# Properties

 Label 1216.2.c.a Level $1216$ Weight $2$ Character orbit 1216.c Analytic conductor $9.710$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(609,1216)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1216.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 3 q^{7} + 2 q^{9}+O(q^{10})$$ q + i * q^3 - 3 * q^7 + 2 * q^9 $$q + i q^{3} - 3 q^{7} + 2 q^{9} - 3 i q^{13} + 3 q^{17} - i q^{19} - 3 i q^{21} + 9 q^{23} + 5 q^{25} + 5 i q^{27} + 9 i q^{29} - 6 q^{31} + 6 i q^{37} + 3 q^{39} + 6 q^{41} + 8 i q^{43} + 2 q^{49} + 3 i q^{51} - 9 i q^{53} + q^{57} + 3 i q^{59} - 6 i q^{61} - 6 q^{63} + 5 i q^{67} + 9 i q^{69} + 11 q^{73} + 5 i q^{75} + 12 q^{79} + q^{81} + 6 i q^{83} - 9 q^{87} + 9 i q^{91} - 6 i q^{93} - 8 q^{97} +O(q^{100})$$ q + i * q^3 - 3 * q^7 + 2 * q^9 - 3*i * q^13 + 3 * q^17 - i * q^19 - 3*i * q^21 + 9 * q^23 + 5 * q^25 + 5*i * q^27 + 9*i * q^29 - 6 * q^31 + 6*i * q^37 + 3 * q^39 + 6 * q^41 + 8*i * q^43 + 2 * q^49 + 3*i * q^51 - 9*i * q^53 + q^57 + 3*i * q^59 - 6*i * q^61 - 6 * q^63 + 5*i * q^67 + 9*i * q^69 + 11 * q^73 + 5*i * q^75 + 12 * q^79 + q^81 + 6*i * q^83 - 9 * q^87 + 9*i * q^91 - 6*i * q^93 - 8 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q - 6 * q^7 + 4 * q^9 $$2 q - 6 q^{7} + 4 q^{9} + 6 q^{17} + 18 q^{23} + 10 q^{25} - 12 q^{31} + 6 q^{39} + 12 q^{41} + 4 q^{49} + 2 q^{57} - 12 q^{63} + 22 q^{73} + 24 q^{79} + 2 q^{81} - 18 q^{87} - 16 q^{97}+O(q^{100})$$ 2 * q - 6 * q^7 + 4 * q^9 + 6 * q^17 + 18 * q^23 + 10 * q^25 - 12 * q^31 + 6 * q^39 + 12 * q^41 + 4 * q^49 + 2 * q^57 - 12 * q^63 + 22 * q^73 + 24 * q^79 + 2 * q^81 - 18 * q^87 - 16 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
609.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 −3.00000 0 2.00000 0
609.2 0 1.00000i 0 0 0 −3.00000 0 2.00000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.c.a 2
4.b odd 2 1 1216.2.c.d yes 2
8.b even 2 1 inner 1216.2.c.a 2
8.d odd 2 1 1216.2.c.d yes 2
16.e even 4 1 4864.2.a.e 1
16.e even 4 1 4864.2.a.m 1
16.f odd 4 1 4864.2.a.d 1
16.f odd 4 1 4864.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.a 2 1.a even 1 1 trivial
1216.2.c.a 2 8.b even 2 1 inner
1216.2.c.d yes 2 4.b odd 2 1
1216.2.c.d yes 2 8.d odd 2 1
4864.2.a.d 1 16.f odd 4 1
4864.2.a.e 1 16.e even 4 1
4864.2.a.l 1 16.f odd 4 1
4864.2.a.m 1 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{5}$$ T5 $$T_{7} + 3$$ T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$(T + 3)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 9$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$(T - 9)^{2}$$
$29$ $$T^{2} + 81$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 81$$
$59$ $$T^{2} + 9$$
$61$ $$T^{2} + 36$$
$67$ $$T^{2} + 25$$
$71$ $$T^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2}$$
$97$ $$(T + 8)^{2}$$