Properties

 Label 336.6.a.c 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 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)

 $$f(q)$$ $$=$$ $$q - 9 q^{3} - 64 q^{5} - 49 q^{7} + 81 q^{9}+O(q^{10})$$ q - 9 * q^3 - 64 * q^5 - 49 * q^7 + 81 * q^9 $$q - 9 q^{3} - 64 q^{5} - 49 q^{7} + 81 q^{9} + 54 q^{11} + 738 q^{13} + 576 q^{15} - 848 q^{17} + 1604 q^{19} + 441 q^{21} + 3670 q^{23} + 971 q^{25} - 729 q^{27} - 4330 q^{29} + 4760 q^{31} - 486 q^{33} + 3136 q^{35} - 2094 q^{37} - 6642 q^{39} - 6116 q^{41} - 7916 q^{43} - 5184 q^{45} - 6572 q^{47} + 2401 q^{49} + 7632 q^{51} - 7894 q^{53} - 3456 q^{55} - 14436 q^{57} + 41664 q^{59} - 26570 q^{61} - 3969 q^{63} - 47232 q^{65} + 41736 q^{67} - 33030 q^{69} - 83574 q^{71} - 42314 q^{73} - 8739 q^{75} - 2646 q^{77} - 508 q^{79} + 6561 q^{81} + 8364 q^{83} + 54272 q^{85} + 38970 q^{87} - 49220 q^{89} - 36162 q^{91} - 42840 q^{93} - 102656 q^{95} + 159670 q^{97} + 4374 q^{99}+O(q^{100})$$ q - 9 * q^3 - 64 * q^5 - 49 * q^7 + 81 * q^9 + 54 * q^11 + 738 * q^13 + 576 * q^15 - 848 * q^17 + 1604 * q^19 + 441 * q^21 + 3670 * q^23 + 971 * q^25 - 729 * q^27 - 4330 * q^29 + 4760 * q^31 - 486 * q^33 + 3136 * q^35 - 2094 * q^37 - 6642 * q^39 - 6116 * q^41 - 7916 * q^43 - 5184 * q^45 - 6572 * q^47 + 2401 * q^49 + 7632 * q^51 - 7894 * q^53 - 3456 * q^55 - 14436 * q^57 + 41664 * q^59 - 26570 * q^61 - 3969 * q^63 - 47232 * q^65 + 41736 * q^67 - 33030 * q^69 - 83574 * q^71 - 42314 * q^73 - 8739 * q^75 - 2646 * q^77 - 508 * q^79 + 6561 * q^81 + 8364 * q^83 + 54272 * q^85 + 38970 * q^87 - 49220 * q^89 - 36162 * q^91 - 42840 * q^93 - 102656 * q^95 + 159670 * q^97 + 4374 * 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 −64.0000 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.c 1
3.b odd 2 1 1008.6.a.z 1
4.b odd 2 1 168.6.a.d 1
12.b even 2 1 504.6.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.d 1 4.b odd 2 1
336.6.a.c 1 1.a even 1 1 trivial
504.6.a.h 1 12.b even 2 1
1008.6.a.z 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} + 64$$ T5 + 64 $$T_{11} - 54$$ T11 - 54

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 9$$
$5$ $$T + 64$$
$7$ $$T + 49$$
$11$ $$T - 54$$
$13$ $$T - 738$$
$17$ $$T + 848$$
$19$ $$T - 1604$$
$23$ $$T - 3670$$
$29$ $$T + 4330$$
$31$ $$T - 4760$$
$37$ $$T + 2094$$
$41$ $$T + 6116$$
$43$ $$T + 7916$$
$47$ $$T + 6572$$
$53$ $$T + 7894$$
$59$ $$T - 41664$$
$61$ $$T + 26570$$
$67$ $$T - 41736$$
$71$ $$T + 83574$$
$73$ $$T + 42314$$
$79$ $$T + 508$$
$83$ $$T - 8364$$
$89$ $$T + 49220$$
$97$ $$T - 159670$$