Properties

Label 483.2.i.f
Level $483$
Weight $2$
Character orbit 483.i
Analytic conductor $3.857$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} + 8 x^{10} - 3 x^{9} + 42 x^{8} - 15 x^{7} + 101 x^{6} + 6 x^{5} + 150 x^{4} - 28 x^{3} + 40 x^{2} + 8 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} + 8 x^{10} - 3 x^{9} + 42 x^{8} - 15 x^{7} + 101 x^{6} + 6 x^{5} + 150 x^{4} - 28 x^{3} + 40 x^{2} + 8 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-102623 \nu^{11} - 165571 \nu^{10} - 222384 \nu^{9} - 1691067 \nu^{8} - 1870676 \nu^{7} - 7686949 \nu^{6} + 7998165 \nu^{5} - 19046634 \nu^{4} + 17156370 \nu^{3} - 13809438 \nu^{2} + 125186588 \nu - 1558388\)\()/32857894\)
\(\beta_{3}\)\(=\)\((\)\(-387955 \nu^{11} - 1012381 \nu^{10} + 78220 \nu^{9} - 9799615 \nu^{8} + 1245446 \nu^{7} - 45151945 \nu^{6} + 57288245 \nu^{5} - 109780220 \nu^{4} + 110445090 \nu^{3} - 86675710 \nu^{2} + 287420068 \nu - 9835940\)\()/65715788\)
\(\beta_{4}\)\(=\)\((\)\(389597 \nu^{11} - 492220 \nu^{10} + 2951205 \nu^{9} - 1391175 \nu^{8} + 14672007 \nu^{7} - 7714631 \nu^{6} + 31662348 \nu^{5} + 10335747 \nu^{4} + 39392916 \nu^{3} + 6247654 \nu^{2} + 1774442 \nu + 62587576\)\()/32857894\)
\(\beta_{5}\)\(=\)\((\)\(782053 \nu^{11} - 392456 \nu^{10} + 5764204 \nu^{9} + 605046 \nu^{8} + 31455051 \nu^{7} + 2941212 \nu^{6} + 71272722 \nu^{5} + 36354666 \nu^{4} + 127643697 \nu^{3} + 17495432 \nu^{2} + 4671880 \nu + 8030866\)\()/32857894\)
\(\beta_{6}\)\(=\)\((\)\(-2035745 \nu^{11} - 2108749 \nu^{10} - 12760488 \nu^{9} - 25032941 \nu^{8} - 80460962 \nu^{7} - 131297395 \nu^{6} - 186501797 \nu^{5} - 371196906 \nu^{4} - 460140910 \nu^{3} - 480064098 \nu^{2} - 162121760 \nu - 38350448\)\()/65715788\)
\(\beta_{7}\)\(=\)\((\)\(-4015433 \nu^{11} + 5579539 \nu^{10} - 32908376 \nu^{9} + 23574707 \nu^{8} - 167438094 \nu^{7} + 123141597 \nu^{6} - 399676309 \nu^{5} + 118452846 \nu^{4} - 529605618 \nu^{3} + 367719518 \nu^{2} - 125626456 \nu + 42936084\)\()/65715788\)
\(\beta_{8}\)\(=\)\((\)\(-3745062 \nu^{11} + 2035588 \nu^{10} - 28003566 \nu^{9} - 2006303 \nu^{8} - 150105166 \nu^{7} - 10377711 \nu^{6} - 339284118 \nu^{5} - 170000221 \nu^{4} - 528040606 \nu^{3} - 82481982 \nu^{2} - 22081378 \nu - 39617942\)\()/32857894\)
\(\beta_{9}\)\(=\)\((\)\(-3881187 \nu^{11} + 6414399 \nu^{10} - 34645649 \nu^{9} + 33305993 \nu^{8} - 178419829 \nu^{7} + 168715213 \nu^{6} - 469846846 \nu^{5} + 251390220 \nu^{4} - 652666392 \nu^{3} + 461865932 \nu^{2} - 356135402 \nu + 53681036\)\()/32857894\)
\(\beta_{10}\)\(=\)\((\)\(12046299 \nu^{11} - 16738617 \nu^{10} + 98725128 \nu^{9} - 70724121 \nu^{8} + 502314282 \nu^{7} - 369424791 \nu^{6} + 1199028927 \nu^{5} - 355358538 \nu^{4} + 1588816854 \nu^{3} - 1037442766 \nu^{2} + 376879368 \nu + 68339112\)\()/65715788\)
\(\beta_{11}\)\(=\)\((\)\(-6206906 \nu^{11} + 4290544 \nu^{10} - 46524859 \nu^{9} + 1905036 \nu^{8} - 245693958 \nu^{7} + 6891144 \nu^{6} - 550437684 \nu^{5} - 257725548 \nu^{4} - 835966001 \nu^{3} - 129075616 \nu^{2} - 34888064 \nu - 130991922\)\()/32857894\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + 3 \beta_{7} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + 4 \beta_{5} + \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{10} - \beta_{9} - 12 \beta_{7} + 5 \beta_{4} - \beta_{3} - \beta_{1} + 5\)
\(\nu^{5}\)\(=\)\(2 \beta_{8} - 7 \beta_{7} - 7 \beta_{6} - 18 \beta_{5} - 2 \beta_{3} - 18 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(-7 \beta_{11} + 9 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 9 \beta_{5} - 23 \beta_{4} - 2 \beta_{2} + 29\)
\(\nu^{7}\)\(=\)\(2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{4} + 18 \beta_{3} - 39 \beta_{2} + 84 \beta_{1} - 2\)
\(\nu^{8}\)\(=\)\(39 \beta_{11} + 105 \beta_{10} + 39 \beta_{9} - 59 \beta_{8} + 236 \beta_{7} + 22 \beta_{6} + 61 \beta_{5} + 59 \beta_{3} + 61 \beta_{1} - 236\)
\(\nu^{9}\)\(=\)\(22 \beta_{11} - 120 \beta_{8} + 203 \beta_{7} + 203 \beta_{6} + 400 \beta_{5} + 24 \beta_{4} + 203 \beta_{2} - 205\)
\(\nu^{10}\)\(=\)\(-483 \beta_{10} - 203 \beta_{9} - 938 \beta_{7} + 483 \beta_{4} - 345 \beta_{3} + 166 \beta_{2} - 373 \beta_{1} + 483\)
\(\nu^{11}\)\(=\)\(-166 \beta_{11} - 194 \beta_{10} - 166 \beta_{9} + 714 \beta_{8} - 1257 \beta_{7} - 1031 \beta_{6} - 1932 \beta_{5} - 714 \beta_{3} - 1932 \beta_{1} + 1257\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
−1.02170 + 1.76964i
−0.682743 + 1.18255i
−0.144978 + 0.251110i
0.296764 0.514011i
0.916322 1.58712i
1.13633 1.96819i
−1.02170 1.76964i
−0.682743 1.18255i
−0.144978 0.251110i
0.296764 + 0.514011i
0.916322 + 1.58712i
1.13633 + 1.96819i
−1.02170 1.76964i 0.500000 0.866025i −1.08774 + 1.88403i −0.971745 1.68311i −2.04340 −1.16166 + 2.37709i 0.358585 −0.500000 0.866025i −1.98566 + 3.43927i
277.2 −0.682743 1.18255i 0.500000 0.866025i 0.0677246 0.117302i 0.663106 + 1.14853i −1.36549 2.35341 1.20891i −2.91593 −0.500000 0.866025i 0.905461 1.56830i
277.3 −0.144978 0.251110i 0.500000 0.866025i 0.957963 1.65924i −1.16311 2.01456i −0.289957 2.35341 1.20891i −1.13545 −0.500000 0.866025i −0.337250 + 0.584135i
277.4 0.296764 + 0.514011i 0.500000 0.866025i 0.823862 1.42697i 1.57560 + 2.72902i 0.593528 −1.69175 2.03420i 2.16503 −0.500000 0.866025i −0.935165 + 1.61975i
277.5 0.916322 + 1.58712i 0.500000 0.866025i −0.679293 + 1.17657i 0.471745 + 0.817086i 1.83264 −1.16166 + 2.37709i 1.17548 −0.500000 0.866025i −0.864540 + 1.49743i
277.6 1.13633 + 1.96819i 0.500000 0.866025i −1.58251 + 2.74099i −2.07560 3.59505i 2.27267 −1.69175 2.03420i −2.64772 −0.500000 0.866025i 4.71716 8.17036i
415.1 −1.02170 + 1.76964i 0.500000 + 0.866025i −1.08774 1.88403i −0.971745 + 1.68311i −2.04340 −1.16166 2.37709i 0.358585 −0.500000 + 0.866025i −1.98566 3.43927i
415.2 −0.682743 + 1.18255i 0.500000 + 0.866025i 0.0677246 + 0.117302i 0.663106 1.14853i −1.36549 2.35341 + 1.20891i −2.91593 −0.500000 + 0.866025i 0.905461 + 1.56830i
415.3 −0.144978 + 0.251110i 0.500000 + 0.866025i 0.957963 + 1.65924i −1.16311 + 2.01456i −0.289957 2.35341 + 1.20891i −1.13545 −0.500000 + 0.866025i −0.337250 0.584135i
415.4 0.296764 0.514011i 0.500000 + 0.866025i 0.823862 + 1.42697i 1.57560 2.72902i 0.593528 −1.69175 + 2.03420i 2.16503 −0.500000 + 0.866025i −0.935165 1.61975i
415.5 0.916322 1.58712i 0.500000 + 0.866025i −0.679293 1.17657i 0.471745 0.817086i 1.83264 −1.16166 2.37709i 1.17548 −0.500000 + 0.866025i −0.864540 1.49743i
415.6 1.13633 1.96819i 0.500000 + 0.866025i −1.58251 2.74099i −2.07560 + 3.59505i 2.27267 −1.69175 + 2.03420i −2.64772 −0.500000 + 0.866025i 4.71716 + 8.17036i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.i.f 12
7.c even 3 1 inner 483.2.i.f 12
7.c even 3 1 3381.2.a.bc 6
7.d odd 6 1 3381.2.a.bd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.f 12 1.a even 1 1 trivial
483.2.i.f 12 7.c even 3 1 inner
3381.2.a.bc 6 7.c even 3 1
3381.2.a.bd 6 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\(T_{2}^{12} - \cdots\)
\(T_{5}^{12} + \cdots\)
\(T_{11}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 8 T + 40 T^{2} - 28 T^{3} + 150 T^{4} + 6 T^{5} + 101 T^{6} - 15 T^{7} + 42 T^{8} - 3 T^{9} + 8 T^{10} - T^{11} + T^{12} \)
$3$ \( ( 1 - T + T^{2} )^{6} \)
$5$ \( 5476 - 5180 T + 8748 T^{2} - 1540 T^{3} + 4044 T^{4} + 58 T^{5} + 1647 T^{6} + 283 T^{7} + 278 T^{8} + 25 T^{9} + 24 T^{10} + 3 T^{11} + T^{12} \)
$7$ \( ( 343 + 49 T + 14 T^{2} - 23 T^{3} + 2 T^{4} + T^{5} + T^{6} )^{2} \)
$11$ \( 65536 + 32768 T + 98304 T^{2} - 53248 T^{3} + 86016 T^{4} - 24576 T^{5} + 19520 T^{6} - 7584 T^{7} + 3360 T^{8} - 776 T^{9} + 144 T^{10} - 14 T^{11} + T^{12} \)
$13$ \( ( -1 - 4 T + 47 T^{2} + 52 T^{3} - 31 T^{4} + T^{6} )^{2} \)
$17$ \( 324 + 1620 T + 6336 T^{2} + 9648 T^{3} + 12772 T^{4} + 8456 T^{5} + 7821 T^{6} + 4253 T^{7} + 3474 T^{8} + 869 T^{9} + 164 T^{10} + 15 T^{11} + T^{12} \)
$19$ \( 16 + 304 T + 6072 T^{2} - 4368 T^{3} + 17148 T^{4} + 1742 T^{5} + 29375 T^{6} - 10133 T^{7} + 4308 T^{8} - 249 T^{9} + 66 T^{10} - T^{11} + T^{12} \)
$23$ \( ( 1 - T + T^{2} )^{6} \)
$29$ \( ( -4768 + 4128 T + 712 T^{2} - 414 T^{3} - 66 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$31$ \( 5184 + 864 T + 14112 T^{2} + 18264 T^{3} + 39424 T^{4} + 28558 T^{5} + 19979 T^{6} + 4399 T^{7} + 1380 T^{8} - 297 T^{9} + 120 T^{10} - 11 T^{11} + T^{12} \)
$37$ \( 179776 - 323936 T + 469216 T^{2} - 300408 T^{3} + 174240 T^{4} - 31742 T^{5} + 19879 T^{6} - 2393 T^{7} + 1806 T^{8} - 27 T^{9} + 64 T^{10} - 5 T^{11} + T^{12} \)
$41$ \( ( 52928 + 10208 T - 12904 T^{2} + 2198 T^{3} - 26 T^{4} - 18 T^{5} + T^{6} )^{2} \)
$43$ \( ( -19796 - 4508 T + 6042 T^{2} + 2959 T^{3} + 503 T^{4} + 37 T^{5} + T^{6} )^{2} \)
$47$ \( 22733824 - 3661824 T + 10411904 T^{2} - 124864 T^{3} + 3842288 T^{4} - 209476 T^{5} + 252981 T^{6} + 8635 T^{7} + 11246 T^{8} + 19 T^{9} + 122 T^{10} + 3 T^{11} + T^{12} \)
$53$ \( 5465349184 - 492360480 T + 823704576 T^{2} - 293072472 T^{3} + 134224832 T^{4} - 28222734 T^{5} + 5125483 T^{6} - 546507 T^{7} + 55678 T^{8} - 3549 T^{9} + 316 T^{10} - 15 T^{11} + T^{12} \)
$59$ \( 5805220864 + 1734739456 T + 1183995136 T^{2} - 108385152 T^{3} + 74451104 T^{4} - 3856704 T^{5} + 1919588 T^{6} - 107812 T^{7} + 33256 T^{8} - 784 T^{9} + 206 T^{10} - 2 T^{11} + T^{12} \)
$61$ \( 9929723904 - 5146619904 T + 2656355328 T^{2} - 433075200 T^{3} + 127088128 T^{4} - 17896192 T^{5} + 4194624 T^{6} - 400576 T^{7} + 52512 T^{8} - 2320 T^{9} + 308 T^{10} - 12 T^{11} + T^{12} \)
$67$ \( 1142372401 - 625349098 T + 324106343 T^{2} - 69729210 T^{3} + 19181214 T^{4} - 2636814 T^{5} + 704459 T^{6} - 74022 T^{7} + 13926 T^{8} - 1018 T^{9} + 175 T^{10} - 10 T^{11} + T^{12} \)
$71$ \( ( -3208 - 8332 T - 5362 T^{2} - 773 T^{3} + 77 T^{4} + 21 T^{5} + T^{6} )^{2} \)
$73$ \( 3064618881 - 4099444668 T + 4866833367 T^{2} - 998324388 T^{3} + 251885962 T^{4} - 14857580 T^{5} + 4360143 T^{6} - 232024 T^{7} + 47594 T^{8} - 1408 T^{9} + 279 T^{10} - 8 T^{11} + T^{12} \)
$79$ \( 14531820304 + 30046830096 T + 62436850056 T^{2} + 780650656 T^{3} + 1440919172 T^{4} - 131819354 T^{5} + 31089447 T^{6} - 1925537 T^{7} + 187110 T^{8} - 6715 T^{9} + 588 T^{10} - 17 T^{11} + T^{12} \)
$83$ \( ( -2224 - 8664 T - 1612 T^{2} + 1222 T^{3} - 80 T^{4} - 12 T^{5} + T^{6} )^{2} \)
$89$ \( 27290176 + 11618176 T + 11423936 T^{2} + 4409568 T^{3} + 3209536 T^{4} + 1069216 T^{5} + 406292 T^{6} + 66072 T^{7} + 14372 T^{8} + 868 T^{9} + 352 T^{10} + 18 T^{11} + T^{12} \)
$97$ \( ( 1152 - 4992 T + 5632 T^{2} - 680 T^{3} - 228 T^{4} + 2 T^{5} + T^{6} )^{2} \)
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