# Properties

 Label 48.22.c.a Level $48$ Weight $22$ Character orbit 48.c Analytic conductor $134.149$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(47,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 48.c (of order $$2$$, degree $$1$$, 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: $$134.149125258$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = 9\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 6561 \beta q^{3} + 63207986 \beta q^{7} - 10460353203 q^{9} +O(q^{10})$$ q - 6561*b * q^3 + 63207986*b * q^7 - 10460353203 * q^9 $$q - 6561 \beta q^{3} + 63207986 \beta q^{7} - 10460353203 q^{9} + 370076825230 q^{13} - 2560945382530 \beta q^{19} + 100773945863478 q^{21} + 476837158203125 q^{25} + 68630377364883 \beta q^{27} - 80612910215030 \beta q^{31} - 57\!\cdots\!90 q^{37} + \cdots + 11\!\cdots\!30 q^{97} +O(q^{100})$$ q - 6561*b * q^3 + 63207986*b * q^7 - 10460353203 * q^9 + 370076825230 * q^13 - 2560945382530*b * q^19 + 100773945863478 * q^21 + 476837158203125 * q^25 + 68630377364883*b * q^27 - 80612910215030*b * q^31 - 57776323439003290 * q^37 - 2428074050334030*b * q^39 + 6405893160250598*b * q^43 - 412299763001531621 * q^49 - 4082974125111377190 * q^57 - 10860691764464843938 * q^61 - 661177858810279158*b * q^63 + 1861754937027440574*b * q^67 + 39098623343480501290 * q^73 - 3128528594970703125*b * q^75 + 580541259680729690*b * q^79 + 109418989131512359209 * q^81 + 23391810788062286780*b * q^91 - 128523016852757274690 * q^93 + 1132068302544444351730 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 20920706406 q^{9}+O(q^{10})$$ 2 * q - 20920706406 * q^9 $$2 q - 20920706406 q^{9} + 740153650460 q^{13} + 201547891726956 q^{21} + 953674316406250 q^{25} - 11\!\cdots\!80 q^{37}+ \cdots + 22\!\cdots\!60 q^{97}+O(q^{100})$$ 2 * q - 20920706406 * q^9 + 740153650460 * q^13 + 201547891726956 * q^21 + 953674316406250 * q^25 - 115552646878006580 * q^37 - 824599526003063242 * q^49 - 8165948250222754380 * q^57 - 21721383528929687876 * q^61 + 78197246686961002580 * q^73 + 218837978263024718418 * q^81 - 257046033705514549380 * q^93 + 2264136605088888703460 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 102276.i 0 0 0 9.85315e8i 0 −1.04604e10 0
47.2 0 102276.i 0 0 0 9.85315e8i 0 −1.04604e10 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.22.c.a 2
3.b odd 2 1 CM 48.22.c.a 2
4.b odd 2 1 inner 48.22.c.a 2
12.b even 2 1 inner 48.22.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.22.c.a 2 1.a even 1 1 trivial
48.22.c.a 2 3.b odd 2 1 CM
48.22.c.a 2 4.b odd 2 1 inner
48.22.c.a 2 12.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{22}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 10460353203$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 97\!\cdots\!28$$
$11$ $$T^{2}$$
$13$ $$(T - 370076825230)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 15\!\cdots\!00$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 15\!\cdots\!00$$
$37$ $$(T + 57\!\cdots\!90)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 99\!\cdots\!72$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 10\!\cdots\!38)^{2}$$
$67$ $$T^{2} + 84\!\cdots\!68$$
$71$ $$T^{2}$$
$73$ $$(T - 39\!\cdots\!90)^{2}$$
$79$ $$T^{2} + 81\!\cdots\!00$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 11\!\cdots\!30)^{2}$$