Properties

Label 1344.4.p.a
Level $1344$
Weight $4$
Character orbit 1344.p
Analytic conductor $79.299$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} - 610 x^{9} + 8375926786 x^{8} - 15937543350 x^{7} + 15140222838 x^{6} - 113574152394 x^{5} + 45261122709708 x^{4} - 121918187031822 x^{3} + 150109311002562 x^{2} + 2604526546410882 x + 22595395600887201\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q -3 \beta_{1} q^{3} -\beta_{7} q^{5} + ( -\beta_{1} + \beta_{5} ) q^{7} -9 q^{9} +O(q^{10})\) \( q -3 \beta_{1} q^{3} -\beta_{7} q^{5} + ( -\beta_{1} + \beta_{5} ) q^{7} -9 q^{9} + \beta_{10} q^{11} + ( -3 - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{13} ) q^{13} -3 \beta_{6} q^{15} + ( -\beta_{4} - \beta_{8} - \beta_{11} ) q^{17} + ( 7 \beta_{1} + \beta_{6} - \beta_{12} ) q^{19} + ( -3 + 3 \beta_{2} ) q^{21} + ( -19 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - \beta_{12} ) q^{23} + ( 6 - 2 \beta_{2} - 2 \beta_{4} + 5 \beta_{7} + \beta_{13} ) q^{25} + 27 \beta_{1} q^{27} + ( 3 \beta_{11} + \beta_{14} ) q^{29} + ( 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{10} + \beta_{15} ) q^{31} -3 \beta_{11} q^{33} + ( -14 \beta_{1} + \beta_{3} - 6 \beta_{6} + \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - \beta_{15} ) q^{35} + ( -3 \beta_{2} + \beta_{4} - 2 \beta_{8} + \beta_{11} - \beta_{14} ) q^{37} + ( 9 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{12} ) q^{39} + ( 6 \beta_{2} - 5 \beta_{4} + \beta_{8} + \beta_{11} + 2 \beta_{14} ) q^{41} + ( 6 \beta_{3} - 5 \beta_{5} - \beta_{9} + 4 \beta_{10} ) q^{43} + 9 \beta_{7} q^{45} + ( 6 \beta_{3} - 4 \beta_{5} - 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{15} ) q^{47} + ( 24 + 3 \beta_{2} - 2 \beta_{4} + \beta_{7} - \beta_{8} + 5 \beta_{11} + 5 \beta_{13} + \beta_{14} ) q^{49} + ( -3 \beta_{5} + 3 \beta_{9} - 3 \beta_{10} ) q^{51} + ( -6 \beta_{2} + 10 \beta_{4} + 4 \beta_{8} - 3 \beta_{11} + \beta_{14} ) q^{53} + ( 7 \beta_{3} - 5 \beta_{5} - 2 \beta_{9} + 5 \beta_{10} - \beta_{15} ) q^{55} + ( 21 - 3 \beta_{7} + 3 \beta_{13} ) q^{57} + ( -106 \beta_{1} + 14 \beta_{3} + 14 \beta_{5} + 12 \beta_{6} + 2 \beta_{12} ) q^{59} + ( -17 + 11 \beta_{2} + 11 \beta_{4} - 27 \beta_{7} - 3 \beta_{13} ) q^{61} + ( 9 \beta_{1} - 9 \beta_{5} ) q^{63} + ( -88 - 16 \beta_{2} - 16 \beta_{4} + 14 \beta_{7} - 8 \beta_{13} ) q^{65} + ( 12 \beta_{3} - 5 \beta_{5} - 7 \beta_{9} + 4 \beta_{10} + 2 \beta_{15} ) q^{67} + ( -57 + 6 \beta_{2} + 6 \beta_{4} + 6 \beta_{7} + 3 \beta_{13} ) q^{69} + ( -41 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} - 22 \beta_{6} + 13 \beta_{12} ) q^{71} + ( 10 \beta_{2} - 6 \beta_{4} + 4 \beta_{8} - 14 \beta_{11} ) q^{73} + ( -18 \beta_{1} + 6 \beta_{3} + 6 \beta_{5} + 15 \beta_{6} + 3 \beta_{12} ) q^{75} + ( 10 - 2 \beta_{2} + 16 \beta_{4} + 13 \beta_{7} - 6 \beta_{8} - 5 \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{77} + ( -86 \beta_{1} + 19 \beta_{3} + 19 \beta_{5} + 6 \beta_{6} - 12 \beta_{12} ) q^{79} + 81 q^{81} + ( 190 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} - 12 \beta_{6} + 10 \beta_{12} ) q^{83} + ( -21 \beta_{2} + 25 \beta_{4} + 4 \beta_{8} - 11 \beta_{11} - 5 \beta_{14} ) q^{85} + ( 9 \beta_{10} - 3 \beta_{15} ) q^{87} + ( 18 \beta_{2} - 21 \beta_{4} - 3 \beta_{8} + 21 \beta_{11} - 2 \beta_{14} ) q^{89} + ( -151 \beta_{1} + 24 \beta_{3} - 3 \beta_{5} - 25 \beta_{6} + 3 \beta_{9} - 6 \beta_{10} + \beta_{12} + 4 \beta_{15} ) q^{91} + ( -6 \beta_{2} + 6 \beta_{4} - 9 \beta_{11} + 3 \beta_{14} ) q^{93} + ( 116 \beta_{1} + 14 \beta_{3} + 14 \beta_{5} + 28 \beta_{6} - 4 \beta_{12} ) q^{95} + ( -10 \beta_{2} + 4 \beta_{4} - 6 \beta_{8} - 12 \beta_{11} - 6 \beta_{14} ) q^{97} -9 \beta_{10} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 144q^{9} + O(q^{10}) \) \( 16q - 144q^{9} - 56q^{13} - 36q^{21} + 80q^{25} + 392q^{49} + 336q^{57} - 184q^{61} - 1536q^{65} - 864q^{69} + 240q^{77} + 1296q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} - 610 x^{9} + 8375926786 x^{8} - 15937543350 x^{7} + 15140222838 x^{6} - 113574152394 x^{5} + 45261122709708 x^{4} - 121918187031822 x^{3} + 150109311002562 x^{2} + 2604526546410882 x + 22595395600887201\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(20\!\cdots\!32\)\( \nu^{15} + \)\(33\!\cdots\!67\)\( \nu^{14} - \)\(33\!\cdots\!01\)\( \nu^{13} - \)\(83\!\cdots\!14\)\( \nu^{12} - \)\(36\!\cdots\!44\)\( \nu^{11} + \)\(59\!\cdots\!13\)\( \nu^{10} - \)\(52\!\cdots\!75\)\( \nu^{9} + \)\(55\!\cdots\!78\)\( \nu^{8} - \)\(17\!\cdots\!34\)\( \nu^{7} + \)\(28\!\cdots\!91\)\( \nu^{6} - \)\(25\!\cdots\!53\)\( \nu^{5} + \)\(50\!\cdots\!24\)\( \nu^{4} - \)\(99\!\cdots\!50\)\( \nu^{3} + \)\(20\!\cdots\!45\)\( \nu^{2} - \)\(25\!\cdots\!91\)\( \nu - \)\(27\!\cdots\!76\)\(\)\()/ \)\(41\!\cdots\!92\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(45\!\cdots\!01\)\( \nu^{15} - \)\(29\!\cdots\!44\)\( \nu^{14} - \)\(12\!\cdots\!48\)\( \nu^{13} - \)\(10\!\cdots\!85\)\( \nu^{12} + \)\(78\!\cdots\!22\)\( \nu^{11} - \)\(51\!\cdots\!78\)\( \nu^{10} - \)\(21\!\cdots\!18\)\( \nu^{9} - \)\(20\!\cdots\!85\)\( \nu^{8} + \)\(36\!\cdots\!67\)\( \nu^{7} - \)\(23\!\cdots\!12\)\( \nu^{6} - \)\(90\!\cdots\!00\)\( \nu^{5} - \)\(93\!\cdots\!31\)\( \nu^{4} + \)\(15\!\cdots\!40\)\( \nu^{3} - \)\(85\!\cdots\!42\)\( \nu^{2} - \)\(13\!\cdots\!24\)\( \nu - \)\(21\!\cdots\!55\)\(\)\()/ \)\(10\!\cdots\!14\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(91\!\cdots\!68\)\( \nu^{15} - \)\(14\!\cdots\!37\)\( \nu^{14} - \)\(56\!\cdots\!70\)\( \nu^{13} - \)\(48\!\cdots\!69\)\( \nu^{12} + \)\(16\!\cdots\!47\)\( \nu^{11} - \)\(25\!\cdots\!57\)\( \nu^{10} - \)\(99\!\cdots\!76\)\( \nu^{9} - \)\(98\!\cdots\!63\)\( \nu^{8} + \)\(77\!\cdots\!66\)\( \nu^{7} - \)\(11\!\cdots\!45\)\( \nu^{6} - \)\(41\!\cdots\!36\)\( \nu^{5} - \)\(46\!\cdots\!37\)\( \nu^{4} + \)\(49\!\cdots\!29\)\( \nu^{3} - \)\(45\!\cdots\!05\)\( \nu^{2} - \)\(57\!\cdots\!52\)\( \nu - \)\(90\!\cdots\!27\)\(\)\()/ \)\(48\!\cdots\!34\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(22\!\cdots\!51\)\( \nu^{15} + \)\(28\!\cdots\!10\)\( \nu^{14} - \)\(15\!\cdots\!60\)\( \nu^{13} - \)\(10\!\cdots\!44\)\( \nu^{12} + \)\(38\!\cdots\!04\)\( \nu^{11} + \)\(50\!\cdots\!82\)\( \nu^{10} - \)\(21\!\cdots\!96\)\( \nu^{9} - \)\(20\!\cdots\!00\)\( \nu^{8} + \)\(16\!\cdots\!81\)\( \nu^{7} + \)\(22\!\cdots\!42\)\( \nu^{6} - \)\(10\!\cdots\!76\)\( \nu^{5} - \)\(97\!\cdots\!80\)\( \nu^{4} + \)\(41\!\cdots\!06\)\( \nu^{3} + \)\(73\!\cdots\!10\)\( \nu^{2} - \)\(31\!\cdots\!14\)\( \nu - \)\(20\!\cdots\!12\)\(\)\()/ \)\(10\!\cdots\!14\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(22\!\cdots\!51\)\( \nu^{15} - \)\(28\!\cdots\!10\)\( \nu^{14} + \)\(15\!\cdots\!60\)\( \nu^{13} + \)\(10\!\cdots\!44\)\( \nu^{12} - \)\(38\!\cdots\!04\)\( \nu^{11} - \)\(50\!\cdots\!82\)\( \nu^{10} + \)\(21\!\cdots\!96\)\( \nu^{9} + \)\(20\!\cdots\!00\)\( \nu^{8} - \)\(16\!\cdots\!81\)\( \nu^{7} - \)\(22\!\cdots\!42\)\( \nu^{6} + \)\(10\!\cdots\!76\)\( \nu^{5} + \)\(97\!\cdots\!80\)\( \nu^{4} - \)\(41\!\cdots\!06\)\( \nu^{3} - \)\(73\!\cdots\!10\)\( \nu^{2} + \)\(23\!\cdots\!42\)\( \nu + \)\(20\!\cdots\!12\)\(\)\()/ \)\(10\!\cdots\!14\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(63\!\cdots\!46\)\( \nu^{15} - \)\(20\!\cdots\!07\)\( \nu^{14} - \)\(12\!\cdots\!92\)\( \nu^{13} - \)\(25\!\cdots\!54\)\( \nu^{12} - \)\(11\!\cdots\!53\)\( \nu^{11} - \)\(35\!\cdots\!44\)\( \nu^{10} - \)\(26\!\cdots\!71\)\( \nu^{9} + \)\(16\!\cdots\!73\)\( \nu^{8} - \)\(54\!\cdots\!14\)\( \nu^{7} - \)\(14\!\cdots\!61\)\( \nu^{6} - \)\(14\!\cdots\!94\)\( \nu^{5} + \)\(15\!\cdots\!14\)\( \nu^{4} - \)\(26\!\cdots\!43\)\( \nu^{3} - \)\(51\!\cdots\!28\)\( \nu^{2} - \)\(70\!\cdots\!95\)\( \nu - \)\(77\!\cdots\!93\)\(\)\()/ \)\(27\!\cdots\!38\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(30\!\cdots\!71\)\( \nu^{15} + \)\(23\!\cdots\!51\)\( \nu^{14} + \)\(51\!\cdots\!31\)\( \nu^{13} + \)\(70\!\cdots\!30\)\( \nu^{12} - \)\(54\!\cdots\!63\)\( \nu^{11} + \)\(34\!\cdots\!09\)\( \nu^{10} + \)\(10\!\cdots\!05\)\( \nu^{9} + \)\(12\!\cdots\!80\)\( \nu^{8} - \)\(25\!\cdots\!01\)\( \nu^{7} + \)\(16\!\cdots\!77\)\( \nu^{6} + \)\(58\!\cdots\!13\)\( \nu^{5} + \)\(47\!\cdots\!16\)\( \nu^{4} - \)\(12\!\cdots\!41\)\( \nu^{3} + \)\(18\!\cdots\!75\)\( \nu^{2} + \)\(30\!\cdots\!87\)\( \nu + \)\(20\!\cdots\!98\)\(\)\()/ \)\(11\!\cdots\!06\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(11\!\cdots\!76\)\( \nu^{15} - \)\(97\!\cdots\!80\)\( \nu^{14} + \)\(31\!\cdots\!83\)\( \nu^{13} + \)\(25\!\cdots\!19\)\( \nu^{12} + \)\(19\!\cdots\!29\)\( \nu^{11} - \)\(15\!\cdots\!26\)\( \nu^{10} + \)\(40\!\cdots\!85\)\( \nu^{9} + \)\(35\!\cdots\!91\)\( \nu^{8} + \)\(95\!\cdots\!44\)\( \nu^{7} - \)\(67\!\cdots\!72\)\( \nu^{6} + \)\(14\!\cdots\!41\)\( \nu^{5} + \)\(14\!\cdots\!65\)\( \nu^{4} + \)\(72\!\cdots\!37\)\( \nu^{3} - \)\(17\!\cdots\!22\)\( \nu^{2} + \)\(20\!\cdots\!77\)\( \nu + \)\(59\!\cdots\!77\)\(\)\()/ \)\(19\!\cdots\!66\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(53\!\cdots\!25\)\( \nu^{15} - \)\(31\!\cdots\!09\)\( \nu^{14} - \)\(10\!\cdots\!17\)\( \nu^{13} + \)\(38\!\cdots\!05\)\( \nu^{12} + \)\(94\!\cdots\!60\)\( \nu^{11} - \)\(54\!\cdots\!91\)\( \nu^{10} - \)\(13\!\cdots\!81\)\( \nu^{9} + \)\(50\!\cdots\!09\)\( \nu^{8} + \)\(45\!\cdots\!89\)\( \nu^{7} - \)\(25\!\cdots\!73\)\( \nu^{6} - \)\(41\!\cdots\!53\)\( \nu^{5} + \)\(16\!\cdots\!37\)\( \nu^{4} + \)\(34\!\cdots\!80\)\( \nu^{3} - \)\(11\!\cdots\!71\)\( \nu^{2} - \)\(63\!\cdots\!79\)\( \nu + \)\(25\!\cdots\!41\)\(\)\()/ \)\(92\!\cdots\!46\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(10\!\cdots\!09\)\( \nu^{15} + \)\(70\!\cdots\!43\)\( \nu^{14} + \)\(37\!\cdots\!35\)\( \nu^{13} + \)\(20\!\cdots\!72\)\( \nu^{12} - \)\(18\!\cdots\!56\)\( \nu^{11} + \)\(99\!\cdots\!54\)\( \nu^{10} + \)\(63\!\cdots\!12\)\( \nu^{9} + \)\(23\!\cdots\!17\)\( \nu^{8} - \)\(85\!\cdots\!35\)\( \nu^{7} + \)\(49\!\cdots\!03\)\( \nu^{6} + \)\(27\!\cdots\!27\)\( \nu^{5} + \)\(14\!\cdots\!58\)\( \nu^{4} - \)\(35\!\cdots\!32\)\( \nu^{3} + \)\(57\!\cdots\!44\)\( \nu^{2} + \)\(89\!\cdots\!54\)\( \nu - \)\(25\!\cdots\!35\)\(\)\()/ \)\(12\!\cdots\!02\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(39\!\cdots\!68\)\( \nu^{15} - \)\(15\!\cdots\!49\)\( \nu^{14} + \)\(14\!\cdots\!19\)\( \nu^{13} + \)\(15\!\cdots\!01\)\( \nu^{12} + \)\(69\!\cdots\!96\)\( \nu^{11} - \)\(10\!\cdots\!61\)\( \nu^{10} + \)\(24\!\cdots\!35\)\( \nu^{9} - \)\(95\!\cdots\!87\)\( \nu^{8} + \)\(31\!\cdots\!06\)\( \nu^{7} - \)\(90\!\cdots\!99\)\( \nu^{6} + \)\(10\!\cdots\!87\)\( \nu^{5} - \)\(92\!\cdots\!31\)\( \nu^{4} + \)\(13\!\cdots\!90\)\( \nu^{3} - \)\(53\!\cdots\!87\)\( \nu^{2} + \)\(38\!\cdots\!59\)\( \nu + \)\(42\!\cdots\!29\)\(\)\()/ \)\(45\!\cdots\!62\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(17\!\cdots\!50\)\( \nu^{15} - \)\(70\!\cdots\!17\)\( \nu^{14} - \)\(72\!\cdots\!11\)\( \nu^{13} - \)\(69\!\cdots\!84\)\( \nu^{12} - \)\(31\!\cdots\!60\)\( \nu^{11} - \)\(12\!\cdots\!97\)\( \nu^{10} - \)\(13\!\cdots\!87\)\( \nu^{9} + \)\(40\!\cdots\!26\)\( \nu^{8} - \)\(14\!\cdots\!36\)\( \nu^{7} - \)\(51\!\cdots\!45\)\( \nu^{6} - \)\(59\!\cdots\!55\)\( \nu^{5} + \)\(40\!\cdots\!30\)\( \nu^{4} - \)\(48\!\cdots\!82\)\( \nu^{3} - \)\(52\!\cdots\!97\)\( \nu^{2} - \)\(14\!\cdots\!51\)\( \nu - \)\(17\!\cdots\!40\)\(\)\()/ \)\(16\!\cdots\!28\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(47\!\cdots\!55\)\( \nu^{15} + \)\(29\!\cdots\!63\)\( \nu^{14} + \)\(17\!\cdots\!79\)\( \nu^{13} + \)\(11\!\cdots\!90\)\( \nu^{12} - \)\(83\!\cdots\!55\)\( \nu^{11} + \)\(41\!\cdots\!57\)\( \nu^{10} + \)\(30\!\cdots\!81\)\( \nu^{9} + \)\(21\!\cdots\!74\)\( \nu^{8} - \)\(38\!\cdots\!69\)\( \nu^{7} + \)\(20\!\cdots\!09\)\( \nu^{6} + \)\(14\!\cdots\!45\)\( \nu^{5} + \)\(85\!\cdots\!20\)\( \nu^{4} - \)\(14\!\cdots\!61\)\( \nu^{3} + \)\(24\!\cdots\!63\)\( \nu^{2} + \)\(37\!\cdots\!11\)\( \nu + \)\(54\!\cdots\!94\)\(\)\()/ \)\(39\!\cdots\!02\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(32\!\cdots\!76\)\( \nu^{15} - \)\(33\!\cdots\!23\)\( \nu^{14} - \)\(18\!\cdots\!39\)\( \nu^{13} - \)\(12\!\cdots\!84\)\( \nu^{12} - \)\(57\!\cdots\!50\)\( \nu^{11} - \)\(61\!\cdots\!45\)\( \nu^{10} - \)\(32\!\cdots\!71\)\( \nu^{9} + \)\(68\!\cdots\!50\)\( \nu^{8} - \)\(25\!\cdots\!42\)\( \nu^{7} - \)\(29\!\cdots\!41\)\( \nu^{6} - \)\(14\!\cdots\!59\)\( \nu^{5} + \)\(71\!\cdots\!38\)\( \nu^{4} - \)\(55\!\cdots\!52\)\( \nu^{3} - \)\(11\!\cdots\!67\)\( \nu^{2} - \)\(20\!\cdots\!03\)\( \nu - \)\(25\!\cdots\!08\)\(\)\()/ \)\(19\!\cdots\!66\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(10\!\cdots\!01\)\( \nu^{15} + \)\(63\!\cdots\!11\)\( \nu^{14} + \)\(44\!\cdots\!12\)\( \nu^{13} + \)\(72\!\cdots\!78\)\( \nu^{12} - \)\(18\!\cdots\!80\)\( \nu^{11} + \)\(88\!\cdots\!20\)\( \nu^{10} + \)\(76\!\cdots\!33\)\( \nu^{9} + \)\(33\!\cdots\!95\)\( \nu^{8} - \)\(85\!\cdots\!53\)\( \nu^{7} + \)\(43\!\cdots\!05\)\( \nu^{6} + \)\(33\!\cdots\!48\)\( \nu^{5} + \)\(21\!\cdots\!14\)\( \nu^{4} - \)\(30\!\cdots\!46\)\( \nu^{3} + \)\(53\!\cdots\!16\)\( \nu^{2} + \)\(80\!\cdots\!47\)\( \nu + \)\(59\!\cdots\!81\)\(\)\()/ \)\(42\!\cdots\!22\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{14} + 5 \beta_{11} - \beta_{8} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 342 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{15} - 7 \beta_{14} - 13 \beta_{13} - 13 \beta_{12} - 7 \beta_{11} + 7 \beta_{10} - 28 \beta_{9} - 28 \beta_{8} + 157 \beta_{7} + 157 \beta_{6} + 41 \beta_{5} - 41 \beta_{4} + 553 \beta_{3} - 553 \beta_{2} - 97 \beta_{1} + 97\)\()/4\)
\(\nu^{4}\)\(=\)\(163 \beta_{15} + 79 \beta_{13} - 843 \beta_{10} - 198 \beta_{9} - 64 \beta_{7} + 269 \beta_{5} - 71 \beta_{4} - 71 \beta_{3} - 71 \beta_{2} - 44560\)
\(\nu^{5}\)\(=\)\((\)\(-1496 \beta_{15} + 1496 \beta_{14} - 2047 \beta_{13} + 2047 \beta_{12} + 417 \beta_{11} + 417 \beta_{10} - 5238 \beta_{9} + 5238 \beta_{8} + 24568 \beta_{7} - 24568 \beta_{6} - 75344 \beta_{5} - 75344 \beta_{4} - 3843 \beta_{3} - 3843 \beta_{2} + 5299 \beta_{1} + 5299\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-92451 \beta_{14} - 81136 \beta_{12} - 523953 \beta_{11} + 129075 \beta_{8} + 78256 \beta_{6} - 36503 \beta_{5} + 151104 \beta_{4} - 36503 \beta_{3} - 22029 \beta_{2} - 25601422 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(1044781 \beta_{15} + 1044781 \beta_{14} + 845557 \beta_{13} + 845557 \beta_{12} - 80373 \beta_{11} + 80373 \beta_{10} + 3483834 \beta_{9} + 3483834 \beta_{8} - 13191643 \beta_{7} - 13191643 \beta_{6} - 6402785 \beta_{5} + 6402785 \beta_{4} - 47510067 \beta_{3} + 47510067 \beta_{2} - 5582015 \beta_{1} + 5582015\)\()/4\)
\(\nu^{8}\)\(=\)\(-12704042 \beta_{15} - 16333579 \beta_{13} + 80518170 \beta_{10} + 20422200 \beta_{9} + 16195387 \beta_{7} - 22134178 \beta_{5} + 7004246 \beta_{4} + 1711978 \beta_{3} + 7004246 \beta_{2} + 3764173504\)
\(\nu^{9}\)\(=\)\((\)\(172651005 \beta_{15} - 172651005 \beta_{14} + 52776727 \beta_{13} - 52776727 \beta_{12} + 61321509 \beta_{11} + 61321509 \beta_{10} + 567056502 \beta_{9} - 567056502 \beta_{8} - 1695155929 \beta_{7} + 1695155929 \beta_{6} + 6469666126 \beta_{5} + 6469666126 \beta_{4} + 503909881 \beta_{3} + 503909881 \beta_{2} + 2274028613 \beta_{1} + 2274028613\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(6902289991 \beta_{14} + 12082786564 \beta_{12} + 49433985051 \beta_{11} - 12879763521 \beta_{8} - 11882130856 \beta_{6} + 5669850665 \beta_{5} - 13074457964 \beta_{4} + 5669850665 \beta_{3} + 194694443 \beta_{2} + 2231044630090 \beta_{1}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-111743533385 \beta_{15} - 111743533385 \beta_{14} + 14565283037 \beta_{13} + 14565283037 \beta_{12} + 68277298119 \beta_{11} - 68277298119 \beta_{10} - 366693656160 \beta_{9} - 366693656160 \beta_{8} + 847639903675 \beta_{7} + 847639903675 \beta_{6} + 703934146923 \beta_{5} - 703934146923 \beta_{4} + 4183549183859 \beta_{3} - 4183549183859 \beta_{2} + 2315920372097 \beta_{1} - 2315920372097\)\()/4\)
\(\nu^{12}\)\(=\)\(930489180369 \beta_{15} + 2139104560648 \beta_{13} - 7592699813517 \beta_{10} - 2032338076482 \beta_{9} - 2069856552361 \beta_{7} + 1921138610019 \beta_{5} - 1115238623711 \beta_{4} + 111199466463 \beta_{3} - 1115238623711 \beta_{2} - 332000524438189\)
\(\nu^{13}\)\(=\)\((\)\(-17880852446077 \beta_{15} + 17880852446077 \beta_{14} + 9101406164066 \beta_{13} - 9101406164066 \beta_{12} - 15330070159428 \beta_{11} - 15330070159428 \beta_{10} - 59056569341982 \beta_{9} + 59056569341982 \beta_{8} + 102765907530157 \beta_{7} - 102765907530157 \beta_{6} - 564754588847145 \beta_{5} - 564754588847145 \beta_{4} - 55604496349610 \beta_{3} - 55604496349610 \beta_{2} - 506243338698578 \beta_{1} - 506243338698578\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-497993622759701 \beta_{14} - 1473849965809096 \beta_{12} - 4669758783234003 \beta_{11} + 1284513331328631 \beta_{8} + 1402161695970004 \beta_{6} - 848793770477567 \beta_{5} + 1119200067718210 \beta_{4} - 848793770477567 \beta_{3} + 165313263610421 \beta_{2} - 198294012114428710 \beta_{1}\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(11361059747960403 \beta_{15} + 11361059747960403 \beta_{14} - 9575336721511049 \beta_{13} - 9575336721511049 \beta_{12} - 12484222759253871 \beta_{11} + 12484222759253871 \beta_{10} + 37936359163892730 \beta_{9} + 37936359163892730 \beta_{8} - 47723909723138317 \beta_{7} - 47723909723138317 \beta_{6} - 74277728152726735 \beta_{5} + 74277728152726735 \beta_{4} - 373221922451418181 \beta_{3} + 373221922451418181 \beta_{2} - 409273246637842709 \beta_{1} + 409273246637842709\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
−11.9644 11.9644i
4.51719 + 4.51719i
5.50749 + 5.50749i
12.6006 + 12.6006i
−12.1800 12.1800i
−2.77393 2.77393i
−6.36140 6.36140i
11.6544 + 11.6544i
−11.9644 + 11.9644i
4.51719 4.51719i
5.50749 5.50749i
12.6006 12.6006i
−12.1800 + 12.1800i
−2.77393 + 2.77393i
−6.36140 + 6.36140i
11.6544 11.6544i
0 3.00000i 0 −16.7103 0 −16.4816 8.44725i 0 −9.00000 0
223.2 0 3.00000i 0 −16.7103 0 16.4816 8.44725i 0 −9.00000 0
223.3 0 3.00000i 0 −4.35176 0 −7.09314 + 17.1081i 0 −9.00000 0
223.4 0 3.00000i 0 −4.35176 0 7.09314 + 17.1081i 0 −9.00000 0
223.5 0 3.00000i 0 10.4276 0 −9.40604 15.9539i 0 −9.00000 0
223.6 0 3.00000i 0 10.4276 0 9.40604 15.9539i 0 −9.00000 0
223.7 0 3.00000i 0 10.6345 0 −18.0158 + 4.29303i 0 −9.00000 0
223.8 0 3.00000i 0 10.6345 0 18.0158 + 4.29303i 0 −9.00000 0
223.9 0 3.00000i 0 −16.7103 0 −16.4816 + 8.44725i 0 −9.00000 0
223.10 0 3.00000i 0 −16.7103 0 16.4816 + 8.44725i 0 −9.00000 0
223.11 0 3.00000i 0 −4.35176 0 −7.09314 17.1081i 0 −9.00000 0
223.12 0 3.00000i 0 −4.35176 0 7.09314 17.1081i 0 −9.00000 0
223.13 0 3.00000i 0 10.4276 0 −9.40604 + 15.9539i 0 −9.00000 0
223.14 0 3.00000i 0 10.4276 0 9.40604 + 15.9539i 0 −9.00000 0
223.15 0 3.00000i 0 10.6345 0 −18.0158 4.29303i 0 −9.00000 0
223.16 0 3.00000i 0 10.6345 0 18.0158 4.29303i 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.p.a 16
4.b odd 2 1 inner 1344.4.p.a 16
7.b odd 2 1 1344.4.p.b yes 16
8.b even 2 1 1344.4.p.b yes 16
8.d odd 2 1 1344.4.p.b yes 16
28.d even 2 1 1344.4.p.b yes 16
56.e even 2 1 inner 1344.4.p.a 16
56.h odd 2 1 inner 1344.4.p.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.p.a 16 1.a even 1 1 trivial
1344.4.p.a 16 4.b odd 2 1 inner
1344.4.p.a 16 56.e even 2 1 inner
1344.4.p.a 16 56.h odd 2 1 inner
1344.4.p.b yes 16 7.b odd 2 1
1344.4.p.b yes 16 8.b even 2 1
1344.4.p.b yes 16 8.d odd 2 1
1344.4.p.b yes 16 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 260 T_{5}^{2} + 804 T_{5} + 8064 \) acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 9 + T^{2} )^{8} \)
$5$ \( ( 8064 + 804 T - 260 T^{2} + T^{4} )^{4} \)
$7$ \( \)\(19\!\cdots\!01\)\( - 319169065190448004 T^{2} + 685032986151892 T^{4} - 3751801668412 T^{6} + 23447488918 T^{8} - 31889788 T^{10} + 49492 T^{12} - 196 T^{14} + T^{16} \)
$11$ \( ( 10053320832 - 532252080 T^{2} + 5417280 T^{4} - 4620 T^{6} + T^{8} )^{2} \)
$13$ \( ( 624672 + 70056 T - 4604 T^{2} + 14 T^{3} + T^{4} )^{4} \)
$17$ \( ( 645802409513088 + 890623697328 T^{2} + 263318832 T^{4} + 28080 T^{6} + T^{8} )^{2} \)
$19$ \( ( 1880891559936 + 15860544768 T^{2} + 26099008 T^{4} + 9728 T^{6} + T^{8} )^{2} \)
$23$ \( ( 4988879424 + 8947979280 T^{2} + 30371760 T^{4} + 18576 T^{6} + T^{8} )^{2} \)
$29$ \( ( 188578587861692928 + 62949711816000 T^{2} + 4835701008 T^{4} + 123804 T^{6} + T^{8} )^{2} \)
$31$ \( ( 631297159314043392 - 106536332946240 T^{2} + 6204160656 T^{4} - 142620 T^{6} + T^{8} )^{2} \)
$37$ \( ( 518142430247067648 + 143519500567296 T^{2} + 9519540480 T^{4} + 208512 T^{6} + T^{8} )^{2} \)
$41$ \( ( \)\(10\!\cdots\!48\)\( + 4348641712095792 T^{2} + 65675342736 T^{4} + 427104 T^{6} + T^{8} )^{2} \)
$43$ \( ( 16734751747540992 - 20708867540160 T^{2} + 4721458608 T^{4} - 168372 T^{6} + T^{8} )^{2} \)
$47$ \( ( 46627091556634755072 - 5437257718901760 T^{2} + 105224799744 T^{4} - 634464 T^{6} + T^{8} )^{2} \)
$53$ \( ( 9338615013218738688 + 16060010828887872 T^{2} + 201114411408 T^{4} + 793308 T^{6} + T^{8} )^{2} \)
$59$ \( ( 28460511679864651776 + 9234955482485760 T^{2} + 143398037248 T^{4} + 683056 T^{6} + T^{8} )^{2} \)
$61$ \( ( -1820784672 + 81687048 T - 394604 T^{2} + 46 T^{3} + T^{4} )^{4} \)
$67$ \( ( \)\(42\!\cdots\!52\)\( - 374260459014855360 T^{2} + 1238711562864 T^{4} - 1819044 T^{6} + T^{8} )^{2} \)
$71$ \( ( 19268222762880576 + 32353327112945040 T^{2} + 996524903952 T^{4} + 1963872 T^{6} + T^{8} )^{2} \)
$73$ \( ( 77717817041551884288 + 20087887061707776 T^{2} + 354545300736 T^{4} + 1728096 T^{6} + T^{8} )^{2} \)
$79$ \( ( 43687244058769859584 + 18914375429190400 T^{2} + 709508650560 T^{4} + 1665844 T^{6} + T^{8} )^{2} \)
$83$ \( ( 4105135542077374464 + 12764678447932416 T^{2} + 284727547648 T^{4} + 1273408 T^{6} + T^{8} )^{2} \)
$89$ \( ( \)\(29\!\cdots\!12\)\( + 2769455779547306544 T^{2} + 5433582012624 T^{4} + 3975840 T^{6} + T^{8} )^{2} \)
$97$ \( ( \)\(55\!\cdots\!28\)\( + 3288616220215913472 T^{2} + 6247014769152 T^{4} + 4368480 T^{6} + T^{8} )^{2} \)
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