# Properties

 Label 54.13.b.b Level $54$ Weight $13$ Character orbit 54.b Analytic conductor $49.356$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,13,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, 13, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("54.53");

S:= CuspForms(chi, 13);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$13$$ 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: $$49.3556661329$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 9235x^{2} + 9236x + 21362886$$ x^4 - 2*x^3 - 9235*x^2 + 9236*x + 21362886 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}\cdot 3^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 2048 q^{4} + (\beta_{3} + 139 \beta_1) q^{5} + ( - \beta_{2} + 67871) q^{7} + 2048 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - 2048 * q^4 + (b3 + 139*b1) * q^5 + (-b2 + 67871) * q^7 + 2048*b1 * q^8 $$q - \beta_1 q^{2} - 2048 q^{4} + (\beta_{3} + 139 \beta_1) q^{5} + ( - \beta_{2} + 67871) q^{7} + 2048 \beta_1 q^{8} + (32 \beta_{2} + 284928) q^{10} + ( - 59 \beta_{3} + 3727 \beta_1) q^{11} + (211 \beta_{2} + 174071) q^{13} + (64 \beta_{3} - 67879 \beta_1) q^{14} + 4194304 q^{16} + (653 \beta_{3} - 355945 \beta_1) q^{17} + (802 \beta_{2} - 21969241) q^{19} + ( - 2048 \beta_{3} - 284672 \beta_1) q^{20} + ( - 1888 \beta_{2} + 7617792) q^{22} + (5411 \beta_{3} + 321665 \beta_1) q^{23} + ( - 8904 \beta_{2} - 226624751) q^{25} + ( - 13504 \beta_{3} - 172383 \beta_1) q^{26} + (2048 \beta_{2} - 138999808) q^{28} + (9106 \beta_{3} + 14836630 \beta_1) q^{29} + ( - 4296 \beta_{2} - 595633534) q^{31} - 4194304 \beta_1 q^{32} + (20896 \beta_{2} - 728808192) q^{34} + (58967 \beta_{3} - 4037467 \beta_1) q^{35} + (144785 \beta_{2} - 143321785) q^{37} + ( - 51328 \beta_{3} + 21975657 \beta_1) q^{38} + ( - 65536 \beta_{2} - 583532544) q^{40} + ( - 177818 \beta_{3} + 44146834 \beta_1) q^{41} + ( - 275352 \beta_{2} - 3779183086) q^{43} + (120832 \beta_{3} - 7632896 \beta_1) q^{44} + (173152 \beta_{2} + 660155136) q^{46} + (120597 \beta_{3} + 140053719 \beta_1) q^{47} + ( - 135742 \beta_{2} - 8372565024) q^{49} + (569856 \beta_{3} + 226553519 \beta_1) q^{50} + ( - 432128 \beta_{2} - 356497408) q^{52} + ( - 967730 \beta_{3} + 114882106 \beta_1) q^{53} + (143640 \beta_{2} + 24376536000) q^{55} + ( - 131072 \beta_{3} + 139016192 \beta_1) q^{56} + (291392 \beta_{2} + 30387749376) q^{58} + ( - 71857 \beta_{3} + 1396558109 \beta_1) q^{59} + (1595109 \beta_{2} + 14590716599) q^{61} + (274944 \beta_{3} + 595599166 \beta_1) q^{62} - 8589934592 q^{64} + (2052815 \beta_{3} + 2866689965 \beta_1) q^{65} + (652332 \beta_{2} + 77243788775) q^{67} + ( - 1337344 \beta_{3} + 728975360 \beta_1) q^{68} + (1886944 \beta_{2} - 8253636864) q^{70} + (3676340 \beta_{3} + 2566182476 \beta_1) q^{71} + ( - 7089354 \beta_{2} - 44717675857) q^{73} + ( - 9266240 \beta_{3} + 144480065 \beta_1) q^{74} + ( - 1642496 \beta_{2} + 44993005568) q^{76} + ( - 4242445 \beta_{3} + 1047871265 \beta_1) q^{77} + ( - 1331243 \beta_{2} - 226274792209) q^{79} + (4194304 \beta_{3} + 583008256 \beta_1) q^{80} + ( - 5690176 \beta_{2} + 90367194624) q^{82} + (15823242 \beta_{3} + 5549991342 \beta_1) q^{83} + (8480472 \beta_{2} - 180129033792) q^{85} + (17622528 \beta_{3} + 3776980270 \beta_1) q^{86} + (3866624 \beta_{2} - 15601238016) q^{88} + ( - 28598335 \beta_{3} + 7210389107 \beta_1) q^{89} + (14146710 \beta_{2} - 170120279255) q^{91} + ( - 11081728 \beta_{3} - 658769920 \beta_1) q^{92} + (3859104 \beta_{2} + 286860889344) q^{94} + ( - 14828233 \beta_{3} + 7750447373 \beta_1) q^{95} + (547846 \beta_{2} - 1417820309089) q^{97} + (8687488 \beta_{3} + 8371479088 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - 2048 * q^4 + (b3 + 139*b1) * q^5 + (-b2 + 67871) * q^7 + 2048*b1 * q^8 + (32*b2 + 284928) * q^10 + (-59*b3 + 3727*b1) * q^11 + (211*b2 + 174071) * q^13 + (64*b3 - 67879*b1) * q^14 + 4194304 * q^16 + (653*b3 - 355945*b1) * q^17 + (802*b2 - 21969241) * q^19 + (-2048*b3 - 284672*b1) * q^20 + (-1888*b2 + 7617792) * q^22 + (5411*b3 + 321665*b1) * q^23 + (-8904*b2 - 226624751) * q^25 + (-13504*b3 - 172383*b1) * q^26 + (2048*b2 - 138999808) * q^28 + (9106*b3 + 14836630*b1) * q^29 + (-4296*b2 - 595633534) * q^31 - 4194304*b1 * q^32 + (20896*b2 - 728808192) * q^34 + (58967*b3 - 4037467*b1) * q^35 + (144785*b2 - 143321785) * q^37 + (-51328*b3 + 21975657*b1) * q^38 + (-65536*b2 - 583532544) * q^40 + (-177818*b3 + 44146834*b1) * q^41 + (-275352*b2 - 3779183086) * q^43 + (120832*b3 - 7632896*b1) * q^44 + (173152*b2 + 660155136) * q^46 + (120597*b3 + 140053719*b1) * q^47 + (-135742*b2 - 8372565024) * q^49 + (569856*b3 + 226553519*b1) * q^50 + (-432128*b2 - 356497408) * q^52 + (-967730*b3 + 114882106*b1) * q^53 + (143640*b2 + 24376536000) * q^55 + (-131072*b3 + 139016192*b1) * q^56 + (291392*b2 + 30387749376) * q^58 + (-71857*b3 + 1396558109*b1) * q^59 + (1595109*b2 + 14590716599) * q^61 + (274944*b3 + 595599166*b1) * q^62 - 8589934592 * q^64 + (2052815*b3 + 2866689965*b1) * q^65 + (652332*b2 + 77243788775) * q^67 + (-1337344*b3 + 728975360*b1) * q^68 + (1886944*b2 - 8253636864) * q^70 + (3676340*b3 + 2566182476*b1) * q^71 + (-7089354*b2 - 44717675857) * q^73 + (-9266240*b3 + 144480065*b1) * q^74 + (-1642496*b2 + 44993005568) * q^76 + (-4242445*b3 + 1047871265*b1) * q^77 + (-1331243*b2 - 226274792209) * q^79 + (4194304*b3 + 583008256*b1) * q^80 + (-5690176*b2 + 90367194624) * q^82 + (15823242*b3 + 5549991342*b1) * q^83 + (8480472*b2 - 180129033792) * q^85 + (17622528*b3 + 3776980270*b1) * q^86 + (3866624*b2 - 15601238016) * q^88 + (-28598335*b3 + 7210389107*b1) * q^89 + (14146710*b2 - 170120279255) * q^91 + (-11081728*b3 - 658769920*b1) * q^92 + (3859104*b2 + 286860889344) * q^94 + (-14828233*b3 + 7750447373*b1) * q^95 + (547846*b2 - 1417820309089) * q^97 + (8687488*b3 + 8371479088*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8192 q^{4} + 271484 q^{7}+O(q^{10})$$ 4 * q - 8192 * q^4 + 271484 * q^7 $$4 q - 8192 q^{4} + 271484 q^{7} + 1139712 q^{10} + 696284 q^{13} + 16777216 q^{16} - 87876964 q^{19} + 30471168 q^{22} - 906499004 q^{25} - 555999232 q^{28} - 2382534136 q^{31} - 2915232768 q^{34} - 573287140 q^{37} - 2334130176 q^{40} - 15116732344 q^{43} + 2640620544 q^{46} - 33490260096 q^{49} - 1425989632 q^{52} + 97506144000 q^{55} + 121550997504 q^{58} + 58362866396 q^{61} - 34359738368 q^{64} + 308975155100 q^{67} - 33014547456 q^{70} - 178870703428 q^{73} + 179972022272 q^{76} - 905099168836 q^{79} + 361468778496 q^{82} - 720516135168 q^{85} - 62404952064 q^{88} - 680481117020 q^{91} + 1147443557376 q^{94} - 5671281236356 q^{97}+O(q^{100})$$ 4 * q - 8192 * q^4 + 271484 * q^7 + 1139712 * q^10 + 696284 * q^13 + 16777216 * q^16 - 87876964 * q^19 + 30471168 * q^22 - 906499004 * q^25 - 555999232 * q^28 - 2382534136 * q^31 - 2915232768 * q^34 - 573287140 * q^37 - 2334130176 * q^40 - 15116732344 * q^43 + 2640620544 * q^46 - 33490260096 * q^49 - 1425989632 * q^52 + 97506144000 * q^55 + 121550997504 * q^58 + 58362866396 * q^61 - 34359738368 * q^64 + 308975155100 * q^67 - 33014547456 * q^70 - 178870703428 * q^73 + 179972022272 * q^76 - 905099168836 * q^79 + 361468778496 * q^82 - 720516135168 * q^85 - 62404952064 * q^88 - 680481117020 * q^91 + 1147443557376 * q^94 - 5671281236356 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 9235x^{2} + 9236x + 21362886$$ :

 $$\beta_{1}$$ $$=$$ $$( 64\nu^{3} - 96\nu^{2} - 295264\nu + 147648 ) / 18489$$ (64*v^3 - 96*v^2 - 295264*v + 147648) / 18489 $$\beta_{2}$$ $$=$$ $$( -288\nu^{3} + 432\nu^{2} + 3991104\nu - 1995624 ) / 6163$$ (-288*v^3 + 432*v^2 + 3991104*v - 1995624) / 6163 $$\beta_{3}$$ $$=$$ $$( 8\nu^{3} + 1996800\nu^{2} - 2033720\nu - 9221259360 ) / 18489$$ (8*v^3 + 1996800*v^2 - 2033720*v - 9221259360) / 18489
 $$\nu$$ $$=$$ $$( 2\beta_{2} + 27\beta _1 + 432 ) / 864$$ (2*b2 + 27*b1 + 432) / 864 $$\nu^{2}$$ $$=$$ $$( 4\beta_{3} + \beta_{2} + 13\beta _1 + 1995192 ) / 432$$ (4*b3 + b2 + 13*b1 + 1995192) / 432 $$\nu^{3}$$ $$=$$ $$( 12\beta_{3} + 9230\beta_{2} + 374205\beta _1 + 5985360 ) / 864$$ (12*b3 + 9230*b2 + 374205*b1 + 5985360) / 864

## 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
 −67.4724 + 1.41421i 68.4724 + 1.41421i 68.4724 − 1.41421i −67.4724 − 1.41421i
45.2548i 0 −2048.00 14467.5i 0 97235.1 92681.9i 0 −654723.
53.2 45.2548i 0 −2048.00 27059.6i 0 38506.9 92681.9i 0 1.22458e6
53.3 45.2548i 0 −2048.00 27059.6i 0 38506.9 92681.9i 0 1.22458e6
53.4 45.2548i 0 −2048.00 14467.5i 0 97235.1 92681.9i 0 −654723.
 $$n$$: 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 54.13.b.b 4
3.b odd 2 1 inner 54.13.b.b 4
9.c even 3 2 162.13.d.c 8
9.d odd 6 2 162.13.d.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.13.b.b 4 1.a even 1 1 trivial
54.13.b.b 4 3.b odd 2 1 inner
162.13.d.c 8 9.c even 3 2
162.13.d.c 8 9.d odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 941530752T_{5}^{2} + 153259847530905600$$ acting on $$S_{13}^{\mathrm{new}}(54, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2048)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 941530752 T^{2} + \cdots + 15\!\cdots\!00$$
$7$ $$(T^{2} - 135742 T + 3744223105)^{2}$$
$11$ $$T^{4} + 3058161293952 T^{2} + \cdots + 21\!\cdots\!00$$
$13$ $$(T^{2} - 348142 T - 38357910879215)^{2}$$
$17$ $$T^{4} + 886383248261760 T^{2} + \cdots + 57\!\cdots\!36$$
$19$ $$(T^{2} + 43938482 T - 71954800437263)^{2}$$
$23$ $$T^{4} + \cdots + 15\!\cdots\!76$$
$29$ $$T^{4} + \cdots + 17\!\cdots\!96$$
$31$ $$(T^{2} + 1191267068 T + 33\!\cdots\!80)^{2}$$
$37$ $$(T^{2} + 286643570 T - 18\!\cdots\!75)^{2}$$
$41$ $$T^{4} + \cdots + 93\!\cdots\!00$$
$43$ $$(T^{2} + 7558366172 T - 51\!\cdots\!48)^{2}$$
$47$ $$T^{4} + \cdots + 11\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 14\!\cdots\!24$$
$59$ $$T^{4} + \cdots + 15\!\cdots\!24$$
$61$ $$(T^{2} - 29181433198 T - 19\!\cdots\!15)^{2}$$
$67$ $$(T^{2} - 154487577550 T + 55\!\cdots\!61)^{2}$$
$71$ $$T^{4} + \cdots + 58\!\cdots\!84$$
$73$ $$(T^{2} + 89435351714 T - 41\!\cdots\!27)^{2}$$
$79$ $$(T^{2} + 452549584418 T + 49\!\cdots\!17)^{2}$$
$83$ $$T^{4} + \cdots + 20\!\cdots\!96$$
$89$ $$T^{4} + \cdots + 60\!\cdots\!44$$
$97$ $$(T^{2} + 2835640618178 T + 20\!\cdots\!45)^{2}$$