# Properties

 Label 338.2.a.e Level $338$ Weight $2$ Character orbit 338.a Self dual yes Analytic conductor $2.699$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + 4 q^{7} + q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 + q^4 - q^5 + 4 * q^7 + q^8 - 3 * q^9 $$q + q^{2} + q^{4} - q^{5} + 4 q^{7} + q^{8} - 3 q^{9} - q^{10} + 4 q^{11} + 4 q^{14} + q^{16} + 3 q^{17} - 3 q^{18} - q^{20} + 4 q^{22} - 4 q^{23} - 4 q^{25} + 4 q^{28} - q^{29} + 4 q^{31} + q^{32} + 3 q^{34} - 4 q^{35} - 3 q^{36} + 3 q^{37} - q^{40} - 9 q^{41} - 8 q^{43} + 4 q^{44} + 3 q^{45} - 4 q^{46} - 8 q^{47} + 9 q^{49} - 4 q^{50} - 9 q^{53} - 4 q^{55} + 4 q^{56} - q^{58} - 4 q^{59} + 7 q^{61} + 4 q^{62} - 12 q^{63} + q^{64} + 4 q^{67} + 3 q^{68} - 4 q^{70} - 8 q^{71} - 3 q^{72} + 11 q^{73} + 3 q^{74} + 16 q^{77} - 4 q^{79} - q^{80} + 9 q^{81} - 9 q^{82} - 3 q^{85} - 8 q^{86} + 4 q^{88} - 6 q^{89} + 3 q^{90} - 4 q^{92} - 8 q^{94} + 2 q^{97} + 9 q^{98} - 12 q^{99}+O(q^{100})$$ q + q^2 + q^4 - q^5 + 4 * q^7 + q^8 - 3 * q^9 - q^10 + 4 * q^11 + 4 * q^14 + q^16 + 3 * q^17 - 3 * q^18 - q^20 + 4 * q^22 - 4 * q^23 - 4 * q^25 + 4 * q^28 - q^29 + 4 * q^31 + q^32 + 3 * q^34 - 4 * q^35 - 3 * q^36 + 3 * q^37 - q^40 - 9 * q^41 - 8 * q^43 + 4 * q^44 + 3 * q^45 - 4 * q^46 - 8 * q^47 + 9 * q^49 - 4 * q^50 - 9 * q^53 - 4 * q^55 + 4 * q^56 - q^58 - 4 * q^59 + 7 * q^61 + 4 * q^62 - 12 * q^63 + q^64 + 4 * q^67 + 3 * q^68 - 4 * q^70 - 8 * q^71 - 3 * q^72 + 11 * q^73 + 3 * q^74 + 16 * q^77 - 4 * q^79 - q^80 + 9 * q^81 - 9 * q^82 - 3 * q^85 - 8 * q^86 + 4 * q^88 - 6 * q^89 + 3 * q^90 - 4 * q^92 - 8 * q^94 + 2 * q^97 + 9 * q^98 - 12 * 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
1.00000 0 1.00000 −1.00000 0 4.00000 1.00000 −3.00000 −1.00000
 $$n$$: 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.e 1
3.b odd 2 1 3042.2.a.e 1
4.b odd 2 1 2704.2.a.h 1
5.b even 2 1 8450.2.a.f 1
13.b even 2 1 338.2.a.c 1
13.c even 3 2 26.2.c.a 2
13.d odd 4 2 338.2.b.b 2
13.e even 6 2 338.2.c.e 2
13.f odd 12 4 338.2.e.b 4
39.d odd 2 1 3042.2.a.k 1
39.f even 4 2 3042.2.b.e 2
39.i odd 6 2 234.2.h.c 2
52.b odd 2 1 2704.2.a.i 1
52.f even 4 2 2704.2.f.g 2
52.j odd 6 2 208.2.i.b 2
65.d even 2 1 8450.2.a.s 1
65.n even 6 2 650.2.e.c 2
65.q odd 12 4 650.2.o.c 4
91.g even 3 2 1274.2.h.b 2
91.h even 3 2 1274.2.e.n 2
91.m odd 6 2 1274.2.h.a 2
91.n odd 6 2 1274.2.g.a 2
91.v odd 6 2 1274.2.e.m 2
104.n odd 6 2 832.2.i.f 2
104.r even 6 2 832.2.i.e 2
156.p even 6 2 1872.2.t.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.c.a 2 13.c even 3 2
208.2.i.b 2 52.j odd 6 2
234.2.h.c 2 39.i odd 6 2
338.2.a.c 1 13.b even 2 1
338.2.a.e 1 1.a even 1 1 trivial
338.2.b.b 2 13.d odd 4 2
338.2.c.e 2 13.e even 6 2
338.2.e.b 4 13.f odd 12 4
650.2.e.c 2 65.n even 6 2
650.2.o.c 4 65.q odd 12 4
832.2.i.e 2 104.r even 6 2
832.2.i.f 2 104.n odd 6 2
1274.2.e.m 2 91.v odd 6 2
1274.2.e.n 2 91.h even 3 2
1274.2.g.a 2 91.n odd 6 2
1274.2.h.a 2 91.m odd 6 2
1274.2.h.b 2 91.g even 3 2
1872.2.t.k 2 156.p even 6 2
2704.2.a.h 1 4.b odd 2 1
2704.2.a.i 1 52.b odd 2 1
2704.2.f.g 2 52.f even 4 2
3042.2.a.e 1 3.b odd 2 1
3042.2.a.k 1 39.d odd 2 1
3042.2.b.e 2 39.f even 4 2
8450.2.a.f 1 5.b even 2 1
8450.2.a.s 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(338))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 4$$
$11$ $$T - 4$$
$13$ $$T$$
$17$ $$T - 3$$
$19$ $$T$$
$23$ $$T + 4$$
$29$ $$T + 1$$
$31$ $$T - 4$$
$37$ $$T - 3$$
$41$ $$T + 9$$
$43$ $$T + 8$$
$47$ $$T + 8$$
$53$ $$T + 9$$
$59$ $$T + 4$$
$61$ $$T - 7$$
$67$ $$T - 4$$
$71$ $$T + 8$$
$73$ $$T - 11$$
$79$ $$T + 4$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T - 2$$