# Properties

 Label 240.4.a.g Level $240$ Weight $4$ Character orbit 240.a Self dual yes Analytic conductor $14.160$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,4,Mod(1,240)]

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, 0]))

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

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

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 240.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: $$14.1604584014$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) 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 240.4.a.g 1
3.b odd 2 1 720.4.a.q 1
4.b odd 2 1 120.4.a.b 1
5.b even 2 1 1200.4.a.p 1
5.c odd 4 2 1200.4.f.t 2
8.b even 2 1 960.4.a.k 1
8.d odd 2 1 960.4.a.bj 1
12.b even 2 1 360.4.a.n 1
20.d odd 2 1 600.4.a.i 1
20.e even 4 2 600.4.f.a 2
60.h even 2 1 1800.4.a.f 1
60.l odd 4 2 1800.4.f.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.b 1 4.b odd 2 1
240.4.a.g 1 1.a even 1 1 trivial
360.4.a.n 1 12.b even 2 1
600.4.a.i 1 20.d odd 2 1
600.4.f.a 2 20.e even 4 2
720.4.a.q 1 3.b odd 2 1
960.4.a.k 1 8.b even 2 1
960.4.a.bj 1 8.d odd 2 1
1200.4.a.p 1 5.b even 2 1
1200.4.f.t 2 5.c odd 4 2
1800.4.a.f 1 60.h even 2 1
1800.4.f.v 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(240))$$:

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

## Hecke characteristic polynomials

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