Properties

Label 4025.2.a.bc
Level 4025
Weight 2
Character orbit 4025.a
Self dual yes
Analytic conductor 32.140
Analytic rank 0
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{7} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{11} ) q^{6} + q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 1 + \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{7} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{11} ) q^{6} + q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 1 + \beta_{5} - \beta_{7} ) q^{9} + \beta_{8} q^{11} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{12} + ( 1 + \beta_{6} ) q^{13} + \beta_{1} q^{14} + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{16} + ( 1 - \beta_{3} - \beta_{4} - \beta_{12} ) q^{17} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{18} + ( \beta_{1} - \beta_{2} - \beta_{7} + \beta_{9} ) q^{19} -\beta_{7} q^{21} + ( 1 + \beta_{1} + \beta_{2} + \beta_{10} ) q^{22} + q^{23} + ( -2 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{12} ) q^{24} + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{26} + ( 1 + \beta_{2} + \beta_{5} - \beta_{7} + \beta_{12} ) q^{27} + ( 1 + \beta_{2} ) q^{28} + ( 1 + \beta_{1} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{29} + ( 1 - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} - \beta_{13} ) q^{31} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{32} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{33} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{34} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{10} - \beta_{13} ) q^{36} + ( 2 + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{37} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{38} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{39} + ( -1 - \beta_{2} - \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{41} + ( -1 + \beta_{1} - \beta_{11} ) q^{42} + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{43} + ( 2 + 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{13} ) q^{44} + \beta_{1} q^{46} + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{47} + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{48} + q^{49} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{51} + ( 2 + \beta_{2} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{52} + ( -\beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{53} + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{12} ) q^{54} + ( \beta_{1} + \beta_{3} ) q^{56} + ( -1 + 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} ) q^{57} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{58} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{59} + ( \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{61} + ( -2 + \beta_{1} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{62} + ( 1 + \beta_{5} - \beta_{7} ) q^{63} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 4 \beta_{11} + 3 \beta_{12} ) q^{64} + ( -2 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{66} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} - 2 \beta_{12} - \beta_{13} ) q^{67} + ( 1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{68} -\beta_{7} q^{69} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + 4 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{71} + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{10} - 2 \beta_{11} ) q^{72} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} ) q^{73} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{74} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{13} ) q^{76} + \beta_{8} q^{77} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} ) q^{78} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{79} + ( -1 + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{81} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{82} + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} + 2 \beta_{13} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{84} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 4 \beta_{7} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{86} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{12} ) q^{87} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{12} + \beta_{13} ) q^{88} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{89} + ( 1 + \beta_{6} ) q^{91} + ( 1 + \beta_{2} ) q^{92} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{93} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{13} ) q^{94} + ( -9 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{11} - \beta_{12} ) q^{96} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{12} ) q^{97} + \beta_{1} q^{98} + ( 2 + \beta_{1} + \beta_{2} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 3q^{2} + 4q^{3} + 17q^{4} - 4q^{6} + 14q^{7} + 3q^{8} + 18q^{9} + O(q^{10}) \) \( 14q + 3q^{2} + 4q^{3} + 17q^{4} - 4q^{6} + 14q^{7} + 3q^{8} + 18q^{9} + 5q^{11} + 17q^{12} + 11q^{13} + 3q^{14} + 23q^{16} + 3q^{17} + 15q^{18} - 2q^{19} + 4q^{21} + 23q^{22} + 14q^{23} - 12q^{24} - 9q^{26} + 25q^{27} + 17q^{28} + 7q^{29} - 3q^{31} + 24q^{32} + 6q^{33} - 14q^{34} + 13q^{36} + 22q^{37} + 20q^{38} - 10q^{39} - 17q^{41} - 4q^{42} + 18q^{43} + 28q^{44} + 3q^{46} + 30q^{47} + 8q^{48} + 14q^{49} + 4q^{51} + 8q^{52} + 11q^{53} + 20q^{54} + 3q^{56} + 18q^{57} + 38q^{58} - 22q^{59} - 8q^{61} - 22q^{62} + 18q^{63} + 29q^{64} - 9q^{66} + 39q^{67} + q^{68} + 4q^{69} - 5q^{71} - 24q^{72} + 18q^{73} + 35q^{74} - 41q^{76} + 5q^{77} - 22q^{78} + 10q^{79} + 2q^{81} - 8q^{82} + 24q^{83} + 17q^{84} - 26q^{86} + 5q^{87} + 58q^{88} + 25q^{89} + 11q^{91} + 17q^{92} + 47q^{93} - 2q^{94} - 117q^{96} + 43q^{97} + 3q^{98} + 55q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 3 x^{13} - 18 x^{12} + 58 x^{11} + 111 x^{10} - 414 x^{9} - 244 x^{8} + 1330 x^{7} - 27 x^{6} - 1853 x^{5} + 539 x^{4} + 891 x^{3} - 218 x^{2} - 133 x - 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\(-4055 \nu^{13} + 7975 \nu^{12} + 81881 \nu^{11} - 152987 \nu^{10} - 615369 \nu^{9} + 1080275 \nu^{8} + 2103680 \nu^{7} - 3407257 \nu^{6} - 3155676 \nu^{5} + 4556819 \nu^{4} + 1597821 \nu^{3} - 1900192 \nu^{2} - 75581 \nu + 142730\)\()/22937\)
\(\beta_{5}\)\(=\)\((\)\(-21776 \nu^{13} + 33409 \nu^{12} + 446796 \nu^{11} - 619227 \nu^{10} - 3443457 \nu^{9} + 4180444 \nu^{8} + 12330084 \nu^{7} - 12373995 \nu^{6} - 20539065 \nu^{5} + 14830170 \nu^{4} + 14277936 \nu^{3} - 4309323 \nu^{2} - 3505754 \nu - 316562\)\()/22937\)
\(\beta_{6}\)\(=\)\((\)\(25090 \nu^{13} - 41963 \nu^{12} - 504992 \nu^{11} + 782303 \nu^{10} + 3774769 \nu^{9} - 5321389 \nu^{8} - 12816147 \nu^{7} + 15921004 \nu^{6} + 19231422 \nu^{5} - 19453226 \nu^{4} - 10653101 \nu^{3} + 6121945 \nu^{2} + 2002366 \nu + 89726\)\()/22937\)
\(\beta_{7}\)\(=\)\((\)\(-33730 \nu^{13} + 54713 \nu^{12} + 686159 \nu^{11} - 1017543 \nu^{10} - 5217903 \nu^{9} + 6900067 \nu^{8} + 18260121 \nu^{7} - 20566123 \nu^{6} - 29115247 \nu^{5} + 25007434 \nu^{4} + 18540140 \nu^{3} - 7714479 \nu^{2} - 4174085 \nu - 320996\)\()/22937\)
\(\beta_{8}\)\(=\)\((\)\(-34523 \nu^{13} + 57715 \nu^{12} + 699434 \nu^{11} - 1075557 \nu^{10} - 5289707 \nu^{9} + 7310378 \nu^{8} + 18364740 \nu^{7} - 21833829 \nu^{6} - 28918074 \nu^{5} + 26520409 \nu^{4} + 18099684 \nu^{3} - 7961510 \nu^{2} - 4230503 \nu - 436758\)\()/22937\)
\(\beta_{9}\)\(=\)\((\)\(42027 \nu^{13} - 75160 \nu^{12} - 839833 \nu^{11} + 1403897 \nu^{10} + 6219514 \nu^{9} - 9565583 \nu^{8} - 20836696 \nu^{7} + 28645999 \nu^{6} + 30567527 \nu^{5} - 34949868 \nu^{4} - 16164725 \nu^{3} + 10894262 \nu^{2} + 3009015 \nu + 131732\)\()/22937\)
\(\beta_{10}\)\(=\)\((\)\(-45854 \nu^{13} + 78020 \nu^{12} + 926777 \nu^{11} - 1457654 \nu^{10} - 6982144 \nu^{9} + 9941128 \nu^{8} + 24081761 \nu^{7} - 29850195 \nu^{6} - 37450710 \nu^{5} + 36707581 \nu^{4} + 22798483 \nu^{3} - 11779454 \nu^{2} - 5051254 \nu - 264833\)\()/22937\)
\(\beta_{11}\)\(=\)\((\)\(-46477 \nu^{13} + 79019 \nu^{12} + 938797 \nu^{11} - 1473873 \nu^{10} - 7064153 \nu^{9} + 10030001 \nu^{8} + 24294777 \nu^{7} - 30025957 \nu^{6} - 37494256 \nu^{5} + 36720610 \nu^{4} + 22338951 \nu^{3} - 11527225 \nu^{2} - 4784149 \nu - 326507\)\()/22937\)
\(\beta_{12}\)\(=\)\((\)\(52513 \nu^{13} - 88882 \nu^{12} - 1064311 \nu^{11} + 1659877 \nu^{10} + 8052400 \nu^{9} - 11315041 \nu^{8} - 27962096 \nu^{7} + 33964264 \nu^{6} + 44003702 \nu^{5} - 41769609 \nu^{4} - 27425832 \nu^{3} + 13388234 \nu^{2} + 6282190 \nu + 362877\)\()/22937\)
\(\beta_{13}\)\(=\)\((\)\(82827 \nu^{13} - 143014 \nu^{12} - 1668584 \nu^{11} + 2672763 \nu^{10} + 12509311 \nu^{9} - 18243758 \nu^{8} - 42782683 \nu^{7} + 54897262 \nu^{6} + 65368699 \nu^{5} - 67878225 \nu^{4} - 38037463 \nu^{3} + 22247589 \nu^{2} + 7829560 \nu + 291069\)\()/22937\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{3} + 7 \beta_{2} + 16\)
\(\nu^{5}\)\(=\)\(2 \beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} + 10 \beta_{3} + \beta_{2} + 29 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(3 \beta_{12} + 14 \beta_{11} + \beta_{10} + \beta_{9} - 9 \beta_{8} - 12 \beta_{7} + \beta_{6} + 11 \beta_{5} + \beta_{4} + 12 \beta_{3} + 46 \beta_{2} + \beta_{1} + 98\)
\(\nu^{7}\)\(=\)\(28 \beta_{12} + 18 \beta_{11} + 16 \beta_{10} + 2 \beta_{9} - \beta_{8} + 9 \beta_{7} + 14 \beta_{6} + \beta_{5} + 11 \beta_{4} + 83 \beta_{3} + 12 \beta_{2} + 182 \beta_{1} + 20\)
\(\nu^{8}\)\(=\)\(-2 \beta_{13} + 47 \beta_{12} + 143 \beta_{11} + 16 \beta_{10} + 17 \beta_{9} - 68 \beta_{8} - 110 \beta_{7} + 17 \beta_{6} + 95 \beta_{5} + 13 \beta_{4} + 114 \beta_{3} + 303 \beta_{2} + 19 \beta_{1} + 643\)
\(\nu^{9}\)\(=\)\(\beta_{13} + 285 \beta_{12} + 215 \beta_{11} + 173 \beta_{10} + 34 \beta_{9} - 18 \beta_{8} + 58 \beta_{7} + 141 \beta_{6} + 17 \beta_{5} + 95 \beta_{4} + 655 \beta_{3} + 114 \beta_{2} + 1195 \beta_{1} + 244\)
\(\nu^{10}\)\(=\)\(-32 \beta_{13} + 517 \beta_{12} + 1293 \beta_{11} + 181 \beta_{10} + 190 \beta_{9} - 500 \beta_{8} - 907 \beta_{7} + 201 \beta_{6} + 757 \beta_{5} + 124 \beta_{4} + 1001 \beta_{3} + 2030 \beta_{2} + 241 \beta_{1} + 4397\)
\(\nu^{11}\)\(=\)\(20 \beta_{13} + 2567 \beta_{12} + 2171 \beta_{11} + 1602 \beta_{10} + 393 \beta_{9} - 221 \beta_{8} + 304 \beta_{7} + 1259 \beta_{6} + 205 \beta_{5} + 768 \beta_{4} + 5081 \beta_{3} + 1013 \beta_{2} + 8084 \beta_{1} + 2489\)
\(\nu^{12}\)\(=\)\(-343 \beta_{13} + 4943 \beta_{12} + 11007 \beta_{11} + 1792 \beta_{10} + 1799 \beta_{9} - 3687 \beta_{8} - 7100 \beta_{7} + 2044 \beta_{6} + 5820 \beta_{5} + 1071 \beta_{4} + 8470 \beta_{3} + 13876 \beta_{2} + 2596 \beta_{1} + 30917\)
\(\nu^{13}\)\(=\)\(252 \beta_{13} + 21770 \beta_{12} + 20092 \beta_{11} + 13735 \beta_{10} + 3872 \beta_{9} - 2314 \beta_{8} + 1138 \beta_{7} + 10635 \beta_{6} + 2154 \beta_{5} + 6065 \beta_{4} + 39173 \beta_{3} + 8771 \beta_{2} + 55926 \beta_{1} + 23306\)

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
−2.60812
−2.33696
−2.04050
−1.51853
−0.689960
−0.256730
−0.0822464
0.744662
1.29908
1.31920
1.63325
2.30520
2.42992
2.80172
−2.60812 2.93022 4.80228 0 −7.64237 1.00000 −7.30867 5.58621 0
1.2 −2.33696 0.00757499 3.46136 0 −0.0177024 1.00000 −3.41514 −2.99994 0
1.3 −2.04050 −1.31355 2.16365 0 2.68030 1.00000 −0.333918 −1.27458 0
1.4 −1.51853 0.630051 0.305932 0 −0.956751 1.00000 2.57249 −2.60304 0
1.5 −0.689960 2.51200 −1.52395 0 −1.73318 1.00000 2.43139 3.31014 0
1.6 −0.256730 −2.72326 −1.93409 0 0.699143 1.00000 1.01000 4.41613 0
1.7 −0.0822464 1.69790 −1.99324 0 −0.139646 1.00000 0.328429 −0.117136 0
1.8 0.744662 −1.85918 −1.44548 0 −1.38446 1.00000 −2.56572 0.456544 0
1.9 1.29908 −2.64602 −0.312383 0 −3.43741 1.00000 −3.00398 4.00145 0
1.10 1.31920 2.92477 −0.259709 0 3.85835 1.00000 −2.98101 5.55425 0
1.11 1.63325 −0.456743 0.667506 0 −0.745975 1.00000 −2.17630 −2.79139 0
1.12 2.30520 0.930494 3.31395 0 2.14497 1.00000 3.02891 −2.13418 0
1.13 2.42992 3.09771 3.90452 0 7.52721 1.00000 4.62785 6.59584 0
1.14 2.80172 −1.73197 5.84965 0 −4.85249 1.00000 10.7857 −0.000289796 0 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.bc yes 14
5.b even 2 1 4025.2.a.z 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.z 14 5.b even 2 1
4025.2.a.bc yes 14 1.a even 1 1 trivial

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

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 10 T^{2} - 20 T^{3} + 43 T^{4} - 74 T^{5} + 136 T^{6} - 226 T^{7} + 385 T^{8} - 609 T^{9} + 951 T^{10} - 1439 T^{11} + 2122 T^{12} - 3131 T^{13} + 4407 T^{14} - 6262 T^{15} + 8488 T^{16} - 11512 T^{17} + 15216 T^{18} - 19488 T^{19} + 24640 T^{20} - 28928 T^{21} + 34816 T^{22} - 37888 T^{23} + 44032 T^{24} - 40960 T^{25} + 40960 T^{26} - 24576 T^{27} + 16384 T^{28} \)
$3$ \( 1 - 4 T + 20 T^{2} - 63 T^{3} + 209 T^{4} - 554 T^{5} + 1508 T^{6} - 3520 T^{7} + 8373 T^{8} - 17727 T^{9} + 38002 T^{10} - 73930 T^{11} + 145099 T^{12} - 260483 T^{13} + 471199 T^{14} - 781449 T^{15} + 1305891 T^{16} - 1996110 T^{17} + 3078162 T^{18} - 4307661 T^{19} + 6103917 T^{20} - 7698240 T^{21} + 9893988 T^{22} - 10904382 T^{23} + 12341241 T^{24} - 11160261 T^{25} + 10628820 T^{26} - 6377292 T^{27} + 4782969 T^{28} \)
$5$ \( \)
$7$ \( ( 1 - T )^{14} \)
$11$ \( 1 - 5 T + 76 T^{2} - 272 T^{3} + 2324 T^{4} - 5905 T^{5} + 38186 T^{6} - 61069 T^{7} + 353283 T^{8} - 56808 T^{9} + 805238 T^{10} + 9768951 T^{11} - 32652216 T^{12} + 199091648 T^{13} - 572672536 T^{14} + 2190008128 T^{15} - 3950918136 T^{16} + 13002473781 T^{17} + 11789489558 T^{18} - 9148985208 T^{19} + 625862384763 T^{20} - 1190062045799 T^{21} + 8185508229866 T^{22} - 13923681115355 T^{23} + 60278574772724 T^{24} - 77604774406192 T^{25} + 238520556630796 T^{26} - 172613560719655 T^{27} + 379749833583241 T^{28} \)
$13$ \( 1 - 11 T + 126 T^{2} - 870 T^{3} + 6084 T^{4} - 33289 T^{5} + 186025 T^{6} - 893691 T^{7} + 4340781 T^{8} - 18757848 T^{9} + 81627572 T^{10} - 323152792 T^{11} + 1299676347 T^{12} - 4804492671 T^{13} + 18074146521 T^{14} - 62458404723 T^{15} + 219645302643 T^{16} - 709966684024 T^{17} + 2331365083892 T^{18} - 6964657657464 T^{19} + 20952120797829 T^{20} - 56077784906247 T^{21} + 151746307374025 T^{22} - 353013179627797 T^{23} + 838731064409316 T^{24} - 1559179542812190 T^{25} + 2935558725432606 T^{26} - 3331626172514783 T^{27} + 3937376385699289 T^{28} \)
$17$ \( 1 - 3 T + 111 T^{2} - 267 T^{3} + 6459 T^{4} - 12043 T^{5} + 256809 T^{6} - 359254 T^{7} + 7809940 T^{8} - 7858562 T^{9} + 193283627 T^{10} - 137606996 T^{11} + 4049124032 T^{12} - 2194603371 T^{13} + 73593772698 T^{14} - 37308257307 T^{15} + 1170196845248 T^{16} - 676063171348 T^{17} + 16143241810667 T^{18} - 11158034265634 T^{19} + 188512965635860 T^{20} - 147415809629942 T^{21} + 1791437292665769 T^{22} - 1428153796653371 T^{23} + 13021304603000091 T^{24} - 9150596314138011 T^{25} + 64671068332503471 T^{26} - 29713734098717811 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 + 2 T + 117 T^{2} + 164 T^{3} + 7048 T^{4} + 6613 T^{5} + 294219 T^{6} + 176952 T^{7} + 9630855 T^{8} + 3751931 T^{9} + 262568065 T^{10} + 70881973 T^{11} + 6136370112 T^{12} + 1299950195 T^{13} + 124727554846 T^{14} + 24699053705 T^{15} + 2215229610432 T^{16} + 486179452807 T^{17} + 34218132798865 T^{18} + 9290152597169 T^{19} + 453092058258255 T^{20} + 158172391959528 T^{21} + 4996886934359979 T^{22} + 2133933745412527 T^{23} + 43211754984981448 T^{24} + 19104402459307916 T^{25} + 258957845530740837 T^{26} + 84105966924514118 T^{27} + 799006685782884121 T^{28} \)
$23$ \( ( 1 - T )^{14} \)
$29$ \( 1 - 7 T + 157 T^{2} - 1077 T^{3} + 13907 T^{4} - 90207 T^{5} + 890241 T^{6} - 5279677 T^{7} + 44297446 T^{8} - 240998825 T^{9} + 1812178993 T^{10} - 9157766584 T^{11} + 63364064656 T^{12} - 300806471186 T^{13} + 1947499885897 T^{14} - 8723387664394 T^{15} + 53289178375696 T^{16} - 223348769217176 T^{17} + 1281719770348033 T^{18} - 4943162808399925 T^{19} + 26349153941538166 T^{20} - 91073775201472193 T^{21} + 445339866920813601 T^{22} - 1308646117045214883 T^{23} + 5850775493505895307 T^{24} - 13139949017665177833 T^{25} + 55548920963258639437 T^{26} - 71824400990710215323 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 + 3 T + 188 T^{2} + 550 T^{3} + 15965 T^{4} + 43204 T^{5} + 779963 T^{6} + 1855224 T^{7} + 21382677 T^{8} + 42888527 T^{9} + 114325175 T^{10} + 171574022 T^{11} - 17281079762 T^{12} - 23731916416 T^{13} - 822309308409 T^{14} - 735689408896 T^{15} - 16607117651282 T^{16} + 5111361689402 T^{17} + 105581699941175 T^{18} + 1227862115650577 T^{19} + 18977204547134037 T^{20} + 51042062001465864 T^{21} + 665223452235594683 T^{22} + 1142297435829629884 T^{23} + 13085365601648487965 T^{24} + 13974662293022657050 T^{25} + \)\(14\!\cdots\!68\)\( T^{26} + 73252638892335127773 T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 - 22 T + 522 T^{2} - 7817 T^{3} + 115272 T^{4} - 1356664 T^{5} + 15428426 T^{6} - 151665750 T^{7} + 1434199673 T^{8} - 12154351467 T^{9} + 99018715088 T^{10} - 736205811706 T^{11} + 5263209408638 T^{12} - 34643930511858 T^{13} + 219325179695320 T^{14} - 1281825428938746 T^{15} + 7205333680425422 T^{16} - 37291032980344018 T^{17} + 185577014088041168 T^{18} - 842830825490534919 T^{19} + 3679763976795264257 T^{20} - 14397914344284294750 T^{21} + 54192029331340558346 T^{22} - \)\(17\!\cdots\!28\)\( T^{23} + \)\(55\!\cdots\!28\)\( T^{24} - \)\(13\!\cdots\!21\)\( T^{25} + \)\(34\!\cdots\!82\)\( T^{26} - \)\(53\!\cdots\!34\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 + 17 T + 378 T^{2} + 5316 T^{3} + 73453 T^{4} + 840084 T^{5} + 9214851 T^{6} + 89272404 T^{7} + 831682317 T^{8} + 7065105151 T^{9} + 57747973055 T^{10} + 438152240282 T^{11} + 3205596096524 T^{12} + 21938127627268 T^{13} + 145091451945897 T^{14} + 899463232717988 T^{15} + 5388607038256844 T^{16} + 30197890552475722 T^{17} + 163181970087869855 T^{18} + 818536242460391351 T^{19} + 3950577701146406397 T^{20} + 17386182218631279924 T^{21} + 73579896232490875971 T^{22} + \)\(27\!\cdots\!24\)\( T^{23} + \)\(98\!\cdots\!53\)\( T^{24} + \)\(29\!\cdots\!56\)\( T^{25} + \)\(85\!\cdots\!18\)\( T^{26} + \)\(15\!\cdots\!57\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( 1 - 18 T + 478 T^{2} - 5787 T^{3} + 89169 T^{4} - 806610 T^{5} + 9355076 T^{6} - 67553931 T^{7} + 677579870 T^{8} - 4226402868 T^{9} + 40623384000 T^{10} - 236903394387 T^{11} + 2211069001608 T^{12} - 12144130144285 T^{13} + 104422411826012 T^{14} - 522197596204255 T^{15} + 4088266583973192 T^{16} - 18835478177527209 T^{17} + 138883265842584000 T^{18} - 621316905115414524 T^{19} + 4283228352964223630 T^{20} - 18362415699238111617 T^{21} + \)\(10\!\cdots\!76\)\( T^{22} - \)\(40\!\cdots\!30\)\( T^{23} + \)\(19\!\cdots\!81\)\( T^{24} - \)\(53\!\cdots\!09\)\( T^{25} + \)\(19\!\cdots\!78\)\( T^{26} - \)\(30\!\cdots\!74\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 - 30 T + 757 T^{2} - 13956 T^{3} + 226130 T^{4} - 3145061 T^{5} + 39751893 T^{6} - 453037275 T^{7} + 4780591274 T^{8} - 46507325725 T^{9} + 423391135294 T^{10} - 3594201351750 T^{11} + 28722881382201 T^{12} - 215242935935997 T^{13} + 1522545273662593 T^{14} - 10116417988991859 T^{15} + 63448844973282009 T^{16} - 373160766942740250 T^{17} + 2066013678462561214 T^{18} - 10666222943951405075 T^{19} + 51531022742384439146 T^{20} - \)\(22\!\cdots\!25\)\( T^{21} + \)\(94\!\cdots\!73\)\( T^{22} - \)\(35\!\cdots\!87\)\( T^{23} + \)\(11\!\cdots\!70\)\( T^{24} - \)\(34\!\cdots\!68\)\( T^{25} + \)\(87\!\cdots\!37\)\( T^{26} - \)\(16\!\cdots\!10\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( 1 - 11 T + 459 T^{2} - 4050 T^{3} + 95108 T^{4} - 659652 T^{5} + 11697724 T^{6} - 59472636 T^{7} + 942701787 T^{8} - 2806666614 T^{9} + 52282455135 T^{10} - 11366484357 T^{11} + 2178795550640 T^{12} + 7016039332924 T^{13} + 94871220818868 T^{14} + 371850084644972 T^{15} + 6120236701747760 T^{16} - 1692208091617089 T^{17} + 412533718876069935 T^{18} - 1173735328328370702 T^{19} + 20894382844021637523 T^{20} - 69863168024671000332 T^{21} + \)\(72\!\cdots\!64\)\( T^{22} - \)\(21\!\cdots\!16\)\( T^{23} + \)\(16\!\cdots\!92\)\( T^{24} - \)\(37\!\cdots\!50\)\( T^{25} + \)\(22\!\cdots\!19\)\( T^{26} - \)\(28\!\cdots\!03\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \)
$59$ \( 1 + 22 T + 614 T^{2} + 9721 T^{3} + 169952 T^{4} + 2196912 T^{5} + 29980987 T^{6} + 333692017 T^{7} + 3847134473 T^{8} + 37895606369 T^{9} + 382954251456 T^{10} + 3387468788676 T^{11} + 30588576474205 T^{12} + 244652170345875 T^{13} + 1993147423237107 T^{14} + 14434478050406625 T^{15} + 106478834706707605 T^{16} + 695714952349488204 T^{17} + 4640394911377127616 T^{18} + 27092489818537260331 T^{19} + \)\(16\!\cdots\!93\)\( T^{20} + \)\(83\!\cdots\!23\)\( T^{21} + \)\(44\!\cdots\!27\)\( T^{22} + \)\(19\!\cdots\!68\)\( T^{23} + \)\(86\!\cdots\!52\)\( T^{24} + \)\(29\!\cdots\!39\)\( T^{25} + \)\(10\!\cdots\!34\)\( T^{26} + \)\(23\!\cdots\!38\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \)
$61$ \( 1 + 8 T + 526 T^{2} + 4031 T^{3} + 137780 T^{4} + 1022326 T^{5} + 23833533 T^{6} + 171989587 T^{7} + 3049610443 T^{8} + 21304378998 T^{9} + 306010094100 T^{10} + 2042085853071 T^{11} + 24878939119392 T^{12} + 155308290678935 T^{13} + 1667120075729314 T^{14} + 9473805731415035 T^{15} + 92574532463257632 T^{16} + 463514689015908651 T^{17} + 4236967107303638100 T^{18} + 17993599696812886398 T^{19} + \)\(15\!\cdots\!23\)\( T^{20} + \)\(54\!\cdots\!27\)\( T^{21} + \)\(45\!\cdots\!73\)\( T^{22} + \)\(11\!\cdots\!66\)\( T^{23} + \)\(98\!\cdots\!80\)\( T^{24} + \)\(17\!\cdots\!91\)\( T^{25} + \)\(13\!\cdots\!46\)\( T^{26} + \)\(12\!\cdots\!48\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \)
$67$ \( 1 - 39 T + 1009 T^{2} - 18287 T^{3} + 273624 T^{4} - 3409810 T^{5} + 38266001 T^{6} - 386317903 T^{7} + 3713668585 T^{8} - 33551472462 T^{9} + 298538213605 T^{10} - 2551754474434 T^{11} + 21842282044670 T^{12} - 180850389231113 T^{13} + 1505220801615634 T^{14} - 12116976078484571 T^{15} + 98050004098523630 T^{16} - 767473330994193142 T^{17} + 6015879665478201205 T^{18} - 45298685347765303434 T^{19} + \)\(33\!\cdots\!65\)\( T^{20} - \)\(23\!\cdots\!69\)\( T^{21} + \)\(15\!\cdots\!41\)\( T^{22} - \)\(92\!\cdots\!70\)\( T^{23} + \)\(49\!\cdots\!76\)\( T^{24} - \)\(22\!\cdots\!21\)\( T^{25} + \)\(82\!\cdots\!49\)\( T^{26} - \)\(21\!\cdots\!93\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \)
$71$ \( 1 + 5 T + 301 T^{2} + 887 T^{3} + 47257 T^{4} + 105001 T^{5} + 5092609 T^{6} + 8882415 T^{7} + 362758064 T^{8} + 426597757 T^{9} + 15099109549 T^{10} + 10059265392 T^{11} + 54382732686 T^{12} - 650360712684 T^{13} - 31591977346715 T^{14} - 46175610600564 T^{15} + 274143355470126 T^{16} + 3600321735716112 T^{17} + 383693755243241869 T^{18} + 769680194250165707 T^{19} + 46469410993032288944 T^{20} + 80786631721694594265 T^{21} + \)\(32\!\cdots\!49\)\( T^{22} + \)\(48\!\cdots\!31\)\( T^{23} + \)\(15\!\cdots\!57\)\( T^{24} + \)\(20\!\cdots\!77\)\( T^{25} + \)\(49\!\cdots\!41\)\( T^{26} + \)\(58\!\cdots\!55\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 - 18 T + 596 T^{2} - 7217 T^{3} + 150439 T^{4} - 1482612 T^{5} + 25965976 T^{6} - 230470980 T^{7} + 3558209497 T^{8} - 28542821925 T^{9} + 392859833814 T^{10} - 2878115746178 T^{11} + 36436831583161 T^{12} - 248126005750359 T^{13} + 2891953096870395 T^{14} - 18113198419776207 T^{15} + 194171875506664969 T^{16} - 1119635953230927026 T^{17} + 11156528239869921174 T^{18} - 59171313316775076525 T^{19} + \)\(53\!\cdots\!33\)\( T^{20} - \)\(25\!\cdots\!60\)\( T^{21} + \)\(20\!\cdots\!56\)\( T^{22} - \)\(87\!\cdots\!56\)\( T^{23} + \)\(64\!\cdots\!11\)\( T^{24} - \)\(22\!\cdots\!09\)\( T^{25} + \)\(13\!\cdots\!16\)\( T^{26} - \)\(30\!\cdots\!94\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 - 10 T + 492 T^{2} - 4862 T^{3} + 133358 T^{4} - 1236539 T^{5} + 25248098 T^{6} - 217682804 T^{7} + 3684169111 T^{8} - 29521156037 T^{9} + 437078923458 T^{10} - 3258625021461 T^{11} + 43548923964458 T^{12} - 302143143647501 T^{13} + 3709096196507328 T^{14} - 23869308348152579 T^{15} + 271788834462182378 T^{16} - 1606629221956109979 T^{17} + 17024259472081900098 T^{18} - 90838262089528330763 T^{19} + \)\(89\!\cdots\!31\)\( T^{20} - \)\(41\!\cdots\!36\)\( T^{21} + \)\(38\!\cdots\!78\)\( T^{22} - \)\(14\!\cdots\!41\)\( T^{23} + \)\(12\!\cdots\!58\)\( T^{24} - \)\(36\!\cdots\!98\)\( T^{25} + \)\(29\!\cdots\!72\)\( T^{26} - \)\(46\!\cdots\!90\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 - 24 T + 731 T^{2} - 13603 T^{3} + 260411 T^{4} - 3964887 T^{5} + 59666939 T^{6} - 778011917 T^{7} + 9968714740 T^{8} - 114820087847 T^{9} + 1303030698647 T^{10} - 13560161394564 T^{11} + 139457980760552 T^{12} - 1330096080532638 T^{13} + 12559576595348166 T^{14} - 110397974684208954 T^{15} + 960726029459442728 T^{16} - 7753524003313565868 T^{17} + 61839649169243591687 T^{18} - \)\(45\!\cdots\!21\)\( T^{19} + \)\(32\!\cdots\!60\)\( T^{20} - \)\(21\!\cdots\!59\)\( T^{21} + \)\(13\!\cdots\!99\)\( T^{22} - \)\(74\!\cdots\!61\)\( T^{23} + \)\(40\!\cdots\!39\)\( T^{24} - \)\(17\!\cdots\!01\)\( T^{25} + \)\(78\!\cdots\!91\)\( T^{26} - \)\(21\!\cdots\!12\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( 1 - 25 T + 794 T^{2} - 14006 T^{3} + 277865 T^{4} - 3948430 T^{5} + 61379150 T^{6} - 746774942 T^{7} + 9908005206 T^{8} - 107104187344 T^{9} + 1271677533784 T^{10} - 12552819431873 T^{11} + 137456776403760 T^{12} - 1264112781649036 T^{13} + 12992236516231040 T^{14} - 112506037566764204 T^{15} + 1088795125894182960 T^{16} - 8849348562068076937 T^{17} + 79787898298961369944 T^{18} - \)\(59\!\cdots\!56\)\( T^{19} + \)\(49\!\cdots\!66\)\( T^{20} - \)\(33\!\cdots\!18\)\( T^{21} + \)\(24\!\cdots\!50\)\( T^{22} - \)\(13\!\cdots\!70\)\( T^{23} + \)\(86\!\cdots\!65\)\( T^{24} - \)\(38\!\cdots\!34\)\( T^{25} + \)\(19\!\cdots\!74\)\( T^{26} - \)\(54\!\cdots\!25\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \)
$97$ \( 1 - 43 T + 1592 T^{2} - 39590 T^{3} + 887293 T^{4} - 16234448 T^{5} + 276616898 T^{6} - 4130832308 T^{7} + 58811863262 T^{8} - 760864401164 T^{9} + 9522564908486 T^{10} - 110245919746819 T^{11} + 1241634039076756 T^{12} - 13016470869116572 T^{13} + 132803983609724832 T^{14} - 1262597674304307484 T^{15} + 11682534673673197204 T^{16} - \)\(10\!\cdots\!87\)\( T^{17} + \)\(84\!\cdots\!66\)\( T^{18} - \)\(65\!\cdots\!48\)\( T^{19} + \)\(48\!\cdots\!98\)\( T^{20} - \)\(33\!\cdots\!04\)\( T^{21} + \)\(21\!\cdots\!78\)\( T^{22} - \)\(12\!\cdots\!16\)\( T^{23} + \)\(65\!\cdots\!57\)\( T^{24} - \)\(28\!\cdots\!70\)\( T^{25} + \)\(11\!\cdots\!72\)\( T^{26} - \)\(28\!\cdots\!11\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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