Properties

Label 70.2.k.a
Level 70
Weight 2
Character orbit 70.k
Analytic conductor 0.559
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 70.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 10 x^{14} + 61 x^{12} + 266 x^{10} + 852 x^{8} + 1438 x^{6} + 1933 x^{4} + 3038 x^{2} + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 171 \nu^{14} + 2802 \nu^{12} + 20266 \nu^{10} + 96110 \nu^{8} + 343988 \nu^{6} + 866714 \nu^{4} + 786719 \nu^{2} + 1302910 \)\()/1020740\)
\(\beta_{3}\)\(=\)\((\)\(-6279 \nu^{15} - 6023 \nu^{14} - 94829 \nu^{13} - 319853 \nu^{12} - 291088 \nu^{11} - 2335656 \nu^{10} - 1140202 \nu^{9} - 13637974 \nu^{8} - 2242730 \nu^{7} - 52276470 \nu^{6} - 2623012 \nu^{5} - 131660244 \nu^{4} - 4504619 \nu^{3} - 120098303 \nu^{2} - 163267041 \nu - 204554077\)\()/ 224562800 \)
\(\beta_{4}\)\(=\)\((\)\(6279 \nu^{15} - 6023 \nu^{14} + 94829 \nu^{13} - 319853 \nu^{12} + 291088 \nu^{11} - 2335656 \nu^{10} + 1140202 \nu^{9} - 13637974 \nu^{8} + 2242730 \nu^{7} - 52276470 \nu^{6} + 2623012 \nu^{5} - 131660244 \nu^{4} + 4504619 \nu^{3} - 120098303 \nu^{2} + 163267041 \nu - 204554077\)\()/ 224562800 \)
\(\beta_{5}\)\(=\)\((\)\( 657161 \nu^{14} + 6218761 \nu^{12} + 35853172 \nu^{10} + 144339958 \nu^{8} + 433352890 \nu^{6} + 516320528 \nu^{4} + 189804581 \nu^{2} + 1019822349 \)\()/ 785969800 \)
\(\beta_{6}\)\(=\)\((\)\( -623003 \nu^{14} - 5366258 \nu^{12} - 30938216 \nu^{10} - 125675214 \nu^{8} - 373946420 \nu^{6} - 445540384 \nu^{4} - 843429583 \nu^{2} - 1273004222 \)\()/ 392984900 \)
\(\beta_{7}\)\(=\)\((\)\(4174573 \nu^{15} + 307671 \nu^{14} + 41450603 \nu^{13} + 4646621 \nu^{12} + 238976156 \nu^{11} + 14263312 \nu^{10} + 995989274 \nu^{9} + 55869898 \nu^{8} + 2888475470 \nu^{7} + 109893770 \nu^{6} + 3441488944 \nu^{5} + 128527588 \nu^{4} + 1618097653 \nu^{3} + 220726331 \nu^{2} + 6797535927 \nu + 8000085009\)\()/ 11003577200 \)
\(\beta_{8}\)\(=\)\((\)\(-4174573 \nu^{15} + 307671 \nu^{14} - 41450603 \nu^{13} + 4646621 \nu^{12} - 238976156 \nu^{11} + 14263312 \nu^{10} - 995989274 \nu^{9} + 55869898 \nu^{8} - 2888475470 \nu^{7} + 109893770 \nu^{6} - 3441488944 \nu^{5} + 128527588 \nu^{4} - 1618097653 \nu^{3} + 220726331 \nu^{2} - 6797535927 \nu + 8000085009\)\()/ 11003577200 \)
\(\beta_{9}\)\(=\)\((\)\(-5413027 \nu^{15} - 6673919 \nu^{14} - 53207257 \nu^{13} - 67697189 \nu^{12} - 316254784 \nu^{11} - 390296228 \nu^{10} - 1397075246 \nu^{9} - 1650388502 \nu^{8} - 4444289310 \nu^{7} - 4717462610 \nu^{6} - 7454251516 \nu^{5} - 5620645072 \nu^{4} - 10077798427 \nu^{3} - 2889806479 \nu^{2} - 15782597033 \nu - 11101746201\)\()/ 11003577200 \)
\(\beta_{10}\)\(=\)\((\)\(-5413027 \nu^{15} + 6673919 \nu^{14} - 53207257 \nu^{13} + 67697189 \nu^{12} - 316254784 \nu^{11} + 390296228 \nu^{10} - 1397075246 \nu^{9} + 1650388502 \nu^{8} - 4444289310 \nu^{7} + 4717462610 \nu^{6} - 7454251516 \nu^{5} + 5620645072 \nu^{4} - 10077798427 \nu^{3} + 2889806479 \nu^{2} - 15782597033 \nu + 11101746201\)\()/ 11003577200 \)
\(\beta_{11}\)\(=\)\((\)\( 26590 \nu^{15} + 257521 \nu^{13} + 1484692 \nu^{11} + 6079906 \nu^{9} + 17945290 \nu^{7} + 21381008 \nu^{5} + 8929484 \nu^{3} + 42231189 \nu \)\()/50016260\)
\(\beta_{12}\)\(=\)\((\)\( -154017 \nu^{15} - 1553547 \nu^{13} - 9597064 \nu^{11} - 41588666 \nu^{9} - 133651610 \nu^{7} - 226254436 \nu^{5} - 303303017 \nu^{3} - 477500443 \nu \)\()/ 239208200 \)
\(\beta_{13}\)\(=\)\((\)\(7738601 \nu^{15} - 12843957 \nu^{14} + 58163261 \nu^{13} - 106723897 \nu^{12} + 313306372 \nu^{11} - 615297844 \nu^{10} + 1167320938 \nu^{9} - 2508526286 \nu^{8} + 3102633990 \nu^{7} - 7437029530 \nu^{6} + 1535163128 \nu^{5} - 8860887056 \nu^{4} + 6928535561 \nu^{3} - 16785607657 \nu^{2} + 9719272549 \nu - 17501784573\)\()/ 11003577200 \)
\(\beta_{14}\)\(=\)\((\)\(-5179651 \nu^{15} - 39396031 \nu^{13} - 212402552 \nu^{11} - 802913678 \nu^{9} - 2161262850 \nu^{7} - 1033713348 \nu^{5} - 4832044771 \nu^{3} - 6839548919 \nu\)\()/ 5501788600 \)
\(\beta_{15}\)\(=\)\((\)\(-850941 \nu^{15} - 939402 \nu^{14} - 7115276 \nu^{13} - 7955037 \nu^{12} - 39448752 \nu^{11} - 44968654 \nu^{10} - 154522158 \nu^{9} - 183171156 \nu^{8} - 427936390 \nu^{7} - 539065950 \nu^{6} - 355475148 \nu^{5} - 642101046 \nu^{4} - 610473801 \nu^{3} - 1214738142 \nu^{2} - 1179772034 \nu - 1821562113\)\()/ 785969800 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} + \beta_{13} + \beta_{7} - 2 \beta_{6} - \beta_{5} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{15} - 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{4} - \beta_{3} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{15} - 5 \beta_{13} + 5 \beta_{8} + 8 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-14 \beta_{15} + 22 \beta_{14} + 14 \beta_{13} + 3 \beta_{12} - 3 \beta_{10} - 3 \beta_{9} - 14 \beta_{7}\)
\(\nu^{6}\)\(=\)\(14 \beta_{15} + 14 \beta_{13} - 14 \beta_{10} + 14 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} - 22 \beta_{6} + 22 \beta_{5} - 14 \beta_{4} - 14 \beta_{3} - 22 \beta_{2} - 7\)
\(\nu^{7}\)\(=\)\(64 \beta_{15} - 94 \beta_{14} - 64 \beta_{13} + 94 \beta_{11} + 64 \beta_{8} + 2 \beta_{4} - 2 \beta_{3} - 7 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-75 \beta_{15} - 75 \beta_{13} + 62 \beta_{10} - 62 \beta_{9} - 75 \beta_{7} + 112 \beta_{6} - 87 \beta_{5} + 112\)
\(\nu^{9}\)\(=\)\(-75 \beta_{15} + 112 \beta_{14} + 75 \beta_{13} + 112 \beta_{12} - 385 \beta_{11} - 75 \beta_{10} - 75 \beta_{9} - 273 \beta_{8} + 198 \beta_{7} - 75 \beta_{4} + 75 \beta_{3} + 112 \beta_{1}\)
\(\nu^{10}\)\(=\)\(337 \beta_{15} + 337 \beta_{13} - 337 \beta_{8} - 486 \beta_{6} + 198 \beta_{4} + 198 \beta_{3} + 273 \beta_{2}\)
\(\nu^{11}\)\(=\)\(-332 \beta_{15} + 456 \beta_{14} + 332 \beta_{13} - 759 \beta_{12} + 535 \beta_{10} + 535 \beta_{9} - 332 \beta_{7}\)
\(\nu^{12}\)\(=\)\(332 \beta_{15} + 332 \beta_{13} - 332 \beta_{10} + 332 \beta_{9} + 2364 \beta_{8} + 2696 \beta_{7} - 456 \beta_{6} + 456 \beta_{5} - 332 \beta_{4} - 332 \beta_{3} - 456 \beta_{2} - 3803\)
\(\nu^{13}\)\(=\)\(1452 \beta_{15} - 2032 \beta_{14} - 1452 \beta_{13} + 2032 \beta_{11} + 1452 \beta_{8} + 2696 \beta_{4} - 2696 \beta_{3} - 3803 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-10647 \beta_{15} - 10647 \beta_{13} - 1244 \beta_{10} + 1244 \beta_{9} - 10647 \beta_{7} + 15030 \beta_{6} + 1771 \beta_{5} + 15030\)
\(\nu^{15}\)\(=\)\(-10647 \beta_{15} + 15030 \beta_{14} + 10647 \beta_{13} + 15030 \beta_{12} + 7801 \beta_{11} - 10647 \beta_{10} - 10647 \beta_{9} + 5503 \beta_{8} - 16150 \beta_{7} - 10647 \beta_{4} + 10647 \beta_{3} + 15030 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(57\)
\(\chi(n)\) \(1 + \beta_{6}\) \(\beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−1.01089 + 0.750919i
0.144868 1.25092i
0.587308 2.01725i
−1.45333 + 1.51725i
−0.587308 + 2.01725i
1.45333 1.51725i
1.01089 0.750919i
−0.144868 + 1.25092i
−0.587308 2.01725i
1.45333 + 1.51725i
1.01089 + 0.750919i
−0.144868 1.25092i
−1.01089 0.750919i
0.144868 + 1.25092i
0.587308 + 2.01725i
−1.45333 1.51725i
−0.965926 0.258819i −0.0749894 0.279864i 0.866025 + 0.500000i 2.20382 + 0.378409i 0.289737i 0.126334 2.64273i −0.707107 0.707107i 2.52538 1.45803i −2.03078 0.935904i
3.2 −0.965926 0.258819i 0.523277 + 1.95290i 0.866025 + 0.500000i −2.03078 + 0.935904i 2.02179i 1.83959 + 1.90155i −0.707107 0.707107i −0.941911 + 0.543813i 2.20382 0.378409i
3.3 0.965926 + 0.258819i −0.752300 2.80762i 0.866025 + 0.500000i −1.38266 + 1.75735i 2.90667i 2.58583 0.559876i 0.707107 + 0.707107i −4.71872 + 2.72435i −1.79038 + 1.33961i
3.4 0.965926 + 0.258819i 0.304013 + 1.13459i 0.866025 + 0.500000i −1.79038 1.33961i 1.17462i −2.55176 0.698943i 0.707107 + 0.707107i 1.40320 0.810140i −1.38266 1.75735i
17.1 −0.258819 + 0.965926i −2.80762 + 0.752300i −0.866025 0.500000i −2.21323 + 0.318742i 2.90667i 0.559876 + 2.58583i 0.707107 0.707107i 4.71872 2.72435i 0.264946 2.22032i
17.2 −0.258819 + 0.965926i 1.13459 0.304013i −0.866025 0.500000i 0.264946 + 2.22032i 1.17462i 0.698943 2.55176i 0.707107 0.707107i −1.40320 + 0.810140i −2.21323 0.318742i
17.3 0.258819 0.965926i −0.279864 + 0.0749894i −0.866025 0.500000i 0.774197 2.09777i 0.289737i 2.64273 + 0.126334i −0.707107 + 0.707107i −2.52538 + 1.45803i −1.82591 1.29076i
17.4 0.258819 0.965926i 1.95290 0.523277i −0.866025 0.500000i −1.82591 + 1.29076i 2.02179i −1.90155 + 1.83959i −0.707107 + 0.707107i 0.941911 0.543813i 0.774197 + 2.09777i
33.1 −0.258819 0.965926i −2.80762 0.752300i −0.866025 + 0.500000i −2.21323 0.318742i 2.90667i 0.559876 2.58583i 0.707107 + 0.707107i 4.71872 + 2.72435i 0.264946 + 2.22032i
33.2 −0.258819 0.965926i 1.13459 + 0.304013i −0.866025 + 0.500000i 0.264946 2.22032i 1.17462i 0.698943 + 2.55176i 0.707107 + 0.707107i −1.40320 0.810140i −2.21323 + 0.318742i
33.3 0.258819 + 0.965926i −0.279864 0.0749894i −0.866025 + 0.500000i 0.774197 + 2.09777i 0.289737i 2.64273 0.126334i −0.707107 0.707107i −2.52538 1.45803i −1.82591 + 1.29076i
33.4 0.258819 + 0.965926i 1.95290 + 0.523277i −0.866025 + 0.500000i −1.82591 1.29076i 2.02179i −1.90155 1.83959i −0.707107 0.707107i 0.941911 + 0.543813i 0.774197 2.09777i
47.1 −0.965926 + 0.258819i −0.0749894 + 0.279864i 0.866025 0.500000i 2.20382 0.378409i 0.289737i 0.126334 + 2.64273i −0.707107 + 0.707107i 2.52538 + 1.45803i −2.03078 + 0.935904i
47.2 −0.965926 + 0.258819i 0.523277 1.95290i 0.866025 0.500000i −2.03078 0.935904i 2.02179i 1.83959 1.90155i −0.707107 + 0.707107i −0.941911 0.543813i 2.20382 + 0.378409i
47.3 0.965926 0.258819i −0.752300 + 2.80762i 0.866025 0.500000i −1.38266 1.75735i 2.90667i 2.58583 + 0.559876i 0.707107 0.707107i −4.71872 2.72435i −1.79038 1.33961i
47.4 0.965926 0.258819i 0.304013 1.13459i 0.866025 0.500000i −1.79038 + 1.33961i 1.17462i −2.55176 + 0.698943i 0.707107 0.707107i 1.40320 + 0.810140i −1.38266 + 1.75735i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.k.a 16
3.b odd 2 1 630.2.bv.c 16
4.b odd 2 1 560.2.ci.c 16
5.b even 2 1 350.2.o.c 16
5.c odd 4 1 inner 70.2.k.a 16
5.c odd 4 1 350.2.o.c 16
7.b odd 2 1 490.2.l.c 16
7.c even 3 1 490.2.g.c 16
7.c even 3 1 490.2.l.c 16
7.d odd 6 1 inner 70.2.k.a 16
7.d odd 6 1 490.2.g.c 16
15.e even 4 1 630.2.bv.c 16
20.e even 4 1 560.2.ci.c 16
21.g even 6 1 630.2.bv.c 16
28.f even 6 1 560.2.ci.c 16
35.f even 4 1 490.2.l.c 16
35.i odd 6 1 350.2.o.c 16
35.k even 12 1 inner 70.2.k.a 16
35.k even 12 1 350.2.o.c 16
35.k even 12 1 490.2.g.c 16
35.l odd 12 1 490.2.g.c 16
35.l odd 12 1 490.2.l.c 16
105.w odd 12 1 630.2.bv.c 16
140.x odd 12 1 560.2.ci.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.k.a 16 1.a even 1 1 trivial
70.2.k.a 16 5.c odd 4 1 inner
70.2.k.a 16 7.d odd 6 1 inner
70.2.k.a 16 35.k even 12 1 inner
350.2.o.c 16 5.b even 2 1
350.2.o.c 16 5.c odd 4 1
350.2.o.c 16 35.i odd 6 1
350.2.o.c 16 35.k even 12 1
490.2.g.c 16 7.c even 3 1
490.2.g.c 16 7.d odd 6 1
490.2.g.c 16 35.k even 12 1
490.2.g.c 16 35.l odd 12 1
490.2.l.c 16 7.b odd 2 1
490.2.l.c 16 7.c even 3 1
490.2.l.c 16 35.f even 4 1
490.2.l.c 16 35.l odd 12 1
560.2.ci.c 16 4.b odd 2 1
560.2.ci.c 16 20.e even 4 1
560.2.ci.c 16 28.f even 6 1
560.2.ci.c 16 140.x odd 12 1
630.2.bv.c 16 3.b odd 2 1
630.2.bv.c 16 15.e even 4 1
630.2.bv.c 16 21.g even 6 1
630.2.bv.c 16 105.w odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(70, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$3$ \( 1 + 3 T^{4} - 24 T^{5} + 48 T^{7} + 5 T^{8} - 72 T^{9} + 288 T^{10} + 120 T^{11} - 570 T^{12} - 720 T^{13} + 2016 T^{14} - 2136 T^{15} - 2570 T^{16} - 6408 T^{17} + 18144 T^{18} - 19440 T^{19} - 46170 T^{20} + 29160 T^{21} + 209952 T^{22} - 157464 T^{23} + 32805 T^{24} + 944784 T^{25} - 4251528 T^{27} + 1594323 T^{28} + 43046721 T^{32} \)
$5$ \( 1 + 12 T + 66 T^{2} + 216 T^{3} + 450 T^{4} + 468 T^{5} - 864 T^{6} - 6252 T^{7} - 18241 T^{8} - 31260 T^{9} - 21600 T^{10} + 58500 T^{11} + 281250 T^{12} + 675000 T^{13} + 1031250 T^{14} + 937500 T^{15} + 390625 T^{16} \)
$7$ \( 1 - 8 T + 32 T^{2} - 80 T^{3} + 73 T^{4} + 160 T^{5} - 416 T^{6} - 696 T^{7} + 4944 T^{8} - 4872 T^{9} - 20384 T^{10} + 54880 T^{11} + 175273 T^{12} - 1344560 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16} \)
$11$ \( ( 1 + 6 T + 5 T^{2} - 18 T^{3} - 29 T^{4} - 252 T^{5} - 544 T^{6} - 516 T^{7} - 6650 T^{8} - 5676 T^{9} - 65824 T^{10} - 335412 T^{11} - 424589 T^{12} - 2898918 T^{13} + 8857805 T^{14} + 116923026 T^{15} + 214358881 T^{16} )^{2} \)
$13$ \( 1 + 714 T^{4} + 266977 T^{8} + 68653746 T^{12} + 13237862628 T^{16} + 1960819639506 T^{20} + 217781340700417 T^{24} + 16634832777451434 T^{28} + 665416609183179841 T^{32} \)
$17$ \( 1 + 36 T + 648 T^{2} + 7776 T^{3} + 70080 T^{4} + 507420 T^{5} + 3088368 T^{6} + 16357284 T^{7} + 77768530 T^{8} + 343436904 T^{9} + 1468660248 T^{10} + 6351048828 T^{11} + 28493424864 T^{12} + 131692528476 T^{13} + 608521516920 T^{14} + 2723900163624 T^{15} + 11569567164915 T^{16} + 46306302781608 T^{17} + 175862718389880 T^{18} + 647005392402588 T^{19} + 2379799338066144 T^{20} + 9017581135777596 T^{21} + 35449888073657112 T^{22} + 140925443446588392 T^{23} + 542494401823131730 T^{24} + 1939775574818354148 T^{25} + 6226131050341877232 T^{26} + 17390245624419136860 T^{27} + 40830166385061650880 T^{28} + 77017998783876566112 T^{29} + \)\(10\!\cdots\!92\)\( T^{30} + \)\(10\!\cdots\!48\)\( T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 90 T^{2} + 4011 T^{4} - 123414 T^{6} + 3034057 T^{8} - 61998180 T^{10} + 1094493006 T^{12} - 18780034728 T^{14} + 344074548210 T^{16} - 6779592536808 T^{18} + 142635423034926 T^{20} - 2916758998496580 T^{22} + 51529098329487337 T^{24} - 756659411140252614 T^{26} + 8877606140374371771 T^{28} - 71910601720459570890 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 + 4 T + 8 T^{2} - 128 T^{3} - 198 T^{4} - 292 T^{5} + 8608 T^{6} + 3564 T^{7} - 17335 T^{8} + 1017836 T^{9} + 5500848 T^{10} - 425132 T^{11} + 20115594 T^{12} + 929862264 T^{13} + 4560471720 T^{14} - 14572270428 T^{15} - 102219015884 T^{16} - 335162219844 T^{17} + 2412489539880 T^{18} + 11313634166088 T^{19} + 5629167940554 T^{20} - 2736295372276 T^{21} + 814322923933872 T^{22} + 3465553913672692 T^{23} - 1357520929846135 T^{24} + 6419308085454132 T^{25} + 356599408527090592 T^{26} - 278220449310866684 T^{27} - 4339095637540023558 T^{28} - 64516654327867825024 T^{29} + 92742690596309998472 T^{30} + \)\(10\!\cdots\!28\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( ( 1 - 70 T^{2} + 2685 T^{4} - 80906 T^{6} + 2505752 T^{8} - 68041946 T^{10} + 1899049485 T^{12} - 41637632470 T^{14} + 500246412961 T^{16} )^{2} \)
$31$ \( ( 1 - 12 T + 156 T^{2} - 1296 T^{3} + 10686 T^{4} - 72996 T^{5} + 489600 T^{6} - 2919012 T^{7} + 17046947 T^{8} - 90489372 T^{9} + 470505600 T^{10} - 2174623836 T^{11} + 9868745406 T^{12} - 37103379696 T^{13} + 138450574236 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} )^{2} \)
$37$ \( 1 - 4 T + 8 T^{2} + 240 T^{3} - 3049 T^{4} + 13544 T^{5} - 984 T^{6} - 514972 T^{7} + 3800209 T^{8} - 12256568 T^{9} - 16248928 T^{10} + 295990824 T^{11} + 119383314 T^{12} - 6433305488 T^{13} + 16607500784 T^{14} + 478842888760 T^{15} - 3828375485890 T^{16} + 17717186884120 T^{17} + 22735668573296 T^{18} - 325866222883664 T^{19} + 223743551149554 T^{20} + 20525174971850568 T^{21} - 41690303687539552 T^{22} - 1163539007448259544 T^{23} + 13348156033105669489 T^{24} - 66926657065750392844 T^{25} - 4731647022459163416 T^{26} + \)\(24\!\cdots\!72\)\( T^{27} - \)\(20\!\cdots\!69\)\( T^{28} + \)\(58\!\cdots\!80\)\( T^{29} + \)\(72\!\cdots\!12\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( ( 1 - 188 T^{2} + 17802 T^{4} - 1146544 T^{6} + 54447827 T^{8} - 1927340464 T^{10} + 50304197322 T^{12} - 893019597308 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 + 4 T + 8 T^{2} + 32 T^{3} + 3425 T^{4} + 20696 T^{5} + 55896 T^{6} + 651228 T^{7} + 6527152 T^{8} + 28002804 T^{9} + 103351704 T^{10} + 1645476872 T^{11} + 11709393425 T^{12} + 4704270176 T^{13} + 50570904392 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 - 12 T + 72 T^{2} - 288 T^{3} - 449 T^{4} + 2880 T^{5} + 39240 T^{6} - 500820 T^{7} + 44673 T^{8} + 37644384 T^{9} - 288946944 T^{10} + 1723474464 T^{11} + 9984376994 T^{12} - 179746348152 T^{13} + 1044056404368 T^{14} - 4096130163936 T^{15} + 8350203627806 T^{16} - 192518117704992 T^{17} + 2306320597248912 T^{18} - 18661805104185096 T^{19} + 48720574714458914 T^{20} + 395270263010401248 T^{21} - 3114621328032504576 T^{22} + 19071515289987429792 T^{23} + 1063721609040849153 T^{24} - \)\(56\!\cdots\!40\)\( T^{25} + \)\(20\!\cdots\!60\)\( T^{26} + \)\(71\!\cdots\!40\)\( T^{27} - \)\(52\!\cdots\!09\)\( T^{28} - \)\(15\!\cdots\!76\)\( T^{29} + \)\(18\!\cdots\!68\)\( T^{30} - \)\(14\!\cdots\!16\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 28 T + 392 T^{2} + 1808 T^{3} - 25321 T^{4} - 567704 T^{5} - 4335448 T^{6} + 1435172 T^{7} + 392677969 T^{8} + 4105212392 T^{9} + 14977011104 T^{10} - 124625519288 T^{11} - 2189877391278 T^{12} - 14281594759536 T^{13} - 14623477702416 T^{14} + 648224117593688 T^{15} + 7103959219709822 T^{16} + 34355878232465464 T^{17} - 41077348866086544 T^{18} - 2126200983015441072 T^{19} - 17279185948208624718 T^{20} - 52117830479026168984 T^{21} + \)\(33\!\cdots\!16\)\( T^{22} + \)\(48\!\cdots\!04\)\( T^{23} + \)\(24\!\cdots\!09\)\( T^{24} + \)\(47\!\cdots\!76\)\( T^{25} - \)\(75\!\cdots\!52\)\( T^{26} - \)\(52\!\cdots\!88\)\( T^{27} - \)\(12\!\cdots\!61\)\( T^{28} + \)\(47\!\cdots\!84\)\( T^{29} + \)\(54\!\cdots\!48\)\( T^{30} + \)\(20\!\cdots\!96\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 320 T^{2} + 54244 T^{4} - 6209920 T^{6} + 532238186 T^{8} - 36082424000 T^{10} + 2045329177232 T^{12} - 105925693186240 T^{14} + 5808829694261683 T^{16} - 368727337981301440 T^{18} + 24783992004353124752 T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(78\!\cdots\!06\)\( T^{24} - \)\(31\!\cdots\!20\)\( T^{26} + \)\(96\!\cdots\!64\)\( T^{28} - \)\(19\!\cdots\!20\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( ( 1 + 6 T + 207 T^{2} + 1170 T^{3} + 23079 T^{4} + 119796 T^{5} + 1969284 T^{6} + 9244740 T^{7} + 135493478 T^{8} + 563929140 T^{9} + 7327705764 T^{10} + 27191415876 T^{11} + 319548164439 T^{12} + 988177672170 T^{13} + 10664717492727 T^{14} + 18856457016126 T^{15} + 191707312997281 T^{16} )^{2} \)
$67$ \( 1 - 32 T + 512 T^{2} - 3896 T^{3} - 10509 T^{4} + 663104 T^{5} - 8249312 T^{6} + 44834240 T^{7} + 165963557 T^{8} - 5505231928 T^{9} + 49681319072 T^{10} - 159136747544 T^{11} - 1412717145882 T^{12} + 21627811976512 T^{13} - 103082665870656 T^{14} - 335467607852040 T^{15} + 7552610469365206 T^{16} - 22476329726086680 T^{17} - 462738087093374784 T^{18} + 6504845613491678656 T^{19} - 28467834145442833722 T^{20} - \)\(21\!\cdots\!08\)\( T^{21} + \)\(44\!\cdots\!68\)\( T^{22} - \)\(33\!\cdots\!44\)\( T^{23} + \)\(67\!\cdots\!37\)\( T^{24} + \)\(12\!\cdots\!80\)\( T^{25} - \)\(15\!\cdots\!88\)\( T^{26} + \)\(80\!\cdots\!32\)\( T^{27} - \)\(85\!\cdots\!49\)\( T^{28} - \)\(21\!\cdots\!52\)\( T^{29} + \)\(18\!\cdots\!48\)\( T^{30} - \)\(78\!\cdots\!76\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( ( 1 - 4 T + 94 T^{2} + 964 T^{3} - 1158 T^{4} + 68444 T^{5} + 473854 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{4} \)
$73$ \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 12064 T^{4} + 83700 T^{5} + 177264 T^{6} - 3613524 T^{7} + 19255890 T^{8} - 96865032 T^{9} - 2225167272 T^{10} - 39441388620 T^{11} + 6056513120 T^{12} - 1050679100940 T^{13} - 2551121523144 T^{14} - 71385024146568 T^{15} + 865631164724531 T^{16} - 5211106762699464 T^{17} - 13594926596834376 T^{18} - 408732031810375980 T^{19} + 171994319201421920 T^{20} - 81764822336595471660 T^{21} - \)\(33\!\cdots\!08\)\( T^{22} - \)\(10\!\cdots\!04\)\( T^{23} + \)\(15\!\cdots\!90\)\( T^{24} - \)\(21\!\cdots\!12\)\( T^{25} + \)\(76\!\cdots\!36\)\( T^{26} + \)\(26\!\cdots\!00\)\( T^{27} + \)\(27\!\cdots\!44\)\( T^{28} + \)\(48\!\cdots\!04\)\( T^{29} + \)\(87\!\cdots\!48\)\( T^{30} + \)\(10\!\cdots\!84\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 + 344 T^{2} + 56836 T^{4} + 6614224 T^{6} + 630756554 T^{8} + 45125337896 T^{10} + 2052833339792 T^{12} + 54520165812136 T^{14} + 1660868031271123 T^{16} + 340260354833540776 T^{18} + 79958024864398923152 T^{20} + \)\(10\!\cdots\!16\)\( T^{22} + \)\(95\!\cdots\!94\)\( T^{24} + \)\(62\!\cdots\!24\)\( T^{26} + \)\(33\!\cdots\!76\)\( T^{28} + \)\(12\!\cdots\!64\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 25638 T^{4} + 318852241 T^{8} - 2957648820318 T^{12} + 22747905510477444 T^{16} - \)\(14\!\cdots\!78\)\( T^{20} + \)\(71\!\cdots\!81\)\( T^{24} - \)\(27\!\cdots\!18\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - 522 T^{2} + 144915 T^{4} - 27754278 T^{6} + 4091335849 T^{8} - 494840689332 T^{10} + 51768593923950 T^{12} - 4926383904774360 T^{14} + 445675116984224850 T^{16} - 39021886909717705560 T^{18} + \)\(32\!\cdots\!50\)\( T^{20} - \)\(24\!\cdots\!52\)\( T^{22} + \)\(16\!\cdots\!69\)\( T^{24} - \)\(86\!\cdots\!78\)\( T^{26} + \)\(35\!\cdots\!15\)\( T^{28} - \)\(10\!\cdots\!02\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( ( 1 + 3868 T^{4} + 90482118 T^{8} + 342431258908 T^{12} + 7837433594376961 T^{16} )^{2} \)
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