Properties

Label 6025.2.a.h
Level 6025
Weight 2
Character orbit 6025.a
Self dual Yes
Analytic conductor 48.110
Analytic rank 0
Dimension 12
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} + \nu^{10} - 23 \nu^{9} - 13 \nu^{8} + 184 \nu^{7} + 54 \nu^{6} - 611 \nu^{5} - 94 \nu^{4} + 768 \nu^{3} + 94 \nu^{2} - 213 \nu - 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{11} + 9 \nu^{10} + 57 \nu^{9} - 119 \nu^{8} - 273 \nu^{7} + 488 \nu^{6} + 538 \nu^{5} - 663 \nu^{4} - 422 \nu^{3} + 168 \nu^{2} + 34 \nu + 7 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{11} - 10 \nu^{10} - 130 \nu^{9} + 172 \nu^{8} + 869 \nu^{7} - 1046 \nu^{6} - 2491 \nu^{5} + 2625 \nu^{4} + 2774 \nu^{3} - 2230 \nu^{2} - 745 \nu + 85 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{11} + 48 \nu^{10} + 160 \nu^{9} - 690 \nu^{8} - 569 \nu^{7} + 3330 \nu^{6} + 597 \nu^{5} - 6481 \nu^{4} - 470 \nu^{3} + 4634 \nu^{2} + 951 \nu - 169 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 13 \nu^{11} - 44 \nu^{10} - 180 \nu^{9} + 662 \nu^{8} + 829 \nu^{7} - 3426 \nu^{6} - 1629 \nu^{5} + 7333 \nu^{4} + 1798 \nu^{3} - 5698 \nu^{2} - 1335 \nu + 141 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( 11 \nu^{11} - 30 \nu^{10} - 158 \nu^{9} + 432 \nu^{8} + 773 \nu^{7} - 2086 \nu^{6} - 1631 \nu^{5} + 4025 \nu^{4} + 1654 \nu^{3} - 2750 \nu^{2} - 741 \nu + 93 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -11 \nu^{11} + 34 \nu^{10} + 150 \nu^{9} - 492 \nu^{8} - 669 \nu^{7} + 2398 \nu^{6} + 1223 \nu^{5} - 4717 \nu^{4} - 1178 \nu^{3} + 3354 \nu^{2} + 737 \nu - 109 \)\()/8\)
\(\beta_{10}\)\(=\)\((\)\( 25 \nu^{11} - 76 \nu^{10} - 340 \nu^{9} + 1086 \nu^{8} + 1513 \nu^{7} - 5178 \nu^{6} - 2777 \nu^{5} + 9809 \nu^{4} + 2718 \nu^{3} - 6602 \nu^{2} - 1643 \nu + 201 \)\()/16\)
\(\beta_{11}\)\(=\)\((\)\( 31 \nu^{11} - 90 \nu^{10} - 450 \nu^{9} + 1340 \nu^{8} + 2237 \nu^{7} - 6822 \nu^{6} - 4819 \nu^{5} + 14249 \nu^{4} + 5014 \nu^{3} - 10758 \nu^{2} - 2497 \nu + 381 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{11} + \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 8 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(11 \beta_{11} - 9 \beta_{10} + 10 \beta_{9} - \beta_{7} - 8 \beta_{6} - 11 \beta_{5} - 9 \beta_{4} + 9 \beta_{3} + \beta_{2} + 29 \beta_{1} - 10\)
\(\nu^{6}\)\(=\)\(20 \beta_{11} + 2 \beta_{10} + 10 \beta_{9} - 14 \beta_{8} - 19 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} - 2 \beta_{4} + 11 \beta_{3} + 56 \beta_{2} + 88\)
\(\nu^{7}\)\(=\)\(93 \beta_{11} - 67 \beta_{10} + 78 \beta_{9} - 3 \beta_{8} - 12 \beta_{7} - 54 \beta_{6} - 94 \beta_{5} - 68 \beta_{4} + 71 \beta_{3} + 11 \beta_{2} + 181 \beta_{1} - 77\)
\(\nu^{8}\)\(=\)\(163 \beta_{11} + 24 \beta_{10} + 76 \beta_{9} - 138 \beta_{8} - 146 \beta_{7} - 78 \beta_{6} - 71 \beta_{5} - 30 \beta_{4} + 98 \beta_{3} + 379 \beta_{2} + 4 \beta_{1} + 558\)
\(\nu^{9}\)\(=\)\(720 \beta_{11} - 478 \beta_{10} + 557 \beta_{9} - 59 \beta_{8} - 106 \beta_{7} - 355 \beta_{6} - 733 \beta_{5} - 499 \beta_{4} + 542 \beta_{3} + 88 \beta_{2} + 1179 \beta_{1} - 544\)
\(\nu^{10}\)\(=\)\(1256 \beta_{11} + 191 \beta_{10} + 522 \beta_{9} - 1189 \beta_{8} - 1056 \beta_{7} - 566 \beta_{6} - 561 \beta_{5} - 323 \beta_{4} + 822 \beta_{3} + 2542 \beta_{2} + 76 \beta_{1} + 3665\)
\(\nu^{11}\)\(=\)\(5372 \beta_{11} - 3384 \beta_{10} + 3821 \beta_{9} - 748 \beta_{8} - 845 \beta_{7} - 2351 \beta_{6} - 5486 \beta_{5} - 3655 \beta_{4} + 4093 \beta_{3} + 630 \beta_{2} + 7881 \beta_{1} - 3713\)

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
2.70063
2.49073
2.01020
1.63125
1.54879
0.115670
0.0822506
−0.342147
−1.28632
−1.32986
−2.02418
−2.59703
−2.70063 2.50808 5.29342 0 −6.77340 −0.354992 −8.89432 3.29045 0
1.2 −2.49073 −1.22208 4.20371 0 3.04385 −0.136122 −5.48885 −1.50653 0
1.3 −2.01020 −0.500591 2.04092 0 1.00629 0.852319 −0.0822476 −2.74941 0
1.4 −1.63125 1.16790 0.660992 0 −1.90514 −5.06139 2.18426 −1.63601 0
1.5 −1.54879 −2.81087 0.398765 0 4.35346 4.24623 2.47998 4.90098 0
1.6 −0.115670 3.28295 −1.98662 0 −0.379739 −3.19647 0.461133 7.77775 0
1.7 −0.0822506 −1.81824 −1.99323 0 0.149552 −0.690569 0.328446 0.306010 0
1.8 0.342147 −2.18519 −1.88294 0 −0.747658 −1.82459 −1.32853 1.77508 0
1.9 1.28632 0.126224 −0.345373 0 0.162365 −1.03110 −3.01691 −2.98407 0
1.10 1.32986 2.18147 −0.231473 0 2.90104 3.83334 −2.96755 1.75880 0
1.11 2.02418 −2.93498 2.09729 0 −5.94092 −0.381245 0.196936 5.61411 0
1.12 2.59703 1.20534 4.74454 0 3.13029 0.744578 7.12764 −1.54716 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

\(T_{2}^{12} + \cdots\)
\(T_{3}^{12} + \cdots\)