# Properties

 Label 27.9.b.c Level $27$ Weight $9$ Character orbit 27.b Analytic conductor $10.999$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,9,Mod(26,27)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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("27.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$27 = 3^{3}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 27.b (of order $$2$$, degree $$1$$, 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: $$10.9992224717$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-30})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 30$$ x^2 + 30 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 3\sqrt{-30}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 14 q^{4} + 26 \beta q^{5} - 679 q^{7} + 242 \beta q^{8}+O(q^{10})$$ q + b * q^2 - 14 * q^4 + 26*b * q^5 - 679 * q^7 + 242*b * q^8 $$q + \beta q^{2} - 14 q^{4} + 26 \beta q^{5} - 679 q^{7} + 242 \beta q^{8} - 7020 q^{10} + 814 \beta q^{11} - 30817 q^{13} - 679 \beta q^{14} - 68924 q^{16} - 7806 \beta q^{17} - 138391 q^{19} - 364 \beta q^{20} - 219780 q^{22} + 18482 \beta q^{23} + 208105 q^{25} - 30817 \beta q^{26} + 9506 q^{28} + 80740 \beta q^{29} + 352214 q^{31} - 6972 \beta q^{32} + 2107620 q^{34} - 17654 \beta q^{35} + 1189991 q^{37} - 138391 \beta q^{38} - 1698840 q^{40} - 66580 \beta q^{41} + 6246086 q^{43} - 11396 \beta q^{44} - 4990140 q^{46} - 146 \beta q^{47} - 5303760 q^{49} + 208105 \beta q^{50} + 431438 q^{52} + 765468 \beta q^{53} - 5714280 q^{55} - 164318 \beta q^{56} - 21799800 q^{58} - 641206 \beta q^{59} + 16580399 q^{61} + 352214 \beta q^{62} - 15762104 q^{64} - 801242 \beta q^{65} + 7667153 q^{67} + 109284 \beta q^{68} + 4766580 q^{70} - 1413192 \beta q^{71} + 24949631 q^{73} + 1189991 \beta q^{74} + 1937474 q^{76} - 552706 \beta q^{77} + 41685089 q^{79} - 1792024 \beta q^{80} + 17976600 q^{82} + 2722060 \beta q^{83} + 54798120 q^{85} + 6246086 \beta q^{86} - 53186760 q^{88} - 45078 \beta q^{89} + 20924743 q^{91} - 258748 \beta q^{92} + 39420 q^{94} - 3598166 \beta q^{95} - 105926089 q^{97} - 5303760 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 14 * q^4 + 26*b * q^5 - 679 * q^7 + 242*b * q^8 - 7020 * q^10 + 814*b * q^11 - 30817 * q^13 - 679*b * q^14 - 68924 * q^16 - 7806*b * q^17 - 138391 * q^19 - 364*b * q^20 - 219780 * q^22 + 18482*b * q^23 + 208105 * q^25 - 30817*b * q^26 + 9506 * q^28 + 80740*b * q^29 + 352214 * q^31 - 6972*b * q^32 + 2107620 * q^34 - 17654*b * q^35 + 1189991 * q^37 - 138391*b * q^38 - 1698840 * q^40 - 66580*b * q^41 + 6246086 * q^43 - 11396*b * q^44 - 4990140 * q^46 - 146*b * q^47 - 5303760 * q^49 + 208105*b * q^50 + 431438 * q^52 + 765468*b * q^53 - 5714280 * q^55 - 164318*b * q^56 - 21799800 * q^58 - 641206*b * q^59 + 16580399 * q^61 + 352214*b * q^62 - 15762104 * q^64 - 801242*b * q^65 + 7667153 * q^67 + 109284*b * q^68 + 4766580 * q^70 - 1413192*b * q^71 + 24949631 * q^73 + 1189991*b * q^74 + 1937474 * q^76 - 552706*b * q^77 + 41685089 * q^79 - 1792024*b * q^80 + 17976600 * q^82 + 2722060*b * q^83 + 54798120 * q^85 + 6246086*b * q^86 - 53186760 * q^88 - 45078*b * q^89 + 20924743 * q^91 - 258748*b * q^92 + 39420 * q^94 - 3598166*b * q^95 - 105926089 * q^97 - 5303760*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 28 q^{4} - 1358 q^{7}+O(q^{10})$$ 2 * q - 28 * q^4 - 1358 * q^7 $$2 q - 28 q^{4} - 1358 q^{7} - 14040 q^{10} - 61634 q^{13} - 137848 q^{16} - 276782 q^{19} - 439560 q^{22} + 416210 q^{25} + 19012 q^{28} + 704428 q^{31} + 4215240 q^{34} + 2379982 q^{37} - 3397680 q^{40} + 12492172 q^{43} - 9980280 q^{46} - 10607520 q^{49} + 862876 q^{52} - 11428560 q^{55} - 43599600 q^{58} + 33160798 q^{61} - 31524208 q^{64} + 15334306 q^{67} + 9533160 q^{70} + 49899262 q^{73} + 3874948 q^{76} + 83370178 q^{79} + 35953200 q^{82} + 109596240 q^{85} - 106373520 q^{88} + 41849486 q^{91} + 78840 q^{94} - 211852178 q^{97}+O(q^{100})$$ 2 * q - 28 * q^4 - 1358 * q^7 - 14040 * q^10 - 61634 * q^13 - 137848 * q^16 - 276782 * q^19 - 439560 * q^22 + 416210 * q^25 + 19012 * q^28 + 704428 * q^31 + 4215240 * q^34 + 2379982 * q^37 - 3397680 * q^40 + 12492172 * q^43 - 9980280 * q^46 - 10607520 * q^49 + 862876 * q^52 - 11428560 * q^55 - 43599600 * q^58 + 33160798 * q^61 - 31524208 * q^64 + 15334306 * q^67 + 9533160 * q^70 + 49899262 * q^73 + 3874948 * q^76 + 83370178 * q^79 + 35953200 * q^82 + 109596240 * q^85 - 106373520 * q^88 + 41849486 * q^91 + 78840 * q^94 - 211852178 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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
 − 5.47723i 5.47723i
16.4317i 0 −14.0000 427.224i 0 −679.000 3976.47i 0 −7020.00
26.2 16.4317i 0 −14.0000 427.224i 0 −679.000 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

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.c 2 1.a even 1 1 trivial
27.9.b.c 2 3.b odd 2 1 inner
81.9.d.c 4 9.c even 3 2
81.9.d.c 4 9.d odd 6 2
432.9.e.f 2 4.b odd 2 1
432.9.e.f 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 270$$ acting on $$S_{9}^{\mathrm{new}}(27, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 270$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 182520$$
$7$ $$(T + 679)^{2}$$
$11$ $$T^{2} + 178900920$$
$13$ $$(T + 30817)^{2}$$
$17$ $$T^{2} + 16452081720$$
$19$ $$(T + 138391)^{2}$$
$23$ $$T^{2} + 92227767480$$
$29$ $$T^{2} + 1760115852000$$
$31$ $$(T - 352214)^{2}$$
$37$ $$(T - 1189991)^{2}$$
$41$ $$T^{2} + 1196882028000$$
$43$ $$(T - 6246086)^{2}$$
$47$ $$T^{2} + 5755320$$
$53$ $$T^{2} + 158204139936480$$
$59$ $$T^{2} + 111009186297720$$
$61$ $$(T - 16580399)^{2}$$
$67$ $$(T - 7667153)^{2}$$
$71$ $$T^{2} + 539220139793280$$
$73$ $$(T - 24949631)^{2}$$
$79$ $$(T - 41685089)^{2}$$
$83$ $$T^{2} + 20\!\cdots\!00$$
$89$ $$T^{2} + 548647042680$$
$97$ $$(T + 105926089)^{2}$$