# Properties

 Label 336.6.a.l 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 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)

 $$f(q)$$ $$=$$ $$q + 9 q^{3} - 34 q^{5} + 49 q^{7} + 81 q^{9}+O(q^{10})$$ q + 9 * q^3 - 34 * q^5 + 49 * q^7 + 81 * q^9 $$q + 9 q^{3} - 34 q^{5} + 49 q^{7} + 81 q^{9} + 340 q^{11} + 454 q^{13} - 306 q^{15} - 798 q^{17} - 892 q^{19} + 441 q^{21} + 3192 q^{23} - 1969 q^{25} + 729 q^{27} - 8242 q^{29} + 2496 q^{31} + 3060 q^{33} - 1666 q^{35} + 9798 q^{37} + 4086 q^{39} + 19834 q^{41} + 17236 q^{43} - 2754 q^{45} - 8928 q^{47} + 2401 q^{49} - 7182 q^{51} + 150 q^{53} - 11560 q^{55} - 8028 q^{57} + 42396 q^{59} + 14758 q^{61} + 3969 q^{63} - 15436 q^{65} + 1676 q^{67} + 28728 q^{69} - 14568 q^{71} + 78378 q^{73} - 17721 q^{75} + 16660 q^{77} + 2272 q^{79} + 6561 q^{81} + 37764 q^{83} + 27132 q^{85} - 74178 q^{87} - 117286 q^{89} + 22246 q^{91} + 22464 q^{93} + 30328 q^{95} + 10002 q^{97} + 27540 q^{99}+O(q^{100})$$ q + 9 * q^3 - 34 * q^5 + 49 * q^7 + 81 * q^9 + 340 * q^11 + 454 * q^13 - 306 * q^15 - 798 * q^17 - 892 * q^19 + 441 * q^21 + 3192 * q^23 - 1969 * q^25 + 729 * q^27 - 8242 * q^29 + 2496 * q^31 + 3060 * q^33 - 1666 * q^35 + 9798 * q^37 + 4086 * q^39 + 19834 * q^41 + 17236 * q^43 - 2754 * q^45 - 8928 * q^47 + 2401 * q^49 - 7182 * q^51 + 150 * q^53 - 11560 * q^55 - 8028 * q^57 + 42396 * q^59 + 14758 * q^61 + 3969 * q^63 - 15436 * q^65 + 1676 * q^67 + 28728 * q^69 - 14568 * q^71 + 78378 * q^73 - 17721 * q^75 + 16660 * q^77 + 2272 * q^79 + 6561 * q^81 + 37764 * q^83 + 27132 * q^85 - 74178 * q^87 - 117286 * q^89 + 22246 * q^91 + 22464 * q^93 + 30328 * q^95 + 10002 * q^97 + 27540 * 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 −34.0000 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.l 1
3.b odd 2 1 1008.6.a.t 1
4.b odd 2 1 21.6.a.b 1
12.b even 2 1 63.6.a.c 1
20.d odd 2 1 525.6.a.c 1
20.e even 4 2 525.6.d.d 2
28.d even 2 1 147.6.a.e 1
28.f even 6 2 147.6.e.e 2
28.g odd 6 2 147.6.e.f 2
84.h odd 2 1 441.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 4.b odd 2 1
63.6.a.c 1 12.b even 2 1
147.6.a.e 1 28.d even 2 1
147.6.e.e 2 28.f even 6 2
147.6.e.f 2 28.g odd 6 2
336.6.a.l 1 1.a even 1 1 trivial
441.6.a.d 1 84.h odd 2 1
525.6.a.c 1 20.d odd 2 1
525.6.d.d 2 20.e even 4 2
1008.6.a.t 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} + 34$$ T5 + 34 $$T_{11} - 340$$ T11 - 340

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 9$$
$5$ $$T + 34$$
$7$ $$T - 49$$
$11$ $$T - 340$$
$13$ $$T - 454$$
$17$ $$T + 798$$
$19$ $$T + 892$$
$23$ $$T - 3192$$
$29$ $$T + 8242$$
$31$ $$T - 2496$$
$37$ $$T - 9798$$
$41$ $$T - 19834$$
$43$ $$T - 17236$$
$47$ $$T + 8928$$
$53$ $$T - 150$$
$59$ $$T - 42396$$
$61$ $$T - 14758$$
$67$ $$T - 1676$$
$71$ $$T + 14568$$
$73$ $$T - 78378$$
$79$ $$T - 2272$$
$83$ $$T - 37764$$
$89$ $$T + 117286$$
$97$ $$T - 10002$$