Properties

Label 6422.2.a.bo
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.2799281781\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + 2005 x^{6} - 10779 x^{5} - 2515 x^{4} + 8205 x^{3} + 686 x^{2} - 2352 x + 392\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + 2005 x^{6} - 10779 x^{5} - 2515 x^{4} + 8205 x^{3} + 686 x^{2} - 2352 x + 392\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(193884074943 \nu^{14} - 201950287980 \nu^{13} - 5788594784501 \nu^{12} + 5299062439584 \nu^{11} + 65988596033239 \nu^{10} - 52893370441157 \nu^{9} - 358319001961414 \nu^{8} + 251948699275524 \nu^{7} + 944978516892045 \nu^{6} - 600449094842537 \nu^{5} - 1104409988684731 \nu^{4} + 730959253960553 \nu^{3} + 365895032920633 \nu^{2} - 413057659876974 \nu + 85125155787042\)\()/ 16434237893906 \)
\(\beta_{3}\)\(=\)\((\)\(273993038595 \nu^{14} - 454917364489 \nu^{13} - 7951800462895 \nu^{12} + 12186175218625 \nu^{11} + 88249328545335 \nu^{10} - 124585987195702 \nu^{9} - 468200188646339 \nu^{8} + 608864922643750 \nu^{7} + 1217594052720517 \nu^{6} - 1465720096410848 \nu^{5} - 1444099340782060 \nu^{4} + 1643143750883146 \nu^{3} + 556343742252924 \nu^{2} - 666749229157753 \nu + 81940496280496\)\()/ 16434237893906 \)
\(\beta_{4}\)\(=\)\((\)\(-738118820283 \nu^{14} + 1081004001560 \nu^{13} + 21481071019325 \nu^{12} - 28454159571020 \nu^{11} - 238763832207155 \nu^{10} + 284894237294333 \nu^{9} + 1265522649849060 \nu^{8} - 1359044988601340 \nu^{7} - 3268919762807557 \nu^{6} + 3185803074970641 \nu^{5} + 3786161154929085 \nu^{4} - 3475248401407507 \nu^{3} - 1302965684854027 \nu^{2} + 1372857398534398 \nu - 264120839469396\)\()/ 32868475787812 \)
\(\beta_{5}\)\(=\)\((\)\(-64988194927 \nu^{14} + 77426247650 \nu^{13} + 1897449624715 \nu^{12} - 1992867835800 \nu^{11} - 21125056496611 \nu^{10} + 19539491510203 \nu^{9} + 111874927884310 \nu^{8} - 91870381768304 \nu^{7} - 287868019827689 \nu^{6} + 215610231747635 \nu^{5} + 333176606135653 \nu^{4} - 241681305631293 \nu^{3} - 122101207661989 \nu^{2} + 103767446150848 \nu - 15343610161320\)\()/ 2347748270558 \)
\(\beta_{6}\)\(=\)\((\)\(-38607285770 \nu^{14} + 50021871766 \nu^{13} + 1121665530434 \nu^{12} - 1305517419943 \nu^{11} - 12415512036371 \nu^{10} + 13008703761993 \nu^{9} + 65303162796476 \nu^{8} - 62273040919082 \nu^{7} - 166633261058502 \nu^{6} + 148929343175049 \nu^{5} + 190414901229259 \nu^{4} - 169760686984571 \nu^{3} - 67114532473162 \nu^{2} + 71434868027679 \nu - 10333663483962\)\()/ 1173874135279 \)
\(\beta_{7}\)\(=\)\((\)\(-708637533972 \nu^{14} + 761490392103 \nu^{13} + 20542837721082 \nu^{12} - 18673109849469 \nu^{11} - 227199761459728 \nu^{10} + 172010466453857 \nu^{9} + 1196932182959133 \nu^{8} - 749096724730526 \nu^{7} - 3072890555515652 \nu^{6} + 1619035306382097 \nu^{5} + 3578304082477239 \nu^{4} - 1700370409410973 \nu^{3} - 1421033485078085 \nu^{2} + 698910448979995 \nu - 47541675350932\)\()/ 16434237893906 \)
\(\beta_{8}\)\(=\)\((\)\(-452379850816 \nu^{14} + 617043986580 \nu^{13} + 13044910106152 \nu^{12} - 15862649785368 \nu^{11} - 143068057586618 \nu^{10} + 153867290344588 \nu^{9} + 743420063021889 \nu^{8} - 702051876445799 \nu^{7} - 1866261467998072 \nu^{6} + 1543606408234078 \nu^{5} + 2097439246563698 \nu^{4} - 1548859004852233 \nu^{3} - 744084199533680 \nu^{2} + 547477038182340 \nu - 78689644084590\)\()/ 8217118946953 \)
\(\beta_{9}\)\(=\)\((\)\(-1203557977309 \nu^{14} + 1313396214410 \nu^{13} + 35288418753293 \nu^{12} - 33303397247476 \nu^{11} - 395540879814035 \nu^{10} + 320593628175049 \nu^{9} + 2119862844174942 \nu^{8} - 1476283493144462 \nu^{7} - 5579671207743691 \nu^{6} + 3405661069585783 \nu^{5} + 6752227522168469 \nu^{4} - 3832894169046687 \nu^{3} - 2791480590276163 \nu^{2} + 1688719828154604 \nu - 162345305353616\)\()/ 16434237893906 \)
\(\beta_{10}\)\(=\)\((\)\(-2559294464187 \nu^{14} + 3047607863230 \nu^{13} + 74715943940201 \nu^{12} - 78460707110958 \nu^{11} - 831910388749235 \nu^{10} + 768469241227507 \nu^{9} + 4409904957645870 \nu^{8} - 3600614910154128 \nu^{7} - 11393074204045365 \nu^{6} + 8394806597986567 \nu^{5} + 13379924724767959 \nu^{4} - 9325836824202817 \nu^{3} - 5237156925585941 \nu^{2} + 3852959296600344 \nu - 412603877563136\)\()/ 32868475787812 \)
\(\beta_{11}\)\(=\)\((\)\(405661145241 \nu^{14} - 586147754198 \nu^{13} - 11722490860231 \nu^{12} + 15394907788986 \nu^{11} + 128950911913821 \nu^{10} - 154255468331797 \nu^{9} - 672748773629942 \nu^{8} + 740905503370676 \nu^{7} + 1695297344154367 \nu^{6} - 1769079252015537 \nu^{5} - 1892953218006637 \nu^{4} + 2000161757183539 \nu^{3} + 604829097442615 \nu^{2} - 839017078741672 \nu + 147404924606680\)\()/ 4695496541116 \)
\(\beta_{12}\)\(=\)\((\)\(3319521574605 \nu^{14} - 4358410626934 \nu^{13} - 96553802732003 \nu^{12} + 113273480977518 \nu^{11} + 1071806270090377 \nu^{10} - 1123238856659845 \nu^{9} - 5668346732766810 \nu^{8} + 5353513365252120 \nu^{7} + 14601751012343479 \nu^{6} - 12792925088177737 \nu^{5} - 16936790106872365 \nu^{4} + 14725928878784715 \nu^{3} + 6086076225300559 \nu^{2} - 6420536938189472 \nu + 982897019051052\)\()/ 32868475787812 \)
\(\beta_{13}\)\(=\)\((\)\(123406974291 \nu^{14} - 177046669743 \nu^{13} - 3570779556163 \nu^{12} + 4614652320035 \nu^{11} + 39390862911450 \nu^{10} - 45617310101636 \nu^{9} - 206817431427508 \nu^{8} + 214086371949073 \nu^{7} + 529254330906825 \nu^{6} - 493229672476407 \nu^{5} - 615547575337017 \nu^{4} + 537435648340022 \nu^{3} + 230098982794981 \nu^{2} - 220752492445392 \nu + 32503592648054\)\()/ 1173874135279 \)
\(\beta_{14}\)\(=\)\((\)\(994365464810 \nu^{14} - 1409343410986 \nu^{13} - 28742233187393 \nu^{12} + 36628072445753 \nu^{11} + 316679440101669 \nu^{10} - 361262705776584 \nu^{9} - 1660124766606934 \nu^{8} + 1694176590194813 \nu^{7} + 4239829127298664 \nu^{6} - 3907018971030303 \nu^{5} - 4922746154491539 \nu^{4} + 4242454481109403 \nu^{3} + 1863316708868465 \nu^{2} - 1710141797654178 \nu + 205796730419665\)\()/ 8217118946953 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{14} - \beta_{13} + \beta_{5} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-2 \beta_{12} + 2 \beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 6 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(8 \beta_{14} - 9 \beta_{13} + 2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{7} + 3 \beta_{6} + 12 \beta_{5} - \beta_{3} + 8 \beta_{2} + 7 \beta_{1} + 28\)
\(\nu^{5}\)\(=\)\(-\beta_{14} - 24 \beta_{12} + 25 \beta_{11} - 3 \beta_{10} - 12 \beta_{9} + 12 \beta_{8} - 12 \beta_{7} + 14 \beta_{6} + 18 \beta_{5} - 4 \beta_{4} + 10 \beta_{3} + 10 \beta_{2} + 43 \beta_{1} - 11\)
\(\nu^{6}\)\(=\)\(61 \beta_{14} - 76 \beta_{13} - 5 \beta_{12} + 34 \beta_{11} - 7 \beta_{10} - 28 \beta_{9} - 11 \beta_{8} + 9 \beta_{7} + 50 \beta_{6} + 130 \beta_{5} + 4 \beta_{4} - 5 \beta_{3} + 66 \beta_{2} + 50 \beta_{1} + 220\)
\(\nu^{7}\)\(=\)\(-17 \beta_{14} - 246 \beta_{12} + 268 \beta_{11} - 58 \beta_{10} - 122 \beta_{9} + 126 \beta_{8} - 127 \beta_{7} + 165 \beta_{6} + 249 \beta_{5} - 64 \beta_{4} + 89 \beta_{3} + 98 \beta_{2} + 338 \beta_{1} - 111\)
\(\nu^{8}\)\(=\)\(459 \beta_{14} - 648 \beta_{13} - 104 \beta_{12} + 450 \beta_{11} - 138 \beta_{10} - 325 \beta_{9} - 104 \beta_{8} + 52 \beta_{7} + 638 \beta_{6} + 1392 \beta_{5} + 68 \beta_{4} + 26 \beta_{3} + 575 \beta_{2} + 368 \beta_{1} + 1794\)
\(\nu^{9}\)\(=\)\(-231 \beta_{14} - 11 \beta_{13} - 2450 \beta_{12} + 2797 \beta_{11} - 798 \beta_{10} - 1235 \beta_{9} + 1254 \beta_{8} - 1299 \beta_{7} + 1860 \beta_{6} + 3101 \beta_{5} - 757 \beta_{4} + 824 \beta_{3} + 985 \beta_{2} + 2755 \beta_{1} - 1071\)
\(\nu^{10}\)\(=\)\(3408 \beta_{14} - 5631 \beta_{13} - 1570 \beta_{12} + 5447 \beta_{11} - 1937 \beta_{10} - 3627 \beta_{9} - 945 \beta_{8} + 106 \beta_{7} + 7408 \beta_{6} + 14870 \beta_{5} + 844 \beta_{4} + 990 \beta_{3} + 5200 \beta_{2} + 2732 \beta_{1} + 14936\)
\(\nu^{11}\)\(=\)\(-2874 \beta_{14} - 361 \beta_{13} - 24318 \beta_{12} + 29103 \beta_{11} - 9648 \beta_{10} - 12765 \beta_{9} + 12117 \beta_{8} - 13081 \beta_{7} + 20571 \beta_{6} + 36546 \beta_{5} - 7945 \beta_{4} + 8132 \beta_{3} + 10048 \beta_{2} + 22828 \beta_{1} - 9967\)
\(\nu^{12}\)\(=\)\(24758 \beta_{14} - 49902 \beta_{13} - 20938 \beta_{12} + 63141 \beta_{11} - 23926 \beta_{10} - 39977 \beta_{9} - 8366 \beta_{8} - 2868 \beta_{7} + 82278 \beta_{6} + 158645 \beta_{5} + 9375 \beta_{4} + 16331 \beta_{3} + 48360 \beta_{2} + 20221 \beta_{1} + 126497\)
\(\nu^{13}\)\(=\)\(-33990 \beta_{14} - 7308 \beta_{13} - 242106 \beta_{12} + 303419 \beta_{11} - 109603 \beta_{10} - 134414 \beta_{9} + 115089 \beta_{8} - 130758 \beta_{7} + 225103 \beta_{6} + 416899 \beta_{5} - 78242 \beta_{4} + 84352 \beta_{3} + 103348 \beta_{2} + 191029 \beta_{1} - 89840\)
\(\nu^{14}\)\(=\)\(173034 \beta_{14} - 450638 \beta_{13} - 261645 \beta_{12} + 713868 \beta_{11} - 278091 \beta_{10} - 437558 \beta_{9} - 71962 \beta_{8} - 65190 \beta_{7} + 892494 \beta_{6} + 1690694 \beta_{5} + 98730 \beta_{4} + 219300 \beta_{3} + 460353 \beta_{2} + 148379 \beta_{1} + 1088558\)

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
−2.98660
−2.65291
−2.60210
−1.54553
−1.43594
−1.35072
−1.02081
0.206045
0.503505
1.01084
1.38264
1.75888
2.61859
2.85344
3.26068
−1.00000 −2.98660 1.00000 3.18940 2.98660 −3.65892 −1.00000 5.91977 −3.18940
1.2 −1.00000 −2.65291 1.00000 0.987196 2.65291 3.23991 −1.00000 4.03794 −0.987196
1.3 −1.00000 −2.60210 1.00000 −0.883013 2.60210 −3.97763 −1.00000 3.77094 0.883013
1.4 −1.00000 −1.54553 1.00000 1.89914 1.54553 −0.915609 −1.00000 −0.611330 −1.89914
1.5 −1.00000 −1.43594 1.00000 2.70512 1.43594 −0.857586 −1.00000 −0.938065 −2.70512
1.6 −1.00000 −1.35072 1.00000 −2.02622 1.35072 1.66961 −1.00000 −1.17555 2.02622
1.7 −1.00000 −1.02081 1.00000 −1.43823 1.02081 −5.08222 −1.00000 −1.95796 1.43823
1.8 −1.00000 0.206045 1.00000 −3.15439 −0.206045 −3.30532 −1.00000 −2.95755 3.15439
1.9 −1.00000 0.503505 1.00000 1.68542 −0.503505 4.13863 −1.00000 −2.74648 −1.68542
1.10 −1.00000 1.01084 1.00000 −0.455213 −1.01084 0.356053 −1.00000 −1.97821 0.455213
1.11 −1.00000 1.38264 1.00000 3.52682 −1.38264 −0.0942941 −1.00000 −1.08830 −3.52682
1.12 −1.00000 1.75888 1.00000 −3.17158 −1.75888 −5.12706 −1.00000 0.0936587 3.17158
1.13 −1.00000 2.61859 1.00000 −3.36169 −2.61859 −2.21822 −1.00000 3.85700 3.36169
1.14 −1.00000 2.85344 1.00000 −2.12918 −2.85344 0.00224701 −1.00000 5.14213 2.12918
1.15 −1.00000 3.26068 1.00000 1.62641 −3.26068 −2.16959 −1.00000 7.63201 −1.62641
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(13\) \(-1\)
\(19\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6422.2.a.bo 15
13.b even 2 1 6422.2.a.bq yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6422.2.a.bo 15 1.a even 1 1 trivial
6422.2.a.bq yes 15 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\(T_{3}^{15} - \cdots\)
\(T_{5}^{15} + \cdots\)
\(T_{7}^{15} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{15} \)
$3$ \( 392 - 2352 T + 686 T^{2} + 8205 T^{3} - 2515 T^{4} - 10779 T^{5} + 2005 T^{6} + 6791 T^{7} - 636 T^{8} - 2208 T^{9} + 85 T^{10} + 373 T^{11} - 4 T^{12} - 31 T^{13} + T^{15} \)
$5$ \( -13117 - 28898 T + 31740 T^{2} + 68705 T^{3} - 30675 T^{4} - 58870 T^{5} + 15120 T^{6} + 24551 T^{7} - 3985 T^{8} - 5492 T^{9} + 555 T^{10} + 666 T^{11} - 38 T^{12} - 41 T^{13} + T^{14} + T^{15} \)
$7$ \( -8 + 3472 T + 39302 T^{2} - 3611 T^{3} - 247883 T^{4} - 346320 T^{5} - 92363 T^{6} + 114893 T^{7} + 85520 T^{8} + 10650 T^{9} - 7769 T^{10} - 2865 T^{11} - 125 T^{12} + 95 T^{13} + 18 T^{14} + T^{15} \)
$11$ \( 219128 - 2177560 T + 447342 T^{2} + 4609823 T^{3} - 2615950 T^{4} - 1875447 T^{5} + 1164698 T^{6} + 385528 T^{7} - 193280 T^{8} - 49638 T^{9} + 13206 T^{10} + 3211 T^{11} - 385 T^{12} - 94 T^{13} + 4 T^{14} + T^{15} \)
$13$ \( T^{15} \)
$17$ \( -3037649 - 11932917 T - 2673364 T^{2} + 20282503 T^{3} + 6411954 T^{4} - 9311373 T^{5} - 2405513 T^{6} + 1794216 T^{7} + 324677 T^{8} - 164938 T^{9} - 16887 T^{10} + 7310 T^{11} + 326 T^{12} - 143 T^{13} - 2 T^{14} + T^{15} \)
$19$ \( ( -1 + T )^{15} \)
$23$ \( -405660352 + 553074896 T + 691491848 T^{2} - 750841657 T^{3} - 37234704 T^{4} + 170875442 T^{5} - 20214077 T^{6} - 14271664 T^{7} + 2806233 T^{8} + 457323 T^{9} - 129206 T^{10} - 3656 T^{11} + 2480 T^{12} - 71 T^{13} - 17 T^{14} + T^{15} \)
$29$ \( 31065433664 + 15261553568 T - 9264141696 T^{2} - 5748812616 T^{3} + 689627080 T^{4} + 785552436 T^{5} + 29120465 T^{6} - 47635655 T^{7} - 5527979 T^{8} + 1266696 T^{9} + 231233 T^{10} - 10227 T^{11} - 3684 T^{12} - 73 T^{13} + 20 T^{14} + T^{15} \)
$31$ \( -29984427256 - 54059336744 T - 25712858122 T^{2} + 3509681729 T^{3} + 4928177459 T^{4} + 499955255 T^{5} - 300627617 T^{6} - 61288074 T^{7} + 6316689 T^{8} + 2426799 T^{9} + 41249 T^{10} - 39524 T^{11} - 3146 T^{12} + 173 T^{13} + 30 T^{14} + T^{15} \)
$37$ \( -522209344 + 509895136 T + 1095817904 T^{2} - 106696776 T^{3} - 685620604 T^{4} - 250023382 T^{5} + 59445131 T^{6} + 59199917 T^{7} + 13757830 T^{8} + 473177 T^{9} - 321946 T^{10} - 53844 T^{11} - 1603 T^{12} + 345 T^{13} + 35 T^{14} + T^{15} \)
$41$ \( -18752 - 242720 T - 737568 T^{2} - 208216 T^{3} + 1213208 T^{4} + 611808 T^{5} - 574435 T^{6} - 381786 T^{7} + 72219 T^{8} + 87458 T^{9} + 8451 T^{10} - 5822 T^{11} - 1442 T^{12} - 38 T^{13} + 15 T^{14} + T^{15} \)
$43$ \( -529101272 + 1509010384 T + 369456626 T^{2} - 1960568555 T^{3} - 1134284546 T^{4} + 71162019 T^{5} + 158054021 T^{6} + 17084597 T^{7} - 7132492 T^{8} - 1285927 T^{9} + 114223 T^{10} + 30486 T^{11} - 499 T^{12} - 292 T^{13} - T^{14} + T^{15} \)
$47$ \( -1176125837312 + 2174295578624 T - 1192827297024 T^{2} + 53315359744 T^{3} + 119334870144 T^{4} - 19995953696 T^{5} - 3460767883 T^{6} + 802415055 T^{7} + 43778530 T^{8} - 13943509 T^{9} - 253918 T^{10} + 124236 T^{11} + 549 T^{12} - 558 T^{13} + T^{15} \)
$53$ \( -1601243116096 - 556780810560 T + 739438260320 T^{2} + 211121982104 T^{3} - 78201270168 T^{4} - 19993808682 T^{5} + 3003569481 T^{6} + 776068248 T^{7} - 47877481 T^{8} - 13981206 T^{9} + 348277 T^{10} + 127064 T^{11} - 1087 T^{12} - 569 T^{13} + T^{14} + T^{15} \)
$59$ \( -27405374296 + 119397510092 T - 34113477412 T^{2} - 61771345813 T^{3} + 26711610254 T^{4} + 1531503335 T^{5} - 1887963946 T^{6} + 93946069 T^{7} + 51058047 T^{8} - 4491661 T^{9} - 629049 T^{10} + 68645 T^{11} + 3486 T^{12} - 438 T^{13} - 7 T^{14} + T^{15} \)
$61$ \( 4856975148389 - 6416627259791 T + 649295177538 T^{2} + 987723453151 T^{3} - 107663899908 T^{4} - 61307698405 T^{5} + 3431593932 T^{6} + 1671282877 T^{7} - 48177061 T^{8} - 23214194 T^{9} + 349854 T^{10} + 172689 T^{11} - 1305 T^{12} - 656 T^{13} + 2 T^{14} + T^{15} \)
$67$ \( 17735387144 + 23680333160 T - 14312795678 T^{2} - 45614130733 T^{3} - 35791689208 T^{4} - 13530024302 T^{5} - 2404472737 T^{6} - 31272623 T^{7} + 62730711 T^{8} + 9160624 T^{9} + 22179 T^{10} - 100607 T^{11} - 7202 T^{12} + 154 T^{13} + 34 T^{14} + T^{15} \)
$71$ \( 72939544 + 1592165192 T + 2613722798 T^{2} + 420381857 T^{3} - 1223821863 T^{4} - 435157189 T^{5} + 163062021 T^{6} + 57974248 T^{7} - 8866634 T^{8} - 2749599 T^{9} + 195311 T^{10} + 53215 T^{11} - 1633 T^{12} - 406 T^{13} + 4 T^{14} + T^{15} \)
$73$ \( -5733317071 - 44210903016 T - 76837487636 T^{2} - 23956096561 T^{3} + 12248825858 T^{4} + 7519655884 T^{5} + 695307829 T^{6} - 264060394 T^{7} - 53586053 T^{8} + 1362050 T^{9} + 966921 T^{10} + 37317 T^{11} - 6060 T^{12} - 392 T^{13} + 12 T^{14} + T^{15} \)
$79$ \( -57872680424 + 188533464552 T - 137804066582 T^{2} - 35703865697 T^{3} + 39109496283 T^{4} + 1775141866 T^{5} - 3402675613 T^{6} - 19668400 T^{7} + 119544015 T^{8} - 1124108 T^{9} - 1755917 T^{10} + 42172 T^{11} + 10944 T^{12} - 392 T^{13} - 23 T^{14} + T^{15} \)
$83$ \( 215836257496 - 2067823694012 T - 848121757340 T^{2} + 738903866339 T^{3} + 45502853112 T^{4} - 50649077515 T^{5} - 1422814823 T^{6} + 1435715263 T^{7} + 27214192 T^{8} - 20676190 T^{9} - 286472 T^{10} + 159749 T^{11} + 1495 T^{12} - 631 T^{13} - 3 T^{14} + T^{15} \)
$89$ \( 2289295683584 + 2254865736704 T + 18255227648 T^{2} - 576021368896 T^{3} - 137089943792 T^{4} + 30543311788 T^{5} + 11923017483 T^{6} - 90547053 T^{7} - 298166165 T^{8} - 11327749 T^{9} + 2846606 T^{10} + 146622 T^{11} - 11583 T^{12} - 656 T^{13} + 17 T^{14} + T^{15} \)
$97$ \( 1664366526976 + 2669698079232 T - 3706771818976 T^{2} - 2726738156104 T^{3} - 250188181460 T^{4} + 134372072396 T^{5} + 24587221237 T^{6} - 1468173699 T^{7} - 514345789 T^{8} - 5205039 T^{9} + 4276553 T^{10} + 152840 T^{11} - 14917 T^{12} - 718 T^{13} + 18 T^{14} + T^{15} \)
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