Properties

Label 1386.2.bk.b
Level $1386$
Weight $2$
Character orbit 1386.bk
Analytic conductor $11.067$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + 130215 x^{8} - 239606 x^{7} + 378750 x^{6} - 477124 x^{5} + 493030 x^{4} - 386266 x^{3} + 223844 x^{2} - 82874 x + 13417\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + ( 1 + \beta_{13} ) q^{4} + ( -1 - \beta_{9} - \beta_{13} ) q^{5} + ( 1 + \beta_{1} + \beta_{8} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{7} + ( \beta_{11} + \beta_{12} ) q^{8} +O(q^{10})\) \( q + \beta_{12} q^{2} + ( 1 + \beta_{13} ) q^{4} + ( -1 - \beta_{9} - \beta_{13} ) q^{5} + ( 1 + \beta_{1} + \beta_{8} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{7} + ( \beta_{11} + \beta_{12} ) q^{8} + ( \beta_{7} - \beta_{11} - \beta_{12} ) q^{10} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{11} + ( -1 - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{13} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{14} ) q^{14} + \beta_{13} q^{16} + ( 1 - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{17} + ( -1 - \beta_{1} - \beta_{4} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{19} + ( \beta_{8} - \beta_{9} - \beta_{13} ) q^{20} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{13} ) q^{22} + ( -2 \beta_{1} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{23} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{25} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{12} - \beta_{14} ) q^{26} + ( 1 + \beta_{1} + \beta_{2} + \beta_{8} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{28} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{29} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{31} + \beta_{11} q^{32} + ( -\beta_{2} - \beta_{3} - \beta_{7} + 2 \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{34} + ( -2 - 4 \beta_{1} - \beta_{2} - 2 \beta_{4} - 3 \beta_{8} - \beta_{9} + \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{35} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 4 \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{37} + ( 1 + \beta_{5} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{38} + ( \beta_{7} + \beta_{10} - \beta_{11} ) q^{40} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{41} + ( 1 + \beta_{1} + 2 \beta_{4} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{13} ) q^{43} + ( -\beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{15} ) q^{44} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 3 \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{46} + ( -3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{47} + ( 1 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{10} + 2 \beta_{11} + \beta_{15} ) q^{49} + ( -1 - \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{50} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{52} + ( 2 - \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{53} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + 4 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{55} + ( -1 - \beta_{13} + \beta_{14} ) q^{56} + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{58} + ( 4 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{59} + ( -\beta_{3} - \beta_{4} - 3 \beta_{7} - \beta_{8} + \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{13} - \beta_{15} ) q^{61} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{62} - q^{64} + ( 8 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{12} + 5 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{65} + ( -4 + \beta_{3} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} - 5 \beta_{12} - 4 \beta_{13} + \beta_{14} - \beta_{15} ) q^{67} + ( -1 - \beta_{2} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{68} + ( 1 + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{70} + ( 2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{71} + ( 6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} + 6 \beta_{13} - 2 \beta_{14} ) q^{73} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{74} + ( 1 - \beta_{1} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{76} + ( 3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 4 \beta_{12} + 3 \beta_{13} ) q^{77} + ( 8 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{79} + ( 1 + \beta_{8} ) q^{80} + ( 1 + \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{82} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{83} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + \beta_{12} - 6 \beta_{13} + \beta_{14} - \beta_{15} ) q^{85} + ( -2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + 3 \beta_{11} + \beta_{12} - \beta_{13} ) q^{86} + ( 1 - \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} - \beta_{14} ) q^{88} + ( 2 + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{13} - 2 \beta_{15} ) q^{89} + ( -6 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} - \beta_{5} - 4 \beta_{8} + \beta_{9} + 3 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} - 3 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{91} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{7} - 2 \beta_{10} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{92} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{11} + \beta_{12} - 3 \beta_{14} + 2 \beta_{15} ) q^{94} + ( 4 + 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{11} - 2 \beta_{14} - 2 \beta_{15} ) q^{95} + ( -5 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - 7 \beta_{13} - \beta_{14} + \beta_{15} ) q^{97} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{4} - 12q^{5} + 6q^{7} + O(q^{10}) \) \( 16q + 8q^{4} - 12q^{5} + 6q^{7} - 2q^{10} + 4q^{11} - 8q^{14} - 8q^{16} + 10q^{19} + 2q^{22} + 4q^{23} + 10q^{25} - 12q^{26} + 6q^{31} - 8q^{35} + 14q^{37} + 12q^{38} + 2q^{40} + 32q^{41} - 4q^{44} - 18q^{46} + 24q^{47} - 6q^{49} + 14q^{55} - 4q^{56} - 28q^{61} - 8q^{62} - 16q^{64} + 72q^{65} - 16q^{67} - 30q^{70} + 56q^{71} + 44q^{73} + 24q^{74} + 20q^{76} + 52q^{77} + 30q^{79} + 12q^{80} - 12q^{82} + 8q^{83} + 12q^{86} - 2q^{88} + 36q^{89} - 8q^{91} + 8q^{92} - 14q^{94} + 72q^{95} - 40q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + 130215 x^{8} - 239606 x^{7} + 378750 x^{6} - 477124 x^{5} + 493030 x^{4} - 386266 x^{3} + 223844 x^{2} - 82874 x + 13417\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-3325577 \nu^{14} + 23279039 \nu^{13} - 218220364 \nu^{12} + 1006694677 \nu^{11} - 4890679917 \nu^{10} + 15780182142 \nu^{9} - 48699356733 \nu^{8} + 110238860505 \nu^{7} - 222125567883 \nu^{6} + 336742486204 \nu^{5} - 415275809861 \nu^{4} + 371398223899 \nu^{3} - 230411582664 \nu^{2} + 86434816533 \nu - 13304183911\)\()/ 3261539424 \)
\(\beta_{2}\)\(=\)\((\)\(14496748 \nu^{15} + 165765669 \nu^{14} - 1387066295 \nu^{13} + 16433893584 \nu^{12} - 89388557825 \nu^{11} + 460987458741 \nu^{10} - 1704964101954 \nu^{9} + 5511712419345 \nu^{8} - 14215412394141 \nu^{7} + 30601359218347 \nu^{6} - 53771673709840 \nu^{5} + 74664957471809 \nu^{4} - 81879734308403 \nu^{3} + 64273622949996 \nu^{2} - 34535277031465 \nu + 8734585660679\)\()/ 316369324128 \)
\(\beta_{3}\)\(=\)\((\)\(14496748 \nu^{15} - 383216889 \nu^{14} + 2455811611 \nu^{13} - 20082727968 \nu^{12} + 79753758709 \nu^{11} - 375114507753 \nu^{10} + 1016663085714 \nu^{9} - 3103062983781 \nu^{8} + 5617482562641 \nu^{7} - 11089350694487 \nu^{6} + 11620451848424 \nu^{5} - 12919822583029 \nu^{4} + 2845093331383 \nu^{3} + 825486570276 \nu^{2} - 5168168247283 \nu + 1933997835005\)\()/ 316369324128 \)
\(\beta_{4}\)\(=\)\((\)\(-28661011 \nu^{15} + 376248067 \nu^{14} - 3019246022 \nu^{13} + 19609879751 \nu^{12} - 91540961349 \nu^{11} + 380021763240 \nu^{10} - 1215316946595 \nu^{9} + 3449204606697 \nu^{8} - 7754522847207 \nu^{7} + 14949580551542 \nu^{6} - 22774230237823 \nu^{5} + 27760712142815 \nu^{4} - 25755258117138 \nu^{3} + 16706236742943 \nu^{2} - 7914233731523 \nu + 1766457326490\)\()/ 316369324128 \)
\(\beta_{5}\)\(=\)\((\)\(-30920939 \nu^{15} - 47258327 \nu^{14} - 211956648 \nu^{13} - 7396232221 \nu^{12} + 30361391909 \nu^{11} - 217943855310 \nu^{10} + 703406190993 \nu^{9} - 2539649789049 \nu^{8} + 5922057053211 \nu^{7} - 13228223786684 \nu^{6} + 21168894176673 \nu^{5} - 29136194430747 \nu^{4} + 28877960180876 \nu^{3} - 21487115641149 \nu^{2} + 10220884783775 \nu - 2047012171496\)\()/ 158184662064 \)
\(\beta_{6}\)\(=\)\((\)\(-30920939 \nu^{15} + 511072412 \nu^{14} - 4120271821 \nu^{13} + 28521203647 \nu^{12} - 134335126050 \nu^{11} + 572900637369 \nu^{10} - 1834246369401 \nu^{9} + 5281654047894 \nu^{8} - 11789222038392 \nu^{7} + 23033657253157 \nu^{6} - 34641417899267 \nu^{5} + 43240535558476 \nu^{4} - 39267317510445 \nu^{3} + 26479017902163 \nu^{2} - 10659357632440 \nu + 1740262265133\)\()/ 158184662064 \)
\(\beta_{7}\)\(=\)\((\)\(7915220 \nu^{15} - 40953259 \nu^{14} + 414028833 \nu^{13} - 1460644352 \nu^{12} + 7486016479 \nu^{11} - 18277094811 \nu^{10} + 59268665334 \nu^{9} - 96106377663 \nu^{8} + 208572400371 \nu^{7} - 200115377413 \nu^{6} + 311565411864 \nu^{5} - 169504852815 \nu^{4} + 313514371381 \nu^{3} - 241017203316 \nu^{2} + 288697703863 \nu - 132462757465\)\()/ 28760847648 \)
\(\beta_{8}\)\(=\)\((\)\(-7979 \nu^{15} + 68912 \nu^{14} - 628546 \nu^{13} + 3357412 \nu^{12} - 16625487 \nu^{11} + 61197354 \nu^{10} - 201214665 \nu^{9} + 526067616 \nu^{8} - 1200045765 \nu^{7} + 2196214414 \nu^{6} - 3408471857 \nu^{5} + 4104840184 \nu^{4} - 3987532800 \nu^{3} + 2721817314 \nu^{2} - 1298790385 \nu + 285645792\)\()/18627492\)
\(\beta_{9}\)\(=\)\((\)\(-7979 \nu^{15} + 68912 \nu^{14} - 628546 \nu^{13} + 3357412 \nu^{12} - 16625487 \nu^{11} + 61197354 \nu^{10} - 201214665 \nu^{9} + 526067616 \nu^{8} - 1200045765 \nu^{7} + 2196214414 \nu^{6} - 3408471857 \nu^{5} + 4104840184 \nu^{4} - 3987532800 \nu^{3} + 2721817314 \nu^{2} - 1317417877 \nu + 304273284\)\()/18627492\)
\(\beta_{10}\)\(=\)\((\)\(-6671 \nu^{15} + 41254 \nu^{14} - 402799 \nu^{13} + 1707247 \nu^{12} - 8612564 \nu^{11} + 26205779 \nu^{10} - 85020497 \nu^{9} + 186738624 \nu^{8} - 410011674 \nu^{7} + 634496219 \nu^{6} - 934157663 \nu^{5} + 967933718 \nu^{4} - 916480415 \nu^{3} + 580147099 \nu^{2} - 304374922 \nu + 82787651\)\()/7787936\)
\(\beta_{11}\)\(=\)\((\)\(24636003 \nu^{15} - 180367241 \nu^{14} + 1683650240 \nu^{13} - 8042787727 \nu^{12} + 39684270259 \nu^{11} - 133306414374 \nu^{10} + 429081442695 \nu^{9} - 1032074337351 \nu^{8} + 2272009170957 \nu^{7} - 3839497364232 \nu^{6} + 5691504149431 \nu^{5} - 6374489557805 \nu^{4} + 5941303854100 \nu^{3} - 3888260852079 \nu^{2} + 1867377806769 \nu - 463004009716\)\()/ 28760847648 \)
\(\beta_{12}\)\(=\)\((\)\(24636003 \nu^{15} - 189172804 \nu^{14} + 1745289181 \nu^{13} - 8640627827 \nu^{12} + 42470004626 \nu^{11} - 147354845901 \nu^{10} + 475256763393 \nu^{9} - 1181342381406 \nu^{8} + 2620154110008 \nu^{7} - 4588380736521 \nu^{6} + 6885584559083 \nu^{5} - 8007122573884 \nu^{4} + 7544240352965 \nu^{3} - 5042857783047 \nu^{2} + 2373229705776 \nu - 503813289929\)\()/ 28760847648 \)
\(\beta_{13}\)\(=\)\((\)\(15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} - 96035346 \nu^{10} + 317610396 \nu^{9} - 791678301 \nu^{8} + 1812360285 \nu^{7} - 3211318028 \nu^{6} + 5028988886 \nu^{5} - 5993427763 \nu^{4} + 5979185161 \nu^{3} - 4172380467 \nu^{2} + 2121539399 \nu - 518381770\)\()/18627492\)
\(\beta_{14}\)\(=\)\((\)\(-731076296 \nu^{15} + 5564679969 \nu^{14} - 51759660791 \nu^{13} + 255254450652 \nu^{12} - 1262781933077 \nu^{11} + 4374603032517 \nu^{10} - 14191982654406 \nu^{9} + 35285931287397 \nu^{8} - 78739325671977 \nu^{7} + 138259928421331 \nu^{6} - 209114172126772 \nu^{5} + 245045998417421 \nu^{4} - 234039065550755 \nu^{3} + 159802254505512 \nu^{2} - 77971936362277 \nu + 18336335470607\)\()/ 316369324128 \)
\(\beta_{15}\)\(=\)\((\)\(791321153 \nu^{15} - 5699211694 \nu^{14} + 53761281981 \nu^{13} - 254911441289 \nu^{12} + 1268932240096 \nu^{11} - 4251240695721 \nu^{10} + 13809836508279 \nu^{9} - 33245971930596 \nu^{8} + 74020750115370 \nu^{7} - 125650206707929 \nu^{6} + 189347503067457 \nu^{5} - 214060374517662 \nu^{4} + 205152860375869 \nu^{3} - 136001913845709 \nu^{2} + 68927049496618 \nu - 16525376998129\)\()/ 316369324128 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(-\beta_{9} + \beta_{8} + 1\)
\(\nu^{2}\)\(=\)\(\beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{7} - 2 \beta_{6} - 2 \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(2 \beta_{15} + \beta_{14} - 5 \beta_{13} - \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} - 12 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 4 \beta_{4} - \beta_{3} - \beta_{2} - 5 \beta_{1} - 14\)
\(\nu^{4}\)\(=\)\(-8 \beta_{15} - 10 \beta_{14} + 2 \beta_{13} + 5 \beta_{12} + 17 \beta_{11} + 10 \beta_{10} + 31 \beta_{9} - 13 \beta_{8} - 16 \beta_{7} + 20 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 15 \beta_{1} + 19\)
\(\nu^{5}\)\(=\)\(-28 \beta_{15} - 22 \beta_{14} + 64 \beta_{13} + 24 \beta_{12} + 23 \beta_{11} + 58 \beta_{10} - 69 \beta_{9} + 119 \beta_{8} + 24 \beta_{7} + 52 \beta_{6} - 8 \beta_{5} + 43 \beta_{4} + 12 \beta_{3} + 12 \beta_{2} + 74 \beta_{1} + 162\)
\(\nu^{6}\)\(=\)\(52 \beta_{15} + 75 \beta_{14} + 61 \beta_{13} - 74 \beta_{12} - 170 \beta_{11} - 65 \beta_{10} - 411 \beta_{9} + 264 \beta_{8} + 228 \beta_{7} - 176 \beta_{6} + 31 \beta_{5} + 149 \beta_{4} + 29 \beta_{3} + 53 \beta_{2} - 76 \beta_{1} - 5\)
\(\nu^{7}\)\(=\)\(325 \beta_{15} + 298 \beta_{14} - 657 \beta_{13} - 450 \beta_{12} - 323 \beta_{11} - 852 \beta_{10} + 373 \beta_{9} - 1066 \beta_{8} - 167 \beta_{7} - 719 \beta_{6} + 219 \beta_{5} - 344 \beta_{4} - 139 \beta_{3} - 55 \beta_{2} - 886 \beta_{1} - 1701\)
\(\nu^{8}\)\(=\)\(-228 \beta_{15} - 448 \beta_{14} - 1368 \beta_{13} + 585 \beta_{12} + 1711 \beta_{11} - 150 \beta_{10} + 5023 \beta_{9} - 3987 \beta_{8} - 3036 \beta_{7} + 1380 \beta_{6} - 128 \beta_{5} - 2090 \beta_{4} - 422 \beta_{3} - 746 \beta_{2} - 67 \beta_{1} - 1786\)
\(\nu^{9}\)\(=\)\(-3506 \beta_{15} - 3490 \beta_{14} + 5934 \beta_{13} + 6780 \beta_{12} + 4769 \beta_{11} + 10592 \beta_{10} + 824 \beta_{9} + 8212 \beta_{8} - 336 \beta_{7} + 9182 \beta_{6} - 3490 \beta_{5} + 1801 \beta_{4} + 1418 \beta_{3} - 544 \beta_{2} + 9812 \beta_{1} + 16429\)
\(\nu^{10}\)\(=\)\(-775 \beta_{15} + 1126 \beta_{14} + 20300 \beta_{13} - 160 \beta_{12} - 15824 \beta_{11} + 13530 \beta_{10} - 56633 \beta_{9} + 52830 \beta_{8} + 37407 \beta_{7} - 8116 \beta_{6} - 3201 \beta_{5} + 25763 \beta_{4} + 5773 \beta_{3} + 7951 \beta_{2} + 10306 \beta_{1} + 36293\)
\(\nu^{11}\)\(=\)\(35787 \beta_{15} + 37693 \beta_{14} - 47572 \beta_{13} - 87239 \beta_{12} - 66603 \beta_{11} - 115639 \beta_{10} - 67910 \beta_{9} - 44026 \beta_{8} + 36821 \beta_{7} - 110918 \beta_{6} + 43269 \beta_{5} + 5326 \beta_{4} - 12756 \beta_{3} + 18132 \beta_{2} - 101639 \beta_{1} - 143099\)
\(\nu^{12}\)\(=\)\(39984 \beta_{15} + 26340 \beta_{14} - 258878 \beta_{13} - 94424 \beta_{12} + 124230 \beta_{11} - 288350 \beta_{10} + 581676 \beta_{9} - 643364 \beta_{8} - 424150 \beta_{7} + 706 \beta_{6} + 98482 \beta_{5} - 289332 \beta_{4} - 73746 \beta_{3} - 66104 \beta_{2} - 215666 \beta_{1} - 542199\)
\(\nu^{13}\)\(=\)\(-342678 \beta_{15} - 375858 \beta_{14} + 321008 \beta_{13} + 984284 \beta_{12} + 862898 \beta_{11} + 1100808 \beta_{10} + 1390953 \beta_{9} - 90875 \beta_{8} - 832560 \beta_{7} + 1270202 \beta_{6} - 441664 \beta_{5} - 348692 \beta_{4} + 98730 \beta_{3} - 303334 \beta_{2} + 963960 \beta_{1} + 1048389\)
\(\nu^{14}\)\(=\)\(-729089 \beta_{15} - 672801 \beta_{14} + 3058731 \beta_{13} + 2237884 \beta_{12} - 659178 \beta_{11} + 4637781 \beta_{10} - 5273480 \beta_{9} + 7301078 \beta_{8} + 4374105 \beta_{7} + 1058408 \beta_{6} - 1805654 \beta_{5} + 2971940 \beta_{4} + 895656 \beta_{3} + 343122 \beta_{2} + 3421446 \beta_{1} + 7051750\)
\(\nu^{15}\)\(=\)\(3012048 \beta_{15} + 3370541 \beta_{14} - 1368219 \beta_{13} - 9668935 \beta_{12} - 10414656 \beta_{11} - 8491987 \beta_{10} - 21648505 \beta_{9} + 7773878 \beta_{8} + 14106296 \beta_{7} - 13734847 \beta_{6} + 3483128 \beta_{5} + 6940034 \beta_{4} - 565701 \beta_{3} + 3983641 \beta_{2} - 7907405 \beta_{1} - 4830300\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(-\beta_{13}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
0.500000 + 2.40229i
0.500000 + 1.35798i
0.500000 1.56688i
0.500000 3.19339i
0.500000 + 3.43554i
0.500000 + 0.921602i
0.500000 0.0286340i
0.500000 3.32851i
0.500000 2.40229i
0.500000 1.35798i
0.500000 + 1.56688i
0.500000 + 3.19339i
0.500000 3.43554i
0.500000 0.921602i
0.500000 + 0.0286340i
0.500000 + 3.32851i
−0.866025 + 0.500000i 0 0.500000 0.866025i −2.83045 + 1.63416i 0 2.54243 + 0.732142i 1.00000i 0 1.63416 2.83045i
703.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −1.92604 + 1.11200i 0 −2.45660 0.982398i 1.00000i 0 1.11200 1.92604i
703.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.606961 0.350429i 0 1.82993 1.91085i 1.00000i 0 −0.350429 + 0.606961i
703.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 2.01555 1.16368i 0 1.31629 + 2.29508i 1.00000i 0 −1.16368 + 2.01555i
703.5 0.866025 0.500000i 0 0.500000 0.866025i −3.72526 + 2.15078i 0 1.43173 + 2.22489i 1.00000i 0 −2.15078 + 3.72526i
703.6 0.866025 0.500000i 0 0.500000 0.866025i −1.54813 + 0.893814i 0 −0.165362 2.64058i 1.00000i 0 −0.893814 + 1.54813i
703.7 0.866025 0.500000i 0 0.500000 0.866025i −0.725202 + 0.418696i 0 −2.44037 + 1.02205i 1.00000i 0 −0.418696 + 0.725202i
703.8 0.866025 0.500000i 0 0.500000 0.866025i 2.13257 1.23124i 0 0.941950 2.47239i 1.00000i 0 1.23124 2.13257i
901.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.83045 1.63416i 0 2.54243 0.732142i 1.00000i 0 1.63416 + 2.83045i
901.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.92604 1.11200i 0 −2.45660 + 0.982398i 1.00000i 0 1.11200 + 1.92604i
901.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.606961 + 0.350429i 0 1.82993 + 1.91085i 1.00000i 0 −0.350429 0.606961i
901.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.01555 + 1.16368i 0 1.31629 2.29508i 1.00000i 0 −1.16368 2.01555i
901.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −3.72526 2.15078i 0 1.43173 2.22489i 1.00000i 0 −2.15078 3.72526i
901.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.54813 0.893814i 0 −0.165362 + 2.64058i 1.00000i 0 −0.893814 1.54813i
901.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.725202 0.418696i 0 −2.44037 1.02205i 1.00000i 0 −0.418696 0.725202i
901.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.13257 + 1.23124i 0 0.941950 + 2.47239i 1.00000i 0 1.23124 + 2.13257i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.bk.b 16
3.b odd 2 1 462.2.p.b yes 16
7.d odd 6 1 1386.2.bk.a 16
11.b odd 2 1 1386.2.bk.a 16
21.g even 6 1 462.2.p.a 16
21.g even 6 1 3234.2.e.a 16
21.h odd 6 1 3234.2.e.b 16
33.d even 2 1 462.2.p.a 16
77.i even 6 1 inner 1386.2.bk.b 16
231.k odd 6 1 462.2.p.b yes 16
231.k odd 6 1 3234.2.e.b 16
231.l even 6 1 3234.2.e.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.p.a 16 21.g even 6 1
462.2.p.a 16 33.d even 2 1
462.2.p.b yes 16 3.b odd 2 1
462.2.p.b yes 16 231.k odd 6 1
1386.2.bk.a 16 7.d odd 6 1
1386.2.bk.a 16 11.b odd 2 1
1386.2.bk.b 16 1.a even 1 1 trivial
1386.2.bk.b 16 77.i even 6 1 inner
3234.2.e.a 16 21.g even 6 1
3234.2.e.a 16 231.l even 6 1
3234.2.e.b 16 21.h odd 6 1
3234.2.e.b 16 231.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\(T_{5}^{16} + \cdots\)
\( T_{13}^{8} - 64 T_{13}^{6} + 8 T_{13}^{5} + 836 T_{13}^{4} - 1168 T_{13}^{3} - 592 T_{13}^{2} + 1216 T_{13} - 128 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( T^{16} \)
$5$ \( 35344 + 29328 T - 66900 T^{2} - 62244 T^{3} + 118381 T^{4} + 154872 T^{5} + 25139 T^{6} - 39396 T^{7} - 11772 T^{8} + 8472 T^{9} + 4381 T^{10} - 312 T^{11} - 484 T^{12} - 12 T^{13} + 47 T^{14} + 12 T^{15} + T^{16} \)
$7$ \( 5764801 - 4941258 T + 2470629 T^{2} - 168070 T^{3} - 328937 T^{4} + 187964 T^{5} - 13916 T^{6} - 28112 T^{7} + 16958 T^{8} - 4016 T^{9} - 284 T^{10} + 548 T^{11} - 137 T^{12} - 10 T^{13} + 21 T^{14} - 6 T^{15} + T^{16} \)
$11$ \( 214358881 - 77948684 T - 15944049 T^{2} + 3221020 T^{3} + 6163861 T^{4} - 1277760 T^{5} - 338074 T^{6} - 29304 T^{7} + 71702 T^{8} - 2664 T^{9} - 2794 T^{10} - 960 T^{11} + 421 T^{12} + 20 T^{13} - 9 T^{14} - 4 T^{15} + T^{16} \)
$13$ \( ( -128 + 1216 T - 592 T^{2} - 1168 T^{3} + 836 T^{4} + 8 T^{5} - 64 T^{6} + T^{8} )^{2} \)
$17$ \( 576 - 6624 T + 69528 T^{2} - 159780 T^{3} + 538417 T^{4} + 880544 T^{5} + 2791005 T^{6} + 1314372 T^{7} + 661234 T^{8} + 134304 T^{9} + 43709 T^{10} + 5340 T^{11} + 2082 T^{12} + 136 T^{13} + 53 T^{14} + T^{16} \)
$19$ \( 2768896 - 3940352 T + 12875776 T^{2} + 2782208 T^{3} + 23727360 T^{4} - 8339456 T^{5} + 6396800 T^{6} - 1621056 T^{7} + 860080 T^{8} - 205120 T^{9} + 72936 T^{10} - 13172 T^{11} + 3489 T^{12} - 530 T^{13} + 107 T^{14} - 10 T^{15} + T^{16} \)
$23$ \( 24039882304 - 37979937888 T + 51979552888 T^{2} - 18938176388 T^{3} + 8371183769 T^{4} - 1395574872 T^{5} + 569006503 T^{6} - 73933044 T^{7} + 25326352 T^{8} - 2171968 T^{9} + 697889 T^{10} - 47696 T^{11} + 13392 T^{12} - 532 T^{13} + 143 T^{14} - 4 T^{15} + T^{16} \)
$29$ \( 12810617856 + 11931674112 T^{2} + 3667420672 T^{4} + 491528720 T^{6} + 33087905 T^{8} + 1203204 T^{10} + 23910 T^{12} + 244 T^{14} + T^{16} \)
$31$ \( 262144 - 1376256 T + 2318336 T^{2} + 473088 T^{3} - 2094848 T^{4} - 308736 T^{5} + 1998848 T^{6} - 866688 T^{7} - 81488 T^{8} + 108960 T^{9} + 3296 T^{10} - 11520 T^{11} + 1441 T^{12} + 294 T^{13} - 37 T^{14} - 6 T^{15} + T^{16} \)
$37$ \( 125622042624 - 34637930496 T + 40451561472 T^{2} - 12484740096 T^{3} + 9620770048 T^{4} - 2631917056 T^{5} + 994476160 T^{6} - 189887424 T^{7} + 54018160 T^{8} - 8801536 T^{9} + 1843896 T^{10} - 199348 T^{11} + 27961 T^{12} - 2062 T^{13} + 263 T^{14} - 14 T^{15} + T^{16} \)
$41$ \( ( 256 - 8064 T + 2848 T^{2} + 6824 T^{3} - 4815 T^{4} + 952 T^{5} + 2 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$43$ \( 23084548096 + 25298960384 T^{2} + 8477803520 T^{4} + 1124077312 T^{6} + 71086992 T^{8} + 2303384 T^{10} + 38761 T^{12} + 318 T^{14} + T^{16} \)
$47$ \( 1344737098384 - 3457746300816 T + 3090667648540 T^{2} - 326590505388 T^{3} - 363709353443 T^{4} + 50476519572 T^{5} + 44065903899 T^{6} - 5594306520 T^{7} - 527953996 T^{8} + 90988980 T^{9} + 5374933 T^{10} - 1115364 T^{11} - 2484 T^{12} + 5976 T^{13} - 57 T^{14} - 24 T^{15} + T^{16} \)
$53$ \( 26488213504 + 61179127808 T + 111058636544 T^{2} + 72713118848 T^{3} + 40171071120 T^{4} + 9183945376 T^{5} + 2778511712 T^{6} + 171246848 T^{7} + 163718908 T^{8} + 2083056 T^{9} + 3016384 T^{10} + 36036 T^{11} + 40865 T^{12} + 232 T^{13} + 235 T^{14} + T^{16} \)
$59$ \( 109280491776 - 294630488064 T + 190208538240 T^{2} + 201062028288 T^{3} + 24074195440 T^{4} - 15710742144 T^{5} - 2352978552 T^{6} + 1019215632 T^{7} + 245570017 T^{8} - 8528784 T^{9} - 4201594 T^{10} + 70032 T^{11} + 52915 T^{12} - 266 T^{14} + T^{16} \)
$61$ \( 3489501184 - 3714447360 T + 11848985344 T^{2} + 9559013504 T^{3} + 16635270224 T^{4} + 2680338048 T^{5} + 1536277900 T^{6} + 219443376 T^{7} + 100720417 T^{8} + 12716608 T^{9} + 3059300 T^{10} + 399224 T^{11} + 69975 T^{12} + 6928 T^{13} + 620 T^{14} + 28 T^{15} + T^{16} \)
$67$ \( 2403352576 - 7497142272 T + 23480118784 T^{2} - 4344754048 T^{3} + 7183184528 T^{4} + 129355824 T^{5} + 1933517692 T^{6} + 56306316 T^{7} + 86888329 T^{8} + 6732460 T^{9} + 3037910 T^{10} + 195332 T^{11} + 38595 T^{12} + 2344 T^{13} + 350 T^{14} + 16 T^{15} + T^{16} \)
$71$ \( ( 4193232 - 2297712 T - 45068 T^{2} + 251544 T^{3} - 58124 T^{4} + 4360 T^{5} + 97 T^{6} - 28 T^{7} + T^{8} )^{2} \)
$73$ \( 119793516544 + 942044872704 T + 7545799049216 T^{2} - 961323204608 T^{3} + 623445557248 T^{4} - 102344933376 T^{5} + 38692855808 T^{6} - 6019252224 T^{7} + 1260078592 T^{8} - 189543936 T^{9} + 30397952 T^{10} - 3545344 T^{11} + 349904 T^{12} - 24272 T^{13} + 1316 T^{14} - 44 T^{15} + T^{16} \)
$79$ \( 1130708969104 + 4406943704592 T + 6164867690476 T^{2} + 1712977494492 T^{3} - 246923168675 T^{4} - 138506030502 T^{5} + 15304968387 T^{6} + 4732344210 T^{7} - 620713636 T^{8} - 102519378 T^{9} + 25951621 T^{10} - 1615398 T^{11} - 32868 T^{12} + 6390 T^{13} + 87 T^{14} - 30 T^{15} + T^{16} \)
$83$ \( ( 5604 + 12516 T - 263 T^{2} - 8396 T^{3} + 1031 T^{4} + 780 T^{5} - 141 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$89$ \( 96546188427264 - 151241105866752 T + 105379288645632 T^{2} - 41364533084160 T^{3} + 9579425505280 T^{4} - 1178899292160 T^{5} + 26915948544 T^{6} + 10521747456 T^{7} - 350878976 T^{8} - 253334784 T^{9} + 37971200 T^{10} - 1587840 T^{11} - 71984 T^{12} + 6336 T^{13} + 256 T^{14} - 36 T^{15} + T^{16} \)
$97$ \( 68597371921 + 325271379408 T^{2} + 446119112140 T^{4} + 163854186512 T^{6} + 4910736438 T^{8} + 58793584 T^{10} + 340460 T^{12} + 944 T^{14} + T^{16} \)
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