# Properties

 Label 338.6.a.e Level $338$ Weight $6$ Character orbit 338.a Self dual yes Analytic conductor $54.210$ 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] = [338,6,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, 6, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$54.2097310968$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 4 q^{3} + 16 q^{4} + 68 q^{5} + 16 q^{6} - 82 q^{7} + 64 q^{8} - 227 q^{9}+O(q^{10})$$ q + 4 * q^2 + 4 * q^3 + 16 * q^4 + 68 * q^5 + 16 * q^6 - 82 * q^7 + 64 * q^8 - 227 * q^9 $$q + 4 q^{2} + 4 q^{3} + 16 q^{4} + 68 q^{5} + 16 q^{6} - 82 q^{7} + 64 q^{8} - 227 q^{9} + 272 q^{10} - 390 q^{11} + 64 q^{12} - 328 q^{14} + 272 q^{15} + 256 q^{16} - 1738 q^{17} - 908 q^{18} - 1074 q^{19} + 1088 q^{20} - 328 q^{21} - 1560 q^{22} - 2104 q^{23} + 256 q^{24} + 1499 q^{25} - 1880 q^{27} - 1312 q^{28} - 1690 q^{29} + 1088 q^{30} - 1430 q^{31} + 1024 q^{32} - 1560 q^{33} - 6952 q^{34} - 5576 q^{35} - 3632 q^{36} - 8852 q^{37} - 4296 q^{38} + 4352 q^{40} + 6760 q^{41} - 1312 q^{42} + 16916 q^{43} - 6240 q^{44} - 15436 q^{45} - 8416 q^{46} + 25158 q^{47} + 1024 q^{48} - 10083 q^{49} + 5996 q^{50} - 6952 q^{51} + 38214 q^{53} - 7520 q^{54} - 26520 q^{55} - 5248 q^{56} - 4296 q^{57} - 6760 q^{58} - 21286 q^{59} + 4352 q^{60} - 5458 q^{61} - 5720 q^{62} + 18614 q^{63} + 4096 q^{64} - 6240 q^{66} + 44542 q^{67} - 27808 q^{68} - 8416 q^{69} - 22304 q^{70} - 17790 q^{71} - 14528 q^{72} - 31064 q^{73} - 35408 q^{74} + 5996 q^{75} - 17184 q^{76} + 31980 q^{77} - 45360 q^{79} + 17408 q^{80} + 47641 q^{81} + 27040 q^{82} - 124546 q^{83} - 5248 q^{84} - 118184 q^{85} + 67664 q^{86} - 6760 q^{87} - 24960 q^{88} + 18744 q^{89} - 61744 q^{90} - 33664 q^{92} - 5720 q^{93} + 100632 q^{94} - 73032 q^{95} + 4096 q^{96} - 121488 q^{97} - 40332 q^{98} + 88530 q^{99}+O(q^{100})$$ q + 4 * q^2 + 4 * q^3 + 16 * q^4 + 68 * q^5 + 16 * q^6 - 82 * q^7 + 64 * q^8 - 227 * q^9 + 272 * q^10 - 390 * q^11 + 64 * q^12 - 328 * q^14 + 272 * q^15 + 256 * q^16 - 1738 * q^17 - 908 * q^18 - 1074 * q^19 + 1088 * q^20 - 328 * q^21 - 1560 * q^22 - 2104 * q^23 + 256 * q^24 + 1499 * q^25 - 1880 * q^27 - 1312 * q^28 - 1690 * q^29 + 1088 * q^30 - 1430 * q^31 + 1024 * q^32 - 1560 * q^33 - 6952 * q^34 - 5576 * q^35 - 3632 * q^36 - 8852 * q^37 - 4296 * q^38 + 4352 * q^40 + 6760 * q^41 - 1312 * q^42 + 16916 * q^43 - 6240 * q^44 - 15436 * q^45 - 8416 * q^46 + 25158 * q^47 + 1024 * q^48 - 10083 * q^49 + 5996 * q^50 - 6952 * q^51 + 38214 * q^53 - 7520 * q^54 - 26520 * q^55 - 5248 * q^56 - 4296 * q^57 - 6760 * q^58 - 21286 * q^59 + 4352 * q^60 - 5458 * q^61 - 5720 * q^62 + 18614 * q^63 + 4096 * q^64 - 6240 * q^66 + 44542 * q^67 - 27808 * q^68 - 8416 * q^69 - 22304 * q^70 - 17790 * q^71 - 14528 * q^72 - 31064 * q^73 - 35408 * q^74 + 5996 * q^75 - 17184 * q^76 + 31980 * q^77 - 45360 * q^79 + 17408 * q^80 + 47641 * q^81 + 27040 * q^82 - 124546 * q^83 - 5248 * q^84 - 118184 * q^85 + 67664 * q^86 - 6760 * q^87 - 24960 * q^88 + 18744 * q^89 - 61744 * q^90 - 33664 * q^92 - 5720 * q^93 + 100632 * q^94 - 73032 * q^95 + 4096 * q^96 - 121488 * q^97 - 40332 * q^98 + 88530 * 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
4.00000 4.00000 16.0000 68.0000 16.0000 −82.0000 64.0000 −227.000 272.000
 $$n$$: 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 338.6.a.e 1
13.b even 2 1 338.6.a.b 1
13.d odd 4 2 26.6.b.b 2
39.f even 4 2 234.6.b.a 2
52.f even 4 2 208.6.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.b 2 13.d odd 4 2
208.6.f.a 2 52.f even 4 2
234.6.b.a 2 39.f even 4 2
338.6.a.b 1 13.b even 2 1
338.6.a.e 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3} - 4$$ T3 - 4 $$T_{5} - 68$$ T5 - 68

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T - 4$$
$5$ $$T - 68$$
$7$ $$T + 82$$
$11$ $$T + 390$$
$13$ $$T$$
$17$ $$T + 1738$$
$19$ $$T + 1074$$
$23$ $$T + 2104$$
$29$ $$T + 1690$$
$31$ $$T + 1430$$
$37$ $$T + 8852$$
$41$ $$T - 6760$$
$43$ $$T - 16916$$
$47$ $$T - 25158$$
$53$ $$T - 38214$$
$59$ $$T + 21286$$
$61$ $$T + 5458$$
$67$ $$T - 44542$$
$71$ $$T + 17790$$
$73$ $$T + 31064$$
$79$ $$T + 45360$$
$83$ $$T + 124546$$
$89$ $$T - 18744$$
$97$ $$T + 121488$$