# Properties

 Label 27.5.d.a Level 27 Weight 5 Character orbit 27.d Analytic conductor 2.791 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 27.d (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.79098900326$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.39400128.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{4}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{4} + ( 2 + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{5} + ( -3 - 5 \beta_{1} - 12 \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{7} + ( -26 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} - 50 \beta_{4} + 2 \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{4} + ( 2 + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{5} + ( -3 - 5 \beta_{1} - 12 \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{7} + ( -26 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} - 50 \beta_{4} + 2 \beta_{5} ) q^{8} + ( 2 - 7 \beta_{1} + 8 \beta_{2} - \beta_{3} ) q^{10} + ( -53 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + 52 \beta_{4} - \beta_{5} ) q^{11} + ( 16 \beta_{1} + 8 \beta_{2} + 5 \beta_{3} + 10 \beta_{4} - 5 \beta_{5} ) q^{13} + ( 256 - 5 \beta_{2} - 7 \beta_{3} + 121 \beta_{4} - 7 \beta_{5} ) q^{14} + ( 26 + 6 \beta_{1} + 27 \beta_{2} + 11 \beta_{4} - 15 \beta_{5} ) q^{16} + ( -88 - 24 \beta_{1} - 17 \beta_{2} + 7 \beta_{3} - 190 \beta_{4} - 14 \beta_{5} ) q^{17} + ( -52 - 9 \beta_{2} + 9 \beta_{3} ) q^{19} + ( -170 - 4 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} + 178 \beta_{4} + 8 \beta_{5} ) q^{20} + ( -134 \beta_{1} - 67 \beta_{2} + 2 \beta_{3} + 56 \beta_{4} - 2 \beta_{5} ) q^{22} + ( 82 + 43 \beta_{2} + 14 \beta_{3} + 55 \beta_{4} + 14 \beta_{5} ) q^{23} + ( -92 + 34 \beta_{1} + 33 \beta_{2} - 57 \beta_{4} + 35 \beta_{5} ) q^{25} + ( -230 + 13 \beta_{1} - 13 \beta_{3} - 434 \beta_{4} + 26 \beta_{5} ) q^{26} + ( 134 + 114 \beta_{1} - 84 \beta_{2} - 30 \beta_{3} ) q^{28} + ( 75 + 64 \beta_{1} - 21 \beta_{2} + 42 \beta_{3} - 96 \beta_{4} - 21 \beta_{5} ) q^{29} + ( 202 \beta_{1} + 101 \beta_{2} - 46 \beta_{3} - 329 \beta_{4} + 46 \beta_{5} ) q^{31} + ( 172 - 124 \beta_{2} + 5 \beta_{3} + 91 \beta_{4} + 5 \beta_{5} ) q^{32} + ( 186 + 12 \beta_{1} + 27 \beta_{2} + 183 \beta_{4} - 3 \beta_{5} ) q^{34} + ( 363 + 191 \beta_{1} + 179 \beta_{2} - 12 \beta_{3} + 750 \beta_{4} + 24 \beta_{5} ) q^{35} + ( 128 - 162 \beta_{1} + 126 \beta_{2} + 36 \beta_{3} ) q^{37} + ( 126 - 88 \beta_{1} + 9 \beta_{2} - 18 \beta_{3} - 117 \beta_{4} + 9 \beta_{5} ) q^{38} + ( -68 \beta_{1} - 34 \beta_{2} + 44 \beta_{3} + 404 \beta_{4} - 44 \beta_{5} ) q^{40} + ( -1638 + 50 \beta_{2} - 36 \beta_{3} - 855 \beta_{4} - 36 \beta_{5} ) q^{41} + ( -261 - 251 \beta_{1} - 417 \beta_{2} - 346 \beta_{4} - 85 \beta_{5} ) q^{43} + ( 464 - 388 \beta_{1} - 339 \beta_{2} + 49 \beta_{3} + 830 \beta_{4} - 98 \beta_{5} ) q^{44} + ( -832 - 157 \beta_{1} + 128 \beta_{2} + 29 \beta_{3} ) q^{46} + ( 1175 - 191 \beta_{1} + 40 \beta_{2} - 80 \beta_{3} - 1135 \beta_{4} + 40 \beta_{5} ) q^{47} + ( 64 \beta_{1} + 32 \beta_{2} + 119 \beta_{3} + 653 \beta_{4} - 119 \beta_{5} ) q^{49} + ( -380 + 438 \beta_{2} - \beta_{3} - 191 \beta_{4} - \beta_{5} ) q^{50} + ( 46 + 180 \beta_{1} + 306 \beta_{2} + 100 \beta_{4} + 54 \beta_{5} ) q^{52} + ( 1554 - 234 \beta_{1} - 222 \beta_{2} + 12 \beta_{3} + 3084 \beta_{4} - 24 \beta_{5} ) q^{53} + ( 365 + 95 \beta_{1} + 29 \beta_{2} - 124 \beta_{3} ) q^{55} + ( -188 + 634 \beta_{1} - 28 \beta_{2} + 56 \beta_{3} + 160 \beta_{4} - 28 \beta_{5} ) q^{56} + ( 238 \beta_{1} + 119 \beta_{2} - 127 \beta_{3} - 2035 \beta_{4} + 127 \beta_{5} ) q^{58} + ( -1918 - 789 \beta_{2} + 13 \beta_{3} - 946 \beta_{4} + 13 \beta_{5} ) q^{59} + ( 1293 + 112 \beta_{1} + 195 \beta_{2} + 1322 \beta_{4} + 29 \beta_{5} ) q^{61} + ( -1376 + 1165 \beta_{1} + 1110 \beta_{2} - 55 \beta_{3} - 2642 \beta_{4} + 110 \beta_{5} ) q^{62} + ( 2074 + 390 \beta_{1} - 501 \beta_{2} + 111 \beta_{3} ) q^{64} + ( 891 - 368 \beta_{1} - 27 \beta_{2} + 54 \beta_{3} - 918 \beta_{4} - 27 \beta_{5} ) q^{65} + ( -722 \beta_{1} - 361 \beta_{2} - 193 \beta_{3} + 1324 \beta_{4} + 193 \beta_{5} ) q^{67} + ( 2252 + 4 \beta_{2} + 127 \beta_{3} + 1253 \beta_{4} + 127 \beta_{5} ) q^{68} + ( -3316 + 67 \beta_{1} - 21 \beta_{2} - 3161 \beta_{4} + 155 \beta_{5} ) q^{70} + ( -442 - 420 \beta_{1} - 572 \beta_{2} - 152 \beta_{3} - 580 \beta_{4} + 304 \beta_{5} ) q^{71} + ( -2734 + 324 \beta_{1} - 297 \beta_{2} - 27 \beta_{3} ) q^{73} + ( -2736 - 988 \beta_{1} - 126 \beta_{2} + 252 \beta_{3} + 2610 \beta_{4} - 126 \beta_{5} ) q^{74} + ( -28 \beta_{1} - 14 \beta_{2} - 29 \beta_{3} + 1335 \beta_{4} + 29 \beta_{5} ) q^{76} + ( 282 + 1208 \beta_{2} - 75 \beta_{3} + 66 \beta_{4} - 75 \beta_{5} ) q^{77} + ( 1911 + 151 \beta_{1} + 474 \beta_{2} + 1739 \beta_{4} - 172 \beta_{5} ) q^{79} + ( -2656 - 764 \beta_{1} - 646 \beta_{2} + 118 \beta_{3} - 5548 \beta_{4} - 236 \beta_{5} ) q^{80} + ( -1072 - 631 \beta_{1} + 545 \beta_{2} + 86 \beta_{3} ) q^{82} + ( -225 + 2053 \beta_{1} + 252 \beta_{2} - 504 \beta_{3} + 477 \beta_{4} + 252 \beta_{5} ) q^{83} + ( -660 \beta_{1} - 330 \beta_{2} + 498 \beta_{3} - 4452 \beta_{4} - 498 \beta_{5} ) q^{85} + ( 7660 - 1253 \beta_{2} - 166 \beta_{3} + 3664 \beta_{4} - 166 \beta_{5} ) q^{86} + ( 4774 - 364 \beta_{1} - 519 \beta_{2} + 4565 \beta_{4} - 209 \beta_{5} ) q^{88} + ( 1630 + 366 \beta_{1} + 734 \beta_{2} + 368 \beta_{3} + 2524 \beta_{4} - 736 \beta_{5} ) q^{89} + ( 6227 - 203 \beta_{1} + 421 \beta_{2} - 218 \beta_{3} ) q^{91} + ( -3390 - 754 \beta_{1} + 96 \beta_{2} - 192 \beta_{3} + 3486 \beta_{4} + 96 \beta_{5} ) q^{92} + ( 1906 \beta_{1} + 953 \beta_{2} + 311 \beta_{3} + 5189 \beta_{4} - 311 \beta_{5} ) q^{94} + ( 3100 + 234 \beta_{2} + 38 \beta_{3} + 1588 \beta_{4} + 38 \beta_{5} ) q^{95} + ( -9261 + 514 \beta_{1} + 906 \beta_{2} - 9139 \beta_{4} + 122 \beta_{5} ) q^{97} + ( -2306 - 1056 \beta_{1} - 1207 \beta_{2} - 151 \beta_{3} - 4310 \beta_{4} + 302 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{2} + 15q^{4} + 12q^{5} + 12q^{7} + O(q^{10})$$ $$6q + 3q^{2} + 15q^{4} + 12q^{5} + 12q^{7} - 36q^{10} - 483q^{11} - 6q^{13} + 1146q^{14} + 15q^{16} - 258q^{19} - 1614q^{20} - 369q^{22} + 282q^{23} - 273q^{25} + 1308q^{28} + 1056q^{29} + 1290q^{31} + 1161q^{32} + 513q^{34} + 12q^{37} + 789q^{38} - 1314q^{40} - 7629q^{41} - 285q^{43} - 5760q^{46} + 9642q^{47} - 1863q^{49} - 3027q^{50} - 240q^{52} + 2016q^{55} + 462q^{56} + 6462q^{58} - 6225q^{59} + 3630q^{61} + 15450q^{64} + 7158q^{65} - 5055q^{67} + 10503q^{68} - 9684q^{70} - 14622q^{73} - 26454q^{74} - 4047q^{76} - 2580q^{77} + 4764q^{79} - 9702q^{82} + 1866q^{83} + 12366q^{85} + 37731q^{86} + 14787q^{88} + 34836q^{91} - 33636q^{92} - 12708q^{94} + 13362q^{95} - 28959q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 11 x^{4} + 14 x^{3} + 98 x^{2} + 20 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 11 \nu^{4} - 121 \nu^{3} + 98 \nu^{2} + 1118 \nu - 220$$$$)/1098$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 11 \nu^{4} + 121 \nu^{3} - 98 \nu^{2} + 529 \nu + 220$$$$)/549$$ $$\beta_{3}$$ $$=$$ $$($$$$-17 \nu^{5} + 187 \nu^{4} - 410 \nu^{3} + 1666 \nu^{2} + 889 \nu + 10534$$$$)/549$$ $$\beta_{4}$$ $$=$$ $$($$$$55 \nu^{5} - 56 \nu^{4} + 616 \nu^{3} + 649 \nu^{2} + 5488 \nu + 22$$$$)/1098$$ $$\beta_{5}$$ $$=$$ $$($$$$347 \nu^{5} - 340 \nu^{4} + 3740 \nu^{3} + 5339 \nu^{2} + 32954 \nu + 7166$$$$)/366$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 19 \beta_{4} + 3 \beta_{2} + 2 \beta_{1} - 20$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} + 11 \beta_{2} - 12 \beta_{1} - 26$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-11 \beta_{5} + 215 \beta_{4} + 11 \beta_{3} - 34 \beta_{2} - 68 \beta_{1}$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-23 \beta_{5} + 503 \beta_{4} - 293 \beta_{2} - 158 \beta_{1} + 526$$$$)/3$$

## Character Values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 −1.28901 − 2.23263i −0.102534 − 0.177594i 1.89154 + 3.27625i −1.28901 + 2.23263i −0.102534 + 0.177594i 1.89154 − 3.27625i
−3.86703 2.23263i 0 1.96929 + 3.41090i −13.8760 + 8.01130i 0 −36.2418 + 62.7727i 53.8574i 0 71.5451
8.2 −0.307601 0.177594i 0 −7.93692 13.7472i 30.0804 17.3669i 0 15.6054 27.0294i 11.3212i 0 −12.3370
8.3 5.67463 + 3.27625i 0 13.4676 + 23.3266i −10.2044 + 5.89150i 0 26.6364 46.1356i 71.6534i 0 −77.2081
17.1 −3.86703 + 2.23263i 0 1.96929 3.41090i −13.8760 8.01130i 0 −36.2418 62.7727i 53.8574i 0 71.5451
17.2 −0.307601 + 0.177594i 0 −7.93692 + 13.7472i 30.0804 + 17.3669i 0 15.6054 + 27.0294i 11.3212i 0 −12.3370
17.3 5.67463 3.27625i 0 13.4676 23.3266i −10.2044 5.89150i 0 26.6364 + 46.1356i 71.6534i 0 −77.2081
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{5}^{\mathrm{new}}(27, [\chi])$$.