# Properties

 Label 338.2.a.d Level $338$ Weight $2$ Character orbit 338.a Self dual yes Analytic conductor $2.699$ 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] = [338,2,Mod(1,338)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$2.69894358832$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - 3 * q^5 - q^6 - 3 * q^7 + q^8 - 2 * q^9 $$q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9} - 3 q^{10} - q^{12} - 3 q^{14} + 3 q^{15} + q^{16} - 3 q^{17} - 2 q^{18} - 6 q^{19} - 3 q^{20} + 3 q^{21} + 6 q^{23} - q^{24} + 4 q^{25} + 5 q^{27} - 3 q^{28} + 3 q^{30} + q^{32} - 3 q^{34} + 9 q^{35} - 2 q^{36} - 3 q^{37} - 6 q^{38} - 3 q^{40} + 3 q^{42} + q^{43} + 6 q^{45} + 6 q^{46} - 3 q^{47} - q^{48} + 2 q^{49} + 4 q^{50} + 3 q^{51} - 6 q^{53} + 5 q^{54} - 3 q^{56} + 6 q^{57} + 6 q^{59} + 3 q^{60} - 8 q^{61} + 6 q^{63} + q^{64} - 12 q^{67} - 3 q^{68} - 6 q^{69} + 9 q^{70} + 15 q^{71} - 2 q^{72} - 6 q^{73} - 3 q^{74} - 4 q^{75} - 6 q^{76} + 10 q^{79} - 3 q^{80} + q^{81} + 6 q^{83} + 3 q^{84} + 9 q^{85} + q^{86} + 6 q^{89} + 6 q^{90} + 6 q^{92} - 3 q^{94} + 18 q^{95} - q^{96} - 12 q^{97} + 2 q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 - 3 * q^5 - q^6 - 3 * q^7 + q^8 - 2 * q^9 - 3 * q^10 - q^12 - 3 * q^14 + 3 * q^15 + q^16 - 3 * q^17 - 2 * q^18 - 6 * q^19 - 3 * q^20 + 3 * q^21 + 6 * q^23 - q^24 + 4 * q^25 + 5 * q^27 - 3 * q^28 + 3 * q^30 + q^32 - 3 * q^34 + 9 * q^35 - 2 * q^36 - 3 * q^37 - 6 * q^38 - 3 * q^40 + 3 * q^42 + q^43 + 6 * q^45 + 6 * q^46 - 3 * q^47 - q^48 + 2 * q^49 + 4 * q^50 + 3 * q^51 - 6 * q^53 + 5 * q^54 - 3 * q^56 + 6 * q^57 + 6 * q^59 + 3 * q^60 - 8 * q^61 + 6 * q^63 + q^64 - 12 * q^67 - 3 * q^68 - 6 * q^69 + 9 * q^70 + 15 * q^71 - 2 * q^72 - 6 * q^73 - 3 * q^74 - 4 * q^75 - 6 * q^76 + 10 * q^79 - 3 * q^80 + q^81 + 6 * q^83 + 3 * q^84 + 9 * q^85 + q^86 + 6 * q^89 + 6 * q^90 + 6 * q^92 - 3 * q^94 + 18 * q^95 - q^96 - 12 * q^97 + 2 * q^98

## 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
1.00000 −1.00000 1.00000 −3.00000 −1.00000 −3.00000 1.00000 −2.00000 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$13$$ $$-1$$

## 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 338.2.a.d 1
3.b odd 2 1 3042.2.a.g 1
4.b odd 2 1 2704.2.a.j 1
5.b even 2 1 8450.2.a.h 1
13.b even 2 1 338.2.a.b 1
13.c even 3 2 338.2.c.b 2
13.d odd 4 2 26.2.b.a 2
13.e even 6 2 338.2.c.f 2
13.f odd 12 4 338.2.e.c 4
39.d odd 2 1 3042.2.a.j 1
39.f even 4 2 234.2.b.b 2
52.b odd 2 1 2704.2.a.k 1
52.f even 4 2 208.2.f.a 2
65.d even 2 1 8450.2.a.u 1
65.f even 4 2 650.2.c.a 2
65.g odd 4 2 650.2.d.b 2
65.k even 4 2 650.2.c.d 2
91.i even 4 2 1274.2.d.c 2
91.z odd 12 4 1274.2.n.d 4
91.bb even 12 4 1274.2.n.c 4
104.j odd 4 2 832.2.f.d 2
104.m even 4 2 832.2.f.b 2
156.l odd 4 2 1872.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 13.d odd 4 2
208.2.f.a 2 52.f even 4 2
234.2.b.b 2 39.f even 4 2
338.2.a.b 1 13.b even 2 1
338.2.a.d 1 1.a even 1 1 trivial
338.2.c.b 2 13.c even 3 2
338.2.c.f 2 13.e even 6 2
338.2.e.c 4 13.f odd 12 4
650.2.c.a 2 65.f even 4 2
650.2.c.d 2 65.k even 4 2
650.2.d.b 2 65.g odd 4 2
832.2.f.b 2 104.m even 4 2
832.2.f.d 2 104.j odd 4 2
1274.2.d.c 2 91.i even 4 2
1274.2.n.c 4 91.bb even 12 4
1274.2.n.d 4 91.z odd 12 4
1872.2.c.f 2 156.l odd 4 2
2704.2.a.j 1 4.b odd 2 1
2704.2.a.k 1 52.b odd 2 1
3042.2.a.g 1 3.b odd 2 1
3042.2.a.j 1 39.d odd 2 1
8450.2.a.h 1 5.b even 2 1
8450.2.a.u 1 65.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{5} + 3$$ T5 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T + 3$$
$7$ $$T + 3$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 3$$
$19$ $$T + 6$$
$23$ $$T - 6$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T + 3$$
$41$ $$T$$
$43$ $$T - 1$$
$47$ $$T + 3$$
$53$ $$T + 6$$
$59$ $$T - 6$$
$61$ $$T + 8$$
$67$ $$T + 12$$
$71$ $$T - 15$$
$73$ $$T + 6$$
$79$ $$T - 10$$
$83$ $$T - 6$$
$89$ $$T - 6$$
$97$ $$T + 12$$