Properties

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

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

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

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

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

Embeddings

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

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

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

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

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

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