# Properties

 Label 54.9.b.c Level $54$ Weight $9$ Character orbit 54.b Analytic conductor $21.998$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}\cdot 3^{6}$$ 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 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{2} - 128 q^{4} + (51 \beta_{2} - 11 \beta_1) q^{5} + ( - 7 \beta_{3} + 77) q^{7} + 256 \beta_{2} q^{8}+O(q^{10})$$ q - 2*b2 * q^2 - 128 * q^4 + (51*b2 - 11*b1) * q^5 + (-7*b3 + 77) * q^7 + 256*b2 * q^8 $$q - 2 \beta_{2} q^{2} - 128 q^{4} + (51 \beta_{2} - 11 \beta_1) q^{5} + ( - 7 \beta_{3} + 77) q^{7} + 256 \beta_{2} q^{8} + ( - 22 \beta_{3} + 3264) q^{10} + ( - 870 \beta_{2} + 409 \beta_1) q^{11} + (106 \beta_{3} - 11170) q^{13} + ( - 154 \beta_{2} + 448 \beta_1) q^{14} + 16384 q^{16} + ( - 324 \beta_{2} + 3098 \beta_1) q^{17} + (946 \beta_{3} - 14128) q^{19} + ( - 6528 \beta_{2} + 1408 \beta_1) q^{20} + (818 \beta_{3} - 55680) q^{22} + (27222 \beta_{2} + 1220 \beta_1) q^{23} + (1122 \beta_{3} + 219184) q^{25} + (22340 \beta_{2} - 6784 \beta_1) q^{26} + (896 \beta_{3} - 9856) q^{28} + ( - 85848 \beta_{2} - 2 \beta_1) q^{29} + (7305 \beta_{3} - 593857) q^{31} - 32768 \beta_{2} q^{32} + (6196 \beta_{3} - 20736) q^{34} + (60060 \beta_{2} - 12271 \beta_1) q^{35} + (14618 \beta_{3} + 1299332) q^{37} + (28256 \beta_{2} - 60544 \beta_1) q^{38} + (2816 \beta_{3} - 417792) q^{40} + (54870 \beta_{2} - 28052 \beta_1) q^{41} + (28038 \beta_{3} - 688774) q^{43} + (111360 \beta_{2} - 52352 \beta_1) q^{44} + (2440 \beta_{3} + 1742208) q^{46} + ( - 17268 \beta_{2} + 136770 \beta_1) q^{47} + ( - 1078 \beta_{3} - 4615800) q^{49} + ( - 438368 \beta_{2} - 71808 \beta_1) q^{50} + ( - 13568 \beta_{3} + 1429760) q^{52} + (344133 \beta_{2} + 379789 \beta_1) q^{53} + ( - 30429 \beta_{3} + 4699611) q^{55} + (19712 \beta_{2} - 57344 \beta_1) q^{56} + ( - 4 \beta_{3} - 5494272) q^{58} + ( - 1017804 \beta_{2} + 455762 \beta_1) q^{59} + ( - 95208 \beta_{3} + 9210596) q^{61} + (1187714 \beta_{2} - 467520 \beta_1) q^{62} - 2097152 q^{64} + ( - 1419684 \beta_{2} + 295862 \beta_1) q^{65} + ( - 61242 \beta_{3} - 13042594) q^{67} + (41472 \beta_{2} - 396544 \beta_1) q^{68} + ( - 24542 \beta_{3} + 3843840) q^{70} + (6071274 \beta_{2} + 38798 \beta_1) q^{71} + ( - 145926 \beta_{3} + 5610233) q^{73} + ( - 2598664 \beta_{2} - 935552 \beta_1) q^{74} + ( - 121088 \beta_{3} + 1808384) q^{76} + ( - 2154117 \beta_{2} + 226373 \beta_1) q^{77} + ( - 138764 \beta_{3} - 12786610) q^{79} + (835584 \beta_{2} - 180224 \beta_1) q^{80} + ( - 56104 \beta_{3} + 3511680) q^{82} + ( - 11181894 \beta_{2} + 462483 \beta_1) q^{83} + ( - 161562 \beta_{3} + 25371630) q^{85} + (1377548 \beta_{2} - 1794432 \beta_1) q^{86} + ( - 104704 \beta_{3} + 7127040) q^{88} + (17830278 \beta_{2} + 114002 \beta_1) q^{89} + (86352 \beta_{3} - 18169466) q^{91} + ( - 3484416 \beta_{2} - 156160 \beta_1) q^{92} + (273540 \beta_{3} - 1105152) q^{94} + ( - 8306502 \beta_{2} + 1699280 \beta_1) q^{95} + (487336 \beta_{3} - 67338805) q^{97} + (9231600 \beta_{2} + 68992 \beta_1) q^{98}+O(q^{100})$$ q - 2*b2 * q^2 - 128 * q^4 + (51*b2 - 11*b1) * q^5 + (-7*b3 + 77) * q^7 + 256*b2 * q^8 + (-22*b3 + 3264) * q^10 + (-870*b2 + 409*b1) * q^11 + (106*b3 - 11170) * q^13 + (-154*b2 + 448*b1) * q^14 + 16384 * q^16 + (-324*b2 + 3098*b1) * q^17 + (946*b3 - 14128) * q^19 + (-6528*b2 + 1408*b1) * q^20 + (818*b3 - 55680) * q^22 + (27222*b2 + 1220*b1) * q^23 + (1122*b3 + 219184) * q^25 + (22340*b2 - 6784*b1) * q^26 + (896*b3 - 9856) * q^28 + (-85848*b2 - 2*b1) * q^29 + (7305*b3 - 593857) * q^31 - 32768*b2 * q^32 + (6196*b3 - 20736) * q^34 + (60060*b2 - 12271*b1) * q^35 + (14618*b3 + 1299332) * q^37 + (28256*b2 - 60544*b1) * q^38 + (2816*b3 - 417792) * q^40 + (54870*b2 - 28052*b1) * q^41 + (28038*b3 - 688774) * q^43 + (111360*b2 - 52352*b1) * q^44 + (2440*b3 + 1742208) * q^46 + (-17268*b2 + 136770*b1) * q^47 + (-1078*b3 - 4615800) * q^49 + (-438368*b2 - 71808*b1) * q^50 + (-13568*b3 + 1429760) * q^52 + (344133*b2 + 379789*b1) * q^53 + (-30429*b3 + 4699611) * q^55 + (19712*b2 - 57344*b1) * q^56 + (-4*b3 - 5494272) * q^58 + (-1017804*b2 + 455762*b1) * q^59 + (-95208*b3 + 9210596) * q^61 + (1187714*b2 - 467520*b1) * q^62 - 2097152 * q^64 + (-1419684*b2 + 295862*b1) * q^65 + (-61242*b3 - 13042594) * q^67 + (41472*b2 - 396544*b1) * q^68 + (-24542*b3 + 3843840) * q^70 + (6071274*b2 + 38798*b1) * q^71 + (-145926*b3 + 5610233) * q^73 + (-2598664*b2 - 935552*b1) * q^74 + (-121088*b3 + 1808384) * q^76 + (-2154117*b2 + 226373*b1) * q^77 + (-138764*b3 - 12786610) * q^79 + (835584*b2 - 180224*b1) * q^80 + (-56104*b3 + 3511680) * q^82 + (-11181894*b2 + 462483*b1) * q^83 + (-161562*b3 + 25371630) * q^85 + (1377548*b2 - 1794432*b1) * q^86 + (-104704*b3 + 7127040) * q^88 + (17830278*b2 + 114002*b1) * q^89 + (86352*b3 - 18169466) * q^91 + (-3484416*b2 - 156160*b1) * q^92 + (273540*b3 - 1105152) * q^94 + (-8306502*b2 + 1699280*b1) * q^95 + (487336*b3 - 67338805) * q^97 + (9231600*b2 + 68992*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 512 q^{4} + 308 q^{7}+O(q^{10})$$ 4 * q - 512 * q^4 + 308 * q^7 $$4 q - 512 q^{4} + 308 q^{7} + 13056 q^{10} - 44680 q^{13} + 65536 q^{16} - 56512 q^{19} - 222720 q^{22} + 876736 q^{25} - 39424 q^{28} - 2375428 q^{31} - 82944 q^{34} + 5197328 q^{37} - 1671168 q^{40} - 2755096 q^{43} + 6968832 q^{46} - 18463200 q^{49} + 5719040 q^{52} + 18798444 q^{55} - 21977088 q^{58} + 36842384 q^{61} - 8388608 q^{64} - 52170376 q^{67} + 15375360 q^{70} + 22440932 q^{73} + 7233536 q^{76} - 51146440 q^{79} + 14046720 q^{82} + 101486520 q^{85} + 28508160 q^{88} - 72677864 q^{91} - 4420608 q^{94} - 269355220 q^{97}+O(q^{100})$$ 4 * q - 512 * q^4 + 308 * q^7 + 13056 * q^10 - 44680 * q^13 + 65536 * q^16 - 56512 * q^19 - 222720 * q^22 + 876736 * q^25 - 39424 * q^28 - 2375428 * q^31 - 82944 * q^34 + 5197328 * q^37 - 1671168 * q^40 - 2755096 * q^43 + 6968832 * q^46 - 18463200 * q^49 + 5719040 * q^52 + 18798444 * q^55 - 21977088 * q^58 + 36842384 * q^61 - 8388608 * q^64 - 52170376 * q^67 + 15375360 * q^70 + 22440932 * q^73 + 7233536 * q^76 - 51146440 * q^79 + 14046720 * q^82 + 101486520 * q^85 + 28508160 * q^88 - 72677864 * q^91 - 4420608 * q^94 - 269355220 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$27\zeta_{8}^{2}$$ 27*v^2 $$\beta_{2}$$ $$=$$ $$4\zeta_{8}^{3} + 4\zeta_{8}$$ 4*v^3 + 4*v $$\beta_{3}$$ $$=$$ $$-108\zeta_{8}^{3} + 108\zeta_{8}$$ -108*v^3 + 108*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + 27\beta_{2} ) / 216$$ (b3 + 27*b2) / 216 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 27$$ (b1) / 27 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + 27\beta_{2} ) / 216$$ (-b3 + 27*b2) / 216

## 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
 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i
11.3137i 0 −128.000 8.50043i 0 −992.145 1448.15i 0 −96.1714
53.2 11.3137i 0 −128.000 585.500i 0 1146.15 1448.15i 0 6624.17
53.3 11.3137i 0 −128.000 585.500i 0 1146.15 1448.15i 0 6624.17
53.4 11.3137i 0 −128.000 8.50043i 0 −992.145 1448.15i 0 −96.1714
 $$n$$: 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.c 4
3.b odd 2 1 inner 54.9.b.c 4
4.b odd 2 1 432.9.e.h 4
9.c even 3 2 162.9.d.e 8
9.d odd 6 2 162.9.d.e 8
12.b even 2 1 432.9.e.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.c 4 1.a even 1 1 trivial
54.9.b.c 4 3.b odd 2 1 inner
162.9.d.e 8 9.c even 3 2
162.9.d.e 8 9.d odd 6 2
432.9.e.h 4 4.b odd 2 1
432.9.e.h 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 342882T_{5}^{2} + 24770529$$ acting on $$S_{9}^{\mathrm{new}}(54, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 128)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 342882 T^{2} + 24770529$$
$7$ $$(T^{2} - 154 T - 1137143)^{2}$$
$11$ $$T^{4} + \cdots + 95\!\cdots\!01$$
$13$ $$(T^{2} + 22340 T - 137344508)^{2}$$
$17$ $$T^{4} + \cdots + 48\!\cdots\!56$$
$19$ $$(T^{2} + 28256 T - 20677000064)^{2}$$
$23$ $$T^{4} + \cdots + 51\!\cdots\!44$$
$29$ $$T^{4} + \cdots + 55\!\cdots\!44$$
$31$ $$(T^{2} + 1187714 T - 892186510751)^{2}$$
$37$ $$(T^{2} + \cdots - 3296601588848)^{2}$$
$41$ $$T^{4} + \cdots + 22\!\cdots\!56$$
$43$ $$(T^{2} + \cdots - 17864418046556)^{2}$$
$47$ $$T^{4} + \cdots + 18\!\cdots\!24$$
$53$ $$T^{4} + \cdots + 10\!\cdots\!21$$
$59$ $$T^{4} + \cdots + 13\!\cdots\!96$$
$61$ $$(T^{2} + \cdots - 126623053147376)^{2}$$
$67$ $$(T^{2} + \cdots + 82615668195844)^{2}$$
$71$ $$T^{4} + \cdots + 13\!\cdots\!56$$
$73$ $$(T^{2} + \cdots - 465280990005839)^{2}$$
$79$ $$(T^{2} + \cdots - 285693688560188)^{2}$$
$83$ $$T^{4} + \cdots + 14\!\cdots\!41$$
$89$ $$T^{4} + \cdots + 10\!\cdots\!84$$
$97$ $$(T^{2} + \cdots - 10\!\cdots\!63)^{2}$$