# Properties

 Label 54.9.b.a Level $54$ Weight $9$ Character orbit 54.b Analytic conductor $21.998$ 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] = [54,9,Mod(53,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, 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("54.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 54.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: $$21.9984449433$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ 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 $$\beta = 8\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 128 q^{4} + 60 \beta q^{5} - 2065 q^{7} - 128 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 128 * q^4 + 60*b * q^5 - 2065 * q^7 - 128*b * q^8 $$q + \beta q^{2} - 128 q^{4} + 60 \beta q^{5} - 2065 q^{7} - 128 \beta q^{8} - 7680 q^{10} + 588 \beta q^{11} + 8063 q^{13} - 2065 \beta q^{14} + 16384 q^{16} - 1908 \beta q^{17} - 226609 q^{19} - 7680 \beta q^{20} - 75264 q^{22} - 32556 \beta q^{23} - 70175 q^{25} + 8063 \beta q^{26} + 264320 q^{28} - 82824 \beta q^{29} + 826370 q^{31} + 16384 \beta q^{32} + 244224 q^{34} - 123900 \beta q^{35} + 1344575 q^{37} - 226609 \beta q^{38} + 983040 q^{40} - 458904 \beta q^{41} - 6147742 q^{43} - 75264 \beta q^{44} + 4167168 q^{46} + 522444 \beta q^{47} - 1500576 q^{49} - 70175 \beta q^{50} - 1032064 q^{52} - 67896 \beta q^{53} - 4515840 q^{55} + 264320 \beta q^{56} + 10601472 q^{58} + 41892 \beta q^{59} - 14985697 q^{61} + 826370 \beta q^{62} - 2097152 q^{64} + 483780 \beta q^{65} - 10023697 q^{67} + 244224 \beta q^{68} + 15859200 q^{70} + 4020336 \beta q^{71} - 23261569 q^{73} + 1344575 \beta q^{74} + 29005952 q^{76} - 1214220 \beta q^{77} + 14267183 q^{79} + 983040 \beta q^{80} + 58739712 q^{82} + 3198936 \beta q^{83} + 14653440 q^{85} - 6147742 \beta q^{86} + 9633792 q^{88} + 10172412 \beta q^{89} - 16650095 q^{91} + 4167168 \beta q^{92} - 66872832 q^{94} - 13596540 \beta q^{95} - 40571617 q^{97} - 1500576 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 128 * q^4 + 60*b * q^5 - 2065 * q^7 - 128*b * q^8 - 7680 * q^10 + 588*b * q^11 + 8063 * q^13 - 2065*b * q^14 + 16384 * q^16 - 1908*b * q^17 - 226609 * q^19 - 7680*b * q^20 - 75264 * q^22 - 32556*b * q^23 - 70175 * q^25 + 8063*b * q^26 + 264320 * q^28 - 82824*b * q^29 + 826370 * q^31 + 16384*b * q^32 + 244224 * q^34 - 123900*b * q^35 + 1344575 * q^37 - 226609*b * q^38 + 983040 * q^40 - 458904*b * q^41 - 6147742 * q^43 - 75264*b * q^44 + 4167168 * q^46 + 522444*b * q^47 - 1500576 * q^49 - 70175*b * q^50 - 1032064 * q^52 - 67896*b * q^53 - 4515840 * q^55 + 264320*b * q^56 + 10601472 * q^58 + 41892*b * q^59 - 14985697 * q^61 + 826370*b * q^62 - 2097152 * q^64 + 483780*b * q^65 - 10023697 * q^67 + 244224*b * q^68 + 15859200 * q^70 + 4020336*b * q^71 - 23261569 * q^73 + 1344575*b * q^74 + 29005952 * q^76 - 1214220*b * q^77 + 14267183 * q^79 + 983040*b * q^80 + 58739712 * q^82 + 3198936*b * q^83 + 14653440 * q^85 - 6147742*b * q^86 + 9633792 * q^88 + 10172412*b * q^89 - 16650095 * q^91 + 4167168*b * q^92 - 66872832 * q^94 - 13596540*b * q^95 - 40571617 * q^97 - 1500576*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 256 q^{4} - 4130 q^{7}+O(q^{10})$$ 2 * q - 256 * q^4 - 4130 * q^7 $$2 q - 256 q^{4} - 4130 q^{7} - 15360 q^{10} + 16126 q^{13} + 32768 q^{16} - 453218 q^{19} - 150528 q^{22} - 140350 q^{25} + 528640 q^{28} + 1652740 q^{31} + 488448 q^{34} + 2689150 q^{37} + 1966080 q^{40} - 12295484 q^{43} + 8334336 q^{46} - 3001152 q^{49} - 2064128 q^{52} - 9031680 q^{55} + 21202944 q^{58} - 29971394 q^{61} - 4194304 q^{64} - 20047394 q^{67} + 31718400 q^{70} - 46523138 q^{73} + 58011904 q^{76} + 28534366 q^{79} + 117479424 q^{82} + 29306880 q^{85} + 19267584 q^{88} - 33300190 q^{91} - 133745664 q^{94} - 81143234 q^{97}+O(q^{100})$$ 2 * q - 256 * q^4 - 4130 * q^7 - 15360 * q^10 + 16126 * q^13 + 32768 * q^16 - 453218 * q^19 - 150528 * q^22 - 140350 * q^25 + 528640 * q^28 + 1652740 * q^31 + 488448 * q^34 + 2689150 * q^37 + 1966080 * q^40 - 12295484 * q^43 + 8334336 * q^46 - 3001152 * q^49 - 2064128 * q^52 - 9031680 * q^55 + 21202944 * q^58 - 29971394 * q^61 - 4194304 * q^64 - 20047394 * q^67 + 31718400 * q^70 - 46523138 * q^73 + 58011904 * q^76 + 28534366 * q^79 + 117479424 * q^82 + 29306880 * q^85 + 19267584 * q^88 - 33300190 * q^91 - 133745664 * q^94 - 81143234 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 − 1.41421i 1.41421i
11.3137i 0 −128.000 678.823i 0 −2065.00 1448.15i 0 −7680.00
53.2 11.3137i 0 −128.000 678.823i 0 −2065.00 1448.15i 0 −7680.00
 $$n$$: 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 54.9.b.a 2
3.b odd 2 1 inner 54.9.b.a 2
4.b odd 2 1 432.9.e.g 2
9.c even 3 2 162.9.d.c 4
9.d odd 6 2 162.9.d.c 4
12.b even 2 1 432.9.e.g 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 460800$$ acting on $$S_{9}^{\mathrm{new}}(54, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 128$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 460800$$
$7$ $$(T + 2065)^{2}$$
$11$ $$T^{2} + 44255232$$
$13$ $$(T - 8063)^{2}$$
$17$ $$T^{2} + 465979392$$
$19$ $$(T + 226609)^{2}$$
$23$ $$T^{2} + 135666321408$$
$29$ $$T^{2} + 878056316928$$
$31$ $$(T - 826370)^{2}$$
$37$ $$(T - 1344575)^{2}$$
$41$ $$T^{2} + 26955888795648$$
$43$ $$(T + 6147742)^{2}$$
$47$ $$T^{2} + 34937309841408$$
$53$ $$T^{2} + 590062952448$$
$59$ $$T^{2} + 224632276992$$
$61$ $$(T + 14985697)^{2}$$
$67$ $$(T + 10023697)^{2}$$
$71$ $$T^{2} + 20\!\cdots\!88$$
$73$ $$(T + 23261569)^{2}$$
$79$ $$(T - 14267183)^{2}$$
$83$ $$T^{2} + 13\!\cdots\!88$$
$89$ $$T^{2} + 13\!\cdots\!32$$
$97$ $$(T + 40571617)^{2}$$