# Properties

 Label 338.2.a.f 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + 3 * q^5 + q^6 + q^7 + q^8 - 2 * q^9 $$q + q^{2} + q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} + q^{8} - 2 q^{9} + 3 q^{10} - 6 q^{11} + q^{12} + q^{14} + 3 q^{15} + q^{16} - 3 q^{17} - 2 q^{18} - 2 q^{19} + 3 q^{20} + q^{21} - 6 q^{22} + q^{24} + 4 q^{25} - 5 q^{27} + q^{28} + 6 q^{29} + 3 q^{30} + 4 q^{31} + q^{32} - 6 q^{33} - 3 q^{34} + 3 q^{35} - 2 q^{36} + 7 q^{37} - 2 q^{38} + 3 q^{40} + q^{42} - q^{43} - 6 q^{44} - 6 q^{45} - 3 q^{47} + q^{48} - 6 q^{49} + 4 q^{50} - 3 q^{51} - 5 q^{54} - 18 q^{55} + q^{56} - 2 q^{57} + 6 q^{58} + 6 q^{59} + 3 q^{60} + 8 q^{61} + 4 q^{62} - 2 q^{63} + q^{64} - 6 q^{66} - 14 q^{67} - 3 q^{68} + 3 q^{70} + 3 q^{71} - 2 q^{72} - 2 q^{73} + 7 q^{74} + 4 q^{75} - 2 q^{76} - 6 q^{77} + 8 q^{79} + 3 q^{80} + q^{81} - 12 q^{83} + q^{84} - 9 q^{85} - q^{86} + 6 q^{87} - 6 q^{88} + 6 q^{89} - 6 q^{90} + 4 q^{93} - 3 q^{94} - 6 q^{95} + q^{96} + 10 q^{97} - 6 q^{98} + 12 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + 3 * q^5 + q^6 + q^7 + q^8 - 2 * q^9 + 3 * q^10 - 6 * q^11 + q^12 + q^14 + 3 * q^15 + q^16 - 3 * q^17 - 2 * q^18 - 2 * q^19 + 3 * q^20 + q^21 - 6 * q^22 + q^24 + 4 * q^25 - 5 * q^27 + q^28 + 6 * q^29 + 3 * q^30 + 4 * q^31 + q^32 - 6 * q^33 - 3 * q^34 + 3 * q^35 - 2 * q^36 + 7 * q^37 - 2 * q^38 + 3 * q^40 + q^42 - q^43 - 6 * q^44 - 6 * q^45 - 3 * q^47 + q^48 - 6 * q^49 + 4 * q^50 - 3 * q^51 - 5 * q^54 - 18 * q^55 + q^56 - 2 * q^57 + 6 * q^58 + 6 * q^59 + 3 * q^60 + 8 * q^61 + 4 * q^62 - 2 * q^63 + q^64 - 6 * q^66 - 14 * q^67 - 3 * q^68 + 3 * q^70 + 3 * q^71 - 2 * q^72 - 2 * q^73 + 7 * q^74 + 4 * q^75 - 2 * q^76 - 6 * q^77 + 8 * q^79 + 3 * q^80 + q^81 - 12 * q^83 + q^84 - 9 * q^85 - q^86 + 6 * q^87 - 6 * q^88 + 6 * q^89 - 6 * q^90 + 4 * q^93 - 3 * q^94 - 6 * q^95 + q^96 + 10 * q^97 - 6 * 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 1.00000 1.00000 3.00000 1.00000 1.00000 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.f 1
3.b odd 2 1 3042.2.a.a 1
4.b odd 2 1 2704.2.a.f 1
5.b even 2 1 8450.2.a.c 1
13.b even 2 1 26.2.a.a 1
13.c even 3 2 338.2.c.a 2
13.d odd 4 2 338.2.b.c 2
13.e even 6 2 338.2.c.d 2
13.f odd 12 4 338.2.e.a 4
39.d odd 2 1 234.2.a.e 1
39.f even 4 2 3042.2.b.a 2
52.b odd 2 1 208.2.a.a 1
52.f even 4 2 2704.2.f.d 2
65.d even 2 1 650.2.a.j 1
65.h odd 4 2 650.2.b.d 2
91.b odd 2 1 1274.2.a.d 1
91.r even 6 2 1274.2.f.p 2
91.s odd 6 2 1274.2.f.r 2
104.e even 2 1 832.2.a.d 1
104.h odd 2 1 832.2.a.i 1
117.n odd 6 2 2106.2.e.b 2
117.t even 6 2 2106.2.e.ba 2
143.d odd 2 1 3146.2.a.n 1
156.h even 2 1 1872.2.a.q 1
195.e odd 2 1 5850.2.a.p 1
195.s even 4 2 5850.2.e.a 2
208.o odd 4 2 3328.2.b.j 2
208.p even 4 2 3328.2.b.m 2
221.b even 2 1 7514.2.a.c 1
247.d odd 2 1 9386.2.a.j 1
260.g odd 2 1 5200.2.a.x 1
312.b odd 2 1 7488.2.a.g 1
312.h even 2 1 7488.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 13.b even 2 1
208.2.a.a 1 52.b odd 2 1
234.2.a.e 1 39.d odd 2 1
338.2.a.f 1 1.a even 1 1 trivial
338.2.b.c 2 13.d odd 4 2
338.2.c.a 2 13.c even 3 2
338.2.c.d 2 13.e even 6 2
338.2.e.a 4 13.f odd 12 4
650.2.a.j 1 65.d even 2 1
650.2.b.d 2 65.h odd 4 2
832.2.a.d 1 104.e even 2 1
832.2.a.i 1 104.h odd 2 1
1274.2.a.d 1 91.b odd 2 1
1274.2.f.p 2 91.r even 6 2
1274.2.f.r 2 91.s odd 6 2
1872.2.a.q 1 156.h even 2 1
2106.2.e.b 2 117.n odd 6 2
2106.2.e.ba 2 117.t even 6 2
2704.2.a.f 1 4.b odd 2 1
2704.2.f.d 2 52.f even 4 2
3042.2.a.a 1 3.b odd 2 1
3042.2.b.a 2 39.f even 4 2
3146.2.a.n 1 143.d odd 2 1
3328.2.b.j 2 208.o odd 4 2
3328.2.b.m 2 208.p even 4 2
5200.2.a.x 1 260.g odd 2 1
5850.2.a.p 1 195.e odd 2 1
5850.2.e.a 2 195.s even 4 2
7488.2.a.g 1 312.b odd 2 1
7488.2.a.h 1 312.h even 2 1
7514.2.a.c 1 221.b even 2 1
8450.2.a.c 1 5.b even 2 1
9386.2.a.j 1 247.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T - 3$$
$7$ $$T - 1$$
$11$ $$T + 6$$
$13$ $$T$$
$17$ $$T + 3$$
$19$ $$T + 2$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 4$$
$37$ $$T - 7$$
$41$ $$T$$
$43$ $$T + 1$$
$47$ $$T + 3$$
$53$ $$T$$
$59$ $$T - 6$$
$61$ $$T - 8$$
$67$ $$T + 14$$
$71$ $$T - 3$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T - 6$$
$97$ $$T - 10$$