# Properties

 Label 1216.2.c.f Level $1216$ Weight $2$ Character orbit 1216.c Analytic conductor $9.710$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \beta_1 q^{3} - \beta_{3} q^{5} + \beta_{2} q^{7} - q^{9}+O(q^{10})$$ q - 2*b1 * q^3 - b3 * q^5 + b2 * q^7 - q^9 $$q - 2 \beta_1 q^{3} - \beta_{3} q^{5} + \beta_{2} q^{7} - q^{9} - 5 \beta_1 q^{11} - 2 \beta_{2} q^{15} + 5 q^{17} + \beta_1 q^{19} - 2 \beta_{3} q^{21} + 2 \beta_{2} q^{23} - 6 q^{25} - 4 \beta_1 q^{27} + 2 \beta_{3} q^{29} - 10 q^{33} - 11 \beta_1 q^{35} + 2 \beta_{3} q^{37} - 6 q^{41} + \beta_1 q^{43} + \beta_{3} q^{45} + 3 \beta_{2} q^{47} + 4 q^{49} - 10 \beta_1 q^{51} + 4 \beta_{3} q^{53} - 5 \beta_{2} q^{55} + 2 q^{57} + 6 \beta_1 q^{59} + 3 \beta_{3} q^{61} - \beta_{2} q^{63} - 8 \beta_1 q^{67} - 4 \beta_{3} q^{69} - 2 \beta_{2} q^{71} - 9 q^{73} + 12 \beta_1 q^{75} - 5 \beta_{3} q^{77} - 4 \beta_{2} q^{79} - 11 q^{81} + 4 \beta_1 q^{83} - 5 \beta_{3} q^{85} + 4 \beta_{2} q^{87} - 4 q^{89} + \beta_{2} q^{95} - 12 q^{97} + 5 \beta_1 q^{99}+O(q^{100})$$ q - 2*b1 * q^3 - b3 * q^5 + b2 * q^7 - q^9 - 5*b1 * q^11 - 2*b2 * q^15 + 5 * q^17 + b1 * q^19 - 2*b3 * q^21 + 2*b2 * q^23 - 6 * q^25 - 4*b1 * q^27 + 2*b3 * q^29 - 10 * q^33 - 11*b1 * q^35 + 2*b3 * q^37 - 6 * q^41 + b1 * q^43 + b3 * q^45 + 3*b2 * q^47 + 4 * q^49 - 10*b1 * q^51 + 4*b3 * q^53 - 5*b2 * q^55 + 2 * q^57 + 6*b1 * q^59 + 3*b3 * q^61 - b2 * q^63 - 8*b1 * q^67 - 4*b3 * q^69 - 2*b2 * q^71 - 9 * q^73 + 12*b1 * q^75 - 5*b3 * q^77 - 4*b2 * q^79 - 11 * q^81 + 4*b1 * q^83 - 5*b3 * q^85 + 4*b2 * q^87 - 4 * q^89 + b2 * q^95 - 12 * q^97 + 5*b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} + 20 q^{17} - 24 q^{25} - 40 q^{33} - 24 q^{41} + 16 q^{49} + 8 q^{57} - 36 q^{73} - 44 q^{81} - 16 q^{89} - 48 q^{97}+O(q^{100})$$ 4 * q - 4 * q^9 + 20 * q^17 - 24 * q^25 - 40 * q^33 - 24 * q^41 + 16 * q^49 + 8 * q^57 - 36 * q^73 - 44 * q^81 - 16 * q^89 - 48 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 2\nu ) / 3$$ (v^3 - 2*v) / 3 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 8\nu ) / 3$$ (-v^3 + 8*v) / 3 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 5$$ 2*v^2 - 5
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 5 ) / 2$$ (b3 + 5) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 4\beta_1$$ b2 + 4*b1

## 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
 1.65831 + 0.500000i −1.65831 + 0.500000i −1.65831 − 0.500000i 1.65831 − 0.500000i
0 2.00000i 0 3.31662i 0 3.31662 0 −1.00000 0
609.2 0 2.00000i 0 3.31662i 0 −3.31662 0 −1.00000 0
609.3 0 2.00000i 0 3.31662i 0 −3.31662 0 −1.00000 0
609.4 0 2.00000i 0 3.31662i 0 3.31662 0 −1.00000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.f 4
4.b odd 2 1 inner 1216.2.c.f 4
8.b even 2 1 inner 1216.2.c.f 4
8.d odd 2 1 inner 1216.2.c.f 4
16.e even 4 1 4864.2.a.r 2
16.e even 4 1 4864.2.a.y 2
16.f odd 4 1 4864.2.a.r 2
16.f odd 4 1 4864.2.a.y 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{5}^{2} + 11$$ T5^2 + 11 $$T_{7}^{2} - 11$$ T7^2 - 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 4)^{2}$$
$5$ $$(T^{2} + 11)^{2}$$
$7$ $$(T^{2} - 11)^{2}$$
$11$ $$(T^{2} + 25)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T - 5)^{4}$$
$19$ $$(T^{2} + 1)^{2}$$
$23$ $$(T^{2} - 44)^{2}$$
$29$ $$(T^{2} + 44)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} + 44)^{2}$$
$41$ $$(T + 6)^{4}$$
$43$ $$(T^{2} + 1)^{2}$$
$47$ $$(T^{2} - 99)^{2}$$
$53$ $$(T^{2} + 176)^{2}$$
$59$ $$(T^{2} + 36)^{2}$$
$61$ $$(T^{2} + 99)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 44)^{2}$$
$73$ $$(T + 9)^{4}$$
$79$ $$(T^{2} - 176)^{2}$$
$83$ $$(T^{2} + 16)^{2}$$
$89$ $$(T + 4)^{4}$$
$97$ $$(T + 12)^{4}$$