# Properties

 Label 1216.2.c.b Level $1216$ Weight $2$ Character orbit 1216.c Analytic conductor $9.710$ Analytic rank $1$ Dimension $2$ CM no 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: $$1$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} - 4 i q^{5} - q^{7} - 6 q^{9} +O(q^{10})$$ q + 3*i * q^3 - 4*i * q^5 - q^7 - 6 * q^9 $$q + 3 i q^{3} - 4 i q^{5} - q^{7} - 6 q^{9} + 5 i q^{13} + 12 q^{15} - 5 q^{17} + i q^{19} - 3 i q^{21} + 3 q^{23} - 11 q^{25} - 9 i q^{27} - 7 i q^{29} - 10 q^{31} + 4 i q^{35} - 2 i q^{37} - 15 q^{39} - 6 q^{41} - 4 i q^{43} + 24 i q^{45} - 8 q^{47} - 6 q^{49} - 15 i q^{51} - 9 i q^{53} - 3 q^{57} + i q^{59} + 2 i q^{61} + 6 q^{63} + 20 q^{65} + 7 i q^{67} + 9 i q^{69} + 12 q^{71} + 11 q^{73} - 33 i q^{75} - 16 q^{79} + 9 q^{81} + 14 i q^{83} + 20 i q^{85} + 21 q^{87} - 4 q^{89} - 5 i q^{91} - 30 i q^{93} + 4 q^{95} - 12 q^{97} +O(q^{100})$$ q + 3*i * q^3 - 4*i * q^5 - q^7 - 6 * q^9 + 5*i * q^13 + 12 * q^15 - 5 * q^17 + i * q^19 - 3*i * q^21 + 3 * q^23 - 11 * q^25 - 9*i * q^27 - 7*i * q^29 - 10 * q^31 + 4*i * q^35 - 2*i * q^37 - 15 * q^39 - 6 * q^41 - 4*i * q^43 + 24*i * q^45 - 8 * q^47 - 6 * q^49 - 15*i * q^51 - 9*i * q^53 - 3 * q^57 + i * q^59 + 2*i * q^61 + 6 * q^63 + 20 * q^65 + 7*i * q^67 + 9*i * q^69 + 12 * q^71 + 11 * q^73 - 33*i * q^75 - 16 * q^79 + 9 * q^81 + 14*i * q^83 + 20*i * q^85 + 21 * q^87 - 4 * q^89 - 5*i * q^91 - 30*i * q^93 + 4 * q^95 - 12 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} - 12 q^{9}+O(q^{10})$$ 2 * q - 2 * q^7 - 12 * q^9 $$2 q - 2 q^{7} - 12 q^{9} + 24 q^{15} - 10 q^{17} + 6 q^{23} - 22 q^{25} - 20 q^{31} - 30 q^{39} - 12 q^{41} - 16 q^{47} - 12 q^{49} - 6 q^{57} + 12 q^{63} + 40 q^{65} + 24 q^{71} + 22 q^{73} - 32 q^{79} + 18 q^{81} + 42 q^{87} - 8 q^{89} + 8 q^{95} - 24 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 - 12 * q^9 + 24 * q^15 - 10 * q^17 + 6 * q^23 - 22 * q^25 - 20 * q^31 - 30 * q^39 - 12 * q^41 - 16 * q^47 - 12 * q^49 - 6 * q^57 + 12 * q^63 + 40 * q^65 + 24 * q^71 + 22 * q^73 - 32 * q^79 + 18 * q^81 + 42 * q^87 - 8 * q^89 + 8 * q^95 - 24 * 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 3.00000i 0 4.00000i 0 −1.00000 0 −6.00000 0
609.2 0 3.00000i 0 4.00000i 0 −1.00000 0 −6.00000 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.b 2
4.b odd 2 1 1216.2.c.c yes 2
8.b even 2 1 inner 1216.2.c.b 2
8.d odd 2 1 1216.2.c.c yes 2
16.e even 4 1 4864.2.a.a 1
16.e even 4 1 4864.2.a.p 1
16.f odd 4 1 4864.2.a.b 1
16.f odd 4 1 4864.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.b 2 1.a even 1 1 trivial
1216.2.c.b 2 8.b even 2 1 inner
1216.2.c.c yes 2 4.b odd 2 1
1216.2.c.c yes 2 8.d odd 2 1
4864.2.a.a 1 16.e even 4 1
4864.2.a.b 1 16.f odd 4 1
4864.2.a.o 1 16.f odd 4 1
4864.2.a.p 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 $$S_{2}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{5}^{2} + 16$$ T5^2 + 16 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2} + 16$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 25$$
$17$ $$(T + 5)^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$(T - 3)^{2}$$
$29$ $$T^{2} + 49$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 81$$
$59$ $$T^{2} + 1$$
$61$ $$T^{2} + 4$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$(T + 4)^{2}$$
$97$ $$(T + 12)^{2}$$