# Properties

 Label 338.8.a.a Level $338$ Weight $8$ Character orbit 338.a Self dual yes Analytic conductor $105.586$ 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] = [338,8,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, 8, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$105.586138614$$ 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{2} - 87 q^{3} + 64 q^{4} - 321 q^{5} + 696 q^{6} + 181 q^{7} - 512 q^{8} + 5382 q^{9}+O(q^{10})$$ q - 8 * q^2 - 87 * q^3 + 64 * q^4 - 321 * q^5 + 696 * q^6 + 181 * q^7 - 512 * q^8 + 5382 * q^9 $$q - 8 q^{2} - 87 q^{3} + 64 q^{4} - 321 q^{5} + 696 q^{6} + 181 q^{7} - 512 q^{8} + 5382 q^{9} + 2568 q^{10} - 7782 q^{11} - 5568 q^{12} - 1448 q^{14} + 27927 q^{15} + 4096 q^{16} + 9069 q^{17} - 43056 q^{18} + 37150 q^{19} - 20544 q^{20} - 15747 q^{21} + 62256 q^{22} + 19008 q^{23} + 44544 q^{24} + 24916 q^{25} - 277965 q^{27} + 11584 q^{28} + 174750 q^{29} - 223416 q^{30} - 29012 q^{31} - 32768 q^{32} + 677034 q^{33} - 72552 q^{34} - 58101 q^{35} + 344448 q^{36} - 323669 q^{37} - 297200 q^{38} + 164352 q^{40} - 795312 q^{41} + 125976 q^{42} - 314137 q^{43} - 498048 q^{44} - 1727622 q^{45} - 152064 q^{46} + 447441 q^{47} - 356352 q^{48} - 790782 q^{49} - 199328 q^{50} - 789003 q^{51} - 1469232 q^{53} + 2223720 q^{54} + 2498022 q^{55} - 92672 q^{56} - 3232050 q^{57} - 1398000 q^{58} - 1627770 q^{59} + 1787328 q^{60} - 2399608 q^{61} + 232096 q^{62} + 974142 q^{63} + 262144 q^{64} - 5416272 q^{66} + 64066 q^{67} + 580416 q^{68} - 1653696 q^{69} + 464808 q^{70} + 322383 q^{71} - 2755584 q^{72} + 4454782 q^{73} + 2589352 q^{74} - 2167692 q^{75} + 2377600 q^{76} - 1408542 q^{77} + 753560 q^{79} - 1314816 q^{80} + 12412521 q^{81} + 6362496 q^{82} + 1219092 q^{83} - 1007808 q^{84} - 2911149 q^{85} + 2513096 q^{86} - 15203250 q^{87} + 3984384 q^{88} - 3390330 q^{89} + 13820976 q^{90} + 1216512 q^{92} + 2524044 q^{93} - 3579528 q^{94} - 11925150 q^{95} + 2850816 q^{96} - 1628774 q^{97} + 6326256 q^{98} - 41882724 q^{99}+O(q^{100})$$ q - 8 * q^2 - 87 * q^3 + 64 * q^4 - 321 * q^5 + 696 * q^6 + 181 * q^7 - 512 * q^8 + 5382 * q^9 + 2568 * q^10 - 7782 * q^11 - 5568 * q^12 - 1448 * q^14 + 27927 * q^15 + 4096 * q^16 + 9069 * q^17 - 43056 * q^18 + 37150 * q^19 - 20544 * q^20 - 15747 * q^21 + 62256 * q^22 + 19008 * q^23 + 44544 * q^24 + 24916 * q^25 - 277965 * q^27 + 11584 * q^28 + 174750 * q^29 - 223416 * q^30 - 29012 * q^31 - 32768 * q^32 + 677034 * q^33 - 72552 * q^34 - 58101 * q^35 + 344448 * q^36 - 323669 * q^37 - 297200 * q^38 + 164352 * q^40 - 795312 * q^41 + 125976 * q^42 - 314137 * q^43 - 498048 * q^44 - 1727622 * q^45 - 152064 * q^46 + 447441 * q^47 - 356352 * q^48 - 790782 * q^49 - 199328 * q^50 - 789003 * q^51 - 1469232 * q^53 + 2223720 * q^54 + 2498022 * q^55 - 92672 * q^56 - 3232050 * q^57 - 1398000 * q^58 - 1627770 * q^59 + 1787328 * q^60 - 2399608 * q^61 + 232096 * q^62 + 974142 * q^63 + 262144 * q^64 - 5416272 * q^66 + 64066 * q^67 + 580416 * q^68 - 1653696 * q^69 + 464808 * q^70 + 322383 * q^71 - 2755584 * q^72 + 4454782 * q^73 + 2589352 * q^74 - 2167692 * q^75 + 2377600 * q^76 - 1408542 * q^77 + 753560 * q^79 - 1314816 * q^80 + 12412521 * q^81 + 6362496 * q^82 + 1219092 * q^83 - 1007808 * q^84 - 2911149 * q^85 + 2513096 * q^86 - 15203250 * q^87 + 3984384 * q^88 - 3390330 * q^89 + 13820976 * q^90 + 1216512 * q^92 + 2524044 * q^93 - 3579528 * q^94 - 11925150 * q^95 + 2850816 * q^96 - 1628774 * q^97 + 6326256 * q^98 - 41882724 * 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
−8.00000 −87.0000 64.0000 −321.000 696.000 181.000 −512.000 5382.00 2568.00
 $$n$$: 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 338.8.a.a 1
13.b even 2 1 26.8.a.b 1
13.d odd 4 2 338.8.b.a 2
39.d odd 2 1 234.8.a.a 1
52.b odd 2 1 208.8.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.b 1 13.b even 2 1
208.8.a.e 1 52.b odd 2 1
234.8.a.a 1 39.d odd 2 1
338.8.a.a 1 1.a even 1 1 trivial
338.8.b.a 2 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3} + 87$$ T3 + 87 $$T_{5} + 321$$ T5 + 321

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 8$$
$3$ $$T + 87$$
$5$ $$T + 321$$
$7$ $$T - 181$$
$11$ $$T + 7782$$
$13$ $$T$$
$17$ $$T - 9069$$
$19$ $$T - 37150$$
$23$ $$T - 19008$$
$29$ $$T - 174750$$
$31$ $$T + 29012$$
$37$ $$T + 323669$$
$41$ $$T + 795312$$
$43$ $$T + 314137$$
$47$ $$T - 447441$$
$53$ $$T + 1469232$$
$59$ $$T + 1627770$$
$61$ $$T + 2399608$$
$67$ $$T - 64066$$
$71$ $$T - 322383$$
$73$ $$T - 4454782$$
$79$ $$T - 753560$$
$83$ $$T - 1219092$$
$89$ $$T + 3390330$$
$97$ $$T + 1628774$$