Properties

Label 4025.2.a.ba
Level 4025
Weight 2
Character orbit 4025.a
Self dual yes
Analytic conductor 32.140
Analytic rank 1
Dimension 14
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - x^{13} - 22 x^{12} + 18 x^{11} + 187 x^{10} - 118 x^{9} - 772 x^{8} + 346 x^{7} + 1581 x^{6} - 443 x^{5} - 1429 x^{4} + 193 x^{3} + 386 x^{2} - 3 x - 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\(-134 \nu^{13} + 37911 \nu^{12} + 9431 \nu^{11} - 833971 \nu^{10} - 178425 \nu^{9} + 6807185 \nu^{8} + 1395439 \nu^{7} - 25174921 \nu^{6} - 5051683 \nu^{5} + 40706850 \nu^{4} + 8180960 \nu^{3} - 22002987 \nu^{2} - 4823059 \nu + 1719962\)\()/1242873\)
\(\beta_{5}\)\(=\)\((\)\(-8617 \nu^{13} - 22078 \nu^{12} + 273594 \nu^{11} + 349093 \nu^{10} - 3046784 \nu^{9} - 1722211 \nu^{8} + 15312112 \nu^{7} + 2054981 \nu^{6} - 35751429 \nu^{5} + 4401504 \nu^{4} + 34327336 \nu^{3} - 7553195 \nu^{2} - 8567221 \nu + 208671\)\()/414291\)
\(\beta_{6}\)\(=\)\((\)\(3427 \nu^{13} - 14217 \nu^{12} - 50538 \nu^{11} + 246032 \nu^{10} + 208975 \nu^{9} - 1485425 \nu^{8} + 41160 \nu^{7} + 3684384 \nu^{6} - 1876665 \nu^{5} - 3337746 \nu^{4} + 3436136 \nu^{3} + 746869 \nu^{2} - 1764856 \nu - 136436\)\()/138097\)
\(\beta_{7}\)\(=\)\((\)\(37777 \nu^{13} + 6483 \nu^{12} - 831559 \nu^{11} - 153367 \nu^{10} + 6791373 \nu^{9} + 1291991 \nu^{8} - 25128557 \nu^{7} - 4839829 \nu^{6} + 40647488 \nu^{5} + 7989474 \nu^{4} - 21977125 \nu^{3} - 4771335 \nu^{2} + 476687 \nu - 670\)\()/1242873\)
\(\beta_{8}\)\(=\)\((\)\(-17266 \nu^{13} + 28793 \nu^{12} + 353631 \nu^{11} - 477956 \nu^{10} - 2836277 \nu^{9} + 2649746 \nu^{8} + 11524948 \nu^{7} - 5168881 \nu^{6} - 25230657 \nu^{5} + 449829 \nu^{4} + 27355117 \nu^{3} + 5168560 \nu^{2} - 9402664 \nu - 1259865\)\()/414291\)
\(\beta_{9}\)\(=\)\((\)\(74561 \nu^{13} - 92562 \nu^{12} - 1571588 \nu^{11} + 1647493 \nu^{10} + 12628452 \nu^{9} - 10583948 \nu^{8} - 48372151 \nu^{7} + 29999800 \nu^{6} + 89700733 \nu^{5} - 36175764 \nu^{4} - 72909281 \nu^{3} + 12237873 \nu^{2} + 22678219 \nu + 3221839\)\()/1242873\)
\(\beta_{10}\)\(=\)\((\)\(92428 \nu^{13} - 123405 \nu^{12} - 1867552 \nu^{11} + 2127977 \nu^{10} + 14235777 \nu^{9} - 12965704 \nu^{8} - 51178406 \nu^{7} + 33005087 \nu^{6} + 87794291 \nu^{5} - 29107302 \nu^{4} - 61333048 \nu^{3} - 3662787 \nu^{2} + 6952400 \nu + 4568996\)\()/1242873\)
\(\beta_{11}\)\(=\)\((\)\(-114541 \nu^{13} + 106461 \nu^{12} + 2641672 \nu^{11} - 2117591 \nu^{10} - 23419830 \nu^{9} + 15754684 \nu^{8} + 99159767 \nu^{7} - 53798933 \nu^{6} - 199755620 \nu^{5} + 81640410 \nu^{4} + 158892526 \nu^{3} - 42632694 \nu^{2} - 23191094 \nu + 1997443\)\()/1242873\)
\(\beta_{12}\)\(=\)\((\)\(-64550 \nu^{13} + 87137 \nu^{12} + 1331797 \nu^{11} - 1563334 \nu^{10} - 10454477 \nu^{9} + 10248993 \nu^{8} + 39148617 \nu^{7} - 30423831 \nu^{6} - 70995548 \nu^{5} + 41000631 \nu^{4} + 53938016 \nu^{3} - 21439823 \nu^{2} - 9671628 \nu + 1835824\)\()/414291\)
\(\beta_{13}\)\(=\)\((\)\(-203639 \nu^{13} + 277101 \nu^{12} + 4169738 \nu^{11} - 4742359 \nu^{10} - 32522883 \nu^{9} + 28618406 \nu^{8} + 121398712 \nu^{7} - 72669595 \nu^{6} - 221496586 \nu^{5} + 69984327 \nu^{4} + 175750652 \nu^{3} - 12281418 \nu^{2} - 40794727 \nu - 2790322\)\()/1242873\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{13} + \beta_{10} + 2 \beta_{9} - \beta_{7} + \beta_{4} + \beta_{3} + 7 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{10} + 2 \beta_{9} - \beta_{6} + \beta_{5} + 10 \beta_{3} + \beta_{2} + 27 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(12 \beta_{13} - \beta_{12} + 9 \beta_{10} + 23 \beta_{9} - \beta_{8} - 10 \beta_{7} + \beta_{6} + 12 \beta_{4} + 12 \beta_{3} + 48 \beta_{2} + \beta_{1} + 76\)
\(\nu^{7}\)\(=\)\(3 \beta_{13} - \beta_{12} - 2 \beta_{11} - 12 \beta_{10} + 29 \beta_{9} - 2 \beta_{7} - 13 \beta_{6} + 15 \beta_{5} + 5 \beta_{4} + 83 \beta_{3} + 13 \beta_{2} + 155 \beta_{1} + 78\)
\(\nu^{8}\)\(=\)\(111 \beta_{13} - 14 \beta_{12} - 2 \beta_{11} + 67 \beta_{10} + 205 \beta_{9} - 17 \beta_{8} - 78 \beta_{7} + 12 \beta_{6} + 4 \beta_{5} + 109 \beta_{4} + 110 \beta_{3} + 333 \beta_{2} + 17 \beta_{1} + 453\)
\(\nu^{9}\)\(=\)\(57 \beta_{13} - 21 \beta_{12} - 33 \beta_{11} - 107 \beta_{10} + 307 \beta_{9} - 6 \beta_{8} - 37 \beta_{7} - 129 \beta_{6} + 161 \beta_{5} + 88 \beta_{4} + 654 \beta_{3} + 130 \beta_{2} + 932 \beta_{1} + 567\)
\(\nu^{10}\)\(=\)\(945 \beta_{13} - 149 \beta_{12} - 38 \beta_{11} + 476 \beta_{10} + 1692 \beta_{9} - 194 \beta_{8} - 571 \beta_{7} + 93 \beta_{6} + 83 \beta_{5} + 906 \beta_{4} + 931 \beta_{3} + 2344 \beta_{2} + 186 \beta_{1} + 2876\)
\(\nu^{11}\)\(=\)\(720 \beta_{13} - 281 \beta_{12} - 370 \beta_{11} - 854 \beta_{10} + 2872 \beta_{9} - 121 \beta_{8} - 444 \beta_{7} - 1155 \beta_{6} + 1513 \beta_{5} + 1043 \beta_{4} + 5055 \beta_{3} + 1186 \beta_{2} + 5815 \beta_{1} + 4049\)
\(\nu^{12}\)\(=\)\(7758 \beta_{13} - 1431 \beta_{12} - 479 \beta_{11} + 3338 \beta_{10} + 13556 \beta_{9} - 1883 \beta_{8} - 4128 \beta_{7} + 546 \beta_{6} + 1117 \beta_{5} + 7294 \beta_{4} + 7643 \beta_{3} + 16718 \beta_{2} + 1716 \beta_{1} + 19068\)
\(\nu^{13}\)\(=\)\(7632 \beta_{13} - 3084 \beta_{12} - 3546 \beta_{11} - 6459 \beta_{10} + 25222 \beta_{9} - 1596 \beta_{8} - 4431 \beta_{7} - 9803 \beta_{6} + 13271 \beta_{5} + 10489 \beta_{4} + 38763 \beta_{3} + 10361 \beta_{2} + 37406 \beta_{1} + 28967\)

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
2.79074
2.52898
2.17410
1.82692
1.31493
0.704319
0.117364
−0.116065
−0.582976
−1.25067
−1.76659
−1.83943
−2.25329
−2.64833
−2.79074 −2.29281 5.78823 0 6.39865 −1.00000 −10.5720 2.25700 0
1.2 −2.52898 1.80521 4.39572 0 −4.56532 −1.00000 −6.05871 0.258773 0
1.3 −2.17410 −1.37249 2.72672 0 2.98394 −1.00000 −1.57997 −1.11626 0
1.4 −1.82692 2.52976 1.33763 0 −4.62167 −1.00000 1.21009 3.39970 0
1.5 −1.31493 −3.36299 −0.270967 0 4.42209 −1.00000 2.98616 8.30972 0
1.6 −0.704319 2.54639 −1.50393 0 −1.79347 −1.00000 2.46789 3.48411 0
1.7 −0.117364 −0.701283 −1.98623 0 0.0823052 −1.00000 0.467839 −2.50820 0
1.8 0.116065 −1.59146 −1.98653 0 −0.184713 −1.00000 −0.462695 −0.467244 0
1.9 0.582976 0.367598 −1.66014 0 0.214301 −1.00000 −2.13377 −2.86487 0
1.10 1.25067 −3.28603 −0.435826 0 −4.10974 −1.00000 −3.04641 7.79801 0
1.11 1.76659 0.991358 1.12083 0 1.75132 −1.00000 −1.55314 −2.01721 0
1.12 1.83943 1.64563 1.38349 0 3.02702 −1.00000 −1.13403 −0.291898 0
1.13 2.25329 −2.73097 3.07734 0 −6.15367 −1.00000 2.42755 4.45818 0
1.14 2.64833 −0.547903 5.01366 0 −1.45103 −1.00000 7.98118 −2.69980 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 4025.2.a.ba 14
5.b even 2 1 4025.2.a.bb yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.ba 14 1.a even 1 1 trivial
4025.2.a.bb yes 14 5.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\(T_{2}^{14} + \cdots\)
\(T_{3}^{14} + \cdots\)
\(T_{11}^{14} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 6 T^{2} + 8 T^{3} + 23 T^{4} + 34 T^{5} + 72 T^{6} + 106 T^{7} + 185 T^{8} + 271 T^{9} + 423 T^{10} + 613 T^{11} + 902 T^{12} + 1285 T^{13} + 1835 T^{14} + 2570 T^{15} + 3608 T^{16} + 4904 T^{17} + 6768 T^{18} + 8672 T^{19} + 11840 T^{20} + 13568 T^{21} + 18432 T^{22} + 17408 T^{23} + 23552 T^{24} + 16384 T^{25} + 24576 T^{26} + 8192 T^{27} + 16384 T^{28} \)
$3$ \( 1 + 6 T + 30 T^{2} + 113 T^{3} + 379 T^{4} + 1128 T^{5} + 3092 T^{6} + 7852 T^{7} + 18729 T^{8} + 42123 T^{9} + 89988 T^{10} + 182742 T^{11} + 354537 T^{12} + 656181 T^{13} + 1163133 T^{14} + 1968543 T^{15} + 3190833 T^{16} + 4934034 T^{17} + 7289028 T^{18} + 10235889 T^{19} + 13653441 T^{20} + 17172324 T^{21} + 20286612 T^{22} + 22202424 T^{23} + 22379571 T^{24} + 20017611 T^{25} + 15943230 T^{26} + 9565938 T^{27} + 4782969 T^{28} \)
$5$ 1
$7$ \( ( 1 + T )^{14} \)
$11$ \( 1 + 3 T + 56 T^{2} + 164 T^{3} + 1664 T^{4} + 5195 T^{5} + 35322 T^{6} + 113563 T^{7} + 588855 T^{8} + 1869624 T^{9} + 8339162 T^{10} + 25257627 T^{11} + 105090824 T^{12} + 298539964 T^{13} + 1203660840 T^{14} + 3283939604 T^{15} + 12715989704 T^{16} + 33617901537 T^{17} + 122093670842 T^{18} + 301104814824 T^{19} + 1043192552655 T^{20} + 2213021600273 T^{21} + 7571584394682 T^{22} + 12249538254745 T^{23} + 43159874536064 T^{24} + 46791113980204 T^{25} + 175751989096376 T^{26} + 103568136431793 T^{27} + 379749833583241 T^{28} \)
$13$ \( 1 + 15 T + 178 T^{2} + 1508 T^{3} + 11300 T^{4} + 71799 T^{5} + 420911 T^{6} + 2223123 T^{7} + 11090475 T^{8} + 51372270 T^{9} + 227840558 T^{10} + 952960696 T^{11} + 3841197605 T^{12} + 14702213917 T^{13} + 54350435451 T^{14} + 191128780921 T^{15} + 649162395245 T^{16} + 2093654649112 T^{17} + 6507354177038 T^{18} + 19074164245110 T^{19} + 53531604544275 T^{20} + 139497671358591 T^{21} + 343350033506831 T^{22} + 761392450482027 T^{23} + 1557800957893700 T^{24} + 2702577874207796 T^{25} + 4147059151801618 T^{26} + 4543126598883795 T^{27} + 3937376385699289 T^{28} \)
$17$ \( 1 + 9 T + 149 T^{2} + 763 T^{3} + 7421 T^{4} + 17067 T^{5} + 157573 T^{6} - 283194 T^{7} + 1387336 T^{8} - 20072042 T^{9} + 28252889 T^{10} - 362444022 T^{11} + 1671275370 T^{12} - 2938264453 T^{13} + 42508089594 T^{14} - 49950495701 T^{15} + 482998581930 T^{16} - 1780687480086 T^{17} + 2359709542169 T^{18} - 28499429337994 T^{19} + 33486918426184 T^{20} - 116205450161562 T^{21} + 1099191027250693 T^{22} + 2023939288174299 T^{23} + 14960690735232029 T^{24} + 26149456882723979 T^{25} + 86810713347234389 T^{26} + 89141202296153433 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 + 4 T + 81 T^{2} + 310 T^{3} + 3782 T^{4} + 16961 T^{5} + 130657 T^{6} + 622874 T^{7} + 3702559 T^{8} + 17984199 T^{9} + 95378049 T^{10} + 417374617 T^{11} + 2176491978 T^{12} + 8565924945 T^{13} + 44678910394 T^{14} + 162752573955 T^{15} + 785713604058 T^{16} + 2862772498003 T^{17} + 12429762723729 T^{18} + 44530657159701 T^{19} + 174190150109479 T^{20} + 556769465557886 T^{21} + 2219021396247937 T^{22} + 5473106042029619 T^{23} + 23187692587003382 T^{24} + 36111980258447890 T^{25} + 179278508444359041 T^{26} + 168211933849028236 T^{27} + 799006685782884121 T^{28} \)
$23$ \( ( 1 - T )^{14} \)
$29$ \( 1 - 11 T + 297 T^{2} - 2577 T^{3} + 39915 T^{4} - 292931 T^{5} + 3378729 T^{6} - 21913721 T^{7} + 207489014 T^{8} - 1219318621 T^{9} + 9932410601 T^{10} - 53425297096 T^{11} + 384489909416 T^{12} - 1893616655986 T^{13} + 12254160692157 T^{14} - 54914883023594 T^{15} + 323356013818856 T^{16} - 1302989570874344 T^{17} + 7025005302285881 T^{18} - 25009625913805529 T^{19} + 123419304378495494 T^{20} - 378008976719935789 T^{21} + 1690197062617306569 T^{22} - 4249592777857282039 T^{23} + 16792529217177522915 T^{24} - 31440713666223921333 T^{25} + \)\(10\!\cdots\!77\)\( T^{26} - \)\(11\!\cdots\!79\)\( T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 + T + 118 T^{2} + 268 T^{3} + 8515 T^{4} + 15570 T^{5} + 479405 T^{6} + 624508 T^{7} + 21239885 T^{8} + 22150327 T^{9} + 849436783 T^{10} + 614045046 T^{11} + 30877887782 T^{12} + 21447171678 T^{13} + 994916163825 T^{14} + 664862322018 T^{15} + 29673650158502 T^{16} + 18293015965386 T^{17} + 784472707272943 T^{18} + 634145056382377 T^{19} + 18850476121516685 T^{20} + 17181847613232388 T^{21} + 408880227804402605 T^{22} + 411664917041647470 T^{23} + 6979134863641520515 T^{24} + 6809471808236494708 T^{25} + 92944208487048871798 T^{26} + 24417546297445042591 T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 + 18 T + 346 T^{2} + 4219 T^{3} + 51108 T^{4} + 501188 T^{5} + 4847622 T^{6} + 40907834 T^{7} + 340648565 T^{8} + 2556544565 T^{9} + 19019833968 T^{10} + 129999203178 T^{11} + 884554435782 T^{12} + 5587696050546 T^{13} + 35215825019648 T^{14} + 206744753870202 T^{15} + 1210955022585558 T^{16} + 6584849638575234 T^{17} + 35646231049300848 T^{18} + 177280916383943705 T^{19} + 874011019408453085 T^{20} + 3883457471065159922 T^{21} + 17027172675375425862 T^{22} + 65135264444415051476 T^{23} + \)\(24\!\cdots\!92\)\( T^{24} + \)\(75\!\cdots\!47\)\( T^{25} + \)\(22\!\cdots\!26\)\( T^{26} + \)\(43\!\cdots\!46\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 + 7 T + 262 T^{2} + 2640 T^{3} + 39667 T^{4} + 420038 T^{5} + 4561507 T^{6} + 42891762 T^{7} + 399809855 T^{8} + 3325823807 T^{9} + 27223537581 T^{10} + 204976865162 T^{11} + 1500708456526 T^{12} + 10205025251938 T^{13} + 67954179567427 T^{14} + 418406035329458 T^{15} + 2522690915420206 T^{16} + 14127210523830202 T^{17} + 76927210778424141 T^{18} + 385317311474377207 T^{19} + 1899138487829095055 T^{20} + 8353353963786668322 T^{21} + 36423292327112045347 T^{22} + \)\(13\!\cdots\!18\)\( T^{23} + \)\(53\!\cdots\!67\)\( T^{24} + \)\(14\!\cdots\!40\)\( T^{25} + \)\(59\!\cdots\!22\)\( T^{26} + \)\(64\!\cdots\!47\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( 1 + 18 T + 366 T^{2} + 4641 T^{3} + 59465 T^{4} + 608854 T^{5} + 6143696 T^{6} + 54168393 T^{7} + 470772018 T^{8} + 3719594892 T^{9} + 29137454200 T^{10} + 211460532837 T^{11} + 1531245359196 T^{12} + 10358137224639 T^{13} + 70183362359220 T^{14} + 445399900659477 T^{15} + 2831272669153404 T^{16} + 16812592584271359 T^{17} + 99615157556414200 T^{18} + 546811853663673156 T^{19} + 2975920839088362882 T^{20} + 14723977351158141051 T^{21} + 71808749292696153296 T^{22} + \)\(30\!\cdots\!22\)\( T^{23} + \)\(12\!\cdots\!85\)\( T^{24} + \)\(43\!\cdots\!87\)\( T^{25} + \)\(14\!\cdots\!66\)\( T^{26} + \)\(30\!\cdots\!74\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 + 10 T + 355 T^{2} + 3310 T^{3} + 63280 T^{4} + 531585 T^{5} + 7379569 T^{6} + 55788965 T^{7} + 631560442 T^{8} + 4349115453 T^{9} + 42803940520 T^{10} + 272778367170 T^{11} + 2436702056939 T^{12} + 14571457354777 T^{13} + 121448028946007 T^{14} + 684858495674519 T^{15} + 5382674843778251 T^{16} + 28320668414690910 T^{17} + 208869575280574120 T^{18} + 997447914012093171 T^{19} + 6807725997596415418 T^{20} + 28263979535701090795 T^{21} + \)\(17\!\cdots\!09\)\( T^{22} + \)\(59\!\cdots\!95\)\( T^{23} + \)\(33\!\cdots\!20\)\( T^{24} + \)\(81\!\cdots\!30\)\( T^{25} + \)\(41\!\cdots\!55\)\( T^{26} + \)\(54\!\cdots\!70\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( 1 + 5 T + 323 T^{2} + 1362 T^{3} + 51888 T^{4} + 196444 T^{5} + 5782396 T^{6} + 21213640 T^{7} + 513898043 T^{8} + 1898767906 T^{9} + 38413174831 T^{10} + 141408233359 T^{11} + 2467213668708 T^{12} + 8804595640936 T^{13} + 138915308808228 T^{14} + 466643568969608 T^{15} + 6930403195400772 T^{16} + 21052433557787843 T^{17} + 303098426153683711 T^{18} + 794056180542247658 T^{19} + 11390221808538370547 T^{20} + 24919899224491776680 T^{21} + \)\(36\!\cdots\!56\)\( T^{22} + \)\(64\!\cdots\!52\)\( T^{23} + \)\(90\!\cdots\!12\)\( T^{24} + \)\(12\!\cdots\!14\)\( T^{25} + \)\(15\!\cdots\!43\)\( T^{26} + \)\(13\!\cdots\!65\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \)
$59$ \( 1 + 24 T + 608 T^{2} + 10083 T^{3} + 158930 T^{4} + 2078978 T^{5} + 25891621 T^{6} + 287913389 T^{7} + 3078654123 T^{8} + 30368747261 T^{9} + 289271481384 T^{10} + 2585269937382 T^{11} + 22339677863663 T^{12} + 182354038243299 T^{13} + 1440211099667985 T^{14} + 10758888256354641 T^{15} + 77764418643410903 T^{16} + 530960154469577778 T^{17} + 3505206966934707624 T^{18} + 21711355347078595039 T^{19} + \)\(12\!\cdots\!43\)\( T^{20} + \)\(71\!\cdots\!91\)\( T^{21} + \)\(38\!\cdots\!41\)\( T^{22} + \)\(18\!\cdots\!42\)\( T^{23} + \)\(81\!\cdots\!30\)\( T^{24} + \)\(30\!\cdots\!97\)\( T^{25} + \)\(10\!\cdots\!48\)\( T^{26} + \)\(25\!\cdots\!96\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \)
$61$ \( 1 + 6 T + 338 T^{2} + 2325 T^{3} + 53576 T^{4} + 388762 T^{5} + 5396527 T^{6} + 38481445 T^{7} + 411017829 T^{8} + 2750062490 T^{9} + 28025837788 T^{10} + 176522824439 T^{11} + 1890525358402 T^{12} + 11391003929697 T^{13} + 121195392493670 T^{14} + 694851239711517 T^{15} + 7034644858613842 T^{16} + 40067327213988659 T^{17} + 388041293904439708 T^{18} + 2322692606572849490 T^{19} + 21175792419125482269 T^{20} + \)\(12\!\cdots\!45\)\( T^{21} + \)\(10\!\cdots\!87\)\( T^{22} + \)\(45\!\cdots\!42\)\( T^{23} + \)\(38\!\cdots\!76\)\( T^{24} + \)\(10\!\cdots\!25\)\( T^{25} + \)\(89\!\cdots\!98\)\( T^{26} + \)\(97\!\cdots\!86\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \)
$67$ \( 1 + 61 T + 2421 T^{2} + 70493 T^{3} + 1678132 T^{4} + 33783902 T^{5} + 594403169 T^{6} + 9280668101 T^{7} + 130520603593 T^{8} + 1666814817226 T^{9} + 19482287693337 T^{10} + 209325348319550 T^{11} + 2076314918972930 T^{12} + 19049840366825643 T^{13} + 161975474676955394 T^{14} + 1276339304577318081 T^{15} + 9320577671269482770 T^{16} + 62957319736632816650 T^{17} + \)\(39\!\cdots\!77\)\( T^{18} + \)\(22\!\cdots\!82\)\( T^{19} + \)\(11\!\cdots\!17\)\( T^{20} + \)\(56\!\cdots\!23\)\( T^{21} + \)\(24\!\cdots\!29\)\( T^{22} + \)\(91\!\cdots\!94\)\( T^{23} + \)\(30\!\cdots\!68\)\( T^{24} + \)\(86\!\cdots\!19\)\( T^{25} + \)\(19\!\cdots\!81\)\( T^{26} + \)\(33\!\cdots\!07\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \)
$71$ \( 1 - 11 T + 485 T^{2} - 4693 T^{3} + 125337 T^{4} - 1125339 T^{5} + 22710833 T^{6} - 189790693 T^{7} + 3161856556 T^{8} - 24639829899 T^{9} + 354752247773 T^{10} - 2568996107968 T^{11} + 32864019371742 T^{12} - 219990301067060 T^{13} + 2546955729242985 T^{14} - 15619311375761260 T^{15} + 165667521652951422 T^{16} - 919471965998934848 T^{17} + 9014850954440436413 T^{18} - 44455904307423165549 T^{19} + \)\(40\!\cdots\!76\)\( T^{20} - \)\(17\!\cdots\!63\)\( T^{21} + \)\(14\!\cdots\!13\)\( T^{22} - \)\(51\!\cdots\!09\)\( T^{23} + \)\(40\!\cdots\!37\)\( T^{24} - \)\(10\!\cdots\!03\)\( T^{25} + \)\(79\!\cdots\!85\)\( T^{26} - \)\(12\!\cdots\!21\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 + 28 T + 870 T^{2} + 15771 T^{3} + 289787 T^{4} + 3900456 T^{5} + 52554836 T^{6} + 550404858 T^{7} + 5817654033 T^{8} + 47280069271 T^{9} + 400851669934 T^{10} + 2374590680106 T^{11} + 16903410188361 T^{12} + 66686657077971 T^{13} + 697274321398641 T^{14} + 4868125966691883 T^{15} + 90078272893775769 T^{16} + 923756142602795802 T^{17} + 11383482328038186094 T^{18} + 98014968520782318703 T^{19} + \)\(88\!\cdots\!37\)\( T^{20} + \)\(60\!\cdots\!26\)\( T^{21} + \)\(42\!\cdots\!16\)\( T^{22} + \)\(22\!\cdots\!28\)\( T^{23} + \)\(12\!\cdots\!63\)\( T^{24} + \)\(49\!\cdots\!67\)\( T^{25} + \)\(19\!\cdots\!70\)\( T^{26} + \)\(46\!\cdots\!24\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 - 6 T + 520 T^{2} - 4426 T^{3} + 142954 T^{4} - 1459383 T^{5} + 28192858 T^{6} - 302539912 T^{7} + 4410079975 T^{8} - 45526968993 T^{9} + 563448300622 T^{10} - 5358116361729 T^{11} + 59262289453086 T^{12} - 512503429802565 T^{13} + 5148351823043328 T^{14} - 40487770954402635 T^{15} + 369855948476709726 T^{16} - 2641760332870504431 T^{17} + 21946356948539250382 T^{18} - \)\(14\!\cdots\!07\)\( T^{19} + \)\(10\!\cdots\!75\)\( T^{20} - \)\(58\!\cdots\!08\)\( T^{21} + \)\(42\!\cdots\!38\)\( T^{22} - \)\(17\!\cdots\!77\)\( T^{23} + \)\(13\!\cdots\!54\)\( T^{24} - \)\(33\!\cdots\!54\)\( T^{25} + \)\(30\!\cdots\!20\)\( T^{26} - \)\(28\!\cdots\!34\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 + 16 T + 673 T^{2} + 9849 T^{3} + 226719 T^{4} + 2965629 T^{5} + 50566245 T^{6} + 592879257 T^{7} + 8390024810 T^{8} + 89206969507 T^{9} + 1104538811693 T^{10} + 10748118243738 T^{11} + 119552096879454 T^{12} + 1068592698071772 T^{13} + 10822676226410490 T^{14} + 88693193939957076 T^{15} + 823594395402558606 T^{16} + 6145634286232219806 T^{17} + 52419557482284947453 T^{18} + \)\(35\!\cdots\!01\)\( T^{19} + \)\(27\!\cdots\!90\)\( T^{20} + \)\(16\!\cdots\!39\)\( T^{21} + \)\(11\!\cdots\!45\)\( T^{22} + \)\(55\!\cdots\!87\)\( T^{23} + \)\(35\!\cdots\!31\)\( T^{24} + \)\(12\!\cdots\!83\)\( T^{25} + \)\(71\!\cdots\!53\)\( T^{26} + \)\(14\!\cdots\!08\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( 1 + 39 T + 1396 T^{2} + 35212 T^{3} + 796031 T^{4} + 15331322 T^{5} + 269092862 T^{6} + 4255162212 T^{7} + 62241133074 T^{8} + 839225063128 T^{9} + 10571593611698 T^{10} + 124113093132237 T^{11} + 1369231973742878 T^{12} + 14152783048988514 T^{13} + 137828187030160824 T^{14} + 1259597691359977746 T^{15} + 10845686464017336638 T^{16} + 87495883152339985653 T^{17} + \)\(66\!\cdots\!18\)\( T^{18} + \)\(46\!\cdots\!72\)\( T^{19} + \)\(30\!\cdots\!14\)\( T^{20} + \)\(18\!\cdots\!48\)\( T^{21} + \)\(10\!\cdots\!22\)\( T^{22} + \)\(53\!\cdots\!98\)\( T^{23} + \)\(24\!\cdots\!31\)\( T^{24} + \)\(97\!\cdots\!68\)\( T^{25} + \)\(34\!\cdots\!16\)\( T^{26} + \)\(85\!\cdots\!91\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \)
$97$ \( 1 + 19 T + 756 T^{2} + 12008 T^{3} + 273069 T^{4} + 3740542 T^{5} + 63239870 T^{6} + 768039638 T^{7} + 10669336340 T^{8} + 117754783874 T^{9} + 1423043481350 T^{10} + 14627973463715 T^{11} + 160696162359694 T^{12} + 1571983664798348 T^{13} + 16266310222546160 T^{14} + 152482415485439756 T^{15} + 1511990191642360846 T^{16} + 13350556425049160195 T^{17} + \)\(12\!\cdots\!50\)\( T^{18} + \)\(10\!\cdots\!18\)\( T^{19} + \)\(88\!\cdots\!60\)\( T^{20} + \)\(62\!\cdots\!94\)\( T^{21} + \)\(49\!\cdots\!70\)\( T^{22} + \)\(28\!\cdots\!14\)\( T^{23} + \)\(20\!\cdots\!81\)\( T^{24} + \)\(85\!\cdots\!24\)\( T^{25} + \)\(52\!\cdots\!96\)\( T^{26} + \)\(12\!\cdots\!63\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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