Properties

 Label 81.4.c.a Level $81$ Weight $4$ Character orbit 81.c Analytic conductor $4.779$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,4,Mod(28,81)]

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([2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 81.c (of order $$3$$, 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: $$4.77915471046$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{2} - \zeta_{6} q^{4} - 15 \zeta_{6} q^{5} + ( - 25 \zeta_{6} + 25) q^{7} - 21 q^{8} +O(q^{10})$$ q + (3*z - 3) * q^2 - z * q^4 - 15*z * q^5 + (-25*z + 25) * q^7 - 21 * q^8 $$q + (3 \zeta_{6} - 3) q^{2} - \zeta_{6} q^{4} - 15 \zeta_{6} q^{5} + ( - 25 \zeta_{6} + 25) q^{7} - 21 q^{8} + 45 q^{10} + ( - 15 \zeta_{6} + 15) q^{11} - 20 \zeta_{6} q^{13} + 75 \zeta_{6} q^{14} + ( - 71 \zeta_{6} + 71) q^{16} + 72 q^{17} + 2 q^{19} + (15 \zeta_{6} - 15) q^{20} + 45 \zeta_{6} q^{22} - 114 \zeta_{6} q^{23} + (100 \zeta_{6} - 100) q^{25} + 60 q^{26} - 25 q^{28} + (30 \zeta_{6} - 30) q^{29} - 101 \zeta_{6} q^{31} + 45 \zeta_{6} q^{32} + (216 \zeta_{6} - 216) q^{34} - 375 q^{35} - 430 q^{37} + (6 \zeta_{6} - 6) q^{38} + 315 \zeta_{6} q^{40} + 30 \zeta_{6} q^{41} + (110 \zeta_{6} - 110) q^{43} - 15 q^{44} + 342 q^{46} + ( - 330 \zeta_{6} + 330) q^{47} - 282 \zeta_{6} q^{49} - 300 \zeta_{6} q^{50} + (20 \zeta_{6} - 20) q^{52} + 621 q^{53} - 225 q^{55} + (525 \zeta_{6} - 525) q^{56} - 90 \zeta_{6} q^{58} + 660 \zeta_{6} q^{59} + ( - 376 \zeta_{6} + 376) q^{61} + 303 q^{62} + 433 q^{64} + (300 \zeta_{6} - 300) q^{65} + 250 \zeta_{6} q^{67} - 72 \zeta_{6} q^{68} + ( - 1125 \zeta_{6} + 1125) q^{70} - 360 q^{71} + 785 q^{73} + ( - 1290 \zeta_{6} + 1290) q^{74} - 2 \zeta_{6} q^{76} - 375 \zeta_{6} q^{77} + (488 \zeta_{6} - 488) q^{79} - 1065 q^{80} - 90 q^{82} + (489 \zeta_{6} - 489) q^{83} - 1080 \zeta_{6} q^{85} - 330 \zeta_{6} q^{86} + (315 \zeta_{6} - 315) q^{88} - 450 q^{89} - 500 q^{91} + (114 \zeta_{6} - 114) q^{92} + 990 \zeta_{6} q^{94} - 30 \zeta_{6} q^{95} + ( - 1105 \zeta_{6} + 1105) q^{97} + 846 q^{98} +O(q^{100})$$ q + (3*z - 3) * q^2 - z * q^4 - 15*z * q^5 + (-25*z + 25) * q^7 - 21 * q^8 + 45 * q^10 + (-15*z + 15) * q^11 - 20*z * q^13 + 75*z * q^14 + (-71*z + 71) * q^16 + 72 * q^17 + 2 * q^19 + (15*z - 15) * q^20 + 45*z * q^22 - 114*z * q^23 + (100*z - 100) * q^25 + 60 * q^26 - 25 * q^28 + (30*z - 30) * q^29 - 101*z * q^31 + 45*z * q^32 + (216*z - 216) * q^34 - 375 * q^35 - 430 * q^37 + (6*z - 6) * q^38 + 315*z * q^40 + 30*z * q^41 + (110*z - 110) * q^43 - 15 * q^44 + 342 * q^46 + (-330*z + 330) * q^47 - 282*z * q^49 - 300*z * q^50 + (20*z - 20) * q^52 + 621 * q^53 - 225 * q^55 + (525*z - 525) * q^56 - 90*z * q^58 + 660*z * q^59 + (-376*z + 376) * q^61 + 303 * q^62 + 433 * q^64 + (300*z - 300) * q^65 + 250*z * q^67 - 72*z * q^68 + (-1125*z + 1125) * q^70 - 360 * q^71 + 785 * q^73 + (-1290*z + 1290) * q^74 - 2*z * q^76 - 375*z * q^77 + (488*z - 488) * q^79 - 1065 * q^80 - 90 * q^82 + (489*z - 489) * q^83 - 1080*z * q^85 - 330*z * q^86 + (315*z - 315) * q^88 - 450 * q^89 - 500 * q^91 + (114*z - 114) * q^92 + 990*z * q^94 - 30*z * q^95 + (-1105*z + 1105) * q^97 + 846 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - q^{4} - 15 q^{5} + 25 q^{7} - 42 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 - q^4 - 15 * q^5 + 25 * q^7 - 42 * q^8 $$2 q - 3 q^{2} - q^{4} - 15 q^{5} + 25 q^{7} - 42 q^{8} + 90 q^{10} + 15 q^{11} - 20 q^{13} + 75 q^{14} + 71 q^{16} + 144 q^{17} + 4 q^{19} - 15 q^{20} + 45 q^{22} - 114 q^{23} - 100 q^{25} + 120 q^{26} - 50 q^{28} - 30 q^{29} - 101 q^{31} + 45 q^{32} - 216 q^{34} - 750 q^{35} - 860 q^{37} - 6 q^{38} + 315 q^{40} + 30 q^{41} - 110 q^{43} - 30 q^{44} + 684 q^{46} + 330 q^{47} - 282 q^{49} - 300 q^{50} - 20 q^{52} + 1242 q^{53} - 450 q^{55} - 525 q^{56} - 90 q^{58} + 660 q^{59} + 376 q^{61} + 606 q^{62} + 866 q^{64} - 300 q^{65} + 250 q^{67} - 72 q^{68} + 1125 q^{70} - 720 q^{71} + 1570 q^{73} + 1290 q^{74} - 2 q^{76} - 375 q^{77} - 488 q^{79} - 2130 q^{80} - 180 q^{82} - 489 q^{83} - 1080 q^{85} - 330 q^{86} - 315 q^{88} - 900 q^{89} - 1000 q^{91} - 114 q^{92} + 990 q^{94} - 30 q^{95} + 1105 q^{97} + 1692 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 - q^4 - 15 * q^5 + 25 * q^7 - 42 * q^8 + 90 * q^10 + 15 * q^11 - 20 * q^13 + 75 * q^14 + 71 * q^16 + 144 * q^17 + 4 * q^19 - 15 * q^20 + 45 * q^22 - 114 * q^23 - 100 * q^25 + 120 * q^26 - 50 * q^28 - 30 * q^29 - 101 * q^31 + 45 * q^32 - 216 * q^34 - 750 * q^35 - 860 * q^37 - 6 * q^38 + 315 * q^40 + 30 * q^41 - 110 * q^43 - 30 * q^44 + 684 * q^46 + 330 * q^47 - 282 * q^49 - 300 * q^50 - 20 * q^52 + 1242 * q^53 - 450 * q^55 - 525 * q^56 - 90 * q^58 + 660 * q^59 + 376 * q^61 + 606 * q^62 + 866 * q^64 - 300 * q^65 + 250 * q^67 - 72 * q^68 + 1125 * q^70 - 720 * q^71 + 1570 * q^73 + 1290 * q^74 - 2 * q^76 - 375 * q^77 - 488 * q^79 - 2130 * q^80 - 180 * q^82 - 489 * q^83 - 1080 * q^85 - 330 * q^86 - 315 * q^88 - 900 * q^89 - 1000 * q^91 - 114 * q^92 + 990 * q^94 - 30 * q^95 + 1105 * q^97 + 1692 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{6}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 2.59808i 0 −0.500000 + 0.866025i −7.50000 + 12.9904i 0 12.5000 + 21.6506i −21.0000 0 45.0000
55.1 −1.50000 + 2.59808i 0 −0.500000 0.866025i −7.50000 12.9904i 0 12.5000 21.6506i −21.0000 0 45.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.4.c.a 2
3.b odd 2 1 81.4.c.c 2
9.c even 3 1 27.4.a.b yes 1
9.c even 3 1 inner 81.4.c.a 2
9.d odd 6 1 27.4.a.a 1
9.d odd 6 1 81.4.c.c 2
36.f odd 6 1 432.4.a.n 1
36.h even 6 1 432.4.a.a 1
45.h odd 6 1 675.4.a.j 1
45.j even 6 1 675.4.a.a 1
45.k odd 12 2 675.4.b.a 2
45.l even 12 2 675.4.b.b 2
63.l odd 6 1 1323.4.a.k 1
63.o even 6 1 1323.4.a.d 1
72.j odd 6 1 1728.4.a.bc 1
72.l even 6 1 1728.4.a.bd 1
72.n even 6 1 1728.4.a.c 1
72.p odd 6 1 1728.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.a 1 9.d odd 6 1
27.4.a.b yes 1 9.c even 3 1
81.4.c.a 2 1.a even 1 1 trivial
81.4.c.a 2 9.c even 3 1 inner
81.4.c.c 2 3.b odd 2 1
81.4.c.c 2 9.d odd 6 1
432.4.a.a 1 36.h even 6 1
432.4.a.n 1 36.f odd 6 1
675.4.a.a 1 45.j even 6 1
675.4.a.j 1 45.h odd 6 1
675.4.b.a 2 45.k odd 12 2
675.4.b.b 2 45.l even 12 2
1323.4.a.d 1 63.o even 6 1
1323.4.a.k 1 63.l odd 6 1
1728.4.a.c 1 72.n even 6 1
1728.4.a.d 1 72.p odd 6 1
1728.4.a.bc 1 72.j odd 6 1
1728.4.a.bd 1 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(81, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 9$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 15T + 225$$
$7$ $$T^{2} - 25T + 625$$
$11$ $$T^{2} - 15T + 225$$
$13$ $$T^{2} + 20T + 400$$
$17$ $$(T - 72)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 114T + 12996$$
$29$ $$T^{2} + 30T + 900$$
$31$ $$T^{2} + 101T + 10201$$
$37$ $$(T + 430)^{2}$$
$41$ $$T^{2} - 30T + 900$$
$43$ $$T^{2} + 110T + 12100$$
$47$ $$T^{2} - 330T + 108900$$
$53$ $$(T - 621)^{2}$$
$59$ $$T^{2} - 660T + 435600$$
$61$ $$T^{2} - 376T + 141376$$
$67$ $$T^{2} - 250T + 62500$$
$71$ $$(T + 360)^{2}$$
$73$ $$(T - 785)^{2}$$
$79$ $$T^{2} + 488T + 238144$$
$83$ $$T^{2} + 489T + 239121$$
$89$ $$(T + 450)^{2}$$
$97$ $$T^{2} - 1105 T + 1221025$$