# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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