47} + 57 q^{49} + 318 q^{51} - 126 q^{53} + 280 q^{55} - 12 q^{57} + 56 q^{59} - 2 q^{61} + 180 q^{63} + 430 q^{65} - 388 q^{67} - 408 q^{69} - 672 q^{71} + 1170 q^{73} - 75 q^{75} - 1120 q^{77} + 408 q^{79} + 81 q^{81} + 668 q^{83} + 530 q^{85} + 618 q^{87} + 66 q^{89} - 1720 q^{91} + 456 q^{93} - 20 q^{95} - 926 q^{97} - 504 q^{99}+O(q^{100})$$ q - 3 * q^3 - 5 * q^5 + 20 * q^7 + 9 * q^9 - 56 * q^11 - 86 * q^13 + 15 * q^15 - 106 * q^17 + 4 * q^19 - 60 * q^21 + 136 * q^23 + 25 * q^25 - 27 * q^27 - 206 * q^29 - 152 * q^31 + 168 * q^33 - 100 * q^35 + 282 * q^37 + 258 * q^39 - 246 * q^41 + 412 * q^43 - 45 * q^45 + 40 * q^47 + 57 * q^49 + 318 * q^51 - 126 * q^53 + 280 * q^55 - 12 * q^57 + 56 * q^59 - 2 * q^61 + 180 * q^63 + 430 * q^65 - 388 * q^67 - 408 * q^69 - 672 * q^71 + 1170 * q^73 - 75 * q^75 - 1120 * q^77 + 408 * q^79 + 81 * q^81 + 668 * q^83 + 530 * q^85 + 618 * q^87 + 66 * q^89 - 1720 * q^91 + 456 * q^93 - 20 * q^95 - 926 * q^97 - 504 * 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 20.0000 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 120.4.a.b 1
3.b odd 2 1 360.4.a.n 1
4.b odd 2 1 240.4.a.g 1
5.b even 2 1 600.4.a.i 1
5.c odd 4 2 600.4.f.a 2
8.b even 2 1 960.4.a.bj 1
8.d odd 2 1 960.4.a.k 1
12.b even 2 1 720.4.a.q 1
15.d odd 2 1 1800.4.a.f 1
15.e even 4 2 1800.4.f.v 2
20.d odd 2 1 1200.4.a.p 1
20.e even 4 2 1200.4.f.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.b 1 1.a even 1 1 trivial
240.4.a.g 1 4.b odd 2 1
360.4.a.n 1 3.b odd 2 1
600.4.a.i 1 5.b even 2 1
600.4.f.a 2 5.c odd 4 2
720.4.a.q 1 12.b even 2 1
960.4.a.k 1 8.d odd 2 1
960.4.a.bj 1 8.b even 2 1
1200.4.a.p 1 20.d odd 2 1
1200.4.f.t 2 20.e even 4 2
1800.4.a.f 1 15.d odd 2 1
1800.4.f.v 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(120))$$:

 $$T_{7} - 20$$ T7 - 20 $$T_{11} + 56$$ T11 + 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 5$$
$7$ $$T - 20$$
$11$ $$T + 56$$
$13$ $$T + 86$$
$17$ $$T + 106$$
$19$ $$T - 4$$
$23$ $$T - 136$$
$29$ $$T + 206$$
$31$ $$T + 152$$
$37$ $$T - 282$$
$41$ $$T + 246$$
$43$ $$T - 412$$
$47$ $$T - 40$$
$53$ $$T + 126$$
$59$ $$T - 56$$
$61$ $$T + 2$$
$67$ $$T + 388$$
$71$ $$T + 672$$
$73$ $$T - 1170$$
$79$ $$T - 408$$
$83$ $$T - 668$$
$89$ $$T - 66$$
$97$ $$T + 926$$