Properties

Label 570.2.u.i
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: \(x^{12} + 24 x^{10} + 216 x^{8} + 905 x^{6} + 1770 x^{4} + 1395 x^{2} + 361\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 24 x^{10} + 216 x^{8} + 905 x^{6} + 1770 x^{4} + 1395 x^{2} + 361\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{11} + 2 \nu^{10} - 27 \nu^{9} + 28 \nu^{8} - 277 \nu^{7} + 38 \nu^{6} - 1304 \nu^{5} - 774 \nu^{4} - 2566 \nu^{3} - 2742 \nu^{2} - 1177 \nu - 1520 \)\()/304\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} - 2 \nu^{10} - 27 \nu^{9} - 28 \nu^{8} - 277 \nu^{7} - 38 \nu^{6} - 1304 \nu^{5} + 774 \nu^{4} - 2566 \nu^{3} + 2742 \nu^{2} - 1177 \nu + 1520 \)\()/304\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{10} + 211 \nu^{8} + 1387 \nu^{6} + 3742 \nu^{4} + 4204 \nu^{2} + 2109 \)\()/304\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{11} + 17 \nu^{10} + 55 \nu^{9} + 333 \nu^{8} + 315 \nu^{7} + 2185 \nu^{6} + 530 \nu^{5} + 5334 \nu^{4} - 24 \nu^{3} + 3464 \nu^{2} + 417 \nu + 171 \)\()/608\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{11} + 11 \nu^{10} - 5 \nu^{9} + 211 \nu^{8} - 829 \nu^{7} + 1387 \nu^{6} - 6690 \nu^{5} + 3742 \nu^{4} - 16628 \nu^{3} + 4204 \nu^{2} - 8827 \nu + 2109 \)\()/608\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} - 23 \nu^{10} - \nu^{9} - 455 \nu^{8} + 239 \nu^{7} - 2983 \nu^{6} + 2078 \nu^{5} - 6926 \nu^{4} + 5460 \nu^{3} - 3028 \nu^{2} + 4065 \nu + 551 \)\()/608\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{11} - 17 \nu^{10} + 55 \nu^{9} - 333 \nu^{8} + 315 \nu^{7} - 2185 \nu^{6} + 530 \nu^{5} - 5334 \nu^{4} - 24 \nu^{3} - 3464 \nu^{2} + 417 \nu - 171 \)\()/608\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{11} - 7 \nu^{10} + 53 \nu^{9} - 155 \nu^{8} + 793 \nu^{7} - 1311 \nu^{6} + 4686 \nu^{5} - 5290 \nu^{4} + 10592 \nu^{3} - 9384 \nu^{2} + 6419 \nu - 3933 \)\()/608\)
\(\beta_{9}\)\(=\)\((\)\( -7 \nu^{11} - 11 \nu^{10} - 167 \nu^{9} - 211 \nu^{8} - 1479 \nu^{7} - 1387 \nu^{6} - 5822 \nu^{5} - 3742 \nu^{4} - 8996 \nu^{3} - 4204 \nu^{2} - 1769 \nu - 1805 \)\()/608\)
\(\beta_{10}\)\(=\)\((\)\( 13 \nu^{11} + 49 \nu^{10} + 277 \nu^{9} + 1009 \nu^{8} + 2109 \nu^{7} + 7201 \nu^{6} + 6882 \nu^{5} + 20994 \nu^{4} + 8948 \nu^{3} + 23356 \nu^{2} + 3515 \nu + 8303 \)\()/608\)
\(\beta_{11}\)\(=\)\((\)\( 9 \nu^{11} + 197 \nu^{9} + 1545 \nu^{7} + 5238 \nu^{5} + 7304 \nu^{3} + 2979 \nu + 152 \)\()/304\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 1\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{11} - 7 \beta_{9} - 4 \beta_{8} + 21 \beta_{7} - 4 \beta_{6} - 7 \beta_{5} + 13 \beta_{4} - \beta_{2} + 3 \beta_{1} + 7\)\()/3\)
\(\nu^{4}\)\(=\)\(-\beta_{11} + 2 \beta_{10} + \beta_{9} - 10 \beta_{8} - 4 \beta_{7} + 8 \beta_{6} - \beta_{5} + 4 \beta_{4} - 7 \beta_{3} - 12 \beta_{2} + 2 \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\((\)\(52 \beta_{11} + 64 \beta_{9} + 16 \beta_{8} - 141 \beta_{7} + 16 \beta_{6} + 46 \beta_{5} - 109 \beta_{4} + 9 \beta_{3} - 23 \beta_{2} - 39 \beta_{1} - 58\)\()/3\)
\(\nu^{6}\)\(=\)\(14 \beta_{11} - 28 \beta_{10} - 14 \beta_{9} + 87 \beta_{8} + 6 \beta_{7} - 59 \beta_{6} + 14 \beta_{5} - 6 \beta_{4} + 50 \beta_{3} + 116 \beta_{2} - 29 \beta_{1} - 167\)
\(\nu^{7}\)\(=\)\((\)\(-424 \beta_{11} - 574 \beta_{9} - 52 \beta_{8} + 1020 \beta_{7} - 52 \beta_{6} - 310 \beta_{5} + 916 \beta_{4} - 132 \beta_{3} + 290 \beta_{2} + 342 \beta_{1} + 499\)\()/3\)
\(\nu^{8}\)\(=\)\(-149 \beta_{11} + 298 \beta_{10} + 149 \beta_{9} - 733 \beta_{8} + 99 \beta_{7} + 435 \beta_{6} - 149 \beta_{5} - 99 \beta_{4} - 384 \beta_{3} - 1037 \beta_{2} + 304 \beta_{1} + 1253\)
\(\nu^{9}\)\(=\)\((\)\(3610 \beta_{11} + 5026 \beta_{9} + 19 \beta_{8} - 7815 \beta_{7} + 19 \beta_{6} + 2128 \beta_{5} - 7777 \beta_{4} + 1449 \beta_{3} - 2639 \beta_{2} - 2658 \beta_{1} - 4318\)\()/3\)
\(\nu^{10}\)\(=\)\(1433 \beta_{11} - 2866 \beta_{10} - 1433 \beta_{9} + 6110 \beta_{8} - 1677 \beta_{7} - 3244 \beta_{6} + 1433 \beta_{5} + 1677 \beta_{4} + 3088 \beta_{3} + 8965 \beta_{2} - 2855 \beta_{1} - 9805\)
\(\nu^{11}\)\(=\)\((\)\(-31045 \beta_{11} - 43375 \beta_{9} + 2114 \beta_{8} + 61974 \beta_{7} + 2114 \beta_{6} - 14785 \beta_{5} + 66202 \beta_{4} - 14295 \beta_{3} + 21848 \beta_{2} + 19734 \beta_{1} + 37210\)\()/3\)

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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
0.918492i
2.88811i
0.918492i
2.88811i
2.57727i
1.89323i
2.57727i
1.89323i
0.728740i
2.01431i
0.728740i
2.01431i
−0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i 0.766044 0.642788i 0.173648 0.984808i −0.814143 + 1.41014i −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 0.487791 0.844879i −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 −0.814143 1.41014i −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 0.487791 + 0.844879i −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 −2.15664 3.73541i −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 0.716948 + 1.24179i −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 −2.15664 + 3.73541i −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 0.716948 1.24179i −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 −1.21767 2.10906i −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 1.48371 + 2.56987i −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 −1.21767 + 2.10906i −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 1.48371 2.56987i −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.i 12
19.e even 9 1 inner 570.2.u.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.i 12 1.a even 1 1 trivial
570.2.u.i 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

$p$ $F_p(T)$
$2$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$3$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$5$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$7$ \( 5041 - 4047 T + 7296 T^{2} - 1153 T^{3} + 3951 T^{4} - 270 T^{5} + 1503 T^{6} + 180 T^{7} + 261 T^{8} + 17 T^{9} + 24 T^{10} + 3 T^{11} + T^{12} \)
$11$ \( 1 + 39 T + 1602 T^{2} - 3077 T^{3} + 8145 T^{4} + 2160 T^{5} + 2547 T^{6} + 882 T^{7} + 675 T^{8} + 217 T^{9} + 66 T^{10} + 9 T^{11} + T^{12} \)
$13$ \( 1682209 + 4027185 T + 4206003 T^{2} + 2384231 T^{3} + 764901 T^{4} + 143388 T^{5} + 24774 T^{6} + 6003 T^{7} + 1116 T^{8} + 239 T^{9} + 54 T^{10} + 3 T^{11} + T^{12} \)
$17$ \( 23104 + 43776 T + 353664 T^{2} + 190352 T^{3} + 720 T^{4} - 21942 T^{5} + 489 T^{6} + 1368 T^{7} - 234 T^{8} - 16 T^{9} + 36 T^{10} - 9 T^{11} + T^{12} \)
$19$ \( 47045881 + 7428297 T + 1563852 T^{2} - 342950 T^{3} - 42237 T^{4} + 513 T^{5} + 1545 T^{6} + 27 T^{7} - 117 T^{8} - 50 T^{9} + 12 T^{10} + 3 T^{11} + T^{12} \)
$23$ \( 361 + 3192 T + 1830 T^{2} - 31855 T^{3} + 72279 T^{4} - 67653 T^{5} + 34557 T^{6} - 11907 T^{7} + 3303 T^{8} - 625 T^{9} + 102 T^{10} - 12 T^{11} + T^{12} \)
$29$ \( 576 + 2592 T + 6048 T^{2} + 11808 T^{3} + 18828 T^{4} + 18162 T^{5} + 9651 T^{6} + 1935 T^{7} + 126 T^{8} + 42 T^{9} + 15 T^{10} + 3 T^{11} + T^{12} \)
$31$ \( 104366656 + 31751328 T + 20202576 T^{2} + 5149232 T^{3} + 2305548 T^{4} + 526032 T^{5} + 153513 T^{6} + 26649 T^{7} + 5949 T^{8} + 854 T^{9} + 141 T^{10} + 12 T^{11} + T^{12} \)
$37$ \( ( 73 + 465 T - 417 T^{2} - 151 T^{3} + 123 T^{4} - 21 T^{5} + T^{6} )^{2} \)
$41$ \( 1923769 - 890454 T + 1843140 T^{2} - 98977 T^{3} - 195714 T^{4} + 58527 T^{5} + 3939 T^{6} - 5904 T^{7} + 2907 T^{8} - 958 T^{9} + 192 T^{10} - 21 T^{11} + T^{12} \)
$43$ \( 46656 - 209952 T + 606528 T^{2} - 1185840 T^{3} + 1215000 T^{4} - 397548 T^{5} - 11799 T^{6} + 6966 T^{7} + 8721 T^{8} - 1359 T^{9} + 144 T^{10} - 9 T^{11} + T^{12} \)
$47$ \( 4915089 + 7961247 T + 5567616 T^{2} + 2018277 T^{3} + 447669 T^{4} + 90396 T^{5} + 10290 T^{6} - 4887 T^{7} - 432 T^{8} + 27 T^{9} + 24 T^{10} + T^{12} \)
$53$ \( 1285437609 - 1761171066 T + 861993441 T^{2} - 217545642 T^{3} + 66611736 T^{4} + 2546748 T^{5} + 980157 T^{6} + 265320 T^{7} + 15651 T^{8} - 684 T^{9} + 42 T^{10} + 18 T^{11} + T^{12} \)
$59$ \( 23409 - 776628 T + 10976715 T^{2} + 81108 T^{3} + 546588 T^{4} + 135432 T^{5} - 63477 T^{6} - 16308 T^{7} + 1836 T^{8} + 966 T^{9} + 162 T^{10} + 15 T^{11} + T^{12} \)
$61$ \( 427331584 + 581462016 T + 213760512 T^{2} - 20800832 T^{3} - 10703520 T^{4} + 4682376 T^{5} + 623913 T^{6} - 202923 T^{7} + 37071 T^{8} - 3200 T^{9} + 330 T^{10} - 9 T^{11} + T^{12} \)
$67$ \( 60341824 - 84826560 T + 270497664 T^{2} - 175332632 T^{3} + 39978072 T^{4} - 3149928 T^{5} + 471849 T^{6} + 8163 T^{7} - 10278 T^{8} + 1084 T^{9} + 96 T^{10} - 18 T^{11} + T^{12} \)
$71$ \( 1254293056 + 224820768 T - 69326880 T^{2} - 22186168 T^{3} + 4666572 T^{4} + 538092 T^{5} - 145527 T^{6} + 15417 T^{7} + 3762 T^{8} - 1060 T^{9} + 186 T^{10} - 12 T^{11} + T^{12} \)
$73$ \( 13809070144 - 4093648032 T + 304979904 T^{2} + 155799760 T^{3} + 20460312 T^{4} - 1044468 T^{5} + 86649 T^{6} - 31014 T^{7} - 6012 T^{8} + 400 T^{9} + 114 T^{10} + 9 T^{11} + T^{12} \)
$79$ \( 1430654976 - 1339877376 T + 7362330624 T^{2} - 1160858304 T^{3} + 121453056 T^{4} + 3378888 T^{5} + 627753 T^{6} + 16821 T^{7} + 42048 T^{8} - 1944 T^{9} - 291 T^{10} + 9 T^{11} + T^{12} \)
$83$ \( 60559303744 - 44930747040 T + 31932754800 T^{2} - 3001043008 T^{3} + 818767260 T^{4} - 68626620 T^{5} + 14985765 T^{6} - 967500 T^{7} + 111645 T^{8} - 4366 T^{9} + 465 T^{10} - 15 T^{11} + T^{12} \)
$89$ \( 86322681 - 85709475 T - 24719877 T^{2} + 22745277 T^{3} + 20784096 T^{4} + 3337236 T^{5} + 327549 T^{6} + 46332 T^{7} + 2700 T^{8} + 237 T^{9} + 123 T^{10} - 3 T^{11} + T^{12} \)
$97$ \( 51623475264 + 23679617760 T + 5282762976 T^{2} + 845357184 T^{3} + 133789140 T^{4} + 25859970 T^{5} + 5637693 T^{6} + 1021050 T^{7} + 140976 T^{8} + 14307 T^{9} + 1050 T^{10} + 48 T^{11} + T^{12} \)
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