# Properties

 Label 336.2.a.a Level $336$ Weight $2$ Character orbit 336.a Self dual yes Analytic conductor $2.683$ 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] = [336,2,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, 2, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$2.68297350792$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{5} + q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^5 + q^7 + q^9 $$q - q^{3} - 2 q^{5} + q^{7} + q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} - q^{21} - q^{25} - q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} + 6 q^{37} + 2 q^{39} + 2 q^{41} + 4 q^{43} - 2 q^{45} + q^{49} + 6 q^{51} + 6 q^{53} + 8 q^{55} + 4 q^{57} - 12 q^{59} - 2 q^{61} + q^{63} + 4 q^{65} - 4 q^{67} - 6 q^{73} + q^{75} - 4 q^{77} + 16 q^{79} + q^{81} + 12 q^{83} + 12 q^{85} + 2 q^{87} - 14 q^{89} - 2 q^{91} + 8 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100})$$ q - q^3 - 2 * q^5 + q^7 + q^9 - 4 * q^11 - 2 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^19 - q^21 - q^25 - q^27 - 2 * q^29 + 4 * q^33 - 2 * q^35 + 6 * q^37 + 2 * q^39 + 2 * q^41 + 4 * q^43 - 2 * q^45 + q^49 + 6 * q^51 + 6 * q^53 + 8 * q^55 + 4 * q^57 - 12 * q^59 - 2 * q^61 + q^63 + 4 * q^65 - 4 * q^67 - 6 * q^73 + q^75 - 4 * q^77 + 16 * q^79 + q^81 + 12 * q^83 + 12 * q^85 + 2 * q^87 - 14 * q^89 - 2 * q^91 + 8 * q^95 + 18 * q^97 - 4 * 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 −1.00000 0 −2.00000 0 1.00000 0 1.00000 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 336.2.a.a 1
3.b odd 2 1 1008.2.a.l 1
4.b odd 2 1 21.2.a.a 1
5.b even 2 1 8400.2.a.bn 1
7.b odd 2 1 2352.2.a.v 1
7.c even 3 2 2352.2.q.x 2
7.d odd 6 2 2352.2.q.e 2
8.b even 2 1 1344.2.a.s 1
8.d odd 2 1 1344.2.a.g 1
12.b even 2 1 63.2.a.a 1
16.e even 4 2 5376.2.c.l 2
16.f odd 4 2 5376.2.c.r 2
20.d odd 2 1 525.2.a.d 1
20.e even 4 2 525.2.d.a 2
21.c even 2 1 7056.2.a.p 1
24.f even 2 1 4032.2.a.h 1
24.h odd 2 1 4032.2.a.k 1
28.d even 2 1 147.2.a.a 1
28.f even 6 2 147.2.e.c 2
28.g odd 6 2 147.2.e.b 2
36.f odd 6 2 567.2.f.g 2
36.h even 6 2 567.2.f.b 2
44.c even 2 1 2541.2.a.j 1
52.b odd 2 1 3549.2.a.c 1
56.e even 2 1 9408.2.a.bv 1
56.h odd 2 1 9408.2.a.m 1
60.h even 2 1 1575.2.a.c 1
60.l odd 4 2 1575.2.d.a 2
68.d odd 2 1 6069.2.a.b 1
76.d even 2 1 7581.2.a.d 1
84.h odd 2 1 441.2.a.f 1
84.j odd 6 2 441.2.e.b 2
84.n even 6 2 441.2.e.a 2
132.d odd 2 1 7623.2.a.g 1
140.c even 2 1 3675.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 4.b odd 2 1
63.2.a.a 1 12.b even 2 1
147.2.a.a 1 28.d even 2 1
147.2.e.b 2 28.g odd 6 2
147.2.e.c 2 28.f even 6 2
336.2.a.a 1 1.a even 1 1 trivial
441.2.a.f 1 84.h odd 2 1
441.2.e.a 2 84.n even 6 2
441.2.e.b 2 84.j odd 6 2
525.2.a.d 1 20.d odd 2 1
525.2.d.a 2 20.e even 4 2
567.2.f.b 2 36.h even 6 2
567.2.f.g 2 36.f odd 6 2
1008.2.a.l 1 3.b odd 2 1
1344.2.a.g 1 8.d odd 2 1
1344.2.a.s 1 8.b even 2 1
1575.2.a.c 1 60.h even 2 1
1575.2.d.a 2 60.l odd 4 2
2352.2.a.v 1 7.b odd 2 1
2352.2.q.e 2 7.d odd 6 2
2352.2.q.x 2 7.c even 3 2
2541.2.a.j 1 44.c even 2 1
3549.2.a.c 1 52.b odd 2 1
3675.2.a.n 1 140.c even 2 1
4032.2.a.h 1 24.f even 2 1
4032.2.a.k 1 24.h odd 2 1
5376.2.c.l 2 16.e even 4 2
5376.2.c.r 2 16.f odd 4 2
6069.2.a.b 1 68.d odd 2 1
7056.2.a.p 1 21.c even 2 1
7581.2.a.d 1 76.d even 2 1
7623.2.a.g 1 132.d odd 2 1
8400.2.a.bn 1 5.b even 2 1
9408.2.a.m 1 56.h odd 2 1
9408.2.a.bv 1 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(336))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 2$$
$7$ $$T - 1$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T - 6$$
$41$ $$T - 2$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T - 16$$
$83$ $$T - 12$$
$89$ $$T + 14$$
$97$ $$T - 18$$