Properties

Label 570.2.m.b
Level $570$
Weight $2$
Character orbit 570.m
Analytic conductor $4.551$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 108 x^{16} + 1318 x^{12} + 4652 x^{8} + 5057 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 108 x^{16} + 1318 x^{12} + 4652 x^{8} + 5057 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -1233 \nu^{19} - 120336 \nu^{15} - 289166 \nu^{11} + 6031344 \nu^{7} + 9961151 \nu^{3} \)\()/5112544\)
\(\beta_{2}\)\(=\)\((\)\( 1843 \nu^{18} + 199306 \nu^{14} + 2457754 \nu^{10} + 8963426 \nu^{6} + 11554103 \nu^{2} \)\()/2556272\)
\(\beta_{3}\)\(=\)\((\)\( 2595 \nu^{18} + 273087 \nu^{14} + 2661844 \nu^{10} + 4366379 \nu^{6} - 932561 \nu^{2} \)\()/1278136\)
\(\beta_{4}\)\(=\)\((\)\( 4296 \nu^{18} - 7207 \nu^{16} + 477582 \nu^{14} - 770625 \nu^{12} + 7084096 \nu^{10} - 8670841 \nu^{8} + 32921622 \nu^{6} - 24076543 \nu^{4} + 42067188 \nu^{2} - 13013696 \)\()/5112544\)
\(\beta_{5}\)\(=\)\((\)\( -3686 \nu^{19} + 4131 \nu^{17} - 398612 \nu^{15} + 450983 \nu^{13} - 4915508 \nu^{11} + 5923921 \nu^{9} - 17926852 \nu^{7} + 21144461 \nu^{5} - 23108206 \nu^{3} + 8864472 \nu \)\()/5112544\)
\(\beta_{6}\)\(=\)\((\)\( -4296 \nu^{18} - 7207 \nu^{16} - 477582 \nu^{14} - 770625 \nu^{12} - 7084096 \nu^{10} - 8670841 \nu^{8} - 32921622 \nu^{6} - 24076543 \nu^{4} - 42067188 \nu^{2} - 13013696 \)\()/5112544\)
\(\beta_{7}\)\(=\)\((\)\( -10283 \nu^{17} - 1090267 \nu^{13} - 11417761 \nu^{9} - 27008625 \nu^{5} - 12050376 \nu \)\()/5112544\)
\(\beta_{8}\)\(=\)\((\)\( -14579 \nu^{18} - 5890 \nu^{16} - 1567849 \nu^{14} - 610604 \nu^{12} - 18501857 \nu^{10} - 5104050 \nu^{8} - 59930247 \nu^{6} - 3630112 \nu^{4} - 54117564 \nu^{2} + 9123648 \)\()/5112544\)
\(\beta_{9}\)\(=\)\((\)\( 14579 \nu^{18} - 5890 \nu^{16} + 1567849 \nu^{14} - 610604 \nu^{12} + 18501857 \nu^{10} - 5104050 \nu^{8} + 59930247 \nu^{6} - 3630112 \nu^{4} + 54117564 \nu^{2} + 9123648 \)\()/5112544\)
\(\beta_{10}\)\(=\)\((\)\( 30391 \nu^{19} + 3256034 \nu^{15} + 37292880 \nu^{11} + 113829150 \nu^{7} + 98273977 \nu^{3} \)\()/10225088\)
\(\beta_{11}\)\(=\)\((\)\( -14918 \nu^{19} + 1504 \nu^{17} - 1619593 \nu^{15} + 147562 \nu^{13} - 20551431 \nu^{11} + 408180 \nu^{9} - 78087767 \nu^{7} - 9194094 \nu^{5} - 88088723 \nu^{3} - 24973328 \nu \)\()/5112544\)
\(\beta_{12}\)\(=\)\((\)\( 14918 \nu^{19} + 1504 \nu^{17} + 1619593 \nu^{15} + 147562 \nu^{13} + 20551431 \nu^{11} + 408180 \nu^{9} + 78087767 \nu^{7} - 9194094 \nu^{5} + 88088723 \nu^{3} - 24973328 \nu \)\()/5112544\)
\(\beta_{13}\)\(=\)\((\)\( 18604 \nu^{19} - 1504 \nu^{17} + 2018205 \nu^{15} - 147562 \nu^{13} + 25466939 \nu^{11} - 408180 \nu^{9} + 96014619 \nu^{7} + 9194094 \nu^{5} + 111196929 \nu^{3} + 19860784 \nu \)\()/5112544\)
\(\beta_{14}\)\(=\)\((\)\( -18604 \nu^{19} - 1504 \nu^{17} - 2018205 \nu^{15} - 147562 \nu^{13} - 25466939 \nu^{11} - 408180 \nu^{9} - 96014619 \nu^{7} + 9194094 \nu^{5} - 111196929 \nu^{3} + 19860784 \nu \)\()/5112544\)
\(\beta_{15}\)\(=\)\((\)\(10893 \nu^{19} + 3076 \nu^{17} + 13621 \nu^{16} + 1169237 \nu^{15} + 319642 \nu^{13} + 1438589 \nu^{12} + 13586349 \nu^{11} + 2746920 \nu^{9} + 14554471 \nu^{8} + 42003395 \nu^{7} + 2932082 \nu^{5} + 32174759 \nu^{4} + 31009358 \nu^{3} - 963320 \nu + 18284064\)\()/5112544\)
\(\beta_{16}\)\(=\)\((\)\( -14579 \nu^{18} + 23836 \nu^{16} - 1567849 \nu^{14} + 2504338 \nu^{12} - 18501857 \nu^{10} + 24041672 \nu^{8} - 59930247 \nu^{6} + 37863114 \nu^{4} - 54117564 \nu^{2} - 8423808 \)\()/5112544\)
\(\beta_{17}\)\(=\)\((\)\(-10893 \nu^{19} - 21341 \nu^{18} - 3076 \nu^{17} - 1169237 \nu^{15} - 2286103 \nu^{14} - 319642 \nu^{13} - 13586349 \nu^{11} - 26164285 \nu^{10} - 2746920 \nu^{9} - 42003395 \nu^{7} - 80789181 \nu^{6} - 2932082 \nu^{5} - 31009358 \nu^{3} - 76262450 \nu^{2} + 963320 \nu\)\()/5112544\)
\(\beta_{18}\)\(=\)\((\)\(-11787 \nu^{18} - 17752 \nu^{17} - 8973 \nu^{16} - 1237829 \nu^{14} - 1889572 \nu^{13} - 946867 \nu^{12} - 11825941 \nu^{10} - 20478392 \nu^{9} - 9468811 \nu^{8} - 17814531 \nu^{6} - 53319220 \nu^{5} - 17116501 \nu^{4} + 12922952 \nu^{2} - 37373624 \nu - 349920\)\()/5112544\)
\(\beta_{19}\)\(=\)\((\)\(-77979 \nu^{19} - 23574 \nu^{18} + 17946 \nu^{16} - 8384792 \nu^{15} - 2475658 \nu^{14} + 1893734 \nu^{12} - 98874134 \nu^{11} - 23651882 \nu^{10} + 18937622 \nu^{8} - 323323904 \nu^{7} - 35629062 \nu^{6} + 34233002 \nu^{4} - 316937591 \nu^{3} + 25845904 \nu^{2} + 699840\)\()/10225088\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{17} + 2 \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 6 \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{19} - 2 \beta_{16} - 5 \beta_{14} + 5 \beta_{13} - 5 \beta_{12} + 5 \beta_{11} + 8 \beta_{10} - 2 \beta_{8} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{16} + 18 \beta_{15} - 18 \beta_{14} + 9 \beta_{13} - 18 \beta_{12} + 9 \beta_{11} - 18 \beta_{10} + 15 \beta_{9} + 11 \beta_{8} + 9 \beta_{7} + 13 \beta_{6} + 9 \beta_{5} + 13 \beta_{4} - 9 \beta_{1} - 38\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-48 \beta_{18} - 24 \beta_{16} + 59 \beta_{14} + 39 \beta_{13} + 63 \beta_{12} + 43 \beta_{11} - 48 \beta_{9} + 24 \beta_{8} + 76 \beta_{7} - 24 \beta_{6} - 20 \beta_{5} + 24 \beta_{4} - 24 \beta_{3} + 48 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-166 \beta_{17} - 166 \beta_{14} + 83 \beta_{13} - 166 \beta_{12} + 83 \beta_{11} - 166 \beta_{10} - 131 \beta_{9} + 131 \beta_{8} + 83 \beta_{7} - 111 \beta_{6} + 83 \beta_{5} + 111 \beta_{4} + 52 \beta_{3} - 330 \beta_{2} - 83 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-492 \beta_{19} + 246 \beta_{16} + 411 \beta_{14} - 411 \beta_{13} + 359 \beta_{12} - 359 \beta_{11} - 976 \beta_{10} + 246 \beta_{8} - 246 \beta_{6} + 246 \beta_{4} - 246 \beta_{3} + 492 \beta_{2} - 186 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-544 \beta_{16} - 1582 \beta_{15} + 1582 \beta_{14} - 791 \beta_{13} + 1582 \beta_{12} - 791 \beta_{11} + 1582 \beta_{10} - 1637 \beta_{9} - 1093 \beta_{8} - 791 \beta_{7} - 1279 \beta_{6} - 791 \beta_{5} - 1279 \beta_{4} + 791 \beta_{1} + 3122\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(4856 \beta_{18} + 2428 \beta_{16} - 5213 \beta_{14} - 3445 \beta_{13} - 5757 \beta_{12} - 3989 \beta_{11} + 4856 \beta_{9} - 2428 \beta_{8} - 7832 \beta_{7} + 2428 \beta_{6} + 1768 \beta_{5} - 2428 \beta_{4} + 2428 \beta_{3} - 4856 \beta_{2}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(15282 \beta_{17} + 15282 \beta_{14} - 7641 \beta_{13} + 15282 \beta_{12} - 7641 \beta_{11} + 15282 \beta_{10} + 12441 \beta_{9} - 12441 \beta_{8} - 7641 \beta_{7} + 10673 \beta_{6} - 7641 \beta_{5} - 10673 \beta_{4} - 5400 \beta_{3} + 30150 \beta_{2} + 7641 \beta_{1}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(47428 \beta_{19} - 23714 \beta_{16} - 38789 \beta_{14} + 38789 \beta_{13} - 33389 \beta_{12} + 33389 \beta_{11} + 93656 \beta_{10} - 23714 \beta_{8} + 23714 \beta_{6} - 23714 \beta_{4} + 23714 \beta_{3} - 47428 \beta_{2} + 17050 \beta_{1}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(52828 \beta_{16} + 148306 \beta_{15} - 148306 \beta_{14} + 74153 \beta_{13} - 148306 \beta_{12} + 74153 \beta_{11} - 148306 \beta_{10} + 156759 \beta_{9} + 103931 \beta_{8} + 74153 \beta_{7} + 120981 \beta_{6} + 74153 \beta_{5} + 120981 \beta_{4} - 74153 \beta_{1} - 292662\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-461824 \beta_{18} - 230912 \beta_{16} + 489771 \beta_{14} + 324415 \beta_{13} + 542599 \beta_{12} + 377243 \beta_{11} - 461824 \beta_{9} + 230912 \beta_{8} + 746292 \beta_{7} - 230912 \beta_{6} - 165356 \beta_{5} + 230912 \beta_{4} - 230912 \beta_{3} + 461824 \beta_{2}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-1441366 \beta_{17} - 1441366 \beta_{14} + 720683 \beta_{13} - 1441366 \beta_{12} + 720683 \beta_{11} - 1441366 \beta_{10} - 1176507 \beta_{9} + 1176507 \beta_{8} + 720683 \beta_{7} - 1011151 \beta_{6} + 720683 \beta_{5} + 1011151 \beta_{4} + 514652 \beta_{3} - 2844682 \beta_{2} - 720683 \beta_{1}\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-4492972 \beta_{19} + 2246486 \beta_{16} + 3668827 \beta_{14} - 3668827 \beta_{13} + 3154175 \beta_{12} - 3154175 \beta_{11} - 8868288 \beta_{10} + 2246486 \beta_{8} - 2246486 \beta_{6} + 2246486 \beta_{4} - 2246486 \beta_{3} + 4492972 \beta_{2} - 1606722 \beta_{1}\)\()/2\)
\(\nu^{16}\)\(=\)\((\)\(-5007624 \beta_{16} - 14014766 \beta_{15} + 14014766 \beta_{14} - 7007383 \beta_{13} + 14014766 \beta_{12} - 7007383 \beta_{11} + 14014766 \beta_{10} - 14842429 \beta_{9} - 9834805 \beta_{8} - 7007383 \beta_{7} - 11441527 \beta_{6} - 7007383 \beta_{5} - 11441527 \beta_{4} + 7007383 \beta_{1} + 27660770\)\()/2\)
\(\nu^{17}\)\(=\)\((\)\(43699624 \beta_{18} + 21849812 \beta_{16} - 46294061 \beta_{14} - 30672573 \beta_{13} - 51301685 \beta_{12} - 35680197 \beta_{11} + 43699624 \beta_{9} - 21849812 \beta_{8} - 70630800 \beta_{7} + 21849812 \beta_{6} + 15621488 \beta_{5} - 21849812 \beta_{4} + 21849812 \beta_{3} - 43699624 \beta_{2}\)\()/2\)
\(\nu^{18}\)\(=\)\((\)\(136287746 \beta_{17} + 136287746 \beta_{14} - 68143873 \beta_{13} + 136287746 \beta_{12} - 68143873 \beta_{11} + 136287746 \beta_{10} + 111270017 \beta_{9} - 111270017 \beta_{8} - 68143873 \beta_{7} + 95648529 \beta_{6} - 68143873 \beta_{5} - 95648529 \beta_{4} - 48707248 \beta_{3} + 268993286 \beta_{2} + 68143873 \beta_{1}\)\()/2\)
\(\nu^{19}\)\(=\)\((\)\(424999300 \beta_{19} - 212499650 \beta_{16} - 346996293 \beta_{14} + 346996293 \beta_{13} - 298289045 \beta_{12} + 298289045 \beta_{11} + 838836392 \beta_{10} - 212499650 \beta_{8} + 212499650 \beta_{6} - 212499650 \beta_{4} + 212499650 \beta_{3} - 424999300 \beta_{2} + 151909234 \beta_{1}\)\()/2\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.339574 0.339574i
2.20512 2.20512i
−1.20277 + 1.20277i
0.922947 0.922947i
−0.850665 + 0.850665i
−0.339574 + 0.339574i
−2.20512 + 2.20512i
1.20277 1.20277i
−0.922947 + 0.922947i
0.850665 0.850665i
0.339574 + 0.339574i
2.20512 + 2.20512i
−1.20277 1.20277i
0.922947 + 0.922947i
−0.850665 0.850665i
−0.339574 0.339574i
−2.20512 2.20512i
1.20277 + 1.20277i
−0.922947 0.922947i
0.850665 + 0.850665i
−0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −1.29975 + 1.81952i −1.00000 −0.728588 0.728588i 0.707107 + 0.707107i 1.00000i −0.367533 2.20566i
37.2 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.114611 2.23313i −1.00000 −1.40368 1.40368i 0.707107 + 0.707107i 1.00000i 1.49802 + 1.66010i
37.3 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.528178 + 2.17279i −1.00000 0.904140 + 0.904140i 0.707107 + 0.707107i 1.00000i −1.90987 1.16292i
37.4 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 1.42113 1.72638i −1.00000 3.40461 + 3.40461i 0.707107 + 0.707107i 1.00000i 0.215841 + 2.22563i
37.5 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 2.23583 0.0328054i −1.00000 −3.17648 3.17648i 0.707107 + 0.707107i 1.00000i −1.55777 + 1.60417i
37.6 0.707107 0.707107i −0.707107 0.707107i 1.00000i −1.29975 + 1.81952i −1.00000 −0.728588 0.728588i −0.707107 0.707107i 1.00000i 0.367533 + 2.20566i
37.7 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0.114611 2.23313i −1.00000 −1.40368 1.40368i −0.707107 0.707107i 1.00000i −1.49802 1.66010i
37.8 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0.528178 + 2.17279i −1.00000 0.904140 + 0.904140i −0.707107 0.707107i 1.00000i 1.90987 + 1.16292i
37.9 0.707107 0.707107i −0.707107 0.707107i 1.00000i 1.42113 1.72638i −1.00000 3.40461 + 3.40461i −0.707107 0.707107i 1.00000i −0.215841 2.22563i
37.10 0.707107 0.707107i −0.707107 0.707107i 1.00000i 2.23583 0.0328054i −1.00000 −3.17648 3.17648i −0.707107 0.707107i 1.00000i 1.55777 1.60417i
493.1 −0.707107 0.707107i 0.707107 0.707107i 1.00000i −1.29975 1.81952i −1.00000 −0.728588 + 0.728588i 0.707107 0.707107i 1.00000i −0.367533 + 2.20566i
493.2 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.114611 + 2.23313i −1.00000 −1.40368 + 1.40368i 0.707107 0.707107i 1.00000i 1.49802 1.66010i
493.3 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.528178 2.17279i −1.00000 0.904140 0.904140i 0.707107 0.707107i 1.00000i −1.90987 + 1.16292i
493.4 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 1.42113 + 1.72638i −1.00000 3.40461 3.40461i 0.707107 0.707107i 1.00000i 0.215841 2.22563i
493.5 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 2.23583 + 0.0328054i −1.00000 −3.17648 + 3.17648i 0.707107 0.707107i 1.00000i −1.55777 1.60417i
493.6 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −1.29975 1.81952i −1.00000 −0.728588 + 0.728588i −0.707107 + 0.707107i 1.00000i 0.367533 2.20566i
493.7 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0.114611 + 2.23313i −1.00000 −1.40368 + 1.40368i −0.707107 + 0.707107i 1.00000i −1.49802 + 1.66010i
493.8 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0.528178 2.17279i −1.00000 0.904140 0.904140i −0.707107 + 0.707107i 1.00000i 1.90987 1.16292i
493.9 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 1.42113 + 1.72638i −1.00000 3.40461 3.40461i −0.707107 + 0.707107i 1.00000i −0.215841 + 2.22563i
493.10 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 2.23583 + 0.0328054i −1.00000 −3.17648 + 3.17648i −0.707107 + 0.707107i 1.00000i 1.55777 + 1.60417i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 493.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.m.b 20
3.b odd 2 1 1710.2.p.c 20
5.c odd 4 1 inner 570.2.m.b 20
15.e even 4 1 1710.2.p.c 20
19.b odd 2 1 inner 570.2.m.b 20
57.d even 2 1 1710.2.p.c 20
95.g even 4 1 inner 570.2.m.b 20
285.j odd 4 1 1710.2.p.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.m.b 20 1.a even 1 1 trivial
570.2.m.b 20 5.c odd 4 1 inner
570.2.m.b 20 19.b odd 2 1 inner
570.2.m.b 20 95.g even 4 1 inner
1710.2.p.c 20 3.b odd 2 1
1710.2.p.c 20 15.e even 4 1
1710.2.p.c 20 57.d even 2 1
1710.2.p.c 20 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{5} \)
$3$ \( ( 1 + T^{4} )^{5} \)
$5$ \( ( 3125 - 3750 T + 3125 T^{2} - 2000 T^{3} + 1030 T^{4} - 492 T^{5} + 206 T^{6} - 80 T^{7} + 25 T^{8} - 6 T^{9} + T^{10} )^{2} \)
$7$ \( ( 3200 + 3200 T + 1600 T^{2} - 320 T^{3} + 1408 T^{4} + 1184 T^{5} + 496 T^{6} + 8 T^{7} + 2 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$11$ \( ( -400 + 360 T - 16 T^{2} - 36 T^{3} + 2 T^{4} + T^{5} )^{4} \)
$13$ \( 4096 + 2416896 T^{4} + 7596800 T^{8} + 203872 T^{12} + 1248 T^{16} + T^{20} \)
$17$ \( ( 12800 - 38400 T + 57600 T^{2} - 35840 T^{3} + 9152 T^{4} + 2816 T^{5} + 544 T^{6} - 64 T^{7} + 18 T^{8} + 6 T^{9} + T^{10} )^{2} \)
$19$ \( 6131066257801 - 985046656378 T^{2} + 104865268749 T^{4} - 8806571896 T^{6} + 598735106 T^{8} - 34351324 T^{10} + 1658546 T^{12} - 67576 T^{14} + 2229 T^{16} - 58 T^{18} + T^{20} \)
$23$ \( ( 156800 - 336000 T + 360000 T^{2} - 84160 T^{3} + 4992 T^{4} + 4288 T^{5} + 1936 T^{6} - 24 T^{7} + 2 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$29$ \( ( -51200 + 224320 T^{2} - 63168 T^{4} + 4320 T^{6} - 112 T^{8} + T^{10} )^{2} \)
$31$ \( ( 204800 + 1583680 T^{2} + 303552 T^{4} + 12544 T^{6} + 192 T^{8} + T^{10} )^{2} \)
$37$ \( 362615934976 + 170038534400 T^{4} + 19652250368 T^{8} + 24627296 T^{12} + 8928 T^{16} + T^{20} \)
$41$ \( ( 51200 + 40720 T^{2} + 11328 T^{4} + 1320 T^{6} + 64 T^{8} + T^{10} )^{2} \)
$43$ \( ( 768800 - 570400 T + 211600 T^{2} + 455040 T^{3} + 275312 T^{4} + 67696 T^{5} + 8664 T^{6} + 224 T^{7} + 18 T^{8} + 6 T^{9} + T^{10} )^{2} \)
$47$ \( ( 2000000 - 1200000 T + 360000 T^{2} + 337600 T^{3} + 145728 T^{4} + 1344 T^{5} + 1456 T^{6} + 968 T^{7} + 242 T^{8} + 22 T^{9} + T^{10} )^{2} \)
$53$ \( 12285236038598656 + 105957764694016 T^{4} + 217470279680 T^{8} + 145756672 T^{12} + 30928 T^{16} + T^{20} \)
$59$ \( ( -3200 + 11280 T^{2} - 7968 T^{4} + 1944 T^{6} - 160 T^{8} + T^{10} )^{2} \)
$61$ \( ( 56000 + 12400 T - 816 T^{2} - 248 T^{3} + T^{5} )^{4} \)
$67$ \( 91832310663479296 + 24051048058454016 T^{4} + 17948433448960 T^{8} + 2217869312 T^{12} + 85248 T^{16} + T^{20} \)
$71$ \( ( 3276800 + 1556480 T^{2} + 233472 T^{4} + 12288 T^{6} + 200 T^{8} + T^{10} )^{2} \)
$73$ \( ( 1468820000 - 551756000 T + 103632400 T^{2} - 4836480 T^{3} + 447888 T^{4} - 167216 T^{5} + 39176 T^{6} - 1312 T^{7} + 2 T^{8} + 2 T^{9} + T^{10} )^{2} \)
$79$ \( ( -2508800 + 3068480 T^{2} - 551872 T^{4} + 28256 T^{6} - 384 T^{8} + T^{10} )^{2} \)
$83$ \( ( 356979200 - 164595200 T + 37945600 T^{2} - 5377280 T^{3} + 1017408 T^{4} - 274624 T^{5} + 58976 T^{6} - 8144 T^{7} + 722 T^{8} - 38 T^{9} + T^{10} )^{2} \)
$89$ \( ( -3699200 + 3528080 T^{2} - 347968 T^{4} + 12680 T^{6} - 192 T^{8} + T^{10} )^{2} \)
$97$ \( 1811741154135310336 + 12349758027534336 T^{4} + 21221664520192 T^{8} + 3014714496 T^{12} + 113680 T^{16} + T^{20} \)
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