Properties

Label 429.2.n.a
Level $429$
Weight $2$
Character orbit 429.n
Analytic conductor $3.426$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} + 9 x^{10} - 15 x^{9} + 29 x^{8} - 26 x^{7} + 43 x^{6} + 24 x^{5} + 16 x^{4} - 17 x^{3} + 14 x^{2} - 5 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} + 9 x^{10} - 15 x^{9} + 29 x^{8} - 26 x^{7} + 43 x^{6} + 24 x^{5} + 16 x^{4} - 17 x^{3} + 14 x^{2} - 5 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 19891 \nu^{11} - 114974 \nu^{10} + 362529 \nu^{9} - 673717 \nu^{8} + 1202532 \nu^{7} - 1315063 \nu^{6} + 1708502 \nu^{5} + 58800 \nu^{4} + 422521 \nu^{3} + 518984 \nu^{2} + 15013142 \nu + 95570 \)\()/3960529\)
\(\beta_{3}\)\(=\)\((\)\(-147786 \nu^{11} - 379469 \nu^{10} + 771926 \nu^{9} - 4107793 \nu^{8} + 4845187 \nu^{7} - 14767041 \nu^{6} + 5123199 \nu^{5} - 30881178 \nu^{4} - 35585636 \nu^{3} - 22422420 \nu^{2} + 8521975 \nu - 5509490\)\()/3960529\)
\(\beta_{4}\)\(=\)\((\)\(279535 \nu^{11} - 503769 \nu^{10} + 1828536 \nu^{9} - 1989137 \nu^{8} + 5491685 \nu^{7} - 978328 \nu^{6} + 10188488 \nu^{5} + 17315912 \nu^{4} + 21684207 \nu^{3} + 16074981 \nu^{2} + 5253803 \nu - 299426\)\()/3960529\)
\(\beta_{5}\)\(=\)\((\)\(-334836 \nu^{11} + 687279 \nu^{10} - 2203888 \nu^{9} + 2614830 \nu^{8} - 6289582 \nu^{7} + 1831517 \nu^{6} - 10607072 \nu^{5} - 17211647 \nu^{4} - 20827076 \nu^{3} - 1340313 \nu^{2} - 1098249 \nu + 279535\)\()/3960529\)
\(\beta_{6}\)\(=\)\((\)\(351124 \nu^{11} - 1148942 \nu^{10} + 3466717 \nu^{9} - 6241964 \nu^{8} + 11978675 \nu^{7} - 12574471 \nu^{6} + 18785684 \nu^{5} + 3002403 \nu^{4} + 5032806 \nu^{3} - 7439428 \nu^{2} + 6962947 \nu - 2574616\)\()/3960529\)
\(\beta_{7}\)\(=\)\((\)\(-1268916 \nu^{11} + 3238544 \nu^{10} - 9630840 \nu^{9} + 13682648 \nu^{8} - 27496876 \nu^{7} + 15163504 \nu^{6} - 36702332 \nu^{5} - 57918271 \nu^{4} - 28130100 \nu^{3} + 14357820 \nu^{2} - 5578580 \nu + 1339344\)\()/3960529\)
\(\beta_{8}\)\(=\)\((\)\(1848387 \nu^{11} - 5413412 \nu^{10} + 15752245 \nu^{9} - 25125343 \nu^{8} + 47506293 \nu^{7} - 37721190 \nu^{6} + 63735272 \nu^{5} + 59672975 \nu^{4} + 16008926 \nu^{3} - 45324008 \nu^{2} + 19529979 \nu + 573314\)\()/3960529\)
\(\beta_{9}\)\(=\)\((\)\(2442867 \nu^{11} - 6489486 \nu^{10} + 19422140 \nu^{9} - 29055593 \nu^{8} + 58085028 \nu^{7} - 39215972 \nu^{6} + 82822330 \nu^{5} + 94141734 \nu^{4} + 58097688 \nu^{3} - 32388227 \nu^{2} + 14829418 \nu - 4061746\)\()/3960529\)
\(\beta_{10}\)\(=\)\((\)\(-3187731 \nu^{11} + 8162528 \nu^{10} - 24567797 \nu^{9} + 35584663 \nu^{8} - 72664621 \nu^{7} + 45047258 \nu^{6} - 106163560 \nu^{5} - 128519563 \nu^{4} - 95356701 \nu^{3} + 39962756 \nu^{2} - 23922975 \nu + 544826\)\()/3960529\)
\(\beta_{11}\)\(=\)\((\)\(3214409 \nu^{11} - 9807301 \nu^{10} + 29011875 \nu^{9} - 48707038 \nu^{8} + 92737202 \nu^{7} - 84418540 \nu^{6} + 134195500 \nu^{5} + 74090403 \nu^{4} + 34758610 \nu^{3} - 74436319 \nu^{2} + 31359047 \nu - 9454239\)\()/3960529\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{10} + \beta_{8} - 4 \beta_{5}\)
\(\nu^{4}\)\(=\)\(4 \beta_{10} + 8 \beta_{9} + 5 \beta_{7} - 4 \beta_{4} - 4 \beta_{3} + 4 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-5 \beta_{11} + 17 \beta_{9} + 16 \beta_{7} + \beta_{6} + 16 \beta_{5} + \beta_{3} - \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(-16 \beta_{11} - 16 \beta_{10} + 23 \beta_{6} + 23 \beta_{5} + 15 \beta_{4} + 26 \beta_{3} - 15 \beta_{2} - 23 \beta_{1} + 15\)
\(\nu^{7}\)\(=\)\(-23 \beta_{10} - 72 \beta_{9} - 65 \beta_{7} + 65 \beta_{6} + 9 \beta_{5} + \beta_{4} - 74 \beta_{1} - 7\)
\(\nu^{8}\)\(=\)\(65 \beta_{11} - 9 \beta_{10} - 146 \beta_{9} - 9 \beta_{8} - 104 \beta_{7} + \beta_{5} - 42 \beta_{4} - 98 \beta_{3} + 65 \beta_{2} - 104 \beta_{1} - 98\)
\(\nu^{9}\)\(=\)\(104 \beta_{11} - \beta_{10} - 50 \beta_{9} - 104 \beta_{8} - 58 \beta_{7} - 267 \beta_{6} - 38 \beta_{4} - 38 \beta_{3} + 103 \beta_{2}\)
\(\nu^{10}\)\(=\)\(58 \beta_{11} + 197 \beta_{9} - 267 \beta_{8} - 13 \beta_{7} - 467 \beta_{6} - 13 \beta_{5} + 210 \beta_{3} + 467 \beta_{1} + 373\)
\(\nu^{11}\)\(=\)\(13 \beta_{11} + 13 \beta_{10} - 328 \beta_{6} - 328 \beta_{5} + 187 \beta_{4} + 232 \beta_{3} - 454 \beta_{2} + 1435 \beta_{1} + 187\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3} - \beta_{4} - \beta_{9}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−0.689843 0.501200i
0.275227 + 0.199964i
1.72363 + 1.25229i
−0.566948 1.74489i
0.135246 + 0.416243i
0.622685 + 1.91643i
−0.689843 + 0.501200i
0.275227 0.199964i
1.72363 1.25229i
−0.566948 + 1.74489i
0.135246 0.416243i
0.622685 1.91643i
−0.689843 + 0.501200i 0.309017 + 0.951057i −0.393353 + 1.21061i −0.139763 0.101543i −0.689843 0.501200i −0.572513 + 1.76202i −0.862402 2.65420i −0.809017 + 0.587785i 0.147308
157.2 0.275227 0.199964i 0.309017 + 0.951057i −0.582270 + 1.79204i 3.18709 + 2.31555i 0.275227 + 0.199964i −0.203890 + 0.627508i 0.408342 + 1.25675i −0.809017 + 0.587785i 1.34020
157.3 1.72363 1.25229i 0.309017 + 0.951057i 0.784639 2.41487i 1.18874 + 0.863672i 1.72363 + 1.25229i 0.349352 1.07520i −0.354958 1.09245i −0.809017 + 0.587785i 3.13053
196.1 −0.566948 + 1.74489i −0.809017 + 0.587785i −1.10516 0.802947i −0.477448 1.46943i −0.566948 1.74489i −0.675271 0.490613i −0.940958 + 0.683646i 0.309017 0.951057i 2.83468
196.2 0.135246 0.416243i −0.809017 + 0.587785i 1.46307 + 1.06298i 0.397042 + 1.22197i 0.135246 + 0.416243i 1.16309 + 0.845038i 1.34849 0.979734i 0.309017 0.951057i 0.562336
196.3 0.622685 1.91643i −0.809017 + 0.587785i −1.66692 1.21109i −0.155663 0.479080i 0.622685 + 1.91643i 2.43923 + 1.77220i −0.0985128 + 0.0715738i 0.309017 0.951057i −1.01505
235.1 −0.689843 0.501200i 0.309017 0.951057i −0.393353 1.21061i −0.139763 + 0.101543i −0.689843 + 0.501200i −0.572513 1.76202i −0.862402 + 2.65420i −0.809017 0.587785i 0.147308
235.2 0.275227 + 0.199964i 0.309017 0.951057i −0.582270 1.79204i 3.18709 2.31555i 0.275227 0.199964i −0.203890 0.627508i 0.408342 1.25675i −0.809017 0.587785i 1.34020
235.3 1.72363 + 1.25229i 0.309017 0.951057i 0.784639 + 2.41487i 1.18874 0.863672i 1.72363 1.25229i 0.349352 + 1.07520i −0.354958 + 1.09245i −0.809017 0.587785i 3.13053
313.1 −0.566948 1.74489i −0.809017 0.587785i −1.10516 + 0.802947i −0.477448 + 1.46943i −0.566948 + 1.74489i −0.675271 + 0.490613i −0.940958 0.683646i 0.309017 + 0.951057i 2.83468
313.2 0.135246 + 0.416243i −0.809017 0.587785i 1.46307 1.06298i 0.397042 1.22197i 0.135246 0.416243i 1.16309 0.845038i 1.34849 + 0.979734i 0.309017 + 0.951057i 0.562336
313.3 0.622685 + 1.91643i −0.809017 0.587785i −1.66692 + 1.21109i −0.155663 + 0.479080i 0.622685 1.91643i 2.43923 1.77220i −0.0985128 0.0715738i 0.309017 + 0.951057i −1.01505
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 313.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.n.a 12
11.c even 5 1 inner 429.2.n.a 12
11.c even 5 1 4719.2.a.bg 6
11.d odd 10 1 4719.2.a.bh 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.n.a 12 1.a even 1 1 trivial
429.2.n.a 12 11.c even 5 1 inner
4719.2.a.bg 6 11.c even 5 1
4719.2.a.bh 6 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T + 14 T^{2} - 17 T^{3} + 16 T^{4} + 24 T^{5} + 43 T^{6} - 26 T^{7} + 29 T^{8} - 15 T^{9} + 9 T^{10} - 3 T^{11} + T^{12} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \)
$5$ \( 1 + 9 T + 34 T^{2} + 19 T^{3} + 87 T^{4} - 125 T^{5} + 180 T^{6} - 148 T^{7} + 108 T^{8} - 57 T^{9} + 30 T^{10} - 8 T^{11} + T^{12} \)
$7$ \( 25 + 25 T + 65 T^{2} + 10 T^{3} + 36 T^{4} - 5 T^{5} + 47 T^{6} - 50 T^{7} + 49 T^{8} - 20 T^{9} + 13 T^{10} - 5 T^{11} + T^{12} \)
$11$ \( 1771561 + 966306 T + 248897 T^{2} + 42592 T^{3} + 19118 T^{4} + 7997 T^{5} + 2613 T^{6} + 727 T^{7} + 158 T^{8} + 32 T^{9} + 17 T^{10} + 6 T^{11} + T^{12} \)
$13$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \)
$17$ \( 10201 - 9696 T - 6181 T^{2} + 19411 T^{3} + 40718 T^{4} + 20743 T^{5} + 14259 T^{6} + 4954 T^{7} + 2027 T^{8} + 502 T^{9} + 108 T^{10} + 14 T^{11} + T^{12} \)
$19$ \( 10201 - 17170 T + 11042 T^{2} + 99 T^{3} + 824 T^{4} - 1634 T^{5} + 1274 T^{6} - 272 T^{7} + 312 T^{8} - 8 T^{9} + 34 T^{10} + 2 T^{11} + T^{12} \)
$23$ \( ( -181 + 14 T + 135 T^{2} - 17 T^{3} - 26 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$29$ \( 24025 + 38750 T + 105790 T^{2} + 64100 T^{3} + 72096 T^{4} - 13148 T^{5} - 2182 T^{6} + 5414 T^{7} + 3230 T^{8} + 359 T^{9} + 78 T^{10} + 12 T^{11} + T^{12} \)
$31$ \( 83631025 - 17512675 T + 26420885 T^{2} - 15880700 T^{3} + 5180521 T^{4} - 127612 T^{5} + 139940 T^{6} - 14737 T^{7} + 1179 T^{8} + 467 T^{9} + 145 T^{10} + 12 T^{11} + T^{12} \)
$37$ \( 28561 + 105456 T + 152438 T^{2} - 68497 T^{3} + 682782 T^{4} - 440104 T^{5} + 143809 T^{6} - 28008 T^{7} + 6928 T^{8} - 1116 T^{9} + 109 T^{10} - 4 T^{11} + T^{12} \)
$41$ \( 15625 + 31250 T + 50000 T^{2} + 68750 T^{3} + 105000 T^{4} + 47500 T^{5} + 30000 T^{6} + 2250 T^{7} + 950 T^{8} - 175 T^{9} + 30 T^{10} + 10 T^{11} + T^{12} \)
$43$ \( ( 9001 - 5662 T - 3025 T^{2} + 1644 T^{3} - 89 T^{4} - 14 T^{5} + T^{6} )^{2} \)
$47$ \( 171583801 + 864534 T + 38474264 T^{2} + 13382764 T^{3} + 4614157 T^{4} - 1993370 T^{5} + 714335 T^{6} - 146953 T^{7} + 25208 T^{8} - 3357 T^{9} + 395 T^{10} - 28 T^{11} + T^{12} \)
$53$ \( 24025 - 164300 T + 539810 T^{2} - 770610 T^{3} + 1021896 T^{4} + 417954 T^{5} + 183410 T^{6} + 80556 T^{7} + 28319 T^{8} + 4416 T^{9} + 465 T^{10} + 29 T^{11} + T^{12} \)
$59$ \( 5041 + 75260 T + 489653 T^{2} + 1061687 T^{3} + 2661079 T^{4} - 617893 T^{5} + 410126 T^{6} - 8876 T^{7} - 2433 T^{8} + 174 T^{9} + 171 T^{10} + 11 T^{11} + T^{12} \)
$61$ \( 10730680921 + 412491398 T - 472331018 T^{2} + 69037029 T^{3} + 132680110 T^{4} + 39614802 T^{5} + 7773832 T^{6} + 991532 T^{7} + 97140 T^{8} + 6854 T^{9} + 452 T^{10} + 18 T^{11} + T^{12} \)
$67$ \( ( 631 + 3852 T + 5050 T^{2} + 2397 T^{3} + 452 T^{4} + 36 T^{5} + T^{6} )^{2} \)
$71$ \( 38023050025 - 2910300375 T + 2004681785 T^{2} - 391037380 T^{3} + 66537771 T^{4} - 6909820 T^{5} + 1390108 T^{6} - 86850 T^{7} + 5884 T^{8} - 390 T^{9} + 142 T^{10} - 10 T^{11} + T^{12} \)
$73$ \( 608658241 + 529020253 T + 157436706 T^{2} - 22793348 T^{3} + 4033262 T^{4} + 2811915 T^{5} + 1034385 T^{6} + 172751 T^{7} + 26673 T^{8} + 2191 T^{9} + 260 T^{10} + 11 T^{11} + T^{12} \)
$79$ \( 39300361 - 43312521 T + 52173714 T^{2} - 26997981 T^{3} + 12101367 T^{4} - 4742835 T^{5} + 1533215 T^{6} - 245953 T^{7} + 23448 T^{8} - 692 T^{9} - 10 T^{10} + 7 T^{11} + T^{12} \)
$83$ \( 52113961 + 47421611 T + 151489333 T^{2} - 20355908 T^{3} + 12719865 T^{4} + 691144 T^{5} - 28792 T^{6} - 84199 T^{7} + 22165 T^{8} - 2037 T^{9} + 183 T^{10} - 16 T^{11} + T^{12} \)
$89$ \( ( -65869 - 34386 T - 3295 T^{2} + 1229 T^{3} + 338 T^{4} + 31 T^{5} + T^{6} )^{2} \)
$97$ \( 252083322241 - 192756161364 T + 122896070978 T^{2} - 40334225187 T^{3} + 8290555082 T^{4} - 1177917834 T^{5} + 124617524 T^{6} - 10187208 T^{7} + 668948 T^{8} - 36206 T^{9} + 1634 T^{10} - 54 T^{11} + T^{12} \)
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