Properties

Label 210.2.j.a
Level 210
Weight 2
Character orbit 210.j
Analytic conductor 1.677
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 210.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 16 x^{10} + 86 x^{8} + 196 x^{6} + 185 x^{4} + 60 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{8} + \nu^{7} + 13 \nu^{6} + 12 \nu^{5} + 47 \nu^{4} + 35 \nu^{3} + 53 \nu^{2} + 22 \nu + 10 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{8} - \nu^{7} + 13 \nu^{6} - 12 \nu^{5} + 47 \nu^{4} - 35 \nu^{3} + 53 \nu^{2} - 22 \nu + 10 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} + \nu^{10} - 15 \nu^{9} + 15 \nu^{8} - 73 \nu^{7} + 71 \nu^{6} - 151 \nu^{5} + 127 \nu^{4} - 152 \nu^{3} + 82 \nu^{2} - 84 \nu + 24 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} + \nu^{10} + 15 \nu^{9} + 15 \nu^{8} + 73 \nu^{7} + 71 \nu^{6} + 151 \nu^{5} + 127 \nu^{4} + 152 \nu^{3} + 82 \nu^{2} + 84 \nu + 24 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{11} + \nu^{10} - 45 \nu^{9} + 15 \nu^{8} - 217 \nu^{7} + 69 \nu^{6} - 421 \nu^{5} + 103 \nu^{4} - 298 \nu^{3} + 4 \nu^{2} - 32 \nu - 28 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{11} + \nu^{10} + 45 \nu^{9} + 15 \nu^{8} + 217 \nu^{7} + 69 \nu^{6} + 421 \nu^{5} + 103 \nu^{4} + 298 \nu^{3} + 4 \nu^{2} + 32 \nu - 28 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{11} + 31 \nu^{9} + 157 \nu^{7} + 321 \nu^{5} + 243 \nu^{3} + 46 \nu \)\()/8\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{11} - 3 \nu^{10} + 49 \nu^{9} - 43 \nu^{8} + 271 \nu^{7} - 191 \nu^{6} + 633 \nu^{5} - 327 \nu^{4} + 588 \nu^{3} - 192 \nu^{2} + 140 \nu - 12 \)\()/16\)
\(\beta_{9}\)\(=\)\((\)\( -3 \nu^{11} + 3 \nu^{10} - 47 \nu^{9} + 47 \nu^{8} - 243 \nu^{7} + 241 \nu^{6} - 515 \nu^{5} + 491 \nu^{4} - 412 \nu^{3} + 342 \nu^{2} - 92 \nu + 48 \)\()/16\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{11} + 3 \nu^{10} + 49 \nu^{9} + 43 \nu^{8} + 271 \nu^{7} + 191 \nu^{6} + 633 \nu^{5} + 327 \nu^{4} + 588 \nu^{3} + 192 \nu^{2} + 140 \nu + 12 \)\()/16\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{11} + 3 \nu^{10} + 47 \nu^{9} + 47 \nu^{8} + 243 \nu^{7} + 241 \nu^{6} + 515 \nu^{5} + 491 \nu^{4} + 412 \nu^{3} + 342 \nu^{2} + 92 \nu + 48 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_{1} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{11} + 5 \beta_{10} + 3 \beta_{9} + 5 \beta_{8} - 14 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + 7 \beta_{4} - 7 \beta_{3} + 12 \beta_{2} - 12 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-13 \beta_{11} + 11 \beta_{10} - 13 \beta_{9} - 11 \beta_{8} + 9 \beta_{6} + 9 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 24 \beta_{2} + 24 \beta_{1} + 42\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(11 \beta_{11} - 33 \beta_{10} - 11 \beta_{9} - 33 \beta_{8} + 110 \beta_{7} - 35 \beta_{6} + 35 \beta_{5} - 49 \beta_{4} + 49 \beta_{3} - 86 \beta_{2} + 86 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(117 \beta_{11} - 93 \beta_{10} + 117 \beta_{9} + 93 \beta_{8} - 77 \beta_{6} - 77 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 210 \beta_{2} - 210 \beta_{1} - 322\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-49 \beta_{11} + 243 \beta_{10} + 49 \beta_{9} + 243 \beta_{8} - 874 \beta_{7} + 267 \beta_{6} - 267 \beta_{5} + 365 \beta_{4} - 365 \beta_{3} + 648 \beta_{2} - 648 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-963 \beta_{11} + 745 \beta_{10} - 963 \beta_{9} - 745 \beta_{8} + 631 \beta_{6} + 631 \beta_{5} + 23 \beta_{4} + 23 \beta_{3} + 1716 \beta_{2} + 1716 \beta_{1} + 2510\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(269 \beta_{11} - 1863 \beta_{10} - 269 \beta_{9} - 1863 \beta_{8} + 6930 \beta_{7} - 2089 \beta_{6} + 2089 \beta_{5} - 2811 \beta_{4} + 2811 \beta_{3} - 5006 \beta_{2} + 5006 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(7707 \beta_{11} - 5887 \beta_{10} + 7707 \beta_{9} + 5887 \beta_{8} - 5059 \beta_{6} - 5059 \beta_{5} - 385 \beta_{4} - 385 \beta_{3} - 13714 \beta_{2} - 13714 \beta_{1} - 19678\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-1747 \beta_{11} + 14513 \beta_{10} + 1747 \beta_{9} + 14513 \beta_{8} - 54798 \beta_{7} + 16453 \beta_{6} - 16453 \beta_{5} + 21955 \beta_{4} - 21955 \beta_{3} + 39116 \beta_{2} - 39116 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0.678294i
1.85804i
1.12212i
2.80721i
1.69093i
0.297931i
0.678294i
1.85804i
1.12212i
2.80721i
1.69093i
0.297931i
−0.707107 0.707107i −1.67762 0.430811i 1.00000i −2.22680 + 0.203331i 0.881625 + 1.49088i 0.707107 0.707107i 0.707107 0.707107i 2.62880 + 1.44547i 1.71837 + 1.43081i
113.2 −0.707107 0.707107i 0.510256 + 1.65519i 1.00000i 1.97503 1.04846i 0.809587 1.53120i 0.707107 0.707107i 0.707107 0.707107i −2.47928 + 1.68914i −2.13793 0.655185i
113.3 −0.707107 0.707107i 1.46025 0.931481i 1.00000i −2.16244 0.569088i −1.69121 0.373900i 0.707107 0.707107i 0.707107 0.707107i 1.26469 2.72040i 1.12667 + 1.93148i
113.4 0.707107 + 0.707107i −0.799269 + 1.53661i 1.00000i −1.91438 + 1.15549i −1.65172 + 0.521378i −0.707107 + 0.707107i −0.707107 + 0.707107i −1.72234 2.45633i −2.17073 0.536610i
113.5 0.707107 + 0.707107i 1.17225 + 1.27508i 1.00000i 1.37462 1.76364i −0.0727133 + 1.73052i −0.707107 + 0.707107i −0.707107 + 0.707107i −0.251664 + 2.98943i 2.21908 0.275081i
113.6 0.707107 + 0.707107i 1.33413 1.10458i 1.00000i 0.953972 + 2.02236i 1.72443 + 0.162311i −0.707107 + 0.707107i −0.707107 + 0.707107i 0.559788 2.94731i −0.755464 + 2.10458i
197.1 −0.707107 + 0.707107i −1.67762 + 0.430811i 1.00000i −2.22680 0.203331i 0.881625 1.49088i 0.707107 + 0.707107i 0.707107 + 0.707107i 2.62880 1.44547i 1.71837 1.43081i
197.2 −0.707107 + 0.707107i 0.510256 1.65519i 1.00000i 1.97503 + 1.04846i 0.809587 + 1.53120i 0.707107 + 0.707107i 0.707107 + 0.707107i −2.47928 1.68914i −2.13793 + 0.655185i
197.3 −0.707107 + 0.707107i 1.46025 + 0.931481i 1.00000i −2.16244 + 0.569088i −1.69121 + 0.373900i 0.707107 + 0.707107i 0.707107 + 0.707107i 1.26469 + 2.72040i 1.12667 1.93148i
197.4 0.707107 0.707107i −0.799269 1.53661i 1.00000i −1.91438 1.15549i −1.65172 0.521378i −0.707107 0.707107i −0.707107 0.707107i −1.72234 + 2.45633i −2.17073 + 0.536610i
197.5 0.707107 0.707107i 1.17225 1.27508i 1.00000i 1.37462 + 1.76364i −0.0727133 1.73052i −0.707107 0.707107i −0.707107 0.707107i −0.251664 2.98943i 2.21908 + 0.275081i
197.6 0.707107 0.707107i 1.33413 + 1.10458i 1.00000i 0.953972 2.02236i 1.72443 0.162311i −0.707107 0.707107i −0.707107 0.707107i 0.559788 + 2.94731i −0.755464 2.10458i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.j.a 12
3.b odd 2 1 210.2.j.b yes 12
5.b even 2 1 1050.2.j.c 12
5.c odd 4 1 210.2.j.b yes 12
5.c odd 4 1 1050.2.j.d 12
15.d odd 2 1 1050.2.j.d 12
15.e even 4 1 inner 210.2.j.a 12
15.e even 4 1 1050.2.j.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.j.a 12 1.a even 1 1 trivial
210.2.j.a 12 15.e even 4 1 inner
210.2.j.b yes 12 3.b odd 2 1
210.2.j.b yes 12 5.c odd 4 1
1050.2.j.c 12 5.b even 2 1
1050.2.j.c 12 15.e even 4 1
1050.2.j.d 12 5.c odd 4 1
1050.2.j.d 12 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{17}^{12} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{3} \)
$3$ \( 1 - 4 T + 8 T^{2} - 4 T^{3} - 7 T^{4} + 24 T^{5} - 32 T^{6} + 72 T^{7} - 63 T^{8} - 108 T^{9} + 648 T^{10} - 972 T^{11} + 729 T^{12} \)
$5$ \( 1 + 4 T - 2 T^{2} - 4 T^{3} + 89 T^{4} + 88 T^{5} - 288 T^{6} + 440 T^{7} + 2225 T^{8} - 500 T^{9} - 1250 T^{10} + 12500 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 + T^{4} )^{3} \)
$11$ \( 1 - 56 T^{2} + 1358 T^{4} - 19128 T^{6} + 195023 T^{8} - 1950544 T^{10} + 21368868 T^{12} - 236015824 T^{14} + 2855331743 T^{16} - 33886418808 T^{18} + 291099360398 T^{20} - 1452495777656 T^{22} + 3138428376721 T^{24} \)
$13$ \( 1 - 32 T^{3} - 14 T^{4} + 272 T^{5} + 512 T^{6} - 5520 T^{7} - 4301 T^{8} + 109488 T^{9} + 220800 T^{10} + 406064 T^{11} - 6852572 T^{12} + 5278832 T^{13} + 37315200 T^{14} + 240545136 T^{15} - 122840861 T^{16} - 2049537360 T^{17} + 2471326208 T^{18} + 17067596624 T^{19} - 11420230094 T^{20} - 339343979936 T^{21} + 23298085122481 T^{24} \)
$17$ \( 1 - 28 T + 392 T^{2} - 3716 T^{3} + 26626 T^{4} - 151924 T^{5} + 720808 T^{6} - 3009708 T^{7} + 12126655 T^{8} - 52482936 T^{9} + 248165456 T^{10} - 1175330632 T^{11} + 5125539580 T^{12} - 19980620744 T^{13} + 71719816784 T^{14} - 257848664568 T^{15} + 1012830352255 T^{16} - 4273354971756 T^{17} + 17398552835752 T^{18} - 62340292556852 T^{19} + 185736517624066 T^{20} - 440672549062852 T^{21} + 790269608976008 T^{22} - 959613096613724 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 - 104 T^{2} + 6018 T^{4} - 249384 T^{6} + 7954675 T^{8} - 203686352 T^{10} + 4270353988 T^{12} - 73530773072 T^{14} + 1036661200675 T^{16} - 11732489987304 T^{18} + 102207082380738 T^{20} - 637630890811304 T^{22} + 2213314919066161 T^{24} \)
$23$ \( 1 + 24 T + 288 T^{2} + 2648 T^{3} + 22078 T^{4} + 163240 T^{5} + 1065248 T^{6} + 6483944 T^{7} + 37525599 T^{8} + 203370768 T^{9} + 1048113088 T^{10} + 5277774096 T^{11} + 25830292740 T^{12} + 121388804208 T^{13} + 554451823552 T^{14} + 2474412134256 T^{15} + 10501201149759 T^{16} + 41732887576792 T^{17} + 157694934685472 T^{18} + 555803705968280 T^{19} + 1728949933033918 T^{20} + 4769452247554024 T^{21} + 11930835229530912 T^{22} + 22867434189934248 T^{23} + 21914624432020321 T^{24} \)
$29$ \( ( 1 - 4 T + 114 T^{2} - 548 T^{3} + 6631 T^{4} - 29256 T^{5} + 241692 T^{6} - 848424 T^{7} + 5576671 T^{8} - 13365172 T^{9} + 80630034 T^{10} - 82044596 T^{11} + 594823321 T^{12} )^{2} \)
$31$ \( ( 1 + 4 T + 28 T^{2} + 76 T^{3} + 807 T^{4} + 1288 T^{5} + 32184 T^{6} + 39928 T^{7} + 775527 T^{8} + 2264116 T^{9} + 25858588 T^{10} + 114516604 T^{11} + 887503681 T^{12} )^{2} \)
$37$ \( 1 + 20 T + 200 T^{2} + 1052 T^{3} + 3634 T^{4} + 30620 T^{5} + 438952 T^{6} + 3655156 T^{7} + 16552975 T^{8} + 43779560 T^{9} + 283860944 T^{10} + 4234271992 T^{11} + 34611552220 T^{12} + 156668063704 T^{13} + 388605632336 T^{14} + 2217566052680 T^{15} + 31022940178975 T^{16} + 253462980492292 T^{17} + 1126230738683368 T^{18} + 2906814077812460 T^{19} + 12764350335548914 T^{20} + 136719750264421004 T^{21} + 961716874483569800 T^{22} + 3558352435589208260 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 212 T^{2} + 22434 T^{4} - 1643716 T^{6} + 97594575 T^{8} - 5014736296 T^{10} + 222514140252 T^{12} - 8429771713576 T^{14} + 275778943846575 T^{16} - 7807822342599556 T^{18} + 179133812590100514 T^{20} - 2845603773752309012 T^{22} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 8 T + 32 T^{2} - 184 T^{3} + 5014 T^{4} - 25096 T^{5} + 57248 T^{6} + 105352 T^{7} + 1869663 T^{8} + 20409392 T^{9} - 214625984 T^{10} + 2505871696 T^{11} - 14493786700 T^{12} + 107752482928 T^{13} - 396843444416 T^{14} + 1622689529744 T^{15} + 6392005734063 T^{16} + 15487633486936 T^{17} + 361885391829152 T^{18} - 6821559864341272 T^{19} + 58604636191891414 T^{20} - 92477040596379112 T^{21} + 691567434025095968 T^{22} - 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 16 T + 128 T^{2} - 48 T^{3} - 3930 T^{4} - 2832 T^{5} + 549504 T^{6} - 4835248 T^{7} + 9727055 T^{8} + 102211168 T^{9} - 484788992 T^{10} - 7463204576 T^{11} + 87433873108 T^{12} - 350770615072 T^{13} - 1070898883328 T^{14} + 10611870095264 T^{15} + 47464925469455 T^{16} - 1108939986406736 T^{17} + 5923221940146816 T^{18} - 1434756677151216 T^{19} - 93578356580720730 T^{20} - 53718262708932816 T^{21} + 6732688926186246272 T^{22} - 39554547441344196848 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 24 T + 288 T^{2} + 2968 T^{3} + 27318 T^{4} + 189144 T^{5} + 1076384 T^{6} + 4855704 T^{7} - 9792929 T^{8} - 439179408 T^{9} - 4825144512 T^{10} - 45292675216 T^{11} - 372958615948 T^{12} - 2400511786448 T^{13} - 13553830934208 T^{14} - 65383712724816 T^{15} - 77270920208849 T^{16} + 2030633528142072 T^{17} + 23857363689477536 T^{18} + 222189563833329528 T^{19} + 1700810222657559798 T^{20} + 9793698340468730744 T^{21} + 50367591465267758112 T^{22} + \)\(22\!\cdots\!28\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( ( 1 - 16 T + 270 T^{2} - 3160 T^{3} + 33193 T^{4} - 284568 T^{5} + 2445376 T^{6} - 16789512 T^{7} + 115544833 T^{8} - 648997640 T^{9} + 3271687470 T^{10} - 11438788784 T^{11} + 42180533641 T^{12} )^{2} \)
$61$ \( ( 1 + 178 T^{2} + 664 T^{3} + 12681 T^{4} + 121240 T^{5} + 672800 T^{6} + 7395640 T^{7} + 47186001 T^{8} + 150715384 T^{9} + 2464559698 T^{10} + 51520374361 T^{12} )^{2} \)
$67$ \( 1 + 64 T^{3} + 2390 T^{4} - 9024 T^{5} + 2048 T^{6} - 508672 T^{7} + 20286015 T^{8} + 4116224 T^{9} + 3266560 T^{10} - 440818304 T^{11} + 142523500468 T^{12} - 29534826368 T^{13} + 14663587840 T^{14} + 1238007878912 T^{15} + 408785942872815 T^{16} - 686770838427904 T^{17} + 185258766682112 T^{18} - 54691861526434752 T^{19} + 970501749360371990 T^{20} + 1741218201362876608 T^{21} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 440 T^{2} + 91446 T^{4} - 12419864 T^{6} + 1305779695 T^{8} - 116298170672 T^{10} + 8924106004404 T^{12} - 586259078357552 T^{14} + 33182057065617295 T^{16} - 1590988104660206744 T^{18} + 59051577418299860406 T^{20} - \)\(14\!\cdots\!40\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 24 T + 288 T^{2} + 2936 T^{3} + 13126 T^{4} - 142696 T^{5} - 2894944 T^{6} - 37628616 T^{7} - 323587345 T^{8} - 985106832 T^{9} + 7253083968 T^{10} + 194887191344 T^{11} + 2401164656404 T^{12} + 14226764968112 T^{13} + 38651684465472 T^{14} - 383223304464144 T^{15} - 9189311407860145 T^{16} - 78006814913505288 T^{17} - 438104110389982816 T^{18} - 1576419579081065512 T^{19} + 10585595166201707206 T^{20} + \)\(17\!\cdots\!68\)\( T^{21} + \)\(12\!\cdots\!12\)\( T^{22} + \)\(75\!\cdots\!48\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 508 T^{2} + 135562 T^{4} - 24649964 T^{6} + 3381685343 T^{8} - 366753014104 T^{10} + 32137748903980 T^{12} - 2288905561023064 T^{14} + 131716918026362783 T^{16} - 5992097027444251244 T^{18} + \)\(20\!\cdots\!82\)\( T^{20} - \)\(48\!\cdots\!08\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 24 T + 288 T^{2} + 3784 T^{3} + 70066 T^{4} + 929320 T^{5} + 9284000 T^{6} + 107957496 T^{7} + 1400664563 T^{8} + 14071243520 T^{9} + 124154702400 T^{10} + 1299362373632 T^{11} + 13371946678180 T^{12} + 107847077011456 T^{13} + 855301744833600 T^{14} + 8045754118570240 T^{15} + 66473188444178723 T^{16} + 425248964460509928 T^{17} + 3035314426357796000 T^{18} + 25218074905680163640 T^{19} + \)\(15\!\cdots\!06\)\( T^{20} + \)\(70\!\cdots\!52\)\( T^{21} + \)\(44\!\cdots\!12\)\( T^{22} + \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( ( 1 - 24 T + 358 T^{2} - 5272 T^{3} + 62767 T^{4} - 696208 T^{5} + 7360436 T^{6} - 61962512 T^{7} + 497177407 T^{8} - 3716596568 T^{9} + 22461722278 T^{10} - 134017426776 T^{11} + 496981290961 T^{12} )^{2} \)
$97$ \( 1 - 8 T + 32 T^{2} + 1240 T^{3} + 7270 T^{4} - 77896 T^{5} + 1159328 T^{6} + 23481432 T^{7} - 27667953 T^{8} + 164181296 T^{9} + 23294478656 T^{10} + 157379895024 T^{11} - 641510857644 T^{12} + 15265849817328 T^{13} + 219177749674304 T^{14} + 149843835964208 T^{15} - 2449423985831793 T^{16} + 201643046305608024 T^{17} + 965687768530327712 T^{18} - 6293863167707090248 T^{19} + 56978142231120506470 T^{20} + \)\(94\!\cdots\!80\)\( T^{21} + \)\(23\!\cdots\!68\)\( T^{22} - \)\(57\!\cdots\!24\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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