# Properties

 Label 3840.2.k.z Level $3840$ Weight $2$ Character orbit 3840.k Analytic conductor $30.663$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1921,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3840.1921");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3840 = 2^{8} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3840.k (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$30.6625543762$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - i q^{3} + i q^{5} + 4 q^{7} - q^{9} +O(q^{10})$$ q - i * q^3 + i * q^5 + 4 * q^7 - q^9 $$q - i q^{3} + i q^{5} + 4 q^{7} - q^{9} - 6 i q^{13} + q^{15} - 2 q^{17} - 4 i q^{19} - 4 i q^{21} - 8 q^{23} - q^{25} + i q^{27} - 6 i q^{29} + 4 i q^{35} + 6 i q^{37} - 6 q^{39} - 10 q^{41} - 4 i q^{43} - i q^{45} - 8 q^{47} + 9 q^{49} + 2 i q^{51} - 10 i q^{53} - 4 q^{57} + 6 i q^{61} - 4 q^{63} + 6 q^{65} + 4 i q^{67} + 8 i q^{69} + 14 q^{73} + i q^{75} - 16 q^{79} + q^{81} - 12 i q^{83} - 2 i q^{85} - 6 q^{87} - 2 q^{89} - 24 i q^{91} + 4 q^{95} + 2 q^{97} +O(q^{100})$$ q - i * q^3 + i * q^5 + 4 * q^7 - q^9 - 6*i * q^13 + q^15 - 2 * q^17 - 4*i * q^19 - 4*i * q^21 - 8 * q^23 - q^25 + i * q^27 - 6*i * q^29 + 4*i * q^35 + 6*i * q^37 - 6 * q^39 - 10 * q^41 - 4*i * q^43 - i * q^45 - 8 * q^47 + 9 * q^49 + 2*i * q^51 - 10*i * q^53 - 4 * q^57 + 6*i * q^61 - 4 * q^63 + 6 * q^65 + 4*i * q^67 + 8*i * q^69 + 14 * q^73 + i * q^75 - 16 * q^79 + q^81 - 12*i * q^83 - 2*i * q^85 - 6 * q^87 - 2 * q^89 - 24*i * q^91 + 4 * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 8 * q^7 - 2 * q^9 $$2 q + 8 q^{7} - 2 q^{9} + 2 q^{15} - 4 q^{17} - 16 q^{23} - 2 q^{25} - 12 q^{39} - 20 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{57} - 8 q^{63} + 12 q^{65} + 28 q^{73} - 32 q^{79} + 2 q^{81} - 12 q^{87} - 4 q^{89} + 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 8 * q^7 - 2 * q^9 + 2 * q^15 - 4 * q^17 - 16 * q^23 - 2 * q^25 - 12 * q^39 - 20 * q^41 - 16 * q^47 + 18 * q^49 - 8 * q^57 - 8 * q^63 + 12 * q^65 + 28 * q^73 - 32 * q^79 + 2 * q^81 - 12 * q^87 - 4 * q^89 + 8 * q^95 + 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$1537$$ $$2561$$ $$2821$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1921.1
 1.00000i − 1.00000i
0 1.00000i 0 1.00000i 0 4.00000 0 −1.00000 0
1921.2 0 1.00000i 0 1.00000i 0 4.00000 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.2.k.z 2
4.b odd 2 1 3840.2.k.a 2
8.b even 2 1 inner 3840.2.k.z 2
8.d odd 2 1 3840.2.k.a 2
16.e even 4 1 240.2.a.a 1
16.e even 4 1 960.2.a.n 1
16.f odd 4 1 120.2.a.a 1
16.f odd 4 1 960.2.a.g 1
48.i odd 4 1 720.2.a.f 1
48.i odd 4 1 2880.2.a.b 1
48.k even 4 1 360.2.a.e 1
48.k even 4 1 2880.2.a.r 1
80.i odd 4 1 1200.2.f.f 2
80.i odd 4 1 4800.2.f.n 2
80.j even 4 1 600.2.f.c 2
80.j even 4 1 4800.2.f.u 2
80.k odd 4 1 600.2.a.a 1
80.k odd 4 1 4800.2.a.bl 1
80.q even 4 1 1200.2.a.r 1
80.q even 4 1 4800.2.a.bh 1
80.s even 4 1 600.2.f.c 2
80.s even 4 1 4800.2.f.u 2
80.t odd 4 1 1200.2.f.f 2
80.t odd 4 1 4800.2.f.n 2
112.j even 4 1 5880.2.a.p 1
144.u even 12 2 3240.2.q.a 2
144.v odd 12 2 3240.2.q.m 2
240.t even 4 1 1800.2.a.c 1
240.z odd 4 1 1800.2.f.g 2
240.bb even 4 1 3600.2.f.l 2
240.bd odd 4 1 1800.2.f.g 2
240.bf even 4 1 3600.2.f.l 2
240.bm odd 4 1 3600.2.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 16.f odd 4 1
240.2.a.a 1 16.e even 4 1
360.2.a.e 1 48.k even 4 1
600.2.a.a 1 80.k odd 4 1
600.2.f.c 2 80.j even 4 1
600.2.f.c 2 80.s even 4 1
720.2.a.f 1 48.i odd 4 1
960.2.a.g 1 16.f odd 4 1
960.2.a.n 1 16.e even 4 1
1200.2.a.r 1 80.q even 4 1
1200.2.f.f 2 80.i odd 4 1
1200.2.f.f 2 80.t odd 4 1
1800.2.a.c 1 240.t even 4 1
1800.2.f.g 2 240.z odd 4 1
1800.2.f.g 2 240.bd odd 4 1
2880.2.a.b 1 48.i odd 4 1
2880.2.a.r 1 48.k even 4 1
3240.2.q.a 2 144.u even 12 2
3240.2.q.m 2 144.v odd 12 2
3600.2.a.bo 1 240.bm odd 4 1
3600.2.f.l 2 240.bb even 4 1
3600.2.f.l 2 240.bf even 4 1
3840.2.k.a 2 4.b odd 2 1
3840.2.k.a 2 8.d odd 2 1
3840.2.k.z 2 1.a even 1 1 trivial
3840.2.k.z 2 8.b even 2 1 inner
4800.2.a.bh 1 80.q even 4 1
4800.2.a.bl 1 80.k odd 4 1
4800.2.f.n 2 80.i odd 4 1
4800.2.f.n 2 80.t odd 4 1
4800.2.f.u 2 80.j even 4 1
4800.2.f.u 2 80.s even 4 1
5880.2.a.p 1 112.j even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3840, [\chi])$$:

 $$T_{7} - 4$$ T7 - 4 $$T_{11}$$ T11 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17} + 2$$ T17 + 2 $$T_{23} + 8$$ T23 + 8 $$T_{31}$$ T31 $$T_{47} + 8$$ T47 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 1$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 8)^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 100$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 36$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$(T - 14)^{2}$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 2)^{2}$$
$97$ $$(T - 2)^{2}$$