# Properties

 Label 336.2.q.f Level $336$ Weight $2$ Character orbit 336.q Analytic conductor $2.683$ 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] = [336,2,Mod(193,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, 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("336.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 336.q (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: $$2.68297350792$$ 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 21) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 + 2*z * q^5 + (z + 2) * q^7 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{5} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} + q^{13} + 2 q^{15} + \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 3) q^{21} + ( - \zeta_{6} + 1) q^{25} - q^{27} + 4 q^{29} + ( - 9 \zeta_{6} + 9) q^{31} + 2 \zeta_{6} q^{33} + (6 \zeta_{6} - 2) q^{35} - 3 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{39} - 10 q^{41} - 5 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} - 6 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + (12 \zeta_{6} - 12) q^{53} - 4 q^{55} + q^{57} + (12 \zeta_{6} - 12) q^{59} - 10 \zeta_{6} q^{61} + ( - 3 \zeta_{6} + 1) q^{63} + 2 \zeta_{6} q^{65} + (5 \zeta_{6} - 5) q^{67} + 6 q^{71} + ( - 3 \zeta_{6} + 3) q^{73} - \zeta_{6} q^{75} + (4 \zeta_{6} - 6) q^{77} - \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 6 q^{83} + ( - 4 \zeta_{6} + 4) q^{87} - 16 \zeta_{6} q^{89} + (\zeta_{6} + 2) q^{91} - 9 \zeta_{6} q^{93} + (2 \zeta_{6} - 2) q^{95} - 6 q^{97} + 2 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + 2*z * q^5 + (z + 2) * q^7 - z * q^9 + (2*z - 2) * q^11 + q^13 + 2 * q^15 + z * q^19 + (-2*z + 3) * q^21 + (-z + 1) * q^25 - q^27 + 4 * q^29 + (-9*z + 9) * q^31 + 2*z * q^33 + (6*z - 2) * q^35 - 3*z * q^37 + (-z + 1) * q^39 - 10 * q^41 - 5 * q^43 + (-2*z + 2) * q^45 - 6*z * q^47 + (5*z + 3) * q^49 + (12*z - 12) * q^53 - 4 * q^55 + q^57 + (12*z - 12) * q^59 - 10*z * q^61 + (-3*z + 1) * q^63 + 2*z * q^65 + (5*z - 5) * q^67 + 6 * q^71 + (-3*z + 3) * q^73 - z * q^75 + (4*z - 6) * q^77 - z * q^79 + (z - 1) * q^81 - 6 * q^83 + (-4*z + 4) * q^87 - 16*z * q^89 + (z + 2) * q^91 - 9*z * q^93 + (2*z - 2) * q^95 - 6 * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + 5 q^{7} - q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 + 5 * q^7 - q^9 $$2 q + q^{3} + 2 q^{5} + 5 q^{7} - q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{15} + q^{19} + 4 q^{21} + q^{25} - 2 q^{27} + 8 q^{29} + 9 q^{31} + 2 q^{33} + 2 q^{35} - 3 q^{37} + q^{39} - 20 q^{41} - 10 q^{43} + 2 q^{45} - 6 q^{47} + 11 q^{49} - 12 q^{53} - 8 q^{55} + 2 q^{57} - 12 q^{59} - 10 q^{61} - q^{63} + 2 q^{65} - 5 q^{67} + 12 q^{71} + 3 q^{73} - q^{75} - 8 q^{77} - q^{79} - q^{81} - 12 q^{83} + 4 q^{87} - 16 q^{89} + 5 q^{91} - 9 q^{93} - 2 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 + 5 * q^7 - q^9 - 2 * q^11 + 2 * q^13 + 4 * q^15 + q^19 + 4 * q^21 + q^25 - 2 * q^27 + 8 * q^29 + 9 * q^31 + 2 * q^33 + 2 * q^35 - 3 * q^37 + q^39 - 20 * q^41 - 10 * q^43 + 2 * q^45 - 6 * q^47 + 11 * q^49 - 12 * q^53 - 8 * q^55 + 2 * q^57 - 12 * q^59 - 10 * q^61 - q^63 + 2 * q^65 - 5 * q^67 + 12 * q^71 + 3 * q^73 - q^75 - 8 * q^77 - q^79 - q^81 - 12 * q^83 + 4 * q^87 - 16 * q^89 + 5 * q^91 - 9 * q^93 - 2 * q^95 - 12 * q^97 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 1.00000 + 1.73205i 0 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 1.00000 1.73205i 0 2.50000 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.q.f 2
3.b odd 2 1 1008.2.s.d 2
4.b odd 2 1 21.2.e.a 2
7.b odd 2 1 2352.2.q.c 2
7.c even 3 1 inner 336.2.q.f 2
7.c even 3 1 2352.2.a.d 1
7.d odd 6 1 2352.2.a.w 1
7.d odd 6 1 2352.2.q.c 2
8.b even 2 1 1344.2.q.c 2
8.d odd 2 1 1344.2.q.m 2
12.b even 2 1 63.2.e.b 2
20.d odd 2 1 525.2.i.e 2
20.e even 4 2 525.2.r.e 4
21.g even 6 1 7056.2.a.m 1
21.h odd 6 1 1008.2.s.d 2
21.h odd 6 1 7056.2.a.bp 1
28.d even 2 1 147.2.e.a 2
28.f even 6 1 147.2.a.b 1
28.f even 6 1 147.2.e.a 2
28.g odd 6 1 21.2.e.a 2
28.g odd 6 1 147.2.a.c 1
36.f odd 6 1 567.2.g.a 2
36.f odd 6 1 567.2.h.f 2
36.h even 6 1 567.2.g.f 2
36.h even 6 1 567.2.h.a 2
56.j odd 6 1 9408.2.a.k 1
56.k odd 6 1 1344.2.q.m 2
56.k odd 6 1 9408.2.a.bg 1
56.m even 6 1 9408.2.a.bz 1
56.p even 6 1 1344.2.q.c 2
56.p even 6 1 9408.2.a.cv 1
84.h odd 2 1 441.2.e.e 2
84.j odd 6 1 441.2.a.a 1
84.j odd 6 1 441.2.e.e 2
84.n even 6 1 63.2.e.b 2
84.n even 6 1 441.2.a.b 1
140.p odd 6 1 525.2.i.e 2
140.p odd 6 1 3675.2.a.a 1
140.s even 6 1 3675.2.a.c 1
140.w even 12 2 525.2.r.e 4
252.o even 6 1 567.2.h.a 2
252.u odd 6 1 567.2.g.a 2
252.bb even 6 1 567.2.g.f 2
252.bl odd 6 1 567.2.h.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 4.b odd 2 1
21.2.e.a 2 28.g odd 6 1
63.2.e.b 2 12.b even 2 1
63.2.e.b 2 84.n even 6 1
147.2.a.b 1 28.f even 6 1
147.2.a.c 1 28.g odd 6 1
147.2.e.a 2 28.d even 2 1
147.2.e.a 2 28.f even 6 1
336.2.q.f 2 1.a even 1 1 trivial
336.2.q.f 2 7.c even 3 1 inner
441.2.a.a 1 84.j odd 6 1
441.2.a.b 1 84.n even 6 1
441.2.e.e 2 84.h odd 2 1
441.2.e.e 2 84.j odd 6 1
525.2.i.e 2 20.d odd 2 1
525.2.i.e 2 140.p odd 6 1
525.2.r.e 4 20.e even 4 2
525.2.r.e 4 140.w even 12 2
567.2.g.a 2 36.f odd 6 1
567.2.g.a 2 252.u odd 6 1
567.2.g.f 2 36.h even 6 1
567.2.g.f 2 252.bb even 6 1
567.2.h.a 2 36.h even 6 1
567.2.h.a 2 252.o even 6 1
567.2.h.f 2 36.f odd 6 1
567.2.h.f 2 252.bl odd 6 1
1008.2.s.d 2 3.b odd 2 1
1008.2.s.d 2 21.h odd 6 1
1344.2.q.c 2 8.b even 2 1
1344.2.q.c 2 56.p even 6 1
1344.2.q.m 2 8.d odd 2 1
1344.2.q.m 2 56.k odd 6 1
2352.2.a.d 1 7.c even 3 1
2352.2.a.w 1 7.d odd 6 1
2352.2.q.c 2 7.b odd 2 1
2352.2.q.c 2 7.d odd 6 1
3675.2.a.a 1 140.p odd 6 1
3675.2.a.c 1 140.s even 6 1
7056.2.a.m 1 21.g even 6 1
7056.2.a.bp 1 21.h odd 6 1
9408.2.a.k 1 56.j odd 6 1
9408.2.a.bg 1 56.k odd 6 1
9408.2.a.bz 1 56.m even 6 1
9408.2.a.cv 1 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2}$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} - 9T + 81$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$(T + 10)^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 3T + 9$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 16T + 256$$
$97$ $$(T + 6)^{2}$$