Properties

Label 25.2.e.a
Level 25
Weight 2
Character orbit 25.e
Analytic conductor 0.200
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 25 = 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 25.e (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.199626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.58140625.2
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{1} ) q^{2} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{3} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{4} \) \( + ( 1 - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{5} \) \( + ( -2 - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{6} \) \( + ( -1 + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{7} \) \( + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{8} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} ) q^{2} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{3} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{4} \) \( + ( 1 - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{5} \) \( + ( -2 - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{6} \) \( + ( -1 + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{7} \) \( + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{8} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{9} \) \( + ( -1 + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{10} \) \( -2 \beta_{4} q^{11} \) \( + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{12} \) \( + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{13} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{14} \) \( + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{15} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{6} + \beta_{7} ) q^{16} \) \( + ( 1 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{17} \) \( + ( 1 - \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{7} ) q^{18} \) \( + ( -2 - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} \) \( + ( -2 + \beta_{2} + 3 \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{20} \) \( + ( 2 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{21} \) \( + ( 2 \beta_{4} + 2 \beta_{5} ) q^{22} \) \( + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} \) \( + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{24} \) \( + ( -3 \beta_{2} + \beta_{5} + 2 \beta_{7} ) q^{25} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{26} \) \( + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{27} \) \( + ( -3 - 3 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{7} ) q^{28} \) \( + ( -3 + 4 \beta_{1} + 5 \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{29} \) \( + ( 3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{30} \) \( + ( \beta_{1} + 4 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{31} \) \( + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - \beta_{6} - \beta_{7} ) q^{32} \) \( + ( 2 \beta_{1} + 2 \beta_{6} ) q^{33} \) \( + ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{34} \) \( + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{35} \) \( + ( 4 + \beta_{1} - 3 \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{36} \) \( + ( 2 + \beta_{1} - 4 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{37} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{38} \) \( + ( -1 - \beta_{1} + 3 \beta_{4} - 2 \beta_{7} ) q^{39} \) \( + ( 3 + \beta_{1} + 3 \beta_{3} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{40} \) \( + ( -1 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} \) \( + ( -2 + \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{42} \) \( + ( 3 + \beta_{1} - \beta_{2} + 5 \beta_{3} - 6 \beta_{4} - \beta_{6} + \beta_{7} ) q^{43} \) \( + ( -2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{44} \) \( + ( -3 + 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{45} \) \( + ( -3 + 2 \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{46} \) \( + ( \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{47} \) \( + ( -3 - 3 \beta_{2} + \beta_{5} + \beta_{6} ) q^{48} \) \( + ( 4 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{49} \) \( + ( -4 - 3 \beta_{1} - 4 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{50} \) \( + ( -4 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{51} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{52} \) \( + ( -2 + 2 \beta_{2} + \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} ) q^{53} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{54} \) \( + ( -4 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{55} \) \( + ( 2 - \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{56} \) \( + ( 1 + \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{57} \) \( + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{58} \) \( + ( 3 - \beta_{1} - 4 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} \) \( + ( -3 \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{60} \) \( + ( -2 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{61} \) \( + ( -3 - \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{62} \) \( + ( 2 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{63} \) \( + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 9 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{64} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{65} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{66} \) \( + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{67} \) \( + ( -4 + 4 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{6} + 4 \beta_{7} ) q^{68} \) \( + ( 4 - \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{69} \) \( + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{70} \) \( + ( -4 + 6 \beta_{1} + \beta_{2} - 6 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{71} \) \( + ( 1 - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} ) q^{72} \) \( + ( -2 + 3 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{73} \) \( + ( -3 + \beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{74} \) \( + ( 5 \beta_{1} - \beta_{2} + 5 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{75} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{76} \) \( + ( -2 + 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{77} \) \( + ( 5 - \beta_{2} + \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{78} \) \( + ( 7 - 6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 8 \beta_{6} + \beta_{7} ) q^{79} \) \( + ( 1 + 4 \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{6} ) q^{80} \) \( + ( 4 - 2 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} ) q^{81} \) \( + ( -1 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{82} \) \( + ( -5 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{83} \) \( + ( -1 + \beta_{1} + 6 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{84} \) \( + ( -3 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{85} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} + 5 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{86} \) \( + ( -3 - \beta_{1} - 6 \beta_{2} + \beta_{3} + 5 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{87} \) \( + ( -2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{88} \) \( + ( -6 - \beta_{1} + 5 \beta_{2} + 7 \beta_{4} - 2 \beta_{7} ) q^{89} \) \( + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} ) q^{90} \) \( + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{91} \) \( + ( 6 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{92} \) \( + ( -5 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} + \beta_{4} + 4 \beta_{6} - 5 \beta_{7} ) q^{93} \) \( + ( -4 + \beta_{1} + 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{94} \) \( + ( 5 - 6 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{95} \) \( + ( 6 - 6 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} - 3 \beta_{5} - 6 \beta_{6} ) q^{96} \) \( + ( -12 + 4 \beta_{2} - 9 \beta_{3} + 7 \beta_{4} + 4 \beta_{5} - 4 \beta_{7} ) q^{97} \) \( + ( 4 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{98} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut 9q^{6} \) \(\mathstrut +\mathstrut 10q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut 9q^{6} \) \(\mathstrut +\mathstrut 10q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 15q^{12} \) \(\mathstrut -\mathstrut 5q^{13} \) \(\mathstrut +\mathstrut 13q^{14} \) \(\mathstrut +\mathstrut 15q^{15} \) \(\mathstrut +\mathstrut 3q^{16} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut -\mathstrut 5q^{19} \) \(\mathstrut -\mathstrut 15q^{20} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 5q^{23} \) \(\mathstrut -\mathstrut 20q^{24} \) \(\mathstrut -\mathstrut 10q^{25} \) \(\mathstrut +\mathstrut 6q^{26} \) \(\mathstrut -\mathstrut 5q^{27} \) \(\mathstrut -\mathstrut 15q^{28} \) \(\mathstrut -\mathstrut 5q^{29} \) \(\mathstrut +\mathstrut 15q^{30} \) \(\mathstrut -\mathstrut 9q^{31} \) \(\mathstrut +\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 13q^{34} \) \(\mathstrut +\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut 23q^{36} \) \(\mathstrut +\mathstrut 30q^{37} \) \(\mathstrut +\mathstrut 15q^{38} \) \(\mathstrut -\mathstrut 3q^{39} \) \(\mathstrut +\mathstrut 10q^{40} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 15q^{42} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 15q^{45} \) \(\mathstrut -\mathstrut 19q^{46} \) \(\mathstrut -\mathstrut 30q^{48} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut -\mathstrut 15q^{50} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 10q^{52} \) \(\mathstrut -\mathstrut 10q^{53} \) \(\mathstrut -\mathstrut 5q^{54} \) \(\mathstrut -\mathstrut 10q^{55} \) \(\mathstrut +\mathstrut 10q^{56} \) \(\mathstrut +\mathstrut 20q^{58} \) \(\mathstrut -\mathstrut 10q^{60} \) \(\mathstrut -\mathstrut 9q^{61} \) \(\mathstrut -\mathstrut 30q^{62} \) \(\mathstrut +\mathstrut 10q^{63} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut +\mathstrut 5q^{65} \) \(\mathstrut +\mathstrut 12q^{66} \) \(\mathstrut +\mathstrut 20q^{67} \) \(\mathstrut +\mathstrut 17q^{69} \) \(\mathstrut +\mathstrut 30q^{70} \) \(\mathstrut +\mathstrut 6q^{71} \) \(\mathstrut +\mathstrut 5q^{72} \) \(\mathstrut +\mathstrut 15q^{73} \) \(\mathstrut -\mathstrut 12q^{74} \) \(\mathstrut -\mathstrut 10q^{75} \) \(\mathstrut -\mathstrut 20q^{76} \) \(\mathstrut +\mathstrut 10q^{77} \) \(\mathstrut +\mathstrut 25q^{78} \) \(\mathstrut +\mathstrut 15q^{79} \) \(\mathstrut +\mathstrut 20q^{80} \) \(\mathstrut +\mathstrut 28q^{81} \) \(\mathstrut -\mathstrut 45q^{83} \) \(\mathstrut +\mathstrut 18q^{84} \) \(\mathstrut -\mathstrut 15q^{85} \) \(\mathstrut -\mathstrut 9q^{86} \) \(\mathstrut -\mathstrut 20q^{87} \) \(\mathstrut -\mathstrut 20q^{88} \) \(\mathstrut -\mathstrut 25q^{89} \) \(\mathstrut -\mathstrut 25q^{90} \) \(\mathstrut +\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 30q^{92} \) \(\mathstrut -\mathstrut 27q^{94} \) \(\mathstrut +\mathstrut 15q^{95} \) \(\mathstrut +\mathstrut 16q^{96} \) \(\mathstrut -\mathstrut 60q^{97} \) \(\mathstrut -\mathstrut 10q^{98} \) \(\mathstrut -\mathstrut 8q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut +\mathstrut \) \(4\) \(x^{6}\mathstrut -\mathstrut \) \(7\) \(x^{5}\mathstrut +\mathstrut \) \(11\) \(x^{4}\mathstrut +\mathstrut \) \(5\) \(x^{3}\mathstrut -\mathstrut \) \(10\) \(x^{2}\mathstrut -\mathstrut \) \(25\) \(x\mathstrut +\mathstrut \) \(25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800 \)\()/1355\)
\(\beta_{3}\)\(=\)\((\)\( 420 \nu^{7} - 776 \nu^{6} + 698 \nu^{5} - 1924 \nu^{4} + 2297 \nu^{3} + 5129 \nu^{2} + 1055 \nu - 10265 \)\()/1355\)
\(\beta_{4}\)\(=\)\((\)\( 728 \nu^{7} - 1327 \nu^{6} + 1246 \nu^{5} - 3353 \nu^{4} + 3584 \nu^{3} + 8547 \nu^{2} + 2190 \nu - 15715 \)\()/1355\)
\(\beta_{5}\)\(=\)\((\)\( -857 \nu^{7} + 1666 \nu^{6} - 1743 \nu^{5} + 4424 \nu^{4} - 4907 \nu^{3} - 9470 \nu^{2} - 2485 \nu + 18200 \)\()/1355\)
\(\beta_{6}\)\(=\)\((\)\( 891 \nu^{7} - 1623 \nu^{6} + 1624 \nu^{5} - 4492 \nu^{4} + 4991 \nu^{3} + 9520 \nu^{2} + 3235 \nu - 18960 \)\()/1355\)
\(\beta_{7}\)\(=\)\((\)\( 955 \nu^{7} - 1829 \nu^{6} + 1942 \nu^{5} - 4891 \nu^{4} + 5723 \nu^{3} + 9646 \nu^{2} + 2415 \nu - 20550 \)\()/1355\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(5\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{5}\)\(=\)\(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(13\)
\(\nu^{6}\)\(=\)\(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut -\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(21\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(21\)
\(\nu^{7}\)\(=\)\(23\) \(\beta_{7}\mathstrut +\mathstrut \) \(38\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\) \(\beta_{4}\mathstrut -\mathstrut \) \(23\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−0.983224 0.644389i
1.17421 + 0.0566033i
−0.357358 + 1.86824i
1.66637 0.917186i
−0.357358 1.86824i
1.66637 + 0.917186i
−0.983224 + 0.644389i
1.17421 0.0566033i
−1.98322 0.644389i 1.29224 1.77862i 1.89991 + 1.38036i −1.22570 + 1.87020i −3.70892 + 2.69469i 0.992398i −0.427051 0.587785i −0.566541 1.74363i 3.63597 2.91920i
4.2 0.174207 + 0.0566033i −0.865190 + 1.19083i −1.59089 1.15585i 0.107666 2.23347i −0.218127 + 0.158479i 3.26086i −0.427051 0.587785i 0.257524 + 0.792578i 0.145178 0.382993i
9.1 −1.35736 + 1.86824i −0.451659 + 0.146753i −1.02988 3.16963i 2.19625 + 0.420099i 0.338893 1.04301i 3.03582i 2.92705 + 0.951057i −2.24459 + 1.63079i −3.76594 + 3.53290i
9.2 0.666375 0.917186i −2.47539 + 0.804303i 0.220859 + 0.679734i −1.07822 1.95894i −0.911842 + 2.80636i 0.407162i 2.92705 + 0.951057i 3.05361 2.21858i −2.51521 0.316463i
14.1 −1.35736 1.86824i −0.451659 0.146753i −1.02988 + 3.16963i 2.19625 0.420099i 0.338893 + 1.04301i 3.03582i 2.92705 0.951057i −2.24459 1.63079i −3.76594 3.53290i
14.2 0.666375 + 0.917186i −2.47539 0.804303i 0.220859 0.679734i −1.07822 + 1.95894i −0.911842 2.80636i 0.407162i 2.92705 0.951057i 3.05361 + 2.21858i −2.51521 + 0.316463i
19.1 −1.98322 + 0.644389i 1.29224 + 1.77862i 1.89991 1.38036i −1.22570 1.87020i −3.70892 2.69469i 0.992398i −0.427051 + 0.587785i −0.566541 + 1.74363i 3.63597 + 2.91920i
19.2 0.174207 0.0566033i −0.865190 1.19083i −1.59089 + 1.15585i 0.107666 + 2.23347i −0.218127 0.158479i 3.26086i −0.427051 + 0.587785i 0.257524 0.792578i 0.145178 + 0.382993i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
25.e Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(25, [\chi])\).