# Properties

 Label 312.2.q.a Level $312$ Weight $2$ Character orbit 312.q Analytic conductor $2.491$ Analytic rank $0$ 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] = [312,2,Mod(217,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("312.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$312 = 2^{3} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 312.q (of order $$3$$, degree $$2$$, 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: $$2.49133254306$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + 3 q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 + 3 * q^5 - z * q^9 $$q + (\zeta_{6} - 1) q^{3} + 3 q^{5} - \zeta_{6} q^{9} + (\zeta_{6} + 3) q^{13} + (3 \zeta_{6} - 3) q^{15} + \zeta_{6} q^{17} + (4 \zeta_{6} - 4) q^{23} + 4 q^{25} + q^{27} + (3 \zeta_{6} - 3) q^{29} + 8 q^{31} + ( - 5 \zeta_{6} + 5) q^{37} + (3 \zeta_{6} - 4) q^{39} + (3 \zeta_{6} - 3) q^{41} - 4 \zeta_{6} q^{43} - 3 \zeta_{6} q^{45} - 8 q^{47} + ( - 7 \zeta_{6} + 7) q^{49} - q^{51} - 13 q^{53} - 12 \zeta_{6} q^{59} - 15 \zeta_{6} q^{61} + (3 \zeta_{6} + 9) q^{65} + (12 \zeta_{6} - 12) q^{67} - 4 \zeta_{6} q^{69} - 8 \zeta_{6} q^{71} + 3 q^{73} + (4 \zeta_{6} - 4) q^{75} - 4 q^{79} + (\zeta_{6} - 1) q^{81} + 12 q^{83} + 3 \zeta_{6} q^{85} - 3 \zeta_{6} q^{87} + (10 \zeta_{6} - 10) q^{89} + (8 \zeta_{6} - 8) q^{93} - 2 \zeta_{6} q^{97} +O(q^{100})$$ q + (z - 1) * q^3 + 3 * q^5 - z * q^9 + (z + 3) * q^13 + (3*z - 3) * q^15 + z * q^17 + (4*z - 4) * q^23 + 4 * q^25 + q^27 + (3*z - 3) * q^29 + 8 * q^31 + (-5*z + 5) * q^37 + (3*z - 4) * q^39 + (3*z - 3) * q^41 - 4*z * q^43 - 3*z * q^45 - 8 * q^47 + (-7*z + 7) * q^49 - q^51 - 13 * q^53 - 12*z * q^59 - 15*z * q^61 + (3*z + 9) * q^65 + (12*z - 12) * q^67 - 4*z * q^69 - 8*z * q^71 + 3 * q^73 + (4*z - 4) * q^75 - 4 * q^79 + (z - 1) * q^81 + 12 * q^83 + 3*z * q^85 - 3*z * q^87 + (10*z - 10) * q^89 + (8*z - 8) * q^93 - 2*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 6 q^{5} - q^{9}+O(q^{10})$$ 2 * q - q^3 + 6 * q^5 - q^9 $$2 q - q^{3} + 6 q^{5} - q^{9} + 7 q^{13} - 3 q^{15} + q^{17} - 4 q^{23} + 8 q^{25} + 2 q^{27} - 3 q^{29} + 16 q^{31} + 5 q^{37} - 5 q^{39} - 3 q^{41} - 4 q^{43} - 3 q^{45} - 16 q^{47} + 7 q^{49} - 2 q^{51} - 26 q^{53} - 12 q^{59} - 15 q^{61} + 21 q^{65} - 12 q^{67} - 4 q^{69} - 8 q^{71} + 6 q^{73} - 4 q^{75} - 8 q^{79} - q^{81} + 24 q^{83} + 3 q^{85} - 3 q^{87} - 10 q^{89} - 8 q^{93} - 2 q^{97}+O(q^{100})$$ 2 * q - q^3 + 6 * q^5 - q^9 + 7 * q^13 - 3 * q^15 + q^17 - 4 * q^23 + 8 * q^25 + 2 * q^27 - 3 * q^29 + 16 * q^31 + 5 * q^37 - 5 * q^39 - 3 * q^41 - 4 * q^43 - 3 * q^45 - 16 * q^47 + 7 * q^49 - 2 * q^51 - 26 * q^53 - 12 * q^59 - 15 * q^61 + 21 * q^65 - 12 * q^67 - 4 * q^69 - 8 * q^71 + 6 * q^73 - 4 * q^75 - 8 * q^79 - q^81 + 24 * q^83 + 3 * q^85 - 3 * q^87 - 10 * q^89 - 8 * q^93 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$$.

 $$n$$ $$79$$ $$145$$ $$157$$ $$209$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
217.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 3.00000 0 0 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 3.00000 0 0 0 −0.500000 + 0.866025i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.q.a 2
3.b odd 2 1 936.2.t.a 2
4.b odd 2 1 624.2.q.f 2
12.b even 2 1 1872.2.t.b 2
13.c even 3 1 inner 312.2.q.a 2
13.c even 3 1 4056.2.a.q 1
13.e even 6 1 4056.2.a.l 1
13.f odd 12 2 4056.2.c.i 2
39.i odd 6 1 936.2.t.a 2
52.i odd 6 1 8112.2.a.b 1
52.j odd 6 1 624.2.q.f 2
52.j odd 6 1 8112.2.a.n 1
156.p even 6 1 1872.2.t.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.a 2 1.a even 1 1 trivial
312.2.q.a 2 13.c even 3 1 inner
624.2.q.f 2 4.b odd 2 1
624.2.q.f 2 52.j odd 6 1
936.2.t.a 2 3.b odd 2 1
936.2.t.a 2 39.i odd 6 1
1872.2.t.b 2 12.b even 2 1
1872.2.t.b 2 156.p even 6 1
4056.2.a.l 1 13.e even 6 1
4056.2.a.q 1 13.c even 3 1
4056.2.c.i 2 13.f odd 12 2
8112.2.a.b 1 52.i odd 6 1
8112.2.a.n 1 52.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(312, [\chi])$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 7T + 13$$
$17$ $$T^{2} - T + 1$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} - 5T + 25$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$(T + 8)^{2}$$
$53$ $$(T + 13)^{2}$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} + 15T + 225$$
$67$ $$T^{2} + 12T + 144$$
$71$ $$T^{2} + 8T + 64$$
$73$ $$(T - 3)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + 10T + 100$$
$97$ $$T^{2} + 2T + 4$$