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.

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

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

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

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 \(S_{8}^{\mathrm{new}}(27, [\chi])\).