Properties

Label 6034.2.a.l
Level 6034
Weight 2
Character orbit 6034.a
Self dual Yes
Analytic conductor 48.182
Analytic rank 1
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6034.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 3 x^{19} - 36 x^{18} + 97 x^{17} + 573 x^{16} - 1292 x^{15} - 5329 x^{14} + 9121 x^{13} + 31784 x^{12} - 36075 x^{11} - 124276 x^{10} + 74594 x^{9} + 312410 x^{8} - 47208 x^{7} - 477646 x^{6} - 101137 x^{5} + 391391 x^{4} + 205294 x^{3} - 112848 x^{2} - 109144 x - 21776\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-3960343 \nu^{19} + 614360943 \nu^{18} - 1462616462 \nu^{17} - 19663940787 \nu^{16} + 43709410711 \nu^{15} + 268014365098 \nu^{14} - 504766627077 \nu^{13} - 2047106484441 \nu^{12} + 2948268091502 \nu^{11} + 9589231744201 \nu^{10} - 9140268418762 \nu^{9} - 28036276822386 \nu^{8} + 13552222306822 \nu^{7} + 49475485804836 \nu^{6} - 3408061802102 \nu^{5} - 47354517541381 \nu^{4} - 13461222095163 \nu^{3} + 17476023229232 \nu^{2} + 11129234398640 \nu + 1809016113384\)\()/ 3040618192 \)
\(\beta_{4}\)\(=\)\((\)\(16644951 \nu^{19} - 81044325 \nu^{18} - 577306280 \nu^{17} + 2574973259 \nu^{16} + 9155450243 \nu^{15} - 32772811792 \nu^{14} - 87337002091 \nu^{13} + 208383070343 \nu^{12} + 530311897740 \nu^{11} - 641981540601 \nu^{10} - 1972951737436 \nu^{9} + 524324073778 \nu^{8} + 4051274477878 \nu^{7} + 1855579562496 \nu^{6} - 3563958517610 \nu^{5} - 4498670133799 \nu^{4} - 389081593375 \nu^{3} + 2608653060254 \nu^{2} + 1778261525636 \nu + 335027839864\)\()/ 1520309096 \)
\(\beta_{5}\)\(=\)\((\)\(34635180 \nu^{19} - 101591649 \nu^{18} - 1492020713 \nu^{17} + 4034068656 \nu^{16} + 27150657839 \nu^{15} - 65128150081 \nu^{14} - 275173394620 \nu^{13} + 554472938489 \nu^{12} + 1715817274811 \nu^{11} - 2697245206372 \nu^{10} - 6808718394193 \nu^{9} + 7567057791828 \nu^{8} + 17092287131486 \nu^{7} - 11605231024486 \nu^{6} - 26098069899884 \nu^{5} + 7824943444266 \nu^{4} + 22030650504305 \nu^{3} + 423891695997 \nu^{2} - 7872031684318 \nu - 2198305565808\)\()/ 760154548 \)
\(\beta_{6}\)\(=\)\((\)\(24749436 \nu^{19} - 91273626 \nu^{18} - 806665861 \nu^{17} + 2893666957 \nu^{16} + 11460917862 \nu^{15} - 37991837591 \nu^{14} - 94984626483 \nu^{13} + 268421167748 \nu^{12} + 510837094381 \nu^{11} - 1104063935083 \nu^{10} - 1833575792166 \nu^{9} + 2655993621793 \nu^{8} + 4292697618894 \nu^{7} - 3510755640666 \nu^{6} - 6173786067610 \nu^{5} + 2015862861242 \nu^{4} + 4863664147150 \nu^{3} + 140670079897 \nu^{2} - 1577973995415 \nu - 434932067360\)\()/ 380077274 \)
\(\beta_{7}\)\(=\)\((\)\(100169175 \nu^{19} - 425556827 \nu^{18} - 3269423778 \nu^{17} + 14088317343 \nu^{16} + 46757370765 \nu^{15} - 195724039610 \nu^{14} - 395224629647 \nu^{13} + 1488124645517 \nu^{12} + 2218857147522 \nu^{11} - 6735921243469 \nu^{10} - 8570078476466 \nu^{9} + 18400675058302 \nu^{8} + 22292286183426 \nu^{7} - 29044895255988 \nu^{6} - 36728041496050 \nu^{5} + 22694772257685 \nu^{4} + 34185117118823 \nu^{3} - 3701357233648 \nu^{2} - 13584964832756 \nu - 3456518219736\)\()/ 1520309096 \)
\(\beta_{8}\)\(=\)\((\)\(264438933 \nu^{19} - 677141369 \nu^{18} - 9871031970 \nu^{17} + 22739279353 \nu^{16} + 159530819887 \nu^{15} - 315571793034 \nu^{14} - 1464415784497 \nu^{13} + 2340509122487 \nu^{12} + 8371072116058 \nu^{11} - 9986677320235 \nu^{10} - 30600553577726 \nu^{9} + 24434559246422 \nu^{8} + 70662011375414 \nu^{7} - 31448906153828 \nu^{6} - 98399132432526 \nu^{5} + 14323866889735 \nu^{4} + 74468175567565 \nu^{3} + 7430391971228 \nu^{2} - 23206861195928 \nu - 6827044858744\)\()/ 3040618192 \)
\(\beta_{9}\)\(=\)\((\)\(84627259 \nu^{19} - 299043217 \nu^{18} - 2760722388 \nu^{17} + 9521282651 \nu^{16} + 38959352751 \nu^{15} - 126057411084 \nu^{14} - 317530122351 \nu^{13} + 905324889811 \nu^{12} + 1669140863184 \nu^{11} - 3841472821561 \nu^{10} - 5893209464936 \nu^{9} + 9778625961422 \nu^{8} + 13891466884386 \nu^{7} - 14287583245004 \nu^{6} - 20850420947606 \nu^{5} + 10123784644597 \nu^{4} + 17888086708617 \nu^{3} - 1039701681626 \nu^{2} - 6630457002476 \nu - 1733210336848\)\()/ 760154548 \)
\(\beta_{10}\)\(=\)\((\)\(112313458 \nu^{19} - 531994539 \nu^{18} - 3327989547 \nu^{17} + 17119353404 \nu^{16} + 41775750129 \nu^{15} - 230447535695 \nu^{14} - 303351747064 \nu^{13} + 1695729550403 \nu^{12} + 1489592807765 \nu^{11} - 7441733031364 \nu^{10} - 5373267279201 \nu^{9} + 19833998059986 \nu^{8} + 14086688762286 \nu^{7} - 31007101914510 \nu^{6} - 24546970398728 \nu^{5} + 25037390039884 \nu^{4} + 24561108636495 \nu^{3} - 6016009624925 \nu^{2} - 10464986875224 \nu - 2401691382064\)\()/ 760154548 \)
\(\beta_{11}\)\(=\)\((\)\(248833715 \nu^{19} - 1291516455 \nu^{18} - 7059152210 \nu^{17} + 41300189627 \nu^{16} + 83884106449 \nu^{15} - 552284871818 \nu^{14} - 577091775883 \nu^{13} + 4035613393209 \nu^{12} + 2766913680226 \nu^{11} - 17571019110121 \nu^{10} - 10240996838690 \nu^{9} + 46386347886198 \nu^{8} + 28224832586138 \nu^{7} - 71711913218188 \nu^{6} - 51199959091866 \nu^{5} + 57289050280281 \nu^{4} + 52250106029851 \nu^{3} - 13916717002464 \nu^{2} - 22319090928660 \nu - 5022246308728\)\()/ 1520309096 \)
\(\beta_{12}\)\(=\)\((\)\(315317385 \nu^{19} - 1293022755 \nu^{18} - 9626405628 \nu^{17} + 40646537601 \nu^{16} + 125166030661 \nu^{15} - 529755791300 \nu^{14} - 933086775793 \nu^{13} + 3730825904025 \nu^{12} + 4524429755200 \nu^{11} - 15440497361475 \nu^{10} - 15072335059492 \nu^{9} + 38096694582042 \nu^{8} + 34290680964482 \nu^{7} - 53852893948784 \nu^{6} - 50203746507230 \nu^{5} + 37945554762711 \nu^{4} + 41819809511415 \nu^{3} - 6866102538522 \nu^{2} - 14788998243232 \nu - 3367570379784\)\()/ 1520309096 \)
\(\beta_{13}\)\(=\)\((\)\(189297804 \nu^{19} - 776958427 \nu^{18} - 6117797523 \nu^{17} + 25168764824 \nu^{16} + 86433528077 \nu^{15} - 340072159095 \nu^{14} - 718699678124 \nu^{13} + 2495980296331 \nu^{12} + 3937430053521 \nu^{11} - 10805281715184 \nu^{10} - 14657547984787 \nu^{9} + 27913124781788 \nu^{8} + 36210148063178 \nu^{7} - 41105211794078 \nu^{6} - 55931170695232 \nu^{5} + 29241168607046 \nu^{4} + 48367939070307 \nu^{3} - 3319275783001 \nu^{2} - 17766040350958 \nu - 4582432715876\)\()/ 760154548 \)
\(\beta_{14}\)\(=\)\((\)\(835590203 \nu^{19} - 3826397139 \nu^{18} - 25262436066 \nu^{17} + 123370536655 \nu^{16} + 327212594709 \nu^{15} - 1666268316042 \nu^{14} - 2478852923031 \nu^{13} + 12324747482549 \nu^{12} + 12707364215778 \nu^{11} - 54481031610765 \nu^{10} - 47030140681646 \nu^{9} + 146507264783378 \nu^{8} + 123486106092610 \nu^{7} - 230993811060164 \nu^{6} - 212352079365906 \nu^{5} + 186645175207745 \nu^{4} + 209047703145903 \nu^{3} - 41734671242616 \nu^{2} - 87948678687112 \nu - 20980523930824\)\()/ 3040618192 \)
\(\beta_{15}\)\(=\)\((\)\(-1027938729 \nu^{19} + 4718179533 \nu^{18} + 30511731298 \nu^{17} - 149948611701 \nu^{16} - 385986008075 \nu^{15} + 1989813546586 \nu^{14} + 2844198628989 \nu^{13} - 14409203636595 \nu^{12} - 14170320998026 \nu^{11} + 62115207658047 \nu^{10} + 51071504508166 \nu^{9} - 162220478086054 \nu^{8} - 130670424420302 \nu^{7} + 247481464312132 \nu^{6} + 218562657331126 \nu^{5} - 193027604258035 \nu^{4} - 208738589070129 \nu^{3} + 41787055500476 \nu^{2} + 85005626154864 \nu + 19966247957256\)\()/ 3040618192 \)
\(\beta_{16}\)\(=\)\((\)\(519967205 \nu^{19} - 2072767045 \nu^{18} - 15916321158 \nu^{17} + 64771735209 \nu^{16} + 208012032923 \nu^{15} - 839298144558 \nu^{14} - 1562543194369 \nu^{13} + 5883644866955 \nu^{12} + 7642558381022 \nu^{11} - 24302200221347 \nu^{10} - 25658775696594 \nu^{9} + 60093801761878 \nu^{8} + 58785626890430 \nu^{7} - 85545372062572 \nu^{6} - 86809450322110 \nu^{5} + 60797505146367 \nu^{4} + 73339444737417 \nu^{3} - 10490555195360 \nu^{2} - 26573508701288 \nu - 6299303348056\)\()/ 1520309096 \)
\(\beta_{17}\)\(=\)\((\)\(-274087579 \nu^{19} + 1083920107 \nu^{18} + 8622426264 \nu^{17} - 34420744897 \nu^{16} - 117006906077 \nu^{15} + 454747815080 \nu^{14} + 921739033949 \nu^{13} - 3260250462441 \nu^{12} - 4750930316416 \nu^{11} + 13806485341891 \nu^{10} + 16739364778226 \nu^{9} - 35051513158072 \nu^{8} - 39808268971858 \nu^{7} + 51180850872076 \nu^{6} + 60341382365326 \nu^{5} - 36936455863561 \nu^{4} - 51989409745755 \nu^{3} + 5630388544830 \nu^{2} + 19202507682118 \nu + 4753242395944\)\()/ 760154548 \)
\(\beta_{18}\)\(=\)\((\)\(-290115955 \nu^{19} + 1153750064 \nu^{18} + 9150086029 \nu^{17} - 36865983801 \nu^{16} - 124471421788 \nu^{15} + 491110264125 \nu^{14} + 983119021817 \nu^{13} - 3560419069126 \nu^{12} - 5090068253835 \nu^{11} + 15306357628247 \nu^{10} + 18105460574843 \nu^{9} - 39653842084616 \nu^{8} - 43818194314372 \nu^{7} + 59478980973710 \nu^{6} + 68183513110990 \nu^{5} - 44579225210911 \nu^{4} - 60749876331836 \nu^{3} + 7613668131281 \nu^{2} + 23329830760960 \nu + 5764506969456\)\()/ 760154548 \)
\(\beta_{19}\)\(=\)\((\)\(-317683195 \nu^{19} + 1370868643 \nu^{18} + 9610163980 \nu^{17} - 43406255205 \nu^{16} - 124377259165 \nu^{15} + 572895705936 \nu^{14} + 935317640093 \nu^{13} - 4117650204609 \nu^{12} - 4686813636124 \nu^{11} + 17575111887427 \nu^{10} + 16601593774334 \nu^{9} - 45314356507148 \nu^{8} - 41023320358782 \nu^{7} + 67960867918620 \nu^{6} + 65906318041966 \nu^{5} - 51620949387853 \nu^{4} - 60574697135363 \nu^{3} + 10163029426582 \nu^{2} + 23834003996814 \nu + 5699083771564\)\()/ 760154548 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{18} + \beta_{17} - \beta_{16} + \beta_{14} + \beta_{12} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + 7 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-\beta_{19} + \beta_{18} - \beta_{16} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{4} + 2 \beta_{3} + 10 \beta_{2} + 10 \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(2 \beta_{19} - 12 \beta_{18} + 12 \beta_{17} - 9 \beta_{16} - \beta_{15} + 10 \beta_{14} - \beta_{13} + 10 \beta_{12} - 10 \beta_{10} + \beta_{9} + 14 \beta_{8} - 11 \beta_{7} - 11 \beta_{5} + 11 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 54 \beta_{1} + 20\)
\(\nu^{6}\)\(=\)\(-16 \beta_{19} + 14 \beta_{18} + 2 \beta_{17} - 12 \beta_{16} + 2 \beta_{15} + 13 \beta_{14} + 15 \beta_{13} + 13 \beta_{12} - 13 \beta_{10} - 13 \beta_{9} - 14 \beta_{8} - 14 \beta_{7} - 3 \beta_{6} + \beta_{5} - 14 \beta_{4} + 27 \beta_{3} + 87 \beta_{2} + 87 \beta_{1} + 201\)
\(\nu^{7}\)\(=\)\(30 \beta_{19} - 117 \beta_{18} + 121 \beta_{17} - 72 \beta_{16} - 17 \beta_{15} + 88 \beta_{14} - 13 \beta_{13} + 88 \beta_{12} - \beta_{11} - 86 \beta_{10} + 13 \beta_{9} + 150 \beta_{8} - 104 \beta_{7} - 104 \beta_{5} + 101 \beta_{4} - 43 \beta_{3} - 36 \beta_{2} + 431 \beta_{1} + 174\)
\(\nu^{8}\)\(=\)\(-184 \beta_{19} + 151 \beta_{18} + 35 \beta_{17} - 113 \beta_{16} + 36 \beta_{15} + 132 \beta_{14} + 170 \beta_{13} + 130 \beta_{12} + 5 \beta_{11} - 132 \beta_{10} - 135 \beta_{9} - 145 \beta_{8} - 155 \beta_{7} - 54 \beta_{6} + 16 \beta_{5} - 156 \beta_{4} + 283 \beta_{3} + 736 \beta_{2} + 730 \beta_{1} + 1650\)
\(\nu^{9}\)\(=\)\(325 \beta_{19} - 1069 \beta_{18} + 1158 \beta_{17} - 573 \beta_{16} - 199 \beta_{15} + 766 \beta_{14} - 121 \beta_{13} + 762 \beta_{12} - 14 \beta_{11} - 727 \beta_{10} + 126 \beta_{9} + 1469 \beta_{8} - 952 \beta_{7} - 12 \beta_{6} - 948 \beta_{5} + 885 \beta_{4} - 454 \beta_{3} - 454 \beta_{2} + 3503 \beta_{1} + 1449\)
\(\nu^{10}\)\(=\)\(-1876 \beta_{19} + 1481 \beta_{18} + 454 \beta_{17} - 993 \beta_{16} + 437 \beta_{15} + 1236 \beta_{14} + 1737 \beta_{13} + 1204 \beta_{12} + 96 \beta_{11} - 1235 \beta_{10} - 1298 \beta_{9} - 1346 \beta_{8} - 1574 \beta_{7} - 689 \beta_{6} + 179 \beta_{5} - 1587 \beta_{4} + 2727 \beta_{3} + 6193 \beta_{2} + 6037 \beta_{1} + 13856\)
\(\nu^{11}\)\(=\)\(3124 \beta_{19} - 9506 \beta_{18} + 10811 \beta_{17} - 4614 \beta_{16} - 2023 \beta_{15} + 6699 \beta_{14} - 979 \beta_{13} + 6603 \beta_{12} - 109 \beta_{11} - 6196 \beta_{10} + 1107 \beta_{9} + 13814 \beta_{8} - 8638 \beta_{7} - 305 \beta_{6} - 8520 \beta_{5} + 7623 \beta_{4} - 4296 \beta_{3} - 4998 \beta_{2} + 28805 \beta_{1} + 11824\)
\(\nu^{12}\)\(=\)\(-18080 \beta_{19} + 13930 \beta_{18} + 5196 \beta_{17} - 8507 \beta_{16} + 4540 \beta_{15} + 11184 \beta_{14} + 16876 \beta_{13} + 10827 \beta_{12} + 1275 \beta_{11} - 11159 \beta_{10} - 12034 \beta_{9} - 11872 \beta_{8} - 15282 \beta_{7} - 7709 \beta_{6} + 1746 \beta_{5} - 15402 \beta_{4} + 25346 \beta_{3} + 52157 \beta_{2} + 49597 \beta_{1} + 117536\)
\(\nu^{13}\)\(=\)\(28391 \beta_{19} - 83477 \beta_{18} + 99476 \beta_{17} - 37639 \beta_{16} - 19267 \beta_{15} + 58810 \beta_{14} - 7258 \beta_{13} + 57364 \beta_{12} - 407 \beta_{11} - 53359 \beta_{10} + 9382 \beta_{9} + 127136 \beta_{8} - 78074 \beta_{7} - 5009 \beta_{6} - 76015 \beta_{5} + 65224 \beta_{4} - 38661 \beta_{3} - 51390 \beta_{2} + 238836 \beta_{1} + 95468\)
\(\nu^{14}\)\(=\)\(-169112 \beta_{19} + 128427 \beta_{18} + 55330 \beta_{17} - 72118 \beta_{16} + 43714 \beta_{15} + 99559 \beta_{14} + 159398 \beta_{13} + 96015 \beta_{12} + 14756 \beta_{11} - 99224 \beta_{10} - 109366 \beta_{9} - 101994 \beta_{8} - 144399 \beta_{7} - 80806 \beta_{6} + 15956 \beta_{5} - 145468 \beta_{4} + 231523 \beta_{3} + 440534 \beta_{2} + 406385 \beta_{1} + 1002422\)
\(\nu^{15}\)\(=\)\(250476 \beta_{19} - 728577 \beta_{18} + 906580 \beta_{17} - 310590 \beta_{16} - 177563 \beta_{15} + 517053 \beta_{14} - 49812 \beta_{13} + 499319 \beta_{12} + 3792 \beta_{11} - 463184 \beta_{10} + 78792 \beta_{9} + 1155425 \beta_{8} - 703530 \beta_{7} - 67724 \beta_{6} - 675304 \beta_{5} + 556960 \beta_{4} - 338913 \beta_{3} - 507490 \beta_{2} + 1992955 \beta_{1} + 766341\)
\(\nu^{16}\)\(=\)\(-1554790 \beta_{19} + 1171964 \beta_{18} + 562008 \beta_{17} - 608905 \beta_{16} + 403928 \beta_{15} + 878774 \beta_{14} + 1479616 \beta_{13} + 844891 \beta_{12} + 160064 \beta_{11} - 875917 \beta_{10} - 982050 \beta_{9} - 864197 \beta_{8} - 1340293 \beta_{7} - 815737 \beta_{6} + 140953 \beta_{5} - 1350710 \beta_{4} + 2096109 \beta_{3} + 3734079 \beta_{2} + 3328578 \beta_{1} + 8579038\)
\(\nu^{17}\)\(=\)\(2173001 \beta_{19} - 6339442 \beta_{18} + 8205746 \beta_{17} - 2588120 \beta_{16} - 1608507 \beta_{15} + 4545938 \beta_{14} - 310890 \beta_{13} + 4351431 \beta_{12} + 112121 \beta_{11} - 4042355 \beta_{10} + 664382 \beta_{9} + 10417814 \beta_{8} - 6321045 \beta_{7} - 820712 \beta_{6} - 5983425 \beta_{5} + 4757646 \beta_{4} - 2926613 \beta_{3} - 4880221 \beta_{2} + 16716643 \beta_{1} + 6131100\)
\(\nu^{18}\)\(=\)\(-14145184 \beta_{19} + 10634440 \beta_{18} + 5523546 \beta_{17} - 5137667 \beta_{16} + 3644424 \beta_{15} + 7719830 \beta_{14} + 13577614 \beta_{13} + 7398958 \beta_{12} + 1674987 \beta_{11} - 7710410 \beta_{10} - 8749493 \beta_{9} - 7271553 \beta_{8} - 12287437 \beta_{7} - 8039128 \beta_{6} + 1222620 \beta_{5} - 12399171 \beta_{4} + 18886903 \beta_{3} + 31768221 \beta_{2} + 27292204 \beta_{1} + 73611932\)
\(\nu^{19}\)\(=\)\(18666972 \beta_{19} - 55075743 \beta_{18} + 73890253 \beta_{17} - 21746507 \beta_{16} - 14440173 \beta_{15} + 39941739 \beta_{14} - 1649264 \beta_{13} + 37946875 \beta_{12} + 1754403 \beta_{11} - 35404964 \beta_{10} + 5661014 \beta_{9} + 93439765 \beta_{8} - 56633328 \beta_{7} - 9278750 \beta_{6} - 52925812 \beta_{5} + 40703498 \beta_{4} - 25037221 \beta_{3} - 46057226 \beta_{2} + 140839698 \beta_{1} + 48954545\)

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
2.98166
2.88014
2.84514
2.80072
2.78441
1.59091
1.42043
1.35617
1.32183
−0.395900
−0.860159
−0.903145
−1.01642
−1.35221
−1.37338
−1.76208
−1.78235
−2.22609
−2.34324
−2.96642
1.00000 −2.98166 1.00000 −1.98359 −2.98166 −1.00000 1.00000 5.89028 −1.98359
1.2 1.00000 −2.88014 1.00000 −1.11289 −2.88014 −1.00000 1.00000 5.29519 −1.11289
1.3 1.00000 −2.84514 1.00000 −4.15918 −2.84514 −1.00000 1.00000 5.09480 −4.15918
1.4 1.00000 −2.80072 1.00000 0.235711 −2.80072 −1.00000 1.00000 4.84406 0.235711
1.5 1.00000 −2.78441 1.00000 1.22741 −2.78441 −1.00000 1.00000 4.75292 1.22741
1.6 1.00000 −1.59091 1.00000 3.74979 −1.59091 −1.00000 1.00000 −0.468990 3.74979
1.7 1.00000 −1.42043 1.00000 −4.18942 −1.42043 −1.00000 1.00000 −0.982389 −4.18942
1.8 1.00000 −1.35617 1.00000 1.14574 −1.35617 −1.00000 1.00000 −1.16079 1.14574
1.9 1.00000 −1.32183 1.00000 1.76551 −1.32183 −1.00000 1.00000 −1.25276 1.76551
1.10 1.00000 0.395900 1.00000 0.151677 0.395900 −1.00000 1.00000 −2.84326 0.151677
1.11 1.00000 0.860159 1.00000 −0.340662 0.860159 −1.00000 1.00000 −2.26013 −0.340662
1.12 1.00000 0.903145 1.00000 −3.69995 0.903145 −1.00000 1.00000 −2.18433 −3.69995
1.13 1.00000 1.01642 1.00000 1.24888 1.01642 −1.00000 1.00000 −1.96689 1.24888
1.14 1.00000 1.35221 1.00000 2.46727 1.35221 −1.00000 1.00000 −1.17152 2.46727
1.15 1.00000 1.37338 1.00000 1.18048 1.37338 −1.00000 1.00000 −1.11382 1.18048
1.16 1.00000 1.76208 1.00000 −1.26930 1.76208 −1.00000 1.00000 0.104927 −1.26930
1.17 1.00000 1.78235 1.00000 −0.422890 1.78235 −1.00000 1.00000 0.176787 −0.422890
1.18 1.00000 2.22609 1.00000 −2.43328 2.22609 −1.00000 1.00000 1.95547 −2.43328
1.19 1.00000 2.34324 1.00000 −0.555415 2.34324 −1.00000 1.00000 2.49080 −0.555415
1.20 1.00000 2.96642 1.00000 −3.00590 2.96642 −1.00000 1.00000 5.79966 −3.00590
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(431\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\(T_{3}^{20} + \cdots\)
\(T_{5}^{20} + \cdots\)
\(T_{11}^{20} + \cdots\)