# Properties

 Label 1216.2.i.a 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 608) 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.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} + 2 q^{7} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + (3*z - 3) * q^3 + (-2*z + 2) * q^5 + 2 * q^7 - 6*z * q^9 $$q + (3 \zeta_{6} - 3) q^{3} + ( - 2 \zeta_{6} + 2) q^{5} + 2 q^{7} - 6 \zeta_{6} q^{9} - 3 q^{11} + 2 \zeta_{6} q^{13} + 6 \zeta_{6} q^{15} + ( - 6 \zeta_{6} + 6) q^{17} + (3 \zeta_{6} + 2) q^{19} + (6 \zeta_{6} - 6) q^{21} - 4 \zeta_{6} q^{23} + \zeta_{6} q^{25} + 9 q^{27} - 6 \zeta_{6} q^{29} + 8 q^{31} + ( - 9 \zeta_{6} + 9) q^{33} + ( - 4 \zeta_{6} + 4) q^{35} + 8 q^{37} - 6 q^{39} + ( - 3 \zeta_{6} + 3) q^{41} + ( - 8 \zeta_{6} + 8) q^{43} - 12 q^{45} + 10 \zeta_{6} q^{47} - 3 q^{49} + 18 \zeta_{6} q^{51} + 2 \zeta_{6} q^{53} + (6 \zeta_{6} - 6) q^{55} + (6 \zeta_{6} - 15) q^{57} + ( - 3 \zeta_{6} + 3) q^{59} + 14 \zeta_{6} q^{61} - 12 \zeta_{6} q^{63} + 4 q^{65} - 7 \zeta_{6} q^{67} + 12 q^{69} + (2 \zeta_{6} - 2) q^{71} + (11 \zeta_{6} - 11) q^{73} - 3 q^{75} - 6 q^{77} + (4 \zeta_{6} - 4) q^{79} + (9 \zeta_{6} - 9) q^{81} - 15 q^{83} - 12 \zeta_{6} q^{85} + 18 q^{87} + 18 \zeta_{6} q^{89} + 4 \zeta_{6} q^{91} + (24 \zeta_{6} - 24) q^{93} + ( - 4 \zeta_{6} + 10) q^{95} + ( - 3 \zeta_{6} + 3) q^{97} + 18 \zeta_{6} q^{99} +O(q^{100})$$ q + (3*z - 3) * q^3 + (-2*z + 2) * q^5 + 2 * q^7 - 6*z * q^9 - 3 * q^11 + 2*z * q^13 + 6*z * q^15 + (-6*z + 6) * q^17 + (3*z + 2) * q^19 + (6*z - 6) * q^21 - 4*z * q^23 + z * q^25 + 9 * q^27 - 6*z * q^29 + 8 * q^31 + (-9*z + 9) * q^33 + (-4*z + 4) * q^35 + 8 * q^37 - 6 * q^39 + (-3*z + 3) * q^41 + (-8*z + 8) * q^43 - 12 * q^45 + 10*z * q^47 - 3 * q^49 + 18*z * q^51 + 2*z * q^53 + (6*z - 6) * q^55 + (6*z - 15) * q^57 + (-3*z + 3) * q^59 + 14*z * q^61 - 12*z * q^63 + 4 * q^65 - 7*z * q^67 + 12 * q^69 + (2*z - 2) * q^71 + (11*z - 11) * q^73 - 3 * q^75 - 6 * q^77 + (4*z - 4) * q^79 + (9*z - 9) * q^81 - 15 * q^83 - 12*z * q^85 + 18 * q^87 + 18*z * q^89 + 4*z * q^91 + (24*z - 24) * q^93 + (-4*z + 10) * q^95 + (-3*z + 3) * q^97 + 18*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 2 q^{5} + 4 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 2 * q^5 + 4 * q^7 - 6 * q^9 $$2 q - 3 q^{3} + 2 q^{5} + 4 q^{7} - 6 q^{9} - 6 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 7 q^{19} - 6 q^{21} - 4 q^{23} + q^{25} + 18 q^{27} - 6 q^{29} + 16 q^{31} + 9 q^{33} + 4 q^{35} + 16 q^{37} - 12 q^{39} + 3 q^{41} + 8 q^{43} - 24 q^{45} + 10 q^{47} - 6 q^{49} + 18 q^{51} + 2 q^{53} - 6 q^{55} - 24 q^{57} + 3 q^{59} + 14 q^{61} - 12 q^{63} + 8 q^{65} - 7 q^{67} + 24 q^{69} - 2 q^{71} - 11 q^{73} - 6 q^{75} - 12 q^{77} - 4 q^{79} - 9 q^{81} - 30 q^{83} - 12 q^{85} + 36 q^{87} + 18 q^{89} + 4 q^{91} - 24 q^{93} + 16 q^{95} + 3 q^{97} + 18 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 2 * q^5 + 4 * q^7 - 6 * q^9 - 6 * q^11 + 2 * q^13 + 6 * q^15 + 6 * q^17 + 7 * q^19 - 6 * q^21 - 4 * q^23 + q^25 + 18 * q^27 - 6 * q^29 + 16 * q^31 + 9 * q^33 + 4 * q^35 + 16 * q^37 - 12 * q^39 + 3 * q^41 + 8 * q^43 - 24 * q^45 + 10 * q^47 - 6 * q^49 + 18 * q^51 + 2 * q^53 - 6 * q^55 - 24 * q^57 + 3 * q^59 + 14 * q^61 - 12 * q^63 + 8 * q^65 - 7 * q^67 + 24 * q^69 - 2 * q^71 - 11 * q^73 - 6 * q^75 - 12 * q^77 - 4 * q^79 - 9 * q^81 - 30 * q^83 - 12 * q^85 + 36 * q^87 + 18 * q^89 + 4 * q^91 - 24 * q^93 + 16 * q^95 + 3 * q^97 + 18 * 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 −1.50000 2.59808i 0 1.00000 + 1.73205i 0 2.00000 0 −3.00000 + 5.19615i 0
961.1 0 −1.50000 + 2.59808i 0 1.00000 1.73205i 0 2.00000 0 −3.00000 5.19615i 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.a 2
4.b odd 2 1 1216.2.i.j 2
8.b even 2 1 608.2.i.b yes 2
8.d odd 2 1 608.2.i.a 2
19.c even 3 1 inner 1216.2.i.a 2
76.g odd 6 1 1216.2.i.j 2
152.k odd 6 1 608.2.i.a 2
152.p even 6 1 608.2.i.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.i.a 2 8.d odd 2 1
608.2.i.a 2 152.k odd 6 1
608.2.i.b yes 2 8.b even 2 1
608.2.i.b yes 2 152.p even 6 1
1216.2.i.a 2 1.a even 1 1 trivial
1216.2.i.a 2 19.c even 3 1 inner
1216.2.i.j 2 4.b odd 2 1
1216.2.i.j 2 76.g odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$(T - 2)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 7T + 19$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$(T - 8)^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} - 10T + 100$$
$53$ $$T^{2} - 2T + 4$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} - 14T + 196$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$T^{2} + 2T + 4$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} + 4T + 16$$
$83$ $$(T + 15)^{2}$$
$89$ $$T^{2} - 18T + 324$$
$97$ $$T^{2} - 3T + 9$$