# Properties

 Label 338.8.b.k Level $338$ Weight $8$ Character orbit 338.b Analytic conductor $105.586$ Analytic rank $0$ Dimension $24$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,Mod(337,338)]

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([1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 338.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$105.586138614$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 216 q^{3} - 1536 q^{4} + 21028 q^{9}+O(q^{10})$$ 24 * q + 216 * q^3 - 1536 * q^4 + 21028 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 216 q^{3} - 1536 q^{4} + 21028 q^{9} - 4672 q^{10} - 13824 q^{12} - 5824 q^{14} + 98304 q^{16} - 11692 q^{17} + 52416 q^{22} - 319864 q^{23} - 740928 q^{25} + 886128 q^{27} - 76820 q^{29} + 477440 q^{30} - 358576 q^{35} - 1345792 q^{36} - 1308480 q^{38} + 299008 q^{40} + 880192 q^{42} - 3067480 q^{43} + 884736 q^{48} - 10349804 q^{49} - 4068272 q^{51} + 8278012 q^{53} + 897968 q^{55} + 372736 q^{56} + 3415424 q^{61} - 9111360 q^{62} - 6291456 q^{64} + 15411264 q^{66} + 748288 q^{68} - 1863776 q^{69} - 3529152 q^{74} - 55849880 q^{75} - 7183368 q^{77} + 33989704 q^{79} - 9874424 q^{81} - 11810560 q^{82} - 27528672 q^{87} - 3354624 q^{88} + 43138080 q^{90} + 20471296 q^{92} - 54295936 q^{94} + 59408472 q^{95}+O(q^{100})$$ 24 * q + 216 * q^3 - 1536 * q^4 + 21028 * q^9 - 4672 * q^10 - 13824 * q^12 - 5824 * q^14 + 98304 * q^16 - 11692 * q^17 + 52416 * q^22 - 319864 * q^23 - 740928 * q^25 + 886128 * q^27 - 76820 * q^29 + 477440 * q^30 - 358576 * q^35 - 1345792 * q^36 - 1308480 * q^38 + 299008 * q^40 + 880192 * q^42 - 3067480 * q^43 + 884736 * q^48 - 10349804 * q^49 - 4068272 * q^51 + 8278012 * q^53 + 897968 * q^55 + 372736 * q^56 + 3415424 * q^61 - 9111360 * q^62 - 6291456 * q^64 + 15411264 * q^66 + 748288 * q^68 - 1863776 * q^69 - 3529152 * q^74 - 55849880 * q^75 - 7183368 * q^77 + 33989704 * q^79 - 9874424 * q^81 - 11810560 * q^82 - 27528672 * q^87 - 3354624 * q^88 + 43138080 * q^90 + 20471296 * q^92 - 54295936 * q^94 + 59408472 * q^95

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1 8.00000i 83.1100 −64.0000 420.243i 664.880i 1052.48i 512.000i 4720.28 3361.94
337.2 8.00000i 83.1100 −64.0000 420.243i 664.880i 1052.48i 512.000i 4720.28 3361.94
337.3 8.00000i 87.2749 −64.0000 309.911i 698.199i 1697.32i 512.000i 5429.90 2479.29
337.4 8.00000i 87.2749 −64.0000 309.911i 698.199i 1697.32i 512.000i 5429.90 2479.29
337.5 8.00000i −67.8928 −64.0000 145.341i 543.142i 229.806i 512.000i 2422.43 −1162.73
337.6 8.00000i −67.8928 −64.0000 145.341i 543.142i 229.806i 512.000i 2422.43 −1162.73
337.7 8.00000i −10.1193 −64.0000 17.6732i 80.9547i 1027.75i 512.000i −2084.60 −141.386
337.8 8.00000i −10.1193 −64.0000 17.6732i 80.9547i 1027.75i 512.000i −2084.60 −141.386
337.9 8.00000i −63.0837 −64.0000 119.855i 504.669i 1111.19i 512.000i 1792.55 958.841
337.10 8.00000i −63.0837 −64.0000 119.855i 504.669i 1111.19i 512.000i 1792.55 958.841
337.11 8.00000i 54.7462 −64.0000 166.556i 437.969i 508.593i 512.000i 810.143 −1332.45
337.12 8.00000i 54.7462 −64.0000 166.556i 437.969i 508.593i 512.000i 810.143 −1332.45
337.13 8.00000i −5.72241 −64.0000 178.768i 45.7793i 920.259i 512.000i −2154.25 −1430.15
337.14 8.00000i −5.72241 −64.0000 178.768i 45.7793i 920.259i 512.000i −2154.25 −1430.15
337.15 8.00000i −53.9339 −64.0000 193.461i 431.471i 94.6847i 512.000i 721.863 1547.69
337.16 8.00000i −53.9339 −64.0000 193.461i 431.471i 94.6847i 512.000i 721.863 1547.69
337.17 8.00000i 51.2719 −64.0000 325.958i 410.175i 1423.39i 512.000i 441.807 −2607.66
337.18 8.00000i 51.2719 −64.0000 325.958i 410.175i 1423.39i 512.000i 441.807 −2607.66
337.19 8.00000i −40.7841 −64.0000 488.619i 326.272i 1508.40i 512.000i −523.661 −3908.95
337.20 8.00000i −40.7841 −64.0000 488.619i 326.272i 1508.40i 512.000i −523.661 −3908.95
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.8.b.k 24
13.b even 2 1 inner 338.8.b.k 24
13.d odd 4 1 338.8.a.q 12
13.d odd 4 1 338.8.a.r yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.8.a.q 12 13.d odd 4 1
338.8.a.r yes 12 13.d odd 4 1
338.8.b.k 24 1.a even 1 1 trivial
338.8.b.k 24 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 108 T_{3}^{11} - 12547 T_{3}^{10} + 1469828 T_{3}^{9} + 55623513 T_{3}^{8} + \cdots + 11\!\cdots\!57$$ acting on $$S_{8}^{\mathrm{new}}(338, [\chi])$$.