# Properties

 Label 48.22.c.c Level $48$ Weight $22$ Character orbit 48.c Analytic conductor $134.149$ Analytic rank $0$ Dimension $28$ CM no 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: $$28$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 109254828 q^{9}+O(q^{10})$$ 28 * q + 109254828 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 109254828 q^{9} + 285248048392 q^{13} + 247146979606248 q^{21} - 31\!\cdots\!84 q^{25}+ \cdots + 16\!\cdots\!20 q^{97}+O(q^{100})$$ 28 * q + 109254828 * q^9 + 285248048392 * q^13 + 247146979606248 * q^21 - 3168527686741684 * q^25 - 7429429202649408 * q^33 + 33089121948065192 * q^37 - 121806830473453440 * q^45 - 569345069725368716 * q^49 + 10547854228195302552 * q^57 - 8280751575178739512 * q^61 - 22754055866538936960 * q^69 - 98548109928833394536 * q^73 + 563980976909609989500 * q^81 - 93841573689641488896 * q^85 - 1293830212233248374968 * q^93 + 1670728237822569097720 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 0 −102247. 2435.85i 0 1.44607e7i 0 1.26694e9i 0 1.04485e10 + 4.98117e8i 0
47.2 0 −102247. + 2435.85i 0 1.44607e7i 0 1.26694e9i 0 1.04485e10 4.98117e8i 0
47.3 0 −102214. 3563.40i 0 3.58965e7i 0 2.03455e8i 0 1.04350e10 + 7.28456e8i 0
47.4 0 −102214. + 3563.40i 0 3.58965e7i 0 2.03455e8i 0 1.04350e10 7.28456e8i 0
47.5 0 −87580.8 52820.0i 0 2.44170e7i 0 1.13340e9i 0 4.88046e9 + 9.25203e9i 0
47.6 0 −87580.8 + 52820.0i 0 2.44170e7i 0 1.13340e9i 0 4.88046e9 9.25203e9i 0
47.7 0 −66724.2 77512.8i 0 3.35931e6i 0 5.38569e8i 0 −1.55613e9 + 1.03440e10i 0
47.8 0 −66724.2 + 77512.8i 0 3.35931e6i 0 5.38569e8i 0 −1.55613e9 1.03440e10i 0
47.9 0 −51369.6 88439.3i 0 1.40544e6i 0 5.11154e8i 0 −5.18267e9 + 9.08619e9i 0
47.10 0 −51369.6 + 88439.3i 0 1.40544e6i 0 5.11154e8i 0 −5.18267e9 9.08619e9i 0
47.11 0 −31002.5 97463.8i 0 2.82711e7i 0 1.02999e8i 0 −8.53805e9 + 6.04324e9i 0
47.12 0 −31002.5 + 97463.8i 0 2.82711e7i 0 1.02999e8i 0 −8.53805e9 6.04324e9i 0
47.13 0 −554.307 102274.i 0 3.49802e7i 0 7.47725e8i 0 −1.04597e10 + 1.13383e8i 0
47.14 0 −554.307 + 102274.i 0 3.49802e7i 0 7.47725e8i 0 −1.04597e10 1.13383e8i 0
47.15 0 554.307 102274.i 0 3.49802e7i 0 7.47725e8i 0 −1.04597e10 1.13383e8i 0
47.16 0 554.307 + 102274.i 0 3.49802e7i 0 7.47725e8i 0 −1.04597e10 + 1.13383e8i 0
47.17 0 31002.5 97463.8i 0 2.82711e7i 0 1.02999e8i 0 −8.53805e9 6.04324e9i 0
47.18 0 31002.5 + 97463.8i 0 2.82711e7i 0 1.02999e8i 0 −8.53805e9 + 6.04324e9i 0
47.19 0 51369.6 88439.3i 0 1.40544e6i 0 5.11154e8i 0 −5.18267e9 9.08619e9i 0
47.20 0 51369.6 + 88439.3i 0 1.40544e6i 0 5.11154e8i 0 −5.18267e9 + 9.08619e9i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.28 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 inner
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.c 28
3.b odd 2 1 inner 48.22.c.c 28
4.b odd 2 1 inner 48.22.c.c 28
12.b even 2 1 inner 48.22.c.c 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.22.c.c 28 1.a even 1 1 trivial
48.22.c.c 28 3.b odd 2 1 inner
48.22.c.c 28 4.b odd 2 1 inner
48.22.c.c 28 12.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{14} + \cdots + 35\!\cdots\!00$$ acting on $$S_{22}^{\mathrm{new}}(48, [\chi])$$.