Properties

 Label 336.2.a.d Level $336$ Weight $2$ Character orbit 336.a Self dual yes Analytic conductor $2.683$ 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] = [336,2,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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{5} + q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 2 * q^5 + q^7 + q^9 $$q + q^{3} - 2 q^{5} + q^{7} + q^{9} + 4 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 4 q^{19} + q^{21} - 8 q^{23} - q^{25} + q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} - 10 q^{37} + 6 q^{39} - 6 q^{41} + 4 q^{43} - 2 q^{45} + q^{49} + 2 q^{51} + 6 q^{53} - 8 q^{55} + 4 q^{57} - 4 q^{59} + 6 q^{61} + q^{63} - 12 q^{65} - 4 q^{67} - 8 q^{69} - 8 q^{71} + 10 q^{73} - q^{75} + 4 q^{77} + q^{81} + 4 q^{83} - 4 q^{85} - 2 q^{87} - 6 q^{89} + 6 q^{91} - 8 q^{95} - 14 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^5 + q^7 + q^9 + 4 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^17 + 4 * q^19 + q^21 - 8 * q^23 - q^25 + q^27 - 2 * q^29 + 4 * q^33 - 2 * q^35 - 10 * q^37 + 6 * q^39 - 6 * q^41 + 4 * q^43 - 2 * q^45 + q^49 + 2 * q^51 + 6 * q^53 - 8 * q^55 + 4 * q^57 - 4 * q^59 + 6 * q^61 + q^63 - 12 * q^65 - 4 * q^67 - 8 * q^69 - 8 * q^71 + 10 * q^73 - q^75 + 4 * q^77 + q^81 + 4 * q^83 - 4 * q^85 - 2 * q^87 - 6 * q^89 + 6 * q^91 - 8 * q^95 - 14 * 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.d 1
3.b odd 2 1 1008.2.a.j 1
4.b odd 2 1 42.2.a.a 1
5.b even 2 1 8400.2.a.k 1
7.b odd 2 1 2352.2.a.l 1
7.c even 3 2 2352.2.q.i 2
7.d odd 6 2 2352.2.q.n 2
8.b even 2 1 1344.2.a.i 1
8.d odd 2 1 1344.2.a.q 1
12.b even 2 1 126.2.a.a 1
16.e even 4 2 5376.2.c.e 2
16.f odd 4 2 5376.2.c.bc 2
20.d odd 2 1 1050.2.a.i 1
20.e even 4 2 1050.2.g.a 2
21.c even 2 1 7056.2.a.k 1
24.f even 2 1 4032.2.a.e 1
24.h odd 2 1 4032.2.a.m 1
28.d even 2 1 294.2.a.g 1
28.f even 6 2 294.2.e.a 2
28.g odd 6 2 294.2.e.c 2
36.f odd 6 2 1134.2.f.g 2
36.h even 6 2 1134.2.f.j 2
44.c even 2 1 5082.2.a.d 1
52.b odd 2 1 7098.2.a.f 1
56.e even 2 1 9408.2.a.n 1
56.h odd 2 1 9408.2.a.bw 1
60.h even 2 1 3150.2.a.bo 1
60.l odd 4 2 3150.2.g.r 2
84.h odd 2 1 882.2.a.b 1
84.j odd 6 2 882.2.g.j 2
84.n even 6 2 882.2.g.h 2
140.c even 2 1 7350.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 4.b odd 2 1
126.2.a.a 1 12.b even 2 1
294.2.a.g 1 28.d even 2 1
294.2.e.a 2 28.f even 6 2
294.2.e.c 2 28.g odd 6 2
336.2.a.d 1 1.a even 1 1 trivial
882.2.a.b 1 84.h odd 2 1
882.2.g.h 2 84.n even 6 2
882.2.g.j 2 84.j odd 6 2
1008.2.a.j 1 3.b odd 2 1
1050.2.a.i 1 20.d odd 2 1
1050.2.g.a 2 20.e even 4 2
1134.2.f.g 2 36.f odd 6 2
1134.2.f.j 2 36.h even 6 2
1344.2.a.i 1 8.b even 2 1
1344.2.a.q 1 8.d odd 2 1
2352.2.a.l 1 7.b odd 2 1
2352.2.q.i 2 7.c even 3 2
2352.2.q.n 2 7.d odd 6 2
3150.2.a.bo 1 60.h even 2 1
3150.2.g.r 2 60.l odd 4 2
4032.2.a.e 1 24.f even 2 1
4032.2.a.m 1 24.h odd 2 1
5082.2.a.d 1 44.c even 2 1
5376.2.c.e 2 16.e even 4 2
5376.2.c.bc 2 16.f odd 4 2
7056.2.a.k 1 21.c even 2 1
7098.2.a.f 1 52.b odd 2 1
7350.2.a.f 1 140.c even 2 1
8400.2.a.k 1 5.b even 2 1
9408.2.a.n 1 56.e even 2 1
9408.2.a.bw 1 56.h odd 2 1

Hecke kernels

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

 $$T_{5} + 2$$ T5 + 2 $$T_{11} - 4$$ T11 - 4 $$T_{13} - 6$$ T13 - 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T - 1$$
$11$ $$T - 4$$
$13$ $$T - 6$$
$17$ $$T - 2$$
$19$ $$T - 4$$
$23$ $$T + 8$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T + 10$$
$41$ $$T + 6$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 4$$
$61$ $$T - 6$$
$67$ $$T + 4$$
$71$ $$T + 8$$
$73$ $$T - 10$$
$79$ $$T$$
$83$ $$T - 4$$
$89$ $$T + 6$$
$97$ $$T + 14$$