# Properties

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

# Related objects

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 60) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 5 q^{5} - 32 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 5 * q^5 - 32 * q^7 + 9 * q^9 $$q + 3 q^{3} + 5 q^{5} - 32 q^{7} + 9 q^{9} - 36 q^{11} - 10 q^{13} + 15 q^{15} - 78 q^{17} - 140 q^{19} - 96 q^{21} + 192 q^{23} + 25 q^{25} + 27 q^{27} + 6 q^{29} + 16 q^{31} - 108 q^{33} - 160 q^{35} - 34 q^{37} - 30 q^{39} - 390 q^{41} + 52 q^{43} + 45 q^{45} - 408 q^{47} + 681 q^{49} - 234 q^{51} - 114 q^{53} - 180 q^{55} - 420 q^{57} - 516 q^{59} - 58 q^{61} - 288 q^{63} - 50 q^{65} + 892 q^{67} + 576 q^{69} + 120 q^{71} - 646 q^{73} + 75 q^{75} + 1152 q^{77} + 1168 q^{79} + 81 q^{81} + 732 q^{83} - 390 q^{85} + 18 q^{87} - 1590 q^{89} + 320 q^{91} + 48 q^{93} - 700 q^{95} + 194 q^{97} - 324 q^{99}+O(q^{100})$$ q + 3 * q^3 + 5 * q^5 - 32 * q^7 + 9 * q^9 - 36 * q^11 - 10 * q^13 + 15 * q^15 - 78 * q^17 - 140 * q^19 - 96 * q^21 + 192 * q^23 + 25 * q^25 + 27 * q^27 + 6 * q^29 + 16 * q^31 - 108 * q^33 - 160 * q^35 - 34 * q^37 - 30 * q^39 - 390 * q^41 + 52 * q^43 + 45 * q^45 - 408 * q^47 + 681 * q^49 - 234 * q^51 - 114 * q^53 - 180 * q^55 - 420 * q^57 - 516 * q^59 - 58 * q^61 - 288 * q^63 - 50 * q^65 + 892 * q^67 + 576 * q^69 + 120 * q^71 - 646 * q^73 + 75 * q^75 + 1152 * q^77 + 1168 * q^79 + 81 * q^81 + 732 * q^83 - 390 * q^85 + 18 * q^87 - 1590 * q^89 + 320 * q^91 + 48 * q^93 - 700 * q^95 + 194 * q^97 - 324 * 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 −32.0000 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.j 1
3.b odd 2 1 720.4.a.c 1
4.b odd 2 1 60.4.a.b 1
5.b even 2 1 1200.4.a.s 1
5.c odd 4 2 1200.4.f.e 2
8.b even 2 1 960.4.a.a 1
8.d odd 2 1 960.4.a.bb 1
12.b even 2 1 180.4.a.c 1
20.d odd 2 1 300.4.a.e 1
20.e even 4 2 300.4.d.d 2
36.f odd 6 2 1620.4.i.a 2
36.h even 6 2 1620.4.i.g 2
60.h even 2 1 900.4.a.b 1
60.l odd 4 2 900.4.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.b 1 4.b odd 2 1
180.4.a.c 1 12.b even 2 1
240.4.a.j 1 1.a even 1 1 trivial
300.4.a.e 1 20.d odd 2 1
300.4.d.d 2 20.e even 4 2
720.4.a.c 1 3.b odd 2 1
900.4.a.b 1 60.h even 2 1
900.4.d.b 2 60.l odd 4 2
960.4.a.a 1 8.b even 2 1
960.4.a.bb 1 8.d odd 2 1
1200.4.a.s 1 5.b even 2 1
1200.4.f.e 2 5.c odd 4 2
1620.4.i.a 2 36.f odd 6 2
1620.4.i.g 2 36.h even 6 2

## Hecke kernels

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

 $$T_{7} + 32$$ T7 + 32 $$T_{11} + 36$$ T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 5$$
$7$ $$T + 32$$
$11$ $$T + 36$$
$13$ $$T + 10$$
$17$ $$T + 78$$
$19$ $$T + 140$$
$23$ $$T - 192$$
$29$ $$T - 6$$
$31$ $$T - 16$$
$37$ $$T + 34$$
$41$ $$T + 390$$
$43$ $$T - 52$$
$47$ $$T + 408$$
$53$ $$T + 114$$
$59$ $$T + 516$$
$61$ $$T + 58$$
$67$ $$T - 892$$
$71$ $$T - 120$$
$73$ $$T + 646$$
$79$ $$T - 1168$$
$83$ $$T - 732$$
$89$ $$T + 1590$$
$97$ $$T - 194$$