# Properties

 Label 336.2.a.e Level $336$ Weight $2$ Character orbit 336.a Self dual yes Analytic conductor $2.683$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,2,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## 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: yes Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} + 2 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + 2 * q^5 - q^7 + q^9 $$q + q^{3} + 2 q^{5} - q^{7} + q^{9} + 6 q^{13} + 2 q^{15} - 2 q^{17} - 4 q^{19} - q^{21} + 4 q^{23} - q^{25} + q^{27} - 10 q^{29} + 8 q^{31} - 2 q^{35} + 6 q^{37} + 6 q^{39} - 2 q^{41} + 4 q^{43} + 2 q^{45} - 8 q^{47} + q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{57} - 12 q^{59} - 2 q^{61} - q^{63} + 12 q^{65} - 12 q^{67} + 4 q^{69} + 12 q^{71} - 14 q^{73} - q^{75} + 8 q^{79} + q^{81} - 12 q^{83} - 4 q^{85} - 10 q^{87} - 2 q^{89} - 6 q^{91} + 8 q^{93} - 8 q^{95} + 10 q^{97}+O(q^{100})$$ q + q^3 + 2 * q^5 - q^7 + q^9 + 6 * q^13 + 2 * q^15 - 2 * q^17 - 4 * q^19 - q^21 + 4 * q^23 - q^25 + q^27 - 10 * q^29 + 8 * q^31 - 2 * q^35 + 6 * q^37 + 6 * q^39 - 2 * q^41 + 4 * q^43 + 2 * q^45 - 8 * q^47 + q^49 - 2 * q^51 - 10 * q^53 - 4 * q^57 - 12 * q^59 - 2 * q^61 - q^63 + 12 * q^65 - 12 * q^67 + 4 * q^69 + 12 * q^71 - 14 * q^73 - q^75 + 8 * q^79 + q^81 - 12 * q^83 - 4 * q^85 - 10 * q^87 - 2 * q^89 - 6 * q^91 + 8 * q^93 - 8 * q^95 + 10 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 2.00000 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.a.e 1
3.b odd 2 1 1008.2.a.b 1
4.b odd 2 1 168.2.a.a 1
5.b even 2 1 8400.2.a.y 1
7.b odd 2 1 2352.2.a.c 1
7.c even 3 2 2352.2.q.d 2
7.d odd 6 2 2352.2.q.w 2
8.b even 2 1 1344.2.a.b 1
8.d odd 2 1 1344.2.a.m 1
12.b even 2 1 504.2.a.e 1
16.e even 4 2 5376.2.c.bb 2
16.f odd 4 2 5376.2.c.d 2
20.d odd 2 1 4200.2.a.t 1
20.e even 4 2 4200.2.t.j 2
21.c even 2 1 7056.2.a.bq 1
24.f even 2 1 4032.2.a.bh 1
24.h odd 2 1 4032.2.a.bc 1
28.d even 2 1 1176.2.a.f 1
28.f even 6 2 1176.2.q.d 2
28.g odd 6 2 1176.2.q.f 2
56.e even 2 1 9408.2.a.be 1
56.h odd 2 1 9408.2.a.da 1
84.h odd 2 1 3528.2.a.v 1
84.j odd 6 2 3528.2.s.g 2
84.n even 6 2 3528.2.s.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 4.b odd 2 1
336.2.a.e 1 1.a even 1 1 trivial
504.2.a.e 1 12.b even 2 1
1008.2.a.b 1 3.b odd 2 1
1176.2.a.f 1 28.d even 2 1
1176.2.q.d 2 28.f even 6 2
1176.2.q.f 2 28.g odd 6 2
1344.2.a.b 1 8.b even 2 1
1344.2.a.m 1 8.d odd 2 1
2352.2.a.c 1 7.b odd 2 1
2352.2.q.d 2 7.c even 3 2
2352.2.q.w 2 7.d odd 6 2
3528.2.a.v 1 84.h odd 2 1
3528.2.s.g 2 84.j odd 6 2
3528.2.s.w 2 84.n even 6 2
4032.2.a.bc 1 24.h odd 2 1
4032.2.a.bh 1 24.f even 2 1
4200.2.a.t 1 20.d odd 2 1
4200.2.t.j 2 20.e even 4 2
5376.2.c.d 2 16.f odd 4 2
5376.2.c.bb 2 16.e even 4 2
7056.2.a.bq 1 21.c even 2 1
8400.2.a.y 1 5.b even 2 1
9408.2.a.be 1 56.e even 2 1
9408.2.a.da 1 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{11}$$ T11 $$T_{13} - 6$$ T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 2$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T + 2$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T + 10$$
$31$ $$T - 8$$
$37$ $$T - 6$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T + 10$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T + 12$$
$71$ $$T - 12$$
$73$ $$T + 14$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T + 2$$
$97$ $$T - 10$$