# Properties

 Label 81.9.d.d Level $81$ Weight $9$ Character orbit 81.d Analytic conductor $32.998$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 14x^{2} + 196$$ x^4 - 14*x^2 + 196 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 3) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 248 \beta_{2} q^{4} + ( - 10 \beta_{3} + 10 \beta_1) q^{5} + ( - 1750 \beta_{2} + 1750) q^{7} - 8 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 248*b2 * q^4 + (-10*b3 + 10*b1) * q^5 + (-1750*b2 + 1750) * q^7 - 8*b3 * q^8 $$q + \beta_1 q^{2} + 248 \beta_{2} q^{4} + ( - 10 \beta_{3} + 10 \beta_1) q^{5} + ( - 1750 \beta_{2} + 1750) q^{7} - 8 \beta_{3} q^{8} + 5040 q^{10} + 310 \beta_1 q^{11} - 25730 \beta_{2} q^{13} + ( - 1750 \beta_{3} + 1750 \beta_1) q^{14} + ( - 67520 \beta_{2} + 67520) q^{16} - 3336 \beta_{3} q^{17} + 18938 q^{19} + 2480 \beta_1 q^{20} + 156240 \beta_{2} q^{22} + ( - 20956 \beta_{3} + 20956 \beta_1) q^{23} + (340225 \beta_{2} - 340225) q^{25} - 25730 \beta_{3} q^{26} + 434000 q^{28} + 20530 \beta_1 q^{29} + 351478 \beta_{2} q^{31} + ( - 65472 \beta_{3} + 65472 \beta_1) q^{32} + ( - 1681344 \beta_{2} + 1681344) q^{34} - 17500 \beta_{3} q^{35} + 1335170 q^{37} + 18938 \beta_1 q^{38} - 40320 \beta_{2} q^{40} + (83540 \beta_{3} - 83540 \beta_1) q^{41} + ( - 3526150 \beta_{2} + 3526150) q^{43} + 76880 \beta_{3} q^{44} + 10561824 q^{46} - 181784 \beta_1 q^{47} + 2702301 \beta_{2} q^{49} + (340225 \beta_{3} - 340225 \beta_1) q^{50} + ( - 6381040 \beta_{2} + 6381040) q^{52} + 294066 \beta_{3} q^{53} + 1562400 q^{55} - 14000 \beta_1 q^{56} + 10347120 \beta_{2} q^{58} + (610910 \beta_{3} - 610910 \beta_1) q^{59} + (753602 \beta_{2} - 753602) q^{61} + 351478 \beta_{3} q^{62} + 15712768 q^{64} - 257300 \beta_1 q^{65} - 2268890 \beta_{2} q^{67} + ( - 827328 \beta_{3} + 827328 \beta_1) q^{68} + ( - 8820000 \beta_{2} + 8820000) q^{70} - 758220 \beta_{3} q^{71} + 27672770 q^{73} + 1335170 \beta_1 q^{74} + 4696624 \beta_{2} q^{76} + ( - 542500 \beta_{3} + 542500 \beta_1) q^{77} + ( - 22980982 \beta_{2} + 22980982) q^{79} - 675200 \beta_{3} q^{80} - 42104160 q^{82} - 2066606 \beta_1 q^{83} - 16813440 \beta_{2} q^{85} + ( - 3526150 \beta_{3} + 3526150 \beta_1) q^{86} + ( - 1249920 \beta_{2} + 1249920) q^{88} - 3234540 \beta_{3} q^{89} - 45027500 q^{91} + 5197088 \beta_1 q^{92} - 91619136 \beta_{2} q^{94} + ( - 189380 \beta_{3} + 189380 \beta_1) q^{95} + (147271010 \beta_{2} - 147271010) q^{97} + 2702301 \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 + 248*b2 * q^4 + (-10*b3 + 10*b1) * q^5 + (-1750*b2 + 1750) * q^7 - 8*b3 * q^8 + 5040 * q^10 + 310*b1 * q^11 - 25730*b2 * q^13 + (-1750*b3 + 1750*b1) * q^14 + (-67520*b2 + 67520) * q^16 - 3336*b3 * q^17 + 18938 * q^19 + 2480*b1 * q^20 + 156240*b2 * q^22 + (-20956*b3 + 20956*b1) * q^23 + (340225*b2 - 340225) * q^25 - 25730*b3 * q^26 + 434000 * q^28 + 20530*b1 * q^29 + 351478*b2 * q^31 + (-65472*b3 + 65472*b1) * q^32 + (-1681344*b2 + 1681344) * q^34 - 17500*b3 * q^35 + 1335170 * q^37 + 18938*b1 * q^38 - 40320*b2 * q^40 + (83540*b3 - 83540*b1) * q^41 + (-3526150*b2 + 3526150) * q^43 + 76880*b3 * q^44 + 10561824 * q^46 - 181784*b1 * q^47 + 2702301*b2 * q^49 + (340225*b3 - 340225*b1) * q^50 + (-6381040*b2 + 6381040) * q^52 + 294066*b3 * q^53 + 1562400 * q^55 - 14000*b1 * q^56 + 10347120*b2 * q^58 + (610910*b3 - 610910*b1) * q^59 + (753602*b2 - 753602) * q^61 + 351478*b3 * q^62 + 15712768 * q^64 - 257300*b1 * q^65 - 2268890*b2 * q^67 + (-827328*b3 + 827328*b1) * q^68 + (-8820000*b2 + 8820000) * q^70 - 758220*b3 * q^71 + 27672770 * q^73 + 1335170*b1 * q^74 + 4696624*b2 * q^76 + (-542500*b3 + 542500*b1) * q^77 + (-22980982*b2 + 22980982) * q^79 - 675200*b3 * q^80 - 42104160 * q^82 - 2066606*b1 * q^83 - 16813440*b2 * q^85 + (-3526150*b3 + 3526150*b1) * q^86 + (-1249920*b2 + 1249920) * q^88 - 3234540*b3 * q^89 - 45027500 * q^91 + 5197088*b1 * q^92 - 91619136*b2 * q^94 + (-189380*b3 + 189380*b1) * q^95 + (147271010*b2 - 147271010) * q^97 + 2702301*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 496 q^{4} + 3500 q^{7}+O(q^{10})$$ 4 * q + 496 * q^4 + 3500 * q^7 $$4 q + 496 q^{4} + 3500 q^{7} + 20160 q^{10} - 51460 q^{13} + 135040 q^{16} + 75752 q^{19} + 312480 q^{22} - 680450 q^{25} + 1736000 q^{28} + 702956 q^{31} + 3362688 q^{34} + 5340680 q^{37} - 80640 q^{40} + 7052300 q^{43} + 42247296 q^{46} + 5404602 q^{49} + 12762080 q^{52} + 6249600 q^{55} + 20694240 q^{58} - 1507204 q^{61} + 62851072 q^{64} - 4537780 q^{67} + 17640000 q^{70} + 110691080 q^{73} + 9393248 q^{76} + 45961964 q^{79} - 168416640 q^{82} - 33626880 q^{85} + 2499840 q^{88} - 180110000 q^{91} - 183238272 q^{94} - 294542020 q^{97}+O(q^{100})$$ 4 * q + 496 * q^4 + 3500 * q^7 + 20160 * q^10 - 51460 * q^13 + 135040 * q^16 + 75752 * q^19 + 312480 * q^22 - 680450 * q^25 + 1736000 * q^28 + 702956 * q^31 + 3362688 * q^34 + 5340680 * q^37 - 80640 * q^40 + 7052300 * q^43 + 42247296 * q^46 + 5404602 * q^49 + 12762080 * q^52 + 6249600 * q^55 + 20694240 * q^58 - 1507204 * q^61 + 62851072 * q^64 - 4537780 * q^67 + 17640000 * q^70 + 110691080 * q^73 + 9393248 * q^76 + 45961964 * q^79 - 168416640 * q^82 - 33626880 * q^85 + 2499840 * q^88 - 180110000 * q^91 - 183238272 * q^94 - 294542020 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 14x^{2} + 196$$ :

 $$\beta_{1}$$ $$=$$ $$6\nu$$ 6*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 14$$ (v^2) / 14 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} ) / 7$$ (3*v^3) / 7
 $$\nu$$ $$=$$ $$( \beta_1 ) / 6$$ (b1) / 6 $$\nu^{2}$$ $$=$$ $$14\beta_{2}$$ 14*b2 $$\nu^{3}$$ $$=$$ $$( 7\beta_{3} ) / 3$$ (7*b3) / 3

## Character values

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

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

## 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
 −3.24037 − 1.87083i 3.24037 + 1.87083i −3.24037 + 1.87083i 3.24037 − 1.87083i
−19.4422 11.2250i 0 124.000 + 214.774i −194.422 + 112.250i 0 875.000 1515.54i 179.600i 0 5040.00
26.2 19.4422 + 11.2250i 0 124.000 + 214.774i 194.422 112.250i 0 875.000 1515.54i 179.600i 0 5040.00
53.1 −19.4422 + 11.2250i 0 124.000 214.774i −194.422 112.250i 0 875.000 + 1515.54i 179.600i 0 5040.00
53.2 19.4422 11.2250i 0 124.000 214.774i 194.422 + 112.250i 0 875.000 + 1515.54i 179.600i 0 5040.00
 $$n$$: 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.d 4
3.b odd 2 1 inner 81.9.d.d 4
9.c even 3 1 3.9.b.a 2
9.c even 3 1 inner 81.9.d.d 4
9.d odd 6 1 3.9.b.a 2
9.d odd 6 1 inner 81.9.d.d 4
36.f odd 6 1 48.9.e.b 2
36.h even 6 1 48.9.e.b 2
45.h odd 6 1 75.9.c.c 2
45.j even 6 1 75.9.c.c 2
45.k odd 12 2 75.9.d.b 4
45.l even 12 2 75.9.d.b 4
72.j odd 6 1 192.9.e.e 2
72.l even 6 1 192.9.e.f 2
72.n even 6 1 192.9.e.e 2
72.p odd 6 1 192.9.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 9.c even 3 1
3.9.b.a 2 9.d odd 6 1
48.9.e.b 2 36.f odd 6 1
48.9.e.b 2 36.h even 6 1
75.9.c.c 2 45.h odd 6 1
75.9.c.c 2 45.j even 6 1
75.9.d.b 4 45.k odd 12 2
75.9.d.b 4 45.l even 12 2
81.9.d.d 4 1.a even 1 1 trivial
81.9.d.d 4 3.b odd 2 1 inner
81.9.d.d 4 9.c even 3 1 inner
81.9.d.d 4 9.d odd 6 1 inner
192.9.e.e 2 72.j odd 6 1
192.9.e.e 2 72.n even 6 1
192.9.e.f 2 72.l even 6 1
192.9.e.f 2 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 504T_{2}^{2} + 254016$$ acting on $$S_{9}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 504 T^{2} + 254016$$
$3$ $$T^{4}$$
$5$ $$T^{4} + \cdots + 2540160000$$
$7$ $$(T^{2} - 1750 T + 3062500)^{2}$$
$11$ $$T^{4} + \cdots + 23\!\cdots\!00$$
$13$ $$(T^{2} + 25730 T + 662032900)^{2}$$
$17$ $$(T^{2} + 5608963584)^{2}$$
$19$ $$(T - 18938)^{4}$$
$23$ $$T^{4} + \cdots + 48\!\cdots\!36$$
$29$ $$T^{4} + \cdots + 45\!\cdots\!00$$
$31$ $$(T^{2} - 351478 T + 123536784484)^{2}$$
$37$ $$(T - 1335170)^{4}$$
$41$ $$T^{4} + \cdots + 12\!\cdots\!00$$
$43$ $$(T^{2} + \cdots + 12433733822500)^{2}$$
$47$ $$T^{4} + \cdots + 27\!\cdots\!76$$
$53$ $$(T^{2} + 43583305427424)^{2}$$
$59$ $$T^{4} + \cdots + 35\!\cdots\!00$$
$61$ $$(T^{2} + 753602 T + 567915974404)^{2}$$
$67$ $$(T^{2} + \cdots + 5147861832100)^{2}$$
$71$ $$(T^{2} + 289748374473600)^{2}$$
$73$ $$(T - 27672770)^{4}$$
$79$ $$(T^{2} + \cdots + 528125533684324)^{2}$$
$83$ $$T^{4} + \cdots + 46\!\cdots\!36$$
$89$ $$(T^{2} + 52\!\cdots\!00)^{2}$$
$97$ $$(T^{2} + \cdots + 21\!\cdots\!00)^{2}$$