Properties

Label 8025.2.a.bc
Level 8025
Weight 2
Character orbit 8025.a
Self dual Yes
Analytic conductor 64.080
Analytic rank 1
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0799476221\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(13\) \(x^{8}\mathstrut +\mathstrut \) \(26\) \(x^{7}\mathstrut +\mathstrut \) \(51\) \(x^{6}\mathstrut -\mathstrut \) \(101\) \(x^{5}\mathstrut -\mathstrut \) \(65\) \(x^{4}\mathstrut +\mathstrut \) \(126\) \(x^{3}\mathstrut +\mathstrut \) \(5\) \(x^{2}\mathstrut -\mathstrut \) \(10\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{7} - \nu^{6} - 11 \nu^{5} + 10 \nu^{4} + 31 \nu^{3} - 20 \nu^{2} - 21 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{9} - \nu^{8} - 14 \nu^{7} + 14 \nu^{6} + 62 \nu^{5} - 60 \nu^{4} - 94 \nu^{3} + 83 \nu^{2} + 21 \nu - 6 \)
\(\beta_{5}\)\(=\)\( \nu^{9} - \nu^{8} - 14 \nu^{7} + 14 \nu^{6} + 62 \nu^{5} - 60 \nu^{4} - 95 \nu^{3} + 83 \nu^{2} + 26 \nu - 6 \)
\(\beta_{6}\)\(=\)\( -\nu^{9} + 2 \nu^{8} + 13 \nu^{7} - 25 \nu^{6} - 52 \nu^{5} + 91 \nu^{4} + 75 \nu^{3} - 104 \nu^{2} - 25 \nu + 6 \)
\(\beta_{7}\)\(=\)\( \nu^{9} - 2 \nu^{8} - 14 \nu^{7} + 27 \nu^{6} + 62 \nu^{5} - 111 \nu^{4} - 96 \nu^{3} + 145 \nu^{2} + 26 \nu - 7 \)
\(\beta_{8}\)\(=\)\( 2 \nu^{9} - 4 \nu^{8} - 26 \nu^{7} + 51 \nu^{6} + 104 \nu^{5} - 192 \nu^{4} - 150 \nu^{3} + 230 \nu^{2} + 51 \nu - 14 \)
\(\beta_{9}\)\(=\)\( 5 \nu^{9} - 9 \nu^{8} - 66 \nu^{7} + 116 \nu^{6} + 269 \nu^{5} - 443 \nu^{4} - 385 \nu^{3} + 536 \nu^{2} + 108 \nu - 29 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(38\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(86\)
\(\nu^{7}\)\(=\)\(12\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut -\mathstrut \) \(79\) \(\beta_{5}\mathstrut +\mathstrut \) \(79\) \(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut -\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(184\) \(\beta_{1}\mathstrut -\mathstrut \) \(16\)
\(\nu^{8}\)\(=\)\(79\) \(\beta_{9}\mathstrut -\mathstrut \) \(66\) \(\beta_{8}\mathstrut -\mathstrut \) \(80\) \(\beta_{7}\mathstrut +\mathstrut \) \(105\) \(\beta_{6}\mathstrut -\mathstrut \) \(77\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(79\) \(\beta_{3}\mathstrut +\mathstrut \) \(250\) \(\beta_{2}\mathstrut -\mathstrut \) \(97\) \(\beta_{1}\mathstrut +\mathstrut \) \(538\)
\(\nu^{9}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(106\) \(\beta_{8}\mathstrut -\mathstrut \) \(92\) \(\beta_{7}\mathstrut +\mathstrut \) \(117\) \(\beta_{6}\mathstrut -\mathstrut \) \(577\) \(\beta_{5}\mathstrut +\mathstrut \) \(580\) \(\beta_{4}\mathstrut -\mathstrut \) \(77\) \(\beta_{3}\mathstrut -\mathstrut \) \(125\) \(\beta_{2}\mathstrut +\mathstrut \) \(1224\) \(\beta_{1}\mathstrut -\mathstrut \) \(171\)

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
2.59935
2.07812
1.91855
1.47926
0.173275
0.148156
−0.329799
−1.67234
−1.72711
−2.66746
−2.59935 1.00000 4.75664 0 −2.59935 4.67552 −7.16549 1.00000 0
1.2 −2.07812 1.00000 2.31860 0 −2.07812 −0.151414 −0.662092 1.00000 0
1.3 −1.91855 1.00000 1.68082 0 −1.91855 −2.92362 0.612371 1.00000 0
1.4 −1.47926 1.00000 0.188212 0 −1.47926 1.25499 2.68011 1.00000 0
1.5 −0.173275 1.00000 −1.96998 0 −0.173275 −1.72833 0.687897 1.00000 0
1.6 −0.148156 1.00000 −1.97805 0 −0.148156 0.985585 0.589370 1.00000 0
1.7 0.329799 1.00000 −1.89123 0 0.329799 0.941793 −1.28333 1.00000 0
1.8 1.67234 1.00000 0.796721 0 1.67234 −4.55690 −2.01229 1.00000 0
1.9 1.72711 1.00000 0.982925 0 1.72711 5.06713 −1.75660 1.00000 0
1.10 2.66746 1.00000 5.11534 0 2.66746 −3.56475 8.31005 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(107\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\(T_{2}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)
\(T_{11}^{10} + \cdots\)
\(T_{13}^{10} - \cdots\)