# Properties

 Label 336.2.q.c Level $336$ Weight $2$ Character orbit 336.q Analytic conductor $2.683$ 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] = [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 84) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{5} + (3 \zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 + 2*z * q^5 + (3*z - 2) * q^7 - z * q^9 $$q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{5} + (3 \zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} - 3 q^{13} - 2 q^{15} + (8 \zeta_{6} - 8) q^{17} - \zeta_{6} q^{19} + ( - 2 \zeta_{6} - 1) q^{21} + 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + q^{27} + 4 q^{29} + ( - 3 \zeta_{6} + 3) q^{31} + 2 \zeta_{6} q^{33} + (2 \zeta_{6} - 6) q^{35} + \zeta_{6} q^{37} + ( - 3 \zeta_{6} + 3) q^{39} + 6 q^{41} - 11 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 6 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} - 8 \zeta_{6} q^{51} + ( - 12 \zeta_{6} + 12) q^{53} + 4 q^{55} + q^{57} + ( - 4 \zeta_{6} + 4) q^{59} + 6 \zeta_{6} q^{61} + ( - \zeta_{6} + 3) q^{63} - 6 \zeta_{6} q^{65} + ( - 13 \zeta_{6} + 13) q^{67} - 8 q^{69} + 10 q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + \zeta_{6} q^{75} + (4 \zeta_{6} + 2) q^{77} - 3 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 2 q^{83} - 16 q^{85} + (4 \zeta_{6} - 4) q^{87} + ( - 9 \zeta_{6} + 6) q^{91} + 3 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{95} + 10 q^{97} - 2 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + 2*z * q^5 + (3*z - 2) * q^7 - z * q^9 + (-2*z + 2) * q^11 - 3 * q^13 - 2 * q^15 + (8*z - 8) * q^17 - z * q^19 + (-2*z - 1) * q^21 + 8*z * q^23 + (-z + 1) * q^25 + q^27 + 4 * q^29 + (-3*z + 3) * q^31 + 2*z * q^33 + (2*z - 6) * q^35 + z * q^37 + (-3*z + 3) * q^39 + 6 * q^41 - 11 * q^43 + (-2*z + 2) * q^45 + 6*z * q^47 + (-3*z - 5) * q^49 - 8*z * q^51 + (-12*z + 12) * q^53 + 4 * q^55 + q^57 + (-4*z + 4) * q^59 + 6*z * q^61 + (-z + 3) * q^63 - 6*z * q^65 + (-13*z + 13) * q^67 - 8 * q^69 + 10 * q^71 + (-11*z + 11) * q^73 + z * q^75 + (4*z + 2) * q^77 - 3*z * q^79 + (z - 1) * q^81 - 2 * q^83 - 16 * q^85 + (4*z - 4) * q^87 + (-9*z + 6) * q^91 + 3*z * q^93 + (-2*z + 2) * q^95 + 10 * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 2 q^{5} - q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 + 2 * q^5 - q^7 - q^9 $$2 q - q^{3} + 2 q^{5} - q^{7} - q^{9} + 2 q^{11} - 6 q^{13} - 4 q^{15} - 8 q^{17} - q^{19} - 4 q^{21} + 8 q^{23} + q^{25} + 2 q^{27} + 8 q^{29} + 3 q^{31} + 2 q^{33} - 10 q^{35} + q^{37} + 3 q^{39} + 12 q^{41} - 22 q^{43} + 2 q^{45} + 6 q^{47} - 13 q^{49} - 8 q^{51} + 12 q^{53} + 8 q^{55} + 2 q^{57} + 4 q^{59} + 6 q^{61} + 5 q^{63} - 6 q^{65} + 13 q^{67} - 16 q^{69} + 20 q^{71} + 11 q^{73} + q^{75} + 8 q^{77} - 3 q^{79} - q^{81} - 4 q^{83} - 32 q^{85} - 4 q^{87} + 3 q^{91} + 3 q^{93} + 2 q^{95} + 20 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - q^3 + 2 * q^5 - q^7 - q^9 + 2 * q^11 - 6 * q^13 - 4 * q^15 - 8 * q^17 - q^19 - 4 * q^21 + 8 * q^23 + q^25 + 2 * q^27 + 8 * q^29 + 3 * q^31 + 2 * q^33 - 10 * q^35 + q^37 + 3 * q^39 + 12 * q^41 - 22 * q^43 + 2 * q^45 + 6 * q^47 - 13 * q^49 - 8 * q^51 + 12 * q^53 + 8 * q^55 + 2 * q^57 + 4 * q^59 + 6 * q^61 + 5 * q^63 - 6 * q^65 + 13 * q^67 - 16 * q^69 + 20 * q^71 + 11 * q^73 + q^75 + 8 * q^77 - 3 * q^79 - q^81 - 4 * q^83 - 32 * q^85 - 4 * q^87 + 3 * q^91 + 3 * q^93 + 2 * q^95 + 20 * 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 −0.500000 + 2.59808i 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 1.00000 1.73205i 0 −0.500000 2.59808i 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.c 2
3.b odd 2 1 1008.2.s.c 2
4.b odd 2 1 84.2.i.a 2
7.b odd 2 1 2352.2.q.q 2
7.c even 3 1 inner 336.2.q.c 2
7.c even 3 1 2352.2.a.o 1
7.d odd 6 1 2352.2.a.k 1
7.d odd 6 1 2352.2.q.q 2
8.b even 2 1 1344.2.q.n 2
8.d odd 2 1 1344.2.q.b 2
12.b even 2 1 252.2.k.a 2
20.d odd 2 1 2100.2.q.b 2
20.e even 4 2 2100.2.bc.a 4
21.g even 6 1 7056.2.a.o 1
21.h odd 6 1 1008.2.s.c 2
21.h odd 6 1 7056.2.a.bs 1
28.d even 2 1 588.2.i.b 2
28.f even 6 1 588.2.a.f 1
28.f even 6 1 588.2.i.b 2
28.g odd 6 1 84.2.i.a 2
28.g odd 6 1 588.2.a.a 1
36.f odd 6 1 2268.2.i.g 2
36.f odd 6 1 2268.2.l.b 2
36.h even 6 1 2268.2.i.b 2
36.h even 6 1 2268.2.l.g 2
56.j odd 6 1 9408.2.a.bx 1
56.k odd 6 1 1344.2.q.b 2
56.k odd 6 1 9408.2.a.cx 1
56.m even 6 1 9408.2.a.i 1
56.p even 6 1 1344.2.q.n 2
56.p even 6 1 9408.2.a.bi 1
84.h odd 2 1 1764.2.k.j 2
84.j odd 6 1 1764.2.a.c 1
84.j odd 6 1 1764.2.k.j 2
84.n even 6 1 252.2.k.a 2
84.n even 6 1 1764.2.a.h 1
140.p odd 6 1 2100.2.q.b 2
140.w even 12 2 2100.2.bc.a 4
252.o even 6 1 2268.2.i.b 2
252.u odd 6 1 2268.2.l.b 2
252.bb even 6 1 2268.2.l.g 2
252.bl odd 6 1 2268.2.i.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 4.b odd 2 1
84.2.i.a 2 28.g odd 6 1
252.2.k.a 2 12.b even 2 1
252.2.k.a 2 84.n even 6 1
336.2.q.c 2 1.a even 1 1 trivial
336.2.q.c 2 7.c even 3 1 inner
588.2.a.a 1 28.g odd 6 1
588.2.a.f 1 28.f even 6 1
588.2.i.b 2 28.d even 2 1
588.2.i.b 2 28.f even 6 1
1008.2.s.c 2 3.b odd 2 1
1008.2.s.c 2 21.h odd 6 1
1344.2.q.b 2 8.d odd 2 1
1344.2.q.b 2 56.k odd 6 1
1344.2.q.n 2 8.b even 2 1
1344.2.q.n 2 56.p even 6 1
1764.2.a.c 1 84.j odd 6 1
1764.2.a.h 1 84.n even 6 1
1764.2.k.j 2 84.h odd 2 1
1764.2.k.j 2 84.j odd 6 1
2100.2.q.b 2 20.d odd 2 1
2100.2.q.b 2 140.p odd 6 1
2100.2.bc.a 4 20.e even 4 2
2100.2.bc.a 4 140.w even 12 2
2268.2.i.b 2 36.h even 6 1
2268.2.i.b 2 252.o even 6 1
2268.2.i.g 2 36.f odd 6 1
2268.2.i.g 2 252.bl odd 6 1
2268.2.l.b 2 36.f odd 6 1
2268.2.l.b 2 252.u odd 6 1
2268.2.l.g 2 36.h even 6 1
2268.2.l.g 2 252.bb even 6 1
2352.2.a.k 1 7.d odd 6 1
2352.2.a.o 1 7.c even 3 1
2352.2.q.q 2 7.b odd 2 1
2352.2.q.q 2 7.d odd 6 1
7056.2.a.o 1 21.g even 6 1
7056.2.a.bs 1 21.h odd 6 1
9408.2.a.i 1 56.m even 6 1
9408.2.a.bi 1 56.p even 6 1
9408.2.a.bx 1 56.j odd 6 1
9408.2.a.cx 1 56.k odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T + 3)^{2}$$
$17$ $$T^{2} + 8T + 64$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} - 3T + 9$$
$37$ $$T^{2} - T + 1$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 11)^{2}$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} - 6T + 36$$
$67$ $$T^{2} - 13T + 169$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} + 3T + 9$$
$83$ $$(T + 2)^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 10)^{2}$$