# Properties

 Label 240.4.a.h 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 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 9 * q^9 $$q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 9 q^{9} - 72 q^{11} - 6 q^{13} - 15 q^{15} + 38 q^{17} - 52 q^{19} - 12 q^{21} - 152 q^{23} + 25 q^{25} + 27 q^{27} - 78 q^{29} - 120 q^{31} - 216 q^{33} + 20 q^{35} - 150 q^{37} - 18 q^{39} + 362 q^{41} + 484 q^{43} - 45 q^{45} - 280 q^{47} - 327 q^{49} + 114 q^{51} - 670 q^{53} + 360 q^{55} - 156 q^{57} - 696 q^{59} + 222 q^{61} - 36 q^{63} + 30 q^{65} + 4 q^{67} - 456 q^{69} - 96 q^{71} + 178 q^{73} + 75 q^{75} + 288 q^{77} + 632 q^{79} + 81 q^{81} + 612 q^{83} - 190 q^{85} - 234 q^{87} + 994 q^{89} + 24 q^{91} - 360 q^{93} + 260 q^{95} + 1634 q^{97} - 648 q^{99}+O(q^{100})$$ q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 9 * q^9 - 72 * q^11 - 6 * q^13 - 15 * q^15 + 38 * q^17 - 52 * q^19 - 12 * q^21 - 152 * q^23 + 25 * q^25 + 27 * q^27 - 78 * q^29 - 120 * q^31 - 216 * q^33 + 20 * q^35 - 150 * q^37 - 18 * q^39 + 362 * q^41 + 484 * q^43 - 45 * q^45 - 280 * q^47 - 327 * q^49 + 114 * q^51 - 670 * q^53 + 360 * q^55 - 156 * q^57 - 696 * q^59 + 222 * q^61 - 36 * q^63 + 30 * q^65 + 4 * q^67 - 456 * q^69 - 96 * q^71 + 178 * q^73 + 75 * q^75 + 288 * q^77 + 632 * q^79 + 81 * q^81 + 612 * q^83 - 190 * q^85 - 234 * q^87 + 994 * q^89 + 24 * q^91 - 360 * q^93 + 260 * q^95 + 1634 * q^97 - 648 * 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 −4.00000 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 240.4.a.h 1
3.b odd 2 1 720.4.a.v 1
4.b odd 2 1 120.4.a.a 1
5.b even 2 1 1200.4.a.k 1
5.c odd 4 2 1200.4.f.a 2
8.b even 2 1 960.4.a.o 1
8.d odd 2 1 960.4.a.bf 1
12.b even 2 1 360.4.a.l 1
20.d odd 2 1 600.4.a.l 1
20.e even 4 2 600.4.f.i 2
60.h even 2 1 1800.4.a.n 1
60.l odd 4 2 1800.4.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.a 1 4.b odd 2 1
240.4.a.h 1 1.a even 1 1 trivial
360.4.a.l 1 12.b even 2 1
600.4.a.l 1 20.d odd 2 1
600.4.f.i 2 20.e even 4 2
720.4.a.v 1 3.b odd 2 1
960.4.a.o 1 8.b even 2 1
960.4.a.bf 1 8.d odd 2 1
1200.4.a.k 1 5.b even 2 1
1200.4.f.a 2 5.c odd 4 2
1800.4.a.n 1 60.h even 2 1
1800.4.f.a 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(240))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 72$$ T11 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T + 4$$
$11$ $$T + 72$$
$13$ $$T + 6$$
$17$ $$T - 38$$
$19$ $$T + 52$$
$23$ $$T + 152$$
$29$ $$T + 78$$
$31$ $$T + 120$$
$37$ $$T + 150$$
$41$ $$T - 362$$
$43$ $$T - 484$$
$47$ $$T + 280$$
$53$ $$T + 670$$
$59$ $$T + 696$$
$61$ $$T - 222$$
$67$ $$T - 4$$
$71$ $$T + 96$$
$73$ $$T - 178$$
$79$ $$T - 632$$
$83$ $$T - 612$$
$89$ $$T - 994$$
$97$ $$T - 1634$$