Properties

Label 252.8.k.e
Level $252$
Weight $8$
Character orbit 252.k
Analytic conductor $78.721$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} - 993183560713460 x^{9} + 4531523101103784409 x^{8} - 25038421018919762700 x^{7} + 77467785793463920028824 x^{6} - 352418565495722251364640 x^{5} + 967025470523891143640719680 x^{4} - 1651641278575894088053178880 x^{3} + 7200793191683961412674394484736 x^{2} + 21939035854717669821876654243840 x + 34494438285389383231614325978300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{13}\cdot 7^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{4} ) q^{5} + ( 35 + 140 \beta_{1} - 3 \beta_{3} + 2 \beta_{9} ) q^{7} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{4} ) q^{5} + ( 35 + 140 \beta_{1} - 3 \beta_{3} + 2 \beta_{9} ) q^{7} -\beta_{10} q^{11} + ( -1765 - 2 \beta_{3} + 2 \beta_{5} + 5 \beta_{7} + 5 \beta_{8} + 7 \beta_{9} ) q^{13} -\beta_{12} q^{17} + ( 5289 - 5267 \beta_{1} - 10 \beta_{3} - 39 \beta_{5} + 22 \beta_{6} + 22 \beta_{7} + 29 \beta_{8} - 29 \beta_{9} ) q^{19} + ( -14 \beta_{2} - 14 \beta_{4} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( -10054 \beta_{1} - 65 \beta_{3} - 33 \beta_{5} + 7 \beta_{6} + 65 \beta_{8} + 33 \beta_{9} ) q^{25} + ( -21 \beta_{4} - 2 \beta_{11} + 5 \beta_{13} - 2 \beta_{15} ) q^{29} + ( 20582 \beta_{1} + 78 \beta_{3} + 5 \beta_{5} - 24 \beta_{6} - 78 \beta_{8} - 5 \beta_{9} ) q^{31} + ( 10 \beta_{2} + 8 \beta_{4} - 19 \beta_{10} - 11 \beta_{11} - 11 \beta_{12} - 6 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{35} + ( -81085 + 80910 \beta_{1} + 113 \beta_{3} + 335 \beta_{5} - 175 \beta_{6} - 175 \beta_{7} - 222 \beta_{8} + 222 \beta_{9} ) q^{37} + ( 100 \beta_{4} + 56 \beta_{11} - 10 \beta_{13} ) q^{41} + ( 83805 + 475 \beta_{3} - 475 \beta_{5} - 70 \beta_{7} - 32 \beta_{8} - 507 \beta_{9} ) q^{43} + ( 178 \beta_{2} + 178 \beta_{4} - 77 \beta_{10} - 77 \beta_{11} + 23 \beta_{12} + 23 \beta_{13} + 7 \beta_{14} + 7 \beta_{15} ) q^{47} + ( 71638 - 115003 \beta_{1} - 854 \beta_{3} - 882 \beta_{5} + 98 \beta_{6} - 539 \beta_{7} + 931 \beta_{8} + 63 \beta_{9} ) q^{49} + ( 441 \beta_{2} + 176 \beta_{10} + 18 \beta_{12} + 4 \beta_{14} ) q^{53} + ( -19761 + 1638 \beta_{3} - 1638 \beta_{5} + 868 \beta_{7} + 933 \beta_{8} - 705 \beta_{9} ) q^{55} + ( -626 \beta_{2} + 84 \beta_{10} + 75 \beta_{12} - 7 \beta_{14} ) q^{59} + ( -539378 + 540456 \beta_{1} + 390 \beta_{3} - 786 \beta_{5} + 1078 \beta_{6} + 1078 \beta_{7} + 1176 \beta_{8} - 1176 \beta_{9} ) q^{61} + ( -1988 \beta_{2} - 1988 \beta_{4} + 252 \beta_{10} + 252 \beta_{11} - 65 \beta_{12} - 65 \beta_{13} + 12 \beta_{14} + 12 \beta_{15} ) q^{65} + ( -488815 \beta_{1} - 5895 \beta_{3} - 5298 \beta_{5} - 1190 \beta_{6} + 5895 \beta_{8} + 5298 \beta_{9} ) q^{67} + ( 882 \beta_{4} - 293 \beta_{11} + 75 \beta_{13} + 19 \beta_{15} ) q^{71} + ( 809240 \beta_{1} - 2145 \beta_{3} - 4389 \beta_{5} + 535 \beta_{6} + 2145 \beta_{8} + 4389 \beta_{9} ) q^{73} + ( 2579 \beta_{2} - 2379 \beta_{4} - 866 \beta_{10} - 494 \beta_{11} - 25 \beta_{12} + \beta_{13} + 6 \beta_{14} - 26 \beta_{15} ) q^{77} + ( -761232 + 762044 \beta_{1} - 1985 \beta_{3} - 8564 \beta_{5} + 812 \beta_{6} + 812 \beta_{7} + 6579 \beta_{8} - 6579 \beta_{9} ) q^{79} + ( 7042 \beta_{4} + 1372 \beta_{11} + 39 \beta_{13} - 7 \beta_{15} ) q^{83} + ( 27132 + 11064 \beta_{3} - 11064 \beta_{5} - 2786 \beta_{7} + 7254 \beta_{8} - 3810 \beta_{9} ) q^{85} + ( 3758 \beta_{2} + 3758 \beta_{4} - 980 \beta_{10} - 980 \beta_{11} - 140 \beta_{12} - 140 \beta_{13} - 84 \beta_{14} - 84 \beta_{15} ) q^{89} + ( 258209 + 1476825 \beta_{1} + 5816 \beta_{3} - 16905 \beta_{5} - 3038 \beta_{6} - 1470 \beta_{7} + 5439 \beta_{8} - 5205 \beta_{9} ) q^{91} + ( 11410 \beta_{2} + 1353 \beta_{10} - 161 \beta_{12} - 49 \beta_{14} ) q^{95} + ( -1697780 + 17080 \beta_{3} - 17080 \beta_{5} - 2905 \beta_{7} + 17595 \beta_{8} + 515 \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 1680q^{7} + O(q^{10}) \) \( 16q + 1680q^{7} - 28280q^{13} + 42224q^{19} - 80460q^{25} + 164752q^{31} - 647980q^{37} + 1341440q^{43} + 230104q^{49} - 323120q^{55} - 4319336q^{61} - 3905760q^{67} + 6471780q^{73} - 6093104q^{79} + 456400q^{85} + 15969856q^{91} - 27141240q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} - 993183560713460 x^{9} + 4531523101103784409 x^{8} - 25038421018919762700 x^{7} + 77467785793463920028824 x^{6} - 352418565495722251364640 x^{5} + 967025470523891143640719680 x^{4} - 1651641278575894088053178880 x^{3} + 7200793191683961412674394484736 x^{2} + 21939035854717669821876654243840 x + 34494438285389383231614325978300416\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(63\!\cdots\!15\)\( \nu^{15} - \)\(65\!\cdots\!00\)\( \nu^{14} + \)\(51\!\cdots\!70\)\( \nu^{13} - \)\(52\!\cdots\!20\)\( \nu^{12} + \)\(30\!\cdots\!25\)\( \nu^{11} - \)\(29\!\cdots\!48\)\( \nu^{10} + \)\(10\!\cdots\!50\)\( \nu^{9} - \)\(94\!\cdots\!00\)\( \nu^{8} + \)\(27\!\cdots\!55\)\( \nu^{7} - \)\(23\!\cdots\!00\)\( \nu^{6} + \)\(44\!\cdots\!60\)\( \nu^{5} - \)\(34\!\cdots\!40\)\( \nu^{4} + \)\(52\!\cdots\!00\)\( \nu^{3} - \)\(46\!\cdots\!80\)\( \nu^{2} + \)\(20\!\cdots\!20\)\( \nu - \)\(15\!\cdots\!72\)\(\)\()/ \)\(18\!\cdots\!28\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(75\!\cdots\!99\)\( \nu^{15} + \)\(16\!\cdots\!32\)\( \nu^{14} - \)\(28\!\cdots\!82\)\( \nu^{13} + \)\(16\!\cdots\!80\)\( \nu^{12} - \)\(88\!\cdots\!37\)\( \nu^{11} + \)\(93\!\cdots\!44\)\( \nu^{10} + \)\(17\!\cdots\!74\)\( \nu^{9} + \)\(32\!\cdots\!52\)\( \nu^{8} + \)\(87\!\cdots\!89\)\( \nu^{7} + \)\(71\!\cdots\!80\)\( \nu^{6} + \)\(27\!\cdots\!68\)\( \nu^{5} + \)\(10\!\cdots\!60\)\( \nu^{4} + \)\(32\!\cdots\!88\)\( \nu^{3} + \)\(97\!\cdots\!24\)\( \nu^{2} + \)\(32\!\cdots\!16\)\( \nu + \)\(51\!\cdots\!36\)\(\)\()/ \)\(12\!\cdots\!80\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(85\!\cdots\!63\)\( \nu^{15} - \)\(17\!\cdots\!60\)\( \nu^{14} - \)\(32\!\cdots\!26\)\( \nu^{13} - \)\(15\!\cdots\!64\)\( \nu^{12} - \)\(21\!\cdots\!93\)\( \nu^{11} - \)\(84\!\cdots\!88\)\( \nu^{10} - \)\(97\!\cdots\!74\)\( \nu^{9} - \)\(27\!\cdots\!04\)\( \nu^{8} - \)\(23\!\cdots\!63\)\( \nu^{7} - \)\(58\!\cdots\!72\)\( \nu^{6} - \)\(42\!\cdots\!56\)\( \nu^{5} - \)\(82\!\cdots\!64\)\( \nu^{4} - \)\(41\!\cdots\!28\)\( \nu^{3} - \)\(72\!\cdots\!08\)\( \nu^{2} - \)\(29\!\cdots\!80\)\( \nu - \)\(33\!\cdots\!64\)\(\)\()/ \)\(12\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(77\!\cdots\!57\)\( \nu^{15} - \)\(15\!\cdots\!52\)\( \nu^{14} - \)\(77\!\cdots\!50\)\( \nu^{13} - \)\(11\!\cdots\!28\)\( \nu^{12} - \)\(43\!\cdots\!59\)\( \nu^{11} - \)\(59\!\cdots\!60\)\( \nu^{10} - \)\(14\!\cdots\!42\)\( \nu^{9} - \)\(16\!\cdots\!80\)\( \nu^{8} - \)\(32\!\cdots\!05\)\( \nu^{7} - \)\(33\!\cdots\!84\)\( \nu^{6} - \)\(49\!\cdots\!00\)\( \nu^{5} - \)\(40\!\cdots\!08\)\( \nu^{4} - \)\(46\!\cdots\!84\)\( \nu^{3} - \)\(34\!\cdots\!80\)\( \nu^{2} - \)\(24\!\cdots\!96\)\( \nu - \)\(11\!\cdots\!84\)\(\)\()/ \)\(68\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(30\!\cdots\!77\)\( \nu^{15} + \)\(21\!\cdots\!16\)\( \nu^{14} + \)\(50\!\cdots\!42\)\( \nu^{13} + \)\(17\!\cdots\!32\)\( \nu^{12} + \)\(30\!\cdots\!23\)\( \nu^{11} + \)\(92\!\cdots\!76\)\( \nu^{10} + \)\(12\!\cdots\!74\)\( \nu^{9} + \)\(27\!\cdots\!08\)\( \nu^{8} + \)\(28\!\cdots\!61\)\( \nu^{7} + \)\(57\!\cdots\!56\)\( \nu^{6} + \)\(48\!\cdots\!72\)\( \nu^{5} + \)\(77\!\cdots\!72\)\( \nu^{4} + \)\(46\!\cdots\!08\)\( \nu^{3} + \)\(67\!\cdots\!76\)\( \nu^{2} + \)\(30\!\cdots\!08\)\( \nu + \)\(29\!\cdots\!20\)\(\)\()/ \)\(12\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(82\!\cdots\!73\)\( \nu^{15} + \)\(26\!\cdots\!04\)\( \nu^{14} - \)\(19\!\cdots\!06\)\( \nu^{13} + \)\(25\!\cdots\!00\)\( \nu^{12} - \)\(81\!\cdots\!43\)\( \nu^{11} + \)\(14\!\cdots\!00\)\( \nu^{10} + \)\(67\!\cdots\!58\)\( \nu^{9} + \)\(47\!\cdots\!16\)\( \nu^{8} + \)\(23\!\cdots\!83\)\( \nu^{7} + \)\(10\!\cdots\!80\)\( \nu^{6} + \)\(60\!\cdots\!64\)\( \nu^{5} + \)\(16\!\cdots\!00\)\( \nu^{4} + \)\(69\!\cdots\!48\)\( \nu^{3} + \)\(15\!\cdots\!36\)\( \nu^{2} + \)\(63\!\cdots\!92\)\( \nu + \)\(77\!\cdots\!20\)\(\)\()/ \)\(30\!\cdots\!72\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(21\!\cdots\!13\)\( \nu^{15} - \)\(28\!\cdots\!60\)\( \nu^{14} - \)\(20\!\cdots\!90\)\( \nu^{13} - \)\(21\!\cdots\!56\)\( \nu^{12} - \)\(11\!\cdots\!51\)\( \nu^{11} - \)\(11\!\cdots\!20\)\( \nu^{10} - \)\(38\!\cdots\!94\)\( \nu^{9} - \)\(31\!\cdots\!60\)\( \nu^{8} - \)\(85\!\cdots\!25\)\( \nu^{7} - \)\(63\!\cdots\!52\)\( \nu^{6} - \)\(13\!\cdots\!80\)\( \nu^{5} - \)\(77\!\cdots\!80\)\( \nu^{4} - \)\(12\!\cdots\!64\)\( \nu^{3} - \)\(66\!\cdots\!40\)\( \nu^{2} - \)\(63\!\cdots\!64\)\( \nu - \)\(16\!\cdots\!44\)\(\)\()/ \)\(24\!\cdots\!16\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(38\!\cdots\!90\)\( \nu^{15} + \)\(18\!\cdots\!03\)\( \nu^{14} - \)\(25\!\cdots\!76\)\( \nu^{13} + \)\(26\!\cdots\!18\)\( \nu^{12} - \)\(12\!\cdots\!42\)\( \nu^{11} + \)\(15\!\cdots\!77\)\( \nu^{10} - \)\(28\!\cdots\!76\)\( \nu^{9} + \)\(60\!\cdots\!66\)\( \nu^{8} - \)\(51\!\cdots\!18\)\( \nu^{7} + \)\(13\!\cdots\!99\)\( \nu^{6} - \)\(41\!\cdots\!96\)\( \nu^{5} + \)\(22\!\cdots\!88\)\( \nu^{4} - \)\(20\!\cdots\!32\)\( \nu^{3} + \)\(20\!\cdots\!12\)\( \nu^{2} + \)\(26\!\cdots\!64\)\( \nu + \)\(12\!\cdots\!32\)\(\)\()/ \)\(38\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(33\!\cdots\!33\)\( \nu^{15} + \)\(25\!\cdots\!44\)\( \nu^{14} + \)\(23\!\cdots\!58\)\( \nu^{13} - \)\(69\!\cdots\!92\)\( \nu^{12} + \)\(11\!\cdots\!47\)\( \nu^{11} - \)\(53\!\cdots\!76\)\( \nu^{10} + \)\(29\!\cdots\!86\)\( \nu^{9} - \)\(28\!\cdots\!08\)\( \nu^{8} + \)\(53\!\cdots\!89\)\( \nu^{7} - \)\(72\!\cdots\!36\)\( \nu^{6} + \)\(52\!\cdots\!68\)\( \nu^{5} - \)\(13\!\cdots\!32\)\( \nu^{4} + \)\(34\!\cdots\!12\)\( \nu^{3} - \)\(12\!\cdots\!76\)\( \nu^{2} - \)\(10\!\cdots\!28\)\( \nu - \)\(82\!\cdots\!00\)\(\)\()/ \)\(30\!\cdots\!20\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(48\!\cdots\!03\)\( \nu^{15} + \)\(11\!\cdots\!36\)\( \nu^{14} - \)\(20\!\cdots\!42\)\( \nu^{13} + \)\(12\!\cdots\!00\)\( \nu^{12} - \)\(71\!\cdots\!05\)\( \nu^{11} + \)\(67\!\cdots\!28\)\( \nu^{10} + \)\(46\!\cdots\!58\)\( \nu^{9} + \)\(23\!\cdots\!88\)\( \nu^{8} + \)\(36\!\cdots\!73\)\( \nu^{7} + \)\(51\!\cdots\!40\)\( \nu^{6} + \)\(13\!\cdots\!12\)\( \nu^{5} + \)\(77\!\cdots\!00\)\( \nu^{4} + \)\(16\!\cdots\!24\)\( \nu^{3} + \)\(70\!\cdots\!80\)\( \nu^{2} + \)\(17\!\cdots\!72\)\( \nu + \)\(36\!\cdots\!52\)\(\)\()/ \)\(72\!\cdots\!20\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(91\!\cdots\!09\)\( \nu^{15} - \)\(21\!\cdots\!44\)\( \nu^{14} - \)\(91\!\cdots\!30\)\( \nu^{13} - \)\(16\!\cdots\!48\)\( \nu^{12} - \)\(50\!\cdots\!15\)\( \nu^{11} - \)\(83\!\cdots\!60\)\( \nu^{10} - \)\(17\!\cdots\!18\)\( \nu^{9} - \)\(23\!\cdots\!80\)\( \nu^{8} - \)\(38\!\cdots\!85\)\( \nu^{7} - \)\(46\!\cdots\!72\)\( \nu^{6} - \)\(57\!\cdots\!60\)\( \nu^{5} - \)\(57\!\cdots\!56\)\( \nu^{4} - \)\(52\!\cdots\!84\)\( \nu^{3} - \)\(47\!\cdots\!40\)\( \nu^{2} - \)\(29\!\cdots\!00\)\( \nu - \)\(14\!\cdots\!40\)\(\)\()/ \)\(80\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!97\)\( \nu^{15} + \)\(20\!\cdots\!28\)\( \nu^{14} - \)\(44\!\cdots\!54\)\( \nu^{13} + \)\(21\!\cdots\!80\)\( \nu^{12} - \)\(16\!\cdots\!07\)\( \nu^{11} + \)\(11\!\cdots\!72\)\( \nu^{10} + \)\(56\!\cdots\!02\)\( \nu^{9} + \)\(41\!\cdots\!80\)\( \nu^{8} + \)\(50\!\cdots\!07\)\( \nu^{7} + \)\(90\!\cdots\!20\)\( \nu^{6} + \)\(22\!\cdots\!20\)\( \nu^{5} + \)\(13\!\cdots\!60\)\( \nu^{4} + \)\(27\!\cdots\!72\)\( \nu^{3} + \)\(12\!\cdots\!84\)\( \nu^{2} + \)\(30\!\cdots\!68\)\( \nu + \)\(64\!\cdots\!88\)\(\)\()/ \)\(36\!\cdots\!60\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(81\!\cdots\!09\)\( \nu^{15} - \)\(19\!\cdots\!24\)\( \nu^{14} - \)\(82\!\cdots\!90\)\( \nu^{13} - \)\(15\!\cdots\!12\)\( \nu^{12} - \)\(45\!\cdots\!59\)\( \nu^{11} - \)\(76\!\cdots\!40\)\( \nu^{10} - \)\(15\!\cdots\!86\)\( \nu^{9} - \)\(21\!\cdots\!20\)\( \nu^{8} - \)\(34\!\cdots\!85\)\( \nu^{7} - \)\(42\!\cdots\!00\)\( \nu^{6} - \)\(52\!\cdots\!80\)\( \nu^{5} - \)\(53\!\cdots\!36\)\( \nu^{4} - \)\(47\!\cdots\!56\)\( \nu^{3} - \)\(43\!\cdots\!00\)\( \nu^{2} - \)\(26\!\cdots\!16\)\( \nu - \)\(14\!\cdots\!24\)\(\)\()/ \)\(20\!\cdots\!80\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(46\!\cdots\!07\)\( \nu^{15} - \)\(64\!\cdots\!88\)\( \nu^{14} + \)\(21\!\cdots\!74\)\( \nu^{13} - \)\(66\!\cdots\!80\)\( \nu^{12} + \)\(85\!\cdots\!97\)\( \nu^{11} - \)\(37\!\cdots\!52\)\( \nu^{10} + \)\(49\!\cdots\!38\)\( \nu^{9} - \)\(12\!\cdots\!00\)\( \nu^{8} - \)\(98\!\cdots\!17\)\( \nu^{7} - \)\(28\!\cdots\!20\)\( \nu^{6} - \)\(76\!\cdots\!40\)\( \nu^{5} - \)\(42\!\cdots\!60\)\( \nu^{4} - \)\(90\!\cdots\!32\)\( \nu^{3} - \)\(38\!\cdots\!24\)\( \nu^{2} - \)\(10\!\cdots\!08\)\( \nu - \)\(20\!\cdots\!08\)\(\)\()/ \)\(72\!\cdots\!20\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(59\!\cdots\!53\)\( \nu^{15} + \)\(12\!\cdots\!28\)\( \nu^{14} + \)\(63\!\cdots\!30\)\( \nu^{13} + \)\(93\!\cdots\!84\)\( \nu^{12} + \)\(35\!\cdots\!03\)\( \nu^{11} + \)\(47\!\cdots\!80\)\( \nu^{10} + \)\(12\!\cdots\!82\)\( \nu^{9} + \)\(13\!\cdots\!40\)\( \nu^{8} + \)\(26\!\cdots\!45\)\( \nu^{7} + \)\(26\!\cdots\!80\)\( \nu^{6} + \)\(40\!\cdots\!60\)\( \nu^{5} + \)\(33\!\cdots\!92\)\( \nu^{4} + \)\(35\!\cdots\!72\)\( \nu^{3} + \)\(27\!\cdots\!00\)\( \nu^{2} + \)\(19\!\cdots\!72\)\( \nu + \)\(92\!\cdots\!88\)\(\)\()/ \)\(80\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{9} - 5 \beta_{8} + \beta_{5} + 63 \beta_{4} - 4 \beta_{3} + 63 \beta_{2}\)\()/126\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{14} - 33 \beta_{12} - 57 \beta_{10} + 1052 \beta_{9} + 2089 \beta_{8} + 210 \beta_{6} - 1052 \beta_{5} - 2089 \beta_{3} - 75 \beta_{2} - 2821224 \beta_{1}\)\()/126\)
\(\nu^{3}\)\(=\)\((\)\(3249 \beta_{15} - 864 \beta_{13} - 16263 \beta_{11} - 37634 \beta_{9} + 619508 \beta_{8} - 63756 \beta_{7} - 657142 \beta_{5} - 3148974 \beta_{4} + 657142 \beta_{3} + 9763992\)\()/252\)
\(\nu^{4}\)\(=\)\((\)\(194508 \beta_{15} + 194508 \beta_{14} + 1807515 \beta_{13} + 1807515 \beta_{12} + 1523958 \beta_{11} + 1523958 \beta_{10} + 44481269 \beta_{9} - 44481269 \beta_{8} - 8008518 \beta_{7} - 8008518 \beta_{6} + 99507457 \beta_{5} + 1419603 \beta_{4} + 55026188 \beta_{3} + 1419603 \beta_{2} + 72481855152 \beta_{1} - 72489863670\)\()/126\)
\(\nu^{5}\)\(=\)\((\)\(163253895 \beta_{14} - 34799100 \beta_{12} - 641970105 \beta_{10} - 28291292084 \beta_{9} - 1671004750 \beta_{8} - 4738521060 \beta_{6} + 28291292084 \beta_{5} + 1671004750 \beta_{3} - 88275468750 \beta_{2} - 226848463380 \beta_{1}\)\()/252\)
\(\nu^{6}\)\(=\)\((\)\(-9066555951 \beta_{15} - 77238171597 \beta_{13} - 25403468157 \beta_{11} - 3847036593817 \beta_{9} - 2263513699340 \beta_{8} + 259376468106 \beta_{7} - 1583522894477 \beta_{5} + 154010063061 \beta_{4} + 1583522894477 \beta_{3} + 2079342683649042\)\()/126\)
\(\nu^{7}\)\(=\)\((\)\(-6559965413589 \beta_{15} - 6559965413589 \beta_{14} + 1037354143224 \beta_{13} + 1037354143224 \beta_{12} + 21290387894643 \beta_{11} + 21290387894643 \beta_{10} + 980378473239862 \beta_{9} - 980378473239862 \beta_{8} + 234143264366172 \beta_{7} + 234143264366172 \beta_{6} - 158006190047134 \beta_{5} + 2697019280692590 \beta_{4} - 1138384663286996 \beta_{3} + 2697019280692590 \beta_{2} - 11070043119630684 \beta_{1} + 11304186383996856\)\()/252\)
\(\nu^{8}\)\(=\)\((\)\(-376064851106226 \beta_{14} - 3022949379396255 \beta_{12} - 91083682319556 \beta_{10} + 86536889975543084 \beta_{9} + 140645163781973809 \beta_{8} + 8454274001180238 \beta_{6} - 86536889975543084 \beta_{5} - 140645163781973809 \beta_{3} + 15309519466391709 \beta_{2} - 64624346034281951232 \beta_{1}\)\()/126\)
\(\nu^{9}\)\(=\)\((\)\(246759249839945283 \beta_{15} - 22201430928268788 \beta_{13} - 694649984643376461 \beta_{11} + 8560301279034334702 \beta_{9} + 43667941343384940212 \beta_{8} - 9906482796013716948 \beta_{7} - 35107640064350605510 \beta_{5} - 87466194110263468878 \beta_{4} + 35107640064350605510 \beta_{3} - 1311745391948870623584\)\()/252\)
\(\nu^{10}\)\(=\)\((\)\(14827134331306300725 \beta_{15} + 14827134331306300725 \beta_{14} + 113598637693014104145 \beta_{13} + 113598637693014104145 \beta_{12} - 16582090983173550165 \beta_{11} - 16582090983173550165 \beta_{10} + 1830307028222558463005 \beta_{9} - 1830307028222558463005 \beta_{8} - 288890412414209948370 \beta_{7} - 288890412414209948370 \beta_{6} + 5055844634204412972745 \beta_{5} - 910780558498118504325 \beta_{4} + 3225537605981854509740 \beta_{3} - 910780558498118504325 \beta_{2} + 2120878281555523942808664 \beta_{1} - 2121167171967938152757034\)\()/126\)
\(\nu^{11}\)\(=\)\((\)\(9069087707878824443985 \beta_{14} - 46980131941007142000 \beta_{12} - 23117280169427570921415 \beta_{10} - 1639185045393697840153172 \beta_{9} - 394192811069079931132222 \beta_{8} - 389477118410518635394380 \beta_{6} + 1639185045393697840153172 \beta_{5} + 394192811069079931132222 \beta_{3} - 2949887920578517956849774 \beta_{2} + 80140072346777186884603260 \beta_{1}\)\()/252\)
\(\nu^{12}\)\(=\)\((\)\(-570344854904722906394712 \beta_{15} - 4179750441340046586172707 \beta_{13} + 1055508452617253616728622 \beta_{11} - 180982553285741181596108641 \beta_{9} - 119270408130012453769394828 \beta_{8} + 10367738460880843114320150 \beta_{7} - 61712145155728727826713813 \beta_{5} + 44856800724328041999794445 \beta_{4} + 61712145155728727826713813 \beta_{3} + 72092590552779166190391323046\)\()/126\)
\(\nu^{13}\)\(=\)\((\)\(-330653118835792624750007871 \beta_{15} - 330653118835792624750007871 \beta_{14} - 31337276809321761224721684 \beta_{13} - 31337276809321761224721684 \beta_{12} + 789249226414454830343716257 \beta_{11} + 789249226414454830343716257 \beta_{10} + 43980269926038923215204989478 \beta_{9} - 43980269926038923215204989478 \beta_{8} + 14738758577922922267835468484 \beta_{7} + 14738758577922922267835468484 \beta_{6} - 16877310414766275399125206414 \beta_{5} + 101999207499498512587927356366 \beta_{4} - 60857580340805198614330195892 \beta_{3} + 101999207499498512587927356366 \beta_{2} - 4009355902253300474451102909228 \beta_{1} + 4024094660831223396718938377712\)\()/252\)
\(\nu^{14}\)\(=\)\((\)\(-21651381551214403546374228027 \beta_{14} - 152025200032722756957028386645 \beta_{12} + 48572773882365353594341701543 \beta_{10} + 4403653259025256386341637790412 \beta_{9} + 6479566073895443218050065765113 \beta_{8} + 386119908152767400137052578842 \beta_{6} - 4403653259025256386341637790412 \beta_{5} - 6479566073895443218050065765113 \beta_{3} + 2007717067393559806442197886133 \beta_{2} - 2504273417306956140820027134950568 \beta_{1}\)\()/126\)
\(\nu^{15}\)\(=\)\((\)\(12029613387009532791986091820365 \beta_{15} + 2479559844207740937408904095000 \beta_{13} - 27552874416468267091471725979035 \beta_{11} + 694680297219135291832406661211870 \beta_{9} + 2245802814289335485625002714126036 \beta_{8} - 546252323953915616654052360606780 \beta_{7} - 1551122517070200193792596052914166 \beta_{5} - 3582671638088609186466052350039150 \beta_{4} + 1551122517070200193792596052914166 \beta_{3} - 181854956667776876195086978795828440\)\()/252\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−77.2060 + 133.725i
−92.9954 + 161.073i
−73.4015 + 127.135i
−50.1630 + 86.8849i
67.3630 116.676i
62.2223 107.772i
68.2530 118.218i
95.9276 166.152i
−77.2060 133.725i
−92.9954 161.073i
−73.4015 127.135i
−50.1630 86.8849i
67.3630 + 116.676i
62.2223 + 107.772i
68.2530 + 118.218i
95.9276 + 166.152i
0 0 0 −173.134 + 299.876i 0 334.186 843.720i 0 0 0
37.2 0 0 0 −161.248 + 279.290i 0 −882.807 + 210.227i 0 0 0
37.3 0 0 0 −135.624 + 234.907i 0 87.0016 + 903.313i 0 0 0
37.4 0 0 0 −117.526 + 203.561i 0 881.619 + 215.154i 0 0 0
37.5 0 0 0 117.526 203.561i 0 881.619 + 215.154i 0 0 0
37.6 0 0 0 135.624 234.907i 0 87.0016 + 903.313i 0 0 0
37.7 0 0 0 161.248 279.290i 0 −882.807 + 210.227i 0 0 0
37.8 0 0 0 173.134 299.876i 0 334.186 843.720i 0 0 0
109.1 0 0 0 −173.134 299.876i 0 334.186 + 843.720i 0 0 0
109.2 0 0 0 −161.248 279.290i 0 −882.807 210.227i 0 0 0
109.3 0 0 0 −135.624 234.907i 0 87.0016 903.313i 0 0 0
109.4 0 0 0 −117.526 203.561i 0 881.619 215.154i 0 0 0
109.5 0 0 0 117.526 + 203.561i 0 881.619 215.154i 0 0 0
109.6 0 0 0 135.624 + 234.907i 0 87.0016 903.313i 0 0 0
109.7 0 0 0 161.248 + 279.290i 0 −882.807 210.227i 0 0 0
109.8 0 0 0 173.134 + 299.876i 0 334.186 + 843.720i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.k.e 16
3.b odd 2 1 inner 252.8.k.e 16
7.c even 3 1 inner 252.8.k.e 16
21.h odd 6 1 inner 252.8.k.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.8.k.e 16 1.a even 1 1 trivial
252.8.k.e 16 3.b odd 2 1 inner
252.8.k.e 16 7.c even 3 1 inner
252.8.k.e 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!50\)\( T_{5}^{10} + \)\(11\!\cdots\!25\)\( T_{5}^{8} + \)\(78\!\cdots\!00\)\( T_{5}^{6} + \)\(40\!\cdots\!00\)\( T_{5}^{4} + \)\(12\!\cdots\!00\)\( T_{5}^{2} + \)\(25\!\cdots\!00\)\( \)">\(T_{5}^{16} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( \)\(25\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{4} + \)\(78\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!25\)\( T^{8} + 10973506102891850 T^{10} + 79038704955 T^{12} + 352730 T^{14} + T^{16} \)
$7$ \( ( \)\(45\!\cdots\!01\)\( - \)\(46\!\cdots\!80\)\( T + 200261639612415626 T^{2} + 450830497404000 T^{3} - 1232165859813 T^{4} + 547428000 T^{5} + 295274 T^{6} - 840 T^{7} + T^{8} )^{2} \)
$11$ \( \)\(12\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{4} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(36\!\cdots\!25\)\( T^{8} + \)\(17\!\cdots\!50\)\( T^{10} + 6701588161920435 T^{12} + 93007250 T^{14} + T^{16} \)
$13$ \( ( -247198114185804 - 527666752620 T - 76586567 T^{2} + 7070 T^{3} + T^{4} )^{4} \)
$17$ \( \)\(93\!\cdots\!00\)\( + \)\(41\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{4} + \)\(20\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!00\)\( T^{8} + \)\(74\!\cdots\!00\)\( T^{10} + 1797407154790106160 T^{12} + 1530777080 T^{14} + T^{16} \)
$19$ \( ( \)\(12\!\cdots\!16\)\( + \)\(12\!\cdots\!72\)\( T + \)\(13\!\cdots\!80\)\( T^{2} - \)\(54\!\cdots\!48\)\( T^{3} + 3147829124233809109 T^{4} - 39307778154232 T^{5} + 2001963805 T^{6} - 21112 T^{7} + T^{8} )^{2} \)
$23$ \( \)\(36\!\cdots\!00\)\( + \)\(41\!\cdots\!00\)\( T^{2} + \)\(45\!\cdots\!00\)\( T^{4} + \)\(25\!\cdots\!00\)\( T^{6} + \)\(98\!\cdots\!00\)\( T^{8} + \)\(18\!\cdots\!00\)\( T^{10} + \)\(26\!\cdots\!00\)\( T^{12} + 19627491800 T^{14} + T^{16} \)
$29$ \( ( \)\(13\!\cdots\!00\)\( - \)\(53\!\cdots\!00\)\( T^{2} + \)\(44\!\cdots\!25\)\( T^{4} - 118720919450 T^{6} + T^{8} )^{2} \)
$31$ \( ( \)\(20\!\cdots\!21\)\( - \)\(98\!\cdots\!16\)\( T + \)\(43\!\cdots\!70\)\( T^{2} - \)\(95\!\cdots\!64\)\( T^{3} + 23464431730137896419 T^{4} - 357403787721344 T^{5} + 7761417570 T^{6} - 82376 T^{7} + T^{8} )^{2} \)
$37$ \( ( \)\(38\!\cdots\!56\)\( + \)\(15\!\cdots\!40\)\( T + \)\(55\!\cdots\!28\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!73\)\( T^{4} + 25474786650872750 T^{5} + 177955551283 T^{6} + 323990 T^{7} + T^{8} )^{2} \)
$41$ \( ( \)\(13\!\cdots\!00\)\( - \)\(51\!\cdots\!00\)\( T^{2} + \)\(73\!\cdots\!40\)\( T^{4} - 448676625600 T^{6} + T^{8} )^{2} \)
$43$ \( ( \)\(50\!\cdots\!16\)\( + 762587418422560 T - 83403429537 T^{2} - 335360 T^{3} + T^{4} )^{4} \)
$47$ \( \)\(10\!\cdots\!00\)\( + \)\(99\!\cdots\!00\)\( T^{2} + \)\(90\!\cdots\!00\)\( T^{4} + \)\(43\!\cdots\!00\)\( T^{6} + \)\(14\!\cdots\!00\)\( T^{8} + \)\(25\!\cdots\!00\)\( T^{10} + \)\(32\!\cdots\!60\)\( T^{12} + 2176158907440 T^{14} + T^{16} \)
$53$ \( \)\(16\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{4} + \)\(50\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!25\)\( T^{8} + \)\(12\!\cdots\!50\)\( T^{10} + \)\(95\!\cdots\!35\)\( T^{12} + 3709369772010 T^{14} + T^{16} \)
$59$ \( \)\(18\!\cdots\!00\)\( + \)\(51\!\cdots\!00\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{4} + \)\(99\!\cdots\!00\)\( T^{6} + \)\(72\!\cdots\!25\)\( T^{8} + \)\(28\!\cdots\!50\)\( T^{10} + \)\(77\!\cdots\!35\)\( T^{12} + 10566106141010 T^{14} + T^{16} \)
$61$ \( ( \)\(80\!\cdots\!96\)\( - \)\(21\!\cdots\!88\)\( T + \)\(76\!\cdots\!80\)\( T^{2} + \)\(67\!\cdots\!52\)\( T^{3} + \)\(78\!\cdots\!44\)\( T^{4} + 847337381041027408 T^{5} + 6530740002780 T^{6} + 2159668 T^{7} + T^{8} )^{2} \)
$67$ \( ( \)\(52\!\cdots\!96\)\( + \)\(10\!\cdots\!80\)\( T + \)\(18\!\cdots\!68\)\( T^{2} + \)\(86\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!33\)\( T^{4} + 62576466833934468000 T^{5} + 20168710544813 T^{6} + 1952880 T^{7} + T^{8} )^{2} \)
$71$ \( ( \)\(91\!\cdots\!00\)\( - \)\(38\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{4} - 22650049642640 T^{6} + T^{8} )^{2} \)
$73$ \( ( \)\(19\!\cdots\!96\)\( - \)\(28\!\cdots\!60\)\( T + \)\(37\!\cdots\!32\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(99\!\cdots\!33\)\( T^{4} - 27595785411171227250 T^{5} + 14791978436687 T^{6} - 3235890 T^{7} + T^{8} )^{2} \)
$79$ \( ( \)\(37\!\cdots\!01\)\( - \)\(69\!\cdots\!52\)\( T + \)\(13\!\cdots\!30\)\( T^{2} - \)\(13\!\cdots\!52\)\( T^{3} + \)\(39\!\cdots\!79\)\( T^{4} - 39904607716315614848 T^{5} + 29847704448330 T^{6} + 3046552 T^{7} + T^{8} )^{2} \)
$83$ \( ( \)\(10\!\cdots\!00\)\( - \)\(20\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!65\)\( T^{4} - 193945394219570 T^{6} + T^{8} )^{2} \)
$89$ \( \)\(84\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{4} + \)\(63\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!00\)\( T^{8} + \)\(40\!\cdots\!00\)\( T^{10} + \)\(44\!\cdots\!60\)\( T^{12} + 251844521642280 T^{14} + T^{16} \)
$97$ \( ( \)\(10\!\cdots\!00\)\( - \)\(71\!\cdots\!00\)\( T - 180110817122075 T^{2} + 6785310 T^{3} + T^{4} )^{4} \)
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