Properties

Label 4010.2.a.l
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 0
Dimension 17
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 3 x^{16} - 24 x^{15} + 70 x^{14} + 228 x^{13} - 638 x^{12} - 1075 x^{11} + 2854 x^{10} + 2544 x^{9} - 6415 x^{8} - 2573 x^{7} + 6456 x^{6} + 485 x^{5} - 1839 x^{4} + 67 x^{3} + 136 x^{2} - 8 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(517292 \nu^{16} - 36703983 \nu^{15} + 120153563 \nu^{14} + 741135014 \nu^{13} - 2742036456 \nu^{12} - 5467625787 \nu^{11} + 22855792597 \nu^{10} + 17099745144 \nu^{9} - 88325683716 \nu^{8} - 16507498270 \nu^{7} + 157200326808 \nu^{6} - 12946666199 \nu^{5} - 107935060143 \nu^{4} + 15957834852 \nu^{3} + 17753806708 \nu^{2} - 2951997384 \nu - 113062883\)\()/ 309515047 \)
\(\beta_{4}\)\(=\)\((\)\(4837565 \nu^{16} - 18840353 \nu^{15} - 132711821 \nu^{14} + 583174545 \nu^{13} + 1249910253 \nu^{12} - 6920964417 \nu^{11} - 3832677742 \nu^{10} + 39522423700 \nu^{9} - 8919225821 \nu^{8} - 108891365197 \nu^{7} + 76557067027 \nu^{6} + 117093675538 \nu^{5} - 129782414532 \nu^{4} - 4830190853 \nu^{3} + 35278488729 \nu^{2} - 5708177003 \nu - 1191999696\)\()/ 309515047 \)
\(\beta_{5}\)\(=\)\((\)\(-13533000 \nu^{16} + 13049122 \nu^{15} + 426684545 \nu^{14} - 396189248 \nu^{13} - 5245102879 \nu^{12} + 4595130637 \nu^{11} + 31725390068 \nu^{10} - 25754222788 \nu^{9} - 96979101208 \nu^{8} + 70835731577 \nu^{7} + 135261243188 \nu^{6} - 80263510040 \nu^{5} - 62713131246 \nu^{4} + 13581871588 \nu^{3} + 9021163238 \nu^{2} - 3427129740 \nu + 30057660\)\()/ 309515047 \)
\(\beta_{6}\)\(=\)\((\)\(29164450 \nu^{16} - 75243059 \nu^{15} - 712942292 \nu^{14} + 1640652448 \nu^{13} + 7106995572 \nu^{12} - 13595084080 \nu^{11} - 37253291795 \nu^{10} + 52998995529 \nu^{9} + 110121572033 \nu^{8} - 97602977359 \nu^{7} - 180587342672 \nu^{6} + 76514749485 \nu^{5} + 140382802694 \nu^{4} - 23538978861 \nu^{3} - 20618036438 \nu^{2} + 2530180189 \nu - 133252285\)\()/ 309515047 \)
\(\beta_{7}\)\(=\)\((\)\(31959553 \nu^{16} - 24013167 \nu^{15} - 1004560849 \nu^{14} + 653567956 \nu^{13} + 12518001396 \nu^{12} - 6839928369 \nu^{11} - 78482109680 \nu^{10} + 34708011918 \nu^{9} + 258464542138 \nu^{8} - 86805681473 \nu^{7} - 422865996766 \nu^{6} + 88683444929 \nu^{5} + 289410490306 \nu^{4} - 5721251743 \nu^{3} - 57121635689 \nu^{2} - 2388994893 \nu + 2506119991\)\()/ 309515047 \)
\(\beta_{8}\)\(=\)\((\)\(-42578708 \nu^{16} + 160512035 \nu^{15} + 898763550 \nu^{14} - 3638281524 \nu^{13} - 7069307355 \nu^{12} + 32010911487 \nu^{11} + 24429820825 \nu^{10} - 137168820096 \nu^{9} - 28457584155 \nu^{8} + 293829879846 \nu^{7} - 25061014027 \nu^{6} - 288436995914 \nu^{5} + 61669212078 \nu^{4} + 99969508807 \nu^{3} - 15556414844 \nu^{2} - 9595715223 \nu + 734658957\)\()/ 309515047 \)
\(\beta_{9}\)\(=\)\((\)\(-42697450 \nu^{16} + 88292181 \nu^{15} + 1139626837 \nu^{14} - 2036841696 \nu^{13} - 12352098451 \nu^{12} + 18190214717 \nu^{11} + 68978681863 \nu^{10} - 78753218317 \nu^{9} - 207100673241 \nu^{8} + 168438708936 \nu^{7} + 315848585860 \nu^{6} - 156468744478 \nu^{5} - 203405448987 \nu^{4} + 34335215026 \nu^{3} + 31186774911 \nu^{2} - 1005069177 \nu - 455720149\)\()/ 309515047 \)
\(\beta_{10}\)\(=\)\((\)\(-80578876 \nu^{16} + 238642459 \nu^{15} + 1945071424 \nu^{14} - 5575538085 \nu^{13} - 18628834335 \nu^{12} + 50928586221 \nu^{11} + 88901872108 \nu^{10} - 228720423838 \nu^{9} - 214756568868 \nu^{8} + 518021909831 \nu^{7} + 227835002030 \nu^{6} - 530055382254 \nu^{5} - 57832557032 \nu^{4} + 158965030720 \nu^{3} - 276287293 \nu^{2} - 10017945931 \nu - 171215480\)\()/ 309515047 \)
\(\beta_{11}\)\(=\)\((\)\(113055721 \nu^{16} - 299359609 \nu^{15} - 2829478710 \nu^{14} + 6970241055 \nu^{13} + 28404799275 \nu^{12} - 63236140377 \nu^{11} - 144528189191 \nu^{10} + 280528180900 \nu^{9} + 384895427290 \nu^{8} - 621335089574 \nu^{7} - 493500671988 \nu^{6} + 605792160984 \nu^{5} + 239307987195 \nu^{4} - 149037962658 \nu^{3} - 38782026641 \nu^{2} + 6224528889 \nu + 1635727447\)\()/ 309515047 \)
\(\beta_{12}\)\(=\)\((\)\(127487096 \nu^{16} - 350251889 \nu^{15} - 3118269472 \nu^{14} + 7992434926 \nu^{13} + 30622076627 \nu^{12} - 70628441588 \nu^{11} - 153317794321 \nu^{10} + 302367912296 \nu^{9} + 408725147379 \nu^{8} - 637341894902 \nu^{7} - 549342878384 \nu^{6} + 579787781632 \nu^{5} + 316758674053 \nu^{4} - 127655594119 \nu^{3} - 61108503632 \nu^{2} + 3861941654 \nu + 2741181689\)\()/ 309515047 \)
\(\beta_{13}\)\(=\)\((\)\(131491739 \nu^{16} - 366833857 \nu^{15} - 3252935367 \nu^{14} + 8598825523 \nu^{13} + 32151835641 \nu^{12} - 78734188184 \nu^{11} - 160079093236 \nu^{10} + 353717142183 \nu^{9} + 412232176906 \nu^{8} - 797062040583 \nu^{7} - 495886805261 \nu^{6} + 797125646421 \nu^{5} + 202932340288 \nu^{4} - 211409732500 \nu^{3} - 29094172573 \nu^{2} + 12492996538 \nu + 1300063361\)\()/ 309515047 \)
\(\beta_{14}\)\(=\)\((\)\(143660013 \nu^{16} - 441098466 \nu^{15} - 3444899317 \nu^{14} + 10415391985 \nu^{13} + 32519190575 \nu^{12} - 96340887561 \nu^{11} - 150418300729 \nu^{10} + 438713481887 \nu^{9} + 337091453304 \nu^{8} - 1005248527892 \nu^{7} - 277759642068 \nu^{6} + 1022958505739 \nu^{5} - 49144481099 \nu^{4} - 272833266410 \nu^{3} + 34234490641 \nu^{2} + 14665985027 \nu - 820821813\)\()/ 309515047 \)
\(\beta_{15}\)\(=\)\((\)\(-171291908 \nu^{16} + 481721894 \nu^{15} + 4204915171 \nu^{14} - 11215905374 \nu^{13} - 41202594024 \nu^{12} + 101813111297 \nu^{11} + 203184397219 \nu^{10} - 452195502624 \nu^{9} - 517697609720 \nu^{8} + 1003050087593 \nu^{7} + 614763282936 \nu^{6} - 979513317111 \nu^{5} - 244022585342 \nu^{4} + 243290631232 \nu^{3} + 27086625562 \nu^{2} - 9885630770 \nu - 456913120\)\()/ 309515047 \)
\(\beta_{16}\)\(=\)\((\)\(377743308 \nu^{16} - 1097265728 \nu^{15} - 9106434527 \nu^{14} + 25302804816 \nu^{13} + 87385432777 \nu^{12} - 226952470094 \nu^{11} - 421019731065 \nu^{10} + 992835992437 \nu^{9} + 1047036217357 \nu^{8} - 2161251749155 \nu^{7} - 1218459358638 \nu^{6} + 2069300872805 \nu^{5} + 489226814098 \nu^{4} - 519085786871 \nu^{3} - 64166588488 \nu^{2} + 26265785258 \nu + 2910598480\)\()/ 309515047 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - \beta_{10} + \beta_{7} + \beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{16} + \beta_{15} + \beta_{14} - \beta_{12} - 2 \beta_{11} + \beta_{8} + 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 8 \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{16} + \beta_{15} + \beta_{14} - \beta_{12} - 11 \beta_{11} - 9 \beta_{10} + \beta_{9} + \beta_{8} + 11 \beta_{7} + \beta_{6} - \beta_{4} + 10 \beta_{3} + 12 \beta_{2} + 30 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(11 \beta_{16} + 9 \beta_{15} + 12 \beta_{14} - 2 \beta_{13} - 11 \beta_{12} - 24 \beta_{11} + 2 \beta_{9} + 11 \beta_{8} + 24 \beta_{7} + \beta_{6} + 11 \beta_{5} - 12 \beta_{4} + 11 \beta_{3} + 62 \beta_{2} + 16 \beta_{1} + 100\)
\(\nu^{7}\)\(=\)\(16 \beta_{16} + 13 \beta_{15} + 16 \beta_{14} - 2 \beta_{13} - 18 \beta_{12} - 99 \beta_{11} - 68 \beta_{10} + 16 \beta_{9} + 17 \beta_{8} + 102 \beta_{7} + 14 \beta_{6} + 5 \beta_{5} - 17 \beta_{4} + 85 \beta_{3} + 118 \beta_{2} + 199 \beta_{1} + 47\)
\(\nu^{8}\)\(=\)\(99 \beta_{16} + 64 \beta_{15} + 117 \beta_{14} - 34 \beta_{13} - 102 \beta_{12} - 233 \beta_{11} - 2 \beta_{10} + 37 \beta_{9} + 100 \beta_{8} + 238 \beta_{7} + 17 \beta_{6} + 98 \beta_{5} - 120 \beta_{4} + 107 \beta_{3} + 489 \beta_{2} + 176 \beta_{1} + 682\)
\(\nu^{9}\)\(=\)\(184 \beta_{16} + 125 \beta_{15} + 195 \beta_{14} - 43 \beta_{13} - 224 \beta_{12} - 849 \beta_{11} - 486 \beta_{10} + 186 \beta_{9} + 202 \beta_{8} + 908 \beta_{7} + 149 \beta_{6} + 90 \beta_{5} - 215 \beta_{4} + 702 \beta_{3} + 1084 \beta_{2} + 1396 \beta_{1} + 532\)
\(\nu^{10}\)\(=\)\(849 \beta_{16} + 424 \beta_{15} + 1080 \beta_{14} - 407 \beta_{13} - 922 \beta_{12} - 2116 \beta_{11} - 38 \beta_{10} + 468 \beta_{9} + 878 \beta_{8} + 2230 \beta_{7} + 216 \beta_{6} + 829 \beta_{5} - 1141 \beta_{4} + 1026 \beta_{3} + 3940 \beta_{2} + 1685 \beta_{1} + 4905\)
\(\nu^{11}\)\(=\)\(1848 \beta_{16} + 1058 \beta_{15} + 2117 \beta_{14} - 622 \beta_{13} - 2383 \beta_{12} - 7176 \beta_{11} - 3392 \beta_{10} + 1920 \beta_{9} + 2081 \beta_{8} + 7968 \beta_{7} + 1443 \beta_{6} + 1112 \beta_{5} - 2381 \beta_{4} + 5794 \beta_{3} + 9644 \beta_{2} + 10140 \beta_{1} + 5326\)
\(\nu^{12}\)\(=\)\(7182 \beta_{16} + 2736 \beta_{15} + 9779 \beta_{14} - 4253 \beta_{13} - 8293 \beta_{12} - 18641 \beta_{11} - 469 \beta_{10} + 5097 \beta_{9} + 7675 \beta_{8} + 20349 \beta_{7} + 2437 \beta_{6} + 6928 \beta_{5} - 10596 \beta_{4} + 9730 \beta_{3} + 32306 \beta_{2} + 15121 \beta_{1} + 36564\)
\(\nu^{13}\)\(=\)\(17293 \beta_{16} + 8332 \beta_{15} + 21517 \beta_{14} - 7563 \beta_{13} - 23325 \beta_{12} - 60345 \beta_{11} - 23398 \beta_{10} + 18715 \beta_{9} + 19951 \beta_{8} + 69425 \beta_{7} + 13414 \beta_{6} + 11771 \beta_{5} - 24430 \beta_{4} + 48084 \beta_{3} + 84323 \beta_{2} + 75513 \beta_{1} + 50206\)
\(\nu^{14}\)\(=\)\(60478 \beta_{16} + 17465 \beta_{15} + 87776 \beta_{14} - 41619 \beta_{13} - 74174 \beta_{12} - 161446 \beta_{11} - 4795 \beta_{10} + 51463 \beta_{9} + 67073 \beta_{8} + 182709 \beta_{7} + 25706 \beta_{6} + 57852 \beta_{5} - 96852 \beta_{4} + 90883 \beta_{3} + 268314 \beta_{2} + 131210 \beta_{1} + 279909\)
\(\nu^{15}\)\(=\)\(155277 \beta_{16} + 62631 \beta_{15} + 209671 \beta_{14} - 83436 \beta_{13} - 217305 \beta_{12} - 506337 \beta_{11} - 160396 \beta_{10} + 176640 \beta_{9} + 183676 \beta_{8} + 602168 \beta_{7} + 122294 \beta_{6} + 114914 \beta_{5} - 238750 \beta_{4} + 401509 \beta_{3} + 729972 \beta_{2} + 573602 \beta_{1} + 457476\)
\(\nu^{16}\)\(=\)\(508006 \beta_{16} + 110854 \beta_{15} + 783879 \beta_{14} - 393189 \beta_{13} - 658089 \beta_{12} - 1383296 \beta_{11} - 44245 \beta_{10} + 497284 \beta_{9} + 585346 \beta_{8} + 1622062 \beta_{7} + 259478 \beta_{6} + 483987 \beta_{5} - 874463 \beta_{4} + 835576 \beta_{3} + 2248610 \beta_{2} + 1117763 \beta_{1} + 2188243\)

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
−2.56168
−2.39414
−2.24738
−1.71266
−1.70497
−0.548857
−0.291319
−0.103424
0.204745
0.278065
0.507814
1.40160
1.51896
2.15213
2.65670
2.89857
2.94584
−1.00000 −2.56168 1.00000 −1.00000 2.56168 0.101571 −1.00000 3.56218 1.00000
1.2 −1.00000 −2.39414 1.00000 −1.00000 2.39414 4.25395 −1.00000 2.73189 1.00000
1.3 −1.00000 −2.24738 1.00000 −1.00000 2.24738 2.64455 −1.00000 2.05073 1.00000
1.4 −1.00000 −1.71266 1.00000 −1.00000 1.71266 −2.30195 −1.00000 −0.0668018 1.00000
1.5 −1.00000 −1.70497 1.00000 −1.00000 1.70497 −1.19610 −1.00000 −0.0930693 1.00000
1.6 −1.00000 −0.548857 1.00000 −1.00000 0.548857 −1.86676 −1.00000 −2.69876 1.00000
1.7 −1.00000 −0.291319 1.00000 −1.00000 0.291319 −2.64183 −1.00000 −2.91513 1.00000
1.8 −1.00000 −0.103424 1.00000 −1.00000 0.103424 1.84281 −1.00000 −2.98930 1.00000
1.9 −1.00000 0.204745 1.00000 −1.00000 −0.204745 0.0646599 −1.00000 −2.95808 1.00000
1.10 −1.00000 0.278065 1.00000 −1.00000 −0.278065 4.59715 −1.00000 −2.92268 1.00000
1.11 −1.00000 0.507814 1.00000 −1.00000 −0.507814 1.00840 −1.00000 −2.74213 1.00000
1.12 −1.00000 1.40160 1.00000 −1.00000 −1.40160 −1.83860 −1.00000 −1.03552 1.00000
1.13 −1.00000 1.51896 1.00000 −1.00000 −1.51896 1.25913 −1.00000 −0.692760 1.00000
1.14 −1.00000 2.15213 1.00000 −1.00000 −2.15213 −4.40972 −1.00000 1.63168 1.00000
1.15 −1.00000 2.65670 1.00000 −1.00000 −2.65670 4.81257 −1.00000 4.05806 1.00000
1.16 −1.00000 2.89857 1.00000 −1.00000 −2.89857 −4.89781 −1.00000 5.40172 1.00000
1.17 −1.00000 2.94584 1.00000 −1.00000 −2.94584 2.56799 −1.00000 5.67796 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{17} - \cdots\)
\(T_{7}^{17} - \cdots\)
\(T_{11}^{17} + \cdots\)