# Properties

 Label 336.2.q.g Level $336$ Weight $2$ Character orbit 336.q Analytic conductor $2.683$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{5} + (\beta_{3} - \beta_1 + 2) q^{7} - \beta_{2} q^{9}+O(q^{10})$$ q + (b2 - 1) * q^3 + (-b3 + b2 + 2*b1 - 1) * q^5 + (b3 - b1 + 2) * q^7 - b2 * q^9 $$q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{5} + (\beta_{3} - \beta_1 + 2) q^{7} - \beta_{2} q^{9} + ( - 2 \beta_{3} + \beta_1 - 1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{13} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{15} + ( - 4 \beta_{2} + 4) q^{17} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{19} + (2 \beta_{2} + \beta_1 - 2) q^{21} + 4 \beta_{2} q^{23} + ( - 2 \beta_{3} + 9 \beta_{2} + \cdots - 10) q^{25}+ \cdots + (\beta_{3} + \beta_{2} + \beta_1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^3 + (-b3 + b2 + 2*b1 - 1) * q^5 + (b3 - b1 + 2) * q^7 - b2 * q^9 + (-2*b3 + b1 - 1) * q^11 + (-b3 - b2 - b1 + 3) * q^13 + (-b3 - b2 - b1) * q^15 + (-4*b2 + 4) * q^17 + (-b3 - 2*b2 + 2*b1 - 1) * q^19 + (2*b2 + b1 - 2) * q^21 + 4*b2 * q^23 + (-2*b3 + 9*b2 + b1 - 10) * q^25 + q^27 + (-b3 - b2 - b1 + 2) * q^29 + (b2 - 1) * q^31 + (b3 - b2 - 2*b1 + 1) * q^33 + (-b3 - 3*b2 + 3*b1 + 8) * q^35 + (b3 - 2*b2 - 2*b1 + 1) * q^37 + (2*b3 + 3*b2 - b1 - 2) * q^39 + (2*b3 + 2*b2 + 2*b1 + 2) * q^41 + (b3 + b2 + b1 + 3) * q^43 + (2*b3 - b1 + 1) * q^45 - 6*b2 * q^47 + (3*b3 - 3*b1 - 1) * q^49 + 4*b2 * q^51 + (2*b3 + 6*b2 - b1 - 5) * q^53 + (-b3 - b2 - b1 - 14) * q^55 + (-b3 - b2 - b1 + 3) * q^57 + (2*b3 - 2*b2 - b1 + 3) * q^59 - 10*b2 * q^61 + (-b3 - 2*b2) * q^63 + (-2*b3 - 12*b2 + 4*b1 - 2) * q^65 + (2*b3 - 3*b2 - b1 + 4) * q^67 - 4 * q^69 - 2 * q^71 + (-2*b3 - b2 + b1) * q^73 + (b3 - 10*b2 - 2*b1 + 1) * q^75 + (-3*b3 + 5*b2 + 2*b1 + 3) * q^77 + (2*b3 + 3*b2 - 4*b1 + 2) * q^79 + (b2 - 1) * q^81 + (b3 + b2 + b1 - 4) * q^83 + (4*b3 + 4*b2 + 4*b1) * q^85 + (2*b3 + 2*b2 - b1 - 1) * q^87 + (2*b3 + 2*b2 - 4*b1 + 2) * q^89 + (b3 + 8*b2 - 4*b1 + 1) * q^91 - b2 * q^93 + (4*b3 + 14*b2 - 2*b1 - 12) * q^95 + (b3 + b2 + b1 + 12) * q^97 + (b3 + b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + q^{5} + 6 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + q^5 + 6 * q^7 - 2 * q^9 $$4 q - 2 q^{3} + q^{5} + 6 q^{7} - 2 q^{9} - q^{11} + 10 q^{13} - 2 q^{15} + 8 q^{17} - 5 q^{19} - 3 q^{21} + 8 q^{23} - 19 q^{25} + 4 q^{27} + 6 q^{29} - 2 q^{31} - q^{33} + 30 q^{35} - 3 q^{37} - 5 q^{39} + 12 q^{41} + 14 q^{43} + q^{45} - 12 q^{47} - 10 q^{49} + 8 q^{51} - 11 q^{53} - 58 q^{55} + 10 q^{57} + 5 q^{59} - 20 q^{61} - 3 q^{63} - 26 q^{65} + 7 q^{67} - 16 q^{69} - 8 q^{71} + q^{73} - 19 q^{75} + 27 q^{77} + 8 q^{79} - 2 q^{81} - 14 q^{83} + 8 q^{85} - 3 q^{87} + 6 q^{89} + 15 q^{91} - 2 q^{93} - 26 q^{95} + 50 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 + q^5 + 6 * q^7 - 2 * q^9 - q^11 + 10 * q^13 - 2 * q^15 + 8 * q^17 - 5 * q^19 - 3 * q^21 + 8 * q^23 - 19 * q^25 + 4 * q^27 + 6 * q^29 - 2 * q^31 - q^33 + 30 * q^35 - 3 * q^37 - 5 * q^39 + 12 * q^41 + 14 * q^43 + q^45 - 12 * q^47 - 10 * q^49 + 8 * q^51 - 11 * q^53 - 58 * q^55 + 10 * q^57 + 5 * q^59 - 20 * q^61 - 3 * q^63 - 26 * q^65 + 7 * q^67 - 16 * q^69 - 8 * q^71 + q^73 - 19 * q^75 + 27 * q^77 + 8 * q^79 - 2 * q^81 - 14 * q^83 + 8 * q^85 - 3 * q^87 + 6 * q^89 + 15 * q^91 - 2 * q^93 - 26 * q^95 + 50 * q^97 + 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20$$ (v^3 + 4*v^2 - 4*v - 5) / 20 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu + 5 ) / 4$$ (-v^3 + 4*v + 5) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5\beta_{2}$$ b3 + 5*b2 $$\nu^{3}$$ $$=$$ $$-4\beta_{3} + 4\beta _1 + 5$$ -4*b3 + 4*b1 + 5

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 −0.500000 + 0.866025i 0 −1.63746 2.83616i 0 1.50000 + 2.17945i 0 −0.500000 0.866025i 0
193.2 0 −0.500000 + 0.866025i 0 2.13746 + 3.70219i 0 1.50000 2.17945i 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 −1.63746 + 2.83616i 0 1.50000 2.17945i 0 −0.500000 + 0.866025i 0
289.2 0 −0.500000 0.866025i 0 2.13746 3.70219i 0 1.50000 + 2.17945i 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.g 4
3.b odd 2 1 1008.2.s.r 4
4.b odd 2 1 168.2.q.c 4
7.b odd 2 1 2352.2.q.bf 4
7.c even 3 1 inner 336.2.q.g 4
7.c even 3 1 2352.2.a.bf 2
7.d odd 6 1 2352.2.a.ba 2
7.d odd 6 1 2352.2.q.bf 4
8.b even 2 1 1344.2.q.x 4
8.d odd 2 1 1344.2.q.w 4
12.b even 2 1 504.2.s.i 4
21.g even 6 1 7056.2.a.ch 2
21.h odd 6 1 1008.2.s.r 4
21.h odd 6 1 7056.2.a.cu 2
28.d even 2 1 1176.2.q.l 4
28.f even 6 1 1176.2.a.n 2
28.f even 6 1 1176.2.q.l 4
28.g odd 6 1 168.2.q.c 4
28.g odd 6 1 1176.2.a.k 2
56.j odd 6 1 9408.2.a.dw 2
56.k odd 6 1 1344.2.q.w 4
56.k odd 6 1 9408.2.a.ec 2
56.m even 6 1 9408.2.a.dj 2
56.p even 6 1 1344.2.q.x 4
56.p even 6 1 9408.2.a.dp 2
84.h odd 2 1 3528.2.s.bk 4
84.j odd 6 1 3528.2.a.bd 2
84.j odd 6 1 3528.2.s.bk 4
84.n even 6 1 504.2.s.i 4
84.n even 6 1 3528.2.a.bk 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 4.b odd 2 1
168.2.q.c 4 28.g odd 6 1
336.2.q.g 4 1.a even 1 1 trivial
336.2.q.g 4 7.c even 3 1 inner
504.2.s.i 4 12.b even 2 1
504.2.s.i 4 84.n even 6 1
1008.2.s.r 4 3.b odd 2 1
1008.2.s.r 4 21.h odd 6 1
1176.2.a.k 2 28.g odd 6 1
1176.2.a.n 2 28.f even 6 1
1176.2.q.l 4 28.d even 2 1
1176.2.q.l 4 28.f even 6 1
1344.2.q.w 4 8.d odd 2 1
1344.2.q.w 4 56.k odd 6 1
1344.2.q.x 4 8.b even 2 1
1344.2.q.x 4 56.p even 6 1
2352.2.a.ba 2 7.d odd 6 1
2352.2.a.bf 2 7.c even 3 1
2352.2.q.bf 4 7.b odd 2 1
2352.2.q.bf 4 7.d odd 6 1
3528.2.a.bd 2 84.j odd 6 1
3528.2.a.bk 2 84.n even 6 1
3528.2.s.bk 4 84.h odd 2 1
3528.2.s.bk 4 84.j odd 6 1
7056.2.a.ch 2 21.g even 6 1
7056.2.a.cu 2 21.h odd 6 1
9408.2.a.dj 2 56.m even 6 1
9408.2.a.dp 2 56.p even 6 1
9408.2.a.dw 2 56.j odd 6 1
9408.2.a.ec 2 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 $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{3} + 15T_{5}^{2} + 14T_{5} + 196$$ T5^4 - T5^3 + 15*T5^2 + 14*T5 + 196 $$T_{11}^{4} + T_{11}^{3} + 15T_{11}^{2} - 14T_{11} + 196$$ T11^4 + T11^3 + 15*T11^2 - 14*T11 + 196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$T^{4} - T^{3} + \cdots + 196$$
$7$ $$(T^{2} - 3 T + 7)^{2}$$
$11$ $$T^{4} + T^{3} + \cdots + 196$$
$13$ $$(T^{2} - 5 T - 8)^{2}$$
$17$ $$(T^{2} - 4 T + 16)^{2}$$
$19$ $$T^{4} + 5 T^{3} + \cdots + 64$$
$23$ $$(T^{2} - 4 T + 16)^{2}$$
$29$ $$(T^{2} - 3 T - 12)^{2}$$
$31$ $$(T^{2} + T + 1)^{2}$$
$37$ $$T^{4} + 3 T^{3} + \cdots + 144$$
$41$ $$(T^{2} - 6 T - 48)^{2}$$
$43$ $$(T^{2} - 7 T - 2)^{2}$$
$47$ $$(T^{2} + 6 T + 36)^{2}$$
$53$ $$T^{4} + 11 T^{3} + \cdots + 256$$
$59$ $$T^{4} - 5 T^{3} + \cdots + 64$$
$61$ $$(T^{2} + 10 T + 100)^{2}$$
$67$ $$T^{4} - 7 T^{3} + \cdots + 4$$
$71$ $$(T + 2)^{4}$$
$73$ $$T^{4} - T^{3} + \cdots + 196$$
$79$ $$T^{4} - 8 T^{3} + \cdots + 1681$$
$83$ $$(T^{2} + 7 T - 2)^{2}$$
$89$ $$T^{4} - 6 T^{3} + \cdots + 2304$$
$97$ $$(T^{2} - 25 T + 142)^{2}$$