Properties

Label 27.8.c.a
Level 27
Weight 8
Character orbit 27.c
Analytic conductor 8.434
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 27.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{21} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{2} + ( -52 + 3 \beta_{6} + 52 \beta_{7} + \beta_{9} ) q^{4} + ( 30 - 30 \beta_{7} - \beta_{11} ) q^{5} + ( 16 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{5} - 16 \beta_{6} - 6 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{7} + ( -472 + 42 \beta_{1} + 10 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{2} + ( -52 + 3 \beta_{6} + 52 \beta_{7} + \beta_{9} ) q^{4} + ( 30 - 30 \beta_{7} - \beta_{11} ) q^{5} + ( 16 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{5} - 16 \beta_{6} - 6 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{7} + ( -472 + 42 \beta_{1} + 10 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{8} + ( -11 - 61 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{10} + ( -13 \beta_{1} - 20 \beta_{2} - \beta_{3} - 2 \beta_{5} + 13 \beta_{6} + 1407 \beta_{7} + 2 \beta_{8} + 20 \beta_{9} - 5 \beta_{10} - \beta_{11} ) q^{11} + ( -190 + 10 \beta_{4} + 251 \beta_{6} + 190 \beta_{7} + 5 \beta_{8} - 9 \beta_{9} - 10 \beta_{10} + 6 \beta_{11} ) q^{13} + ( 2701 - 8 \beta_{4} - 8 \beta_{6} - 2701 \beta_{7} + 22 \beta_{8} - 23 \beta_{9} + 8 \beta_{10} + 16 \beta_{11} ) q^{14} + ( 1292 \beta_{1} + 44 \beta_{2} + 39 \beta_{3} + 11 \beta_{5} - 1292 \beta_{6} - 1416 \beta_{7} - 11 \beta_{8} - 44 \beta_{9} - 26 \beta_{10} + 39 \beta_{11} ) q^{16} + ( -2497 + 97 \beta_{1} - 33 \beta_{2} - 16 \beta_{3} - 24 \beta_{4} + 21 \beta_{5} ) q^{17} + ( 1131 - 1792 \beta_{1} + 2 \beta_{2} - 33 \beta_{3} - 2 \beta_{4} + 20 \beta_{5} ) q^{19} + ( 287 \beta_{1} + 93 \beta_{2} + 17 \beta_{3} - 15 \beta_{5} - 287 \beta_{6} + 6952 \beta_{7} + 15 \beta_{8} - 93 \beta_{9} - 6 \beta_{10} + 17 \beta_{11} ) q^{20} + ( -4163 - 14 \beta_{4} + 3326 \beta_{6} + 4163 \beta_{7} - \beta_{8} + 69 \beta_{9} + 14 \beta_{10} - 75 \beta_{11} ) q^{22} + ( 8968 + 29 \beta_{4} + 705 \beta_{6} - 8968 \beta_{7} - 64 \beta_{8} + 332 \beta_{9} - 29 \beta_{10} - 104 \beta_{11} ) q^{23} + ( 1654 \beta_{1} - 244 \beta_{2} - 201 \beta_{3} - 20 \beta_{5} - 1654 \beta_{6} + 1579 \beta_{7} + 20 \beta_{8} + 244 \beta_{9} + 98 \beta_{10} - 201 \beta_{11} ) q^{25} + ( -45109 - 815 \beta_{1} - 18 \beta_{2} + 103 \beta_{3} + 90 \beta_{4} - 99 \beta_{5} ) q^{26} + ( 4568 - 3579 \beta_{1} + 3 \beta_{2} + 117 \beta_{3} + 6 \beta_{4} - 135 \beta_{5} ) q^{28} + ( -333 \beta_{1} + 155 \beta_{2} - 120 \beta_{3} + 137 \beta_{5} + 333 \beta_{6} + 68830 \beta_{7} - 137 \beta_{8} - 155 \beta_{9} + 86 \beta_{10} - 120 \beta_{11} ) q^{29} + ( 1302 - 109 \beta_{4} - 353 \beta_{6} - 1302 \beta_{7} - 116 \beta_{8} - 196 \beta_{9} + 109 \beta_{10} + 354 \beta_{11} ) q^{31} + ( 173456 - 30 \beta_{4} - 2852 \beta_{6} - 173456 \beta_{7} - 75 \beta_{8} - 1236 \beta_{9} + 30 \beta_{10} + 319 \beta_{11} ) q^{32} + ( -614 \beta_{1} + 508 \beta_{2} + 465 \beta_{3} - 233 \beta_{5} + 614 \beta_{6} - 18004 \beta_{7} + 233 \beta_{8} - 508 \beta_{9} + 86 \beta_{10} + 465 \beta_{11} ) q^{34} + ( -185512 - 2332 \beta_{1} - 193 \beta_{2} - 303 \beta_{3} + 251 \beta_{4} + 5 \beta_{5} ) q^{35} + ( 16326 + 9491 \beta_{1} - 757 \beta_{2} + 3 \beta_{3} + 262 \beta_{4} - 199 \beta_{5} ) q^{37} + ( -4444 \beta_{1} - 1732 \beta_{2} + 423 \beta_{3} - 319 \beta_{5} + 4444 \beta_{6} + 323318 \beta_{7} + 319 \beta_{8} + 1732 \beta_{9} + 134 \beta_{10} + 423 \beta_{11} ) q^{38} + ( 44544 + 48 \beta_{4} - 12366 \beta_{6} - 44544 \beta_{7} + 330 \beta_{8} - 254 \beta_{9} - 48 \beta_{10} - 630 \beta_{11} ) q^{40} + ( 287339 + 266 \beta_{4} - 2257 \beta_{6} - 287339 \beta_{7} - 7 \beta_{8} + 659 \beta_{9} - 266 \beta_{10} - 243 \beta_{11} ) q^{41} + ( -31098 \beta_{1} - 251 \beta_{2} - 162 \beta_{3} + 537 \beta_{5} + 31098 \beta_{6} + 52345 \beta_{7} - 537 \beta_{8} + 251 \beta_{9} - 201 \beta_{10} - 162 \beta_{11} ) q^{43} + ( -422780 + 15195 \beta_{1} + 1997 \beta_{2} + 140 \beta_{3} - 988 \beta_{4} + 224 \beta_{5} ) q^{44} + ( -126193 + 28576 \beta_{1} + 2375 \beta_{2} - 1020 \beta_{3} - 1012 \beta_{4} + 1306 \beta_{5} ) q^{46} + ( 12862 \beta_{1} + 3555 \beta_{2} - 563 \beta_{3} + 477 \beta_{5} - 12862 \beta_{6} + 278570 \beta_{7} - 477 \beta_{8} - 3555 \beta_{9} - 927 \beta_{10} - 563 \beta_{11} ) q^{47} + ( -52809 + 840 \beta_{4} - 45675 \beta_{6} + 52809 \beta_{7} - 567 \beta_{8} + 1771 \beta_{9} - 840 \beta_{10} - 504 \beta_{11} ) q^{49} + ( 286914 - 1202 \beta_{4} + 22424 \beta_{6} - 286914 \beta_{7} + 1285 \beta_{8} + 3616 \beta_{9} + 1202 \beta_{10} - 1367 \beta_{11} ) q^{50} + ( 7609 \beta_{1} - 2141 \beta_{2} - 1515 \beta_{3} + 469 \beta_{5} - 7609 \beta_{6} + 117912 \beta_{7} - 469 \beta_{8} + 2141 \beta_{9} - 1486 \beta_{10} - 1515 \beta_{11} ) q^{52} + ( -246738 - 20189 \beta_{1} - 3541 \beta_{2} + 1671 \beta_{3} + 134 \beta_{4} + 545 \beta_{5} ) q^{53} + ( 12818 + 28903 \beta_{1} - 178 \beta_{2} + 2076 \beta_{3} + 1181 \beta_{4} - 590 \beta_{5} ) q^{55} + ( -7090 \beta_{1} - 1394 \beta_{2} - 1108 \beta_{3} - 1160 \beta_{5} + 7090 \beta_{6} + 294312 \beta_{7} + 1160 \beta_{8} + 1394 \beta_{9} + 268 \beta_{10} - 1108 \beta_{11} ) q^{56} + ( -128865 - 822 \beta_{4} + 59289 \beta_{6} + 128865 \beta_{7} + 1713 \beta_{8} + 490 \beta_{9} + 822 \beta_{10} + 3357 \beta_{11} ) q^{58} + ( 337263 + 1575 \beta_{4} - 9912 \beta_{6} - 337263 \beta_{7} - 801 \beta_{8} - 2421 \beta_{9} - 1575 \beta_{10} + 4118 \beta_{11} ) q^{59} + ( -5475 \beta_{1} + 9073 \beta_{2} + 2934 \beta_{3} - 933 \beta_{5} + 5475 \beta_{6} - 457148 \beta_{7} + 933 \beta_{8} - 9073 \beta_{9} + 2022 \beta_{10} + 2934 \beta_{11} ) q^{61} + ( 84749 - 2260 \beta_{1} - 2611 \beta_{2} - 4274 \beta_{3} + 200 \beta_{4} - 820 \beta_{5} ) q^{62} + ( 530336 - 137092 \beta_{1} - 7876 \beta_{2} + 849 \beta_{3} - 638 \beta_{4} - 763 \beta_{5} ) q^{64} + ( 16725 \beta_{1} - 1345 \beta_{2} + 5382 \beta_{3} + 1445 \beta_{5} - 16725 \beta_{6} + 238240 \beta_{7} - 1445 \beta_{8} + 1345 \beta_{9} + 1340 \beta_{10} + 5382 \beta_{11} ) q^{65} + ( 628461 - 373 \beta_{4} - 15137 \beta_{6} - 628461 \beta_{7} - 2834 \beta_{8} - 13000 \beta_{9} + 373 \beta_{10} - 2301 \beta_{11} ) q^{67} + ( -383708 - 538 \beta_{4} - 55578 \beta_{6} + 383708 \beta_{7} - 1837 \beta_{8} - 5734 \beta_{9} + 538 \beta_{10} - 2107 \beta_{11} ) q^{68} + ( 157818 \beta_{1} - 10973 \beta_{2} - 972 \beta_{3} - 2250 \beta_{5} - 157818 \beta_{6} + 237933 \beta_{7} + 2250 \beta_{8} + 10973 \beta_{9} + 2016 \beta_{10} - 972 \beta_{11} ) q^{70} + ( -181330 + 41121 \beta_{1} + 9949 \beta_{2} + 539 \beta_{3} + 3646 \beta_{4} - 1265 \beta_{5} ) q^{71} + ( -887113 + 2217 \beta_{1} + 11271 \beta_{2} - 6084 \beta_{3} + 1464 \beta_{4} - 3363 \beta_{5} ) q^{73} + ( -82678 \beta_{1} - 3544 \beta_{2} - 5922 \beta_{3} + 326 \beta_{5} + 82678 \beta_{6} - 1683466 \beta_{7} - 326 \beta_{8} + 3544 \beta_{9} + 1760 \beta_{10} - 5922 \beta_{11} ) q^{74} + ( -970860 - 2134 \beta_{4} + 298648 \beta_{6} + 970860 \beta_{7} - 107 \beta_{8} + 20424 \beta_{9} + 2134 \beta_{10} - 4533 \beta_{11} ) q^{76} + ( -571480 + 1708 \beta_{4} + 14748 \beta_{6} + 571480 \beta_{7} - 3608 \beta_{8} - 8864 \beta_{9} - 1708 \beta_{10} - 8421 \beta_{11} ) q^{77} + ( -82075 \beta_{1} - 15072 \beta_{2} - 1482 \beta_{3} + 1568 \beta_{5} + 82075 \beta_{6} + 961154 \beta_{7} - 1568 \beta_{8} + 15072 \beta_{9} + 1021 \beta_{10} - 1482 \beta_{11} ) q^{79} + ( 3123576 - 100024 \beta_{1} + 3784 \beta_{2} + 12392 \beta_{3} - 8 \beta_{4} - 1760 \beta_{5} ) q^{80} + ( 665079 - 236379 \beta_{1} - 2108 \beta_{2} + 1206 \beta_{3} - 768 \beta_{4} + 1398 \beta_{5} ) q^{82} + ( -4646 \beta_{1} - 2987 \beta_{2} - 6453 \beta_{3} + 1135 \beta_{5} + 4646 \beta_{6} - 1602512 \beta_{7} - 1135 \beta_{8} + 2987 \beta_{9} - 2441 \beta_{10} - 6453 \beta_{11} ) q^{83} + ( -351986 + 598 \beta_{4} - 135151 \beta_{6} + 351986 \beta_{7} + 2015 \beta_{8} + 9581 \beta_{9} - 598 \beta_{10} - 237 \beta_{11} ) q^{85} + ( -5635661 - 2170 \beta_{4} + 157650 \beta_{6} + 5635661 \beta_{7} + 4955 \beta_{8} + 24035 \beta_{9} + 2170 \beta_{10} + 13225 \beta_{11} ) q^{86} + ( 276390 \beta_{1} + 45158 \beta_{2} - 1359 \beta_{3} + 3009 \beta_{5} - 276390 \beta_{6} - 2603712 \beta_{7} - 3009 \beta_{8} - 45158 \beta_{9} - 5538 \beta_{10} - 1359 \beta_{11} ) q^{88} + ( 2096188 + 84733 \beta_{1} - 6167 \beta_{2} - 13625 \beta_{3} - 6818 \beta_{4} + 4219 \beta_{5} ) q^{89} + ( 1262778 + 269779 \beta_{1} - 28368 \beta_{2} + 5532 \beta_{3} - 2389 \beta_{4} + 8296 \beta_{5} ) q^{91} + ( 364011 \beta_{1} + 31909 \beta_{2} + 19959 \beta_{3} - 2957 \beta_{5} - 364011 \beta_{6} - 6337792 \beta_{7} + 2957 \beta_{8} - 31909 \beta_{9} - 5246 \beta_{10} + 19959 \beta_{11} ) q^{92} + ( 2007755 + 7784 \beta_{4} - 176420 \beta_{6} - 2007755 \beta_{7} + 1222 \beta_{8} - 59533 \beta_{9} - 7784 \beta_{10} + 14268 \beta_{11} ) q^{94} + ( -1857108 - 4316 \beta_{4} - 233458 \beta_{6} + 1857108 \beta_{7} + 11050 \beta_{8} + 14758 \beta_{9} + 4316 \beta_{10} + 754 \beta_{11} ) q^{95} + ( -460167 \beta_{1} - 17543 \beta_{2} + 1089 \beta_{3} + 651 \beta_{5} + 460167 \beta_{6} + 1436665 \beta_{7} - 651 \beta_{8} + 17543 \beta_{9} - 8406 \beta_{10} + 1089 \beta_{11} ) q^{97} + ( 8053744 + 138958 \beta_{1} - 54404 \beta_{2} - 11459 \beta_{3} - 3458 \beta_{4} + 7147 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 9q^{2} - 321q^{4} + 180q^{5} - 84q^{7} - 5922q^{8} + O(q^{10}) \) \( 12q + 9q^{2} - 321q^{4} + 180q^{5} - 84q^{7} - 5922q^{8} + 252q^{10} + 8460q^{11} - 1848q^{13} + 16272q^{14} - 12417q^{16} - 30564q^{17} + 24432q^{19} + 40788q^{20} - 35001q^{22} + 51588q^{23} + 4746q^{25} - 536472q^{26} + 75516q^{28} + 414648q^{29} + 8196q^{31} + 1048977q^{32} - 106623q^{34} - 2210616q^{35} + 139344q^{37} + 1952685q^{38} + 305496q^{40} + 1731582q^{41} + 408372q^{43} - 5169114q^{44} - 1684008q^{46} + 1631484q^{47} - 179010q^{49} + 1654461q^{50} + 681594q^{52} - 2835648q^{53} - 16056q^{55} + 1784466q^{56} - 948384q^{58} + 2055636q^{59} - 2723196q^{61} + 1026828q^{62} + 7178178q^{64} + 1387620q^{65} + 3806556q^{67} - 2142639q^{68} + 953442q^{70} - 2408400q^{71} - 10670052q^{73} - 9846504q^{74} - 6727827q^{76} - 3478824q^{77} + 6020916q^{79} + 38072448q^{80} + 9403002q^{82} - 9605052q^{83} - 1698624q^{85} - 34278561q^{86} - 16459029q^{88} + 24630264q^{89} + 13570104q^{91} - 39143394q^{92} + 12602808q^{94} - 10422072q^{95} + 9977226q^{97} + 95833314q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} - 8342916 x^{5} + 72369348 x^{4} - 131054670 x^{3} + 513267363 x^{2} - 449098992 x + 754412211\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{10} - 5 \nu^{9} + 631 \nu^{8} - 2494 \nu^{7} + 213005 \nu^{6} - 630307 \nu^{5} + 25967781 \nu^{4} - 50887950 \nu^{3} + 943236207 \nu^{2} - 917896869 \nu + 3098815785 \)\()/ 479986560 \)
\(\beta_{2}\)\(=\)\( 3 \nu^{2} - 3 \nu + 180 \)
\(\beta_{3}\)\(=\)\((\)\(443 \nu^{10} - 2215 \nu^{9} + 183791 \nu^{8} - 721874 \nu^{7} + 26320567 \nu^{6} - 76444445 \nu^{5} + 1475805645 \nu^{4} - 2825041638 \nu^{3} + 24728370921 \nu^{2} - 23328471195 \nu + 18976459761\)\()/ 239993280 \)
\(\beta_{4}\)\(=\)\((\)\(5029 \nu^{10} - 25145 \nu^{9} + 1832911 \nu^{8} - 7180774 \nu^{7} + 238629713 \nu^{6} - 690862039 \nu^{5} + 13240426797 \nu^{4} - 25337744142 \nu^{3} + 277992051363 \nu^{2} - 265437133713 \nu + 1000917153009\)\()/ 479986560 \)
\(\beta_{5}\)\(=\)\((\)\(2699 \nu^{10} - 13495 \nu^{9} + 873305 \nu^{8} - 3412250 \nu^{7} + 97211743 \nu^{6} - 279749033 \nu^{5} + 4393795275 \nu^{4} - 8325296130 \nu^{3} + 70650764829 \nu^{2} - 66534176943 \nu + 182728260423\)\()/ 159995520 \)
\(\beta_{6}\)\(=\)\((\)\(1958197 \nu^{11} - 9956688 \nu^{10} + 724465158 \nu^{9} - 2684366423 \nu^{8} + 95608029027 \nu^{7} - 153624091098 \nu^{6} + 5117486083294 \nu^{5} + 7856435046999 \nu^{4} + 83127551924445 \nu^{3} + 593549030077326 \nu^{2} - 1383240867349824 \nu + 2867387988937761\)\()/ 780837815928960 \)
\(\beta_{7}\)\(=\)\((\)\(3916394 \nu^{11} - 21540167 \nu^{10} + 1457064271 \nu^{9} - 6395237967 \nu^{8} + 195273274808 \nu^{7} - 653762799151 \nu^{6} + 11260349921425 \nu^{5} - 26531282326773 \nu^{4} + 249039162917340 \nu^{3} - 347350112267085 \nu^{2} + 1069258078504611 \nu - 87187467749373\)\()/ 780837815928960 \)
\(\beta_{8}\)\(=\)\((\)\(499913231 \nu^{11} + 3836540593 \nu^{10} + 180673452721 \nu^{9} + 1190431529238 \nu^{8} + 23137472449883 \nu^{7} + 131460986193239 \nu^{6} + 1138718367264907 \nu^{5} + 6430487977844142 \nu^{4} + 11192627445604809 \nu^{3} + 129378126521627865 \nu^{2} - 210913006206368529 \nu + 477199570249662840\)\()/ 780837815928960 \)
\(\beta_{9}\)\(=\)\((\)\(-77730382 \nu^{11} + 427517101 \nu^{10} - 28848101381 \nu^{9} + 126610077957 \nu^{8} - 3831102185704 \nu^{7} + 12821003239205 \nu^{6} - 217142495718683 \nu^{5} + 511097017128423 \nu^{4} - 4697389961111028 \nu^{3} + 6671454296702895 \nu^{2} - 20396503412810145 \nu + 16868076430785471\)\()/ 86759757325440 \)
\(\beta_{10}\)\(=\)\((\)\(868001755 \nu^{11} - 683443683 \nu^{10} + 299680949037 \nu^{9} + 86084626714 \nu^{8} + 37001818591575 \nu^{7} + 50673438197739 \nu^{6} + 1976398128186919 \nu^{5} + 4779161322029130 \nu^{4} + 43805614789010805 \nu^{3} + 135414409007778237 \nu^{2} + 346760628858611451 \nu + 547729371474062220\)\()/ 780837815928960 \)
\(\beta_{11}\)\(=\)\((\)\(-674669385 \nu^{11} + 3350347411 \nu^{10} - 236441521415 \nu^{9} + 894769481542 \nu^{8} - 29116328059993 \nu^{7} + 77705950762349 \nu^{6} - 1485854186786545 \nu^{5} + 2424301324746570 \nu^{4} - 27207425320413891 \nu^{3} + 20568158469989127 \nu^{2} - 80291612552969745 \nu + 27536223844018512\)\()/ 390418907964480 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + \beta_{2} + \beta_{1} - 179\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{11} + 4 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 155 \beta_{7} + 578 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 289 \beta_{1} - 728\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{11} + 8 \beta_{10} + 22 \beta_{9} + 4 \beta_{8} - 313 \beta_{7} + 1162 \beta_{6} - 7 \beta_{5} + 10 \beta_{4} - 35 \beta_{3} - 388 \beta_{2} - 1264 \beta_{1} + 51556\)\()/9\)
\(\nu^{5}\)\(=\)\((\)\(1107 \beta_{11} - 1658 \beta_{10} - 6592 \beta_{9} - 1039 \beta_{8} - 203622 \beta_{7} - 184787 \beta_{6} + 482 \beta_{5} + 934 \beta_{4} + 306 \beta_{3} + 446 \beta_{2} + 87271 \beta_{1} + 491337\)\()/27\)
\(\nu^{6}\)\(=\)\((\)\(1117 \beta_{11} - 1678 \beta_{10} - 6647 \beta_{9} - 1049 \beta_{8} - 202838 \beta_{7} - 187695 \beta_{6} + 1447 \beta_{5} - 1708 \beta_{4} + 6395 \beta_{3} + 48607 \beta_{2} + 262089 \beta_{1} - 5339585\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(-51457 \beta_{11} + 69106 \beta_{10} + 341729 \beta_{9} + 46673 \beta_{8} + 18594995 \beta_{7} + 6888837 \beta_{6} - 20064 \beta_{5} - 43590 \beta_{4} - 5012 \beta_{3} - 9042 \beta_{2} - 2849597 \beta_{1} - 28796399\)\()/9\)
\(\nu^{8}\)\(=\)\((\)\(-633150 \beta_{11} + 852820 \beta_{10} + 4193960 \beta_{9} + 574790 \beta_{8} + 225977475 \beta_{7} + 85301920 \beta_{6} - 723235 \beta_{5} + 718702 \beta_{4} - 2734119 \beta_{3} - 18450472 \beta_{2} - 130478744 \beta_{1} + 1698678117\)\()/27\)
\(\nu^{9}\)\(=\)\((\)\(6852619 \beta_{11} - 8449306 \beta_{10} - 48596144 \beta_{9} - 5813303 \beta_{8} - 3307887749 \beta_{7} - 794054829 \beta_{6} + 2233204 \beta_{5} + 5996690 \beta_{4} - 324652 \beta_{3} - 1207100 \beta_{2} + 261709638 \beta_{1} + 4488984514\)\()/9\)
\(\nu^{10}\)\(=\)\((\)\(35853809 \beta_{11} - 44390366 \beta_{10} - 253512259 \beta_{9} - 30510853 \beta_{8} - 17105800729 \beta_{7} - 4184848629 \beta_{6} + 37184439 \beta_{5} - 31302588 \beta_{4} + 121061899 \beta_{3} + 787237092 \beta_{2} + 6515774374 \beta_{1} - 60531727202\)\()/9\)
\(\nu^{11}\)\(=\)\((\)\(-2716894845 \beta_{11} + 3079935034 \beta_{10} + 19761824426 \beta_{9} + 2103885437 \beta_{8} + 1508121936912 \beta_{7} + 280038035629 \beta_{6} - 671485801 \beta_{5} - 2471711132 \beta_{4} + 447117795 \beta_{3} + 1723787465 \beta_{2} - 63292817522 \beta_{1} - 1972568702727\)\()/27\)

Character Values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{7}\)

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
0.500000 + 9.80854i
0.500000 + 9.08282i
0.500000 + 1.48508i
0.500000 2.70685i
0.500000 6.17443i
0.500000 11.4952i
0.500000 9.80854i
0.500000 9.08282i
0.500000 1.48508i
0.500000 + 2.70685i
0.500000 + 6.17443i
0.500000 + 11.4952i
−7.74445 13.4138i 0 −55.9529 + 96.9133i −52.7641 + 91.3900i 0 761.419 + 1318.82i −249.280 0 1634.51
10.2 −7.11595 12.3252i 0 −37.2735 + 64.5595i 145.304 251.673i 0 −555.940 962.916i −760.739 0 −4135.89
10.3 −0.536120 0.928588i 0 63.4251 109.856i −47.9866 + 83.1153i 0 −189.000 327.358i −273.261 0 102.906
10.4 3.09420 + 5.35931i 0 44.8519 77.6857i −167.952 + 290.901i 0 442.025 + 765.610i 1347.24 0 −2078.70
10.5 6.09721 + 10.5607i 0 −10.3519 + 17.9301i 246.026 426.130i 0 −382.311 662.182i 1308.41 0 6000.29
10.6 10.7051 + 18.5418i 0 −165.199 + 286.133i −32.6274 + 56.5123i 0 −118.194 204.717i −4333.37 0 −1397.12
19.1 −7.74445 + 13.4138i 0 −55.9529 96.9133i −52.7641 91.3900i 0 761.419 1318.82i −249.280 0 1634.51
19.2 −7.11595 + 12.3252i 0 −37.2735 64.5595i 145.304 + 251.673i 0 −555.940 + 962.916i −760.739 0 −4135.89
19.3 −0.536120 + 0.928588i 0 63.4251 + 109.856i −47.9866 83.1153i 0 −189.000 + 327.358i −273.261 0 102.906
19.4 3.09420 5.35931i 0 44.8519 + 77.6857i −167.952 290.901i 0 442.025 765.610i 1347.24 0 −2078.70
19.5 6.09721 10.5607i 0 −10.3519 17.9301i 246.026 + 426.130i 0 −382.311 + 662.182i 1308.41 0 6000.29
19.6 10.7051 18.5418i 0 −165.199 286.133i −32.6274 56.5123i 0 −118.194 + 204.717i −4333.37 0 −1397.12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{8}^{\mathrm{new}}(27, [\chi])\).