Properties

Label 6025.2.a.g
Level 6025
Weight 2
Character orbit 6025.a
Self dual Yes
Analytic conductor 48.110
Analytic rank 0
Dimension 11
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{2} \) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{3} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{6} \) \( + ( 1 + \beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} ) q^{7} \) \( + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{8} \) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{2} \) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{3} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{6} \) \( + ( 1 + \beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} ) q^{7} \) \( + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{8} \) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{9} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{11} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} ) q^{12} \) \( + ( 1 - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{13} \) \( + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{14} \) \( + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{16} \) \( + ( 1 - \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{17} \) \( + ( 2 - 3 \beta_{1} + 3 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} - 2 \beta_{10} ) q^{18} \) \( + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{10} ) q^{19} \) \( + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{21} \) \( + ( -1 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{22} \) \( + ( 3 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{23} \) \( + ( 3 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{24} \) \( + ( -2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{9} + \beta_{10} ) q^{26} \) \( + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{27} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{28} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + \beta_{8} + \beta_{9} ) q^{29} \) \( + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{31} \) \( + ( -2 + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{32} \) \( + ( 2 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{33} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + 3 \beta_{10} ) q^{34} \) \( + ( 5 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} ) q^{36} \) \( + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{8} ) q^{37} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{38} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{39} \) \( + ( 2 - 4 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{41} \) \( + ( 2 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - 4 \beta_{10} ) q^{42} \) \( + ( 2 + 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{43} \) \( + ( -\beta_{2} + \beta_{4} + \beta_{9} ) q^{44} \) \( + ( 1 - 3 \beta_{1} + 2 \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{46} \) \( + ( 3 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} ) q^{47} \) \( + ( -2 - 4 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} - 3 \beta_{10} ) q^{48} \) \( + ( 1 - \beta_{1} + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{49} \) \( + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{51} \) \( + ( 2 \beta_{1} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{52} \) \( + ( -3 + \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{9} ) q^{53} \) \( + ( 3 - 5 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{54} \) \( + ( 4 - 4 \beta_{1} + 3 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{56} \) \( + ( -\beta_{2} - 7 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{57} \) \( + ( 3 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{58} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{59} \) \( + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{61} \) \( + ( 3 + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{62} \) \( + ( 3 - \beta_{1} - \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{63} \) \( + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{64} \) \( + ( -4 - 5 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{66} \) \( + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{67} \) \( + ( 1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} + 2 \beta_{10} ) q^{68} \) \( + ( 2 \beta_{1} + \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{69} \) \( + ( -3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{9} ) q^{71} \) \( + ( 8 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} - 3 \beta_{8} - \beta_{10} ) q^{72} \) \( + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{73} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{74} \) \( + ( -3 + 4 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{76} \) \( + ( 2 + \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + 2 \beta_{10} ) q^{77} \) \( + ( 3 + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 4 \beta_{9} + 3 \beta_{10} ) q^{78} \) \( + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{79} \) \( + ( 5 + 3 \beta_{1} + 5 \beta_{3} - \beta_{4} + 5 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} ) q^{81} \) \( + ( 6 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{82} \) \( + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{83} \) \( + ( 7 - 5 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} - 7 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 6 \beta_{10} ) q^{84} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{10} ) q^{86} \) \( + ( 1 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{87} \) \( + ( 1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{88} \) \( + ( 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{89} \) \( + ( -3 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 4 \beta_{9} + 3 \beta_{10} ) q^{91} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{9} - \beta_{10} ) q^{92} \) \( + ( 3 - \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} ) q^{93} \) \( + ( -1 + \beta_{1} - \beta_{2} - 5 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + 3 \beta_{10} ) q^{94} \) \( + ( -1 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{96} \) \( + ( 3 - 5 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} - 3 \beta_{8} ) q^{97} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} + 5 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{8} - 3 \beta_{9} - 3 \beta_{10} ) q^{98} \) \( + ( 2 + 7 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - 4 \beta_{6} + \beta_{7} - 4 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(11q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 3q^{11} \) \(\mathstrut +\mathstrut 28q^{12} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut -\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 6q^{18} \) \(\mathstrut -\mathstrut 33q^{19} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 6q^{22} \) \(\mathstrut +\mathstrut 31q^{23} \) \(\mathstrut +\mathstrut 32q^{24} \) \(\mathstrut -\mathstrut 20q^{26} \) \(\mathstrut +\mathstrut 32q^{27} \) \(\mathstrut +\mathstrut q^{28} \) \(\mathstrut +\mathstrut q^{29} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 7q^{32} \) \(\mathstrut +\mathstrut 35q^{33} \) \(\mathstrut +\mathstrut 9q^{34} \) \(\mathstrut +\mathstrut 33q^{36} \) \(\mathstrut +\mathstrut 23q^{37} \) \(\mathstrut -\mathstrut 20q^{38} \) \(\mathstrut +\mathstrut 14q^{39} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut +\mathstrut 26q^{42} \) \(\mathstrut +\mathstrut 19q^{43} \) \(\mathstrut +\mathstrut 6q^{46} \) \(\mathstrut +\mathstrut 35q^{47} \) \(\mathstrut -\mathstrut 16q^{48} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 3q^{51} \) \(\mathstrut +\mathstrut 3q^{52} \) \(\mathstrut -\mathstrut 14q^{53} \) \(\mathstrut +\mathstrut 9q^{54} \) \(\mathstrut +\mathstrut 33q^{56} \) \(\mathstrut -\mathstrut q^{57} \) \(\mathstrut +\mathstrut 11q^{58} \) \(\mathstrut -\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut 9q^{61} \) \(\mathstrut +\mathstrut 23q^{62} \) \(\mathstrut +\mathstrut 31q^{63} \) \(\mathstrut +\mathstrut 18q^{64} \) \(\mathstrut -\mathstrut 36q^{66} \) \(\mathstrut +\mathstrut 54q^{67} \) \(\mathstrut -\mathstrut q^{68} \) \(\mathstrut +\mathstrut 17q^{69} \) \(\mathstrut -\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 64q^{72} \) \(\mathstrut -\mathstrut 17q^{73} \) \(\mathstrut +\mathstrut 8q^{74} \) \(\mathstrut -\mathstrut 31q^{76} \) \(\mathstrut +\mathstrut 18q^{77} \) \(\mathstrut -\mathstrut 15q^{78} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 43q^{81} \) \(\mathstrut +\mathstrut 61q^{82} \) \(\mathstrut +\mathstrut 29q^{83} \) \(\mathstrut +\mathstrut 69q^{84} \) \(\mathstrut +\mathstrut 5q^{86} \) \(\mathstrut -\mathstrut 5q^{87} \) \(\mathstrut +\mathstrut 14q^{88} \) \(\mathstrut -\mathstrut 5q^{89} \) \(\mathstrut -\mathstrut 54q^{91} \) \(\mathstrut +\mathstrut 6q^{92} \) \(\mathstrut +\mathstrut 25q^{93} \) \(\mathstrut -\mathstrut 19q^{94} \) \(\mathstrut +\mathstrut 9q^{96} \) \(\mathstrut -\mathstrut 6q^{97} \) \(\mathstrut +\mathstrut 29q^{98} \) \(\mathstrut +\mathstrut 60q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(2\) \(x^{10}\mathstrut -\mathstrut \) \(11\) \(x^{9}\mathstrut +\mathstrut \) \(15\) \(x^{8}\mathstrut +\mathstrut \) \(43\) \(x^{7}\mathstrut -\mathstrut \) \(28\) \(x^{6}\mathstrut -\mathstrut \) \(62\) \(x^{5}\mathstrut +\mathstrut \) \(14\) \(x^{4}\mathstrut +\mathstrut \) \(31\) \(x^{3}\mathstrut +\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(5\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{10} - 2 \nu^{9} - 11 \nu^{8} + 15 \nu^{7} + 43 \nu^{6} - 28 \nu^{5} - 62 \nu^{4} + 14 \nu^{3} + 31 \nu^{2} + \nu - 5 \)
\(\beta_{3}\)\(=\)\( -4 \nu^{10} + 9 \nu^{9} + 42 \nu^{8} - 71 \nu^{7} - 157 \nu^{6} + 155 \nu^{5} + 220 \nu^{4} - 118 \nu^{3} - 109 \nu^{2} + 26 \nu + 17 \)
\(\beta_{4}\)\(=\)\( 4 \nu^{10} - 9 \nu^{9} - 42 \nu^{8} + 71 \nu^{7} + 157 \nu^{6} - 155 \nu^{5} - 220 \nu^{4} + 118 \nu^{3} + 110 \nu^{2} - 27 \nu - 19 \)
\(\beta_{5}\)\(=\)\( 8 \nu^{10} - 19 \nu^{9} - 80 \nu^{8} + 148 \nu^{7} + 280 \nu^{6} - 315 \nu^{5} - 349 \nu^{4} + 221 \nu^{3} + 132 \nu^{2} - 37 \nu - 17 \)
\(\beta_{6}\)\(=\)\( -9 \nu^{10} + 23 \nu^{9} + 88 \nu^{8} - 187 \nu^{7} - 303 \nu^{6} + 441 \nu^{5} + 390 \nu^{4} - 366 \nu^{3} - 174 \nu^{2} + 78 \nu + 30 \)
\(\beta_{7}\)\(=\)\( 13 \nu^{10} - 30 \nu^{9} - 133 \nu^{8} + 235 \nu^{7} + 477 \nu^{6} - 505 \nu^{5} - 611 \nu^{4} + 367 \nu^{3} + 242 \nu^{2} - 68 \nu - 32 \)
\(\beta_{8}\)\(=\)\( 14 \nu^{10} - 34 \nu^{9} - 140 \nu^{8} + 271 \nu^{7} + 492 \nu^{6} - 609 \nu^{5} - 630 \nu^{4} + 471 \nu^{3} + 261 \nu^{2} - 93 \nu - 39 \)
\(\beta_{9}\)\(=\)\( 18 \nu^{10} - 43 \nu^{9} - 182 \nu^{8} + 342 \nu^{7} + 649 \nu^{6} - 764 \nu^{5} - 849 \nu^{4} + 588 \nu^{3} + 364 \nu^{2} - 117 \nu - 53 \)
\(\beta_{10}\)\(=\)\( -24 \nu^{10} + 58 \nu^{9} + 240 \nu^{8} - 460 \nu^{7} - 843 \nu^{6} + 1023 \nu^{5} + 1074 \nu^{4} - 780 \nu^{3} - 435 \nu^{2} + 151 \nu + 61 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{5}\)\(=\)\(10\) \(\beta_{10}\mathstrut +\mathstrut \) \(11\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(40\) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{6}\)\(=\)\(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(27\) \(\beta_{9}\mathstrut -\mathstrut \) \(17\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut +\mathstrut \) \(50\) \(\beta_{4}\mathstrut +\mathstrut \) \(63\) \(\beta_{3}\mathstrut -\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(86\) \(\beta_{1}\mathstrut +\mathstrut \) \(67\)
\(\nu^{7}\)\(=\)\(87\) \(\beta_{10}\mathstrut +\mathstrut \) \(104\) \(\beta_{9}\mathstrut -\mathstrut \) \(52\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut -\mathstrut \) \(32\) \(\beta_{6}\mathstrut +\mathstrut \) \(89\) \(\beta_{5}\mathstrut +\mathstrut \) \(103\) \(\beta_{4}\mathstrut +\mathstrut \) \(144\) \(\beta_{3}\mathstrut -\mathstrut \) \(113\) \(\beta_{2}\mathstrut +\mathstrut \) \(290\) \(\beta_{1}\mathstrut +\mathstrut \) \(138\)
\(\nu^{8}\)\(=\)\(196\) \(\beta_{10}\mathstrut +\mathstrut \) \(283\) \(\beta_{9}\mathstrut -\mathstrut \) \(203\) \(\beta_{8}\mathstrut +\mathstrut \) \(52\) \(\beta_{7}\mathstrut -\mathstrut \) \(99\) \(\beta_{6}\mathstrut +\mathstrut \) \(211\) \(\beta_{5}\mathstrut +\mathstrut \) \(377\) \(\beta_{4}\mathstrut +\mathstrut \) \(516\) \(\beta_{3}\mathstrut -\mathstrut \) \(247\) \(\beta_{2}\mathstrut +\mathstrut \) \(711\) \(\beta_{1}\mathstrut +\mathstrut \) \(503\)
\(\nu^{9}\)\(=\)\(754\) \(\beta_{10}\mathstrut +\mathstrut \) \(950\) \(\beta_{9}\mathstrut -\mathstrut \) \(622\) \(\beta_{8}\mathstrut +\mathstrut \) \(203\) \(\beta_{7}\mathstrut -\mathstrut \) \(374\) \(\beta_{6}\mathstrut +\mathstrut \) \(791\) \(\beta_{5}\mathstrut +\mathstrut \) \(907\) \(\beta_{4}\mathstrut +\mathstrut \) \(1341\) \(\beta_{3}\mathstrut -\mathstrut \) \(893\) \(\beta_{2}\mathstrut +\mathstrut \) \(2209\) \(\beta_{1}\mathstrut +\mathstrut \) \(1232\)
\(\nu^{10}\)\(=\)\(1956\) \(\beta_{10}\mathstrut +\mathstrut \) \(2710\) \(\beta_{9}\mathstrut -\mathstrut \) \(2112\) \(\beta_{8}\mathstrut +\mathstrut \) \(622\) \(\beta_{7}\mathstrut -\mathstrut \) \(1155\) \(\beta_{6}\mathstrut +\mathstrut \) \(2122\) \(\beta_{5}\mathstrut +\mathstrut \) \(2963\) \(\beta_{4}\mathstrut +\mathstrut \) \(4332\) \(\beta_{3}\mathstrut -\mathstrut \) \(2248\) \(\beta_{2}\mathstrut +\mathstrut \) \(5815\) \(\beta_{1}\mathstrut +\mathstrut \) \(3987\)

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
−0.324121
2.91366
−0.385940
2.52552
−0.514683
1.27000
−1.19664
0.745266
−1.54550
0.582513
−2.07008
−1.76114 0.777505 1.10163 0 −1.36930 −0.840819 1.58216 −2.39549 0
1.2 −1.57045 3.37997 0.466308 0 −5.30806 2.36397 2.40858 8.42418 0
1.3 −1.20514 −0.933583 −0.547644 0 1.12510 2.09289 3.07026 −2.12842 0
1.4 −1.12957 1.80145 −0.724078 0 −2.03485 −2.49744 3.07703 0.245209 0
1.5 −0.428259 −2.33128 −1.81659 0 0.998390 3.85771 1.63449 2.43486 0
1.6 0.517395 −0.462298 −1.73230 0 −0.239191 3.59765 −1.93107 −2.78628 0
1.7 1.36096 −1.34441 −0.147774 0 −1.82970 −1.74128 −2.92305 −1.19255 0
1.8 1.59654 1.29420 0.548929 0 2.06623 2.14214 −2.31669 −1.32506 0
1.9 1.89846 0.0586609 1.60416 0 0.111366 −2.85624 −0.751482 −2.99656 0
1.10 2.13419 3.13727 2.55475 0 6.69551 −1.41692 1.18395 6.84244 0
1.11 2.58701 2.62253 4.69261 0 6.78451 4.29833 6.96581 3.87767 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

\(T_{2}^{11} - \cdots\)
\(T_{3}^{11} - \cdots\)