# Properties

 Label 1216.2.i.f Level $1216$ Weight $2$ Character orbit 1216.i 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(577,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1216.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.i (of order $$3$$, degree $$2$$, 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: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + (4 \zeta_{6} - 4) q^{5} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z + 1) * q^3 + (4*z - 4) * q^5 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + (4 \zeta_{6} - 4) q^{5} + 2 \zeta_{6} q^{9} - 3 q^{11} + 2 \zeta_{6} q^{13} + 4 \zeta_{6} q^{15} + (2 \zeta_{6} - 2) q^{17} + ( - 5 \zeta_{6} + 2) q^{19} - 6 \zeta_{6} q^{23} - 11 \zeta_{6} q^{25} + 5 q^{27} - 4 \zeta_{6} q^{29} - 10 q^{31} + (3 \zeta_{6} - 3) q^{33} - 2 q^{37} + 2 q^{39} + (9 \zeta_{6} - 9) q^{41} + (4 \zeta_{6} - 4) q^{43} - 8 q^{45} + 12 \zeta_{6} q^{47} - 7 q^{49} + 2 \zeta_{6} q^{51} - 2 \zeta_{6} q^{53} + ( - 12 \zeta_{6} + 12) q^{55} + ( - 2 \zeta_{6} - 3) q^{57} + (\zeta_{6} - 1) q^{59} - 8 \zeta_{6} q^{61} - 8 q^{65} + 9 \zeta_{6} q^{67} - 6 q^{69} + ( - 6 \zeta_{6} + 6) q^{71} + ( - 9 \zeta_{6} + 9) q^{73} - 11 q^{75} + ( - 4 \zeta_{6} + 4) q^{79} + (\zeta_{6} - 1) q^{81} + 5 q^{83} - 8 \zeta_{6} q^{85} - 4 q^{87} + 18 \zeta_{6} q^{89} + (10 \zeta_{6} - 10) q^{93} + (8 \zeta_{6} + 12) q^{95} + (\zeta_{6} - 1) q^{97} - 6 \zeta_{6} q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (4*z - 4) * q^5 + 2*z * q^9 - 3 * q^11 + 2*z * q^13 + 4*z * q^15 + (2*z - 2) * q^17 + (-5*z + 2) * q^19 - 6*z * q^23 - 11*z * q^25 + 5 * q^27 - 4*z * q^29 - 10 * q^31 + (3*z - 3) * q^33 - 2 * q^37 + 2 * q^39 + (9*z - 9) * q^41 + (4*z - 4) * q^43 - 8 * q^45 + 12*z * q^47 - 7 * q^49 + 2*z * q^51 - 2*z * q^53 + (-12*z + 12) * q^55 + (-2*z - 3) * q^57 + (z - 1) * q^59 - 8*z * q^61 - 8 * q^65 + 9*z * q^67 - 6 * q^69 + (-6*z + 6) * q^71 + (-9*z + 9) * q^73 - 11 * q^75 + (-4*z + 4) * q^79 + (z - 1) * q^81 + 5 * q^83 - 8*z * q^85 - 4 * q^87 + 18*z * q^89 + (10*z - 10) * q^93 + (8*z + 12) * q^95 + (z - 1) * q^97 - 6*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - 4 * q^5 + 2 * q^9 $$2 q + q^{3} - 4 q^{5} + 2 q^{9} - 6 q^{11} + 2 q^{13} + 4 q^{15} - 2 q^{17} - q^{19} - 6 q^{23} - 11 q^{25} + 10 q^{27} - 4 q^{29} - 20 q^{31} - 3 q^{33} - 4 q^{37} + 4 q^{39} - 9 q^{41} - 4 q^{43} - 16 q^{45} + 12 q^{47} - 14 q^{49} + 2 q^{51} - 2 q^{53} + 12 q^{55} - 8 q^{57} - q^{59} - 8 q^{61} - 16 q^{65} + 9 q^{67} - 12 q^{69} + 6 q^{71} + 9 q^{73} - 22 q^{75} + 4 q^{79} - q^{81} + 10 q^{83} - 8 q^{85} - 8 q^{87} + 18 q^{89} - 10 q^{93} + 32 q^{95} - q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q + q^3 - 4 * q^5 + 2 * q^9 - 6 * q^11 + 2 * q^13 + 4 * q^15 - 2 * q^17 - q^19 - 6 * q^23 - 11 * q^25 + 10 * q^27 - 4 * q^29 - 20 * q^31 - 3 * q^33 - 4 * q^37 + 4 * q^39 - 9 * q^41 - 4 * q^43 - 16 * q^45 + 12 * q^47 - 14 * q^49 + 2 * q^51 - 2 * q^53 + 12 * q^55 - 8 * q^57 - q^59 - 8 * q^61 - 16 * q^65 + 9 * q^67 - 12 * q^69 + 6 * q^71 + 9 * q^73 - 22 * q^75 + 4 * q^79 - q^81 + 10 * q^83 - 8 * q^85 - 8 * q^87 + 18 * q^89 - 10 * q^93 + 32 * q^95 - q^97 - 6 * q^99

## Character values

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

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 −2.00000 3.46410i 0 0 0 1.00000 1.73205i 0
961.1 0 0.500000 0.866025i 0 −2.00000 + 3.46410i 0 0 0 1.00000 + 1.73205i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.i.f 2
4.b odd 2 1 1216.2.i.b 2
8.b even 2 1 152.2.i.b 2
8.d odd 2 1 304.2.i.d 2
19.c even 3 1 inner 1216.2.i.f 2
24.f even 2 1 2736.2.s.a 2
24.h odd 2 1 1368.2.s.a 2
76.g odd 6 1 1216.2.i.b 2
152.k odd 6 1 304.2.i.d 2
152.k odd 6 1 5776.2.a.e 1
152.l odd 6 1 2888.2.a.a 1
152.o even 6 1 5776.2.a.j 1
152.p even 6 1 152.2.i.b 2
152.p even 6 1 2888.2.a.d 1
456.u even 6 1 2736.2.s.a 2
456.x odd 6 1 1368.2.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.b 2 8.b even 2 1
152.2.i.b 2 152.p even 6 1
304.2.i.d 2 8.d odd 2 1
304.2.i.d 2 152.k odd 6 1
1216.2.i.b 2 4.b odd 2 1
1216.2.i.b 2 76.g odd 6 1
1216.2.i.f 2 1.a even 1 1 trivial
1216.2.i.f 2 19.c even 3 1 inner
1368.2.s.a 2 24.h odd 2 1
1368.2.s.a 2 456.x odd 6 1
2736.2.s.a 2 24.f even 2 1
2736.2.s.a 2 456.u even 6 1
2888.2.a.a 1 152.l odd 6 1
2888.2.a.d 1 152.p even 6 1
5776.2.a.e 1 152.k odd 6 1
5776.2.a.j 1 152.o even 6 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{5}^{2} + 4T_{5} + 16$$ T5^2 + 4*T5 + 16 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$T^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} + T + 19$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 4T + 16$$
$31$ $$(T + 10)^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$T^{2} - 12T + 144$$
$53$ $$T^{2} + 2T + 4$$
$59$ $$T^{2} + T + 1$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 9T + 81$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$T^{2} - 9T + 81$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$(T - 5)^{2}$$
$89$ $$T^{2} - 18T + 324$$
$97$ $$T^{2} + T + 1$$