Properties

Label 4010.2.a.k
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 1
Dimension 15
CM No
Inner twists 1

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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: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 7 x^{14} - 7 x^{13} + 133 x^{12} - 99 x^{11} - 941 x^{10} + 1290 x^{9} + 3031 x^{8} - 5452 x^{7} - 4098 x^{6} + 9986 x^{5} + 850 x^{4} - 7216 x^{3} + 1688 x^{2} + 1441 x - 512\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(58437627 \nu^{14} - 534122576 \nu^{13} + 115180567 \nu^{12} + 10329491164 \nu^{11} - 16005728457 \nu^{10} - 76582202190 \nu^{9} + 150516117320 \nu^{8} + 276187879617 \nu^{7} - 577175461581 \nu^{6} - 500355197485 \nu^{5} + 991851100607 \nu^{4} + 406892526083 \nu^{3} - 654395507655 \nu^{2} - 84064848569 \nu + 117611134796\)\()/ 2559277300 \)
\(\beta_{3}\)\(=\)\((\)\(20542792 \nu^{14} - 146543981 \nu^{13} - 164096893 \nu^{12} + 2932664074 \nu^{11} - 1662959242 \nu^{10} - 22602084955 \nu^{9} + 23806335705 \nu^{8} + 84798973897 \nu^{7} - 101567638106 \nu^{6} - 158908340525 \nu^{5} + 181723177022 \nu^{4} + 132349309753 \nu^{3} - 121796392900 \nu^{2} - 30937083869 \nu + 23702683016\)\()/ 255927730 \)
\(\beta_{4}\)\(=\)\((\)\(129009841 \nu^{14} - 883037733 \nu^{13} - 1197107414 \nu^{12} + 17651323112 \nu^{11} - 7047514031 \nu^{10} - 135332605845 \nu^{9} + 123099367535 \nu^{8} + 501129474486 \nu^{7} - 540954449473 \nu^{6} - 914114037680 \nu^{5} + 976805539231 \nu^{4} + 725876949064 \nu^{3} - 659616482965 \nu^{2} - 162102470402 \nu + 125216059318\)\()/ 1279638650 \)
\(\beta_{5}\)\(=\)\((\)\(-58921429 \nu^{14} + 416412812 \nu^{13} + 480260011 \nu^{12} - 8324831388 \nu^{11} + 4686009699 \nu^{10} + 63902469990 \nu^{9} - 68500177060 \nu^{8} - 237554782959 \nu^{7} + 294875044927 \nu^{6} + 437620917495 \nu^{5} - 530713167269 \nu^{4} - 354801041001 \nu^{3} + 355957229305 \nu^{2} + 80515602803 \nu - 67211679492\)\()/ 511855460 \)
\(\beta_{6}\)\(=\)\((\)\(355634499 \nu^{14} - 2493292912 \nu^{13} - 2946774221 \nu^{12} + 49230821268 \nu^{11} - 25215809709 \nu^{10} - 372735449930 \nu^{9} + 372473016340 \nu^{8} + 1365149106929 \nu^{7} - 1568270885997 \nu^{6} - 2470692471645 \nu^{5} + 2745269900859 \nu^{4} + 1948867431071 \nu^{3} - 1792658570635 \nu^{2} - 414042980353 \nu + 330809703252\)\()/ 2559277300 \)
\(\beta_{7}\)\(=\)\((\)\(-357030859 \nu^{14} + 2078381242 \nu^{13} + 4813725111 \nu^{12} - 41262341138 \nu^{11} - 10365332281 \nu^{10} + 312320702630 \nu^{9} - 116638375190 \nu^{8} - 1130828000189 \nu^{7} + 709414616177 \nu^{6} + 1983178061645 \nu^{5} - 1413799065769 \nu^{4} - 1469719755011 \nu^{3} + 1004956026735 \nu^{2} + 305717451873 \nu - 192795588882\)\()/ 1279638650 \)
\(\beta_{8}\)\(=\)\((\)\(-91199013 \nu^{14} + 582879604 \nu^{13} + 1011640657 \nu^{12} - 11599830506 \nu^{11} + 1677789343 \nu^{10} + 88300145100 \nu^{9} - 62212304580 \nu^{8} - 323204937023 \nu^{7} + 295531472449 \nu^{6} + 577941929785 \nu^{5} - 550172826353 \nu^{4} - 442326826767 \nu^{3} + 380121257205 \nu^{2} + 92712343001 \nu - 72900927894\)\()/ 255927730 \)
\(\beta_{9}\)\(=\)\((\)\(-1161424079 \nu^{14} + 7214691402 \nu^{13} + 13595344791 \nu^{12} - 142730708728 \nu^{11} + 6317989589 \nu^{10} + 1078958237680 \nu^{9} - 673308909890 \nu^{8} - 3920595017759 \nu^{7} + 3327978734537 \nu^{6} + 6964442880895 \nu^{5} - 6271311559839 \nu^{4} - 5309427250941 \nu^{3} + 4356518712135 \nu^{2} + 1117219399263 \nu - 835598044392\)\()/ 2559277300 \)
\(\beta_{10}\)\(=\)\((\)\(-1173296833 \nu^{14} + 7231451704 \nu^{13} + 14132908607 \nu^{12} - 143965640456 \nu^{11} + 153207403 \nu^{10} + 1095593363710 \nu^{9} - 646517369380 \nu^{8} - 4006364293143 \nu^{7} + 3286835904099 \nu^{6} + 7154897368415 \nu^{5} - 6275827862253 \nu^{4} - 5476071340057 \nu^{3} + 4396936210545 \nu^{2} + 1159659716951 \nu - 845081247084\)\()/ 2559277300 \)
\(\beta_{11}\)\(=\)\((\)\(-371844021 \nu^{14} + 2450736548 \nu^{13} + 3766297784 \nu^{12} - 48621859147 \nu^{11} + 13776069811 \nu^{10} + 369500503845 \nu^{9} - 304198536685 \nu^{8} - 1355253643316 \nu^{7} + 1376687553763 \nu^{6} + 2447389181955 \nu^{5} - 2507543406986 \nu^{4} - 1918045318359 \nu^{3} + 1690697670940 \nu^{2} + 408737447487 \nu - 316360212508\)\()/ 639819325 \)
\(\beta_{12}\)\(=\)\((\)\(-2202353061 \nu^{14} + 13831059568 \nu^{13} + 25351582419 \nu^{12} - 274524650952 \nu^{11} + 20706458751 \nu^{10} + 2084529565470 \nu^{9} - 1341833245860 \nu^{8} - 7620298033831 \nu^{7} + 6526394884583 \nu^{6} + 13651845662955 \nu^{5} - 12187022546401 \nu^{4} - 10539653148669 \nu^{3} + 8354197042965 \nu^{2} + 2248242866867 \nu - 1574486438628\)\()/ 2559277300 \)
\(\beta_{13}\)\(=\)\((\)\(-1275103963 \nu^{14} + 8052863369 \nu^{13} + 14434590052 \nu^{12} - 159745442766 \nu^{11} + 17211325133 \nu^{10} + 1212073313185 \nu^{9} - 819621745905 \nu^{8} - 4427108700148 \nu^{7} + 3943631782139 \nu^{6} + 7925804013640 \nu^{5} - 7355352227083 \nu^{4} - 6119225069202 \nu^{3} + 5055329192595 \nu^{2} + 1302859800136 \nu - 960771925824\)\()/ 1279638650 \)
\(\beta_{14}\)\(=\)\((\)\(-1426055511 \nu^{14} + 8868477043 \nu^{13} + 16772024394 \nu^{12} - 176015990502 \nu^{11} + 6774287801 \nu^{10} + 1335565850995 \nu^{9} - 823368041735 \nu^{8} - 4872274029956 \nu^{7} + 4085111050933 \nu^{6} + 8687133324980 \nu^{5} - 7706263306751 \nu^{4} - 6637555814944 \nu^{3} + 5345561232915 \nu^{2} + 1390583026792 \nu - 1018945552978\)\()/ 1279638650 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{14} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-3 \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-17 \beta_{14} - 3 \beta_{13} + 5 \beta_{12} + 13 \beta_{11} + 5 \beta_{10} + 15 \beta_{9} + 16 \beta_{8} - 2 \beta_{7} + 12 \beta_{6} - 12 \beta_{5} + 7 \beta_{4} + 2 \beta_{3} + 11 \beta_{2} + 18 \beta_{1} + 17\)
\(\nu^{5}\)\(=\)\(-64 \beta_{14} - 8 \beta_{13} + 26 \beta_{12} + 38 \beta_{11} + 25 \beta_{10} + 47 \beta_{9} + 62 \beta_{8} - 16 \beta_{7} + 36 \beta_{6} - 36 \beta_{5} + 45 \beta_{4} + 14 \beta_{3} + 34 \beta_{2} + 92 \beta_{1} + 24\)
\(\nu^{6}\)\(=\)\(-291 \beta_{14} - 63 \beta_{13} + 113 \beta_{12} + 190 \beta_{11} + 118 \beta_{10} + 239 \beta_{9} + 272 \beta_{8} - 46 \beta_{7} + 166 \beta_{6} - 176 \beta_{5} + 175 \beta_{4} + 36 \beta_{3} + 148 \beta_{2} + 305 \beta_{1} + 165\)
\(\nu^{7}\)\(=\)\(-1180 \beta_{14} - 218 \beta_{13} + 503 \beta_{12} + 688 \beta_{11} + 503 \beta_{10} + 896 \beta_{9} + 1128 \beta_{8} - 240 \beta_{7} + 614 \beta_{6} - 648 \beta_{5} + 850 \beta_{4} + 180 \beta_{3} + 552 \beta_{2} + 1360 \beta_{1} + 468\)
\(\nu^{8}\)\(=\)\(-5098 \beta_{14} - 1146 \beta_{13} + 2134 \beta_{12} + 3105 \beta_{11} + 2248 \beta_{10} + 4040 \beta_{9} + 4785 \beta_{8} - 843 \beta_{7} + 2647 \beta_{6} - 2917 \beta_{5} + 3467 \beta_{4} + 569 \beta_{3} + 2310 \beta_{2} + 5265 \beta_{1} + 2333\)
\(\nu^{9}\)\(=\)\(-21215 \beta_{14} - 4459 \beta_{13} + 9167 \beta_{12} + 12348 \beta_{11} + 9418 \beta_{10} + 16284 \beta_{9} + 20152 \beta_{8} - 3831 \beta_{7} + 10663 \beta_{6} - 11692 \beta_{5} + 15430 \beta_{4} + 2536 \beta_{3} + 9290 \beta_{2} + 22573 \beta_{1} + 8640\)
\(\nu^{10}\)\(=\)\(-90328 \beta_{14} - 20460 \beta_{13} + 38737 \beta_{12} + 53525 \beta_{11} + 40822 \beta_{10} + 70440 \beta_{9} + 85094 \beta_{8} - 14902 \beta_{7} + 45240 \beta_{6} - 50749 \beta_{5} + 64270 \beta_{4} + 9244 \beta_{3} + 38858 \beta_{2} + 92417 \beta_{1} + 38650\)
\(\nu^{11}\)\(=\)\(-379511 \beta_{14} - 83664 \beta_{13} + 164653 \beta_{12} + 220960 \beta_{11} + 171450 \beta_{10} + 292222 \beta_{9} + 359487 \beta_{8} - 64557 \beta_{7} + 187961 \beta_{6} - 210223 \beta_{5} + 277076 \beta_{4} + 39605 \beta_{3} + 160979 \beta_{2} + 392001 \beta_{1} + 156303\)
\(\nu^{12}\)\(=\)\(-1608230 \beta_{14} - 364938 \beta_{13} + 695590 \beta_{12} + 942964 \beta_{11} + 732462 \beta_{10} + 1245514 \beta_{9} + 1518069 \beta_{8} - 263329 \beta_{7} + 794776 \beta_{6} - 898311 \beta_{5} + 1164318 \beta_{4} + 156360 \beta_{3} + 676249 \beta_{2} + 1637252 \beta_{1} + 674056\)
\(\nu^{13}\)\(=\)\(-6781632 \beta_{14} - 1522673 \beta_{13} + 2946121 \beta_{12} + 3949413 \beta_{11} + 3086141 \beta_{10} + 5225929 \beta_{9} + 6416439 \beta_{8} - 1122120 \beta_{7} + 3338232 \beta_{6} - 3768020 \beta_{5} + 4959102 \beta_{4} + 662156 \beta_{3} + 2834799 \beta_{2} + 6925183 \beta_{1} + 2805548\)
\(\nu^{14}\)\(=\)\(-28690546 \beta_{14} - 6514636 \beta_{13} + 12448701 \beta_{12} + 16753801 \beta_{11} + 13101570 \beta_{10} + 22155352 \beta_{9} + 27107353 \beta_{8} - 4672966 \beta_{7} + 14107397 \beta_{6} - 15996587 \beta_{5} + 20913321 \beta_{4} + 2716879 \beta_{3} + 11944811 \beta_{2} + 29136294 \beta_{1} + 11945030\)

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
−2.07374
0.679191
2.09866
4.22659
−2.62108
−1.71302
1.95650
1.81559
2.79590
−1.99921
0.698269
−0.560392
−1.05167
2.25248
0.495927
−1.00000 −3.34293 1.00000 −1.00000 3.34293 3.45446 −1.00000 8.17517 1.00000
1.2 −1.00000 −3.00420 1.00000 −1.00000 3.00420 −4.38135 −1.00000 6.02524 1.00000
1.3 −1.00000 −2.86623 1.00000 −1.00000 2.86623 0.981408 −1.00000 5.21525 1.00000
1.4 −1.00000 −2.24378 1.00000 −1.00000 2.24378 −2.96737 −1.00000 2.03457 1.00000
1.5 −1.00000 −1.88769 1.00000 −1.00000 1.88769 1.11871 −1.00000 0.563364 1.00000
1.6 −1.00000 −1.21181 1.00000 −1.00000 1.21181 0.441519 −1.00000 −1.53151 1.00000
1.7 −1.00000 −0.918908 1.00000 −1.00000 0.918908 −3.92602 −1.00000 −2.15561 1.00000
1.8 −1.00000 −0.876107 1.00000 −1.00000 0.876107 4.75824 −1.00000 −2.23244 1.00000
1.9 −1.00000 −0.154802 1.00000 −1.00000 0.154802 −4.41535 −1.00000 −2.97604 1.00000
1.10 −1.00000 0.271531 1.00000 −1.00000 −0.271531 1.75980 −1.00000 −2.92627 1.00000
1.11 −1.00000 1.23027 1.00000 −1.00000 −1.23027 3.50539 −1.00000 −1.48643 1.00000
1.12 −1.00000 1.40727 1.00000 −1.00000 −1.40727 −2.31473 −1.00000 −1.01959 1.00000
1.13 −1.00000 1.72412 1.00000 −1.00000 −1.72412 −1.94532 −1.00000 −0.0274146 1.00000
1.14 −1.00000 2.71977 1.00000 −1.00000 −2.71977 −0.594829 −1.00000 4.39715 1.00000
1.15 −1.00000 3.15350 1.00000 −1.00000 −3.15350 −0.474552 −1.00000 6.94456 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
\(2\) \(1\)
\(5\) \(1\)
\(401\) \(1\)

Hecke kernels

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

\(T_{3}^{15} + \cdots\)
\(T_{7}^{15} + \cdots\)
\(T_{11}^{15} + \cdots\)