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: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 2 x^{10} - 11 x^{9} + 15 x^{8} + 43 x^{7} - 28 x^{6} - 62 x^{5} + 14 x^{4} + 31 x^{3} + x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1205)
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 4q^{2} + 8q^{3} + 6q^{4} + 7q^{6} + 9q^{7} + 12q^{8} + 9q^{9} + O(q^{10}) \) \( 11q + 4q^{2} + 8q^{3} + 6q^{4} + 7q^{6} + 9q^{7} + 12q^{8} + 9q^{9} - 3q^{11} + 28q^{12} + 9q^{13} + 2q^{14} - 16q^{16} + 4q^{17} + 6q^{18} - 33q^{19} + 2q^{21} - 6q^{22} + 31q^{23} + 32q^{24} - 20q^{26} + 32q^{27} + q^{28} + q^{29} + 6q^{31} - 7q^{32} + 35q^{33} + 9q^{34} + 33q^{36} + 23q^{37} - 20q^{38} + 14q^{39} + 8q^{41} + 26q^{42} + 19q^{43} + 6q^{46} + 35q^{47} - 16q^{48} + 4q^{49} - 3q^{51} + 3q^{52} - 14q^{53} + 9q^{54} + 33q^{56} - q^{57} + 11q^{58} - 6q^{59} + 9q^{61} + 23q^{62} + 31q^{63} + 18q^{64} - 36q^{66} + 54q^{67} - q^{68} + 17q^{69} - 5q^{71} + 64q^{72} - 17q^{73} + 8q^{74} - 31q^{76} + 18q^{77} - 15q^{78} - 16q^{79} + 43q^{81} + 61q^{82} + 29q^{83} + 69q^{84} + 5q^{86} - 5q^{87} + 14q^{88} - 5q^{89} - 54q^{91} + 6q^{92} + 25q^{93} - 19q^{94} + 9q^{96} - 6q^{97} + 29q^{98} + 60q^{99} + O(q^{100}) \)

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6025.2.a.g 11
5.b even 2 1 1205.2.a.b 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.a.b 11 5.b even 2 1
6025.2.a.g 11 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(241\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 16 T^{2} - 44 T^{3} + 119 T^{4} - 265 T^{5} + 569 T^{6} - 1075 T^{7} + 1955 T^{8} - 3212 T^{9} + 5073 T^{10} - 7317 T^{11} + 10146 T^{12} - 12848 T^{13} + 15640 T^{14} - 17200 T^{15} + 18208 T^{16} - 16960 T^{17} + 15232 T^{18} - 11264 T^{19} + 8192 T^{20} - 4096 T^{21} + 2048 T^{22} \)
$3$ \( 1 - 8 T + 44 T^{2} - 184 T^{3} + 644 T^{4} - 1954 T^{5} + 5282 T^{6} - 12921 T^{7} + 28973 T^{8} - 59994 T^{9} + 115445 T^{10} - 206986 T^{11} + 346335 T^{12} - 539946 T^{13} + 782271 T^{14} - 1046601 T^{15} + 1283526 T^{16} - 1424466 T^{17} + 1408428 T^{18} - 1207224 T^{19} + 866052 T^{20} - 472392 T^{21} + 177147 T^{22} \)
$5$ 1
$7$ \( 1 - 9 T + 77 T^{2} - 443 T^{3} + 2422 T^{4} - 10808 T^{5} + 45823 T^{6} - 168895 T^{7} + 593821 T^{8} - 1863877 T^{9} + 5589939 T^{10} - 15122740 T^{11} + 39129573 T^{12} - 91329973 T^{13} + 203680603 T^{14} - 405516895 T^{15} + 770147161 T^{16} - 1271550392 T^{17} + 1994621146 T^{18} - 2553806843 T^{19} + 3107227739 T^{20} - 2542277241 T^{21} + 1977326743 T^{22} \)
$11$ \( 1 + 3 T + 62 T^{2} + 154 T^{3} + 1657 T^{4} + 2555 T^{5} + 22142 T^{6} - 7625 T^{7} + 93562 T^{8} - 937944 T^{9} - 1275223 T^{10} - 15528862 T^{11} - 14027453 T^{12} - 113491224 T^{13} + 124531022 T^{14} - 111637625 T^{15} + 3565991242 T^{16} + 4526338355 T^{17} + 32290242347 T^{18} + 33011267674 T^{19} + 146192756842 T^{20} + 77812273803 T^{21} + 285311670611 T^{22} \)
$13$ \( 1 - 9 T + 97 T^{2} - 587 T^{3} + 3646 T^{4} - 16217 T^{5} + 71982 T^{6} - 244837 T^{7} + 834242 T^{8} - 2260088 T^{9} + 6932271 T^{10} - 20417578 T^{11} + 90119523 T^{12} - 381954872 T^{13} + 1832829674 T^{14} - 6992789557 T^{15} + 26726412726 T^{16} - 78276361553 T^{17} + 228781092982 T^{18} - 478833933227 T^{19} + 1028636439181 T^{20} - 1240726426641 T^{21} + 1792160394037 T^{22} \)
$17$ \( 1 - 4 T + 106 T^{2} - 449 T^{3} + 6030 T^{4} - 25167 T^{5} + 232247 T^{6} - 915996 T^{7} + 6617860 T^{8} - 23891579 T^{9} + 144618675 T^{10} - 466504364 T^{11} + 2458517475 T^{12} - 6904666331 T^{13} + 32513546180 T^{14} - 76504901916 T^{15} + 329757528679 T^{16} - 607470199023 T^{17} + 2474342198190 T^{18} - 3132115091009 T^{19} + 12570314908682 T^{20} - 8063975601796 T^{21} + 34271896307633 T^{22} \)
$19$ \( 1 + 33 T + 583 T^{2} + 7144 T^{3} + 68237 T^{4} + 542625 T^{5} + 3756768 T^{6} + 23329258 T^{7} + 132524391 T^{8} + 695866631 T^{9} + 3391438531 T^{10} + 15344291290 T^{11} + 64437332089 T^{12} + 251207853791 T^{13} + 908984797869 T^{14} + 3040292231818 T^{15} + 9302129488032 T^{16} + 25528271177625 T^{17} + 60995125854143 T^{18} + 121330574364904 T^{19} + 188126927805157 T^{20} + 202325186507433 T^{21} + 116490258898219 T^{22} \)
$23$ \( 1 - 31 T + 645 T^{2} - 9693 T^{3} + 119003 T^{4} - 1220599 T^{5} + 10860307 T^{6} - 84638686 T^{7} + 587470399 T^{8} - 3645456586 T^{9} + 20391619600 T^{10} - 102819393318 T^{11} + 469007250800 T^{12} - 1928446533994 T^{13} + 7147752344633 T^{14} - 23685374528926 T^{15} + 69900660937301 T^{16} - 180692458077511 T^{17} + 405184442669341 T^{18} - 759068380328733 T^{19} + 1161743466643635 T^{20} - 1284221847623119 T^{21} + 952809757913927 T^{22} \)
$29$ \( 1 - T + 165 T^{2} - 481 T^{3} + 13537 T^{4} - 58070 T^{5} + 790904 T^{6} - 3714933 T^{7} + 36720806 T^{8} - 160589230 T^{9} + 1356513220 T^{10} - 5258726656 T^{11} + 39338883380 T^{12} - 135055542430 T^{13} + 895583737534 T^{14} - 2627501527173 T^{15} + 16222349788696 T^{16} - 34541390250470 T^{17} + 233511575594933 T^{18} - 240618524634241 T^{19} + 2393679086018385 T^{20} - 420707233300201 T^{21} + 12200509765705829 T^{22} \)
$31$ \( 1 - 6 T + 167 T^{2} - 640 T^{3} + 12503 T^{4} - 33462 T^{5} + 660282 T^{6} - 1410885 T^{7} + 28528780 T^{8} - 49501876 T^{9} + 1017551600 T^{10} - 1515235802 T^{11} + 31544099600 T^{12} - 47571302836 T^{13} + 849900884980 T^{14} - 1302981926085 T^{15} + 18903313080582 T^{16} - 29697648173622 T^{17} + 343990214229833 T^{18} - 545850263962240 T^{19} + 4415416900832057 T^{20} - 4917769721884806 T^{21} + 25408476896404831 T^{22} \)
$37$ \( 1 - 23 T + 452 T^{2} - 6195 T^{3} + 76439 T^{4} - 786870 T^{5} + 7479142 T^{6} - 62919285 T^{7} + 496649263 T^{8} - 3562483986 T^{9} + 24196899884 T^{10} - 151017785580 T^{11} + 895285295708 T^{12} - 4877040576834 T^{13} + 25156775118739 T^{14} - 117920870094885 T^{15} + 518633301244894 T^{16} - 2018893139449830 T^{17} + 7256497756169387 T^{18} - 21759810217040595 T^{19} + 58742706387374804 T^{20} - 110597440565610527 T^{21} + 177917621779460413 T^{22} \)
$41$ \( 1 - 8 T + 248 T^{2} - 1581 T^{3} + 27398 T^{4} - 146428 T^{5} + 1949997 T^{6} - 9676579 T^{7} + 112075607 T^{8} - 551199103 T^{9} + 5581442406 T^{10} - 25768211940 T^{11} + 228839138646 T^{12} - 926565692143 T^{13} + 7724362910047 T^{14} - 27343699551619 T^{15} + 225919244381397 T^{16} - 695548263801148 T^{17} + 5335877595791638 T^{18} - 12624166787240301 T^{19} + 81190719729702328 T^{20} - 107381274481219208 T^{21} + 550329031716248441 T^{22} \)
$43$ \( 1 - 19 T + 467 T^{2} - 6416 T^{3} + 94986 T^{4} - 1036284 T^{5} + 11574715 T^{6} - 104788953 T^{7} + 953247821 T^{8} - 7321184820 T^{9} + 56082791347 T^{10} - 368610934200 T^{11} + 2411560027921 T^{12} - 13536870732180 T^{13} + 75789874504247 T^{14} - 358252577305353 T^{15} + 1701580830318745 T^{16} - 6550727385869916 T^{17} + 25818962594609502 T^{18} - 74991492981088016 T^{19} + 234710749774505681 T^{20} - 410618163952400731 T^{21} + 929293739471222707 T^{22} \)
$47$ \( 1 - 35 T + 880 T^{2} - 15376 T^{3} + 224009 T^{4} - 2672909 T^{5} + 28152005 T^{6} - 258452490 T^{7} + 2180451303 T^{8} - 16759642682 T^{9} + 123192609095 T^{10} - 854553467168 T^{11} + 5790052627465 T^{12} - 37022050684538 T^{13} + 226380995631369 T^{14} - 1261165704855690 T^{15} + 6456521783789035 T^{16} - 28811861665822061 T^{17} + 113488138591796167 T^{18} - 366122343711237136 T^{19} + 984834816330434960 T^{20} - 1840969628254051715 T^{21} + 2472159215084012303 T^{22} \)
$53$ \( 1 + 14 T + 282 T^{2} + 2766 T^{3} + 33849 T^{4} + 298024 T^{5} + 3100616 T^{6} + 26287378 T^{7} + 243443677 T^{8} + 1835387119 T^{9} + 15238389718 T^{10} + 103865986416 T^{11} + 807634655054 T^{12} + 5155602417271 T^{13} + 36243164300729 T^{14} + 207420056648818 T^{15} + 1296663636723688 T^{16} + 6605511561109096 T^{17} + 39762797372342613 T^{18} + 172210303677824526 T^{19} + 930533332888201506 T^{20} + 2448424585117182686 T^{21} + 9269035929372191597 T^{22} \)
$59$ \( 1 + 6 T + 348 T^{2} + 1596 T^{3} + 61335 T^{4} + 232670 T^{5} + 7399972 T^{6} + 24581802 T^{7} + 680201535 T^{8} + 2028486741 T^{9} + 49647828250 T^{10} + 133465942558 T^{11} + 2929221866750 T^{12} + 7061162345421 T^{13} + 139699111056765 T^{14} + 297866568864522 T^{15} + 5290419794719628 T^{16} + 9814144762251470 T^{17} + 152641438821373365 T^{18} + 234341378416496316 T^{19} + 3014722544891918772 T^{20} + 3066700519803848406 T^{21} + 30155888444737842659 T^{22} \)
$61$ \( 1 - 9 T + 523 T^{2} - 4167 T^{3} + 130382 T^{4} - 923100 T^{5} + 20475161 T^{6} - 128835705 T^{7} + 2253750195 T^{8} - 12545722717 T^{9} + 182770561509 T^{10} - 890638865574 T^{11} + 11149004252049 T^{12} - 46682634229957 T^{13} + 511558473011295 T^{14} - 1783838686552905 T^{15} + 17293245242979461 T^{16} - 47558457572639100 T^{17} + 409757096446090022 T^{18} - 798844373259669927 T^{19} + 6116038406552255743 T^{20} - 6420086204965943409 T^{21} + 43513917611435838661 T^{22} \)
$67$ \( 1 - 54 T + 1793 T^{2} - 43376 T^{3} + 848860 T^{4} - 14000247 T^{5} + 200942236 T^{6} - 2552265686 T^{7} + 29093829595 T^{8} - 299932159848 T^{9} + 2814333707634 T^{10} - 24095922393102 T^{11} + 188560358411478 T^{12} - 1346395465557672 T^{13} + 8750347470480985 T^{14} - 51431014662734006 T^{15} + 271297157880319252 T^{16} - 1266439693586395743 T^{17} + 5144695653294481780 T^{18} - 17613591581696860016 T^{19} + 48781316172556839971 T^{20} - 98433241445795118246 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \)
$71$ \( 1 + 5 T + 369 T^{2} + 2726 T^{3} + 77887 T^{4} + 632327 T^{5} + 11819152 T^{6} + 93566238 T^{7} + 1360132324 T^{8} + 10053683216 T^{9} + 121983570726 T^{10} + 816591365980 T^{11} + 8660833521546 T^{12} + 50680617091856 T^{13} + 486806320215164 T^{14} + 2377675392426078 T^{15} + 21324460942330352 T^{16} + 81001268230914167 T^{17} + 708391623776599817 T^{18} + 1760324126175944486 T^{19} + 16918096765107692439 T^{20} + 16276217755049406005 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \)
$73$ \( 1 + 17 T + 578 T^{2} + 8389 T^{3} + 169543 T^{4} + 2083597 T^{5} + 31678731 T^{6} + 336054025 T^{7} + 4171190164 T^{8} + 38460101207 T^{9} + 405026306682 T^{10} + 3250225364632 T^{11} + 29566920387786 T^{12} + 204953879332103 T^{13} + 1622663884028788 T^{14} + 9543343190970025 T^{15} + 65672277338388483 T^{16} + 315319539893081533 T^{17} + 1873009087123262671 T^{18} + 6765393710899445509 T^{19} + 34027777117378853714 T^{20} + 73059639104960480033 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \)
$79$ \( 1 + 16 T + 643 T^{2} + 8618 T^{3} + 185961 T^{4} + 2117687 T^{5} + 32353783 T^{6} + 320137040 T^{7} + 3904045847 T^{8} + 34623954136 T^{9} + 365890566412 T^{10} + 2989968768694 T^{11} + 28905354746548 T^{12} + 216088097762776 T^{13} + 1924846860359033 T^{14} + 12469363639100240 T^{15} + 99554415012007417 T^{16} + 514783144419899927 T^{17} + 3571178118975113799 T^{18} + 13074443723774742698 T^{19} + 77064576216823579117 T^{20} + \)\(15\!\cdots\!16\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \)
$83$ \( 1 - 29 T + 828 T^{2} - 15713 T^{3} + 281415 T^{4} - 4136872 T^{5} + 57745871 T^{6} - 709662065 T^{7} + 8324128788 T^{8} - 88511841411 T^{9} + 899942775938 T^{10} - 8382930168216 T^{11} + 74695250402854 T^{12} - 609758075480379 T^{13} + 4759628627304156 T^{14} - 33679370082292865 T^{15} + 227463332834435053 T^{16} - 1352510476259761768 T^{17} + 7636491789245882205 T^{18} - 35390267843600751233 T^{19} + \)\(15\!\cdots\!84\)\( T^{20} - \)\(44\!\cdots\!21\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \)
$89$ \( 1 + 5 T + 195 T^{2} + 1230 T^{3} + 33329 T^{4} + 250880 T^{5} + 4285411 T^{6} + 36248750 T^{7} + 498318379 T^{8} + 4149499905 T^{9} + 49565915508 T^{10} + 406334801638 T^{11} + 4411366480212 T^{12} + 32868188747505 T^{13} + 351299009325251 T^{14} + 2274327808448750 T^{15} + 23929989787398539 T^{16} + 124682666276295680 T^{17} + 1474186160733086041 T^{18} + 4842004231013559630 T^{19} + 68319498722959615755 T^{20} + \)\(15\!\cdots\!05\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \)
$97$ \( 1 + 6 T + 496 T^{2} + 4042 T^{3} + 136231 T^{4} + 1232695 T^{5} + 26746490 T^{6} + 239578498 T^{7} + 4080758420 T^{8} + 33830250458 T^{9} + 495002175583 T^{10} + 3696666740452 T^{11} + 48015211031551 T^{12} + 318308826559322 T^{13} + 3724398029456660 T^{14} + 21209712170999938 T^{15} + 229681210310447930 T^{16} + 1026800425615953655 T^{17} + 11007231092737812103 T^{18} + 31678906588471676362 T^{19} + \)\(37\!\cdots\!32\)\( T^{20} + \)\(44\!\cdots\!94\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \)
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