# Properties

 Label 81.9.d.e Level $81$ Weight $9$ Character orbit 81.d Analytic conductor $32.998$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 27) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + (608 \beta_1 + 608) q^{4} + (28 \beta_{3} - 28 \beta_{2}) q^{5} + 1967 \beta_1 q^{7} + 352 \beta_{2} q^{8}+O(q^{10})$$ q + b3 * q^2 + (608*b1 + 608) * q^4 + (28*b3 - 28*b2) * q^5 + 1967*b1 * q^7 + 352*b2 * q^8 $$q + \beta_{3} q^{2} + (608 \beta_1 + 608) q^{4} + (28 \beta_{3} - 28 \beta_{2}) q^{5} + 1967 \beta_1 q^{7} + 352 \beta_{2} q^{8} + 24192 q^{10} - 428 \beta_{3} q^{11} + (45505 \beta_1 + 45505) q^{13} + ( - 1967 \beta_{3} + 1967 \beta_{2}) q^{14} + 148480 \beta_1 q^{16} + 2028 \beta_{2} q^{17} + 152399 q^{19} + 17024 \beta_{3} q^{20} + ( - 369792 \beta_1 - 369792) q^{22} + (4468 \beta_{3} - 4468 \beta_{2}) q^{23} - 286751 \beta_1 q^{25} + 45505 \beta_{2} q^{26} - 1195936 q^{28} - 20024 \beta_{3} q^{29} + (164350 \beta_1 + 164350) q^{31} + ( - 58368 \beta_{3} + 58368 \beta_{2}) q^{32} + 1752192 \beta_1 q^{34} + 55076 \beta_{2} q^{35} - 663937 q^{37} + 152399 \beta_{3} q^{38} + (8515584 \beta_1 + 8515584) q^{40} + (31912 \beta_{3} - 31912 \beta_{2}) q^{41} + 575330 \beta_1 q^{43} - 260224 \beta_{2} q^{44} + 3860352 q^{46} - 314156 \beta_{3} q^{47} + (1895712 \beta_1 + 1895712) q^{49} + (286751 \beta_{3} - 286751 \beta_{2}) q^{50} + 27667040 \beta_1 q^{52} - 353016 \beta_{2} q^{53} - 10354176 q^{55} - 692384 \beta_{3} q^{56} + ( - 17300736 \beta_1 - 17300736) q^{58} + (171460 \beta_{3} - 171460 \beta_{2}) q^{59} - 19212961 \beta_1 q^{61} + 164350 \beta_{2} q^{62} - 12419072 q^{64} + 1274140 \beta_{3} q^{65} + (598033 \beta_1 + 598033) q^{67} + ( - 1233024 \beta_{3} + 1233024 \beta_{2}) q^{68} + 47585664 \beta_1 q^{70} - 995856 \beta_{2} q^{71} + 12850175 q^{73} - 663937 \beta_{3} q^{74} + (92658592 \beta_1 + 92658592) q^{76} + (841876 \beta_{3} - 841876 \beta_{2}) q^{77} - 23584657 \beta_1 q^{79} + 4157440 \beta_{2} q^{80} + 27571968 q^{82} + 1138024 \beta_{3} q^{83} + (49061376 \beta_1 + 49061376) q^{85} + ( - 575330 \beta_{3} + 575330 \beta_{2}) q^{86} - 130166784 \beta_1 q^{88} + 962268 \beta_{2} q^{89} - 89508335 q^{91} + 2716544 \beta_{3} q^{92} + ( - 271430784 \beta_1 - 271430784) q^{94} + (4267172 \beta_{3} - 4267172 \beta_{2}) q^{95} + 136489631 \beta_1 q^{97} + 1895712 \beta_{2} q^{98}+O(q^{100})$$ q + b3 * q^2 + (608*b1 + 608) * q^4 + (28*b3 - 28*b2) * q^5 + 1967*b1 * q^7 + 352*b2 * q^8 + 24192 * q^10 - 428*b3 * q^11 + (45505*b1 + 45505) * q^13 + (-1967*b3 + 1967*b2) * q^14 + 148480*b1 * q^16 + 2028*b2 * q^17 + 152399 * q^19 + 17024*b3 * q^20 + (-369792*b1 - 369792) * q^22 + (4468*b3 - 4468*b2) * q^23 - 286751*b1 * q^25 + 45505*b2 * q^26 - 1195936 * q^28 - 20024*b3 * q^29 + (164350*b1 + 164350) * q^31 + (-58368*b3 + 58368*b2) * q^32 + 1752192*b1 * q^34 + 55076*b2 * q^35 - 663937 * q^37 + 152399*b3 * q^38 + (8515584*b1 + 8515584) * q^40 + (31912*b3 - 31912*b2) * q^41 + 575330*b1 * q^43 - 260224*b2 * q^44 + 3860352 * q^46 - 314156*b3 * q^47 + (1895712*b1 + 1895712) * q^49 + (286751*b3 - 286751*b2) * q^50 + 27667040*b1 * q^52 - 353016*b2 * q^53 - 10354176 * q^55 - 692384*b3 * q^56 + (-17300736*b1 - 17300736) * q^58 + (171460*b3 - 171460*b2) * q^59 - 19212961*b1 * q^61 + 164350*b2 * q^62 - 12419072 * q^64 + 1274140*b3 * q^65 + (598033*b1 + 598033) * q^67 + (-1233024*b3 + 1233024*b2) * q^68 + 47585664*b1 * q^70 - 995856*b2 * q^71 + 12850175 * q^73 - 663937*b3 * q^74 + (92658592*b1 + 92658592) * q^76 + (841876*b3 - 841876*b2) * q^77 - 23584657*b1 * q^79 + 4157440*b2 * q^80 + 27571968 * q^82 + 1138024*b3 * q^83 + (49061376*b1 + 49061376) * q^85 + (-575330*b3 + 575330*b2) * q^86 - 130166784*b1 * q^88 + 962268*b2 * q^89 - 89508335 * q^91 + 2716544*b3 * q^92 + (-271430784*b1 - 271430784) * q^94 + (4267172*b3 - 4267172*b2) * q^95 + 136489631*b1 * q^97 + 1895712*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 1216 q^{4} - 3934 q^{7}+O(q^{10})$$ 4 * q + 1216 * q^4 - 3934 * q^7 $$4 q + 1216 q^{4} - 3934 q^{7} + 96768 q^{10} + 91010 q^{13} - 296960 q^{16} + 609596 q^{19} - 739584 q^{22} + 573502 q^{25} - 4783744 q^{28} + 328700 q^{31} - 3504384 q^{34} - 2655748 q^{37} + 17031168 q^{40} - 1150660 q^{43} + 15441408 q^{46} + 3791424 q^{49} - 55334080 q^{52} - 41416704 q^{55} - 34601472 q^{58} + 38425922 q^{61} - 49676288 q^{64} + 1196066 q^{67} - 95171328 q^{70} + 51400700 q^{73} + 185317184 q^{76} + 47169314 q^{79} + 110287872 q^{82} + 98122752 q^{85} + 260333568 q^{88} - 358033340 q^{91} - 542861568 q^{94} - 272979262 q^{97}+O(q^{100})$$ 4 * q + 1216 * q^4 - 3934 * q^7 + 96768 * q^10 + 91010 * q^13 - 296960 * q^16 + 609596 * q^19 - 739584 * q^22 + 573502 * q^25 - 4783744 * q^28 + 328700 * q^31 - 3504384 * q^34 - 2655748 * q^37 + 17031168 * q^40 - 1150660 * q^43 + 15441408 * q^46 + 3791424 * q^49 - 55334080 * q^52 - 41416704 * q^55 - 34601472 * q^58 + 38425922 * q^61 - 49676288 * q^64 + 1196066 * q^67 - 95171328 * q^70 + 51400700 * q^73 + 185317184 * q^76 + 47169314 * q^79 + 110287872 * q^82 + 98122752 * q^85 + 260333568 * q^88 - 358033340 * q^91 - 542861568 * q^94 - 272979262 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$6\nu^{3} + 24\nu$$ 6*v^3 + 24*v $$\beta_{3}$$ $$=$$ $$-6\nu^{3} + 12\nu$$ -6*v^3 + 12*v
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 36$$ (b3 + b2) / 36 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + \beta_{2} ) / 18$$ (-2*b3 + b2) / 18

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−25.4558 14.6969i 0 304.000 + 526.543i −712.764 + 411.514i 0 −983.500 + 1703.47i 10346.6i 0 24192.0
26.2 25.4558 + 14.6969i 0 304.000 + 526.543i 712.764 411.514i 0 −983.500 + 1703.47i 10346.6i 0 24192.0
53.1 −25.4558 + 14.6969i 0 304.000 526.543i −712.764 411.514i 0 −983.500 1703.47i 10346.6i 0 24192.0
53.2 25.4558 14.6969i 0 304.000 526.543i 712.764 + 411.514i 0 −983.500 1703.47i 10346.6i 0 24192.0
 $$n$$: 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
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.e 4
3.b odd 2 1 inner 81.9.d.e 4
9.c even 3 1 27.9.b.b 2
9.c even 3 1 inner 81.9.d.e 4
9.d odd 6 1 27.9.b.b 2
9.d odd 6 1 inner 81.9.d.e 4
36.f odd 6 1 432.9.e.d 2
36.h even 6 1 432.9.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.b 2 9.c even 3 1
27.9.b.b 2 9.d odd 6 1
81.9.d.e 4 1.a even 1 1 trivial
81.9.d.e 4 3.b odd 2 1 inner
81.9.d.e 4 9.c even 3 1 inner
81.9.d.e 4 9.d odd 6 1 inner
432.9.e.d 2 36.f odd 6 1
432.9.e.d 2 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 864T_{2}^{2} + 746496$$ acting on $$S_{9}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 864 T^{2} + 746496$$
$3$ $$T^{4}$$
$5$ $$T^{4} + \cdots + 458838245376$$
$7$ $$(T^{2} + 1967 T + 3869089)^{2}$$
$11$ $$T^{4} + \cdots + 25\!\cdots\!76$$
$13$ $$(T^{2} - 45505 T + 2070705025)^{2}$$
$17$ $$(T^{2} + 3553445376)^{2}$$
$19$ $$(T - 152399)^{4}$$
$23$ $$T^{4} + \cdots + 29\!\cdots\!96$$
$29$ $$T^{4} + \cdots + 12\!\cdots\!96$$
$31$ $$(T^{2} - 164350 T + 27010922500)^{2}$$
$37$ $$(T + 663937)^{4}$$
$41$ $$T^{4} + \cdots + 77\!\cdots\!56$$
$43$ $$(T^{2} + 575330 T + 331004608900)^{2}$$
$47$ $$T^{4} + \cdots + 72\!\cdots\!16$$
$53$ $$(T^{2} + 107671935965184)^{2}$$
$59$ $$T^{4} + \cdots + 64\!\cdots\!00$$
$61$ $$(T^{2} + \cdots + 369137870387521)^{2}$$
$67$ $$(T^{2} - 598033 T + 357643469089)^{2}$$
$71$ $$(T^{2} + 856854005243904)^{2}$$
$73$ $$(T - 12850175)^{4}$$
$79$ $$(T^{2} + \cdots + 556236045807649)^{2}$$
$83$ $$T^{4} + \cdots + 12\!\cdots\!96$$
$89$ $$(T^{2} + 800029184103936)^{2}$$
$97$ $$(T^{2} + \cdots + 18\!\cdots\!61)^{2}$$