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 $1$

Related objects

Downloads

Learn more about

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:

Atkin-Lehner signs

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

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

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

$p$ $F_p(T)$
$2$ \( 19 + 11 T - 120 T^{2} - 31 T^{3} + 185 T^{4} + 15 T^{5} - 121 T^{6} + 7 T^{7} + 36 T^{8} - 6 T^{9} - 4 T^{10} + T^{11} \)
$3$ \( -4 + 65 T + 72 T^{2} - 298 T^{3} - 69 T^{4} + 371 T^{5} - 58 T^{6} - 148 T^{7} + 56 T^{8} + 11 T^{9} - 8 T^{10} + T^{11} \)
$5$ \( T^{11} \)
$7$ \( 9356 + 10127 T - 10186 T^{2} - 8767 T^{3} + 5251 T^{4} + 2605 T^{5} - 1435 T^{6} - 273 T^{7} + 187 T^{8} - 9 T^{10} + T^{11} \)
$11$ \( 2368 - 7011 T - 1558 T^{2} + 12305 T^{3} - 3225 T^{4} - 5380 T^{5} + 1708 T^{6} + 843 T^{7} - 176 T^{8} - 59 T^{9} + 3 T^{10} + T^{11} \)
$13$ \( -38 - 681 T + 13378 T^{2} - 40567 T^{3} + 24679 T^{4} + 13638 T^{5} - 8404 T^{6} - 267 T^{7} + 583 T^{8} - 46 T^{9} - 9 T^{10} + T^{11} \)
$17$ \( 11574 - 612393 T - 120326 T^{2} + 244560 T^{3} + 38418 T^{4} - 36506 T^{5} - 4563 T^{6} + 2528 T^{7} + 231 T^{8} - 81 T^{9} - 4 T^{10} + T^{11} \)
$19$ \( -42248 - 1191399 T + 9335337 T^{2} + 7704974 T^{3} + 1732338 T^{4} - 164395 T^{5} - 126308 T^{6} - 15572 T^{7} + 874 T^{8} + 374 T^{9} + 33 T^{10} + T^{11} \)
$23$ \( -8444 - 15191 T + 109608 T^{2} - 168883 T^{3} + 110886 T^{4} - 23500 T^{5} - 11052 T^{6} + 8764 T^{7} - 2563 T^{8} + 392 T^{9} - 31 T^{10} + T^{11} \)
$29$ \( 87814 + 1166813 T + 826887 T^{2} - 400731 T^{3} - 481723 T^{4} - 88405 T^{5} + 24087 T^{6} + 7476 T^{7} - 191 T^{8} - 154 T^{9} - T^{10} + T^{11} \)
$31$ \( -2664644 + 2823361 T + 2566716 T^{2} - 4058823 T^{3} + 1449547 T^{4} - 13627 T^{5} - 76552 T^{6} + 8194 T^{7} + 1220 T^{8} - 174 T^{9} - 6 T^{10} + T^{11} \)
$37$ \( 16646506 + 21809433 T - 2372291 T^{2} - 5737958 T^{3} + 397705 T^{4} + 488436 T^{5} - 55195 T^{6} - 13841 T^{7} + 2315 T^{8} + 45 T^{9} - 23 T^{10} + T^{11} \)
$41$ \( 27218 + 464965 T - 1275943 T^{2} - 244631 T^{3} + 759889 T^{4} + 35830 T^{5} - 98540 T^{6} + 9850 T^{7} + 1699 T^{8} - 203 T^{9} - 8 T^{10} + T^{11} \)
$43$ \( 1342504 + 9827029 T - 4346831 T^{2} - 2367323 T^{3} + 840289 T^{4} + 175931 T^{5} - 58765 T^{6} - 4387 T^{7} + 1754 T^{8} - 6 T^{9} - 19 T^{10} + T^{11} \)
$47$ \( 22906712 + 11144191 T - 38280098 T^{2} + 19887717 T^{3} - 2313582 T^{4} - 1055487 T^{5} + 402442 T^{6} - 51035 T^{7} + 1074 T^{8} + 363 T^{9} - 35 T^{10} + T^{11} \)
$53$ \( 200270398 + 229524139 T - 30224435 T^{2} - 72596754 T^{3} - 17304274 T^{4} + 466834 T^{5} + 501650 T^{6} + 22931 T^{7} - 4654 T^{8} - 301 T^{9} + 14 T^{10} + T^{11} \)
$59$ \( -228526492 - 42370177 T + 89545905 T^{2} + 8607780 T^{3} - 8283558 T^{4} - 1038090 T^{5} + 210368 T^{6} + 29711 T^{7} - 1944 T^{8} - 301 T^{9} + 6 T^{10} + T^{11} \)
$61$ \( 7806014 - 3196689 T - 5543590 T^{2} + 1135363 T^{3} + 1052205 T^{4} - 131249 T^{5} - 61719 T^{6} + 6979 T^{7} + 1323 T^{8} - 148 T^{9} - 9 T^{10} + T^{11} \)
$67$ \( -78262892 - 190051801 T - 18614189 T^{2} + 55905980 T^{3} - 6403572 T^{4} - 3014876 T^{5} + 765079 T^{6} - 34803 T^{7} - 7196 T^{8} + 1056 T^{9} - 54 T^{10} + T^{11} \)
$71$ \( 171774332 - 413516785 T - 265270764 T^{2} + 96698187 T^{3} + 9552506 T^{4} - 4226351 T^{5} - 33866 T^{6} + 63900 T^{7} - 824 T^{8} - 412 T^{9} + 5 T^{10} + T^{11} \)
$73$ \( 1586331898 - 1146454197 T + 117556241 T^{2} + 73726882 T^{3} - 13126291 T^{4} - 1747677 T^{5} + 355176 T^{6} + 24273 T^{7} - 4021 T^{8} - 225 T^{9} + 17 T^{10} + T^{11} \)
$79$ \( 7778248 - 14180807 T + 4881367 T^{2} + 3990121 T^{3} - 2379038 T^{4} - 96652 T^{5} + 166071 T^{6} + 3392 T^{7} - 4022 T^{8} - 226 T^{9} + 16 T^{10} + T^{11} \)
$83$ \( 169996516 + 255125105 T - 49693332 T^{2} - 92038012 T^{3} + 14654201 T^{4} + 4226641 T^{5} - 695775 T^{6} - 33985 T^{7} + 8357 T^{8} - 85 T^{9} - 29 T^{10} + T^{11} \)
$89$ \( -13243411022 - 29544792621 T + 3098919430 T^{2} + 1614051215 T^{3} - 79106820 T^{4} - 29057104 T^{5} + 761295 T^{6} + 225658 T^{7} - 3220 T^{8} - 784 T^{9} + 5 T^{10} + T^{11} \)
$97$ \( 4514064014 - 3816423271 T - 485357577 T^{2} + 342411691 T^{3} + 8971668 T^{4} - 10024852 T^{5} + 71993 T^{6} + 117219 T^{7} - 1778 T^{8} - 571 T^{9} + 6 T^{10} + T^{11} \)
show more
show less