Properties

Label 4010.2.a.h
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(2\) \(x^{8}\mathstrut -\mathstrut \) \(8\) \(x^{7}\mathstrut +\mathstrut \) \(16\) \(x^{6}\mathstrut +\mathstrut \) \(17\) \(x^{5}\mathstrut -\mathstrut \) \(36\) \(x^{4}\mathstrut -\mathstrut \) \(4\) \(x^{3}\mathstrut +\mathstrut \) \(17\) \(x^{2}\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + 3 \nu + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + \nu^{3} + 5 \nu^{2} - 3 \nu - 3 \)
\(\beta_{4}\)\(=\)\( -\nu^{5} + \nu^{4} + 5 \nu^{3} - 4 \nu^{2} - 4 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{7} - \nu^{6} - 8 \nu^{5} + 8 \nu^{4} + 17 \nu^{3} - 17 \nu^{2} - 5 \nu + 3 \)
\(\beta_{6}\)\(=\)\( \nu^{8} - \nu^{7} - 9 \nu^{6} + 7 \nu^{5} + 25 \nu^{4} - 13 \nu^{3} - 21 \nu^{2} + 4 \nu + 4 \)
\(\beta_{7}\)\(=\)\( \nu^{8} - 2 \nu^{7} - 8 \nu^{6} + 16 \nu^{5} + 16 \nu^{4} - 34 \nu^{3} + 9 \nu \)
\(\beta_{8}\)\(=\)\( \nu^{8} - 2 \nu^{7} - 7 \nu^{6} + 15 \nu^{5} + 10 \nu^{4} - 30 \nu^{3} + 9 \nu^{2} + 8 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{5}\)\(=\)\(6\) \(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)
\(\nu^{6}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(21\)
\(\nu^{7}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(30\) \(\beta_{7}\mathstrut -\mathstrut \) \(31\) \(\beta_{6}\mathstrut +\mathstrut \) \(32\) \(\beta_{5}\mathstrut +\mathstrut \) \(22\) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(73\) \(\beta_{1}\mathstrut -\mathstrut \) \(19\)
\(\nu^{8}\)\(=\)\(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(39\) \(\beta_{7}\mathstrut -\mathstrut \) \(48\) \(\beta_{6}\mathstrut +\mathstrut \) \(50\) \(\beta_{5}\mathstrut +\mathstrut \) \(38\) \(\beta_{4}\mathstrut +\mathstrut \) \(56\) \(\beta_{3}\mathstrut +\mathstrut \) \(94\) \(\beta_{2}\mathstrut +\mathstrut \) \(49\) \(\beta_{1}\mathstrut +\mathstrut \) \(80\)

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
−1.87174
−0.589646
0.446506
2.07126
2.23442
1.34139
0.793701
−0.408238
−2.01764
1.00000 −2.53021 1.00000 1.00000 −2.53021 1.34700 1.00000 3.40195 1.00000
1.2 1.00000 −2.18512 1.00000 1.00000 −2.18512 −0.791922 1.00000 1.77477 1.00000
1.3 1.00000 −1.84294 1.00000 1.00000 −1.84294 −0.0938987 1.00000 0.396437 1.00000
1.4 1.00000 −0.922780 1.00000 1.00000 −0.922780 −2.45091 1.00000 −2.14848 1.00000
1.5 1.00000 −0.375168 1.00000 1.00000 −0.375168 3.43840 1.00000 −2.85925 1.00000
1.6 1.00000 0.453213 1.00000 1.00000 0.453213 −1.57101 1.00000 −2.79460 1.00000
1.7 1.00000 0.706772 1.00000 1.00000 0.706772 −4.28122 1.00000 −2.50047 1.00000
1.8 1.00000 1.13041 1.00000 1.00000 1.13041 −0.772637 1.00000 −1.72217 1.00000
1.9 1.00000 1.56583 1.00000 1.00000 1.56583 −1.82380 1.00000 −0.548185 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(401\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)