# Properties

 Label 336.6.a.b Level $336$ Weight $6$ Character orbit 336.a Self dual yes Analytic conductor $53.889$ 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] = [336,6,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, 6, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$53.8889634572$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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)

 $$f(q)$$ $$=$$ $$q - 9 q^{3} - 72 q^{5} - 49 q^{7} + 81 q^{9}+O(q^{10})$$ q - 9 * q^3 - 72 * q^5 - 49 * q^7 + 81 * q^9 $$q - 9 q^{3} - 72 q^{5} - 49 q^{7} + 81 q^{9} + 414 q^{11} - 1054 q^{13} + 648 q^{15} - 1848 q^{17} - 236 q^{19} + 441 q^{21} - 2898 q^{23} + 2059 q^{25} - 729 q^{27} - 6522 q^{29} - 6200 q^{31} - 3726 q^{33} + 3528 q^{35} + 9650 q^{37} + 9486 q^{39} + 8484 q^{41} + 10804 q^{43} - 5832 q^{45} - 60 q^{47} + 2401 q^{49} + 16632 q^{51} + 22506 q^{53} - 29808 q^{55} + 2124 q^{57} + 28176 q^{59} - 35194 q^{61} - 3969 q^{63} + 75888 q^{65} + 28216 q^{67} + 26082 q^{69} + 6642 q^{71} - 52090 q^{73} - 18531 q^{75} - 20286 q^{77} - 43340 q^{79} + 6561 q^{81} - 25716 q^{83} + 133056 q^{85} + 58698 q^{87} + 98724 q^{89} + 51646 q^{91} + 55800 q^{93} + 16992 q^{95} - 148954 q^{97} + 33534 q^{99}+O(q^{100})$$ q - 9 * q^3 - 72 * q^5 - 49 * q^7 + 81 * q^9 + 414 * q^11 - 1054 * q^13 + 648 * q^15 - 1848 * q^17 - 236 * q^19 + 441 * q^21 - 2898 * q^23 + 2059 * q^25 - 729 * q^27 - 6522 * q^29 - 6200 * q^31 - 3726 * q^33 + 3528 * q^35 + 9650 * q^37 + 9486 * q^39 + 8484 * q^41 + 10804 * q^43 - 5832 * q^45 - 60 * q^47 + 2401 * q^49 + 16632 * q^51 + 22506 * q^53 - 29808 * q^55 + 2124 * q^57 + 28176 * q^59 - 35194 * q^61 - 3969 * q^63 + 75888 * q^65 + 28216 * q^67 + 26082 * q^69 + 6642 * q^71 - 52090 * q^73 - 18531 * q^75 - 20286 * q^77 - 43340 * q^79 + 6561 * q^81 - 25716 * q^83 + 133056 * q^85 + 58698 * q^87 + 98724 * q^89 + 51646 * q^91 + 55800 * q^93 + 16992 * q^95 - 148954 * q^97 + 33534 * q^99

## 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 −9.00000 0 −72.0000 0 −49.0000 0 81.0000 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 336.6.a.b 1
3.b odd 2 1 1008.6.a.ba 1
4.b odd 2 1 42.6.a.c 1
12.b even 2 1 126.6.a.l 1
20.d odd 2 1 1050.6.a.g 1
20.e even 4 2 1050.6.g.b 2
28.d even 2 1 294.6.a.c 1
28.f even 6 2 294.6.e.n 2
28.g odd 6 2 294.6.e.l 2
84.h odd 2 1 882.6.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.c 1 4.b odd 2 1
126.6.a.l 1 12.b even 2 1
294.6.a.c 1 28.d even 2 1
294.6.e.l 2 28.g odd 6 2
294.6.e.n 2 28.f even 6 2
336.6.a.b 1 1.a even 1 1 trivial
882.6.a.n 1 84.h odd 2 1
1008.6.a.ba 1 3.b odd 2 1
1050.6.a.g 1 20.d odd 2 1
1050.6.g.b 2 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} + 72$$ T5 + 72 $$T_{11} - 414$$ T11 - 414

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 9$$
$5$ $$T + 72$$
$7$ $$T + 49$$
$11$ $$T - 414$$
$13$ $$T + 1054$$
$17$ $$T + 1848$$
$19$ $$T + 236$$
$23$ $$T + 2898$$
$29$ $$T + 6522$$
$31$ $$T + 6200$$
$37$ $$T - 9650$$
$41$ $$T - 8484$$
$43$ $$T - 10804$$
$47$ $$T + 60$$
$53$ $$T - 22506$$
$59$ $$T - 28176$$
$61$ $$T + 35194$$
$67$ $$T - 28216$$
$71$ $$T - 6642$$
$73$ $$T + 52090$$
$79$ $$T + 43340$$
$83$ $$T + 25716$$
$89$ $$T - 98724$$
$97$ $$T + 148954$$