# Properties

 Label 81.4.c.g Level $81$ Weight $4$ Character orbit 81.c Analytic conductor $4.779$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_1 + 1) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 7 \beta_1) q^{4} + ( - 2 \beta_{3} + 2 \beta_{2} + 7 \beta_1) q^{5} + (6 \beta_{3} - 8 \beta_1 - 8) q^{7} + ( - 5 \beta_{2} - 41) q^{8}+O(q^{10})$$ q + (b3 + b1 + 1) * q^2 + (3*b3 - 3*b2 + 7*b1) * q^4 + (-2*b3 + 2*b2 + 7*b1) * q^5 + (6*b3 - 8*b1 - 8) * q^7 + (-5*b2 - 41) * q^8 $$q + (\beta_{3} + \beta_1 + 1) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 7 \beta_1) q^{4} + ( - 2 \beta_{3} + 2 \beta_{2} + 7 \beta_1) q^{5} + (6 \beta_{3} - 8 \beta_1 - 8) q^{7} + ( - 5 \beta_{2} - 41) q^{8} + ( - 3 \beta_{2} + 21) q^{10} + ( - 2 \beta_{3} - 20 \beta_1 - 20) q^{11} + ( - 6 \beta_{3} + 6 \beta_{2} + 35 \beta_1) q^{13} + (4 \beta_{3} - 4 \beta_{2} + 76 \beta_1) q^{14} + ( - 27 \beta_{3} - 55 \beta_1 - 55) q^{16} + (24 \beta_{2} - 21) q^{17} + (18 \beta_{2} + 56) q^{19} + ( - \beta_{3} + 35 \beta_1 + 35) q^{20} + ( - 24 \beta_{3} + 24 \beta_{2} - 48 \beta_1) q^{22} + (22 \beta_{3} - 22 \beta_{2} - 68 \beta_1) q^{23} + (24 \beta_{3} + 20 \beta_1 + 20) q^{25} + ( - 23 \beta_{2} + 49) q^{26} + ( - 36 \beta_{2} - 196) q^{28} + ( - 26 \beta_{3} - 107 \beta_1 - 107) q^{29} + (48 \beta_{3} - 48 \beta_{2} - 28 \beta_1) q^{31} + ( - 69 \beta_{3} + 69 \beta_{2} - 105 \beta_1) q^{32} + (27 \beta_{3} + 315 \beta_1 + 315) q^{34} + ( - 46 \beta_{2} + 224) q^{35} + (18 \beta_{2} + 83) q^{37} + (92 \beta_{3} + 308 \beta_1 + 308) q^{38} + (57 \beta_{3} - 57 \beta_{2} - 147 \beta_1) q^{40} + (28 \beta_{3} - 28 \beta_{2} - 350 \beta_1) q^{41} + (6 \beta_{3} - 188 \beta_1 - 188) q^{43} + (80 \beta_{2} + 224) q^{44} + (24 \beta_{2} - 240) q^{46} + (4 \beta_{3} - 140 \beta_1 - 140) q^{47} + ( - 60 \beta_{3} + 60 \beta_{2} + 225 \beta_1) q^{49} + (68 \beta_{3} - 68 \beta_{2} + 356 \beta_1) q^{50} + ( - 45 \beta_{3} + 7 \beta_1 + 7) q^{52} - 162 q^{53} + ( - 30 \beta_{2} + 84) q^{55} + ( - 236 \beta_{3} - 92 \beta_1 - 92) q^{56} + ( - 159 \beta_{3} + 159 \beta_{2} - 471 \beta_1) q^{58} + ( - 92 \beta_{3} + 92 \beta_{2} - 56 \beta_1) q^{59} + ( - 138 \beta_{3} - 35 \beta_1 - 35) q^{61} + ( - 68 \beta_{2} - 644) q^{62} + (27 \beta_{2} + 631) q^{64} + (100 \beta_{3} - 413 \beta_1 - 413) q^{65} + ( - 78 \beta_{3} + 78 \beta_{2} + 440 \beta_1) q^{67} + (177 \beta_{3} - 177 \beta_{2} + 861 \beta_1) q^{68} + (132 \beta_{3} - 420 \beta_1 - 420) q^{70} + (114 \beta_{2} - 120) q^{71} + ( - 108 \beta_{2} - 133) q^{73} + (119 \beta_{3} + 335 \beta_1 + 335) q^{74} + (348 \beta_{3} - 348 \beta_{2} + 1148 \beta_1) q^{76} + ( - 116 \beta_{3} + 116 \beta_{2} - 8 \beta_1) q^{77} + (6 \beta_{3} - 692 \beta_1 - 692) q^{79} + (25 \beta_{2} - 371) q^{80} + (294 \beta_{2} - 42) q^{82} + (100 \beta_{3} + 532 \beta_1 + 532) q^{83} + (162 \beta_{3} - 162 \beta_{2} - 819 \beta_1) q^{85} + ( - 176 \beta_{3} + 176 \beta_{2} - 104 \beta_1) q^{86} + (192 \beta_{3} + 960 \beta_1 + 960) q^{88} + ( - 240 \beta_{2} - 357) q^{89} + ( - 222 \beta_{2} + 784) q^{91} + ( - 16 \beta_{3} - 448 \beta_1 - 448) q^{92} + ( - 132 \beta_{3} + 132 \beta_{2} - 84 \beta_1) q^{94} + ( - 22 \beta_{3} + 22 \beta_{2} - 112 \beta_1) q^{95} + (60 \beta_{3} + 406 \beta_1 + 406) q^{97} + ( - 105 \beta_{2} + 615) q^{98}+O(q^{100})$$ q + (b3 + b1 + 1) * q^2 + (3*b3 - 3*b2 + 7*b1) * q^4 + (-2*b3 + 2*b2 + 7*b1) * q^5 + (6*b3 - 8*b1 - 8) * q^7 + (-5*b2 - 41) * q^8 + (-3*b2 + 21) * q^10 + (-2*b3 - 20*b1 - 20) * q^11 + (-6*b3 + 6*b2 + 35*b1) * q^13 + (4*b3 - 4*b2 + 76*b1) * q^14 + (-27*b3 - 55*b1 - 55) * q^16 + (24*b2 - 21) * q^17 + (18*b2 + 56) * q^19 + (-b3 + 35*b1 + 35) * q^20 + (-24*b3 + 24*b2 - 48*b1) * q^22 + (22*b3 - 22*b2 - 68*b1) * q^23 + (24*b3 + 20*b1 + 20) * q^25 + (-23*b2 + 49) * q^26 + (-36*b2 - 196) * q^28 + (-26*b3 - 107*b1 - 107) * q^29 + (48*b3 - 48*b2 - 28*b1) * q^31 + (-69*b3 + 69*b2 - 105*b1) * q^32 + (27*b3 + 315*b1 + 315) * q^34 + (-46*b2 + 224) * q^35 + (18*b2 + 83) * q^37 + (92*b3 + 308*b1 + 308) * q^38 + (57*b3 - 57*b2 - 147*b1) * q^40 + (28*b3 - 28*b2 - 350*b1) * q^41 + (6*b3 - 188*b1 - 188) * q^43 + (80*b2 + 224) * q^44 + (24*b2 - 240) * q^46 + (4*b3 - 140*b1 - 140) * q^47 + (-60*b3 + 60*b2 + 225*b1) * q^49 + (68*b3 - 68*b2 + 356*b1) * q^50 + (-45*b3 + 7*b1 + 7) * q^52 - 162 * q^53 + (-30*b2 + 84) * q^55 + (-236*b3 - 92*b1 - 92) * q^56 + (-159*b3 + 159*b2 - 471*b1) * q^58 + (-92*b3 + 92*b2 - 56*b1) * q^59 + (-138*b3 - 35*b1 - 35) * q^61 + (-68*b2 - 644) * q^62 + (27*b2 + 631) * q^64 + (100*b3 - 413*b1 - 413) * q^65 + (-78*b3 + 78*b2 + 440*b1) * q^67 + (177*b3 - 177*b2 + 861*b1) * q^68 + (132*b3 - 420*b1 - 420) * q^70 + (114*b2 - 120) * q^71 + (-108*b2 - 133) * q^73 + (119*b3 + 335*b1 + 335) * q^74 + (348*b3 - 348*b2 + 1148*b1) * q^76 + (-116*b3 + 116*b2 - 8*b1) * q^77 + (6*b3 - 692*b1 - 692) * q^79 + (25*b2 - 371) * q^80 + (294*b2 - 42) * q^82 + (100*b3 + 532*b1 + 532) * q^83 + (162*b3 - 162*b2 - 819*b1) * q^85 + (-176*b3 + 176*b2 - 104*b1) * q^86 + (192*b3 + 960*b1 + 960) * q^88 + (-240*b2 - 357) * q^89 + (-222*b2 + 784) * q^91 + (-16*b3 - 448*b1 - 448) * q^92 + (-132*b3 + 132*b2 - 84*b1) * q^94 + (-22*b3 + 22*b2 - 112*b1) * q^95 + (60*b3 + 406*b1 + 406) * q^97 + (-105*b2 + 615) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} - 17 q^{4} - 12 q^{5} - 10 q^{7} - 174 q^{8}+O(q^{10})$$ 4 * q + 3 * q^2 - 17 * q^4 - 12 * q^5 - 10 * q^7 - 174 * q^8 $$4 q + 3 q^{2} - 17 q^{4} - 12 q^{5} - 10 q^{7} - 174 q^{8} + 78 q^{10} - 42 q^{11} - 64 q^{13} - 156 q^{14} - 137 q^{16} - 36 q^{17} + 260 q^{19} + 69 q^{20} + 120 q^{22} + 114 q^{23} + 64 q^{25} + 150 q^{26} - 856 q^{28} - 240 q^{29} + 8 q^{31} + 279 q^{32} + 657 q^{34} + 804 q^{35} + 368 q^{37} + 708 q^{38} + 237 q^{40} + 672 q^{41} - 370 q^{43} + 1056 q^{44} - 912 q^{46} - 276 q^{47} - 390 q^{49} - 780 q^{50} - 31 q^{52} - 648 q^{53} + 276 q^{55} - 420 q^{56} + 1101 q^{58} + 204 q^{59} - 208 q^{61} - 2712 q^{62} + 2578 q^{64} - 726 q^{65} - 802 q^{67} - 1899 q^{68} - 708 q^{70} - 252 q^{71} - 748 q^{73} + 789 q^{74} - 2644 q^{76} + 132 q^{77} - 1378 q^{79} - 1434 q^{80} + 420 q^{82} + 1164 q^{83} + 1476 q^{85} + 384 q^{86} + 2112 q^{88} - 1908 q^{89} + 2692 q^{91} - 912 q^{92} + 300 q^{94} + 246 q^{95} + 872 q^{97} + 2250 q^{98}+O(q^{100})$$ 4 * q + 3 * q^2 - 17 * q^4 - 12 * q^5 - 10 * q^7 - 174 * q^8 + 78 * q^10 - 42 * q^11 - 64 * q^13 - 156 * q^14 - 137 * q^16 - 36 * q^17 + 260 * q^19 + 69 * q^20 + 120 * q^22 + 114 * q^23 + 64 * q^25 + 150 * q^26 - 856 * q^28 - 240 * q^29 + 8 * q^31 + 279 * q^32 + 657 * q^34 + 804 * q^35 + 368 * q^37 + 708 * q^38 + 237 * q^40 + 672 * q^41 - 370 * q^43 + 1056 * q^44 - 912 * q^46 - 276 * q^47 - 390 * q^49 - 780 * q^50 - 31 * q^52 - 648 * q^53 + 276 * q^55 - 420 * q^56 + 1101 * q^58 + 204 * q^59 - 208 * q^61 - 2712 * q^62 + 2578 * q^64 - 726 * q^65 - 802 * q^67 - 1899 * q^68 - 708 * q^70 - 252 * q^71 - 748 * q^73 + 789 * q^74 - 2644 * q^76 + 132 * q^77 - 1378 * q^79 - 1434 * q^80 + 420 * q^82 + 1164 * q^83 + 1476 * q^85 + 384 * q^86 + 2112 * q^88 - 1908 * q^89 + 2692 * q^91 - 912 * q^92 + 300 * q^94 + 246 * q^95 + 872 * q^97 + 2250 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20$$ (v^3 + 4*v^2 - 4*v - 25) / 20 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5$$ (-v^3 + v^2 + 9*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10$$ (3*v^3 + 2*v^2 + 8*v - 25) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3$$ (b3 + b2 - 2*b1 - 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3$$ (-b3 + 2*b2 + 14*b1 + 13) / 3 $$\nu^{3}$$ $$=$$ $$( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3$$ (8*b3 - 4*b2 - 4*b1 + 19) / 3

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{1}$$

## 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
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
−1.13746 1.97014i 0 1.41238 2.44631i −6.77492 + 11.7345i 0 −13.8248 23.9452i −24.6254 0 30.8248
28.2 2.63746 + 4.56821i 0 −9.91238 + 17.1687i 0.774917 1.34220i 0 8.82475 + 15.2849i −62.3746 0 8.17525
55.1 −1.13746 + 1.97014i 0 1.41238 + 2.44631i −6.77492 11.7345i 0 −13.8248 + 23.9452i −24.6254 0 30.8248
55.2 2.63746 4.56821i 0 −9.91238 17.1687i 0.774917 + 1.34220i 0 8.82475 15.2849i −62.3746 0 8.17525
 $$n$$: 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.g 4
3.b odd 2 1 81.4.c.d 4
9.c even 3 1 81.4.a.b 2
9.c even 3 1 inner 81.4.c.g 4
9.d odd 6 1 81.4.a.e yes 2
9.d odd 6 1 81.4.c.d 4
36.f odd 6 1 1296.4.a.t 2
36.h even 6 1 1296.4.a.k 2
45.h odd 6 1 2025.4.a.h 2
45.j even 6 1 2025.4.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.4.a.b 2 9.c even 3 1
81.4.a.e yes 2 9.d odd 6 1
81.4.c.d 4 3.b odd 2 1
81.4.c.d 4 9.d odd 6 1
81.4.c.g 4 1.a even 1 1 trivial
81.4.c.g 4 9.c even 3 1 inner
1296.4.a.k 2 36.h even 6 1
1296.4.a.t 2 36.f odd 6 1
2025.4.a.h 2 45.h odd 6 1
2025.4.a.o 2 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3T_{2}^{3} + 21T_{2}^{2} + 36T_{2} + 144$$ acting on $$S_{4}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + \cdots + 144$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 12 T^{3} + \cdots + 441$$
$7$ $$T^{4} + 10 T^{3} + \cdots + 238144$$
$11$ $$T^{4} + 42 T^{3} + \cdots + 147456$$
$13$ $$T^{4} + 64 T^{3} + \cdots + 261121$$
$17$ $$(T^{2} + 18 T - 8127)^{2}$$
$19$ $$(T^{2} - 130 T - 392)^{2}$$
$23$ $$T^{4} - 114 T^{3} + \cdots + 13307904$$
$29$ $$T^{4} + 240 T^{3} + \cdots + 22724289$$
$31$ $$T^{4} + \cdots + 1076889856$$
$37$ $$(T^{2} - 184 T + 3847)^{2}$$
$41$ $$T^{4} + \cdots + 10347772176$$
$43$ $$T^{4} + \cdots + 1136498944$$
$47$ $$T^{4} + 276 T^{3} + \cdots + 354041856$$
$53$ $$(T + 162)^{4}$$
$59$ $$T^{4} + \cdots + 12145803264$$
$61$ $$T^{4} + \cdots + 67892034721$$
$67$ $$T^{4} + \cdots + 5491402816$$
$71$ $$(T^{2} + 126 T - 181224)^{2}$$
$73$ $$(T^{2} + 374 T - 131243)^{2}$$
$79$ $$T^{4} + \cdots + 224873227264$$
$83$ $$T^{4} + \cdots + 38503858176$$
$89$ $$(T^{2} + 954 T - 593271)^{2}$$
$97$ $$T^{4} + \cdots + 19264329616$$