Properties

Label 6025.2.a.i
Level 6025
Weight 2
Character orbit 6025.a
Self dual yes
Analytic conductor 48.110
Analytic rank 1
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2 x^{14} - 16 x^{13} + 31 x^{12} + 99 x^{11} - 184 x^{10} - 296 x^{9} + 519 x^{8} + 437 x^{7} - 699 x^{6} - 297 x^{5} + 394 x^{4} + 89 x^{3} - 57 x^{2} - 17 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1205)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 2 x^{14} - 16 x^{13} + 31 x^{12} + 99 x^{11} - 184 x^{10} - 296 x^{9} + 519 x^{8} + 437 x^{7} - 699 x^{6} - 297 x^{5} + 394 x^{4} + 89 x^{3} - 57 x^{2} - 17 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( -\nu^{14} + \nu^{13} + 16 \nu^{12} - 14 \nu^{11} - 98 \nu^{10} + 73 \nu^{9} + 284 \nu^{8} - 172 \nu^{7} - 385 \nu^{6} + 172 \nu^{5} + 198 \nu^{4} - 45 \nu^{3} - 9 \nu^{2} - 13 \nu - 3 \)
\(\beta_{5}\)\(=\)\( -\nu^{14} + \nu^{13} + 16 \nu^{12} - 15 \nu^{11} - 97 \nu^{10} + 86 \nu^{9} + 274 \nu^{8} - 233 \nu^{7} - 354 \nu^{6} + 295 \nu^{5} + 170 \nu^{4} - 143 \nu^{3} - 11 \nu^{2} + 11 \nu + 3 \)
\(\beta_{6}\)\(=\)\( \nu^{13} - \nu^{12} - 17 \nu^{11} + 14 \nu^{10} + 113 \nu^{9} - 71 \nu^{8} - 367 \nu^{7} + 153 \nu^{6} + 589 \nu^{5} - 119 \nu^{4} - 409 \nu^{3} + 9 \nu^{2} + 85 \nu + 10 \)
\(\beta_{7}\)\(=\)\( \nu^{14} - \nu^{13} - 17 \nu^{12} + 14 \nu^{11} + 113 \nu^{10} - 71 \nu^{9} - 367 \nu^{8} + 153 \nu^{7} + 589 \nu^{6} - 119 \nu^{5} - 409 \nu^{4} + 9 \nu^{3} + 85 \nu^{2} + 11 \nu - 1 \)
\(\beta_{8}\)\(=\)\( -\nu^{14} + 2 \nu^{13} + 16 \nu^{12} - 30 \nu^{11} - 99 \nu^{10} + 170 \nu^{9} + 293 \nu^{8} - 448 \nu^{7} - 408 \nu^{6} + 546 \nu^{5} + 210 \nu^{4} - 272 \nu^{3} - 8 \nu^{2} + 34 \nu + 1 \)
\(\beta_{9}\)\(=\)\( -\nu^{14} + 2 \nu^{13} + 15 \nu^{12} - 31 \nu^{11} - 84 \nu^{10} + 186 \nu^{9} + 212 \nu^{8} - 538 \nu^{7} - 223 \nu^{6} + 754 \nu^{5} + 55 \nu^{4} - 440 \nu^{3} + 17 \nu^{2} + 63 \nu + 6 \)
\(\beta_{10}\)\(=\)\( \nu^{14} - 2 \nu^{13} - 16 \nu^{12} + 31 \nu^{11} + 98 \nu^{10} - 183 \nu^{9} - 283 \nu^{8} + 509 \nu^{7} + 376 \nu^{6} - 667 \nu^{5} - 175 \nu^{4} + 359 \nu^{3} - 3 \nu^{2} - 46 \nu - 2 \)
\(\beta_{11}\)\(=\)\( \nu^{14} - 2 \nu^{13} - 16 \nu^{12} + 31 \nu^{11} + 98 \nu^{10} - 183 \nu^{9} - 283 \nu^{8} + 509 \nu^{7} + 376 \nu^{6} - 667 \nu^{5} - 176 \nu^{4} + 359 \nu^{3} + 2 \nu^{2} - 46 \nu - 5 \)
\(\beta_{12}\)\(=\)\( \nu^{14} - 18 \nu^{12} - 3 \nu^{11} + 128 \nu^{10} + 40 \nu^{9} - 450 \nu^{8} - 193 \nu^{7} + 794 \nu^{6} + 397 \nu^{5} - 624 \nu^{4} - 306 \nu^{3} + 160 \nu^{2} + 57 \nu \)
\(\beta_{13}\)\(=\)\( \nu^{14} - 3 \nu^{13} - 15 \nu^{12} + 47 \nu^{11} + 84 \nu^{10} - 281 \nu^{9} - 210 \nu^{8} + 793 \nu^{7} + 204 \nu^{6} - 1053 \nu^{5} - 2 \nu^{4} + 564 \nu^{3} - 54 \nu^{2} - 67 \nu - 6 \)
\(\beta_{14}\)\(=\)\( -2 \nu^{13} + 2 \nu^{12} + 33 \nu^{11} - 29 \nu^{10} - 210 \nu^{9} + 157 \nu^{8} + 641 \nu^{7} - 386 \nu^{6} - 942 \nu^{5} + 413 \nu^{4} + 572 \nu^{3} - 147 \nu^{2} - 84 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{11} + \beta_{10} + 5 \beta_{2} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{14} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 6 \beta_{3} + 17 \beta_{1}\)
\(\nu^{6}\)\(=\)\(2 \beta_{14} + 2 \beta_{12} - 9 \beta_{11} + 8 \beta_{10} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + 22 \beta_{2} + 2 \beta_{1} + 28\)
\(\nu^{7}\)\(=\)\(11 \beta_{14} - \beta_{13} + 10 \beta_{12} - 11 \beta_{11} + 10 \beta_{10} + 9 \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + 32 \beta_{3} + 76 \beta_{1} + 1\)
\(\nu^{8}\)\(=\)\(22 \beta_{14} - \beta_{13} + 21 \beta_{12} - 61 \beta_{11} + 51 \beta_{10} - \beta_{9} + 11 \beta_{8} + \beta_{7} + \beta_{6} - 10 \beta_{5} + 11 \beta_{4} + 13 \beta_{3} + 95 \beta_{2} + 22 \beta_{1} + 118\)
\(\nu^{9}\)\(=\)\(84 \beta_{14} - 11 \beta_{13} + 72 \beta_{12} - 86 \beta_{11} + 73 \beta_{10} - \beta_{9} + 61 \beta_{8} + 13 \beta_{7} + \beta_{6} - 12 \beta_{5} + 13 \beta_{4} + 168 \beta_{3} + \beta_{2} + 351 \beta_{1} + 11\)
\(\nu^{10}\)\(=\)\(170 \beta_{14} - 13 \beta_{13} + 156 \beta_{12} - 374 \beta_{11} + 301 \beta_{10} - 14 \beta_{9} + 86 \beta_{8} + 16 \beta_{7} + 13 \beta_{6} - 71 \beta_{5} + 85 \beta_{4} + 112 \beta_{3} + 412 \beta_{2} + 170 \beta_{1} + 510\)
\(\nu^{11}\)\(=\)\(556 \beta_{14} - 85 \beta_{13} + 457 \beta_{12} - 585 \beta_{11} + 473 \beta_{10} - 17 \beta_{9} + 374 \beta_{8} + 114 \beta_{7} + 16 \beta_{6} - 98 \beta_{5} + 115 \beta_{4} + 885 \beta_{3} + 15 \beta_{2} + 1662 \beta_{1} + 86\)
\(\nu^{12}\)\(=\)\(1144 \beta_{14} - 115 \beta_{13} + 1012 \beta_{12} - 2188 \beta_{11} + 1712 \beta_{10} - 129 \beta_{9} + 585 \beta_{8} + 163 \beta_{7} + 114 \beta_{6} - 444 \beta_{5} + 572 \beta_{4} + 815 \beta_{3} + 1806 \beta_{2} + 1145 \beta_{1} + 2242\)
\(\nu^{13}\)\(=\)\(3428 \beta_{14} - 573 \beta_{13} + 2727 \beta_{12} - 3700 \beta_{11} + 2887 \beta_{10} - 180 \beta_{9} + 2188 \beta_{8} + 846 \beta_{7} + 163 \beta_{6} - 684 \beta_{5} + 863 \beta_{4} + 4697 \beta_{3} + 145 \beta_{2} + 8042 \beta_{1} + 587\)
\(\nu^{14}\)\(=\)\(7178 \beta_{14} - 864 \beta_{13} + 6135 \beta_{12} - 12481 \beta_{11} + 9542 \beta_{10} - 991 \beta_{9} + 3700 \beta_{8} + 1351 \beta_{7} + 846 \beta_{6} - 2617 \beta_{5} + 3590 \beta_{4} + 5425 \beta_{3} + 8019 \beta_{2} + 7194 \beta_{1} + 10003\)

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
2.34497
2.14166
1.96562
1.92974
1.09783
1.07739
0.596683
−0.0849802
−0.227272
−0.324166
−1.00750
−1.39546
−1.93741
−2.06795
−2.10915
−2.34497 3.01587 3.49887 0 −7.07210 −1.32864 −3.51479 6.09545 0
1.2 −2.14166 0.668121 2.58670 0 −1.43089 3.26618 −1.25651 −2.55361 0
1.3 −1.96562 0.551502 1.86368 0 −1.08405 −1.56703 0.267954 −2.69585 0
1.4 −1.92974 −0.860854 1.72391 0 1.66123 4.27878 0.532780 −2.25893 0
1.5 −1.09783 −1.80555 −0.794772 0 1.98218 0.928281 3.06818 0.259998 0
1.6 −1.07739 −0.702822 −0.839239 0 0.757210 −0.460803 3.05896 −2.50604 0
1.7 −0.596683 3.22200 −1.64397 0 −1.92252 −0.0914365 2.17430 7.38130 0
1.8 0.0849802 1.30288 −1.99278 0 0.110719 −0.925444 −0.339307 −1.30250 0
1.9 0.227272 −1.31252 −1.94835 0 −0.298298 −4.34237 −0.897348 −1.27730 0
1.10 0.324166 1.44456 −1.89492 0 0.468276 −1.26380 −1.26260 −0.913251 0
1.11 1.00750 2.74497 −0.984941 0 2.76556 1.78457 −3.00733 4.53484 0
1.12 1.39546 −2.59963 −0.0526942 0 −3.62768 1.49578 −2.86445 3.75808 0
1.13 1.93741 1.04448 1.75357 0 2.02359 2.21578 −0.477443 −1.90906 0
1.14 2.06795 −1.49139 2.27640 0 −3.08411 2.20594 0.571582 −0.775761 0
1.15 2.10915 1.77838 2.44853 0 3.75087 −3.19580 0.946028 0.162626 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6025.2.a.i 15
5.b even 2 1 1205.2.a.c 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.a.c 15 5.b even 2 1
6025.2.a.i 15 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(241\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

\(T_{2}^{15} + \cdots\)
\(T_{3}^{15} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 14 T^{2} + 25 T^{3} + 103 T^{4} + 168 T^{5} + 530 T^{6} + 801 T^{7} + 2121 T^{8} + 2987 T^{9} + 6933 T^{10} + 9114 T^{11} + 19011 T^{12} + 23293 T^{13} + 44373 T^{14} + 50485 T^{15} + 88746 T^{16} + 93172 T^{17} + 152088 T^{18} + 145824 T^{19} + 221856 T^{20} + 191168 T^{21} + 271488 T^{22} + 205056 T^{23} + 271360 T^{24} + 172032 T^{25} + 210944 T^{26} + 102400 T^{27} + 114688 T^{28} + 32768 T^{29} + 32768 T^{30} \)
$3$ \( 1 - 7 T + 44 T^{2} - 194 T^{3} + 773 T^{4} - 2617 T^{5} + 8164 T^{6} - 22967 T^{7} + 60449 T^{8} - 147310 T^{9} + 339467 T^{10} - 734507 T^{11} + 1513337 T^{12} - 2950637 T^{13} + 5499816 T^{14} - 9736601 T^{15} + 16499448 T^{16} - 26555733 T^{17} + 40860099 T^{18} - 59495067 T^{19} + 82490481 T^{20} - 107388990 T^{21} + 132201963 T^{22} - 150686487 T^{23} + 160692012 T^{24} - 154531233 T^{25} + 136934631 T^{26} - 103099554 T^{27} + 70150212 T^{28} - 33480783 T^{29} + 14348907 T^{30} \)
$5$ 1
$7$ \( 1 - 3 T + 69 T^{2} - 184 T^{3} + 2285 T^{4} - 5435 T^{5} + 48514 T^{6} - 103450 T^{7} + 746705 T^{8} - 1437121 T^{9} + 8946471 T^{10} - 15668832 T^{11} + 87418730 T^{12} - 140495069 T^{13} + 717670379 T^{14} - 1064822293 T^{15} + 5023692653 T^{16} - 6884258381 T^{17} + 29984624390 T^{18} - 37620865632 T^{19} + 150363338097 T^{20} - 169075848529 T^{21} + 614943675815 T^{22} - 596368663450 T^{23} + 1957714889998 T^{24} - 1535252978315 T^{25} + 4518191607755 T^{26} - 2546796844984 T^{27} + 6685341718083 T^{28} - 2034669218547 T^{29} + 4747561509943 T^{30} \)
$11$ \( 1 + 10 T + 140 T^{2} + 1028 T^{3} + 8603 T^{4} + 51007 T^{5} + 324685 T^{6} + 1638350 T^{7} + 8671398 T^{8} + 38435840 T^{9} + 176810244 T^{10} + 701777246 T^{11} + 2875783160 T^{12} + 10334097764 T^{13} + 38247516248 T^{14} + 125038989415 T^{15} + 420722678728 T^{16} + 1250425829444 T^{17} + 3827667385960 T^{18} + 10274720658686 T^{19} + 28475466606444 T^{20} + 68091435146240 T^{21} + 168981015635058 T^{22} + 351194872686350 T^{23} + 765590246052335 T^{24} + 1322990216623207 T^{25} + 2454536302266433 T^{26} + 3226304371269188 T^{27} + 4833179700150340 T^{28} + 3797498335832410 T^{29} + 4177248169415651 T^{30} \)
$13$ \( 1 - 8 T + 135 T^{2} - 856 T^{3} + 8138 T^{4} - 42812 T^{5} + 297820 T^{6} - 1347080 T^{7} + 7616486 T^{8} - 30614433 T^{9} + 149729256 T^{10} - 552303058 T^{11} + 2443009380 T^{12} - 8488918786 T^{13} + 35043186281 T^{14} - 116137458321 T^{15} + 455561421653 T^{16} - 1434627274834 T^{17} + 5367291607860 T^{18} - 15774327639538 T^{19} + 55593424648008 T^{20} - 147770020734297 T^{21} + 477923201251262 T^{22} - 1098854539644680 T^{23} + 3158232003266860 T^{24} - 5901997753039388 T^{25} + 14584601286673106 T^{26} - 19943160864843736 T^{27} + 40888139389954155 T^{28} - 31499011085594312 T^{29} + 51185893014090757 T^{30} \)
$17$ \( 1 - T + 126 T^{2} - 269 T^{3} + 8190 T^{4} - 24298 T^{5} + 373120 T^{6} - 1283146 T^{7} + 13229934 T^{8} - 48031043 T^{9} + 380644474 T^{10} - 1384341208 T^{11} + 9095231436 T^{12} - 31895449037 T^{13} + 182878337902 T^{14} - 598744414275 T^{15} + 3108931744334 T^{16} - 9217784771693 T^{17} + 44684872045068 T^{18} - 115621562033368 T^{19} + 540460720920218 T^{20} - 1159352614554467 T^{21} + 5428753561437582 T^{22} - 8950915257389386 T^{23} + 44247508478560640 T^{24} - 48984619793109802 T^{25} + 280686830759514270 T^{26} - 156725381814805709 T^{27} + 1247976832146148062 T^{28} - 168377826559400929 T^{29} + 2862423051509815793 T^{30} \)
$19$ \( 1 + 30 T + 617 T^{2} + 9225 T^{3} + 114598 T^{4} + 1205122 T^{5} + 11195374 T^{6} + 92718239 T^{7} + 698401371 T^{8} + 4807897553 T^{9} + 30589810478 T^{10} + 180290400740 T^{11} + 991023236386 T^{12} + 5083321480560 T^{13} + 24432500370967 T^{14} + 109938656945393 T^{15} + 464217507048373 T^{16} + 1835079054482160 T^{17} + 6797428378371574 T^{18} + 23495625314837540 T^{19} + 75743399134765322 T^{20} + 226191776138629193 T^{21} + 624281248015754169 T^{22} + 1574686057107004799 T^{23} + 3612609461834874346 T^{24} + 7388682830733656722 T^{25} + 13349550689218100962 T^{26} + 20417830128385335225 T^{27} + 25946690796212605403 T^{28} + 23970200573486523630 T^{29} + 15181127029874798299 T^{30} \)
$23$ \( 1 + 19 T + 345 T^{2} + 4344 T^{3} + 48872 T^{4} + 467208 T^{5} + 4038140 T^{6} + 31447888 T^{7} + 225466890 T^{8} + 1495229706 T^{9} + 9276905985 T^{10} + 54168988017 T^{11} + 300138423930 T^{12} + 1586908275369 T^{13} + 8051845774445 T^{14} + 39328563854155 T^{15} + 185192452812235 T^{16} + 839474477670201 T^{17} + 3651784203956310 T^{18} + 15158703775665297 T^{19} + 59709348898212855 T^{20} + 221347658786918634 T^{21} + 767675404527949830 T^{22} + 2462715094286536528 T^{23} + 7273306608360198820 T^{24} + 19354797451106521992 T^{25} + 46565718488769440344 T^{26} + 95197128532696274424 T^{27} + \)\(17\!\cdots\!35\)\( T^{28} + \)\(22\!\cdots\!71\)\( T^{29} + \)\(26\!\cdots\!07\)\( T^{30} \)
$29$ \( 1 + 12 T + 339 T^{2} + 3188 T^{3} + 50639 T^{4} + 382964 T^{5} + 4444766 T^{6} + 27152996 T^{7} + 256269972 T^{8} + 1242735270 T^{9} + 10314402824 T^{10} + 38009283932 T^{11} + 309955761183 T^{12} + 831254958912 T^{13} + 8114646936272 T^{14} + 18821183700381 T^{15} + 235324761151888 T^{16} + 699085420444992 T^{17} + 7559511059492187 T^{18} + 26883244348708892 T^{19} + 211560253169084776 T^{20} + 739207920425231670 T^{21} + 4420625318710893348 T^{22} + 13583188850144381156 T^{23} + 64480869190579351654 T^{24} + \)\(16\!\cdots\!64\)\( T^{25} + \)\(61\!\cdots\!31\)\( T^{26} + \)\(11\!\cdots\!08\)\( T^{27} + \)\(34\!\cdots\!71\)\( T^{28} + \)\(35\!\cdots\!72\)\( T^{29} + \)\(86\!\cdots\!49\)\( T^{30} \)
$31$ \( 1 + 22 T + 440 T^{2} + 5691 T^{3} + 69881 T^{4} + 688745 T^{5} + 6648869 T^{6} + 55558941 T^{7} + 460139509 T^{8} + 3403260763 T^{9} + 25026167067 T^{10} + 166971287488 T^{11} + 1109762228609 T^{12} + 6751797338530 T^{13} + 40964110917559 T^{14} + 228369831804309 T^{15} + 1269887438444329 T^{16} + 6488477242327330 T^{17} + 33060926552490719 T^{18} + 154201490392205248 T^{19} + 716477915912370117 T^{20} + 3020406454565368603 T^{21} + 12659640748342011499 T^{22} + 47385722828613309981 T^{23} + \)\(17\!\cdots\!99\)\( T^{24} + \)\(56\!\cdots\!45\)\( T^{25} + \)\(17\!\cdots\!11\)\( T^{26} + \)\(44\!\cdots\!51\)\( T^{27} + \)\(10\!\cdots\!40\)\( T^{28} + \)\(16\!\cdots\!62\)\( T^{29} + \)\(23\!\cdots\!51\)\( T^{30} \)
$37$ \( 1 - 12 T + 314 T^{2} - 3412 T^{3} + 50098 T^{4} - 494865 T^{5} + 5446581 T^{6} - 48638912 T^{7} + 448085578 T^{8} - 3616250406 T^{9} + 29279798873 T^{10} - 214642130278 T^{11} + 1561641890826 T^{12} - 10455854594328 T^{13} + 69101820742268 T^{14} - 423482062973499 T^{15} + 2556767367463916 T^{16} - 14314064939635032 T^{17} + 79101846696009378 T^{18} - 402273909523946758 T^{19} + 2030377114017960461 T^{20} - 9278309168231172054 T^{21} + 42537605035765287874 T^{22} - \)\(17\!\cdots\!52\)\( T^{23} + \)\(70\!\cdots\!37\)\( T^{24} - \)\(23\!\cdots\!85\)\( T^{25} + \)\(89\!\cdots\!74\)\( T^{26} - \)\(22\!\cdots\!72\)\( T^{27} + \)\(76\!\cdots\!58\)\( T^{28} - \)\(10\!\cdots\!68\)\( T^{29} + \)\(33\!\cdots\!93\)\( T^{30} \)
$41$ \( 1 + 13 T + 341 T^{2} + 3629 T^{3} + 52543 T^{4} + 466994 T^{5} + 4839600 T^{6} + 36354346 T^{7} + 298170361 T^{8} + 1902608617 T^{9} + 13180820992 T^{10} + 71589476876 T^{11} + 455476595943 T^{12} + 2172102398188 T^{13} + 14836124371336 T^{14} + 73017536046731 T^{15} + 608281099224776 T^{16} + 3651304131354028 T^{17} + 31391902468987503 T^{18} + 202294751766602636 T^{19} + 1527079846194171392 T^{20} + 9037589260574844697 T^{21} + 58069952149390641041 T^{22} + \)\(29\!\cdots\!66\)\( T^{23} + \)\(15\!\cdots\!00\)\( T^{24} + \)\(62\!\cdots\!94\)\( T^{25} + \)\(28\!\cdots\!63\)\( T^{26} + \)\(81\!\cdots\!49\)\( T^{27} + \)\(31\!\cdots\!61\)\( T^{28} + \)\(49\!\cdots\!93\)\( T^{29} + \)\(15\!\cdots\!01\)\( T^{30} \)
$43$ \( 1 - 25 T + 664 T^{2} - 11070 T^{3} + 178891 T^{4} - 2300836 T^{5} + 28352101 T^{6} - 301544277 T^{7} + 3080085239 T^{8} - 28211205837 T^{9} + 249502562636 T^{10} - 2021210853870 T^{11} + 15897696311733 T^{12} - 115971363872945 T^{13} + 824791787040357 T^{14} - 5473357413730351 T^{15} + 35466046842735351 T^{16} - 214431051801075305 T^{17} + 1263978140656955631 T^{18} - 6910117688421609870 T^{19} + 36678983257628335748 T^{20} - \)\(17\!\cdots\!13\)\( T^{21} + \)\(83\!\cdots\!73\)\( T^{22} - \)\(35\!\cdots\!77\)\( T^{23} + \)\(14\!\cdots\!43\)\( T^{24} - \)\(49\!\cdots\!64\)\( T^{25} + \)\(16\!\cdots\!37\)\( T^{26} - \)\(44\!\cdots\!70\)\( T^{27} + \)\(11\!\cdots\!52\)\( T^{28} - \)\(18\!\cdots\!25\)\( T^{29} + \)\(31\!\cdots\!07\)\( T^{30} \)
$47$ \( 1 + 16 T + 492 T^{2} + 5699 T^{3} + 99873 T^{4} + 871873 T^{5} + 11279555 T^{6} + 72810887 T^{7} + 783290515 T^{8} + 3220474720 T^{9} + 33378181673 T^{10} + 17288882580 T^{11} + 707170695241 T^{12} - 7584884037603 T^{13} - 4437797603730 T^{14} - 537960786887469 T^{15} - 208576487375310 T^{16} - 16755008839065027 T^{17} + 73420583092006343 T^{18} + 84364231836856980 T^{19} + 7655119309441456711 T^{20} + 34714190468480982880 T^{21} + \)\(39\!\cdots\!45\)\( T^{22} + \)\(17\!\cdots\!07\)\( T^{23} + \)\(12\!\cdots\!85\)\( T^{24} + \)\(45\!\cdots\!77\)\( T^{25} + \)\(24\!\cdots\!19\)\( T^{26} + \)\(66\!\cdots\!59\)\( T^{27} + \)\(26\!\cdots\!84\)\( T^{28} + \)\(41\!\cdots\!04\)\( T^{29} + \)\(12\!\cdots\!43\)\( T^{30} \)
$53$ \( 1 - 4 T + 464 T^{2} - 1139 T^{3} + 104229 T^{4} - 117368 T^{5} + 15276402 T^{6} + 996280 T^{7} + 1653955482 T^{8} + 1832413844 T^{9} + 141924674600 T^{10} + 273324058399 T^{11} + 10108497538498 T^{12} + 24328445769833 T^{13} + 616167868168061 T^{14} + 1516066725600641 T^{15} + 32656897012907233 T^{16} + 68338604167460897 T^{17} + 1504922788038966746 T^{18} + 2156658289640199919 T^{19} + 59352259263211577800 T^{20} + 40614282176195069876 T^{21} + \)\(19\!\cdots\!34\)\( T^{22} + 62028084363030737080 T^{23} + \)\(50\!\cdots\!66\)\( T^{24} - \)\(20\!\cdots\!32\)\( T^{25} + \)\(96\!\cdots\!13\)\( T^{26} - \)\(55\!\cdots\!99\)\( T^{27} + \)\(12\!\cdots\!72\)\( T^{28} - \)\(55\!\cdots\!76\)\( T^{29} + \)\(73\!\cdots\!57\)\( T^{30} \)
$59$ \( 1 + 50 T + 1656 T^{2} + 41091 T^{3} + 846567 T^{4} + 14990210 T^{5} + 235271476 T^{6} + 3323944602 T^{7} + 42862165094 T^{8} + 508599205506 T^{9} + 5594191512530 T^{10} + 57296999928431 T^{11} + 548677398738234 T^{12} + 4924051700103359 T^{13} + 41498895626633429 T^{14} + 328708825551825739 T^{15} + 2448434841971372311 T^{16} + 17140623968059792679 T^{17} + \)\(11\!\cdots\!86\)\( T^{18} + \)\(69\!\cdots\!91\)\( T^{19} + \)\(39\!\cdots\!70\)\( T^{20} + \)\(21\!\cdots\!46\)\( T^{21} + \)\(10\!\cdots\!86\)\( T^{22} + \)\(48\!\cdots\!42\)\( T^{23} + \)\(20\!\cdots\!64\)\( T^{24} + \)\(76\!\cdots\!10\)\( T^{25} + \)\(25\!\cdots\!53\)\( T^{26} + \)\(73\!\cdots\!71\)\( T^{27} + \)\(17\!\cdots\!24\)\( T^{28} + \)\(30\!\cdots\!50\)\( T^{29} + \)\(36\!\cdots\!99\)\( T^{30} \)
$61$ \( 1 + 41 T + 1266 T^{2} + 28143 T^{3} + 537421 T^{4} + 8723591 T^{5} + 127846442 T^{6} + 1684406586 T^{7} + 20552628279 T^{8} + 231552538619 T^{9} + 2451586546111 T^{10} + 24315296310024 T^{11} + 228555938207149 T^{12} + 2028097124596243 T^{13} + 17129570660768330 T^{14} + 137038654279098775 T^{15} + 1044903810306868130 T^{16} + 7546549400622620203 T^{17} + 51877855410196887169 T^{18} + \)\(33\!\cdots\!84\)\( T^{19} + \)\(20\!\cdots\!11\)\( T^{20} + \)\(11\!\cdots\!59\)\( T^{21} + \)\(64\!\cdots\!59\)\( T^{22} + \)\(32\!\cdots\!66\)\( T^{23} + \)\(14\!\cdots\!22\)\( T^{24} + \)\(62\!\cdots\!91\)\( T^{25} + \)\(23\!\cdots\!81\)\( T^{26} + \)\(74\!\cdots\!03\)\( T^{27} + \)\(20\!\cdots\!46\)\( T^{28} + \)\(40\!\cdots\!81\)\( T^{29} + \)\(60\!\cdots\!01\)\( T^{30} \)
$67$ \( 1 - 43 T + 1412 T^{2} - 34612 T^{3} + 725254 T^{4} - 13148340 T^{5} + 213596249 T^{6} - 3140878835 T^{7} + 42386968766 T^{8} - 528391319525 T^{9} + 6126124618550 T^{10} - 66308482555692 T^{11} + 672588399620473 T^{12} - 6405417740114407 T^{13} + 57392366612773961 T^{14} - 484028819246150401 T^{15} + 3845288563055855387 T^{16} - 28753920235373573023 T^{17} + \)\(20\!\cdots\!99\)\( T^{18} - \)\(13\!\cdots\!32\)\( T^{19} + \)\(82\!\cdots\!50\)\( T^{20} - \)\(47\!\cdots\!25\)\( T^{21} + \)\(25\!\cdots\!18\)\( T^{22} - \)\(12\!\cdots\!35\)\( T^{23} + \)\(58\!\cdots\!03\)\( T^{24} - \)\(23\!\cdots\!60\)\( T^{25} + \)\(88\!\cdots\!82\)\( T^{26} - \)\(28\!\cdots\!32\)\( T^{27} + \)\(77\!\cdots\!44\)\( T^{28} - \)\(15\!\cdots\!47\)\( T^{29} + \)\(24\!\cdots\!43\)\( T^{30} \)
$71$ \( 1 + 14 T + 714 T^{2} + 9134 T^{3} + 256359 T^{4} + 2978414 T^{5} + 60611852 T^{6} + 640396335 T^{7} + 10494156375 T^{8} + 101038970433 T^{9} + 1405479787448 T^{10} + 12341749470266 T^{11} + 150201975448272 T^{12} + 1201578068635484 T^{13} + 13032406954284030 T^{14} + 94635638066606051 T^{15} + 925300893754166130 T^{16} + 6057155043991474844 T^{17} + 53758939234666479792 T^{18} + \)\(31\!\cdots\!46\)\( T^{19} + \)\(25\!\cdots\!48\)\( T^{20} + \)\(12\!\cdots\!93\)\( T^{21} + \)\(95\!\cdots\!25\)\( T^{22} + \)\(41\!\cdots\!35\)\( T^{23} + \)\(27\!\cdots\!12\)\( T^{24} + \)\(96\!\cdots\!14\)\( T^{25} + \)\(59\!\cdots\!89\)\( T^{26} + \)\(14\!\cdots\!94\)\( T^{27} + \)\(83\!\cdots\!54\)\( T^{28} + \)\(11\!\cdots\!34\)\( T^{29} + \)\(58\!\cdots\!51\)\( T^{30} \)
$73$ \( 1 - 10 T + 494 T^{2} - 3751 T^{3} + 122454 T^{4} - 787530 T^{5} + 21586985 T^{6} - 125494338 T^{7} + 3021032393 T^{8} - 16172684699 T^{9} + 348789318111 T^{10} - 1732253844491 T^{11} + 34136454539357 T^{12} - 158366641271012 T^{13} + 2879386341403827 T^{14} - 12464863590107445 T^{15} + 210195202922479371 T^{16} - 843935831333222948 T^{17} + 13279661135537042069 T^{18} - 49192962149031940331 T^{19} + \)\(72\!\cdots\!23\)\( T^{20} - \)\(24\!\cdots\!11\)\( T^{21} + \)\(33\!\cdots\!21\)\( T^{22} - \)\(10\!\cdots\!78\)\( T^{23} + \)\(12\!\cdots\!05\)\( T^{24} - \)\(33\!\cdots\!70\)\( T^{25} + \)\(38\!\cdots\!58\)\( T^{26} - \)\(85\!\cdots\!71\)\( T^{27} + \)\(82\!\cdots\!02\)\( T^{28} - \)\(12\!\cdots\!90\)\( T^{29} + \)\(89\!\cdots\!57\)\( T^{30} \)
$79$ \( 1 + 44 T + 1760 T^{2} + 47139 T^{3} + 1146715 T^{4} + 22849120 T^{5} + 419964256 T^{6} + 6763660759 T^{7} + 101700966064 T^{8} + 1382466434370 T^{9} + 17690936696558 T^{10} + 208066457594283 T^{11} + 2315894010741594 T^{12} + 23901530554084313 T^{13} + 234140016077896982 T^{14} + 2135073976875200373 T^{15} + 18497061270153861578 T^{16} + \)\(14\!\cdots\!33\)\( T^{17} + \)\(11\!\cdots\!66\)\( T^{18} + \)\(81\!\cdots\!23\)\( T^{19} + \)\(54\!\cdots\!42\)\( T^{20} + \)\(33\!\cdots\!70\)\( T^{21} + \)\(19\!\cdots\!76\)\( T^{22} + \)\(10\!\cdots\!99\)\( T^{23} + \)\(50\!\cdots\!64\)\( T^{24} + \)\(21\!\cdots\!20\)\( T^{25} + \)\(85\!\cdots\!85\)\( T^{26} + \)\(27\!\cdots\!99\)\( T^{27} + \)\(82\!\cdots\!40\)\( T^{28} + \)\(16\!\cdots\!64\)\( T^{29} + \)\(29\!\cdots\!99\)\( T^{30} \)
$83$ \( 1 + 7 T + 555 T^{2} + 4899 T^{3} + 155129 T^{4} + 1499352 T^{5} + 29597218 T^{6} + 276408647 T^{7} + 4274594676 T^{8} + 35670610542 T^{9} + 487071844767 T^{10} + 3584592585017 T^{11} + 45880931425585 T^{12} + 310684877413560 T^{13} + 3884329781286276 T^{14} + 25729978545945169 T^{15} + 322399371846760908 T^{16} + 2140308120502014840 T^{17} + 26234120137040970395 T^{18} + \)\(17\!\cdots\!57\)\( T^{19} + \)\(19\!\cdots\!81\)\( T^{20} + \)\(11\!\cdots\!98\)\( T^{21} + \)\(11\!\cdots\!52\)\( T^{22} + \)\(62\!\cdots\!27\)\( T^{23} + \)\(55\!\cdots\!54\)\( T^{24} + \)\(23\!\cdots\!48\)\( T^{25} + \)\(19\!\cdots\!43\)\( T^{26} + \)\(52\!\cdots\!39\)\( T^{27} + \)\(49\!\cdots\!65\)\( T^{28} + \)\(51\!\cdots\!03\)\( T^{29} + \)\(61\!\cdots\!07\)\( T^{30} \)
$89$ \( 1 - 4 T + 765 T^{2} - 3907 T^{3} + 286678 T^{4} - 1784983 T^{5} + 70653364 T^{6} - 512993615 T^{7} + 12961873059 T^{8} - 104848697801 T^{9} + 1895706212342 T^{10} - 16279343143646 T^{11} + 230359494603497 T^{12} - 1995366970479321 T^{13} + 23824090802589747 T^{14} - 197140730599035623 T^{15} + 2120344081430487483 T^{16} - 15805301773166701641 T^{17} + \)\(16\!\cdots\!93\)\( T^{18} - \)\(10\!\cdots\!86\)\( T^{19} + \)\(10\!\cdots\!58\)\( T^{20} - \)\(52\!\cdots\!61\)\( T^{21} + \)\(57\!\cdots\!11\)\( T^{22} - \)\(20\!\cdots\!15\)\( T^{23} + \)\(24\!\cdots\!76\)\( T^{24} - \)\(55\!\cdots\!83\)\( T^{25} + \)\(79\!\cdots\!42\)\( T^{26} - \)\(96\!\cdots\!47\)\( T^{27} + \)\(16\!\cdots\!85\)\( T^{28} - \)\(78\!\cdots\!64\)\( T^{29} + \)\(17\!\cdots\!49\)\( T^{30} \)
$97$ \( 1 + 9 T + 911 T^{2} + 8593 T^{3} + 418332 T^{4} + 4024990 T^{5} + 127456591 T^{6} + 1224782348 T^{7} + 28721075133 T^{8} + 270358448786 T^{9} + 5062473091192 T^{10} + 45799947473449 T^{11} + 720603157804507 T^{12} + 6142121503270985 T^{13} + 84329245944514849 T^{14} + 662618094351958005 T^{15} + 8179936856617940353 T^{16} + 57791221224276697865 T^{17} + \)\(65\!\cdots\!11\)\( T^{18} + \)\(40\!\cdots\!69\)\( T^{19} + \)\(43\!\cdots\!44\)\( T^{20} + \)\(22\!\cdots\!94\)\( T^{21} + \)\(23\!\cdots\!29\)\( T^{22} + \)\(95\!\cdots\!28\)\( T^{23} + \)\(96\!\cdots\!47\)\( T^{24} + \)\(29\!\cdots\!10\)\( T^{25} + \)\(29\!\cdots\!96\)\( T^{26} + \)\(59\!\cdots\!13\)\( T^{27} + \)\(61\!\cdots\!47\)\( T^{28} + \)\(58\!\cdots\!21\)\( T^{29} + \)\(63\!\cdots\!93\)\( T^{30} \)
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