# Properties

 Label 1216.3.e.a Level $1216$ Weight $3$ Character orbit 1216.e Self dual yes Analytic conductor $33.134$ Analytic rank $0$ Dimension $1$ CM discriminant -19 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(1025,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1216.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$33.1336001462$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) 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)

 $$f(q)$$ $$=$$ $$q + 9 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10})$$ q + 9 * q^5 - 5 * q^7 + 9 * q^9 $$q + 9 q^{5} - 5 q^{7} + 9 q^{9} - 3 q^{11} + 15 q^{17} + 19 q^{19} - 30 q^{23} + 56 q^{25} - 45 q^{35} + 85 q^{43} + 81 q^{45} + 75 q^{47} - 24 q^{49} - 27 q^{55} - 103 q^{61} - 45 q^{63} - 25 q^{73} + 15 q^{77} + 81 q^{81} - 90 q^{83} + 135 q^{85} + 171 q^{95} - 27 q^{99}+O(q^{100})$$ q + 9 * q^5 - 5 * q^7 + 9 * q^9 - 3 * q^11 + 15 * q^17 + 19 * q^19 - 30 * q^23 + 56 * q^25 - 45 * q^35 + 85 * q^43 + 81 * q^45 + 75 * q^47 - 24 * q^49 - 27 * q^55 - 103 * q^61 - 45 * q^63 - 25 * q^73 + 15 * q^77 + 81 * q^81 - 90 * q^83 + 135 * q^85 + 171 * q^95 - 27 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 0
0 0 0 9.00000 0 −5.00000 0 9.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.3.e.a 1
4.b odd 2 1 1216.3.e.b 1
8.b even 2 1 19.3.b.a 1
8.d odd 2 1 304.3.e.a 1
19.b odd 2 1 CM 1216.3.e.a 1
24.f even 2 1 2736.3.o.a 1
24.h odd 2 1 171.3.c.a 1
40.f even 2 1 475.3.c.a 1
40.i odd 4 2 475.3.d.a 2
76.d even 2 1 1216.3.e.b 1
152.b even 2 1 304.3.e.a 1
152.g odd 2 1 19.3.b.a 1
152.l odd 6 2 361.3.d.a 2
152.p even 6 2 361.3.d.a 2
152.s odd 18 6 361.3.f.a 6
152.t even 18 6 361.3.f.a 6
456.l odd 2 1 2736.3.o.a 1
456.p even 2 1 171.3.c.a 1
760.b odd 2 1 475.3.c.a 1
760.t even 4 2 475.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 8.b even 2 1
19.3.b.a 1 152.g odd 2 1
171.3.c.a 1 24.h odd 2 1
171.3.c.a 1 456.p even 2 1
304.3.e.a 1 8.d odd 2 1
304.3.e.a 1 152.b even 2 1
361.3.d.a 2 152.l odd 6 2
361.3.d.a 2 152.p even 6 2
361.3.f.a 6 152.s odd 18 6
361.3.f.a 6 152.t even 18 6
475.3.c.a 1 40.f even 2 1
475.3.c.a 1 760.b odd 2 1
475.3.d.a 2 40.i odd 4 2
475.3.d.a 2 760.t even 4 2
1216.3.e.a 1 1.a even 1 1 trivial
1216.3.e.a 1 19.b odd 2 1 CM
1216.3.e.b 1 4.b odd 2 1
1216.3.e.b 1 76.d even 2 1
2736.3.o.a 1 24.f even 2 1
2736.3.o.a 1 456.l odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5} - 9$$ T5 - 9 $$T_{7} + 5$$ T7 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 9$$
$7$ $$T + 5$$
$11$ $$T + 3$$
$13$ $$T$$
$17$ $$T - 15$$
$19$ $$T - 19$$
$23$ $$T + 30$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T - 85$$
$47$ $$T - 75$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 103$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T + 25$$
$79$ $$T$$
$83$ $$T + 90$$
$89$ $$T$$
$97$ $$T$$