Properties

Label 300.11.k.c
Level $300$
Weight $11$
Character orbit 300.k
Analytic conductor $190.607$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 300.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(190.607175802\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 63831600 x^{13} + 120528248672 x^{12} - 17600989215600 x^{11} + 2037236579280000 x^{10} - 3343396831784087400 x^{9} + 4317664424723438821024 x^{8} - 1093166020823780215242000 x^{7} + 122767222860388422207360000 x^{6} + 17394213048729389505235839600 x^{5} + 3322978970397292826674840245897 x^{4} - 837815701799476622071156318691400 x^{3} + 126631812340115852451399515174580000 x^{2} + 31278039886234045787552155897225466400 x + 3862835732371990949484385042694585728656\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{12}\cdot 5^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 81 \beta_{3} q^{3} + ( -143 \beta_{2} - \beta_{8} ) q^{7} -19683 \beta_{1} q^{9} +O(q^{10})\) \( q + 81 \beta_{3} q^{3} + ( -143 \beta_{2} - \beta_{8} ) q^{7} -19683 \beta_{1} q^{9} + ( -20694 - \beta_{4} - \beta_{5} ) q^{11} + ( -8265 \beta_{3} + \beta_{12} + \beta_{15} ) q^{13} + ( -121158 \beta_{2} - 20 \beta_{8} - 7 \beta_{9} - 4 \beta_{14} ) q^{17} + ( -332815 \beta_{1} + 3 \beta_{6} - 13 \beta_{7} + 4 \beta_{11} ) q^{19} + ( 34749 + 81 \beta_{5} ) q^{21} + ( -1109406 \beta_{3} - 188 \beta_{12} - 4 \beta_{13} - 19 \beta_{15} ) q^{23} + 1594323 \beta_{2} q^{27} + ( -4077402 \beta_{1} + 223 \beta_{6} - 57 \beta_{7} - 5 \beta_{11} ) q^{29} + ( -8800301 + 198 \beta_{4} + 393 \beta_{5} ) q^{31} + ( -1676214 \beta_{3} - 324 \beta_{12} + 81 \beta_{13} ) q^{33} + ( -5892364 \beta_{2} - 4354 \beta_{8} - 18 \beta_{9} + 150 \beta_{14} ) q^{37} + ( 2008395 \beta_{1} + 162 \beta_{6} + 81 \beta_{7} - 81 \beta_{11} ) q^{39} + ( 1847100 - 563 \beta_{4} + 2953 \beta_{5} - 48 \beta_{10} ) q^{41} + ( -27860081 \beta_{3} - 4878 \beta_{12} - 261 \beta_{13} + 454 \beta_{15} ) q^{43} + ( 16621278 \beta_{2} - 6794 \beta_{8} + 338 \beta_{9} - 602 \beta_{14} ) q^{47} + ( 70914698 \beta_{1} - 46 \beta_{6} + 1140 \beta_{7} + 300 \beta_{11} ) q^{49} + ( 29441394 - 1377 \beta_{4} + 1863 \beta_{5} + 324 \beta_{10} ) q^{51} + ( 2095428 \beta_{3} - 4170 \beta_{12} + 6 \beta_{13} - 1063 \beta_{15} ) q^{53} + ( 26958015 \beta_{2} + 2430 \beta_{8} - 729 \beta_{9} + 972 \beta_{14} ) q^{57} + ( 300376338 \beta_{1} + 4180 \beta_{6} - 808 \beta_{7} - 678 \beta_{11} ) q^{59} + ( 223553927 + 6147 \beta_{4} + 3380 \beta_{5} - 531 \beta_{10} ) q^{61} + ( 2814669 \beta_{3} + 19683 \beta_{12} ) q^{63} + ( 223872123 \beta_{2} + 29420 \beta_{8} + 3429 \beta_{9} - 496 \beta_{14} ) q^{67} + ( 269585658 \beta_{1} - 17091 \beta_{6} - 2511 \beta_{7} + 1539 \beta_{11} ) q^{69} + ( 115824540 - 4899 \beta_{4} - 62203 \beta_{5} - 487 \beta_{10} ) q^{71} + ( -582671708 \beta_{3} + 48004 \beta_{12} + 3492 \beta_{13} - 1066 \beta_{15} ) q^{73} + ( 450719442 \beta_{2} + 105136 \beta_{8} - 9340 \beta_{9} - 661 \beta_{14} ) q^{77} + ( 1065427126 \beta_{1} - 5766 \beta_{6} - 13630 \beta_{7} - 464 \beta_{11} ) q^{79} -387420489 q^{81} + ( -1673532546 \beta_{3} - 54248 \beta_{12} - 3106 \beta_{13} + 8292 \beta_{15} ) q^{83} + ( 330269562 \beta_{2} + 57996 \beta_{8} - 5022 \beta_{9} - 1215 \beta_{14} ) q^{87} + ( 3318477180 \beta_{1} + 53174 \beta_{6} + 28386 \beta_{7} - 4426 \beta_{11} ) q^{89} + ( 439579515 - 5493 \beta_{4} + 63863 \beta_{5} + 6774 \beta_{10} ) q^{91} + ( -712824381 \beta_{3} + 111537 \beta_{12} - 16038 \beta_{13} ) q^{93} + ( 284374349 \beta_{2} - 106334 \beta_{8} + 24426 \beta_{9} + 7472 \beta_{14} ) q^{97} + ( 407320002 \beta_{1} - 19683 \beta_{6} + 19683 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 331104q^{11} + 555984q^{21} - 140804816q^{31} + 29553600q^{41} + 471062304q^{51} + 3576862832q^{61} + 1853192640q^{71} - 6198727824q^{81} + 7033272240q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 63831600 x^{13} + 120528248672 x^{12} - 17600989215600 x^{11} + 2037236579280000 x^{10} - 3343396831784087400 x^{9} + 4317664424723438821024 x^{8} - 1093166020823780215242000 x^{7} + 122767222860388422207360000 x^{6} + 17394213048729389505235839600 x^{5} + 3322978970397292826674840245897 x^{4} - 837815701799476622071156318691400 x^{3} + 126631812340115852451399515174580000 x^{2} + 31278039886234045787552155897225466400 x + 3862835732371990949484385042694585728656\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(29\!\cdots\!05\)\( \nu^{15} + \)\(45\!\cdots\!33\)\( \nu^{14} + \)\(28\!\cdots\!00\)\( \nu^{13} + \)\(19\!\cdots\!00\)\( \nu^{12} - \)\(39\!\cdots\!40\)\( \nu^{11} + \)\(10\!\cdots\!44\)\( \nu^{10} - \)\(10\!\cdots\!00\)\( \nu^{9} + \)\(10\!\cdots\!00\)\( \nu^{8} - \)\(14\!\cdots\!60\)\( \nu^{7} + \)\(52\!\cdots\!96\)\( \nu^{6} - \)\(75\!\cdots\!00\)\( \nu^{5} - \)\(85\!\cdots\!00\)\( \nu^{4} - \)\(60\!\cdots\!45\)\( \nu^{3} + \)\(76\!\cdots\!37\)\( \nu^{2} - \)\(76\!\cdots\!00\)\( \nu - \)\(50\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!80\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(41\!\cdots\!25\)\( \nu^{15} - \)\(12\!\cdots\!65\)\( \nu^{14} + \)\(10\!\cdots\!34\)\( \nu^{13} - \)\(87\!\cdots\!00\)\( \nu^{12} + \)\(59\!\cdots\!00\)\( \nu^{11} - \)\(29\!\cdots\!20\)\( \nu^{10} + \)\(14\!\cdots\!92\)\( \nu^{9} - \)\(16\!\cdots\!00\)\( \nu^{8} + \)\(23\!\cdots\!00\)\( \nu^{7} - \)\(13\!\cdots\!80\)\( \nu^{6} + \)\(55\!\cdots\!48\)\( \nu^{5} - \)\(59\!\cdots\!00\)\( \nu^{4} + \)\(16\!\cdots\!25\)\( \nu^{3} + \)\(70\!\cdots\!15\)\( \nu^{2} + \)\(40\!\cdots\!66\)\( \nu - \)\(17\!\cdots\!00\)\(\)\()/ \)\(69\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(55\!\cdots\!67\)\( \nu^{15} + \)\(37\!\cdots\!50\)\( \nu^{14} + \)\(25\!\cdots\!00\)\( \nu^{13} - \)\(36\!\cdots\!20\)\( \nu^{12} + \)\(63\!\cdots\!76\)\( \nu^{11} - \)\(57\!\cdots\!00\)\( \nu^{10} + \)\(79\!\cdots\!00\)\( \nu^{9} - \)\(18\!\cdots\!60\)\( \nu^{8} + \)\(21\!\cdots\!04\)\( \nu^{7} - \)\(46\!\cdots\!00\)\( \nu^{6} + \)\(39\!\cdots\!00\)\( \nu^{5} + \)\(12\!\cdots\!60\)\( \nu^{4} - \)\(18\!\cdots\!37\)\( \nu^{3} - \)\(90\!\cdots\!50\)\( \nu^{2} + \)\(41\!\cdots\!00\)\( \nu + \)\(88\!\cdots\!20\)\(\)\()/ \)\(13\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(97\!\cdots\!13\)\( \nu^{15} + \)\(59\!\cdots\!80\)\( \nu^{14} - \)\(19\!\cdots\!68\)\( \nu^{13} - \)\(14\!\cdots\!60\)\( \nu^{12} - \)\(43\!\cdots\!76\)\( \nu^{11} + \)\(12\!\cdots\!40\)\( \nu^{10} - \)\(11\!\cdots\!84\)\( \nu^{9} - \)\(80\!\cdots\!00\)\( \nu^{8} - \)\(25\!\cdots\!64\)\( \nu^{7} + \)\(56\!\cdots\!60\)\( \nu^{6} - \)\(14\!\cdots\!16\)\( \nu^{5} - \)\(66\!\cdots\!00\)\( \nu^{4} - \)\(10\!\cdots\!43\)\( \nu^{3} + \)\(49\!\cdots\!20\)\( \nu^{2} + \)\(18\!\cdots\!28\)\( \nu + \)\(21\!\cdots\!00\)\(\)\()/ \)\(99\!\cdots\!38\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(55\!\cdots\!67\)\( \nu^{15} + \)\(45\!\cdots\!20\)\( \nu^{14} + \)\(98\!\cdots\!88\)\( \nu^{13} - \)\(26\!\cdots\!20\)\( \nu^{12} + \)\(68\!\cdots\!16\)\( \nu^{11} - \)\(45\!\cdots\!40\)\( \nu^{10} + \)\(65\!\cdots\!44\)\( \nu^{9} - \)\(12\!\cdots\!60\)\( \nu^{8} + \)\(24\!\cdots\!24\)\( \nu^{7} - \)\(43\!\cdots\!60\)\( \nu^{6} + \)\(13\!\cdots\!56\)\( \nu^{5} + \)\(19\!\cdots\!60\)\( \nu^{4} + \)\(46\!\cdots\!63\)\( \nu^{3} - \)\(43\!\cdots\!20\)\( \nu^{2} - \)\(12\!\cdots\!48\)\( \nu - \)\(71\!\cdots\!80\)\(\)\()/ \)\(34\!\cdots\!72\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(10\!\cdots\!29\)\( \nu^{15} + \)\(23\!\cdots\!20\)\( \nu^{14} + \)\(57\!\cdots\!56\)\( \nu^{13} + \)\(68\!\cdots\!40\)\( \nu^{12} - \)\(14\!\cdots\!92\)\( \nu^{11} + \)\(48\!\cdots\!60\)\( \nu^{10} - \)\(49\!\cdots\!72\)\( \nu^{9} + \)\(38\!\cdots\!20\)\( \nu^{8} - \)\(52\!\cdots\!88\)\( \nu^{7} + \)\(22\!\cdots\!40\)\( \nu^{6} - \)\(33\!\cdots\!28\)\( \nu^{5} - \)\(39\!\cdots\!20\)\( \nu^{4} + \)\(19\!\cdots\!19\)\( \nu^{3} + \)\(42\!\cdots\!80\)\( \nu^{2} - \)\(18\!\cdots\!76\)\( \nu - \)\(13\!\cdots\!40\)\(\)\()/ \)\(37\!\cdots\!36\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(74\!\cdots\!47\)\( \nu^{15} - \)\(15\!\cdots\!14\)\( \nu^{14} - \)\(43\!\cdots\!32\)\( \nu^{13} + \)\(42\!\cdots\!20\)\( \nu^{12} - \)\(83\!\cdots\!56\)\( \nu^{11} + \)\(17\!\cdots\!68\)\( \nu^{10} - \)\(49\!\cdots\!16\)\( \nu^{9} + \)\(24\!\cdots\!60\)\( \nu^{8} - \)\(28\!\cdots\!84\)\( \nu^{7} + \)\(10\!\cdots\!92\)\( \nu^{6} - \)\(17\!\cdots\!84\)\( \nu^{5} - \)\(27\!\cdots\!60\)\( \nu^{4} + \)\(60\!\cdots\!17\)\( \nu^{3} + \)\(20\!\cdots\!74\)\( \nu^{2} - \)\(11\!\cdots\!28\)\( \nu - \)\(80\!\cdots\!20\)\(\)\()/ \)\(17\!\cdots\!62\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(27\!\cdots\!25\)\( \nu^{15} - \)\(36\!\cdots\!65\)\( \nu^{14} - \)\(27\!\cdots\!60\)\( \nu^{13} - \)\(71\!\cdots\!72\)\( \nu^{12} + \)\(30\!\cdots\!00\)\( \nu^{11} + \)\(92\!\cdots\!80\)\( \nu^{10} - \)\(27\!\cdots\!80\)\( \nu^{9} - \)\(22\!\cdots\!96\)\( \nu^{8} + \)\(10\!\cdots\!00\)\( \nu^{7} + \)\(53\!\cdots\!00\)\( \nu^{6} - \)\(87\!\cdots\!20\)\( \nu^{5} + \)\(92\!\cdots\!16\)\( \nu^{4} - \)\(54\!\cdots\!75\)\( \nu^{3} - \)\(33\!\cdots\!45\)\( \nu^{2} - \)\(96\!\cdots\!40\)\( \nu - \)\(30\!\cdots\!88\)\(\)\()/ \)\(17\!\cdots\!38\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(17\!\cdots\!25\)\( \nu^{15} + \)\(15\!\cdots\!35\)\( \nu^{14} - \)\(25\!\cdots\!52\)\( \nu^{13} - \)\(34\!\cdots\!28\)\( \nu^{12} + \)\(93\!\cdots\!00\)\( \nu^{11} + \)\(26\!\cdots\!80\)\( \nu^{10} - \)\(34\!\cdots\!96\)\( \nu^{9} - \)\(10\!\cdots\!04\)\( \nu^{8} + \)\(53\!\cdots\!00\)\( \nu^{7} + \)\(11\!\cdots\!40\)\( \nu^{6} - \)\(12\!\cdots\!64\)\( \nu^{5} + \)\(13\!\cdots\!84\)\( \nu^{4} - \)\(15\!\cdots\!75\)\( \nu^{3} - \)\(43\!\cdots\!25\)\( \nu^{2} - \)\(13\!\cdots\!08\)\( \nu - \)\(35\!\cdots\!12\)\(\)\()/ \)\(56\!\cdots\!54\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(13\!\cdots\!87\)\( \nu^{15} + \)\(11\!\cdots\!20\)\( \nu^{14} - \)\(38\!\cdots\!32\)\( \nu^{13} - \)\(86\!\cdots\!20\)\( \nu^{12} + \)\(15\!\cdots\!76\)\( \nu^{11} - \)\(92\!\cdots\!40\)\( \nu^{10} + \)\(11\!\cdots\!84\)\( \nu^{9} - \)\(43\!\cdots\!60\)\( \nu^{8} + \)\(55\!\cdots\!64\)\( \nu^{7} - \)\(95\!\cdots\!60\)\( \nu^{6} + \)\(10\!\cdots\!16\)\( \nu^{5} + \)\(36\!\cdots\!60\)\( \nu^{4} + \)\(95\!\cdots\!43\)\( \nu^{3} - \)\(97\!\cdots\!20\)\( \nu^{2} - \)\(26\!\cdots\!28\)\( \nu + \)\(77\!\cdots\!20\)\(\)\()/ \)\(10\!\cdots\!18\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(37\!\cdots\!33\)\( \nu^{15} - \)\(14\!\cdots\!80\)\( \nu^{14} + \)\(16\!\cdots\!88\)\( \nu^{13} - \)\(23\!\cdots\!80\)\( \nu^{12} + \)\(45\!\cdots\!84\)\( \nu^{11} - \)\(83\!\cdots\!40\)\( \nu^{10} + \)\(12\!\cdots\!44\)\( \nu^{9} - \)\(13\!\cdots\!40\)\( \nu^{8} + \)\(16\!\cdots\!76\)\( \nu^{7} - \)\(47\!\cdots\!60\)\( \nu^{6} + \)\(71\!\cdots\!56\)\( \nu^{5} + \)\(17\!\cdots\!40\)\( \nu^{4} + \)\(18\!\cdots\!37\)\( \nu^{3} - \)\(43\!\cdots\!20\)\( \nu^{2} + \)\(11\!\cdots\!52\)\( \nu + \)\(71\!\cdots\!80\)\(\)\()/ \)\(17\!\cdots\!62\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(66\!\cdots\!45\)\( \nu^{15} - \)\(86\!\cdots\!15\)\( \nu^{14} - \)\(77\!\cdots\!20\)\( \nu^{13} + \)\(44\!\cdots\!92\)\( \nu^{12} - \)\(73\!\cdots\!60\)\( \nu^{11} + \)\(21\!\cdots\!80\)\( \nu^{10} - \)\(85\!\cdots\!60\)\( \nu^{9} + \)\(23\!\cdots\!56\)\( \nu^{8} - \)\(24\!\cdots\!40\)\( \nu^{7} + \)\(39\!\cdots\!00\)\( \nu^{6} - \)\(25\!\cdots\!40\)\( \nu^{5} - \)\(11\!\cdots\!76\)\( \nu^{4} - \)\(11\!\cdots\!05\)\( \nu^{3} + \)\(16\!\cdots\!05\)\( \nu^{2} - \)\(26\!\cdots\!80\)\( \nu - \)\(53\!\cdots\!32\)\(\)\()/ \)\(18\!\cdots\!18\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(78\!\cdots\!58\)\( \nu^{15} - \)\(81\!\cdots\!85\)\( \nu^{14} - \)\(67\!\cdots\!80\)\( \nu^{13} + \)\(49\!\cdots\!08\)\( \nu^{12} - \)\(90\!\cdots\!84\)\( \nu^{11} + \)\(40\!\cdots\!20\)\( \nu^{10} - \)\(97\!\cdots\!40\)\( \nu^{9} + \)\(25\!\cdots\!44\)\( \nu^{8} - \)\(30\!\cdots\!16\)\( \nu^{7} + \)\(51\!\cdots\!00\)\( \nu^{6} - \)\(35\!\cdots\!60\)\( \nu^{5} - \)\(15\!\cdots\!24\)\( \nu^{4} - \)\(13\!\cdots\!82\)\( \nu^{3} + \)\(17\!\cdots\!95\)\( \nu^{2} - \)\(37\!\cdots\!20\)\( \nu - \)\(75\!\cdots\!68\)\(\)\()/ \)\(56\!\cdots\!54\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(96\!\cdots\!75\)\( \nu^{15} - \)\(28\!\cdots\!15\)\( \nu^{14} - \)\(78\!\cdots\!52\)\( \nu^{13} - \)\(72\!\cdots\!28\)\( \nu^{12} + \)\(13\!\cdots\!00\)\( \nu^{11} - \)\(49\!\cdots\!20\)\( \nu^{10} + \)\(68\!\cdots\!04\)\( \nu^{9} - \)\(48\!\cdots\!04\)\( \nu^{8} + \)\(51\!\cdots\!00\)\( \nu^{7} - \)\(22\!\cdots\!60\)\( \nu^{6} + \)\(43\!\cdots\!36\)\( \nu^{5} - \)\(48\!\cdots\!16\)\( \nu^{4} + \)\(27\!\cdots\!75\)\( \nu^{3} - \)\(96\!\cdots\!75\)\( \nu^{2} + \)\(93\!\cdots\!92\)\( \nu - \)\(23\!\cdots\!12\)\(\)\()/ \)\(18\!\cdots\!18\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(99\!\cdots\!92\)\( \nu^{15} + \)\(21\!\cdots\!35\)\( \nu^{14} + \)\(28\!\cdots\!80\)\( \nu^{13} + \)\(55\!\cdots\!92\)\( \nu^{12} - \)\(13\!\cdots\!16\)\( \nu^{11} + \)\(40\!\cdots\!80\)\( \nu^{10} - \)\(19\!\cdots\!60\)\( \nu^{9} + \)\(24\!\cdots\!56\)\( \nu^{8} - \)\(49\!\cdots\!84\)\( \nu^{7} + \)\(18\!\cdots\!00\)\( \nu^{6} - \)\(20\!\cdots\!40\)\( \nu^{5} - \)\(48\!\cdots\!76\)\( \nu^{4} + \)\(74\!\cdots\!32\)\( \nu^{3} + \)\(71\!\cdots\!55\)\( \nu^{2} - \)\(23\!\cdots\!80\)\( \nu - \)\(49\!\cdots\!32\)\(\)\()/ \)\(18\!\cdots\!18\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{10} - 10 \beta_{7} + 26 \beta_{6} - 26 \beta_{5} - 10 \beta_{4} - 3600 \beta_{2}\)\()/7200\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{15} + \beta_{14} - 3 \beta_{13} + 30 \beta_{12} + 180 \beta_{11} - 3 \beta_{9} + 30 \beta_{8} + 180 \beta_{7} - 7040 \beta_{6} + 330885600 \beta_{1}\)\()/2400\)
\(\nu^{3}\)\(=\)\((\)\(-4860 \beta_{15} + 3240 \beta_{13} - 188460 \beta_{12} - 12503 \beta_{11} - 12503 \beta_{10} - 4691770 \beta_{7} + 13259702 \beta_{6} + 13259702 \beta_{5} + 4691770 \beta_{4} - 2977959600 \beta_{3} - 172345320000 \beta_{1} + 172345320000\)\()/14400\)
\(\nu^{4}\)\(=\)\((\)\(6253 \beta_{15} + 6253 \beta_{14} - 784041 \beta_{13} + 7407600 \beta_{12} + 19497390 \beta_{10} + 784041 \beta_{9} - 7407600 \beta_{8} - 1108910120 \beta_{5} - 78009690 \beta_{4} + 28724220000 \beta_{3} - 28724220000 \beta_{2} - 36158474601600\)\()/1200\)
\(\nu^{5}\)\(=\)\((\)\(-877370400 \beta_{14} + 18225566993 \beta_{11} - 18225566993 \beta_{10} - 1462598100 \beta_{9} + 50485707900 \beta_{8} + 576636105670 \beta_{7} - 1960175810342 \beta_{6} + 1960175810342 \beta_{5} + 576636105670 \beta_{4} + 542369674130400 \beta_{2} + 39602225735100000 \beta_{1} + 39602225735100000\)\()/7200\)
\(\nu^{6}\)\(=\)\((\)\(-109353214359 \beta_{15} + 109353214359 \beta_{14} + 1189699824123 \beta_{13} - 12841202577690 \beta_{12} - 18799869170580 \beta_{11} + 1189699824123 \beta_{9} - 12841202577690 \beta_{8} - 133118732755380 \beta_{7} + 1253158985153440 \beta_{6} - 79203589743600000 \beta_{3} - 79203589743600000 \beta_{2} - 36497876107246935600 \beta_{1}\)\()/4800\)
\(\nu^{7}\)\(=\)\((\)\(148044363536475 \beta_{15} - 398778651424125 \beta_{13} + 10119099318948900 \beta_{12} + 3421605030857996 \beta_{11} + 3421605030857996 \beta_{10} + 73989285199839490 \beta_{7} - 297207260513238074 \beta_{6} - 297207260513238074 \beta_{5} - 73989285199839490 \beta_{4} + 95804077348551157200 \beta_{3} + 6851429748682387785000 \beta_{1} - 6851429748682387785000\)\()/3600\)
\(\nu^{8}\)\(=\)\((\)\(-27372074774626099 \beta_{15} - 27372074774626099 \beta_{14} + 206420712749354103 \beta_{13} - 2556624396367308750 \beta_{12} - 2478452239122385860 \beta_{10} - 206420712749354103 \beta_{9} + 2556624396367308750 \beta_{8} + 176145361756230779680 \beta_{5} + 23258144517368176260 \beta_{4} - 18269924905897912800000 \beta_{3} + 18269924905897912800000 \beta_{2} + 4883237090952946824921600\)\()/2400\)
\(\nu^{9}\)\(=\)\((\)\(200743972241067568260 \beta_{14} - 4386658943994705681037 \beta_{11} + 4386658943994705681037 \beta_{10} + 694859400472914881640 \beta_{9} - 14761179211976770096260 \beta_{8} - 78889919186307949848830 \beta_{7} + 356902852832595889145458 \beta_{6} - 356902852832595889145458 \beta_{5} - 78889919186307949848830 \beta_{4} - 131840503603163455144179600 \beta_{2} - 8679936302412416101773720000 \beta_{1} - 8679936302412416101773720000\)\()/14400\)
\(\nu^{10}\)\(=\)\((\)\(5483015745874479839621 \beta_{15} - 5483015745874479839621 \beta_{14} - 34696252006678675955037 \beta_{13} + 475315054759499727485640 \beta_{12} + 341061852872602827864840 \beta_{11} - 34696252006678675955037 \beta_{9} + 475315054759499727485640 \beta_{8} + 3731487868837922153398890 \beta_{7} - 24977879622556149073707220 \beta_{6} + 3616434590597434684912050000 \beta_{3} + 3616434590597434684912050000 \beta_{2} + 674226328184878895375472832800 \beta_{1}\)\()/1200\)
\(\nu^{11}\)\(=\)\((\)\(-33762363234081888681851400 \beta_{15} + 134384586578648560393143300 \beta_{13} - 2573236135804520393069890500 \beta_{12} - 669515049838532880834195931 \beta_{11} - 669515049838532880834195931 \beta_{10} - 10823755198545786235017398690 \beta_{7} + 52954565840236939182374194594 \beta_{6} + 52954565840236939182374194594 \beta_{5} + 10823755198545786235017398690 \beta_{4} - 22247656042145506281765623181600 \beta_{3} - 1322498304604217496001272461820000 \beta_{1} + 1322498304604217496001272461820000\)\()/7200\)
\(\nu^{12}\)\(=\)\((\)\(1606691369622676078191914865 \beta_{15} + 1606691369622676078191914865 \beta_{14} - 9193700229915727977446885805 \beta_{13} + 134666189930079943240843753110 \beta_{12} + 76756343790783950901283276428 \beta_{10} + 9193700229915727977446885805 \beta_{9} - 134666189930079943240843753110 \beta_{8} - 5714063758723838431490176189984 \beta_{5} - 917684523391083339954700067628 \beta_{4} + 1057903170895406354579922708000000 \beta_{3} - 1057903170895406354579922708000000 \beta_{2} - 151804158194255045498797157869861200\)\()/960\)
\(\nu^{13}\)\(=\)\((\)\(-5612149284382744011696497116095 \beta_{14} + 99992636781119381213275302996754 \beta_{11} - 99992636781119381213275302996754 \beta_{10} - 24236875675381127010767275464405 \beta_{9} + 436417423541446508477848549363320 \beta_{8} + 1515492120032492098921862265395510 \beta_{7} - 7793548107041264470677201158468626 \beta_{6} + 7793548107041264470677201158468626 \beta_{5} + 1515492120032492098921862265395510 \beta_{4} + 3699792533761986217739050630110186000 \beta_{2} + 197511882499971704197457464959909945000 \beta_{1} + 197511882499971704197457464959909945000\)\()/3600\)
\(\nu^{14}\)\(=\)\((\)\(-1399714157085423173207143255831913 \beta_{15} + 1399714157085423173207143255831913 \beta_{14} + 7537883102567204458088197895941461 \beta_{13} - 115233387455136957943465578194010570 \beta_{12} - 54659726511234136364312790948795780 \beta_{11} + 7537883102567204458088197895941461 \beta_{9} - 115233387455136957943465578194010570 \beta_{8} - 688953876288039628134304981035846780 \beta_{7} + 4108719343492937011391935347735065840 \beta_{6} - 921601784019507279125850727144641720000 \beta_{3} - 921601784019507279125850727144641720000 \beta_{2} - 108090266803317492165949285375436236761600 \beta_{1}\)\()/2400\)
\(\nu^{15}\)\(=\)\((\)\(1475576911063523527642168376047232508 \beta_{15} - 6691504498426846018254368994920201592 \beta_{13} + 116133802973397281119485149181676761948 \beta_{12} + 23622176802875340343017430748297351659 \beta_{11} + 23622176802875340343017430748297351659 \beta_{10} + 344124617470843263998204514686463154130 \beta_{7} - 1825856355657096387419672531144364604798 \beta_{6} - 1825856355657096387419672531144364604798 \beta_{5} - 344124617470843263998204514686463154130 \beta_{4} + 972657013587520647743390181665606677616880 \beta_{3} + 46669050170379022315988753460917483373720000 \beta_{1} - 46669050170379022315988753460917483373720000\)\()/2880\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−381.708 381.708i
297.652 + 297.652i
−97.2366 97.2366i
178.843 + 178.843i
180.068 + 180.068i
−96.0119 96.0119i
298.877 + 298.877i
−380.484 380.484i
−381.708 + 381.708i
297.652 297.652i
−97.2366 + 97.2366i
178.843 178.843i
180.068 180.068i
−96.0119 + 96.0119i
298.877 298.877i
−380.484 + 380.484i
0 −99.2043 + 99.2043i 0 0 0 −17583.1 17583.1i 0 19683.0i 0
157.2 0 −99.2043 + 99.2043i 0 0 0 −7200.85 7200.85i 0 19683.0i 0
157.3 0 −99.2043 + 99.2043i 0 0 0 6761.61 + 6761.61i 0 19683.0i 0
157.4 0 −99.2043 + 99.2043i 0 0 0 17321.8 + 17321.8i 0 19683.0i 0
157.5 0 99.2043 99.2043i 0 0 0 −17321.8 17321.8i 0 19683.0i 0
157.6 0 99.2043 99.2043i 0 0 0 −6761.61 6761.61i 0 19683.0i 0
157.7 0 99.2043 99.2043i 0 0 0 7200.85 + 7200.85i 0 19683.0i 0
157.8 0 99.2043 99.2043i 0 0 0 17583.1 + 17583.1i 0 19683.0i 0
193.1 0 −99.2043 99.2043i 0 0 0 −17583.1 + 17583.1i 0 19683.0i 0
193.2 0 −99.2043 99.2043i 0 0 0 −7200.85 + 7200.85i 0 19683.0i 0
193.3 0 −99.2043 99.2043i 0 0 0 6761.61 6761.61i 0 19683.0i 0
193.4 0 −99.2043 99.2043i 0 0 0 17321.8 17321.8i 0 19683.0i 0
193.5 0 99.2043 + 99.2043i 0 0 0 −17321.8 + 17321.8i 0 19683.0i 0
193.6 0 99.2043 + 99.2043i 0 0 0 −6761.61 + 6761.61i 0 19683.0i 0
193.7 0 99.2043 + 99.2043i 0 0 0 7200.85 7200.85i 0 19683.0i 0
193.8 0 99.2043 + 99.2043i 0 0 0 17583.1 17583.1i 0 19683.0i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.11.k.c 16
5.b even 2 1 inner 300.11.k.c 16
5.c odd 4 2 inner 300.11.k.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.11.k.c 16 1.a even 1 1 trivial
300.11.k.c 16 5.b even 2 1 inner
300.11.k.c 16 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + \)\(76\!\cdots\!36\)\( T_{7}^{12} + \)\(15\!\cdots\!86\)\( T_{7}^{8} + \)\(26\!\cdots\!16\)\( T_{7}^{4} + \)\(12\!\cdots\!61\)\( \) acting on \(S_{11}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 387420489 + T^{4} )^{4} \)
$5$ \( T^{16} \)
$7$ \( \)\(12\!\cdots\!61\)\( + \)\(26\!\cdots\!16\)\( T^{4} + \)\(15\!\cdots\!86\)\( T^{8} + 761551649611358436 T^{12} + T^{16} \)
$11$ \( ( -29795605046854126704 - 2463897306212064 T - 49435383384 T^{2} + 82776 T^{3} + T^{4} )^{4} \)
$13$ \( \)\(28\!\cdots\!25\)\( + \)\(16\!\cdots\!00\)\( T^{4} + \)\(10\!\cdots\!50\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{12} + T^{16} \)
$17$ \( \)\(89\!\cdots\!16\)\( + \)\(31\!\cdots\!76\)\( T^{4} + \)\(29\!\cdots\!76\)\( T^{8} + \)\(96\!\cdots\!56\)\( T^{12} + T^{16} \)
$19$ \( ( \)\(27\!\cdots\!25\)\( + \)\(19\!\cdots\!00\)\( T^{2} + \)\(43\!\cdots\!50\)\( T^{4} + 36889114896100 T^{6} + T^{8} )^{2} \)
$23$ \( \)\(11\!\cdots\!16\)\( + \)\(15\!\cdots\!76\)\( T^{4} + \)\(15\!\cdots\!76\)\( T^{8} + \)\(23\!\cdots\!56\)\( T^{12} + T^{16} \)
$29$ \( ( \)\(46\!\cdots\!56\)\( + \)\(22\!\cdots\!56\)\( T^{2} + \)\(27\!\cdots\!96\)\( T^{4} + 663137482008816 T^{6} + T^{8} )^{2} \)
$31$ \( ( -\)\(42\!\cdots\!99\)\( - \)\(61\!\cdots\!96\)\( T - 1862193870583194 T^{2} + 35201204 T^{3} + T^{4} )^{4} \)
$37$ \( \)\(18\!\cdots\!96\)\( + \)\(31\!\cdots\!36\)\( T^{4} + \)\(67\!\cdots\!16\)\( T^{8} + \)\(46\!\cdots\!76\)\( T^{12} + T^{16} \)
$41$ \( ( \)\(84\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T - 32804571011691600 T^{2} - 7388400 T^{3} + T^{4} )^{4} \)
$43$ \( \)\(94\!\cdots\!41\)\( + \)\(28\!\cdots\!76\)\( T^{4} + \)\(15\!\cdots\!26\)\( T^{8} + \)\(75\!\cdots\!56\)\( T^{12} + T^{16} \)
$47$ \( \)\(58\!\cdots\!56\)\( + \)\(58\!\cdots\!56\)\( T^{4} + \)\(19\!\cdots\!96\)\( T^{8} + \)\(26\!\cdots\!16\)\( T^{12} + T^{16} \)
$53$ \( \)\(75\!\cdots\!56\)\( + \)\(31\!\cdots\!56\)\( T^{4} + \)\(14\!\cdots\!96\)\( T^{8} + \)\(18\!\cdots\!16\)\( T^{12} + T^{16} \)
$59$ \( ( \)\(26\!\cdots\!96\)\( + \)\(28\!\cdots\!36\)\( T^{2} + \)\(15\!\cdots\!16\)\( T^{4} + 1140330406880610576 T^{6} + T^{8} )^{2} \)
$61$ \( ( \)\(29\!\cdots\!41\)\( + \)\(99\!\cdots\!68\)\( T - 1839482961885061626 T^{2} - 894215708 T^{3} + T^{4} )^{4} \)
$67$ \( \)\(79\!\cdots\!21\)\( + \)\(29\!\cdots\!36\)\( T^{4} + \)\(28\!\cdots\!66\)\( T^{8} + \)\(27\!\cdots\!76\)\( T^{12} + T^{16} \)
$71$ \( ( -\)\(18\!\cdots\!00\)\( + \)\(90\!\cdots\!00\)\( T - 9815254566778357200 T^{2} - 463298160 T^{3} + T^{4} )^{4} \)
$73$ \( \)\(16\!\cdots\!16\)\( + \)\(90\!\cdots\!76\)\( T^{4} + \)\(14\!\cdots\!76\)\( T^{8} + \)\(74\!\cdots\!56\)\( T^{12} + T^{16} \)
$79$ \( ( \)\(66\!\cdots\!76\)\( + \)\(42\!\cdots\!04\)\( T^{2} + \)\(60\!\cdots\!56\)\( T^{4} + 23158946155517676304 T^{6} + T^{8} )^{2} \)
$83$ \( \)\(79\!\cdots\!56\)\( + \)\(19\!\cdots\!56\)\( T^{4} + \)\(42\!\cdots\!96\)\( T^{8} + \)\(24\!\cdots\!16\)\( T^{12} + T^{16} \)
$89$ \( ( \)\(10\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T^{2} + \)\(74\!\cdots\!00\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + T^{8} )^{2} \)
$97$ \( \)\(26\!\cdots\!61\)\( + \)\(21\!\cdots\!16\)\( T^{4} + \)\(49\!\cdots\!86\)\( T^{8} + \)\(13\!\cdots\!36\)\( T^{12} + T^{16} \)
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