Properties

Label 300.8.i.c
Level $300$
Weight $8$
Character orbit 300.i
Analytic conductor $93.716$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 300.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(93.7155076452\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} + 132 x^{14} - 784 x^{13} + 5524236 x^{12} - 33135588 x^{11} - 49457570 x^{10} + 551013008 x^{9} + 10299424335549 x^{8} - 41201367881396 x^{7} - 506171935260830 x^{6} + 1662723272081664 x^{5} + 7501118196131608896 x^{4} - 15005055667263471212 x^{3} - 559914287081811190962 x^{2} + 567417098903075834864 x + 1831773137685647789586721\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{12}\cdot 5^{8} \)
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.

\(f(q)\) \(=\) \( q -\beta_{6} q^{3} + ( 6 \beta_{4} + 3 \beta_{5} - \beta_{11} ) q^{7} + ( -573 \beta_{1} - \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{6} q^{3} + ( 6 \beta_{4} + 3 \beta_{5} - \beta_{11} ) q^{7} + ( -573 \beta_{1} - \beta_{7} ) q^{9} + \beta_{8} q^{11} + ( -16 \beta_{6} + 8 \beta_{10} - 26 \beta_{12} ) q^{13} + ( -\beta_{3} - 60 \beta_{4} + 15 \beta_{5} ) q^{17} + ( -7172 \beta_{1} + 8 \beta_{7} - 4 \beta_{9} ) q^{19} + ( 13494 - 3 \beta_{8} + 9 \beta_{13} + \beta_{15} ) q^{21} + ( \beta_{2} + 24 \beta_{6} + 6 \beta_{10} ) q^{23} + ( 3 \beta_{3} + 489 \beta_{4} + 138 \beta_{5} + 117 \beta_{11} ) q^{27} + ( -36 \beta_{7} - 9 \beta_{9} + 26 \beta_{14} ) q^{29} + ( 54544 - 44 \beta_{13} - 22 \beta_{15} ) q^{31} + ( -3 \beta_{2} - 312 \beta_{6} - 105 \beta_{10} + 612 \beta_{12} ) q^{33} + ( -1464 \beta_{4} - 732 \beta_{5} - 322 \beta_{11} ) q^{37} + ( 65568 \beta_{1} - 24 \beta_{7} - 26 \beta_{9} - 78 \beta_{14} ) q^{39} + ( 44 \beta_{8} + 252 \beta_{13} - 63 \beta_{15} ) q^{41} + ( 1994 \beta_{6} - 997 \beta_{10} - 1326 \beta_{12} ) q^{43} + ( 7 \beta_{3} - 1632 \beta_{4} + 408 \beta_{5} ) q^{47} + ( 460795 \beta_{1} + 70 \beta_{7} - 35 \beta_{9} ) q^{49} + ( 123660 - 117 \beta_{8} + 60 \beta_{13} - 204 \beta_{15} ) q^{51} + ( -8 \beta_{2} - 7248 \beta_{6} - 1812 \beta_{10} ) q^{53} + ( -36 \beta_{3} + 6308 \beta_{4} + 7092 \beta_{5} - 1404 \beta_{11} ) q^{57} + ( -432 \beta_{7} - 108 \beta_{9} + 91 \beta_{14} ) q^{59} + ( 797054 - 302 \beta_{13} - 151 \beta_{15} ) q^{61} + ( 39 \beta_{2} - 12930 \beta_{6} + 1122 \beta_{10} - 666 \beta_{12} ) q^{63} + ( -5038 \beta_{4} - 2519 \beta_{5} + 6266 \beta_{11} ) q^{67} + ( 49140 \beta_{1} - 87 \beta_{7} - 204 \beta_{9} + 117 \beta_{14} ) q^{69} + ( -46 \beta_{8} + 720 \beta_{13} - 180 \beta_{15} ) q^{71} + ( 38632 \beta_{6} - 19316 \beta_{10} + 6292 \beta_{12} ) q^{73} + ( 14 \beta_{3} - 34728 \beta_{4} + 8682 \beta_{5} ) q^{77} + ( -1300936 \beta_{1} + 444 \beta_{7} - 222 \beta_{9} ) q^{79} + ( 668511 + 702 \beta_{8} + 312 \beta_{13} + 495 \beta_{15} ) q^{81} + ( -6 \beta_{2} - 87588 \beta_{6} - 21897 \beta_{10} ) q^{83} + ( 159 \beta_{3} + 1632 \beta_{4} + 20679 \beta_{5} - 12753 \beta_{11} ) q^{87} + ( 504 \beta_{7} + 126 \beta_{9} - 872 \beta_{14} ) q^{89} + ( -2468544 + 416 \beta_{13} + 208 \beta_{15} ) q^{91} + ( -198 \beta_{2} - 59296 \beta_{6} - 39006 \beta_{10} - 7722 \beta_{12} ) q^{93} + ( 43688 \beta_{4} + 21844 \beta_{5} - 4412 \beta_{11} ) q^{97} + ( -1437480 \beta_{1} + 108 \beta_{7} + 1224 \beta_{9} + 1485 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 215904q^{21} + 872704q^{31} + 1978560q^{51} + 12752864q^{61} + 10696176q^{81} - 39496704q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 132 x^{14} - 784 x^{13} + 5524236 x^{12} - 33135588 x^{11} - 49457570 x^{10} + 551013008 x^{9} + 10299424335549 x^{8} - 41201367881396 x^{7} - 506171935260830 x^{6} + 1662723272081664 x^{5} + 7501118196131608896 x^{4} - 15005055667263471212 x^{3} - 559914287081811190962 x^{2} + 567417098903075834864 x + 1831773137685647789586721\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-127077663550772 \nu^{14} + 889543644855404 \nu^{13} - 18575791285636901 \nu^{12} + 99890680330701154 \nu^{11} - 729761235766306475869 \nu^{10} + 3647911715052036672562 \nu^{9} - 9117910351391012676864 \nu^{8} + 14585233568523692863986 \nu^{7} - 1492651709261143673520544469 \nu^{6} + 4477919399634508344917218354 \nu^{5} + 35031584283286251950704707794 \nu^{4} - 77526357845150826006522794058 \nu^{3} - 1611312411360702025050329347065403 \nu^{2} + 1611351920858187960485940331725082 \nu + 51696753313793325928362747543089343\)\()/ \)\(73\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(22\!\cdots\!41\)\( \nu^{14} - \)\(15\!\cdots\!87\)\( \nu^{13} + \)\(46\!\cdots\!92\)\( \nu^{12} - \)\(27\!\cdots\!21\)\( \nu^{11} + \)\(10\!\cdots\!59\)\( \nu^{10} - \)\(50\!\cdots\!76\)\( \nu^{9} + \)\(18\!\cdots\!87\)\( \nu^{8} - \)\(72\!\cdots\!97\)\( \nu^{7} + \)\(10\!\cdots\!81\)\( \nu^{6} - \)\(30\!\cdots\!94\)\( \nu^{5} + \)\(24\!\cdots\!23\)\( \nu^{4} - \)\(47\!\cdots\!31\)\( \nu^{3} + \)\(24\!\cdots\!06\)\( \nu^{2} - \)\(24\!\cdots\!83\)\( \nu + \)\(95\!\cdots\!97\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(19\!\cdots\!95\)\( \nu^{14} + \)\(13\!\cdots\!65\)\( \nu^{13} + \)\(77\!\cdots\!55\)\( \nu^{12} - \)\(48\!\cdots\!75\)\( \nu^{11} - \)\(87\!\cdots\!31\)\( \nu^{10} + \)\(44\!\cdots\!75\)\( \nu^{9} + \)\(47\!\cdots\!60\)\( \nu^{8} - \)\(19\!\cdots\!85\)\( \nu^{7} - \)\(12\!\cdots\!35\)\( \nu^{6} + \)\(38\!\cdots\!61\)\( \nu^{5} + \)\(55\!\cdots\!60\)\( \nu^{4} - \)\(11\!\cdots\!25\)\( \nu^{3} - \)\(54\!\cdots\!10\)\( \nu^{2} + \)\(54\!\cdots\!80\)\( \nu + \)\(18\!\cdots\!67\)\(\)\()/ \)\(48\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(36\!\cdots\!52\)\( \nu^{15} - \)\(94\!\cdots\!10\)\( \nu^{14} + \)\(21\!\cdots\!40\)\( \nu^{13} - \)\(11\!\cdots\!75\)\( \nu^{12} - \)\(14\!\cdots\!58\)\( \nu^{11} - \)\(49\!\cdots\!13\)\( \nu^{10} + \)\(89\!\cdots\!90\)\( \nu^{9} + \)\(25\!\cdots\!70\)\( \nu^{8} - \)\(32\!\cdots\!90\)\( \nu^{7} - \)\(76\!\cdots\!57\)\( \nu^{6} + \)\(10\!\cdots\!94\)\( \nu^{5} + \)\(80\!\cdots\!40\)\( \nu^{4} - \)\(19\!\cdots\!70\)\( \nu^{3} - \)\(32\!\cdots\!25\)\( \nu^{2} + \)\(32\!\cdots\!66\)\( \nu + \)\(43\!\cdots\!15\)\(\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(36\!\cdots\!52\)\( \nu^{15} + \)\(51\!\cdots\!40\)\( \nu^{14} + \)\(11\!\cdots\!90\)\( \nu^{13} - \)\(85\!\cdots\!25\)\( \nu^{12} - \)\(14\!\cdots\!08\)\( \nu^{11} + \)\(25\!\cdots\!17\)\( \nu^{10} + \)\(51\!\cdots\!40\)\( \nu^{9} - \)\(56\!\cdots\!30\)\( \nu^{8} - \)\(29\!\cdots\!40\)\( \nu^{7} + \)\(39\!\cdots\!93\)\( \nu^{6} + \)\(66\!\cdots\!64\)\( \nu^{5} - \)\(69\!\cdots\!60\)\( \nu^{4} - \)\(16\!\cdots\!20\)\( \nu^{3} + \)\(16\!\cdots\!75\)\( \nu^{2} + \)\(27\!\cdots\!66\)\( \nu - \)\(23\!\cdots\!95\)\(\)\()/ \)\(36\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(23\!\cdots\!32\)\( \nu^{15} + \)\(36\!\cdots\!63\)\( \nu^{14} + \)\(68\!\cdots\!19\)\( \nu^{13} + \)\(10\!\cdots\!16\)\( \nu^{12} + \)\(97\!\cdots\!25\)\( \nu^{11} + \)\(56\!\cdots\!85\)\( \nu^{10} + \)\(98\!\cdots\!12\)\( \nu^{9} + \)\(56\!\cdots\!01\)\( \nu^{8} + \)\(11\!\cdots\!09\)\( \nu^{7} + \)\(92\!\cdots\!55\)\( \nu^{6} + \)\(21\!\cdots\!74\)\( \nu^{5} + \)\(91\!\cdots\!89\)\( \nu^{4} + \)\(33\!\cdots\!07\)\( \nu^{3} + \)\(17\!\cdots\!98\)\( \nu^{2} + \)\(12\!\cdots\!75\)\( \nu + \)\(38\!\cdots\!71\)\(\)\()/ \)\(23\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(21\!\cdots\!94\)\( \nu^{15} + \)\(88\!\cdots\!83\)\( \nu^{14} - \)\(61\!\cdots\!87\)\( \nu^{13} - \)\(39\!\cdots\!02\)\( \nu^{12} - \)\(34\!\cdots\!95\)\( \nu^{11} + \)\(30\!\cdots\!35\)\( \nu^{10} - \)\(23\!\cdots\!58\)\( \nu^{9} - \)\(29\!\cdots\!03\)\( \nu^{8} - \)\(77\!\cdots\!83\)\( \nu^{7} + \)\(13\!\cdots\!25\)\( \nu^{6} - \)\(20\!\cdots\!68\)\( \nu^{5} - \)\(39\!\cdots\!51\)\( \nu^{4} - \)\(59\!\cdots\!11\)\( \nu^{3} - \)\(18\!\cdots\!56\)\( \nu^{2} - \)\(16\!\cdots\!65\)\( \nu - \)\(11\!\cdots\!67\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(40\!\cdots\!14\)\( \nu^{15} - \)\(30\!\cdots\!55\)\( \nu^{14} + \)\(68\!\cdots\!91\)\( \nu^{13} - \)\(43\!\cdots\!74\)\( \nu^{12} - \)\(85\!\cdots\!21\)\( \nu^{11} + \)\(47\!\cdots\!01\)\( \nu^{10} - \)\(21\!\cdots\!30\)\( \nu^{9} + \)\(94\!\cdots\!39\)\( \nu^{8} - \)\(43\!\cdots\!61\)\( \nu^{7} + \)\(15\!\cdots\!31\)\( \nu^{6} - \)\(12\!\cdots\!48\)\( \nu^{5} + \)\(30\!\cdots\!95\)\( \nu^{4} - \)\(51\!\cdots\!37\)\( \nu^{3} + \)\(77\!\cdots\!28\)\( \nu^{2} - \)\(25\!\cdots\!03\)\( \nu + \)\(12\!\cdots\!15\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(21\!\cdots\!94\)\( \nu^{15} - \)\(17\!\cdots\!01\)\( \nu^{14} - \)\(42\!\cdots\!99\)\( \nu^{13} + \)\(18\!\cdots\!26\)\( \nu^{12} - \)\(48\!\cdots\!07\)\( \nu^{11} - \)\(54\!\cdots\!33\)\( \nu^{10} - \)\(19\!\cdots\!94\)\( \nu^{9} + \)\(90\!\cdots\!89\)\( \nu^{8} - \)\(12\!\cdots\!91\)\( \nu^{7} - \)\(16\!\cdots\!43\)\( \nu^{6} - \)\(19\!\cdots\!80\)\( \nu^{5} + \)\(93\!\cdots\!17\)\( \nu^{4} - \)\(85\!\cdots\!87\)\( \nu^{3} + \)\(39\!\cdots\!28\)\( \nu^{2} - \)\(21\!\cdots\!61\)\( \nu + \)\(22\!\cdots\!29\)\(\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(47\!\cdots\!64\)\( \nu^{15} + \)\(56\!\cdots\!83\)\( \nu^{14} - \)\(18\!\cdots\!01\)\( \nu^{13} + \)\(11\!\cdots\!76\)\( \nu^{12} - \)\(21\!\cdots\!79\)\( \nu^{11} + \)\(22\!\cdots\!21\)\( \nu^{10} - \)\(36\!\cdots\!48\)\( \nu^{9} + \)\(58\!\cdots\!61\)\( \nu^{8} - \)\(29\!\cdots\!71\)\( \nu^{7} + \)\(23\!\cdots\!59\)\( \nu^{6} - \)\(55\!\cdots\!54\)\( \nu^{5} + \)\(93\!\cdots\!49\)\( \nu^{4} - \)\(12\!\cdots\!33\)\( \nu^{3} + \)\(40\!\cdots\!98\)\( \nu^{2} - \)\(25\!\cdots\!17\)\( \nu + \)\(38\!\cdots\!11\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(14\!\cdots\!52\)\( \nu^{15} + \)\(11\!\cdots\!90\)\( \nu^{14} + \)\(15\!\cdots\!40\)\( \nu^{13} - \)\(10\!\cdots\!25\)\( \nu^{12} - \)\(72\!\cdots\!58\)\( \nu^{11} + \)\(40\!\cdots\!57\)\( \nu^{10} + \)\(87\!\cdots\!90\)\( \nu^{9} - \)\(39\!\cdots\!30\)\( \nu^{8} - \)\(11\!\cdots\!90\)\( \nu^{7} + \)\(40\!\cdots\!93\)\( \nu^{6} + \)\(13\!\cdots\!74\)\( \nu^{5} - \)\(33\!\cdots\!60\)\( \nu^{4} - \)\(51\!\cdots\!70\)\( \nu^{3} + \)\(77\!\cdots\!25\)\( \nu^{2} + \)\(56\!\cdots\!66\)\( \nu - \)\(28\!\cdots\!75\)\(\)\()/ \)\(92\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(22\!\cdots\!52\)\( \nu^{15} - \)\(17\!\cdots\!90\)\( \nu^{14} + \)\(13\!\cdots\!40\)\( \nu^{13} - \)\(87\!\cdots\!95\)\( \nu^{12} + \)\(11\!\cdots\!98\)\( \nu^{11} - \)\(62\!\cdots\!57\)\( \nu^{10} + \)\(54\!\cdots\!70\)\( \nu^{9} - \)\(24\!\cdots\!70\)\( \nu^{8} + \)\(16\!\cdots\!10\)\( \nu^{7} - \)\(58\!\cdots\!73\)\( \nu^{6} + \)\(82\!\cdots\!86\)\( \nu^{5} - \)\(20\!\cdots\!20\)\( \nu^{4} + \)\(67\!\cdots\!30\)\( \nu^{3} - \)\(10\!\cdots\!25\)\( \nu^{2} + \)\(32\!\cdots\!54\)\( \nu - \)\(16\!\cdots\!05\)\(\)\()/ \)\(92\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(73\!\cdots\!10\)\( \nu^{15} + \)\(10\!\cdots\!05\)\( \nu^{14} + \)\(20\!\cdots\!19\)\( \nu^{13} + \)\(29\!\cdots\!74\)\( \nu^{12} + \)\(24\!\cdots\!51\)\( \nu^{11} + \)\(56\!\cdots\!73\)\( \nu^{10} + \)\(12\!\cdots\!90\)\( \nu^{9} + \)\(11\!\cdots\!91\)\( \nu^{8} + \)\(99\!\cdots\!51\)\( \nu^{7} + \)\(65\!\cdots\!99\)\( \nu^{6} - \)\(46\!\cdots\!08\)\( \nu^{5} + \)\(10\!\cdots\!15\)\( \nu^{4} - \)\(10\!\cdots\!13\)\( \nu^{3} + \)\(13\!\cdots\!32\)\( \nu^{2} - \)\(73\!\cdots\!87\)\( \nu + \)\(38\!\cdots\!99\)\(\)\()/ \)\(24\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(54\!\cdots\!50\)\( \nu^{15} + \)\(41\!\cdots\!25\)\( \nu^{14} - \)\(88\!\cdots\!49\)\( \nu^{13} + \)\(56\!\cdots\!06\)\( \nu^{12} - \)\(35\!\cdots\!21\)\( \nu^{11} + \)\(19\!\cdots\!37\)\( \nu^{10} - \)\(31\!\cdots\!70\)\( \nu^{9} + \)\(13\!\cdots\!99\)\( \nu^{8} - \)\(71\!\cdots\!01\)\( \nu^{7} + \)\(24\!\cdots\!31\)\( \nu^{6} - \)\(18\!\cdots\!12\)\( \nu^{5} + \)\(45\!\cdots\!35\)\( \nu^{4} - \)\(58\!\cdots\!17\)\( \nu^{3} + \)\(88\!\cdots\!28\)\( \nu^{2} + \)\(23\!\cdots\!57\)\( \nu - \)\(11\!\cdots\!49\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(73\!\cdots\!10\)\( \nu^{15} + \)\(36\!\cdots\!35\)\( \nu^{14} - \)\(53\!\cdots\!99\)\( \nu^{13} + \)\(65\!\cdots\!26\)\( \nu^{12} - \)\(30\!\cdots\!11\)\( \nu^{11} + \)\(15\!\cdots\!67\)\( \nu^{10} - \)\(11\!\cdots\!30\)\( \nu^{9} + \)\(24\!\cdots\!69\)\( \nu^{8} - \)\(24\!\cdots\!71\)\( \nu^{7} + \)\(14\!\cdots\!01\)\( \nu^{6} - \)\(16\!\cdots\!32\)\( \nu^{5} + \)\(21\!\cdots\!85\)\( \nu^{4} + \)\(39\!\cdots\!73\)\( \nu^{3} + \)\(23\!\cdots\!28\)\( \nu^{2} + \)\(69\!\cdots\!67\)\( \nu - \)\(30\!\cdots\!99\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-4 \beta_{15} + 12 \beta_{14} + 16 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + 36 \beta_{10} - 4 \beta_{9} - 12 \beta_{8} - 16 \beta_{7} - 72 \beta_{6} - 36 \beta_{5} - 72 \beta_{4} + 1440\)\()/2880\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{15} + 12 \beta_{14} + 16 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + 36 \beta_{10} - 124 \beta_{9} - 12 \beta_{8} + 224 \beta_{7} - 72 \beta_{6} - 348 \beta_{5} + 1176 \beta_{4} + 24 \beta_{3} - 3247200 \beta_{1} - 36000\)\()/2880\)
\(\nu^{3}\)\(=\)\((\)\(-3275 \beta_{15} - 6675 \beta_{14} + 13100 \beta_{13} - 6561 \beta_{12} + 6669 \beta_{11} + 72144 \beta_{10} + 3335 \beta_{9} - 6225 \beta_{8} + 13880 \beta_{7} - 144288 \beta_{6} + 72342 \beta_{5} + 146088 \beta_{4} + 18 \beta_{3} - 2435400 \beta_{1} - 27360\)\()/1440\)
\(\nu^{4}\)\(=\)\((\)\(-283696 \beta_{15} - 26712 \beta_{14} - 488816 \beta_{13} - 26235 \beta_{12} + 26685 \beta_{11} - 886740 \beta_{10} + 22704 \beta_{9} - 24888 \beta_{8} + 36816 \beta_{7} - 5278200 \beta_{6} + 297828 \beta_{5} + 550728 \beta_{4} - 576 \beta_{3} + 60240 \beta_{2} + 243540000 \beta_{1} - 3972006720\)\()/2880\)
\(\nu^{5}\)\(=\)\((\)\(10613830 \beta_{15} - 13509990 \beta_{14} - 46514320 \beta_{13} - 2093229 \beta_{12} - 3151629 \beta_{11} - 343252284 \beta_{10} + 9638130 \beta_{9} + 16734990 \beta_{8} + 38415720 \beta_{7} + 668875368 \beta_{6} + 328755984 \beta_{5} + 657394968 \beta_{4} - 1500 \beta_{3} + 150600 \beta_{2} + 616968000 \beta_{1} - 9929925360\)\()/2880\)
\(\nu^{6}\)\(=\)\((\)\(21160307 \beta_{15} - 10115796 \beta_{14} - 8534978 \beta_{13} - 1553526 \beta_{12} - 2380401 \beta_{11} - 237786696 \beta_{10} + 124793107 \beta_{9} + 12566796 \beta_{8} - 206368622 \beta_{7} + 581348592 \beta_{6} - 1019694708 \beta_{5} + 5557004208 \beta_{4} - 30000492 \beta_{3} - 903600 \beta_{2} + 1293418615500 \beta_{1} + 186206125500\)\()/720\)
\(\nu^{7}\)\(=\)\((\)\(13185881800 \beta_{15} + 23022350100 \beta_{14} - 51663833200 \beta_{13} - 43975205937 \beta_{12} + 44415598383 \beta_{11} - 634831558152 \beta_{10} - 16848910260 \beta_{9} + 15078770400 \beta_{8} - 77272735680 \beta_{7} + 1271205671304 \beta_{6} - 709014588120 \beta_{5} - 1311679252728 \beta_{4} - 420001596 \beta_{3} - 13177500 \beta_{2} + 18105695546400 \beta_{1} + 2641640431680\)\()/2880\)
\(\nu^{8}\)\(=\)\((\)\(265210410416 \beta_{15} + 30759388752 \beta_{14} + 426689854336 \beta_{13} - 58623961935 \beta_{12} + 59235629985 \beta_{11} + 3318089103500 \beta_{10} - 79608829184 \beta_{9} + 20026814448 \beta_{8} + 10988071264 \beta_{7} + 18343528745960 \beta_{6} - 734190478508 \beta_{5} - 2602751384728 \beta_{4} + 4480385696 \beta_{3} - 73527153280 \beta_{2} - 604366103880000 \beta_{1} + 2340591862918560\)\()/960\)
\(\nu^{9}\)\(=\)\((\)\(-13499174754773 \beta_{15} + 8315363193369 \beta_{14} + 64034289011192 \beta_{13} + 64482795692073 \beta_{12} + 58120206802353 \beta_{11} + 686674299316968 \beta_{10} - 8445602785883 \beta_{9} - 16915672608369 \beta_{8} - 31529172278972 \beta_{7} - 1204740241387776 \beta_{6} - 561727386202338 \beta_{5} - 1131430822810776 \beta_{4} + 31502605050 \beta_{3} - 496268435880 \beta_{2} - 4133786987325600 \beta_{1} + 15791049300616920\)\()/1440\)
\(\nu^{10}\)\(=\)\((\)\(-571923207396520 \beta_{15} + 82461262578060 \beta_{14} - 231187507188920 \beta_{13} + 646146952696701 \beta_{12} + 579869199564201 \beta_{11} + 4356221165607996 \beta_{10} - 1201859761862920 \beta_{9} - 169606977414060 \beta_{8} + 1922852120609720 \beta_{7} - 22203548183067192 \beta_{6} + 19756521055186044 \beta_{5} - 112684807466123352 \beta_{4} + 388753003436520 \beta_{3} + 39722295621600 \beta_{2} - 9741819141537714000 \beta_{1} - 3970963705702230000\)\()/2880\)
\(\nu^{11}\)\(=\)\((\)\(-19967617936705750 \beta_{15} - 51896876595944250 \beta_{14} + 65832619332655000 \beta_{13} + 229813019230996863 \beta_{12} - 283761716334839037 \beta_{11} + 1869923179167693048 \beta_{10} + 43669120605733450 \beta_{9} - 16848646450565250 \beta_{8} + 211651665821557600 \beta_{7} - 3817138576574721096 \beta_{6} + 2560243232946401076 \beta_{5} + 4283545690663316808 \beta_{4} + 2137559351140752 \beta_{3} + 227570733967500 \beta_{2} - 53504020027852218000 \beta_{1} - 22129773784599949920\)\()/2880\)
\(\nu^{12}\)\(=\)\((\)\(-436052253977733442 \beta_{15} - 78071702496132924 \beta_{14} - 715963819693841732 \beta_{13} + 342941899289119785 \beta_{12} - 427236481707660135 \beta_{11} - 8883606091718936340 \beta_{10} + 309086020611578808 \beta_{9} - 24806302932702576 \beta_{8} - 168366630606639768 \beta_{7} - 52370630394383293920 \beta_{6} + 1970738236706241636 \beta_{5} + 13996317891782363616 \beta_{4} - 25696339546374252 \beta_{3} + 166456745697291300 \beta_{2} + 2039684247271435050000 \beta_{1} - 3401029687728582627480\)\()/720\)
\(\nu^{13}\)\(=\)\((\)\(71032215297707471968 \beta_{15} - 12920582727114629904 \beta_{14} - 347910384421000919872 \beta_{13} - 557273473142073257709 \beta_{12} - 396826793995780425429 \beta_{11} - 4649163381743910575484 \beta_{10} + 20678891870658875208 \beta_{9} + 81615804539392079904 \beta_{8} + 45712519367006413392 \beta_{7} + 7475755692236163558648 \beta_{6} + 2954133924287738551284 \beta_{5} + 6180573570994182178968 \beta_{4} - 695891580986386200 \beta_{3} + 4324893313670741760 \beta_{2} + 53727144905852578274400 \beta_{1} - 88138332223415892660960\)\()/2880\)
\(\nu^{14}\)\(=\)\((\)\(2495849132528586359252 \beta_{15} - 85704984282846216756 \beta_{14} + 1552432499469863912992 \beta_{13} - 3921697881540511298469 \beta_{12} - 2751849209151570572469 \beta_{11} - 13515418227810508978764 \beta_{10} + 2430635780910372658132 \beta_{9} + 572809883976797399556 \beta_{8} - 4279083648103017705152 \beta_{7} + 129465264766380728789688 \beta_{6} - 60645105594449251670220 \beta_{5} + 367231722933355847769624 \beta_{4} - 1114845342901192687512 \beta_{3} - 243284772870815828880 \beta_{2} + 18873501650667020810296800 \beta_{1} + 16013957804460283708368000\)\()/2880\)
\(\nu^{15}\)\(=\)\((\)\(7755245763692026128875 \beta_{15} + 64915893156064773679875 \beta_{14} + 12796143577264372922500 \beta_{13} - 314446689066447580961061 \beta_{12} + 515663113930844925111489 \beta_{11} - 2215684596413578582184256 \beta_{10} - 57851971746334811002275 \beta_{9} - 369606243832889424375 \beta_{8} - 283589369792517344539200 \beta_{7} + 4831444017375664678228512 \beta_{6} - 4138252727877283095479214 \beta_{5} - 6446949917308027131678408 \beta_{4} - 4174532358869708680410 \beta_{3} - 950155478388755730000 \beta_{2} + 70304301950170865717151000 \beta_{1} + 60823046839845090456930240\)\()/1440\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
27.4835 + 30.5542i
−19.4848 16.4140i
20.4848 + 16.4140i
−26.4835 23.4128i
−26.4835 30.5542i
20.4848 + 23.5555i
−19.4848 23.5555i
27.4835 + 23.4128i
27.4835 30.5542i
−19.4848 + 16.4140i
20.4848 16.4140i
−26.4835 + 23.4128i
−26.4835 + 30.5542i
20.4848 23.5555i
−19.4848 + 23.5555i
27.4835 23.4128i
0 −46.0669 + 8.05216i 0 0 0 −565.187 565.187i 0 2057.33 741.876i 0
257.2 0 −40.5650 23.2698i 0 0 0 208.115 + 208.115i 0 1104.03 + 1887.88i 0
257.3 0 −23.2698 40.5650i 0 0 0 −208.115 208.115i 0 −1104.03 + 1887.88i 0
257.4 0 −8.05216 + 46.0669i 0 0 0 −565.187 565.187i 0 −2057.33 741.876i 0
257.5 0 8.05216 46.0669i 0 0 0 565.187 + 565.187i 0 −2057.33 741.876i 0
257.6 0 23.2698 + 40.5650i 0 0 0 208.115 + 208.115i 0 −1104.03 + 1887.88i 0
257.7 0 40.5650 + 23.2698i 0 0 0 −208.115 208.115i 0 1104.03 + 1887.88i 0
257.8 0 46.0669 8.05216i 0 0 0 565.187 + 565.187i 0 2057.33 741.876i 0
293.1 0 −46.0669 8.05216i 0 0 0 −565.187 + 565.187i 0 2057.33 + 741.876i 0
293.2 0 −40.5650 + 23.2698i 0 0 0 208.115 208.115i 0 1104.03 1887.88i 0
293.3 0 −23.2698 + 40.5650i 0 0 0 −208.115 + 208.115i 0 −1104.03 1887.88i 0
293.4 0 −8.05216 46.0669i 0 0 0 −565.187 + 565.187i 0 −2057.33 + 741.876i 0
293.5 0 8.05216 + 46.0669i 0 0 0 565.187 565.187i 0 −2057.33 + 741.876i 0
293.6 0 23.2698 40.5650i 0 0 0 208.115 208.115i 0 −1104.03 1887.88i 0
293.7 0 40.5650 23.2698i 0 0 0 −208.115 + 208.115i 0 1104.03 1887.88i 0
293.8 0 46.0669 + 8.05216i 0 0 0 565.187 565.187i 0 2057.33 + 741.876i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.8.i.c 16
3.b odd 2 1 inner 300.8.i.c 16
5.b even 2 1 inner 300.8.i.c 16
5.c odd 4 2 inner 300.8.i.c 16
15.d odd 2 1 inner 300.8.i.c 16
15.e even 4 2 inner 300.8.i.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.8.i.c 16 1.a even 1 1 trivial
300.8.i.c 16 3.b odd 2 1 inner
300.8.i.c 16 5.b even 2 1 inner
300.8.i.c 16 5.c odd 4 2 inner
300.8.i.c 16 15.d odd 2 1 inner
300.8.i.c 16 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 415661203008 T_{7}^{4} + \)\(30\!\cdots\!16\)\( \) acting on \(S_{8}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( \)\(52\!\cdots\!21\)\( - 61173549603433732284 T^{4} + 11211740220006 T^{8} - 2674044 T^{12} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( \)\(30\!\cdots\!16\)\( + 415661203008 T^{4} + T^{8} )^{2} \)
$11$ \( ( 273576376627200 + 54637920 T^{2} + T^{4} )^{4} \)
$13$ \( ( \)\(43\!\cdots\!76\)\( + 38009942833102848 T^{4} + T^{8} )^{2} \)
$17$ \( ( \)\(41\!\cdots\!00\)\( + 1475216350887744000 T^{4} + T^{8} )^{2} \)
$19$ \( ( 891908011738521856 + 2094567968 T^{2} + T^{4} )^{4} \)
$23$ \( ( \)\(32\!\cdots\!00\)\( + 1558411553255220000 T^{4} + T^{8} )^{2} \)
$29$ \( ( \)\(44\!\cdots\!00\)\( - 44758861920 T^{2} + T^{4} )^{4} \)
$31$ \( ( -27149305664 - 109088 T + T^{2} )^{8} \)
$37$ \( ( \)\(71\!\cdots\!56\)\( + \)\(26\!\cdots\!68\)\( T^{4} + T^{8} )^{2} \)
$41$ \( ( \)\(23\!\cdots\!00\)\( + 980340950880 T^{2} + T^{4} )^{4} \)
$43$ \( ( \)\(27\!\cdots\!16\)\( + \)\(15\!\cdots\!08\)\( T^{4} + T^{8} )^{2} \)
$47$ \( ( \)\(12\!\cdots\!00\)\( + \)\(91\!\cdots\!00\)\( T^{4} + T^{8} )^{2} \)
$53$ \( ( \)\(68\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T^{4} + T^{8} )^{2} \)
$59$ \( ( \)\(14\!\cdots\!00\)\( - 2402493590880 T^{2} + T^{4} )^{4} \)
$61$ \( ( -783848281484 - 1594108 T + T^{2} )^{8} \)
$67$ \( ( \)\(22\!\cdots\!56\)\( + \)\(87\!\cdots\!68\)\( T^{4} + T^{8} )^{2} \)
$71$ \( ( \)\(41\!\cdots\!00\)\( + 6188564085120 T^{2} + T^{4} )^{4} \)
$73$ \( ( \)\(91\!\cdots\!96\)\( + \)\(66\!\cdots\!28\)\( T^{4} + T^{8} )^{2} \)
$79$ \( ( \)\(18\!\cdots\!16\)\( + 9519780699392 T^{2} + T^{4} )^{4} \)
$83$ \( ( \)\(16\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T^{4} + T^{8} )^{2} \)
$89$ \( ( \)\(36\!\cdots\!00\)\( - 40870985973120 T^{2} + T^{4} )^{4} \)
$97$ \( ( \)\(18\!\cdots\!56\)\( + \)\(91\!\cdots\!68\)\( T^{4} + T^{8} )^{2} \)
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