# Properties

 Label 336.6.a.r Level $336$ Weight $6$ Character orbit 336.a Self dual yes Analytic conductor $53.889$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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)

 $$f(q)$$ $$=$$ $$q + 9 q^{3} + 78 q^{5} - 49 q^{7} + 81 q^{9}+O(q^{10})$$ q + 9 * q^3 + 78 * q^5 - 49 * q^7 + 81 * q^9 $$q + 9 q^{3} + 78 q^{5} - 49 q^{7} + 81 q^{9} - 444 q^{11} - 442 q^{13} + 702 q^{15} - 126 q^{17} - 2684 q^{19} - 441 q^{21} - 4200 q^{23} + 2959 q^{25} + 729 q^{27} - 5442 q^{29} - 80 q^{31} - 3996 q^{33} - 3822 q^{35} - 5434 q^{37} - 3978 q^{39} + 7962 q^{41} + 11524 q^{43} + 6318 q^{45} + 13920 q^{47} + 2401 q^{49} - 1134 q^{51} - 9594 q^{53} - 34632 q^{55} - 24156 q^{57} - 27492 q^{59} + 49478 q^{61} - 3969 q^{63} - 34476 q^{65} + 59356 q^{67} - 37800 q^{69} - 32040 q^{71} - 61846 q^{73} + 26631 q^{75} + 21756 q^{77} + 65776 q^{79} + 6561 q^{81} - 40188 q^{83} - 9828 q^{85} - 48978 q^{87} - 7974 q^{89} + 21658 q^{91} - 720 q^{93} - 209352 q^{95} - 143662 q^{97} - 35964 q^{99}+O(q^{100})$$ q + 9 * q^3 + 78 * q^5 - 49 * q^7 + 81 * q^9 - 444 * q^11 - 442 * q^13 + 702 * q^15 - 126 * q^17 - 2684 * q^19 - 441 * q^21 - 4200 * q^23 + 2959 * q^25 + 729 * q^27 - 5442 * q^29 - 80 * q^31 - 3996 * q^33 - 3822 * q^35 - 5434 * q^37 - 3978 * q^39 + 7962 * q^41 + 11524 * q^43 + 6318 * q^45 + 13920 * q^47 + 2401 * q^49 - 1134 * q^51 - 9594 * q^53 - 34632 * q^55 - 24156 * q^57 - 27492 * q^59 + 49478 * q^61 - 3969 * q^63 - 34476 * q^65 + 59356 * q^67 - 37800 * q^69 - 32040 * q^71 - 61846 * q^73 + 26631 * q^75 + 21756 * q^77 + 65776 * q^79 + 6561 * q^81 - 40188 * q^83 - 9828 * q^85 - 48978 * q^87 - 7974 * q^89 + 21658 * q^91 - 720 * q^93 - 209352 * q^95 - 143662 * q^97 - 35964 * 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 78.0000 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.r 1
3.b odd 2 1 1008.6.a.c 1
4.b odd 2 1 21.6.a.a 1
12.b even 2 1 63.6.a.d 1
20.d odd 2 1 525.6.a.d 1
20.e even 4 2 525.6.d.b 2
28.d even 2 1 147.6.a.b 1
28.f even 6 2 147.6.e.i 2
28.g odd 6 2 147.6.e.j 2
84.h odd 2 1 441.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 4.b odd 2 1
63.6.a.d 1 12.b even 2 1
147.6.a.b 1 28.d even 2 1
147.6.e.i 2 28.f even 6 2
147.6.e.j 2 28.g odd 6 2
336.6.a.r 1 1.a even 1 1 trivial
441.6.a.j 1 84.h odd 2 1
525.6.a.d 1 20.d odd 2 1
525.6.d.b 2 20.e even 4 2
1008.6.a.c 1 3.b odd 2 1

## Hecke kernels

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

 $$T_{5} - 78$$ T5 - 78 $$T_{11} + 444$$ T11 + 444

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 9$$
$5$ $$T - 78$$
$7$ $$T + 49$$
$11$ $$T + 444$$
$13$ $$T + 442$$
$17$ $$T + 126$$
$19$ $$T + 2684$$
$23$ $$T + 4200$$
$29$ $$T + 5442$$
$31$ $$T + 80$$
$37$ $$T + 5434$$
$41$ $$T - 7962$$
$43$ $$T - 11524$$
$47$ $$T - 13920$$
$53$ $$T + 9594$$
$59$ $$T + 27492$$
$61$ $$T - 49478$$
$67$ $$T - 59356$$
$71$ $$T + 32040$$
$73$ $$T + 61846$$
$79$ $$T - 65776$$
$83$ $$T + 40188$$
$89$ $$T + 7974$$
$97$ $$T + 143662$$