# Properties

 Label 336.6.a.h 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 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)

 $$f(q)$$ $$=$$ $$q - 9 q^{3} + 26 q^{5} + 49 q^{7} + 81 q^{9}+O(q^{10})$$ q - 9 * q^3 + 26 * q^5 + 49 * q^7 + 81 * q^9 $$q - 9 q^{3} + 26 q^{5} + 49 q^{7} + 81 q^{9} - 664 q^{11} + 318 q^{13} - 234 q^{15} + 1582 q^{17} - 236 q^{19} - 441 q^{21} - 2212 q^{23} - 2449 q^{25} - 729 q^{27} - 4954 q^{29} + 7128 q^{31} + 5976 q^{33} + 1274 q^{35} + 4358 q^{37} - 2862 q^{39} + 10542 q^{41} + 8452 q^{43} + 2106 q^{45} - 5352 q^{47} + 2401 q^{49} - 14238 q^{51} - 33354 q^{53} - 17264 q^{55} + 2124 q^{57} + 15436 q^{59} - 36762 q^{61} + 3969 q^{63} + 8268 q^{65} - 40972 q^{67} + 19908 q^{69} + 9092 q^{71} - 73454 q^{73} + 22041 q^{75} - 32536 q^{77} - 89400 q^{79} + 6561 q^{81} + 6428 q^{83} + 41132 q^{85} + 44586 q^{87} - 122658 q^{89} + 15582 q^{91} - 64152 q^{93} - 6136 q^{95} + 21370 q^{97} - 53784 q^{99}+O(q^{100})$$ q - 9 * q^3 + 26 * q^5 + 49 * q^7 + 81 * q^9 - 664 * q^11 + 318 * q^13 - 234 * q^15 + 1582 * q^17 - 236 * q^19 - 441 * q^21 - 2212 * q^23 - 2449 * q^25 - 729 * q^27 - 4954 * q^29 + 7128 * q^31 + 5976 * q^33 + 1274 * q^35 + 4358 * q^37 - 2862 * q^39 + 10542 * q^41 + 8452 * q^43 + 2106 * q^45 - 5352 * q^47 + 2401 * q^49 - 14238 * q^51 - 33354 * q^53 - 17264 * q^55 + 2124 * q^57 + 15436 * q^59 - 36762 * q^61 + 3969 * q^63 + 8268 * q^65 - 40972 * q^67 + 19908 * q^69 + 9092 * q^71 - 73454 * q^73 + 22041 * q^75 - 32536 * q^77 - 89400 * q^79 + 6561 * q^81 + 6428 * q^83 + 41132 * q^85 + 44586 * q^87 - 122658 * q^89 + 15582 * q^91 - 64152 * q^93 - 6136 * q^95 + 21370 * q^97 - 53784 * 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 26.0000 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.h 1
3.b odd 2 1 1008.6.a.j 1
4.b odd 2 1 42.6.a.d 1
12.b even 2 1 126.6.a.i 1
20.d odd 2 1 1050.6.a.k 1
20.e even 4 2 1050.6.g.i 2
28.d even 2 1 294.6.a.b 1
28.f even 6 2 294.6.e.p 2
28.g odd 6 2 294.6.e.i 2
84.h odd 2 1 882.6.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.d 1 4.b odd 2 1
126.6.a.i 1 12.b even 2 1
294.6.a.b 1 28.d even 2 1
294.6.e.i 2 28.g odd 6 2
294.6.e.p 2 28.f even 6 2
336.6.a.h 1 1.a even 1 1 trivial
882.6.a.s 1 84.h odd 2 1
1008.6.a.j 1 3.b odd 2 1
1050.6.a.k 1 20.d odd 2 1
1050.6.g.i 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} - 26$$ T5 - 26 $$T_{11} + 664$$ T11 + 664

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 9$$
$5$ $$T - 26$$
$7$ $$T - 49$$
$11$ $$T + 664$$
$13$ $$T - 318$$
$17$ $$T - 1582$$
$19$ $$T + 236$$
$23$ $$T + 2212$$
$29$ $$T + 4954$$
$31$ $$T - 7128$$
$37$ $$T - 4358$$
$41$ $$T - 10542$$
$43$ $$T - 8452$$
$47$ $$T + 5352$$
$53$ $$T + 33354$$
$59$ $$T - 15436$$
$61$ $$T + 36762$$
$67$ $$T + 40972$$
$71$ $$T - 9092$$
$73$ $$T + 73454$$
$79$ $$T + 89400$$
$83$ $$T - 6428$$
$89$ $$T + 122658$$
$97$ $$T - 21370$$