Properties

Label 252.4.b.g
Level $252$
Weight $4$
Character orbit 252.b
Analytic conductor $14.868$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 358 x^{14} - 2828 x^{13} + 52557 x^{12} - 549972 x^{11} + 4434734 x^{10} - 37785264 x^{9} + 272741368 x^{8} - 1739202044 x^{7} + 9778426658 x^{6} - 39463975388 x^{5} + 101978126949 x^{4} - 176540053420 x^{3} + 219245087130 x^{2} - 139977817400 x + 52705588025\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 3 + \beta_{5} ) q^{4} + \beta_{11} q^{5} + ( 1 + \beta_{5} + \beta_{9} ) q^{7} + ( 2 \beta_{2} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 3 + \beta_{5} ) q^{4} + \beta_{11} q^{5} + ( 1 + \beta_{5} + \beta_{9} ) q^{7} + ( 2 \beta_{2} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{8} -\beta_{1} q^{10} + ( 2 \beta_{2} - 2 \beta_{6} + \beta_{7} ) q^{11} -\beta_{14} q^{13} + ( -2 \beta_{2} - 3 \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{10} ) q^{14} + ( -18 + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{9} ) q^{16} + ( \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{17} + ( 2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{9} - \beta_{13} ) q^{19} + ( \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{20} + ( -17 + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{9} ) q^{22} + ( 13 \beta_{2} - 13 \beta_{6} - 7 \beta_{7} ) q^{23} + ( -86 - 13 \beta_{4} + 13 \beta_{5} ) q^{25} + ( 2 \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{15} ) q^{26} + ( -51 + 2 \beta_{3} - 15 \beta_{4} + 9 \beta_{5} + 4 \beta_{9} + \beta_{13} + \beta_{14} ) q^{28} + ( 21 \beta_{2} + 7 \beta_{6} + 2 \beta_{8} ) q^{29} + ( 3 - 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{9} + 3 \beta_{13} ) q^{31} + ( -12 \beta_{2} + 4 \beta_{6} + 18 \beta_{7} - 2 \beta_{8} ) q^{32} + ( 2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{9} - 2 \beta_{13} - 2 \beta_{14} ) q^{34} + ( -\beta_{2} + \beta_{6} - 14 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + \beta_{12} + 2 \beta_{15} ) q^{35} + ( -51 - 11 \beta_{4} + 11 \beta_{5} ) q^{37} + ( -3 \beta_{7} + 2 \beta_{8} - 5 \beta_{10} + 7 \beta_{11} + \beta_{12} - \beta_{15} ) q^{38} + ( 4 - 2 \beta_{1} + 4 \beta_{3} + 4 \beta_{9} + 4 \beta_{13} - 4 \beta_{14} ) q^{40} + ( -7 \beta_{7} - 7 \beta_{10} - \beta_{11} + 7 \beta_{12} ) q^{41} + ( 15 + 13 \beta_{3} + 28 \beta_{4} + 2 \beta_{5} - 13 \beta_{9} ) q^{43} + ( -22 \beta_{2} - 14 \beta_{6} + 9 \beta_{7} - \beta_{8} ) q^{44} + ( -151 - 7 \beta_{3} - 21 \beta_{4} + 40 \beta_{5} + 7 \beta_{9} ) q^{46} + ( 2 \beta_{7} - 4 \beta_{8} + 6 \beta_{10} + 2 \beta_{12} - 4 \beta_{15} ) q^{47} + ( -40 - 4 \beta_{1} - 31 \beta_{4} + 31 \beta_{5} - 4 \beta_{13} - \beta_{14} ) q^{49} + ( -73 \beta_{2} + 26 \beta_{6} + 13 \beta_{7} - 13 \beta_{8} ) q^{50} + ( 10 + 2 \beta_{1} + 10 \beta_{3} + 10 \beta_{9} - 10 \beta_{13} - 2 \beta_{14} ) q^{52} + ( 27 \beta_{2} + 9 \beta_{6} + 22 \beta_{8} ) q^{53} + ( 4 - 3 \beta_{1} + 4 \beta_{3} + 4 \beta_{9} + 3 \beta_{13} ) q^{55} + ( -36 \beta_{2} + 40 \beta_{6} - 3 \beta_{7} - 8 \beta_{8} - 7 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} + \beta_{15} ) q^{56} + ( 156 - 2 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} + 2 \beta_{9} ) q^{58} + ( 4 \beta_{7} - 8 \beta_{8} + 12 \beta_{10} + 4 \beta_{12} ) q^{59} + ( -8 \beta_{1} - 8 \beta_{13} - \beta_{14} ) q^{61} + ( 3 \beta_{8} - 3 \beta_{10} - 21 \beta_{11} - 3 \beta_{12} + 3 \beta_{15} ) q^{62} + ( 68 + 20 \beta_{3} + 44 \beta_{4} - 48 \beta_{5} - 20 \beta_{9} ) q^{64} + ( 72 \beta_{2} + 24 \beta_{6} + 56 \beta_{8} ) q^{65} + ( 21 - 17 \beta_{3} + 4 \beta_{4} + 38 \beta_{5} + 17 \beta_{9} ) q^{67} + ( \beta_{7} + 4 \beta_{8} - 3 \beta_{10} + 13 \beta_{11} - 5 \beta_{12} - 3 \beta_{15} ) q^{68} + ( -45 - 2 \beta_{1} - 25 \beta_{3} - 45 \beta_{4} + 28 \beta_{5} + 5 \beta_{9} + 10 \beta_{13} - 6 \beta_{14} ) q^{70} + ( -39 \beta_{2} + 39 \beta_{6} - 77 \beta_{7} ) q^{71} + ( 16 \beta_{1} + 16 \beta_{13} - 2 \beta_{14} ) q^{73} + ( -40 \beta_{2} + 22 \beta_{6} + 11 \beta_{7} - 11 \beta_{8} ) q^{74} + ( 8 - 6 \beta_{1} + 8 \beta_{3} + 8 \beta_{9} + 8 \beta_{14} ) q^{76} + ( 3 \beta_{2} + \beta_{6} - 4 \beta_{7} - 10 \beta_{8} - 4 \beta_{10} - 7 \beta_{11} + 4 \beta_{12} ) q^{77} + ( -37 + 5 \beta_{3} - 32 \beta_{4} - 42 \beta_{5} - 5 \beta_{9} ) q^{79} + ( 6 \beta_{7} + 8 \beta_{8} - 2 \beta_{10} - 34 \beta_{11} - 14 \beta_{12} - 2 \beta_{15} ) q^{80} + ( -14 + \beta_{1} - 14 \beta_{3} - 14 \beta_{9} + 14 \beta_{13} + 14 \beta_{14} ) q^{82} + ( 6 \beta_{7} - 12 \beta_{8} + 18 \beta_{10} + 6 \beta_{12} + 4 \beta_{15} ) q^{83} + ( 317 + 99 \beta_{4} - 99 \beta_{5} ) q^{85} + ( -4 \beta_{2} - 90 \beta_{6} + 93 \beta_{7} + 11 \beta_{8} ) q^{86} + ( -222 + 10 \beta_{3} + 22 \beta_{4} - 24 \beta_{5} - 10 \beta_{9} ) q^{88} + ( \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{89} + ( -47 + 17 \beta_{1} - 19 \beta_{3} - 60 \beta_{4} - 28 \beta_{5} + 13 \beta_{9} - 17 \beta_{13} ) q^{91} + ( -170 \beta_{2} - 10 \beta_{6} - 9 \beta_{7} - 47 \beta_{8} ) q^{92} + ( -36 + 4 \beta_{1} - 36 \beta_{3} - 36 \beta_{9} - 28 \beta_{13} + 4 \beta_{14} ) q^{94} + ( 134 \beta_{2} - 134 \beta_{6} - 86 \beta_{7} ) q^{95} + ( 8 \beta_{1} + 8 \beta_{13} + 10 \beta_{14} ) q^{97} + ( -9 \beta_{2} + 62 \beta_{6} + 37 \beta_{7} - 30 \beta_{8} + 5 \beta_{10} + 35 \beta_{11} - 7 \beta_{12} + 3 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 40q^{4} + O(q^{10}) \) \( 16q + 40q^{4} - 304q^{16} - 312q^{22} - 1376q^{25} - 816q^{28} - 816q^{37} - 2568q^{46} - 640q^{49} + 2336q^{58} + 1120q^{64} - 424q^{70} + 5072q^{85} - 3536q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} + 358 x^{14} - 2828 x^{13} + 52557 x^{12} - 549972 x^{11} + 4434734 x^{10} - 37785264 x^{9} + 272741368 x^{8} - 1739202044 x^{7} + 9778426658 x^{6} - 39463975388 x^{5} + 101978126949 x^{4} - 176540053420 x^{3} + 219245087130 x^{2} - 139977817400 x + 52705588025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(43\!\cdots\!52\)\( \nu^{15} + \)\(37\!\cdots\!63\)\( \nu^{14} - \)\(45\!\cdots\!16\)\( \nu^{13} + \)\(14\!\cdots\!31\)\( \nu^{12} - \)\(17\!\cdots\!04\)\( \nu^{11} + \)\(25\!\cdots\!94\)\( \nu^{10} - \)\(28\!\cdots\!88\)\( \nu^{9} + \)\(25\!\cdots\!28\)\( \nu^{8} - \)\(19\!\cdots\!56\)\( \nu^{7} + \)\(15\!\cdots\!48\)\( \nu^{6} - \)\(98\!\cdots\!76\)\( \nu^{5} + \)\(57\!\cdots\!26\)\( \nu^{4} - \)\(25\!\cdots\!88\)\( \nu^{3} + \)\(73\!\cdots\!25\)\( \nu^{2} - \)\(11\!\cdots\!80\)\( \nu + \)\(10\!\cdots\!25\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(19\!\cdots\!11\)\( \nu^{15} - \)\(57\!\cdots\!92\)\( \nu^{14} + \)\(70\!\cdots\!88\)\( \nu^{13} - \)\(47\!\cdots\!92\)\( \nu^{12} + \)\(10\!\cdots\!40\)\( \nu^{11} - \)\(97\!\cdots\!56\)\( \nu^{10} + \)\(79\!\cdots\!64\)\( \nu^{9} - \)\(67\!\cdots\!96\)\( \nu^{8} + \)\(47\!\cdots\!92\)\( \nu^{7} - \)\(30\!\cdots\!12\)\( \nu^{6} + \)\(16\!\cdots\!58\)\( \nu^{5} - \)\(65\!\cdots\!32\)\( \nu^{4} + \)\(15\!\cdots\!87\)\( \nu^{3} - \)\(25\!\cdots\!60\)\( \nu^{2} + \)\(25\!\cdots\!70\)\( \nu - \)\(71\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(96\!\cdots\!34\)\( \nu^{15} + \)\(58\!\cdots\!51\)\( \nu^{14} - \)\(34\!\cdots\!02\)\( \nu^{13} + \)\(34\!\cdots\!72\)\( \nu^{12} - \)\(53\!\cdots\!98\)\( \nu^{11} + \)\(62\!\cdots\!28\)\( \nu^{10} - \)\(50\!\cdots\!36\)\( \nu^{9} + \)\(43\!\cdots\!86\)\( \nu^{8} - \)\(31\!\cdots\!42\)\( \nu^{7} + \)\(20\!\cdots\!66\)\( \nu^{6} - \)\(11\!\cdots\!22\)\( \nu^{5} + \)\(50\!\cdots\!62\)\( \nu^{4} - \)\(14\!\cdots\!26\)\( \nu^{3} + \)\(25\!\cdots\!65\)\( \nu^{2} - \)\(27\!\cdots\!60\)\( \nu + \)\(12\!\cdots\!00\)\(\)\()/ \)\(56\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(42\!\cdots\!00\)\( \nu^{15} - \)\(24\!\cdots\!99\)\( \nu^{14} + \)\(15\!\cdots\!00\)\( \nu^{13} - \)\(14\!\cdots\!49\)\( \nu^{12} + \)\(23\!\cdots\!48\)\( \nu^{11} - \)\(26\!\cdots\!26\)\( \nu^{10} + \)\(21\!\cdots\!20\)\( \nu^{9} - \)\(17\!\cdots\!52\)\( \nu^{8} + \)\(13\!\cdots\!04\)\( \nu^{7} - \)\(84\!\cdots\!36\)\( \nu^{6} + \)\(48\!\cdots\!40\)\( \nu^{5} - \)\(20\!\cdots\!54\)\( \nu^{4} + \)\(52\!\cdots\!08\)\( \nu^{3} - \)\(84\!\cdots\!29\)\( \nu^{2} + \)\(89\!\cdots\!60\)\( \nu - \)\(30\!\cdots\!55\)\(\)\()/ \)\(22\!\cdots\!20\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(42\!\cdots\!20\)\( \nu^{15} - \)\(24\!\cdots\!11\)\( \nu^{14} + \)\(15\!\cdots\!40\)\( \nu^{13} - \)\(14\!\cdots\!61\)\( \nu^{12} + \)\(23\!\cdots\!92\)\( \nu^{11} - \)\(26\!\cdots\!54\)\( \nu^{10} + \)\(21\!\cdots\!00\)\( \nu^{9} - \)\(18\!\cdots\!68\)\( \nu^{8} + \)\(13\!\cdots\!36\)\( \nu^{7} - \)\(84\!\cdots\!64\)\( \nu^{6} + \)\(48\!\cdots\!40\)\( \nu^{5} - \)\(20\!\cdots\!26\)\( \nu^{4} + \)\(52\!\cdots\!52\)\( \nu^{3} - \)\(84\!\cdots\!01\)\( \nu^{2} + \)\(89\!\cdots\!40\)\( \nu - \)\(48\!\cdots\!75\)\(\)\()/ \)\(22\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(41\!\cdots\!21\)\( \nu^{15} + \)\(15\!\cdots\!80\)\( \nu^{14} - \)\(14\!\cdots\!08\)\( \nu^{13} + \)\(11\!\cdots\!56\)\( \nu^{12} - \)\(21\!\cdots\!88\)\( \nu^{11} + \)\(22\!\cdots\!60\)\( \nu^{10} - \)\(18\!\cdots\!04\)\( \nu^{9} + \)\(15\!\cdots\!28\)\( \nu^{8} - \)\(11\!\cdots\!36\)\( \nu^{7} + \)\(70\!\cdots\!40\)\( \nu^{6} - \)\(39\!\cdots\!18\)\( \nu^{5} + \)\(15\!\cdots\!76\)\( \nu^{4} - \)\(40\!\cdots\!65\)\( \nu^{3} + \)\(65\!\cdots\!20\)\( \nu^{2} - \)\(68\!\cdots\!50\)\( \nu + \)\(32\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(69\!\cdots\!10\)\( \nu^{15} + \)\(23\!\cdots\!12\)\( \nu^{14} - \)\(25\!\cdots\!99\)\( \nu^{13} + \)\(18\!\cdots\!72\)\( \nu^{12} - \)\(36\!\cdots\!54\)\( \nu^{11} + \)\(36\!\cdots\!96\)\( \nu^{10} - \)\(29\!\cdots\!72\)\( \nu^{9} + \)\(25\!\cdots\!96\)\( \nu^{8} - \)\(18\!\cdots\!32\)\( \nu^{7} + \)\(11\!\cdots\!72\)\( \nu^{6} - \)\(64\!\cdots\!44\)\( \nu^{5} + \)\(25\!\cdots\!52\)\( \nu^{4} - \)\(64\!\cdots\!44\)\( \nu^{3} + \)\(10\!\cdots\!40\)\( \nu^{2} - \)\(10\!\cdots\!75\)\( \nu + \)\(45\!\cdots\!00\)\(\)\()/ \)\(29\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(23\!\cdots\!97\)\( \nu^{15} - \)\(30\!\cdots\!56\)\( \nu^{14} + \)\(83\!\cdots\!11\)\( \nu^{13} - \)\(44\!\cdots\!20\)\( \nu^{12} + \)\(11\!\cdots\!02\)\( \nu^{11} - \)\(99\!\cdots\!88\)\( \nu^{10} + \)\(76\!\cdots\!28\)\( \nu^{9} - \)\(67\!\cdots\!60\)\( \nu^{8} + \)\(45\!\cdots\!20\)\( \nu^{7} - \)\(28\!\cdots\!56\)\( \nu^{6} + \)\(15\!\cdots\!86\)\( \nu^{5} - \)\(50\!\cdots\!20\)\( \nu^{4} + \)\(94\!\cdots\!11\)\( \nu^{3} - \)\(14\!\cdots\!60\)\( \nu^{2} + \)\(96\!\cdots\!85\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(17\!\cdots\!06\)\( \nu^{15} - \)\(94\!\cdots\!54\)\( \nu^{14} + \)\(61\!\cdots\!18\)\( \nu^{13} - \)\(57\!\cdots\!23\)\( \nu^{12} + \)\(92\!\cdots\!82\)\( \nu^{11} - \)\(10\!\cdots\!42\)\( \nu^{10} + \)\(83\!\cdots\!24\)\( \nu^{9} - \)\(70\!\cdots\!74\)\( \nu^{8} + \)\(51\!\cdots\!78\)\( \nu^{7} - \)\(32\!\cdots\!54\)\( \nu^{6} + \)\(18\!\cdots\!98\)\( \nu^{5} - \)\(76\!\cdots\!08\)\( \nu^{4} + \)\(18\!\cdots\!34\)\( \nu^{3} - \)\(28\!\cdots\!90\)\( \nu^{2} + \)\(29\!\cdots\!40\)\( \nu - \)\(72\!\cdots\!25\)\(\)\()/ \)\(56\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(94\!\cdots\!79\)\( \nu^{15} + \)\(19\!\cdots\!24\)\( \nu^{14} - \)\(33\!\cdots\!47\)\( \nu^{13} + \)\(20\!\cdots\!36\)\( \nu^{12} - \)\(46\!\cdots\!46\)\( \nu^{11} + \)\(43\!\cdots\!72\)\( \nu^{10} - \)\(33\!\cdots\!76\)\( \nu^{9} + \)\(29\!\cdots\!68\)\( \nu^{8} - \)\(20\!\cdots\!56\)\( \nu^{7} + \)\(12\!\cdots\!84\)\( \nu^{6} - \)\(67\!\cdots\!82\)\( \nu^{5} + \)\(23\!\cdots\!56\)\( \nu^{4} - \)\(48\!\cdots\!49\)\( \nu^{3} + \)\(61\!\cdots\!60\)\( \nu^{2} - \)\(66\!\cdots\!65\)\( \nu + \)\(28\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(62\!\cdots\!59\)\( \nu^{15} - \)\(14\!\cdots\!20\)\( \nu^{14} + \)\(22\!\cdots\!27\)\( \nu^{13} - \)\(14\!\cdots\!64\)\( \nu^{12} + \)\(31\!\cdots\!02\)\( \nu^{11} - \)\(29\!\cdots\!20\)\( \nu^{10} + \)\(23\!\cdots\!76\)\( \nu^{9} - \)\(20\!\cdots\!32\)\( \nu^{8} + \)\(14\!\cdots\!44\)\( \nu^{7} - \)\(89\!\cdots\!00\)\( \nu^{6} + \)\(48\!\cdots\!42\)\( \nu^{5} - \)\(18\!\cdots\!44\)\( \nu^{4} + \)\(41\!\cdots\!85\)\( \nu^{3} - \)\(64\!\cdots\!60\)\( \nu^{2} + \)\(74\!\cdots\!25\)\( \nu - \)\(32\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(17\!\cdots\!49\)\( \nu^{15} - \)\(39\!\cdots\!76\)\( \nu^{14} + \)\(62\!\cdots\!07\)\( \nu^{13} - \)\(38\!\cdots\!12\)\( \nu^{12} + \)\(87\!\cdots\!98\)\( \nu^{11} - \)\(82\!\cdots\!68\)\( \nu^{10} + \)\(65\!\cdots\!76\)\( \nu^{9} - \)\(56\!\cdots\!56\)\( \nu^{8} + \)\(39\!\cdots\!72\)\( \nu^{7} - \)\(24\!\cdots\!36\)\( \nu^{6} + \)\(13\!\cdots\!02\)\( \nu^{5} - \)\(50\!\cdots\!52\)\( \nu^{4} + \)\(11\!\cdots\!31\)\( \nu^{3} - \)\(19\!\cdots\!80\)\( \nu^{2} + \)\(24\!\cdots\!85\)\( \nu - \)\(10\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(11\!\cdots\!36\)\( \nu^{15} + \)\(58\!\cdots\!29\)\( \nu^{14} - \)\(40\!\cdots\!28\)\( \nu^{13} + \)\(36\!\cdots\!73\)\( \nu^{12} - \)\(59\!\cdots\!72\)\( \nu^{11} + \)\(67\!\cdots\!02\)\( \nu^{10} - \)\(52\!\cdots\!04\)\( \nu^{9} + \)\(44\!\cdots\!24\)\( \nu^{8} - \)\(32\!\cdots\!08\)\( \nu^{7} + \)\(20\!\cdots\!84\)\( \nu^{6} - \)\(11\!\cdots\!08\)\( \nu^{5} + \)\(44\!\cdots\!58\)\( \nu^{4} - \)\(98\!\cdots\!84\)\( \nu^{3} + \)\(12\!\cdots\!75\)\( \nu^{2} - \)\(14\!\cdots\!40\)\( \nu + \)\(13\!\cdots\!75\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(24\!\cdots\!56\)\( \nu^{15} - \)\(13\!\cdots\!14\)\( \nu^{14} + \)\(86\!\cdots\!08\)\( \nu^{13} - \)\(81\!\cdots\!38\)\( \nu^{12} + \)\(12\!\cdots\!32\)\( \nu^{11} - \)\(14\!\cdots\!82\)\( \nu^{10} + \)\(11\!\cdots\!44\)\( \nu^{9} - \)\(99\!\cdots\!44\)\( \nu^{8} + \)\(72\!\cdots\!28\)\( \nu^{7} - \)\(45\!\cdots\!94\)\( \nu^{6} + \)\(25\!\cdots\!88\)\( \nu^{5} - \)\(10\!\cdots\!98\)\( \nu^{4} + \)\(26\!\cdots\!84\)\( \nu^{3} - \)\(42\!\cdots\!50\)\( \nu^{2} + \)\(55\!\cdots\!40\)\( \nu - \)\(24\!\cdots\!00\)\(\)\()/ \)\(21\!\cdots\!25\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(69\!\cdots\!57\)\( \nu^{15} + \)\(12\!\cdots\!00\)\( \nu^{14} - \)\(24\!\cdots\!86\)\( \nu^{13} + \)\(14\!\cdots\!32\)\( \nu^{12} - \)\(33\!\cdots\!36\)\( \nu^{11} + \)\(30\!\cdots\!00\)\( \nu^{10} - \)\(23\!\cdots\!68\)\( \nu^{9} + \)\(20\!\cdots\!16\)\( \nu^{8} - \)\(14\!\cdots\!32\)\( \nu^{7} + \)\(88\!\cdots\!00\)\( \nu^{6} - \)\(47\!\cdots\!06\)\( \nu^{5} + \)\(16\!\cdots\!72\)\( \nu^{4} - \)\(30\!\cdots\!45\)\( \nu^{3} + \)\(32\!\cdots\!00\)\( \nu^{2} - \)\(18\!\cdots\!00\)\( \nu - \)\(15\!\cdots\!00\)\(\)\()/ \)\(14\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{2} + 2\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} + 10 \beta_{11} + 3 \beta_{10} - 10 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + 9 \beta_{6} - 50 \beta_{5} + 42 \beta_{4} - 18 \beta_{3} - \beta_{2} - 2 \beta_{1} - 368\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(12 \beta_{15} + 36 \beta_{14} + 66 \beta_{13} - 215 \beta_{12} + 462 \beta_{11} + 47 \beta_{10} + 114 \beta_{9} - 224 \beta_{8} - 841 \beta_{7} + 1564 \beta_{6} + 508 \beta_{5} - 388 \beta_{4} + 186 \beta_{3} + 1628 \beta_{2} + 30 \beta_{1} + 4462\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(-36 \beta_{15} + 804 \beta_{14} + 1196 \beta_{13} + 820 \beta_{12} - 1636 \beta_{11} - 220 \beta_{10} + 12 \beta_{9} + 748 \beta_{8} + 3026 \beta_{7} - 5621 \beta_{6} + 4782 \beta_{5} + 482 \beta_{4} + 4124 \beta_{3} - 6203 \beta_{2} + 660 \beta_{1} + 39046\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-3240 \beta_{15} - 15240 \beta_{14} - 22450 \beta_{13} + 25851 \beta_{12} - 33578 \beta_{11} + 23669 \beta_{10} - 2210 \beta_{9} + 31972 \beta_{8} + 91613 \beta_{7} - 227570 \beta_{6} - 95040 \beta_{5} + 1000 \beta_{4} - 76810 \beta_{3} - 400182 \beta_{2} - 11790 \beta_{1} - 773946\)\()/16\)
\(\nu^{6}\)\(=\)\((\)\(18750 \beta_{15} - 141186 \beta_{14} - 173172 \beta_{13} - 161927 \beta_{12} + 233324 \beta_{11} - 118573 \beta_{10} + 329416 \beta_{9} - 193358 \beta_{8} - 579503 \beta_{7} + 1383882 \beta_{6} - 295256 \beta_{5} - 791320 \beta_{4} - 529368 \beta_{3} + 2309638 \beta_{2} - 146164 \beta_{1} - 2110944\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(266168 \beta_{15} + 2077460 \beta_{14} + 2633750 \beta_{13} - 805420 \beta_{12} - 489540 \beta_{11} - 3194548 \beta_{10} - 4009054 \beta_{9} - 2303488 \beta_{8} - 199550 \beta_{7} + 11144103 \beta_{6} + 5780738 \beta_{5} + 9715806 \beta_{4} + 8237250 \beta_{3} + 36486257 \beta_{2} + 2074506 \beta_{1} + 43322176\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-4847584 \beta_{15} + 17684688 \beta_{14} + 16913192 \beta_{13} + 21114556 \beta_{12} - 7210088 \beta_{11} + 51939460 \beta_{10} - 77871896 \beta_{9} + 44097328 \beta_{8} + 37622634 \beta_{7} - 239964527 \beta_{6} - 36621690 \beta_{5} + 174287130 \beta_{4} + 30924808 \beta_{3} - 651794481 \beta_{2} + 23156120 \beta_{1} - 366498678\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(-48703440 \beta_{15} - 883011696 \beta_{14} - 946725174 \beta_{13} - 212327525 \beta_{12} + 1001801038 \beta_{11} + 946190405 \beta_{10} + 3072715818 \beta_{9} + 431180796 \beta_{8} - 1847708687 \beta_{7} - 349264568 \beta_{6} + 235058804 \beta_{5} - 6991690844 \beta_{4} - 2265848622 \beta_{3} - 8478589616 \beta_{2} - 1053796170 \beta_{1} + 4634714062\)\()/16\)
\(\nu^{10}\)\(=\)\((\)\(904335150 \beta_{15} - 834604638 \beta_{14} - 11176738 \beta_{13} - 284385156 \beta_{12} - 7899378702 \beta_{11} - 13336545744 \beta_{10} + 9554100242 \beta_{9} - 8147035402 \beta_{8} + 12131540892 \beta_{7} + 27081904205 \beta_{6} + 14644364698 \beta_{5} - 21017586322 \beta_{4} + 4518974522 \beta_{3} + 138383558427 \beta_{2} - 1879735766 \beta_{1} + 127307491120\)\()/8\)
\(\nu^{11}\)\(=\)\((\)\(-1709516380 \beta_{15} + 140781230964 \beta_{14} + 118134494434 \beta_{13} + 78604043615 \beta_{12} - 182664208222 \beta_{11} - 52853769879 \beta_{10} - 735967210550 \beta_{9} + 14926146072 \beta_{8} + 372005735461 \beta_{7} - 412006960442 \beta_{6} - 573841693696 \beta_{5} + 1648637691752 \beta_{4} + 113302270114 \beta_{3} + 104468702290 \beta_{2} + 200813588734 \beta_{1} - 5309532618958\)\()/16\)
\(\nu^{12}\)\(=\)\((\)\(-59587475706 \beta_{15} - 133295785114 \beta_{14} - 202402059216 \beta_{13} - 366139102959 \beta_{12} + 1494678265780 \beta_{11} + 1263659388555 \beta_{10} + 16326659088 \beta_{9} + 540495443954 \beta_{8} - 2741166180499 \beta_{7} - 13462740024 \beta_{6} - 958027896676 \beta_{5} - 63233604956 \beta_{4} - 790626717184 \beta_{3} - 10929941133512 \beta_{2} - 99589169360 \beta_{1} - 7757738479936\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(849995603080 \beta_{15} - 7099600357616 \beta_{14} - 2257804253224 \beta_{13} - 4421761574763 \beta_{12} + 3090593743942 \beta_{11} - 8381128713477 \beta_{10} + 64907719966128 \beta_{9} - 7687785843640 \beta_{8} - 9318926453931 \beta_{7} + 44583240471617 \beta_{6} + 90412717133934 \beta_{5} - 144437042900254 \beta_{4} + 22219850150368 \beta_{3} + 111275340793575 \beta_{2} - 13826805247320 \beta_{1} + 791181188266770\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(6936535769800 \beta_{15} + 85734367697660 \beta_{14} + 85620421150406 \beta_{13} + 242965843464895 \beta_{12} - 681091059946938 \beta_{11} - 347445975883375 \beta_{10} - 344653095116662 \beta_{9} - 63263013712996 \beta_{8} + 1323361309955667 \beta_{7} - 919686719917747 \beta_{6} - 122143046100018 \beta_{5} + 775326991511386 \beta_{4} + 171467748656722 \beta_{3} + 2197410711886891 \beta_{2} + 108616297696162 \beta_{1} - 1297266920300076\)\()/8\)
\(\nu^{15}\)\(=\)\((\)\(-345292836168636 \beta_{15} - 220267946319092 \beta_{14} - 2466449448558502 \beta_{13} - 1230338357585255 \beta_{12} + 6405173991855366 \beta_{11} + 6431225539341103 \beta_{10} - 16028645404405462 \beta_{9} + 3138728928356680 \beta_{8} - 11398692250537553 \beta_{7} - 4212910709054100 \beta_{6} - 37034519133421588 \beta_{5} + 35345314731978412 \beta_{4} - 17363388468424622 \beta_{3} - 59275303729666004 \beta_{2} + 1967418289897926 \beta_{1} - 314693562872341418\)\()/16\)

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.929907 + 6.47387i
0.929907 12.9280i
0.929907 + 12.9280i
0.929907 6.47387i
4.06402 0.600106i
4.06402 + 1.75957i
4.06402 1.75957i
4.06402 + 0.600106i
0.467111 0.600106i
0.467111 + 1.75957i
0.467111 1.75957i
0.467111 + 0.600106i
−4.46104 + 6.47387i
−4.46104 12.9280i
−4.46104 + 12.9280i
−4.46104 6.47387i
−2.69547 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 −16.6272 8.15690i −13.6452 18.0502i 0 −8.83121 + 27.7765i
55.2 −2.69547 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 16.6272 8.15690i −13.6452 18.0502i 0 8.83121 27.7765i
55.3 −2.69547 + 0.856992i 0 6.53113 4.62000i 10.3049i 0 16.6272 + 8.15690i −13.6452 + 18.0502i 0 8.83121 + 27.7765i
55.4 −2.69547 + 0.856992i 0 6.53113 4.62000i 10.3049i 0 −16.6272 + 8.15690i −13.6452 + 18.0502i 0 −8.83121 27.7765i
55.5 −1.79845 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 5.15121 + 17.7895i 19.8950 10.7792i 0 −38.7945 + 31.9604i
55.6 −1.79845 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 −5.15121 + 17.7895i 19.8950 10.7792i 0 38.7945 31.9604i
55.7 −1.79845 + 2.18302i 0 −1.53113 7.85211i 17.7710i 0 −5.15121 17.7895i 19.8950 + 10.7792i 0 38.7945 + 31.9604i
55.8 −1.79845 + 2.18302i 0 −1.53113 7.85211i 17.7710i 0 5.15121 17.7895i 19.8950 + 10.7792i 0 −38.7945 31.9604i
55.9 1.79845 2.18302i 0 −1.53113 7.85211i 17.7710i 0 5.15121 17.7895i −19.8950 10.7792i 0 −38.7945 31.9604i
55.10 1.79845 2.18302i 0 −1.53113 7.85211i 17.7710i 0 −5.15121 17.7895i −19.8950 10.7792i 0 38.7945 + 31.9604i
55.11 1.79845 + 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 −5.15121 + 17.7895i −19.8950 + 10.7792i 0 38.7945 31.9604i
55.12 1.79845 + 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 5.15121 + 17.7895i −19.8950 + 10.7792i 0 −38.7945 + 31.9604i
55.13 2.69547 0.856992i 0 6.53113 4.62000i 10.3049i 0 −16.6272 + 8.15690i 13.6452 18.0502i 0 −8.83121 27.7765i
55.14 2.69547 0.856992i 0 6.53113 4.62000i 10.3049i 0 16.6272 + 8.15690i 13.6452 18.0502i 0 8.83121 + 27.7765i
55.15 2.69547 + 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 16.6272 8.15690i 13.6452 + 18.0502i 0 8.83121 27.7765i
55.16 2.69547 + 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 −16.6272 8.15690i 13.6452 + 18.0502i 0 −8.83121 + 27.7765i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.b.g 16
3.b odd 2 1 inner 252.4.b.g 16
4.b odd 2 1 inner 252.4.b.g 16
7.b odd 2 1 inner 252.4.b.g 16
12.b even 2 1 inner 252.4.b.g 16
21.c even 2 1 inner 252.4.b.g 16
28.d even 2 1 inner 252.4.b.g 16
84.h odd 2 1 inner 252.4.b.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.b.g 16 1.a even 1 1 trivial
252.4.b.g 16 3.b odd 2 1 inner
252.4.b.g 16 4.b odd 2 1 inner
252.4.b.g 16 7.b odd 2 1 inner
252.4.b.g 16 12.b even 2 1 inner
252.4.b.g 16 21.c even 2 1 inner
252.4.b.g 16 28.d even 2 1 inner
252.4.b.g 16 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{4} + 422 T_{5}^{2} + 33536 \)
\( T_{11}^{4} + 442 T_{11}^{2} + 37856 \)
\( T_{19}^{4} - 12028 T_{19}^{2} + 23005696 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4096 - 640 T^{2} + 88 T^{4} - 10 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 33536 + 422 T^{2} + T^{4} )^{4} \)
$7$ \( ( 13841287201 + 18823840 T^{2} - 8162 T^{4} + 160 T^{6} + T^{8} )^{2} \)
$11$ \( ( 37856 + 442 T^{2} + T^{4} )^{4} \)
$13$ \( ( 12609536 + 7648 T^{2} + T^{4} )^{4} \)
$17$ \( ( 8585216 + 6118 T^{2} + T^{4} )^{4} \)
$19$ \( ( 23005696 - 12028 T^{2} + T^{4} )^{4} \)
$23$ \( ( 29353856 + 26842 T^{2} + T^{4} )^{4} \)
$29$ \( ( 18866176 - 10448 T^{2} + T^{4} )^{4} \)
$31$ \( ( 950745600 - 64080 T^{2} + T^{4} )^{4} \)
$37$ \( ( -5264 + 102 T + T^{2} )^{8} \)
$41$ \( ( 14785888256 + 255782 T^{2} + T^{4} )^{4} \)
$43$ \( ( 115302656 + 222172 T^{2} + T^{4} )^{4} \)
$47$ \( ( 70613401600 - 535680 T^{2} + T^{4} )^{4} \)
$53$ \( ( 10418845696 - 293712 T^{2} + T^{4} )^{4} \)
$59$ \( ( 11298144256 - 536192 T^{2} + T^{4} )^{4} \)
$61$ \( ( 46920083456 + 436192 T^{2} + T^{4} )^{4} \)
$67$ \( ( 25756457216 + 503548 T^{2} + T^{4} )^{4} \)
$71$ \( ( 30079110656 + 1021658 T^{2} + T^{4} )^{4} \)
$73$ \( ( 852404633600 + 1988480 T^{2} + T^{4} )^{4} \)
$79$ \( ( 21224784896 + 508828 T^{2} + T^{4} )^{4} \)
$83$ \( ( 477346594816 - 1392768 T^{2} + T^{4} )^{4} \)
$89$ \( ( 3353600 + 5270 T^{2} + T^{4} )^{4} \)
$97$ \( ( 39543504896 + 919168 T^{2} + T^{4} )^{4} \)
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