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

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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:

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 \(S_{5}^{\mathrm{new}}(27, [\chi])\).