Properties

 Label 27.4.c.a Level $27$ Weight $4$ Character orbit 27.c Analytic conductor $1.593$ 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] = [27,4,Mod(10,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("27.10");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$27 = 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 27.c (of order $$3$$, 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: $$1.59305157015$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 9) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_1) q^{2} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + (\beta_{3} - \beta_{2} - 8 \beta_1 + 7) q^{5} + ( - 3 \beta_{3} - 2 \beta_1) q^{7} + (\beta_{2} - 16) q^{8}+O(q^{10})$$ q + (b3 + b1) * q^2 + (3*b3 - 3*b2 + b1 - 4) * q^4 + (b3 - b2 - 8*b1 + 7) * q^5 + (-3*b3 - 2*b1) * q^7 + (b2 - 16) * q^8 $$q + (\beta_{3} + \beta_1) q^{2} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + (\beta_{3} - \beta_{2} - 8 \beta_1 + 7) q^{5} + ( - 3 \beta_{3} - 2 \beta_1) q^{7} + (\beta_{2} - 16) q^{8} + (6 \beta_{2} + 6) q^{10} + ( - 8 \beta_{3} + 37 \beta_1) q^{11} + ( - 15 \beta_{3} + 15 \beta_{2} + 2 \beta_1 + 13) q^{13} + ( - 8 \beta_{3} + 8 \beta_{2} - 26 \beta_1 + 34) q^{14} + (9 \beta_{3} - \beta_1) q^{16} + ( - 9 \beta_{2} - 54) q^{17} + ( - 27 \beta_{2} - 52) q^{19} + (20 \beta_{3} - 16 \beta_1) q^{20} + (21 \beta_{3} - 21 \beta_{2} - 27 \beta_1 + 6) q^{22} + (19 \beta_{3} - 19 \beta_{2} - 26 \beta_1 + 7) q^{23} + (15 \beta_{3} + 53 \beta_1) q^{25} + (28 \beta_{2} + 146) q^{26} + (18 \beta_{2} + 92) q^{28} + (\beta_{3} - 26 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_{2} + 20 \beta_1 - 23) q^{31} + (9 \beta_{3} - 9 \beta_{2} + 207 \beta_1 - 216) q^{32} + ( - 63 \beta_{3} - 117 \beta_1) q^{34} + ( - 19 \beta_{2} - 11) q^{35} + (54 \beta_{2} + 2) q^{37} + ( - 79 \beta_{3} - 241 \beta_1) q^{38} + ( - 24 \beta_{3} + 24 \beta_{2} + 144 \beta_1 - 120) q^{40} + ( - 98 \beta_{3} + 98 \beta_{2} - 17 \beta_1 + 115) q^{41} + (6 \beta_{3} - 47 \beta_1) q^{43} + ( - 79 \beta_{2} + 76) q^{44} + ( - 12 \beta_{2} - 138) q^{46} + (91 \beta_{3} + 154 \beta_1) q^{47} + (21 \beta_{3} - 21 \beta_{2} - 267 \beta_1 + 246) q^{49} + (83 \beta_{3} - 83 \beta_{2} + 173 \beta_1 - 256) q^{50} + (54 \beta_{3} + 358 \beta_1) q^{52} + (162 \beta_{2} + 54) q^{53} + ( - 93 \beta_{2} + 267) q^{55} + (46 \beta_{3} + 10 \beta_1) q^{56} + ( - 24 \beta_{3} + 24 \beta_{2} - 18 \beta_1 + 42) q^{58} + (136 \beta_{3} - 136 \beta_{2} - 467 \beta_1 + 331) q^{59} + (105 \beta_{3} - 272 \beta_1) q^{61} + ( - 26 \beta_{2} - 70) q^{62} + ( - 153 \beta_{2} - 440) q^{64} + ( - 107 \beta_{3} + 136 \beta_1) q^{65} + (66 \beta_{3} - 66 \beta_{2} + 461 \beta_1 - 527) q^{67} + ( - 171 \beta_{3} + 171 \beta_{2} - 261 \beta_1 + 432) q^{68} + ( - 30 \beta_{3} - 144 \beta_1) q^{70} + ( - 144 \beta_{2} - 756) q^{71} + (243 \beta_{2} - 106) q^{73} + (56 \beta_{3} + 380 \beta_1) q^{74} + ( - 183 \beta_{3} + 183 \beta_{2} - 673 \beta_1 + 856) q^{76} + ( - 71 \beta_{3} + 71 \beta_{2} + 118 \beta_1 - 47) q^{77} + ( - 309 \beta_{3} + 556 \beta_1) q^{79} + (64 \beta_{2} - 16) q^{80} + (213 \beta_{2} + 1014) q^{82} + ( - 107 \beta_{3} + 460 \beta_1) q^{83} + (18 \beta_{3} - 18 \beta_{2} + 288 \beta_1 - 306) q^{85} + ( - 35 \beta_{3} + 35 \beta_{2} + \beta_1 + 34) q^{86} + (165 \beta_{3} - 693 \beta_1) q^{88} + ( - 72 \beta_{2} + 162) q^{89} + ( - 69 \beta_{2} - 425) q^{91} + (2 \beta_{3} - 430 \beta_1) q^{92} + (336 \beta_{3} - 336 \beta_{2} + 882 \beta_1 - 1218) q^{94} + (164 \beta_{3} - 164 \beta_{2} - 16 \beta_1 - 148) q^{95} + ( - 102 \beta_{3} - 317 \beta_1) q^{97} + (225 \beta_{2} + 324) q^{98}+O(q^{100})$$ q + (b3 + b1) * q^2 + (3*b3 - 3*b2 + b1 - 4) * q^4 + (b3 - b2 - 8*b1 + 7) * q^5 + (-3*b3 - 2*b1) * q^7 + (b2 - 16) * q^8 + (6*b2 + 6) * q^10 + (-8*b3 + 37*b1) * q^11 + (-15*b3 + 15*b2 + 2*b1 + 13) * q^13 + (-8*b3 + 8*b2 - 26*b1 + 34) * q^14 + (9*b3 - b1) * q^16 + (-9*b2 - 54) * q^17 + (-27*b2 - 52) * q^19 + (20*b3 - 16*b1) * q^20 + (21*b3 - 21*b2 - 27*b1 + 6) * q^22 + (19*b3 - 19*b2 - 26*b1 + 7) * q^23 + (15*b3 + 53*b1) * q^25 + (28*b2 + 146) * q^26 + (18*b2 + 92) * q^28 + (b3 - 26*b1) * q^29 + (3*b3 - 3*b2 + 20*b1 - 23) * q^31 + (9*b3 - 9*b2 + 207*b1 - 216) * q^32 + (-63*b3 - 117*b1) * q^34 + (-19*b2 - 11) * q^35 + (54*b2 + 2) * q^37 + (-79*b3 - 241*b1) * q^38 + (-24*b3 + 24*b2 + 144*b1 - 120) * q^40 + (-98*b3 + 98*b2 - 17*b1 + 115) * q^41 + (6*b3 - 47*b1) * q^43 + (-79*b2 + 76) * q^44 + (-12*b2 - 138) * q^46 + (91*b3 + 154*b1) * q^47 + (21*b3 - 21*b2 - 267*b1 + 246) * q^49 + (83*b3 - 83*b2 + 173*b1 - 256) * q^50 + (54*b3 + 358*b1) * q^52 + (162*b2 + 54) * q^53 + (-93*b2 + 267) * q^55 + (46*b3 + 10*b1) * q^56 + (-24*b3 + 24*b2 - 18*b1 + 42) * q^58 + (136*b3 - 136*b2 - 467*b1 + 331) * q^59 + (105*b3 - 272*b1) * q^61 + (-26*b2 - 70) * q^62 + (-153*b2 - 440) * q^64 + (-107*b3 + 136*b1) * q^65 + (66*b3 - 66*b2 + 461*b1 - 527) * q^67 + (-171*b3 + 171*b2 - 261*b1 + 432) * q^68 + (-30*b3 - 144*b1) * q^70 + (-144*b2 - 756) * q^71 + (243*b2 - 106) * q^73 + (56*b3 + 380*b1) * q^74 + (-183*b3 + 183*b2 - 673*b1 + 856) * q^76 + (-71*b3 + 71*b2 + 118*b1 - 47) * q^77 + (-309*b3 + 556*b1) * q^79 + (64*b2 - 16) * q^80 + (213*b2 + 1014) * q^82 + (-107*b3 + 460*b1) * q^83 + (18*b3 - 18*b2 + 288*b1 - 306) * q^85 + (-35*b3 + 35*b2 + b1 + 34) * q^86 + (165*b3 - 693*b1) * q^88 + (-72*b2 + 162) * q^89 + (-69*b2 - 425) * q^91 + (2*b3 - 430*b1) * q^92 + (336*b3 - 336*b2 + 882*b1 - 1218) * q^94 + (164*b3 - 164*b2 - 16*b1 - 148) * q^95 + (-102*b3 - 317*b1) * q^97 + (225*b2 + 324) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} - 5 q^{4} + 15 q^{5} - 7 q^{7} - 66 q^{8}+O(q^{10})$$ 4 * q + 3 * q^2 - 5 * q^4 + 15 * q^5 - 7 * q^7 - 66 * q^8 $$4 q + 3 q^{2} - 5 q^{4} + 15 q^{5} - 7 q^{7} - 66 q^{8} + 12 q^{10} + 66 q^{11} + 11 q^{13} + 60 q^{14} + 7 q^{16} - 198 q^{17} - 154 q^{19} - 12 q^{20} + 33 q^{22} + 33 q^{23} + 121 q^{25} + 528 q^{26} + 332 q^{28} - 51 q^{29} - 43 q^{31} - 423 q^{32} - 297 q^{34} - 6 q^{35} - 100 q^{37} - 561 q^{38} - 264 q^{40} + 132 q^{41} - 88 q^{43} + 462 q^{44} - 528 q^{46} + 399 q^{47} + 513 q^{49} - 429 q^{50} + 770 q^{52} - 108 q^{53} + 1254 q^{55} + 66 q^{56} + 60 q^{58} + 798 q^{59} - 439 q^{61} - 228 q^{62} - 1454 q^{64} + 165 q^{65} - 988 q^{67} + 693 q^{68} - 318 q^{70} - 2736 q^{71} - 910 q^{73} + 816 q^{74} + 1529 q^{76} - 165 q^{77} + 803 q^{79} - 192 q^{80} + 3630 q^{82} + 813 q^{83} - 594 q^{85} + 33 q^{86} - 1221 q^{88} + 792 q^{89} - 1562 q^{91} - 858 q^{92} - 2100 q^{94} - 132 q^{95} - 736 q^{97} + 846 q^{98}+O(q^{100})$$ 4 * q + 3 * q^2 - 5 * q^4 + 15 * q^5 - 7 * q^7 - 66 * q^8 + 12 * q^10 + 66 * q^11 + 11 * q^13 + 60 * q^14 + 7 * q^16 - 198 * q^17 - 154 * q^19 - 12 * q^20 + 33 * q^22 + 33 * q^23 + 121 * q^25 + 528 * q^26 + 332 * q^28 - 51 * q^29 - 43 * q^31 - 423 * q^32 - 297 * q^34 - 6 * q^35 - 100 * q^37 - 561 * q^38 - 264 * q^40 + 132 * q^41 - 88 * q^43 + 462 * q^44 - 528 * q^46 + 399 * q^47 + 513 * q^49 - 429 * q^50 + 770 * q^52 - 108 * q^53 + 1254 * q^55 + 66 * q^56 + 60 * q^58 + 798 * q^59 - 439 * q^61 - 228 * q^62 - 1454 * q^64 + 165 * q^65 - 988 * q^67 + 693 * q^68 - 318 * q^70 - 2736 * q^71 - 910 * q^73 + 816 * q^74 + 1529 * q^76 - 165 * q^77 + 803 * q^79 - 192 * q^80 + 3630 * q^82 + 813 * q^83 - 594 * q^85 + 33 * q^86 - 1221 * q^88 + 792 * q^89 - 1562 * q^91 - 858 * q^92 - 2100 * q^94 - 132 * q^95 - 736 * q^97 + 846 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
−0.686141 1.18843i 0 3.05842 5.29734i 5.18614 8.98266i 0 2.55842 + 4.43132i −19.3723 0 −14.2337
10.2 2.18614 + 3.78651i 0 −5.55842 + 9.62747i 2.31386 4.00772i 0 −6.05842 10.4935i −13.6277 0 20.2337
19.1 −0.686141 + 1.18843i 0 3.05842 + 5.29734i 5.18614 + 8.98266i 0 2.55842 4.43132i −19.3723 0 −14.2337
19.2 2.18614 3.78651i 0 −5.55842 9.62747i 2.31386 + 4.00772i 0 −6.05842 + 10.4935i −13.6277 0 20.2337
 $$n$$: 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
9.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.4.c.a 4
3.b odd 2 1 9.4.c.a 4
4.b odd 2 1 432.4.i.c 4
9.c even 3 1 inner 27.4.c.a 4
9.c even 3 1 81.4.a.a 2
9.d odd 6 1 9.4.c.a 4
9.d odd 6 1 81.4.a.d 2
12.b even 2 1 144.4.i.c 4
15.d odd 2 1 225.4.e.b 4
15.e even 4 2 225.4.k.b 8
36.f odd 6 1 432.4.i.c 4
36.f odd 6 1 1296.4.a.i 2
36.h even 6 1 144.4.i.c 4
36.h even 6 1 1296.4.a.u 2
45.h odd 6 1 225.4.e.b 4
45.h odd 6 1 2025.4.a.g 2
45.j even 6 1 2025.4.a.n 2
45.l even 12 2 225.4.k.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 3.b odd 2 1
9.4.c.a 4 9.d odd 6 1
27.4.c.a 4 1.a even 1 1 trivial
27.4.c.a 4 9.c even 3 1 inner
81.4.a.a 2 9.c even 3 1
81.4.a.d 2 9.d odd 6 1
144.4.i.c 4 12.b even 2 1
144.4.i.c 4 36.h even 6 1
225.4.e.b 4 15.d odd 2 1
225.4.e.b 4 45.h odd 6 1
225.4.k.b 8 15.e even 4 2
225.4.k.b 8 45.l even 12 2
432.4.i.c 4 4.b odd 2 1
432.4.i.c 4 36.f odd 6 1
1296.4.a.i 2 36.f odd 6 1
1296.4.a.u 2 36.h even 6 1
2025.4.a.g 2 45.h odd 6 1
2025.4.a.n 2 45.j even 6 1

Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(27, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 15 T^{3} + 177 T^{2} + \cdots + 2304$$
$7$ $$T^{4} + 7 T^{3} + 111 T^{2} + \cdots + 3844$$
$11$ $$T^{4} - 66 T^{3} + 3795 T^{2} + \cdots + 314721$$
$13$ $$T^{4} - 11 T^{3} + 1947 T^{2} + \cdots + 3334276$$
$17$ $$(T^{2} + 99 T + 1782)^{2}$$
$19$ $$(T^{2} + 77 T - 4532)^{2}$$
$23$ $$T^{4} - 33 T^{3} + 3795 T^{2} + \cdots + 7322436$$
$29$ $$T^{4} + 51 T^{3} + 1959 T^{2} + \cdots + 412164$$
$31$ $$T^{4} + 43 T^{3} + 1461 T^{2} + \cdots + 150544$$
$37$ $$(T^{2} + 50 T - 23432)^{2}$$
$41$ $$T^{4} - 132 T^{3} + \cdots + 5606565129$$
$43$ $$T^{4} + 88 T^{3} + 6105 T^{2} + \cdots + 2686321$$
$47$ $$T^{4} - 399 T^{3} + \cdots + 813276324$$
$53$ $$(T^{2} + 54 T - 215784)^{2}$$
$59$ $$T^{4} - 798 T^{3} + \cdots + 43678881$$
$61$ $$T^{4} + 439 T^{3} + \cdots + 1829786176$$
$67$ $$T^{4} + 988 T^{3} + \cdots + 43305193801$$
$71$ $$(T^{2} + 1368 T + 296784)^{2}$$
$73$ $$(T^{2} + 455 T - 435398)^{2}$$
$79$ $$T^{4} - 803 T^{3} + \cdots + 392522298256$$
$83$ $$T^{4} - 813 T^{3} + \cdots + 5010940944$$
$89$ $$(T^{2} - 396 T - 3564)^{2}$$
$97$ $$T^{4} + 736 T^{3} + \cdots + 2459267281$$