Properties

Label 8030.2.a.bg
Level 8030
Weight 2
Character orbit 8030.a
Self dual Yes
Analytic conductor 64.120
Analytic rank 0
Dimension 15
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8030.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} + \beta_{1} q^{6} + \beta_{12} q^{7} + q^{8} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} + \beta_{1} q^{6} + \beta_{12} q^{7} + q^{8} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} - q^{10} - q^{11} + \beta_{1} q^{12} + ( \beta_{3} - \beta_{6} ) q^{13} + \beta_{12} q^{14} -\beta_{1} q^{15} + q^{16} + ( -\beta_{6} + \beta_{7} - \beta_{8} + \beta_{13} ) q^{17} + ( 1 + \beta_{1} + \beta_{2} ) q^{18} + ( 2 - \beta_{1} - \beta_{4} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{19} - q^{20} + ( \beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{21} - q^{22} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{23} + \beta_{1} q^{24} + q^{25} + ( \beta_{3} - \beta_{6} ) q^{26} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{27} + \beta_{12} q^{28} + ( 2 + \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{29} -\beta_{1} q^{30} + ( 1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} ) q^{31} + q^{32} -\beta_{1} q^{33} + ( -\beta_{6} + \beta_{7} - \beta_{8} + \beta_{13} ) q^{34} -\beta_{12} q^{35} + ( 1 + \beta_{1} + \beta_{2} ) q^{36} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{12} + \beta_{14} ) q^{37} + ( 2 - \beta_{1} - \beta_{4} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{38} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{39} - q^{40} + ( 2 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{8} - \beta_{11} + \beta_{12} ) q^{41} + ( \beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{42} + ( 2 - \beta_{1} - \beta_{6} - \beta_{9} - 2 \beta_{11} + \beta_{14} ) q^{43} - q^{44} + ( -1 - \beta_{1} - \beta_{2} ) q^{45} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{46} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{47} + \beta_{1} q^{48} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{49} + q^{50} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{51} + ( \beta_{3} - \beta_{6} ) q^{52} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{14} ) q^{53} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{54} + q^{55} + \beta_{12} q^{56} + ( -1 + 4 \beta_{1} - \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{57} + ( 2 + \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{58} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{59} -\beta_{1} q^{60} + ( 2 + \beta_{1} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{61} + ( 1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} ) q^{62} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{63} + q^{64} + ( -\beta_{3} + \beta_{6} ) q^{65} -\beta_{1} q^{66} + ( 4 - \beta_{1} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{67} + ( -\beta_{6} + \beta_{7} - \beta_{8} + \beta_{13} ) q^{68} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} ) q^{69} -\beta_{12} q^{70} + ( 4 + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} ) q^{72} - q^{73} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{12} + \beta_{14} ) q^{74} + \beta_{1} q^{75} + ( 2 - \beta_{1} - \beta_{4} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{76} -\beta_{12} q^{77} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{78} + ( 3 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{79} - q^{80} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{81} + ( 2 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{8} - \beta_{11} + \beta_{12} ) q^{82} + ( 4 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{13} + \beta_{14} ) q^{83} + ( \beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{84} + ( \beta_{6} - \beta_{7} + \beta_{8} - \beta_{13} ) q^{85} + ( 2 - \beta_{1} - \beta_{6} - \beta_{9} - 2 \beta_{11} + \beta_{14} ) q^{86} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{87} - q^{88} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{89} + ( -1 - \beta_{1} - \beta_{2} ) q^{90} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{91} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{92} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{93} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{94} + ( -2 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{12} - \beta_{14} ) q^{95} + \beta_{1} q^{96} + ( 5 - 3 \beta_{1} - \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} + 2 \beta_{13} + 2 \beta_{14} ) q^{97} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{98} + ( -1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} + 7q^{3} + 15q^{4} - 15q^{5} + 7q^{6} + 3q^{7} + 15q^{8} + 16q^{9} + O(q^{10}) \) \( 15q + 15q^{2} + 7q^{3} + 15q^{4} - 15q^{5} + 7q^{6} + 3q^{7} + 15q^{8} + 16q^{9} - 15q^{10} - 15q^{11} + 7q^{12} - q^{13} + 3q^{14} - 7q^{15} + 15q^{16} + 2q^{17} + 16q^{18} + 23q^{19} - 15q^{20} + 20q^{21} - 15q^{22} + 7q^{24} + 15q^{25} - q^{26} + 19q^{27} + 3q^{28} + 23q^{29} - 7q^{30} + 9q^{31} + 15q^{32} - 7q^{33} + 2q^{34} - 3q^{35} + 16q^{36} + 11q^{37} + 23q^{38} + 7q^{39} - 15q^{40} + 27q^{41} + 20q^{42} + 7q^{43} - 15q^{44} - 16q^{45} - 18q^{47} + 7q^{48} + 16q^{49} + 15q^{50} + 21q^{51} - q^{52} - 19q^{53} + 19q^{54} + 15q^{55} + 3q^{56} + 11q^{57} + 23q^{58} + 2q^{59} - 7q^{60} + 31q^{61} + 9q^{62} + 20q^{63} + 15q^{64} + q^{65} - 7q^{66} + 49q^{67} + 2q^{68} + 33q^{69} - 3q^{70} + 32q^{71} + 16q^{72} - 15q^{73} + 11q^{74} + 7q^{75} + 23q^{76} - 3q^{77} + 7q^{78} + 36q^{79} - 15q^{80} + 23q^{81} + 27q^{82} + 33q^{83} + 20q^{84} - 2q^{85} + 7q^{86} + 29q^{87} - 15q^{88} + 6q^{89} - 16q^{90} + 33q^{91} + 20q^{93} - 18q^{94} - 23q^{95} + 7q^{96} + 30q^{97} + 16q^{98} - 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 7 x^{14} - 6 x^{13} + 136 x^{12} - 149 x^{11} - 876 x^{10} + 1631 x^{9} + 2142 x^{8} - 5473 x^{7} - 1914 x^{6} + 7517 x^{5} + 392 x^{4} - 3966 x^{3} - 79 x^{2} + 491 x - 32\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-43181499 \nu^{14} + 311865932 \nu^{13} + 175296873 \nu^{12} - 5835678177 \nu^{11} + 7886278683 \nu^{10} + 34729700996 \nu^{9} - 77534359554 \nu^{8} - 68269172316 \nu^{7} + 240444356295 \nu^{6} + 21463419837 \nu^{5} - 294306936351 \nu^{4} + 35543489014 \nu^{3} + 118461897267 \nu^{2} - 4110529434 \nu - 1043521072\)\()/ 1472051974 \)
\(\beta_{4}\)\(=\)\((\)\(151113864 \nu^{14} - 856502009 \nu^{13} - 2141104338 \nu^{12} + 18174114793 \nu^{11} + 3073362381 \nu^{10} - 137864112717 \nu^{9} + 59761294616 \nu^{8} + 469144714654 \nu^{7} - 231069643498 \nu^{6} - 777906330437 \nu^{5} + 223470183643 \nu^{4} + 546519767435 \nu^{3} - 5531474020 \nu^{2} - 61603367457 \nu + 10563926796\)\()/ 1472051974 \)
\(\beta_{5}\)\(=\)\((\)\(-240146793 \nu^{14} + 1349405164 \nu^{13} + 3310464644 \nu^{12} - 28093591614 \nu^{11} - 3268616532 \nu^{10} + 206295429485 \nu^{9} - 103633474516 \nu^{8} - 664474700726 \nu^{7} + 381947309693 \nu^{6} + 1026239863819 \nu^{5} - 365120483074 \nu^{4} - 678211848007 \nu^{3} + 3929112525 \nu^{2} + 81511550913 \nu + 637595355\)\()/ 736025987 \)
\(\beta_{6}\)\(=\)\((\)\(-393129354 \nu^{14} + 2205793563 \nu^{13} + 5420958086 \nu^{12} - 45934140616 \nu^{11} - 5198415872 \nu^{10} + 337253098892 \nu^{9} - 172983208075 \nu^{8} - 1084416017902 \nu^{7} + 647239619364 \nu^{6} + 1664981512591 \nu^{5} - 651430192655 \nu^{4} - 1091846609004 \nu^{3} + 53669647963 \nu^{2} + 135203428354 \nu - 4867233006\)\()/ 736025987 \)
\(\beta_{7}\)\(=\)\((\)\(-795515309 \nu^{14} + 4429334644 \nu^{13} + 11257805713 \nu^{12} - 92926760279 \nu^{11} - 16038219349 \nu^{10} + 690770757964 \nu^{9} - 314951595170 \nu^{8} - 2266703207886 \nu^{7} + 1221631289563 \nu^{6} + 3569723934171 \nu^{5} - 1214745294905 \nu^{4} - 2404280908366 \nu^{3} + 55691400415 \nu^{2} + 313899580988 \nu - 13759637204\)\()/ 1472051974 \)
\(\beta_{8}\)\(=\)\((\)\(-994410369 \nu^{14} + 5597120447 \nu^{13} + 13686695073 \nu^{12} - 116666528026 \nu^{11} - 12670511954 \nu^{10} + 858064956493 \nu^{9} - 440072471098 \nu^{8} - 2768090362598 \nu^{7} + 1637062417475 \nu^{6} + 4271090527082 \nu^{5} - 1622169591770 \nu^{4} - 2811385228067 \nu^{3} + 105082979077 \nu^{2} + 339270075981 \nu - 18366549708\)\()/ 1472051974 \)
\(\beta_{9}\)\(=\)\((\)\(1069761995 \nu^{14} - 5991781794 \nu^{13} - 14929676803 \nu^{12} + 125312551991 \nu^{11} + 17595342509 \nu^{10} - 926724131214 \nu^{9} + 448071389116 \nu^{8} + 3015600682420 \nu^{7} - 1697713859601 \nu^{6} - 4697245275867 \nu^{5} + 1673615011297 \nu^{4} + 3107314800418 \nu^{3} - 72165090085 \nu^{2} - 362635958812 \nu + 12810965580\)\()/ 1472051974 \)
\(\beta_{10}\)\(=\)\((\)\(-717823592 \nu^{14} + 4066794759 \nu^{13} + 9723141475 \nu^{12} - 84590402393 \nu^{11} - 5728278799 \nu^{10} + 619816571261 \nu^{9} - 344658277806 \nu^{8} - 1985869886386 \nu^{7} + 1278303931904 \nu^{6} + 3034165596685 \nu^{5} - 1335474480560 \nu^{4} - 1981981811698 \nu^{3} + 185958680180 \nu^{2} + 248067908447 \nu - 19927820364\)\()/ 736025987 \)
\(\beta_{11}\)\(=\)\((\)\(-1615094007 \nu^{14} + 9152836614 \nu^{13} + 21956198693 \nu^{12} - 190633964393 \nu^{11} - 14788558357 \nu^{10} + 1400283950824 \nu^{9} - 757944304578 \nu^{8} - 4507662531962 \nu^{7} + 2796518477973 \nu^{6} + 6936568275017 \nu^{5} - 2822674553407 \nu^{4} - 4556519727970 \nu^{3} + 257871227455 \nu^{2} + 554035957086 \nu - 31966415456\)\()/ 1472051974 \)
\(\beta_{12}\)\(=\)\((\)\(1750079193 \nu^{14} - 9894268181 \nu^{13} - 23804879799 \nu^{12} + 205776820544 \nu^{11} + 16491866164 \nu^{10} - 1507631502367 \nu^{9} + 816350417674 \nu^{8} + 4830835828988 \nu^{7} - 3012709029649 \nu^{6} - 7384044873836 \nu^{5} + 3054395671594 \nu^{4} + 4812736302735 \nu^{3} - 310791830327 \nu^{2} - 581915785975 \nu + 38701296324\)\()/ 1472051974 \)
\(\beta_{13}\)\(=\)\((\)\(917646164 \nu^{14} - 5144440308 \nu^{13} - 12683340797 \nu^{12} + 107209799902 \nu^{11} + 12588455386 \nu^{10} - 788006691504 \nu^{9} + 402428700994 \nu^{8} + 2537244016427 \nu^{7} - 1516698439194 \nu^{6} - 3897390863397 \nu^{5} + 1548275603127 \nu^{4} + 2552288392238 \nu^{3} - 158107913614 \nu^{2} - 315720809991 \nu + 20954746371\)\()/ 736025987 \)
\(\beta_{14}\)\(=\)\((\)\(-3971985085 \nu^{14} + 22439444825 \nu^{13} + 54093078921 \nu^{12} - 466825906562 \nu^{11} - 38345255506 \nu^{10} + 3421770842673 \nu^{9} - 1851633389436 \nu^{8} - 10971407637564 \nu^{7} + 6863796132185 \nu^{6} + 16782339092388 \nu^{5} - 7028205018024 \nu^{4} - 10961865196113 \nu^{3} + 791520538587 \nu^{2} + 1358762823395 \nu - 92707558986\)\()/ 1472051974 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - \beta_{12} + \beta_{10} + \beta_{6} + \beta_{2} + 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + 10 \beta_{2} + 10 \beta_{1} + 29\)
\(\nu^{5}\)\(=\)\(-13 \beta_{14} - \beta_{13} - 15 \beta_{12} + 11 \beta_{10} + \beta_{9} - 2 \beta_{8} + 3 \beta_{7} + 10 \beta_{6} + \beta_{5} + 2 \beta_{3} + 12 \beta_{2} + 71 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-6 \beta_{14} + 10 \beta_{13} - 21 \beta_{12} - 14 \beta_{11} + 2 \beta_{10} - 11 \beta_{9} - 11 \beta_{8} + 17 \beta_{7} + 13 \beta_{6} - \beta_{5} - 2 \beta_{4} + 18 \beta_{3} + 92 \beta_{2} + 103 \beta_{1} + 239\)
\(\nu^{7}\)\(=\)\(-144 \beta_{14} - 26 \beta_{13} - 174 \beta_{12} + 2 \beta_{11} + 103 \beta_{10} + 15 \beta_{9} - 32 \beta_{8} + 51 \beta_{7} + 83 \beta_{6} + 17 \beta_{5} - 4 \beta_{4} + 44 \beta_{3} + 125 \beta_{2} + 660 \beta_{1} + 119\)
\(\nu^{8}\)\(=\)\(-133 \beta_{14} + 60 \beta_{13} - 299 \beta_{12} - 148 \beta_{11} + 42 \beta_{10} - 96 \beta_{9} - 94 \beta_{8} + 224 \beta_{7} + 120 \beta_{6} - 9 \beta_{5} - 36 \beta_{4} + 254 \beta_{3} + 842 \beta_{2} + 1099 \beta_{1} + 2064\)
\(\nu^{9}\)\(=\)\(-1550 \beta_{14} - 436 \beta_{13} - 1884 \beta_{12} + 42 \beta_{11} + 954 \beta_{10} + 188 \beta_{9} - 361 \beta_{8} + 676 \beta_{7} + 620 \beta_{6} + 238 \beta_{5} - 85 \beta_{4} + 683 \beta_{3} + 1270 \beta_{2} + 6322 \beta_{1} + 1316\)
\(\nu^{10}\)\(=\)\(-2067 \beta_{14} + 47 \beta_{13} - 3737 \beta_{12} - 1433 \beta_{11} + 641 \beta_{10} - 763 \beta_{9} - 736 \beta_{8} + 2702 \beta_{7} + 876 \beta_{6} + 32 \beta_{5} - 480 \beta_{4} + 3242 \beta_{3} + 7777 \beta_{2} + 11887 \beta_{1} + 18300\)
\(\nu^{11}\)\(=\)\(-16565 \beta_{14} - 6067 \beta_{13} - 20005 \beta_{12} + 533 \beta_{11} + 8984 \beta_{10} + 2175 \beta_{9} - 3575 \beta_{8} + 8254 \beta_{7} + 3998 \beta_{6} + 3081 \beta_{5} - 1237 \beta_{4} + 9207 \beta_{3} + 12903 \beta_{2} + 61894 \beta_{1} + 14655\)
\(\nu^{12}\)\(=\)\(-27851 \beta_{14} - 5541 \beta_{13} - 44232 \beta_{12} - 13497 \beta_{11} + 8626 \beta_{10} - 5625 \beta_{9} - 5533 \beta_{8} + 31355 \beta_{7} + 4206 \beta_{6} + 2376 \beta_{5} - 5706 \beta_{4} + 39155 \beta_{3} + 72745 \beta_{2} + 128952 \beta_{1} + 165526\)
\(\nu^{13}\)\(=\)\(-176606 \beta_{14} - 76562 \beta_{13} - 211771 \beta_{12} + 5037 \beta_{11} + 86425 \beta_{10} + 23902 \beta_{9} - 33323 \beta_{8} + 97094 \beta_{7} + 17774 \beta_{6} + 38126 \beta_{5} - 15482 \beta_{4} + 115233 \beta_{3} + 131835 \beta_{2} + 616111 \beta_{1} + 162718\)
\(\nu^{14}\)\(=\)\(-348250 \beta_{14} - 115865 \beta_{13} - 509659 \beta_{12} - 126936 \beta_{11} + 108588 \beta_{10} - 37570 \beta_{9} - 40806 \beta_{8} + 356710 \beta_{7} - 11548 \beta_{6} + 50097 \beta_{5} - 64005 \beta_{4} + 457075 \beta_{3} + 689628 \beta_{2} + 1397316 \beta_{1} + 1523442\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.92483
−2.60943
−1.49337
−1.29384
−0.765121
−0.490398
0.0684817
0.348096
1.35747
1.35961
2.15389
2.24233
2.74529
3.03129
3.27053
1.00000 −2.92483 1.00000 −1.00000 −2.92483 0.830562 1.00000 5.55462 −1.00000
1.2 1.00000 −2.60943 1.00000 −1.00000 −2.60943 −1.48886 1.00000 3.80914 −1.00000
1.3 1.00000 −1.49337 1.00000 −1.00000 −1.49337 −0.532219 1.00000 −0.769851 −1.00000
1.4 1.00000 −1.29384 1.00000 −1.00000 −1.29384 −4.27001 1.00000 −1.32597 −1.00000
1.5 1.00000 −0.765121 1.00000 −1.00000 −0.765121 2.95797 1.00000 −2.41459 −1.00000
1.6 1.00000 −0.490398 1.00000 −1.00000 −0.490398 −1.29403 1.00000 −2.75951 −1.00000
1.7 1.00000 0.0684817 1.00000 −1.00000 0.0684817 −0.682969 1.00000 −2.99531 −1.00000
1.8 1.00000 0.348096 1.00000 −1.00000 0.348096 4.26660 1.00000 −2.87883 −1.00000
1.9 1.00000 1.35747 1.00000 −1.00000 1.35747 0.608019 1.00000 −1.15727 −1.00000
1.10 1.00000 1.35961 1.00000 −1.00000 1.35961 −4.65260 1.00000 −1.15147 −1.00000
1.11 1.00000 2.15389 1.00000 −1.00000 2.15389 1.68510 1.00000 1.63923 −1.00000
1.12 1.00000 2.24233 1.00000 −1.00000 2.24233 4.69629 1.00000 2.02806 −1.00000
1.13 1.00000 2.74529 1.00000 −1.00000 2.74529 −3.29124 1.00000 4.53661 −1.00000
1.14 1.00000 3.03129 1.00000 −1.00000 3.03129 3.50775 1.00000 6.18873 −1.00000
1.15 1.00000 3.27053 1.00000 −1.00000 3.27053 0.659654 1.00000 7.69640 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)
\(73\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\(T_{3}^{15} - \cdots\)
\(T_{7}^{15} - \cdots\)