Properties

Label 150.2.h.b
Level 150
Weight 2
Character orbit 150.h
Analytic conductor 1.198
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 150.h (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + 6259 x^{8} - 11958 x^{7} - 15752 x^{6} + 14670 x^{5} + 18271 x^{4} - 10440 x^{3} + 1135 x^{2} + 21080 x + 11105\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-216062652991087486517 \nu^{15} - 222689963418511649230 \nu^{14} + 11692850809057402771308 \nu^{13} - 4830492818371367785816 \nu^{12} - 200158344597704877408533 \nu^{11} + 142199510139739938929725 \nu^{10} + 1696861859405690275607183 \nu^{9} - 1203647771351280118822772 \nu^{8} - 7616568797732981879968331 \nu^{7} + 3146071202213920505926562 \nu^{6} + 19090952137993148860203137 \nu^{5} - 132407938274590801116617 \nu^{4} - 24907817546557276804454900 \nu^{3} + 3697181661562006774932655 \nu^{2} - 2518379054720136093673150 \nu - 36766787356071608007667585\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-265732875567395073657 \nu^{15} + 1990174463411413335455 \nu^{14} + 15780868218717044920838 \nu^{13} - 82635447635890840435206 \nu^{12} - 310570846147264249920748 \nu^{11} + 1241691800858697588974155 \nu^{10} + 3010621887147134394227693 \nu^{9} - 9299193782108641569228897 \nu^{8} - 15187750120575825142764691 \nu^{7} + 37039168692149977669870742 \nu^{6} + 41504619394117749813534187 \nu^{5} - 78953875227996269190814922 \nu^{4} - 49901182956943689204688600 \nu^{3} + 104941588005445229750025955 \nu^{2} - 24344868121453920674255425 \nu - 91429957070907564671351610\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{3}\)\(=\)\((\)\(702375526475979478783 \nu^{15} - 3304044074082752777785 \nu^{14} - 25795447997562939235427 \nu^{13} + 114953003304995515650009 \nu^{12} + 368633707984163431786287 \nu^{11} - 1482042781994674721659060 \nu^{10} - 2812846670822251548242917 \nu^{9} + 9176301900507627049359998 \nu^{8} + 12537511618422589653976589 \nu^{7} - 26170693115741185316682238 \nu^{6} - 40039221089263675036570378 \nu^{5} + 32685303511358818643459718 \nu^{4} + 72705390025681627618678875 \nu^{3} - 35152535417255483308930345 \nu^{2} - 30211511665347684930563925 \nu + 49504201727591742141925840\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{4}\)\(=\)\((\)\(2296292574754742295778 \nu^{15} - 18777167771030970954085 \nu^{14} - 5742569515547694072792 \nu^{13} + 376031136320790011131049 \nu^{12} - 424999022122650057110578 \nu^{11} - 3188418937056165399714235 \nu^{10} + 5262312369001650000411893 \nu^{9} + 12651496691912944253088318 \nu^{8} - 21294724748266832493330656 \nu^{7} - 31019540923068154312589818 \nu^{6} + 37963741339531390922508492 \nu^{5} + 52805059704158952082659513 \nu^{4} - 32920117107035471964771150 \nu^{3} - 38527298141204740926854120 \nu^{2} + 44218902932902401592517525 \nu - 10609792203969280509060435\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{5}\)\(=\)\((\)\(102309325228650587446 \nu^{15} - 425968439454909023504 \nu^{14} - 2318583371847014159396 \nu^{13} + 9842957759035774370007 \nu^{12} + 23588922138538677715666 \nu^{11} - 96439403899597794479889 \nu^{10} - 132289916480158900957958 \nu^{9} + 469952553315433857842339 \nu^{8} + 521913912002145220106872 \nu^{7} - 1201274203759528179191999 \nu^{6} - 1439503235730439393390400 \nu^{5} + 1561882257402276944726861 \nu^{4} + 1751624753872932949676820 \nu^{3} - 1056349544697974528792295 \nu^{2} + 648738016230841178303300 \nu + 2136427940493114503422100\)\()/ \)\(12\!\cdots\!55\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-3180944021456305418003 \nu^{15} + 12108798860040862926030 \nu^{14} + 76658914050049775735647 \nu^{13} - 279795475163391518337309 \nu^{12} - 843282840396799929838402 \nu^{11} + 2772410277587660664967595 \nu^{10} + 5120683136367422331041022 \nu^{9} - 13952896772257730298480513 \nu^{8} - 20136486077065159637670929 \nu^{7} + 38669929503840247602021988 \nu^{6} + 48847780672259798172146818 \nu^{5} - 57387686031561159003451668 \nu^{4} - 56558593741097731772595025 \nu^{3} + 52324310540791207677909920 \nu^{2} - 9360398580013835322450250 \nu - 54250932322302201472419340\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{7}\)\(=\)\((\)\(3754084957894051064894 \nu^{15} - 15083470458469574064070 \nu^{14} - 89761163865409232604666 \nu^{13} + 347746317243990280362907 \nu^{12} + 998563896155868111731741 \nu^{11} - 3429976210567685569153495 \nu^{10} - 6202656224864865816953881 \nu^{9} + 17064490132080048666621894 \nu^{8} + 24890198037032388014893387 \nu^{7} - 46215997398926518335783349 \nu^{6} - 60890899573543787233277629 \nu^{5} + 60090136619662887556863749 \nu^{4} + 77529726343056299821308475 \nu^{3} - 40267892959128470588816910 \nu^{2} - 36825642304869580862835650 \nu + 68909263042569036125855870\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-3973936680890235791447 \nu^{15} + 19337803181486775124430 \nu^{14} + 73342780711804225674553 \nu^{13} - 416649754235775402919971 \nu^{12} - 558606750221222446608433 \nu^{11} + 3759928827840084662619945 \nu^{10} + 1783092343379774194213858 \nu^{9} - 16064794943644897425418542 \nu^{8} - 4220420776903807591816446 \nu^{7} + 35426011517516138530176972 \nu^{6} + 10251508388168254787967127 \nu^{5} - 38968246371604410909723242 \nu^{4} - 1287569082167687474870500 \nu^{3} + 19285624254963832677634030 \nu^{2} - 37529598314145175899342950 \nu - 3543808081464618883649710\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{9}\)\(=\)\((\)\(161168803239414780910 \nu^{15} - 661637860075799467724 \nu^{14} - 3827536126025598552754 \nu^{13} + 15994665739743758299833 \nu^{12} + 40151303735551147100534 \nu^{11} - 164810455922834007902603 \nu^{10} - 224574784032425709435836 \nu^{9} + 865180285576144364779891 \nu^{8} + 794296728242089068127598 \nu^{7} - 2456515842661058092401641 \nu^{6} - 1829541259881354437224758 \nu^{5} + 3808469691277882506316428 \nu^{4} + 1780293400073412257719180 \nu^{3} - 3317734863222409081306840 \nu^{2} + 1107814885746452854956660 \nu + 2686892356943722466767320\)\()/ \)\(12\!\cdots\!55\)\( \)
\(\beta_{10}\)\(=\)\((\)\(164194301487618633410 \nu^{15} - 719766109199837227876 \nu^{14} - 3474010950605366952410 \nu^{13} + 16401138606042229504145 \nu^{12} + 31059323892781834514230 \nu^{11} - 157188061964900911061375 \nu^{10} - 130154545341889076617028 \nu^{9} + 733989392081838127266340 \nu^{8} + 317412747752376679716350 \nu^{7} - 1736893910036273955685895 \nu^{6} - 552527714012012839873230 \nu^{5} + 1984064979606023800874943 \nu^{4} + 47852376229771545163160 \nu^{3} - 1332489074800974413015230 \nu^{2} + 2233536035317308402213160 \nu + 1025511991580005661458970\)\()/ \)\(12\!\cdots\!55\)\( \)
\(\beta_{11}\)\(=\)\((\)\(838230261608330797746 \nu^{15} - 4786382082628259634787 \nu^{14} - 13869542625980282893647 \nu^{13} + 106085643722406510377459 \nu^{12} + 90720060109888652320392 \nu^{11} - 1002540084989697030577461 \nu^{10} - 234297600527699664967938 \nu^{9} + 4699509630594445307398133 \nu^{8} + 1109399877527339581698789 \nu^{7} - 12618721201525299962308298 \nu^{6} - 6217938413303324609066387 \nu^{5} + 20000504884661427693942799 \nu^{4} + 10638060647839174881601105 \nu^{3} - 18066278410794456581746620 \nu^{2} + 3148595090397016687536785 \nu + 17988954213833427432880420\)\()/ \)\(60\!\cdots\!75\)\( \)
\(\beta_{12}\)\(=\)\((\)\(908669311439016539378 \nu^{15} - 5174938240432228690153 \nu^{14} - 15283113254819993188609 \nu^{13} + 120512535486778277787298 \nu^{12} + 82750556881744744497304 \nu^{11} - 1186706887477185726218359 \nu^{10} + 90637788457715278790657 \nu^{9} + 5770412096942933836755541 \nu^{8} - 1561116277429088140696022 \nu^{7} - 14822422913804415887485051 \nu^{6} + 1820874604516396099156933 \nu^{5} + 20462537822506322798652590 \nu^{4} + 1296008565761348543038545 \nu^{3} - 11744901364273461227406290 \nu^{2} + 12987872084176881105529190 \nu + 4308719570906786186270775\)\()/ \)\(60\!\cdots\!75\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-6533138943972814245933 \nu^{15} + 31619240120523939425865 \nu^{14} + 125897227911835136338677 \nu^{13} - 708040835635361844344624 \nu^{12} - 1007374201550559641412517 \nu^{11} + 6724682741769758012533880 \nu^{10} + 3470026494541648226365932 \nu^{9} - 31333706626178340309002363 \nu^{8} - 7246181075197585096864139 \nu^{7} + 77592234545423436266661168 \nu^{6} + 11636953568620512039028188 \nu^{5} - 98305813076082701994022623 \nu^{4} + 10820382804042389176367050 \nu^{3} + 67691544638516502492240320 \nu^{2} - 84912508264192029168955175 \nu - 61955069294991105282252615\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-9361640845971301396397 \nu^{15} + 52366780732480415686465 \nu^{14} + 159762414534900794176883 \nu^{13} - 1190582299946926644582166 \nu^{12} - 1024978252277545106715408 \nu^{11} + 11575346968422089033871410 \nu^{10} + 1598877636542195838733293 \nu^{9} - 55828465749972912641212972 \nu^{8} + 613642773665136143529419 \nu^{7} + 145008784517773349507323362 \nu^{6} + 9914427702507185104901877 \nu^{5} - 183985152384127705242957177 \nu^{4} - 15715753986137640451935575 \nu^{3} + 81809532792011322933042280 \nu^{2} - 134474994846987328679889500 \nu - 86034467893737742078614385\)\()/ \)\(30\!\cdots\!75\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-1965198321477020977800 \nu^{15} + 9926479283735303812942 \nu^{14} + 40235447863223328373118 \nu^{13} - 237493265324739844946336 \nu^{12} - 348172097985448217050033 \nu^{11} + 2410564961185097493334441 \nu^{10} + 1429316307775823908404854 \nu^{9} - 12215850222197537282508032 \nu^{8} - 3751634380400785138202611 \nu^{7} + 32442715594977735558622037 \nu^{6} + 8620102019274457312300996 \nu^{5} - 42222846117366993415189833 \nu^{4} - 4541857374012864958803405 \nu^{3} + 21432392262237651257289005 \nu^{2} - 32435795750138591078265085 \nu - 25977835089457219631295215\)\()/ \)\(60\!\cdots\!75\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{15} - \beta_{14} - \beta_{13} + 4 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_{1} - 1\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} - 6 \beta_{8} - \beta_{7} - 7 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 13 \beta_{1} + 16\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(17 \beta_{15} - 6 \beta_{14} - 26 \beta_{13} + 29 \beta_{12} - 23 \beta_{11} - 53 \beta_{10} - 2 \beta_{9} - 34 \beta_{8} + 6 \beta_{7} + 12 \beta_{6} + 19 \beta_{5} - 17 \beta_{4} - 8 \beta_{3} - 19 \beta_{2} - 13 \beta_{1} + 4\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(-12 \beta_{15} + 31 \beta_{14} - 89 \beta_{13} + 21 \beta_{12} - 77 \beta_{11} - 107 \beta_{10} - 78 \beta_{9} - 6 \beta_{8} - 11 \beta_{7} - 77 \beta_{6} + 111 \beta_{5} + 22 \beta_{4} + 53 \beta_{3} - 21 \beta_{2} + 153 \beta_{1} + 66\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(102 \beta_{15} - 31 \beta_{14} - 356 \beta_{13} + 179 \beta_{12} - 158 \beta_{11} - 593 \beta_{10} - 97 \beta_{9} - 244 \beta_{8} - 74 \beta_{7} - 108 \beta_{6} + 159 \beta_{5} - 177 \beta_{4} + 32 \beta_{3} - 99 \beta_{2} + 112 \beta_{1} + 69\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(-67 \beta_{15} + 396 \beta_{14} - 1679 \beta_{13} + 411 \beta_{12} - 757 \beta_{11} - 1647 \beta_{10} - 1303 \beta_{9} + 284 \beta_{8} - 86 \beta_{7} - 877 \beta_{6} + 1561 \beta_{5} - 3 \beta_{4} + 508 \beta_{3} - 276 \beta_{2} + 1658 \beta_{1} - 104\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(322 \beta_{15} + 209 \beta_{14} - 4661 \beta_{13} + 1249 \beta_{12} - 1198 \beta_{11} - 6368 \beta_{10} - 2737 \beta_{9} - 1044 \beta_{8} - 1629 \beta_{7} - 3763 \beta_{6} + 1859 \beta_{5} - 1652 \beta_{4} + 1272 \beta_{3} - 399 \beta_{2} + 3612 \beta_{1} + 659\)\()/5\)
\(\nu^{8}\)\(=\)\((\)\(-722 \beta_{15} + 4821 \beta_{14} - 22869 \beta_{13} + 6081 \beta_{12} - 6637 \beta_{11} - 22292 \beta_{10} - 17653 \beta_{9} + 5054 \beta_{8} - 1361 \beta_{7} - 11897 \beta_{6} + 17361 \beta_{5} - 2488 \beta_{4} + 4463 \beta_{3} - 2791 \beta_{2} + 17423 \beta_{1} - 6464\)\()/5\)
\(\nu^{9}\)\(=\)\((\)\(-4648 \beta_{15} + 10089 \beta_{14} - 61441 \beta_{13} + 10719 \beta_{12} - 10283 \beta_{11} - 69933 \beta_{10} - 49987 \beta_{9} + 6701 \beta_{8} - 21274 \beta_{7} - 64498 \beta_{6} + 23899 \beta_{5} - 13622 \beta_{4} + 20527 \beta_{3} + 561 \beta_{2} + 58037 \beta_{1} + 1354\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(-17042 \beta_{15} + 61161 \beta_{14} - 277069 \beta_{13} + 72921 \beta_{12} - 50587 \beta_{11} - 273867 \beta_{10} - 221948 \beta_{9} + 71349 \beta_{8} - 28166 \beta_{7} - 175417 \beta_{6} + 166606 \beta_{5} - 42628 \beta_{4} + 45218 \beta_{3} - 18106 \beta_{2} + 182258 \beta_{1} - 101659\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(-134263 \beta_{15} + 202349 \beta_{14} - 797386 \beta_{13} + 111324 \beta_{12} - 86558 \beta_{11} - 792838 \beta_{10} - 749067 \beta_{9} + 261221 \beta_{8} - 241464 \beta_{7} - 907323 \beta_{6} + 302154 \beta_{5} - 100422 \beta_{4} + 265777 \beta_{3} + 51356 \beta_{2} + 747912 \beta_{1} - 106706\)\()/5\)
\(\nu^{12}\)\(=\)\((\)\(-378617 \beta_{15} + 805976 \beta_{14} - 3193009 \beta_{13} + 749601 \beta_{12} - 291027 \beta_{11} - 3144842 \beta_{10} - 2726573 \beta_{9} + 1034699 \beta_{8} - 507941 \beta_{7} - 2581267 \beta_{6} + 1434546 \beta_{5} - 498313 \beta_{4} + 553503 \beta_{3} + 29859 \beta_{2} + 1963313 \beta_{1} - 1309799\)\()/5\)
\(\nu^{13}\)\(=\)\((\)\(-2303643 \beta_{15} + 3165794 \beta_{14} - 9996001 \beta_{13} + 1254869 \beta_{12} - 527138 \beta_{11} - 9072163 \beta_{10} - 10035822 \beta_{9} + 4755751 \beta_{8} - 2651744 \beta_{7} - 11712898 \beta_{6} + 3497229 \beta_{5} - 655547 \beta_{4} + 3100882 \beta_{3} + 1093781 \beta_{2} + 8564457 \beta_{1} - 2885261\)\()/5\)
\(\nu^{14}\)\(=\)\((\)\(-6955377 \beta_{15} + 10771406 \beta_{14} - 35942739 \beta_{13} + 6786036 \beta_{12} - 208367 \beta_{11} - 34294982 \beta_{10} - 33244988 \beta_{9} + 15468999 \beta_{8} - 7800891 \beta_{7} - 36336197 \beta_{6} + 11168716 \beta_{5} - 4560408 \beta_{4} + 7312018 \beta_{3} + 3446899 \beta_{2} + 21728638 \beta_{1} - 16212534\)\()/5\)
\(\nu^{15}\)\(=\)\((\)\(-33676153 \beta_{15} + 43741979 \beta_{14} - 119837181 \beta_{13} + 13712654 \beta_{12} + 1164252 \beta_{11} - 101919263 \beta_{10} - 125083137 \beta_{9} + 70977541 \beta_{8} - 29532179 \beta_{7} - 145033883 \beta_{6} + 35437464 \beta_{5} - 3225587 \beta_{4} + 34156632 \beta_{3} + 18966791 \beta_{2} + 90702602 \beta_{1} - 51277636\)\()/5\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.80334 0.309017i
3.42137 0.309017i
2.32349 + 0.309017i
−0.705457 + 0.309017i
−1.80334 + 0.309017i
3.42137 + 0.309017i
2.32349 0.309017i
−0.705457 0.309017i
2.17199 0.809017i
−2.79002 0.809017i
0.543374 + 0.809017i
−1.16141 + 0.809017i
2.17199 + 0.809017i
−2.79002 + 0.809017i
0.543374 0.809017i
−1.16141 0.809017i
−0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i −1.73558 + 1.40988i −0.809017 0.587785i 2.61995i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.21496 1.87720i
19.2 −0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i 2.23558 0.0466062i −0.809017 0.587785i 3.52206i −0.587785 + 0.809017i −0.309017 + 0.951057i −2.11176 + 0.735158i
19.3 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i −1.47959 1.67655i −0.809017 0.587785i 3.23143i 0.587785 0.809017i −0.309017 + 0.951057i −1.92526 1.13727i
19.4 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i 1.97959 + 1.03982i −0.809017 0.587785i 0.329315i 0.587785 0.809017i −0.309017 + 0.951057i 2.20402 + 0.377200i
79.1 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i −1.73558 1.40988i −0.809017 + 0.587785i 2.61995i −0.587785 0.809017i −0.309017 0.951057i 1.21496 + 1.87720i
79.2 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i 2.23558 + 0.0466062i −0.809017 + 0.587785i 3.52206i −0.587785 0.809017i −0.309017 0.951057i −2.11176 0.735158i
79.3 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i −1.47959 + 1.67655i −0.809017 + 0.587785i 3.23143i 0.587785 + 0.809017i −0.309017 0.951057i −1.92526 + 1.13727i
79.4 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 1.97959 1.03982i −0.809017 + 0.587785i 0.329315i 0.587785 + 0.809017i −0.309017 0.951057i 2.20402 0.377200i
109.1 −0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i −1.53938 1.62182i 0.309017 0.951057i 4.63137i 0.951057 + 0.309017i 0.809017 0.587785i 2.21691 0.292102i
109.2 −0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 2.03938 0.917020i 0.309017 0.951057i 4.80694i 0.951057 + 0.309017i 0.809017 0.587785i −0.456833 + 2.18890i
109.3 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i −1.36682 1.76969i 0.309017 0.951057i 0.533559i −0.951057 0.309017i 0.809017 0.587785i −2.23511 + 0.0655797i
109.4 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i 1.86682 + 1.23085i 0.309017 0.951057i 2.70913i −0.951057 0.309017i 0.809017 0.587785i 2.09307 0.786811i
139.1 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i −1.53938 + 1.62182i 0.309017 + 0.951057i 4.63137i 0.951057 0.309017i 0.809017 + 0.587785i 2.21691 + 0.292102i
139.2 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 2.03938 + 0.917020i 0.309017 + 0.951057i 4.80694i 0.951057 0.309017i 0.809017 + 0.587785i −0.456833 2.18890i
139.3 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i −1.36682 + 1.76969i 0.309017 + 0.951057i 0.533559i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.23511 0.0655797i
139.4 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 1.86682 1.23085i 0.309017 + 0.951057i 2.70913i −0.951057 + 0.309017i 0.809017 + 0.587785i 2.09307 + 0.786811i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
25.e Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).