Properties

 Label 338.4.a.d Level $338$ Weight $4$ Character orbit 338.a Self dual yes Analytic conductor $19.943$ 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] = [338,4,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, 4, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$19.9426455819$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 2 q^{5} - 6 q^{6} + 5 q^{7} + 8 q^{8} - 18 q^{9}+O(q^{10})$$ q + 2 * q^2 - 3 * q^3 + 4 * q^4 - 2 * q^5 - 6 * q^6 + 5 * q^7 + 8 * q^8 - 18 * q^9 $$q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 2 q^{5} - 6 q^{6} + 5 q^{7} + 8 q^{8} - 18 q^{9} - 4 q^{10} - 13 q^{11} - 12 q^{12} + 10 q^{14} + 6 q^{15} + 16 q^{16} + 27 q^{17} - 36 q^{18} - 75 q^{19} - 8 q^{20} - 15 q^{21} - 26 q^{22} - 187 q^{23} - 24 q^{24} - 121 q^{25} + 135 q^{27} + 20 q^{28} - 13 q^{29} + 12 q^{30} + 104 q^{31} + 32 q^{32} + 39 q^{33} + 54 q^{34} - 10 q^{35} - 72 q^{36} - 423 q^{37} - 150 q^{38} - 16 q^{40} - 195 q^{41} - 30 q^{42} + 199 q^{43} - 52 q^{44} + 36 q^{45} - 374 q^{46} - 388 q^{47} - 48 q^{48} - 318 q^{49} - 242 q^{50} - 81 q^{51} + 618 q^{53} + 270 q^{54} + 26 q^{55} + 40 q^{56} + 225 q^{57} - 26 q^{58} - 491 q^{59} + 24 q^{60} + 175 q^{61} + 208 q^{62} - 90 q^{63} + 64 q^{64} + 78 q^{66} - 817 q^{67} + 108 q^{68} + 561 q^{69} - 20 q^{70} - 79 q^{71} - 144 q^{72} - 230 q^{73} - 846 q^{74} + 363 q^{75} - 300 q^{76} - 65 q^{77} + 764 q^{79} - 32 q^{80} + 81 q^{81} - 390 q^{82} + 732 q^{83} - 60 q^{84} - 54 q^{85} + 398 q^{86} + 39 q^{87} - 104 q^{88} + 1041 q^{89} + 72 q^{90} - 748 q^{92} - 312 q^{93} - 776 q^{94} + 150 q^{95} - 96 q^{96} + 97 q^{97} - 636 q^{98} + 234 q^{99}+O(q^{100})$$ q + 2 * q^2 - 3 * q^3 + 4 * q^4 - 2 * q^5 - 6 * q^6 + 5 * q^7 + 8 * q^8 - 18 * q^9 - 4 * q^10 - 13 * q^11 - 12 * q^12 + 10 * q^14 + 6 * q^15 + 16 * q^16 + 27 * q^17 - 36 * q^18 - 75 * q^19 - 8 * q^20 - 15 * q^21 - 26 * q^22 - 187 * q^23 - 24 * q^24 - 121 * q^25 + 135 * q^27 + 20 * q^28 - 13 * q^29 + 12 * q^30 + 104 * q^31 + 32 * q^32 + 39 * q^33 + 54 * q^34 - 10 * q^35 - 72 * q^36 - 423 * q^37 - 150 * q^38 - 16 * q^40 - 195 * q^41 - 30 * q^42 + 199 * q^43 - 52 * q^44 + 36 * q^45 - 374 * q^46 - 388 * q^47 - 48 * q^48 - 318 * q^49 - 242 * q^50 - 81 * q^51 + 618 * q^53 + 270 * q^54 + 26 * q^55 + 40 * q^56 + 225 * q^57 - 26 * q^58 - 491 * q^59 + 24 * q^60 + 175 * q^61 + 208 * q^62 - 90 * q^63 + 64 * q^64 + 78 * q^66 - 817 * q^67 + 108 * q^68 + 561 * q^69 - 20 * q^70 - 79 * q^71 - 144 * q^72 - 230 * q^73 - 846 * q^74 + 363 * q^75 - 300 * q^76 - 65 * q^77 + 764 * q^79 - 32 * q^80 + 81 * q^81 - 390 * q^82 + 732 * q^83 - 60 * q^84 - 54 * q^85 + 398 * q^86 + 39 * q^87 - 104 * q^88 + 1041 * q^89 + 72 * q^90 - 748 * q^92 - 312 * q^93 - 776 * q^94 + 150 * q^95 - 96 * q^96 + 97 * q^97 - 636 * q^98 + 234 * 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
2.00000 −3.00000 4.00000 −2.00000 −6.00000 5.00000 8.00000 −18.0000 −4.00000
 $$n$$: 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 338.4.a.d 1
13.b even 2 1 338.4.a.a 1
13.c even 3 2 338.4.c.d 2
13.d odd 4 2 338.4.b.a 2
13.e even 6 2 26.4.c.a 2
13.f odd 12 4 338.4.e.d 4
39.h odd 6 2 234.4.h.b 2
52.i odd 6 2 208.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 13.e even 6 2
208.4.i.a 2 52.i odd 6 2
234.4.h.b 2 39.h odd 6 2
338.4.a.a 1 13.b even 2 1
338.4.a.d 1 1.a even 1 1 trivial
338.4.b.a 2 13.d odd 4 2
338.4.c.d 2 13.c even 3 2
338.4.e.d 4 13.f odd 12 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3} + 3$$ T3 + 3 $$T_{5} + 2$$ T5 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 3$$
$5$ $$T + 2$$
$7$ $$T - 5$$
$11$ $$T + 13$$
$13$ $$T$$
$17$ $$T - 27$$
$19$ $$T + 75$$
$23$ $$T + 187$$
$29$ $$T + 13$$
$31$ $$T - 104$$
$37$ $$T + 423$$
$41$ $$T + 195$$
$43$ $$T - 199$$
$47$ $$T + 388$$
$53$ $$T - 618$$
$59$ $$T + 491$$
$61$ $$T - 175$$
$67$ $$T + 817$$
$71$ $$T + 79$$
$73$ $$T + 230$$
$79$ $$T - 764$$
$83$ $$T - 732$$
$89$ $$T - 1041$$
$97$ $$T - 97$$