Properties

 Label 338.2.a.a 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 3 q^{3} + q^{4} + q^{5} + 3 q^{6} - q^{7} - q^{8} + 6 q^{9}+O(q^{10})$$ q - q^2 - 3 * q^3 + q^4 + q^5 + 3 * q^6 - q^7 - q^8 + 6 * q^9 $$q - q^{2} - 3 q^{3} + q^{4} + q^{5} + 3 q^{6} - q^{7} - q^{8} + 6 q^{9} - q^{10} + 2 q^{11} - 3 q^{12} + q^{14} - 3 q^{15} + q^{16} - 3 q^{17} - 6 q^{18} - 6 q^{19} + q^{20} + 3 q^{21} - 2 q^{22} - 4 q^{23} + 3 q^{24} - 4 q^{25} - 9 q^{27} - q^{28} + 2 q^{29} + 3 q^{30} - 4 q^{31} - q^{32} - 6 q^{33} + 3 q^{34} - q^{35} + 6 q^{36} - 3 q^{37} + 6 q^{38} - q^{40} - 3 q^{42} - 5 q^{43} + 2 q^{44} + 6 q^{45} + 4 q^{46} - 13 q^{47} - 3 q^{48} - 6 q^{49} + 4 q^{50} + 9 q^{51} + 12 q^{53} + 9 q^{54} + 2 q^{55} + q^{56} + 18 q^{57} - 2 q^{58} + 10 q^{59} - 3 q^{60} - 8 q^{61} + 4 q^{62} - 6 q^{63} + q^{64} + 6 q^{66} + 2 q^{67} - 3 q^{68} + 12 q^{69} + q^{70} + 5 q^{71} - 6 q^{72} + 10 q^{73} + 3 q^{74} + 12 q^{75} - 6 q^{76} - 2 q^{77} - 4 q^{79} + q^{80} + 9 q^{81} + 3 q^{84} - 3 q^{85} + 5 q^{86} - 6 q^{87} - 2 q^{88} - 6 q^{89} - 6 q^{90} - 4 q^{92} + 12 q^{93} + 13 q^{94} - 6 q^{95} + 3 q^{96} - 14 q^{97} + 6 q^{98} + 12 q^{99}+O(q^{100})$$ q - q^2 - 3 * q^3 + q^4 + q^5 + 3 * q^6 - q^7 - q^8 + 6 * q^9 - q^10 + 2 * q^11 - 3 * q^12 + q^14 - 3 * q^15 + q^16 - 3 * q^17 - 6 * q^18 - 6 * q^19 + q^20 + 3 * q^21 - 2 * q^22 - 4 * q^23 + 3 * q^24 - 4 * q^25 - 9 * q^27 - q^28 + 2 * q^29 + 3 * q^30 - 4 * q^31 - q^32 - 6 * q^33 + 3 * q^34 - q^35 + 6 * q^36 - 3 * q^37 + 6 * q^38 - q^40 - 3 * q^42 - 5 * q^43 + 2 * q^44 + 6 * q^45 + 4 * q^46 - 13 * q^47 - 3 * q^48 - 6 * q^49 + 4 * q^50 + 9 * q^51 + 12 * q^53 + 9 * q^54 + 2 * q^55 + q^56 + 18 * q^57 - 2 * q^58 + 10 * q^59 - 3 * q^60 - 8 * q^61 + 4 * q^62 - 6 * q^63 + q^64 + 6 * q^66 + 2 * q^67 - 3 * q^68 + 12 * q^69 + q^70 + 5 * q^71 - 6 * q^72 + 10 * q^73 + 3 * q^74 + 12 * q^75 - 6 * q^76 - 2 * q^77 - 4 * q^79 + q^80 + 9 * q^81 + 3 * q^84 - 3 * q^85 + 5 * q^86 - 6 * q^87 - 2 * q^88 - 6 * q^89 - 6 * q^90 - 4 * q^92 + 12 * q^93 + 13 * q^94 - 6 * q^95 + 3 * q^96 - 14 * 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 −3.00000 1.00000 1.00000 3.00000 −1.00000 −1.00000 6.00000 −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.a 1
3.b odd 2 1 3042.2.a.l 1
4.b odd 2 1 2704.2.a.n 1
5.b even 2 1 8450.2.a.y 1
13.b even 2 1 26.2.a.b 1
13.c even 3 2 338.2.c.g 2
13.d odd 4 2 338.2.b.a 2
13.e even 6 2 338.2.c.c 2
13.f odd 12 4 338.2.e.d 4
39.d odd 2 1 234.2.a.b 1
39.f even 4 2 3042.2.b.f 2
52.b odd 2 1 208.2.a.d 1
52.f even 4 2 2704.2.f.j 2
65.d even 2 1 650.2.a.g 1
65.h odd 4 2 650.2.b.a 2
91.b odd 2 1 1274.2.a.o 1
91.r even 6 2 1274.2.f.l 2
91.s odd 6 2 1274.2.f.a 2
104.e even 2 1 832.2.a.j 1
104.h odd 2 1 832.2.a.a 1
117.n odd 6 2 2106.2.e.t 2
117.t even 6 2 2106.2.e.h 2
143.d odd 2 1 3146.2.a.a 1
156.h even 2 1 1872.2.a.m 1
195.e odd 2 1 5850.2.a.bn 1
195.s even 4 2 5850.2.e.v 2
208.o odd 4 2 3328.2.b.k 2
208.p even 4 2 3328.2.b.g 2
221.b even 2 1 7514.2.a.i 1
247.d odd 2 1 9386.2.a.f 1
260.g odd 2 1 5200.2.a.c 1
312.b odd 2 1 7488.2.a.w 1
312.h even 2 1 7488.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 13.b even 2 1
208.2.a.d 1 52.b odd 2 1
234.2.a.b 1 39.d odd 2 1
338.2.a.a 1 1.a even 1 1 trivial
338.2.b.a 2 13.d odd 4 2
338.2.c.c 2 13.e even 6 2
338.2.c.g 2 13.c even 3 2
338.2.e.d 4 13.f odd 12 4
650.2.a.g 1 65.d even 2 1
650.2.b.a 2 65.h odd 4 2
832.2.a.a 1 104.h odd 2 1
832.2.a.j 1 104.e even 2 1
1274.2.a.o 1 91.b odd 2 1
1274.2.f.a 2 91.s odd 6 2
1274.2.f.l 2 91.r even 6 2
1872.2.a.m 1 156.h even 2 1
2106.2.e.h 2 117.t even 6 2
2106.2.e.t 2 117.n odd 6 2
2704.2.a.n 1 4.b odd 2 1
2704.2.f.j 2 52.f even 4 2
3042.2.a.l 1 3.b odd 2 1
3042.2.b.f 2 39.f even 4 2
3146.2.a.a 1 143.d odd 2 1
3328.2.b.g 2 208.p even 4 2
3328.2.b.k 2 208.o odd 4 2
5200.2.a.c 1 260.g odd 2 1
5850.2.a.bn 1 195.e odd 2 1
5850.2.e.v 2 195.s even 4 2
7488.2.a.v 1 312.h even 2 1
7488.2.a.w 1 312.b odd 2 1
7514.2.a.i 1 221.b even 2 1
8450.2.a.y 1 5.b even 2 1
9386.2.a.f 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(338))$$:

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

Hecke characteristic polynomials

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