# Properties

 Label 48.19.e.a Level $48$ Weight $19$ Character orbit 48.e Self dual yes Analytic conductor $98.585$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,19,Mod(17,48)]

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([0, 0, 1]))

N = Newforms(chi, 19, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 19);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$19$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (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: yes Analytic conductor: $$98.5853461007$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) 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)

 $$f(q)$$ $$=$$ $$q + 19683 q^{3} - 77549186 q^{7} + 387420489 q^{9}+O(q^{10})$$ q + 19683 * q^3 - 77549186 * q^7 + 387420489 * q^9 $$q + 19683 q^{3} - 77549186 q^{7} + 387420489 q^{9} - 7197541846 q^{13} - 308559680858 q^{19} - 1526400628038 q^{21} + 3814697265625 q^{25} + 7625597484987 q^{27} + 50018992173358 q^{31} - 23240947030054 q^{37} - 141669216154818 q^{39} + 730385642547286 q^{43} + 43\!\cdots\!47 q^{49}+ \cdots + 14\!\cdots\!66 q^{97}+O(q^{100})$$ q + 19683 * q^3 - 77549186 * q^7 + 387420489 * q^9 - 7197541846 * q^13 - 308559680858 * q^19 - 1526400628038 * q^21 + 3814697265625 * q^25 + 7625597484987 * q^27 + 50018992173358 * q^31 - 23240947030054 * q^37 - 141669216154818 * q^39 + 730385642547286 * q^43 + 4385462651352147 * q^49 - 6073380198328014 * q^57 - 9487161099916918 * q^61 - 30044143561671954 * q^63 + 41747295001607494 * q^67 - 29908998244279726 * q^73 + 75084686279296875 * q^75 - 140655567501204338 * q^79 + 150094635296999121 * q^81 + 558163511358237356 * q^91 + 984523822948205514 * q^93 + 140873967896062466 * q^97

## Character values

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

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$1$$ $$0$$ $$0$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0
0 19683.0 0 0 0 −7.75492e7 0 3.87420e8 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 $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.19.e.a 1
3.b odd 2 1 CM 48.19.e.a 1
4.b odd 2 1 3.19.b.a 1
12.b even 2 1 3.19.b.a 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{19}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 19683$$
$5$ $$T$$
$7$ $$T + 77549186$$
$11$ $$T$$
$13$ $$T + 7197541846$$
$17$ $$T$$
$19$ $$T + 308559680858$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 50018992173358$$
$37$ $$T + 23240947030054$$
$41$ $$T$$
$43$ $$T - 730385642547286$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 9487161099916918$$
$67$ $$T - 41\!\cdots\!94$$
$71$ $$T$$
$73$ $$T + 29\!\cdots\!26$$
$79$ $$T + 14\!\cdots\!38$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 14\!\cdots\!66$$