# 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$27 = 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 27.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.59305157015$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ 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

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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 7 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{7} + ( -16 + \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 7 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{7} + ( -16 + \beta_{2} ) q^{8} + ( 6 + 6 \beta_{2} ) q^{10} + ( 37 \beta_{1} - 8 \beta_{3} ) q^{11} + ( 13 + 2 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} ) q^{13} + ( 34 - 26 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{14} + ( -\beta_{1} + 9 \beta_{3} ) q^{16} + ( -54 - 9 \beta_{2} ) q^{17} + ( -52 - 27 \beta_{2} ) q^{19} + ( -16 \beta_{1} + 20 \beta_{3} ) q^{20} + ( 6 - 27 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{22} + ( 7 - 26 \beta_{1} - 19 \beta_{2} + 19 \beta_{3} ) q^{23} + ( 53 \beta_{1} + 15 \beta_{3} ) q^{25} + ( 146 + 28 \beta_{2} ) q^{26} + ( 92 + 18 \beta_{2} ) q^{28} + ( -26 \beta_{1} + \beta_{3} ) q^{29} + ( -23 + 20 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -216 + 207 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{32} + ( -117 \beta_{1} - 63 \beta_{3} ) q^{34} + ( -11 - 19 \beta_{2} ) q^{35} + ( 2 + 54 \beta_{2} ) q^{37} + ( -241 \beta_{1} - 79 \beta_{3} ) q^{38} + ( -120 + 144 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{40} + ( 115 - 17 \beta_{1} + 98 \beta_{2} - 98 \beta_{3} ) q^{41} + ( -47 \beta_{1} + 6 \beta_{3} ) q^{43} + ( 76 - 79 \beta_{2} ) q^{44} + ( -138 - 12 \beta_{2} ) q^{46} + ( 154 \beta_{1} + 91 \beta_{3} ) q^{47} + ( 246 - 267 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{49} + ( -256 + 173 \beta_{1} - 83 \beta_{2} + 83 \beta_{3} ) q^{50} + ( 358 \beta_{1} + 54 \beta_{3} ) q^{52} + ( 54 + 162 \beta_{2} ) q^{53} + ( 267 - 93 \beta_{2} ) q^{55} + ( 10 \beta_{1} + 46 \beta_{3} ) q^{56} + ( 42 - 18 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{58} + ( 331 - 467 \beta_{1} - 136 \beta_{2} + 136 \beta_{3} ) q^{59} + ( -272 \beta_{1} + 105 \beta_{3} ) q^{61} + ( -70 - 26 \beta_{2} ) q^{62} + ( -440 - 153 \beta_{2} ) q^{64} + ( 136 \beta_{1} - 107 \beta_{3} ) q^{65} + ( -527 + 461 \beta_{1} - 66 \beta_{2} + 66 \beta_{3} ) q^{67} + ( 432 - 261 \beta_{1} + 171 \beta_{2} - 171 \beta_{3} ) q^{68} + ( -144 \beta_{1} - 30 \beta_{3} ) q^{70} + ( -756 - 144 \beta_{2} ) q^{71} + ( -106 + 243 \beta_{2} ) q^{73} + ( 380 \beta_{1} + 56 \beta_{3} ) q^{74} + ( 856 - 673 \beta_{1} + 183 \beta_{2} - 183 \beta_{3} ) q^{76} + ( -47 + 118 \beta_{1} + 71 \beta_{2} - 71 \beta_{3} ) q^{77} + ( 556 \beta_{1} - 309 \beta_{3} ) q^{79} + ( -16 + 64 \beta_{2} ) q^{80} + ( 1014 + 213 \beta_{2} ) q^{82} + ( 460 \beta_{1} - 107 \beta_{3} ) q^{83} + ( -306 + 288 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{85} + ( 34 + \beta_{1} + 35 \beta_{2} - 35 \beta_{3} ) q^{86} + ( -693 \beta_{1} + 165 \beta_{3} ) q^{88} + ( 162 - 72 \beta_{2} ) q^{89} + ( -425 - 69 \beta_{2} ) q^{91} + ( -430 \beta_{1} + 2 \beta_{3} ) q^{92} + ( -1218 + 882 \beta_{1} - 336 \beta_{2} + 336 \beta_{3} ) q^{94} + ( -148 - 16 \beta_{1} - 164 \beta_{2} + 164 \beta_{3} ) q^{95} + ( -317 \beta_{1} - 102 \beta_{3} ) q^{97} + ( 324 + 225 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} - 5 q^{4} + 15 q^{5} - 7 q^{7} - 66 q^{8} + O(q^{10})$$ $$4 q + 3 q^{2} - 5 q^{4} + 15 q^{5} - 7 q^{7} - 66 q^{8} + 12 q^{10} + 66 q^{11} + 11 q^{13} + 60 q^{14} + 7 q^{16} - 198 q^{17} - 154 q^{19} - 12 q^{20} + 33 q^{22} + 33 q^{23} + 121 q^{25} + 528 q^{26} + 332 q^{28} - 51 q^{29} - 43 q^{31} - 423 q^{32} - 297 q^{34} - 6 q^{35} - 100 q^{37} - 561 q^{38} - 264 q^{40} + 132 q^{41} - 88 q^{43} + 462 q^{44} - 528 q^{46} + 399 q^{47} + 513 q^{49} - 429 q^{50} + 770 q^{52} - 108 q^{53} + 1254 q^{55} + 66 q^{56} + 60 q^{58} + 798 q^{59} - 439 q^{61} - 228 q^{62} - 1454 q^{64} + 165 q^{65} - 988 q^{67} + 693 q^{68} - 318 q^{70} - 2736 q^{71} - 910 q^{73} + 816 q^{74} + 1529 q^{76} - 165 q^{77} + 803 q^{79} - 192 q^{80} + 3630 q^{82} + 813 q^{83} - 594 q^{85} + 33 q^{86} - 1221 q^{88} + 792 q^{89} - 1562 q^{91} - 858 q^{92} - 2100 q^{94} - 132 q^{95} - 736 q^{97} + 846 q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11$$$$)/3$$

## 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.

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

$p$ $F_p(T)$
$2$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$2304 - 720 T + 177 T^{2} - 15 T^{3} + T^{4}$$
$7$ $$3844 - 434 T + 111 T^{2} + 7 T^{3} + T^{4}$$
$11$ $$314721 - 37026 T + 3795 T^{2} - 66 T^{3} + T^{4}$$
$13$ $$3334276 + 20086 T + 1947 T^{2} - 11 T^{3} + T^{4}$$
$17$ $$( 1782 + 99 T + T^{2} )^{2}$$
$19$ $$( -4532 + 77 T + T^{2} )^{2}$$
$23$ $$7322436 + 89298 T + 3795 T^{2} - 33 T^{3} + T^{4}$$
$29$ $$412164 + 32742 T + 1959 T^{2} + 51 T^{3} + T^{4}$$
$31$ $$150544 + 16684 T + 1461 T^{2} + 43 T^{3} + T^{4}$$
$37$ $$( -23432 + 50 T + T^{2} )^{2}$$
$41$ $$5606565129 + 9883764 T + 92301 T^{2} - 132 T^{3} + T^{4}$$
$43$ $$2686321 + 144232 T + 6105 T^{2} + 88 T^{3} + T^{4}$$
$47$ $$813276324 + 11378682 T + 187719 T^{2} - 399 T^{3} + T^{4}$$
$53$ $$( -215784 + 54 T + T^{2} )^{2}$$
$59$ $$43678881 - 5273982 T + 630195 T^{2} - 798 T^{3} + T^{4}$$
$61$ $$1829786176 - 18778664 T + 235497 T^{2} + 439 T^{3} + T^{4}$$
$67$ $$43305193801 + 205601812 T + 768045 T^{2} + 988 T^{3} + T^{4}$$
$71$ $$( 296784 + 1368 T + T^{2} )^{2}$$
$73$ $$( -435398 + 455 T + T^{2} )^{2}$$
$79$ $$392522298256 + 503092348 T + 1271325 T^{2} - 803 T^{3} + T^{4}$$
$83$ $$5010940944 - 57550644 T + 590181 T^{2} - 813 T^{3} + T^{4}$$
$89$ $$( -3564 - 396 T + T^{2} )^{2}$$
$97$ $$2459267281 + 36498976 T + 492105 T^{2} + 736 T^{3} + T^{4}$$