# Properties

 Label 81.9.d.c 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: $$\Q(\sqrt{-3}, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 10x^{2} + 100$$ x^4 + 10*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + (14 \beta_1 + 14) q^{4} + ( - 26 \beta_{3} + 26 \beta_{2}) q^{5} - 679 \beta_1 q^{7} - 242 \beta_{2} q^{8}+O(q^{10})$$ q + b3 * q^2 + (14*b1 + 14) * q^4 + (-26*b3 + 26*b2) * q^5 - 679*b1 * q^7 - 242*b2 * q^8 $$q + \beta_{3} q^{2} + (14 \beta_1 + 14) q^{4} + ( - 26 \beta_{3} + 26 \beta_{2}) q^{5} - 679 \beta_1 q^{7} - 242 \beta_{2} q^{8} - 7020 q^{10} + 814 \beta_{3} q^{11} + (30817 \beta_1 + 30817) q^{13} + (679 \beta_{3} - 679 \beta_{2}) q^{14} - 68924 \beta_1 q^{16} + 7806 \beta_{2} q^{17} - 138391 q^{19} - 364 \beta_{3} q^{20} + (219780 \beta_1 + 219780) q^{22} + ( - 18482 \beta_{3} + 18482 \beta_{2}) q^{23} + 208105 \beta_1 q^{25} + 30817 \beta_{2} q^{26} + 9506 q^{28} + 80740 \beta_{3} q^{29} + ( - 352214 \beta_1 - 352214) q^{31} + (6972 \beta_{3} - 6972 \beta_{2}) q^{32} + 2107620 \beta_1 q^{34} + 17654 \beta_{2} q^{35} + 1189991 q^{37} - 138391 \beta_{3} q^{38} + (1698840 \beta_1 + 1698840) q^{40} + (66580 \beta_{3} - 66580 \beta_{2}) q^{41} + 6246086 \beta_1 q^{43} + 11396 \beta_{2} q^{44} - 4990140 q^{46} - 146 \beta_{3} q^{47} + (5303760 \beta_1 + 5303760) q^{49} + ( - 208105 \beta_{3} + 208105 \beta_{2}) q^{50} + 431438 \beta_1 q^{52} - 765468 \beta_{2} q^{53} - 5714280 q^{55} - 164318 \beta_{3} q^{56} + (21799800 \beta_1 + 21799800) q^{58} + (641206 \beta_{3} - 641206 \beta_{2}) q^{59} + 16580399 \beta_1 q^{61} - 352214 \beta_{2} q^{62} - 15762104 q^{64} - 801242 \beta_{3} q^{65} + ( - 7667153 \beta_1 - 7667153) q^{67} + ( - 109284 \beta_{3} + 109284 \beta_{2}) q^{68} + 4766580 \beta_1 q^{70} + 1413192 \beta_{2} q^{71} + 24949631 q^{73} + 1189991 \beta_{3} q^{74} + ( - 1937474 \beta_1 - 1937474) q^{76} + (552706 \beta_{3} - 552706 \beta_{2}) q^{77} + 41685089 \beta_1 q^{79} + 1792024 \beta_{2} q^{80} + 17976600 q^{82} + 2722060 \beta_{3} q^{83} + ( - 54798120 \beta_1 - 54798120) q^{85} + ( - 6246086 \beta_{3} + 6246086 \beta_{2}) q^{86} - 53186760 \beta_1 q^{88} + 45078 \beta_{2} q^{89} + 20924743 q^{91} - 258748 \beta_{3} q^{92} + ( - 39420 \beta_1 - 39420) q^{94} + (3598166 \beta_{3} - 3598166 \beta_{2}) q^{95} - 105926089 \beta_1 q^{97} + 5303760 \beta_{2} q^{98}+O(q^{100})$$ q + b3 * q^2 + (14*b1 + 14) * q^4 + (-26*b3 + 26*b2) * q^5 - 679*b1 * q^7 - 242*b2 * q^8 - 7020 * q^10 + 814*b3 * q^11 + (30817*b1 + 30817) * q^13 + (679*b3 - 679*b2) * q^14 - 68924*b1 * q^16 + 7806*b2 * q^17 - 138391 * q^19 - 364*b3 * q^20 + (219780*b1 + 219780) * q^22 + (-18482*b3 + 18482*b2) * q^23 + 208105*b1 * q^25 + 30817*b2 * q^26 + 9506 * q^28 + 80740*b3 * q^29 + (-352214*b1 - 352214) * q^31 + (6972*b3 - 6972*b2) * q^32 + 2107620*b1 * q^34 + 17654*b2 * q^35 + 1189991 * q^37 - 138391*b3 * q^38 + (1698840*b1 + 1698840) * q^40 + (66580*b3 - 66580*b2) * q^41 + 6246086*b1 * q^43 + 11396*b2 * q^44 - 4990140 * q^46 - 146*b3 * q^47 + (5303760*b1 + 5303760) * q^49 + (-208105*b3 + 208105*b2) * q^50 + 431438*b1 * q^52 - 765468*b2 * q^53 - 5714280 * q^55 - 164318*b3 * q^56 + (21799800*b1 + 21799800) * q^58 + (641206*b3 - 641206*b2) * q^59 + 16580399*b1 * q^61 - 352214*b2 * q^62 - 15762104 * q^64 - 801242*b3 * q^65 + (-7667153*b1 - 7667153) * q^67 + (-109284*b3 + 109284*b2) * q^68 + 4766580*b1 * q^70 + 1413192*b2 * q^71 + 24949631 * q^73 + 1189991*b3 * q^74 + (-1937474*b1 - 1937474) * q^76 + (552706*b3 - 552706*b2) * q^77 + 41685089*b1 * q^79 + 1792024*b2 * q^80 + 17976600 * q^82 + 2722060*b3 * q^83 + (-54798120*b1 - 54798120) * q^85 + (-6246086*b3 + 6246086*b2) * q^86 - 53186760*b1 * q^88 + 45078*b2 * q^89 + 20924743 * q^91 - 258748*b3 * q^92 + (-39420*b1 - 39420) * q^94 + (3598166*b3 - 3598166*b2) * q^95 - 105926089*b1 * q^97 + 5303760*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 28 q^{4} + 1358 q^{7}+O(q^{10})$$ 4 * q + 28 * q^4 + 1358 * q^7 $$4 q + 28 q^{4} + 1358 q^{7} - 28080 q^{10} + 61634 q^{13} + 137848 q^{16} - 553564 q^{19} + 439560 q^{22} - 416210 q^{25} + 38024 q^{28} - 704428 q^{31} - 4215240 q^{34} + 4759964 q^{37} + 3397680 q^{40} - 12492172 q^{43} - 19960560 q^{46} + 10607520 q^{49} - 862876 q^{52} - 22857120 q^{55} + 43599600 q^{58} - 33160798 q^{61} - 63048416 q^{64} - 15334306 q^{67} - 9533160 q^{70} + 99798524 q^{73} - 3874948 q^{76} - 83370178 q^{79} + 71906400 q^{82} - 109596240 q^{85} + 106373520 q^{88} + 83698972 q^{91} - 78840 q^{94} + 211852178 q^{97}+O(q^{100})$$ 4 * q + 28 * q^4 + 1358 * q^7 - 28080 * q^10 + 61634 * q^13 + 137848 * q^16 - 553564 * q^19 + 439560 * q^22 - 416210 * q^25 + 38024 * q^28 - 704428 * q^31 - 4215240 * q^34 + 4759964 * q^37 + 3397680 * q^40 - 12492172 * q^43 - 19960560 * q^46 + 10607520 * q^49 - 862876 * q^52 - 22857120 * q^55 + 43599600 * q^58 - 33160798 * q^61 - 63048416 * q^64 - 15334306 * q^67 - 9533160 * q^70 + 99798524 * q^73 - 3874948 * q^76 - 83370178 * q^79 + 71906400 * q^82 - 109596240 * q^85 + 106373520 * q^88 + 83698972 * q^91 - 78840 * q^94 + 211852178 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 10$$ (v^2) / 10 $$\beta_{2}$$ $$=$$ $$( 3\nu^{3} + 60\nu ) / 10$$ (3*v^3 + 60*v) / 10 $$\beta_{3}$$ $$=$$ $$( -3\nu^{3} + 30\nu ) / 10$$ (-3*v^3 + 30*v) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 9$$ (b3 + b2) / 9 $$\nu^{2}$$ $$=$$ $$10\beta_1$$ 10*b1 $$\nu^{3}$$ $$=$$ $$( -20\beta_{3} + 10\beta_{2} ) / 9$$ (-20*b3 + 10*b2) / 9

## 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
 −1.58114 − 2.73861i 1.58114 + 2.73861i −1.58114 + 2.73861i 1.58114 − 2.73861i
−14.2302 8.21584i 0 7.00000 + 12.1244i 369.986 213.612i 0 339.500 588.031i 3976.47i 0 −7020.00
26.2 14.2302 + 8.21584i 0 7.00000 + 12.1244i −369.986 + 213.612i 0 339.500 588.031i 3976.47i 0 −7020.00
53.1 −14.2302 + 8.21584i 0 7.00000 12.1244i 369.986 + 213.612i 0 339.500 + 588.031i 3976.47i 0 −7020.00
53.2 14.2302 8.21584i 0 7.00000 12.1244i −369.986 213.612i 0 339.500 + 588.031i 3976.47i 0 −7020.00
 $$n$$: 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.c 4
3.b odd 2 1 inner 81.9.d.c 4
9.c even 3 1 27.9.b.c 2
9.c even 3 1 inner 81.9.d.c 4
9.d odd 6 1 27.9.b.c 2
9.d odd 6 1 inner 81.9.d.c 4
36.f odd 6 1 432.9.e.f 2
36.h even 6 1 432.9.e.f 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 270T_{2}^{2} + 72900$$ acting on $$S_{9}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 270 T^{2} + 72900$$
$3$ $$T^{4}$$
$5$ $$T^{4} + \cdots + 33313550400$$
$7$ $$(T^{2} - 679 T + 461041)^{2}$$
$11$ $$T^{4} + \cdots + 32\!\cdots\!00$$
$13$ $$(T^{2} - 30817 T + 949687489)^{2}$$
$17$ $$(T^{2} + 16452081720)^{2}$$
$19$ $$(T + 138391)^{4}$$
$23$ $$T^{4} + \cdots + 85\!\cdots\!00$$
$29$ $$T^{4} + \cdots + 30\!\cdots\!00$$
$31$ $$(T^{2} + 352214 T + 124054701796)^{2}$$
$37$ $$(T - 1189991)^{4}$$
$41$ $$T^{4} + \cdots + 14\!\cdots\!00$$
$43$ $$(T^{2} + \cdots + 39013590319396)^{2}$$
$47$ $$T^{4} + \cdots + 33123708302400$$
$53$ $$(T^{2} + 158204139936480)^{2}$$
$59$ $$T^{4} + \cdots + 12\!\cdots\!00$$
$61$ $$(T^{2} + \cdots + 274909630999201)^{2}$$
$67$ $$(T^{2} + \cdots + 58785235125409)^{2}$$
$71$ $$(T^{2} + 539220139793280)^{2}$$
$73$ $$(T - 24949631)^{4}$$
$79$ $$(T^{2} + \cdots + 17\!\cdots\!21)^{2}$$
$83$ $$T^{4} + \cdots + 40\!\cdots\!00$$
$89$ $$(T^{2} + 548647042680)^{2}$$
$97$ $$(T^{2} + \cdots + 11\!\cdots\!21)^{2}$$