# Properties

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

# Learn more about

## Newspace parameters

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

## Newform invariants

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

## $q$-expansion

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

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

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

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

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.34497 2.14166 1.96562 1.92974 1.09783 1.07739 0.596683 −0.0849802 −0.227272 −0.324166 −1.00750 −1.39546 −1.93741 −2.06795 −2.10915
−2.34497 3.01587 3.49887 0 −7.07210 −1.32864 −3.51479 6.09545 0
1.2 −2.14166 0.668121 2.58670 0 −1.43089 3.26618 −1.25651 −2.55361 0
1.3 −1.96562 0.551502 1.86368 0 −1.08405 −1.56703 0.267954 −2.69585 0
1.4 −1.92974 −0.860854 1.72391 0 1.66123 4.27878 0.532780 −2.25893 0
1.5 −1.09783 −1.80555 −0.794772 0 1.98218 0.928281 3.06818 0.259998 0
1.6 −1.07739 −0.702822 −0.839239 0 0.757210 −0.460803 3.05896 −2.50604 0
1.7 −0.596683 3.22200 −1.64397 0 −1.92252 −0.0914365 2.17430 7.38130 0
1.8 0.0849802 1.30288 −1.99278 0 0.110719 −0.925444 −0.339307 −1.30250 0
1.9 0.227272 −1.31252 −1.94835 0 −0.298298 −4.34237 −0.897348 −1.27730 0
1.10 0.324166 1.44456 −1.89492 0 0.468276 −1.26380 −1.26260 −0.913251 0
1.11 1.00750 2.74497 −0.984941 0 2.76556 1.78457 −3.00733 4.53484 0
1.12 1.39546 −2.59963 −0.0526942 0 −3.62768 1.49578 −2.86445 3.75808 0
1.13 1.93741 1.04448 1.75357 0 2.02359 2.21578 −0.477443 −1.90906 0
1.14 2.06795 −1.49139 2.27640 0 −3.08411 2.20594 0.571582 −0.775761 0
1.15 2.10915 1.77838 2.44853 0 3.75087 −3.19580 0.946028 0.162626 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$241$$ $$1$$

## Inner twists

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

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{15} + \cdots$$ $$T_{3}^{15} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 17 T + 57 T^{2} + 89 T^{3} - 394 T^{4} - 297 T^{5} + 699 T^{6} + 437 T^{7} - 519 T^{8} - 296 T^{9} + 184 T^{10} + 99 T^{11} - 31 T^{12} - 16 T^{13} + 2 T^{14} + T^{15}$$
$3$ $$-191 + 447 T + 721 T^{2} - 2131 T^{3} - 611 T^{4} + 3617 T^{5} - 499 T^{6} - 2749 T^{7} + 949 T^{8} + 970 T^{9} - 484 T^{10} - 133 T^{11} + 100 T^{12} - T^{13} - 7 T^{14} + T^{15}$$
$5$ $$T^{15}$$
$7$ $$-241 - 3172 T - 5244 T^{2} + 8197 T^{3} + 14696 T^{4} - 8475 T^{5} - 14084 T^{6} + 5181 T^{7} + 6226 T^{8} - 1991 T^{9} - 1298 T^{10} + 416 T^{11} + 110 T^{12} - 36 T^{13} - 3 T^{14} + T^{15}$$
$11$ $$20041 + 66359 T - 46243 T^{2} - 266533 T^{3} - 7464 T^{4} + 338234 T^{5} + 74209 T^{6} - 150635 T^{7} - 50568 T^{8} + 18797 T^{9} + 8481 T^{10} - 527 T^{11} - 512 T^{12} - 25 T^{13} + 10 T^{14} + T^{15}$$
$13$ $$-17551 - 147967 T - 338543 T^{2} - 65839 T^{3} + 515260 T^{4} + 279384 T^{5} - 295781 T^{6} - 130656 T^{7} + 97584 T^{8} + 12886 T^{9} - 13380 T^{10} + 533 T^{11} + 600 T^{12} - 60 T^{13} - 8 T^{14} + T^{15}$$
$17$ $$-574859 - 320121 T + 8873889 T^{2} - 4178942 T^{3} - 13713675 T^{4} - 1949876 T^{5} + 4166233 T^{6} + 1301136 T^{7} - 318770 T^{8} - 142575 T^{9} + 8325 T^{10} + 6354 T^{11} - 31 T^{12} - 129 T^{13} - T^{14} + T^{15}$$
$19$ $$77731 + 809612 T + 2324421 T^{2} - 517493 T^{3} - 8108117 T^{4} - 2092049 T^{5} + 4154115 T^{6} + 2719031 T^{7} + 365509 T^{8} - 163928 T^{9} - 64268 T^{10} - 5311 T^{11} + 1245 T^{12} + 332 T^{13} + 30 T^{14} + T^{15}$$
$23$ $$250937 - 2404746 T + 5952482 T^{2} - 2008070 T^{3} - 7907583 T^{4} + 5596003 T^{5} + 2780084 T^{6} - 1782908 T^{7} - 465578 T^{8} + 190424 T^{9} + 42191 T^{10} - 6673 T^{11} - 1774 T^{12} + 19 T^{14} + T^{15}$$
$29$ $$116809 + 1444420 T + 2156742 T^{2} - 4967005 T^{3} - 5690636 T^{4} + 5846919 T^{5} + 2834122 T^{6} - 1479824 T^{7} - 587012 T^{8} + 115385 T^{9} + 50624 T^{10} - 1474 T^{11} - 1684 T^{12} - 96 T^{13} + 12 T^{14} + T^{15}$$
$31$ $$-21531245891 - 28963912421 T - 7277745624 T^{2} + 6580310295 T^{3} + 4293208883 T^{4} + 527325092 T^{5} - 224629774 T^{6} - 69316528 T^{7} - 257675 T^{8} + 2111523 T^{9} + 199627 T^{10} - 20949 T^{11} - 3857 T^{12} - 25 T^{13} + 22 T^{14} + T^{15}$$
$37$ $$-3773372033 + 6957819995 T - 2738896827 T^{2} - 1870786789 T^{3} + 1411958548 T^{4} + 43867218 T^{5} - 185888905 T^{6} + 14263131 T^{7} + 9871186 T^{8} - 931590 T^{9} - 244893 T^{10} + 22274 T^{11} + 2804 T^{12} - 241 T^{13} - 12 T^{14} + T^{15}$$
$41$ $$-8686116791 + 25706705 T + 6982555384 T^{2} - 3946869 T^{3} - 1963303989 T^{4} - 1752142 T^{5} + 241231053 T^{6} + 3919347 T^{7} - 13848678 T^{8} - 551203 T^{9} + 364207 T^{10} + 22080 T^{11} - 3833 T^{12} - 274 T^{13} + 13 T^{14} + T^{15}$$
$43$ $$-336240191 - 1118496962 T + 1944490637 T^{2} - 230184430 T^{3} - 700690434 T^{4} + 227236965 T^{5} + 69195241 T^{6} - 35514290 T^{7} - 68267 T^{8} + 1675073 T^{9} - 148041 T^{10} - 25875 T^{11} + 3980 T^{12} + 19 T^{13} - 25 T^{14} + T^{15}$$
$47$ $$-176160469 - 773316628 T - 932975726 T^{2} + 39726897 T^{3} + 627245293 T^{4} + 254029038 T^{5} - 62513397 T^{6} - 53471588 T^{7} - 6003789 T^{8} + 1737709 T^{9} + 379125 T^{10} - 1929 T^{11} - 4829 T^{12} - 213 T^{13} + 16 T^{14} + T^{15}$$
$53$ $$-27449089409 + 46420605845 T - 12947198063 T^{2} - 9178972167 T^{3} + 4460444524 T^{4} + 340626657 T^{5} - 416308316 T^{6} + 23802894 T^{7} + 15487646 T^{8} - 1710840 T^{9} - 258136 T^{10} + 37343 T^{11} + 1829 T^{12} - 331 T^{13} - 4 T^{14} + T^{15}$$
$59$ $$4073980277 - 7992475897 T - 13867347235 T^{2} + 8771931181 T^{3} + 6949750202 T^{4} - 1353634751 T^{5} - 1038795356 T^{6} - 7377568 T^{7} + 47280836 T^{8} + 4065108 T^{9} - 700368 T^{10} - 110295 T^{11} - 209 T^{12} + 771 T^{13} + 50 T^{14} + T^{15}$$
$61$ $$-416282129347 - 678098169085 T - 185142686302 T^{2} + 97067002349 T^{3} + 43626834819 T^{4} - 1842137971 T^{5} - 2870977746 T^{6} - 243563701 T^{7} + 63595228 T^{8} + 11018266 T^{9} - 129888 T^{10} - 131627 T^{11} - 6871 T^{12} + 351 T^{13} + 41 T^{14} + T^{15}$$
$67$ $$-602438539877 + 300283480954 T + 138660366045 T^{2} - 76008374965 T^{3} - 12123345606 T^{4} + 7334025707 T^{5} + 488828325 T^{6} - 343963403 T^{7} - 5757617 T^{8} + 8374646 T^{9} - 183371 T^{10} - 100588 T^{11} + 5722 T^{12} + 407 T^{13} - 43 T^{14} + T^{15}$$
$71$ $$236049251731 - 307080454775 T + 45530160689 T^{2} + 66050096733 T^{3} - 15674539666 T^{4} - 5255528626 T^{5} + 1238988951 T^{6} + 194332209 T^{7} - 40133269 T^{8} - 4077242 T^{9} + 630444 T^{10} + 51027 T^{11} - 4782 T^{12} - 351 T^{13} + 14 T^{14} + T^{15}$$
$73$ $$416966690521 + 1397517898998 T - 421898973955 T^{2} - 460395694842 T^{3} + 172754523356 T^{4} + 12624963431 T^{5} - 9830267893 T^{6} + 350909714 T^{7} + 186927996 T^{8} - 12609062 T^{9} - 1604984 T^{10} + 133258 T^{11} + 6469 T^{12} - 601 T^{13} - 10 T^{14} + T^{15}$$
$79$ $$-1484049491 + 1325530706 T + 6921356479 T^{2} - 2271667975 T^{3} - 7268360498 T^{4} - 3309724089 T^{5} - 294317454 T^{6} + 157654518 T^{7} + 43678545 T^{8} + 1853596 T^{9} - 694144 T^{10} - 99115 T^{11} - 1525 T^{12} + 575 T^{13} + 44 T^{14} + T^{15}$$
$83$ $$43476792713 - 47425742843 T - 182476469251 T^{2} + 63847221236 T^{3} + 46672086160 T^{4} - 16446086450 T^{5} - 2411914424 T^{6} + 1038480784 T^{7} + 13876991 T^{8} - 20756309 T^{9} + 333119 T^{10} + 176294 T^{11} - 3235 T^{12} - 690 T^{13} + 7 T^{14} + T^{15}$$
$89$ $$2543711203 - 15039262155 T - 11139090057 T^{2} + 29431433868 T^{3} - 1863608481 T^{4} - 5920891733 T^{5} + 241273554 T^{6} + 385699757 T^{7} - 3341523 T^{8} - 9999148 T^{9} - 51975 T^{10} + 114463 T^{11} + 1077 T^{12} - 570 T^{13} - 4 T^{14} + T^{15}$$
$97$ $$3592948942717 - 3150077125084 T - 145904463222 T^{2} + 452074119218 T^{3} - 15308878746 T^{4} - 25989696058 T^{5} + 1294308146 T^{6} + 771986244 T^{7} - 38426924 T^{8} - 12734793 T^{9} + 543175 T^{10} + 116371 T^{11} - 3629 T^{12} - 544 T^{13} + 9 T^{14} + T^{15}$$
show more
show less