# Properties

 Label 240.4.a.e Level $240$ Weight $4$ Character orbit 240.a Self dual yes Analytic conductor $14.160$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 5 q^{5} + 24 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 5 * q^5 + 24 * q^7 + 9 * q^9 $$q - 3 q^{3} + 5 q^{5} + 24 q^{7} + 9 q^{9} - 52 q^{11} + 22 q^{13} - 15 q^{15} - 14 q^{17} + 20 q^{19} - 72 q^{21} + 168 q^{23} + 25 q^{25} - 27 q^{27} + 230 q^{29} + 288 q^{31} + 156 q^{33} + 120 q^{35} - 34 q^{37} - 66 q^{39} + 122 q^{41} + 188 q^{43} + 45 q^{45} - 256 q^{47} + 233 q^{49} + 42 q^{51} - 338 q^{53} - 260 q^{55} - 60 q^{57} - 100 q^{59} + 742 q^{61} + 216 q^{63} + 110 q^{65} + 84 q^{67} - 504 q^{69} + 328 q^{71} - 38 q^{73} - 75 q^{75} - 1248 q^{77} + 240 q^{79} + 81 q^{81} - 1212 q^{83} - 70 q^{85} - 690 q^{87} + 330 q^{89} + 528 q^{91} - 864 q^{93} + 100 q^{95} + 866 q^{97} - 468 q^{99}+O(q^{100})$$ q - 3 * q^3 + 5 * q^5 + 24 * q^7 + 9 * q^9 - 52 * q^11 + 22 * q^13 - 15 * q^15 - 14 * q^17 + 20 * q^19 - 72 * q^21 + 168 * q^23 + 25 * q^25 - 27 * q^27 + 230 * q^29 + 288 * q^31 + 156 * q^33 + 120 * q^35 - 34 * q^37 - 66 * q^39 + 122 * q^41 + 188 * q^43 + 45 * q^45 - 256 * q^47 + 233 * q^49 + 42 * q^51 - 338 * q^53 - 260 * q^55 - 60 * q^57 - 100 * q^59 + 742 * q^61 + 216 * q^63 + 110 * q^65 + 84 * q^67 - 504 * q^69 + 328 * q^71 - 38 * q^73 - 75 * q^75 - 1248 * q^77 + 240 * q^79 + 81 * q^81 - 1212 * q^83 - 70 * q^85 - 690 * q^87 + 330 * q^89 + 528 * q^91 - 864 * q^93 + 100 * q^95 + 866 * q^97 - 468 * 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 24.0000 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.e 1
3.b odd 2 1 720.4.a.n 1
4.b odd 2 1 15.4.a.a 1
5.b even 2 1 1200.4.a.t 1
5.c odd 4 2 1200.4.f.b 2
8.b even 2 1 960.4.a.ba 1
8.d odd 2 1 960.4.a.b 1
12.b even 2 1 45.4.a.c 1
20.d odd 2 1 75.4.a.b 1
20.e even 4 2 75.4.b.b 2
28.d even 2 1 735.4.a.e 1
36.f odd 6 2 405.4.e.g 2
36.h even 6 2 405.4.e.i 2
44.c even 2 1 1815.4.a.e 1
60.h even 2 1 225.4.a.f 1
60.l odd 4 2 225.4.b.e 2
84.h odd 2 1 2205.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 4.b odd 2 1
45.4.a.c 1 12.b even 2 1
75.4.a.b 1 20.d odd 2 1
75.4.b.b 2 20.e even 4 2
225.4.a.f 1 60.h even 2 1
225.4.b.e 2 60.l odd 4 2
240.4.a.e 1 1.a even 1 1 trivial
405.4.e.g 2 36.f odd 6 2
405.4.e.i 2 36.h even 6 2
720.4.a.n 1 3.b odd 2 1
735.4.a.e 1 28.d even 2 1
960.4.a.b 1 8.d odd 2 1
960.4.a.ba 1 8.b even 2 1
1200.4.a.t 1 5.b even 2 1
1200.4.f.b 2 5.c odd 4 2
1815.4.a.e 1 44.c even 2 1
2205.4.a.l 1 84.h odd 2 1

## Hecke kernels

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

 $$T_{7} - 24$$ T7 - 24 $$T_{11} + 52$$ T11 + 52

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 5$$
$7$ $$T - 24$$
$11$ $$T + 52$$
$13$ $$T - 22$$
$17$ $$T + 14$$
$19$ $$T - 20$$
$23$ $$T - 168$$
$29$ $$T - 230$$
$31$ $$T - 288$$
$37$ $$T + 34$$
$41$ $$T - 122$$
$43$ $$T - 188$$
$47$ $$T + 256$$
$53$ $$T + 338$$
$59$ $$T + 100$$
$61$ $$T - 742$$
$67$ $$T - 84$$
$71$ $$T - 328$$
$73$ $$T + 38$$
$79$ $$T - 240$$
$83$ $$T + 1212$$
$89$ $$T - 330$$
$97$ $$T - 866$$