Properties

 Label 81.9.d.g Level $81$ Weight $9$ Character orbit 81.d Analytic conductor $32.998$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(26,81)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("81.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 81.d (of order $$6$$, degree $$2$$, 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: $$32.9976674150$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 2048 q^{4} - 3692 q^{7}+O(q^{10})$$ 32 * q + 2048 * q^4 - 3692 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 2048 q^{4} - 3692 q^{7} + 21504 q^{10} - 63860 q^{13} - 95116 q^{16} + 370216 q^{19} + 691980 q^{22} + 541712 q^{25} - 1994264 q^{28} - 571136 q^{31} - 1027656 q^{34} + 8708536 q^{37} + 2973768 q^{40} - 5453084 q^{43} - 8468088 q^{46} - 14602560 q^{49} + 25384960 q^{52} - 28905144 q^{55} - 1213068 q^{58} - 31037996 q^{61} + 48506000 q^{64} + 62356828 q^{67} - 11075124 q^{70} - 42547328 q^{73} - 69593876 q^{76} + 151045036 q^{79} - 80402280 q^{82} - 160995564 q^{85} - 366976068 q^{88} + 747361592 q^{91} + 544741584 q^{94} + 133878688 q^{97}+O(q^{100})$$ 32 * q + 2048 * q^4 - 3692 * q^7 + 21504 * q^10 - 63860 * q^13 - 95116 * q^16 + 370216 * q^19 + 691980 * q^22 + 541712 * q^25 - 1994264 * q^28 - 571136 * q^31 - 1027656 * q^34 + 8708536 * q^37 + 2973768 * q^40 - 5453084 * q^43 - 8468088 * q^46 - 14602560 * q^49 + 25384960 * q^52 - 28905144 * q^55 - 1213068 * q^58 - 31037996 * q^61 + 48506000 * q^64 + 62356828 * q^67 - 11075124 * q^70 - 42547328 * q^73 - 69593876 * q^76 + 151045036 * q^79 - 80402280 * q^82 - 160995564 * q^85 - 366976068 * q^88 + 747361592 * q^91 + 544741584 * q^94 + 133878688 * q^97

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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1 −25.8080 14.9003i 0 316.037 + 547.392i −153.960 + 88.8891i 0 −1695.41 + 2936.53i 11207.2i 0 5297.89
26.2 −22.2403 12.8404i 0 201.753 + 349.447i 176.989 102.185i 0 1851.21 3206.39i 3788.08i 0 −5248.38
26.3 −20.8866 12.0589i 0 162.833 + 282.035i 378.029 218.255i 0 −275.376 + 476.966i 1680.18i 0 −10527.6
26.4 −19.4167 11.2102i 0 123.338 + 213.628i −935.412 + 540.060i 0 503.318 871.773i 209.045i 0 24216.8
26.5 −12.9399 7.47088i 0 −16.3720 28.3571i 943.515 544.739i 0 −654.245 + 1133.19i 4314.34i 0 −16278.7
26.6 −8.67350 5.00765i 0 −77.8470 134.835i −514.754 + 297.193i 0 −2204.76 + 3818.75i 4123.23i 0 5952.96
26.7 −7.89343 4.55727i 0 −86.4626 149.758i 157.475 90.9184i 0 436.536 756.103i 3909.46i 0 −1657.36
26.8 −5.00808 2.89142i 0 −111.279 192.742i −542.191 + 313.034i 0 1115.72 1932.48i 2767.43i 0 3620.45
26.9 5.00808 + 2.89142i 0 −111.279 192.742i 542.191 313.034i 0 1115.72 1932.48i 2767.43i 0 3620.45
26.10 7.89343 + 4.55727i 0 −86.4626 149.758i −157.475 + 90.9184i 0 436.536 756.103i 3909.46i 0 −1657.36
26.11 8.67350 + 5.00765i 0 −77.8470 134.835i 514.754 297.193i 0 −2204.76 + 3818.75i 4123.23i 0 5952.96
26.12 12.9399 + 7.47088i 0 −16.3720 28.3571i −943.515 + 544.739i 0 −654.245 + 1133.19i 4314.34i 0 −16278.7
26.13 19.4167 + 11.2102i 0 123.338 + 213.628i 935.412 540.060i 0 503.318 871.773i 209.045i 0 24216.8
26.14 20.8866 + 12.0589i 0 162.833 + 282.035i −378.029 + 218.255i 0 −275.376 + 476.966i 1680.18i 0 −10527.6
26.15 22.2403 + 12.8404i 0 201.753 + 349.447i −176.989 + 102.185i 0 1851.21 3206.39i 3788.08i 0 −5248.38
26.16 25.8080 + 14.9003i 0 316.037 + 547.392i 153.960 88.8891i 0 −1695.41 + 2936.53i 11207.2i 0 5297.89
53.1 −25.8080 + 14.9003i 0 316.037 547.392i −153.960 88.8891i 0 −1695.41 2936.53i 11207.2i 0 5297.89
53.2 −22.2403 + 12.8404i 0 201.753 349.447i 176.989 + 102.185i 0 1851.21 + 3206.39i 3788.08i 0 −5248.38
53.3 −20.8866 + 12.0589i 0 162.833 282.035i 378.029 + 218.255i 0 −275.376 476.966i 1680.18i 0 −10527.6
53.4 −19.4167 + 11.2102i 0 123.338 213.628i −935.412 540.060i 0 503.318 + 871.773i 209.045i 0 24216.8
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 26.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.9.d.g 32
3.b odd 2 1 inner 81.9.d.g 32
9.c even 3 1 81.9.b.b 16
9.c even 3 1 inner 81.9.d.g 32
9.d odd 6 1 81.9.b.b 16
9.d odd 6 1 inner 81.9.d.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.9.b.b 16 9.c even 3 1
81.9.b.b 16 9.d odd 6 1
81.9.d.g 32 1.a even 1 1 trivial
81.9.d.g 32 3.b odd 2 1 inner
81.9.d.g 32 9.c even 3 1 inner
81.9.d.g 32 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - 3072 T_{2}^{30} + 5659875 T_{2}^{28} - 6840915480 T_{2}^{26} + 6132249070245 T_{2}^{24} + \cdots + 11\!\cdots\!96$$ acting on $$S_{9}^{\mathrm{new}}(81, [\chi])$$.