# Properties

 Label 3072.2.a.l Level $3072$ Weight $2$ Character orbit 3072.a Self dual yes Analytic conductor $24.530$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3072.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: $$24.5300435009$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 768) 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + (\beta_{2} + \beta_1) q^{5} + \beta_{2} q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (b2 + b1) * q^5 + b2 * q^7 + q^9 $$q - q^{3} + (\beta_{2} + \beta_1) q^{5} + \beta_{2} q^{7} + q^{9} + (\beta_{3} - 2) q^{11} + 3 \beta_1 q^{13} + ( - \beta_{2} - \beta_1) q^{15} - \beta_{3} q^{17} + ( - \beta_{3} - 4) q^{19} - \beta_{2} q^{21} + 2 \beta_1 q^{23} + (2 \beta_{3} + 3) q^{25} - q^{27} + (3 \beta_{2} + \beta_1) q^{29} + 3 \beta_{2} q^{31} + ( - \beta_{3} + 2) q^{33} + (\beta_{3} + 6) q^{35} + (2 \beta_{2} - 3 \beta_1) q^{37} - 3 \beta_1 q^{39} + ( - \beta_{3} + 8) q^{41} + \beta_{3} q^{43} + (\beta_{2} + \beta_1) q^{45} - 2 \beta_1 q^{47} - q^{49} + \beta_{3} q^{51} + (\beta_{2} - 5 \beta_1) q^{53} + 4 \beta_1 q^{55} + (\beta_{3} + 4) q^{57} - 4 \beta_{3} q^{59} + ( - 2 \beta_{2} + 3 \beta_1) q^{61} + \beta_{2} q^{63} + (3 \beta_{3} + 6) q^{65} + ( - 2 \beta_{3} - 8) q^{67} - 2 \beta_1 q^{69} + ( - 2 \beta_{2} + 8 \beta_1) q^{71} + 4 q^{73} + ( - 2 \beta_{3} - 3) q^{75} + ( - 2 \beta_{2} + 6 \beta_1) q^{77} - \beta_{2} q^{79} + q^{81} + ( - \beta_{3} - 2) q^{83} + ( - 2 \beta_{2} - 6 \beta_1) q^{85} + ( - 3 \beta_{2} - \beta_1) q^{87} + ( - 2 \beta_{3} + 2) q^{89} + 3 \beta_{3} q^{91} - 3 \beta_{2} q^{93} + ( - 6 \beta_{2} - 10 \beta_1) q^{95} + ( - 2 \beta_{3} + 8) q^{97} + (\beta_{3} - 2) q^{99}+O(q^{100})$$ q - q^3 + (b2 + b1) * q^5 + b2 * q^7 + q^9 + (b3 - 2) * q^11 + 3*b1 * q^13 + (-b2 - b1) * q^15 - b3 * q^17 + (-b3 - 4) * q^19 - b2 * q^21 + 2*b1 * q^23 + (2*b3 + 3) * q^25 - q^27 + (3*b2 + b1) * q^29 + 3*b2 * q^31 + (-b3 + 2) * q^33 + (b3 + 6) * q^35 + (2*b2 - 3*b1) * q^37 - 3*b1 * q^39 + (-b3 + 8) * q^41 + b3 * q^43 + (b2 + b1) * q^45 - 2*b1 * q^47 - q^49 + b3 * q^51 + (b2 - 5*b1) * q^53 + 4*b1 * q^55 + (b3 + 4) * q^57 - 4*b3 * q^59 + (-2*b2 + 3*b1) * q^61 + b2 * q^63 + (3*b3 + 6) * q^65 + (-2*b3 - 8) * q^67 - 2*b1 * q^69 + (-2*b2 + 8*b1) * q^71 + 4 * q^73 + (-2*b3 - 3) * q^75 + (-2*b2 + 6*b1) * q^77 - b2 * q^79 + q^81 + (-b3 - 2) * q^83 + (-2*b2 - 6*b1) * q^85 + (-3*b2 - b1) * q^87 + (-2*b3 + 2) * q^89 + 3*b3 * q^91 - 3*b2 * q^93 + (-6*b2 - 10*b1) * q^95 + (-2*b3 + 8) * q^97 + (b3 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 4 * q^9 $$4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} - 16 q^{19} + 12 q^{25} - 4 q^{27} + 8 q^{33} + 24 q^{35} + 32 q^{41} - 4 q^{49} + 16 q^{57} + 24 q^{65} - 32 q^{67} + 16 q^{73} - 12 q^{75} + 4 q^{81} - 8 q^{83} + 8 q^{89} + 32 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 - 16 * q^19 + 12 * q^25 - 4 * q^27 + 8 * q^33 + 24 * q^35 + 32 * q^41 - 4 * q^49 + 16 * q^57 + 24 * q^65 - 32 * q^67 + 16 * q^73 - 12 * q^75 + 4 * q^81 - 8 * q^83 + 8 * q^89 + 32 * q^97 - 8 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{24} + \zeta_{24}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5\nu$$ -v^3 + 5*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 4$$ 2*v^2 - 4
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 4 ) / 2$$ (b3 + 4) / 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{2} + 5\beta_1 ) / 2$$ (3*b2 + 5*b1) / 2

## 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
 −1.93185 −0.517638 0.517638 1.93185
0 −1.00000 0 −3.86370 0 −2.44949 0 1.00000 0
1.2 0 −1.00000 0 −1.03528 0 −2.44949 0 1.00000 0
1.3 0 −1.00000 0 1.03528 0 2.44949 0 1.00000 0
1.4 0 −1.00000 0 3.86370 0 2.44949 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.2.a.l 4
3.b odd 2 1 9216.2.a.bi 4
4.b odd 2 1 3072.2.a.r 4
8.b even 2 1 3072.2.a.r 4
8.d odd 2 1 inner 3072.2.a.l 4
12.b even 2 1 9216.2.a.bc 4
16.e even 4 2 3072.2.d.h 8
16.f odd 4 2 3072.2.d.h 8
24.f even 2 1 9216.2.a.bi 4
24.h odd 2 1 9216.2.a.bc 4
32.g even 8 2 768.2.j.e 8
32.g even 8 2 768.2.j.f yes 8
32.h odd 8 2 768.2.j.e 8
32.h odd 8 2 768.2.j.f yes 8
96.o even 8 2 2304.2.k.e 8
96.o even 8 2 2304.2.k.l 8
96.p odd 8 2 2304.2.k.e 8
96.p odd 8 2 2304.2.k.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.e 8 32.g even 8 2
768.2.j.e 8 32.h odd 8 2
768.2.j.f yes 8 32.g even 8 2
768.2.j.f yes 8 32.h odd 8 2
2304.2.k.e 8 96.o even 8 2
2304.2.k.e 8 96.p odd 8 2
2304.2.k.l 8 96.o even 8 2
2304.2.k.l 8 96.p odd 8 2
3072.2.a.l 4 1.a even 1 1 trivial
3072.2.a.l 4 8.d odd 2 1 inner
3072.2.a.r 4 4.b odd 2 1
3072.2.a.r 4 8.b even 2 1
3072.2.d.h 8 16.e even 4 2
3072.2.d.h 8 16.f odd 4 2
9216.2.a.bc 4 12.b even 2 1
9216.2.a.bc 4 24.h odd 2 1
9216.2.a.bi 4 3.b odd 2 1
9216.2.a.bi 4 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3072))$$:

 $$T_{5}^{4} - 16T_{5}^{2} + 16$$ T5^4 - 16*T5^2 + 16 $$T_{7}^{2} - 6$$ T7^2 - 6 $$T_{11}^{2} + 4T_{11} - 8$$ T11^2 + 4*T11 - 8 $$T_{19}^{2} + 8T_{19} + 4$$ T19^2 + 8*T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T + 1)^{4}$$
$5$ $$T^{4} - 16T^{2} + 16$$
$7$ $$(T^{2} - 6)^{2}$$
$11$ $$(T^{2} + 4 T - 8)^{2}$$
$13$ $$(T^{2} - 18)^{2}$$
$17$ $$(T^{2} - 12)^{2}$$
$19$ $$(T^{2} + 8 T + 4)^{2}$$
$23$ $$(T^{2} - 8)^{2}$$
$29$ $$T^{4} - 112T^{2} + 2704$$
$31$ $$(T^{2} - 54)^{2}$$
$37$ $$T^{4} - 84T^{2} + 36$$
$41$ $$(T^{2} - 16 T + 52)^{2}$$
$43$ $$(T^{2} - 12)^{2}$$
$47$ $$(T^{2} - 8)^{2}$$
$53$ $$T^{4} - 112T^{2} + 1936$$
$59$ $$(T^{2} - 192)^{2}$$
$61$ $$T^{4} - 84T^{2} + 36$$
$67$ $$(T^{2} + 16 T + 16)^{2}$$
$71$ $$T^{4} - 304 T^{2} + 10816$$
$73$ $$(T - 4)^{4}$$
$79$ $$(T^{2} - 6)^{2}$$
$83$ $$(T^{2} + 4 T - 8)^{2}$$
$89$ $$(T^{2} - 4 T - 44)^{2}$$
$97$ $$(T^{2} - 16 T + 16)^{2}$$