Properties

Label 570.2.u.j
Level $570$
Weight $2$
Character orbit 570.u
Analytic conductor $4.551$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 \)\()/218\)
\(\beta_{2}\)\(=\)\((\)\( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706 \)\()/218\)
\(\beta_{3}\)\(=\)\((\)\( -27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 \)\()/218\)
\(\beta_{4}\)\(=\)\((\)\( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524 \)\()/218\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555 \)\()/218\)
\(\beta_{6}\)\(=\)\((\)\( -27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83 \)\()/218\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350 \)\()/218\)
\(\beta_{9}\)\(=\)\((\)\( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201 \)\()/218\)
\(\beta_{10}\)\(=\)\((\)\( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194 \)\()/218\)
\(\beta_{11}\)\(=\)\((\)\( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469 \)\()/218\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_{1} - 7\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} - 2 \beta_{4} - 31 \beta_{3} - \beta_{2} - 7 \beta_{1} + 32\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 29 \beta_{1} + 77\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} - 17 \beta_{4} + 161 \beta_{3} + 17 \beta_{2} + 104 \beta_{1} - 121\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} - 46 \beta_{5} - 56 \beta_{4} + 212 \beta_{3} + 149 \beta_{2} - 82 \beta_{1} - 535\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + 515 \beta_{5} + 151 \beta_{4} - 724 \beta_{3} - 19 \beta_{2} - 829 \beta_{1} + 257\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + 812 \beta_{5} + 640 \beta_{4} - 1951 \beta_{3} - 1108 \beta_{2} - 406 \beta_{1} + 3359\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} - 2308 \beta_{5} - 521 \beta_{4} + 2375 \beta_{3} - 913 \beta_{2} + 4949 \beta_{1} + 1451\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} - 7273 \beta_{5} - 5168 \beta_{4} + 14021 \beta_{3} + 6587 \beta_{2} + 7973 \beta_{1} - 18736\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.500000 + 1.96356i
0.500000 0.677980i
0.500000 1.96356i
0.500000 + 0.677980i
0.500000 + 0.168222i
0.500000 + 1.80139i
0.500000 0.168222i
0.500000 1.80139i
0.500000 2.42499i
0.500000 + 1.74095i
0.500000 + 2.42499i
0.500000 1.74095i
−0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i 0.766044 0.642788i −0.173648 + 0.984808i −0.897714 + 1.55489i −0.500000 0.866025i −0.939693 + 0.342020i −0.939693 + 0.342020i
61.2 −0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i 0.766044 0.642788i −0.173648 + 0.984808i −0.308023 + 0.533512i −0.500000 0.866025i −0.939693 + 0.342020i −0.939693 + 0.342020i
271.1 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i 0.766044 + 0.642788i −0.173648 0.984808i −0.897714 1.55489i −0.500000 + 0.866025i −0.939693 0.342020i −0.939693 0.342020i
271.2 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i 0.766044 + 0.642788i −0.173648 0.984808i −0.308023 0.533512i −0.500000 + 0.866025i −0.939693 0.342020i −0.939693 0.342020i
301.1 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i 0.173648 0.984808i 0.939693 + 0.342020i −0.965167 1.67172i −0.500000 + 0.866025i 0.766044 0.642788i 0.766044 0.642788i
301.2 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i 0.173648 0.984808i 0.939693 + 0.342020i 2.05756 + 3.56380i −0.500000 + 0.866025i 0.766044 0.642788i 0.766044 0.642788i
481.1 0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i 0.173648 + 0.984808i 0.939693 0.342020i −0.965167 + 1.67172i −0.500000 0.866025i 0.766044 + 0.642788i 0.766044 + 0.642788i
481.2 0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i 0.173648 + 0.984808i 0.939693 0.342020i 2.05756 3.56380i −0.500000 0.866025i 0.766044 + 0.642788i 0.766044 + 0.642788i
511.1 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i −0.939693 + 0.342020i −0.766044 + 0.642788i −0.284816 0.493316i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.173648 + 0.984808i
511.2 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i −0.939693 + 0.342020i −0.766044 + 0.642788i 1.89816 + 3.28770i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.173648 + 0.984808i
541.1 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −0.939693 0.342020i −0.766044 0.642788i −0.284816 + 0.493316i −0.500000 0.866025i 0.173648 0.984808i 0.173648 0.984808i
541.2 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i −0.939693 0.342020i −0.766044 0.642788i 1.89816 3.28770i −0.500000 0.866025i 0.173648 0.984808i 0.173648 0.984808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.u.j 12
19.e even 9 1 inner 570.2.u.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.j 12 1.a even 1 1 trivial
570.2.u.j 12 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\(T_{7}^{12} - \cdots\)
\(T_{11}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$3$ \( ( 1 - T^{3} + T^{6} )^{2} \)
$5$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$7$ \( 361 + 1425 T + 3972 T^{2} + 5879 T^{3} + 6579 T^{4} + 3672 T^{5} + 1857 T^{6} + 342 T^{7} + 189 T^{8} + 11 T^{9} + 24 T^{10} - 3 T^{11} + T^{12} \)
$11$ \( 927369 + 78003 T + 483246 T^{2} + 104355 T^{3} + 207765 T^{4} + 32724 T^{5} + 25731 T^{6} - 486 T^{7} + 1755 T^{8} - 15 T^{9} + 54 T^{10} - 3 T^{11} + T^{12} \)
$13$ \( 47961 - 135999 T + 198045 T^{2} - 148797 T^{3} + 127773 T^{4} - 66690 T^{5} + 18546 T^{6} - 3213 T^{7} + 486 T^{8} - 81 T^{9} + 30 T^{10} - 9 T^{11} + T^{12} \)
$17$ \( 19855936 - 54969216 T + 53294976 T^{2} - 19966544 T^{3} + 5743872 T^{4} - 1337382 T^{5} + 226089 T^{6} - 29844 T^{7} + 3510 T^{8} - 260 T^{9} + 48 T^{10} - 9 T^{11} + T^{12} \)
$19$ \( 47045881 - 22284891 T + 9383112 T^{2} - 3690142 T^{3} + 1153395 T^{4} - 308655 T^{5} + 78033 T^{6} - 16245 T^{7} + 3195 T^{8} - 538 T^{9} + 72 T^{10} - 9 T^{11} + T^{12} \)
$23$ \( 178409449 - 263426754 T + 153991770 T^{2} - 44062235 T^{3} + 7197093 T^{4} - 973521 T^{5} + 238347 T^{6} - 45711 T^{7} + 7569 T^{8} - 1277 T^{9} + 150 T^{10} - 12 T^{11} + T^{12} \)
$29$ \( 42094144 - 53487072 T + 28874016 T^{2} - 9909920 T^{3} + 2853900 T^{4} - 520398 T^{5} + 138891 T^{6} - 40365 T^{7} + 10908 T^{8} - 2342 T^{9} + 333 T^{10} - 27 T^{11} + T^{12} \)
$31$ \( 36675136 - 49780320 T + 43295952 T^{2} - 23583184 T^{3} + 9619164 T^{4} - 2778912 T^{5} + 623937 T^{6} - 99567 T^{7} + 13509 T^{8} - 1366 T^{9} + 159 T^{10} - 12 T^{11} + T^{12} \)
$37$ \( ( 85483 + 6729 T - 7833 T^{2} - 1295 T^{3} + 57 T^{4} + 21 T^{5} + T^{6} )^{2} \)
$41$ \( 998001 - 161838 T + 4681638 T^{2} + 2953179 T^{3} + 861192 T^{4} + 159651 T^{5} + 43713 T^{6} + 19278 T^{7} + 7263 T^{8} + 1890 T^{9} + 306 T^{10} + 27 T^{11} + T^{12} \)
$43$ \( 16192576 + 18494304 T + 89889408 T^{2} + 34108784 T^{3} + 20263320 T^{4} + 8552268 T^{5} + 1665153 T^{6} + 129186 T^{7} + 1485 T^{8} + 737 T^{9} + 264 T^{10} + 27 T^{11} + T^{12} \)
$47$ \( 26286129 - 7890453 T - 9127944 T^{2} - 2746323 T^{3} + 7633071 T^{4} - 2634966 T^{5} + 161358 T^{6} - 549 T^{7} + 10494 T^{8} + 975 T^{9} + 174 T^{10} + 6 T^{11} + T^{12} \)
$53$ \( 16922627569 - 10028146656 T + 2374169427 T^{2} + 729750860 T^{3} + 197815572 T^{4} + 29733318 T^{5} + 1967103 T^{6} - 19872 T^{7} - 19269 T^{8} - 2278 T^{9} + 66 T^{10} + 18 T^{11} + T^{12} \)
$59$ \( 289034001 + 58755456 T + 194226903 T^{2} + 97857828 T^{3} + 20303784 T^{4} + 2366496 T^{5} + 212787 T^{6} - 5076 T^{7} - 6156 T^{8} - 726 T^{9} + 54 T^{10} + 15 T^{11} + T^{12} \)
$61$ \( 21827584 + 114819072 T + 182195712 T^{2} + 37226816 T^{3} - 3051936 T^{4} - 3160512 T^{5} + 38961 T^{6} + 30987 T^{7} + 15615 T^{8} + 1808 T^{9} + 210 T^{10} + 9 T^{11} + T^{12} \)
$67$ \( 5184 - 171072 T + 43058304 T^{2} - 61651800 T^{3} + 37057176 T^{4} + 625968 T^{5} - 174231 T^{6} - 272673 T^{7} + 70362 T^{8} - 9156 T^{9} + 792 T^{10} - 42 T^{11} + T^{12} \)
$71$ \( 1455575104 - 626761056 T + 27195936 T^{2} - 8311192 T^{3} + 25721100 T^{4} - 11881404 T^{5} + 2975025 T^{6} - 513369 T^{7} + 64854 T^{8} - 5992 T^{9} + 426 T^{10} - 24 T^{11} + T^{12} \)
$73$ \( 504990784 + 898790112 T + 489675072 T^{2} + 92415952 T^{3} + 51984072 T^{4} - 37503324 T^{5} + 5924313 T^{6} - 379746 T^{7} + 33930 T^{8} + 334 T^{9} + 108 T^{10} - 15 T^{11} + T^{12} \)
$79$ \( 1478656 + 16430592 T^{2} - 20836160 T^{3} + 8684640 T^{4} - 2179008 T^{5} + 499113 T^{6} + 726273 T^{7} + 182484 T^{8} + 21988 T^{9} + 1503 T^{10} + 57 T^{11} + T^{12} \)
$83$ \( 17207142976 - 942893088 T + 2042919024 T^{2} - 594776576 T^{3} + 251291916 T^{4} - 43655148 T^{5} + 7127733 T^{6} - 676944 T^{7} + 71667 T^{8} - 5114 T^{9} + 453 T^{10} - 21 T^{11} + T^{12} \)
$89$ \( 17461243881 - 1395012537 T + 1523837313 T^{2} - 535353579 T^{3} - 43043958 T^{4} + 10184166 T^{5} + 3498909 T^{6} + 942066 T^{7} + 183132 T^{8} + 21417 T^{9} + 1479 T^{10} + 57 T^{11} + T^{12} \)
$97$ \( 399252114496 + 84793621344 T + 99223693728 T^{2} + 6627032128 T^{3} + 3094922628 T^{4} - 108650790 T^{5} - 18185535 T^{6} + 1068552 T^{7} + 155232 T^{8} - 3725 T^{9} - 534 T^{10} + 6 T^{11} + T^{12} \)
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