Properties

Label 201.2.d.a
Level 201
Weight 2
Character orbit 201.d
Analytic conductor 1.605
Analytic rank 0
Dimension 20
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 201.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{7} q^{2} + \beta_{5} q^{3} + ( 1 + \beta_{16} ) q^{4} -\beta_{11} q^{5} + \beta_{8} q^{6} -\beta_{18} q^{7} + ( -\beta_{7} + \beta_{13} ) q^{8} + ( -1 + \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{7} q^{2} + \beta_{5} q^{3} + ( 1 + \beta_{16} ) q^{4} -\beta_{11} q^{5} + \beta_{8} q^{6} -\beta_{18} q^{7} + ( -\beta_{7} + \beta_{13} ) q^{8} + ( -1 + \beta_{6} ) q^{9} + ( -1 + \beta_{15} ) q^{10} + ( \beta_{7} - \beta_{14} ) q^{11} + ( \beta_{5} + \beta_{11} + \beta_{14} + \beta_{19} ) q^{12} + \beta_{17} q^{13} + ( -\beta_{6} - \beta_{8} - \beta_{9} + \beta_{12} ) q^{14} + ( \beta_{2} - \beta_{3} + \beta_{12} ) q^{15} + ( 1 - \beta_{15} ) q^{16} + ( -\beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{12} ) q^{17} + ( 2 \beta_{7} + \beta_{10} - \beta_{13} ) q^{18} + ( 2 - \beta_{3} + \beta_{8} + \beta_{16} ) q^{19} + ( -\beta_{1} - \beta_{5} + 2 \beta_{7} - \beta_{13} - \beta_{14} ) q^{20} + ( 1 + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{15} + \beta_{16} ) q^{21} + ( -2 - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{15} - \beta_{16} ) q^{22} + ( \beta_{4} - \beta_{8} + \beta_{12} ) q^{23} + ( \beta_{2} - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{12} - \beta_{15} ) q^{24} + ( -\beta_{2} - \beta_{6} - \beta_{16} ) q^{25} -\beta_{12} q^{26} + ( -2 \beta_{1} - 2 \beta_{5} - \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{27} + ( -\beta_{1} - 3 \beta_{5} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} - \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{28} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} ) q^{29} + ( -\beta_{5} - \beta_{10} - \beta_{11} - \beta_{17} - \beta_{19} ) q^{30} + ( \beta_{1} + \beta_{5} + \beta_{11} + \beta_{14} + \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{31} + ( \beta_{1} + \beta_{5} + 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{32} + ( \beta_{3} - \beta_{8} - \beta_{9} - \beta_{15} - \beta_{16} ) q^{33} + ( 2 \beta_{1} + 2 \beta_{5} + \beta_{7} + 2 \beta_{10} + 2 \beta_{11} - \beta_{13} + \beta_{14} + \beta_{17} + 2 \beta_{19} ) q^{34} + ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{35} + ( -3 - \beta_{2} + \beta_{9} - \beta_{12} + \beta_{15} - 2 \beta_{16} ) q^{36} + ( 1 - \beta_{2} + 2 \beta_{3} - \beta_{6} - 2 \beta_{8} - \beta_{15} - \beta_{16} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{5} - 4 \beta_{7} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{38} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{12} + \beta_{15} + 2 \beta_{16} ) q^{39} + ( -3 - \beta_{2} + 2 \beta_{3} - \beta_{6} - 2 \beta_{8} - 3 \beta_{16} ) q^{40} + ( -\beta_{1} - \beta_{5} + \beta_{13} + \beta_{14} ) q^{41} + ( \beta_{1} - 3 \beta_{7} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{18} - \beta_{19} ) q^{42} + ( \beta_{1} + \beta_{5} + \beta_{7} + 2 \beta_{10} + \beta_{11} - \beta_{13} + \beta_{17} - \beta_{18} ) q^{43} + ( -2 \beta_{1} - 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{11} - \beta_{13} ) q^{44} + ( \beta_{1} - \beta_{7} + \beta_{11} - \beta_{14} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{45} + ( -2 \beta_{5} - \beta_{11} - \beta_{14} - \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{46} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{47} + ( \beta_{5} + \beta_{10} + \beta_{11} + \beta_{17} + \beta_{19} ) q^{48} + ( -2 - 2 \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{8} + 3 \beta_{15} ) q^{49} + ( \beta_{1} + \beta_{5} + \beta_{7} + \beta_{11} ) q^{50} + ( -2 \beta_{1} - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{18} ) q^{51} + ( \beta_{1} - \beta_{5} - \beta_{17} ) q^{52} + ( \beta_{1} + \beta_{5} + \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{53} + ( -2 + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - 3 \beta_{8} - \beta_{9} + 2 \beta_{12} + \beta_{15} ) q^{54} + ( 2 \beta_{2} + 2 \beta_{6} - \beta_{15} + 3 \beta_{16} ) q^{55} + ( \beta_{2} + 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{8} - \beta_{9} + 2 \beta_{12} ) q^{56} + ( 2 \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} + \beta_{19} ) q^{57} + ( -2 \beta_{5} - \beta_{11} - \beta_{14} - 2 \beta_{19} ) q^{58} + ( \beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} + 2 \beta_{12} ) q^{59} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + \beta_{12} - \beta_{16} ) q^{60} + ( -2 \beta_{5} - \beta_{7} - 2 \beta_{10} - \beta_{11} + \beta_{13} - \beta_{17} ) q^{61} + ( \beta_{2} - 2 \beta_{4} + 3 \beta_{8} + \beta_{9} - 2 \beta_{12} ) q^{62} + ( 2 \beta_{1} + 2 \beta_{5} - \beta_{7} + \beta_{10} + \beta_{13} - \beta_{18} + 2 \beta_{19} ) q^{63} + ( -1 + \beta_{2} - 2 \beta_{3} + \beta_{6} + 2 \beta_{8} - \beta_{16} ) q^{64} + ( 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{8} + 2 \beta_{9} - \beta_{12} ) q^{65} + ( -\beta_{1} - 2 \beta_{5} + \beta_{7} - \beta_{10} - \beta_{18} - \beta_{19} ) q^{66} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{15} - \beta_{17} + \beta_{18} ) q^{67} + ( -\beta_{2} + \beta_{3} + 3 \beta_{6} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{12} ) q^{68} + ( -\beta_{1} - \beta_{7} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{17} - \beta_{18} ) q^{69} + ( -\beta_{1} + 3 \beta_{5} + \beta_{11} + \beta_{14} + \beta_{18} + 2 \beta_{19} ) q^{70} + ( -\beta_{2} + \beta_{6} + \beta_{12} ) q^{71} + ( \beta_{1} + 4 \beta_{7} - \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{72} + ( -1 + \beta_{3} - \beta_{8} + 2 \beta_{15} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{5} - \beta_{7} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{74} + ( -\beta_{1} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{17} - \beta_{18} ) q^{75} + ( 8 + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{8} - 3 \beta_{15} + 3 \beta_{16} ) q^{76} + ( 3 \beta_{3} + 2 \beta_{4} + \beta_{8} - \beta_{12} ) q^{77} + ( 2 \beta_{1} + \beta_{5} - 2 \beta_{7} - \beta_{11} + \beta_{14} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{78} + ( -3 \beta_{1} + \beta_{5} - \beta_{7} - 2 \beta_{10} - \beta_{11} + \beta_{13} ) q^{79} + ( -\beta_{1} - \beta_{5} + 4 \beta_{7} - \beta_{11} ) q^{80} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{12} - 2 \beta_{15} - \beta_{16} ) q^{81} + ( -1 + \beta_{2} + \beta_{6} - 2 \beta_{15} + \beta_{16} ) q^{82} + ( -\beta_{2} - \beta_{3} - 2 \beta_{8} - \beta_{9} + 2 \beta_{12} ) q^{83} + ( 7 - \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{8} - 2 \beta_{15} + 2 \beta_{16} ) q^{84} + ( -2 \beta_{5} - \beta_{11} - \beta_{14} + \beta_{17} - 2 \beta_{19} ) q^{85} + ( -2 \beta_{2} - \beta_{3} + 3 \beta_{6} + \beta_{9} - 2 \beta_{12} ) q^{86} + ( -\beta_{1} - 3 \beta_{7} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} - \beta_{18} ) q^{87} + ( -4 + 2 \beta_{2} + 2 \beta_{6} + \beta_{15} - \beta_{16} ) q^{88} + ( 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} + \beta_{12} ) q^{89} + ( 5 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} + \beta_{12} - \beta_{15} + \beta_{16} ) q^{90} + ( 3 \beta_{2} + 3 \beta_{6} - \beta_{15} + \beta_{16} ) q^{91} + ( -\beta_{2} - \beta_{3} - 2 \beta_{8} - \beta_{9} ) q^{92} + ( -1 + \beta_{2} - 2 \beta_{3} + \beta_{6} + 2 \beta_{8} + \beta_{9} - 2 \beta_{12} + \beta_{15} - 2 \beta_{16} ) q^{93} + ( -\beta_{1} + 3 \beta_{5} + \beta_{11} + \beta_{14} + \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{94} + ( 3 \beta_{7} - \beta_{11} - 2 \beta_{13} ) q^{95} + ( 2 - 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{12} + 2 \beta_{15} + \beta_{16} ) q^{96} + ( \beta_{1} - \beta_{5} + \beta_{17} + 2 \beta_{18} ) q^{97} + ( \beta_{1} + \beta_{5} + 3 \beta_{7} - 3 \beta_{11} - \beta_{13} - 2 \beta_{14} ) q^{98} + ( \beta_{1} - 2 \beta_{7} - 2 \beta_{11} + 2 \beta_{13} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 12q^{4} - 4q^{6} - 14q^{9} + O(q^{10}) \) \( 20q + 12q^{4} - 4q^{6} - 14q^{9} - 12q^{10} + 2q^{15} + 12q^{16} + 24q^{19} + 12q^{21} - 28q^{22} + 2q^{24} - 4q^{25} + 10q^{33} - 44q^{36} + 24q^{37} - 8q^{39} - 32q^{40} - 48q^{49} - 26q^{54} - 8q^{55} - 38q^{60} - 16q^{64} + 32q^{67} + 4q^{73} + 116q^{76} - 30q^{81} - 32q^{82} + 90q^{84} - 40q^{88} + 74q^{90} + 20q^{91} - 2q^{93} + 30q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 7 x^{18} + 32 x^{16} + 128 x^{14} + 423 x^{12} + 1186 x^{10} + 3807 x^{8} + 10368 x^{6} + 23328 x^{4} + 45927 x^{2} + 59049\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{18} + 60 \nu^{16} + 196 \nu^{14} + 780 \nu^{12} + 2203 \nu^{10} + 7153 \nu^{8} + 18308 \nu^{6} + 68940 \nu^{4} + 115668 \nu^{2} + 229635 \)\()/31104\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{18} - 365 \nu^{16} - 1477 \nu^{14} - 5845 \nu^{12} - 16650 \nu^{10} - 47795 \nu^{8} - 121005 \nu^{6} - 383373 \nu^{4} - 779301 \nu^{2} - 1482786 \)\()/209952\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{19} - 7 \nu^{17} - 32 \nu^{15} - 128 \nu^{13} - 423 \nu^{11} - 1186 \nu^{9} - 3807 \nu^{7} - 10368 \nu^{5} - 23328 \nu^{3} - 45927 \nu \)\()/19683\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{18} - 7 \nu^{16} - 32 \nu^{14} - 128 \nu^{12} - 423 \nu^{10} - 1186 \nu^{8} - 3807 \nu^{6} - 10368 \nu^{4} - 23328 \nu^{2} - 39366 \)\()/6561\)
\(\beta_{7}\)\(=\)\((\)\( -35 \nu^{19} - 236 \nu^{17} - 580 \nu^{15} - 2716 \nu^{13} - 7785 \nu^{11} - 21683 \nu^{9} - 68868 \nu^{7} - 198108 \nu^{5} - 196020 \nu^{3} - 566433 \nu \)\()/279936\)
\(\beta_{8}\)\(=\)\((\)\( 35 \nu^{18} + 236 \nu^{16} + 580 \nu^{14} + 2716 \nu^{12} + 7785 \nu^{10} + 21683 \nu^{8} + 68868 \nu^{6} + 198108 \nu^{4} + 196020 \nu^{2} + 566433 \)\()/93312\)
\(\beta_{9}\)\(=\)\((\)\( 73 \nu^{18} + 814 \nu^{16} + 2486 \nu^{14} + 13262 \nu^{12} + 34293 \nu^{10} + 100969 \nu^{8} + 344454 \nu^{6} + 957150 \nu^{4} + 1547910 \nu^{2} + 4739229 \)\()/139968\)
\(\beta_{10}\)\(=\)\((\)\( 5 \nu^{19} + 76 \nu^{17} + 276 \nu^{15} + 1052 \nu^{13} + 3079 \nu^{11} + 9269 \nu^{9} + 27188 \nu^{7} + 83772 \nu^{5} + 164484 \nu^{3} + 266895 \nu \)\()/31104\)
\(\beta_{11}\)\(=\)\((\)\( -511 \nu^{19} - 2524 \nu^{17} - 7604 \nu^{15} - 30092 \nu^{13} - 86877 \nu^{11} - 214735 \nu^{9} - 851796 \nu^{7} - 1657260 \nu^{5} - 2732292 \nu^{3} - 4756725 \nu \)\()/2519424\)
\(\beta_{12}\)\(=\)\((\)\( 269 \nu^{18} + 2072 \nu^{16} + 6448 \nu^{14} + 25576 \nu^{12} + 73179 \nu^{10} + 203933 \nu^{8} + 673056 \nu^{6} + 1759320 \nu^{4} + 2507760 \nu^{2} + 5058531 \)\()/419904\)
\(\beta_{13}\)\(=\)\((\)\( 23 \nu^{19} + 188 \nu^{17} + 736 \nu^{15} + 3700 \nu^{13} + 9081 \nu^{11} + 27143 \nu^{9} + 90180 \nu^{7} + 237384 \nu^{5} + 494748 \nu^{3} + 1301265 \nu \)\()/104976\)
\(\beta_{14}\)\(=\)\((\)\( 619 \nu^{19} + 4108 \nu^{17} + 15236 \nu^{15} + 79484 \nu^{13} + 198369 \nu^{11} + 614011 \nu^{9} + 2177892 \nu^{7} + 4978908 \nu^{5} + 10068948 \nu^{3} + 28087641 \nu \)\()/2519424\)
\(\beta_{15}\)\(=\)\((\)\( -92 \nu^{18} - 401 \nu^{16} - 1729 \nu^{14} - 6673 \nu^{12} - 18261 \nu^{10} - 51764 \nu^{8} - 180873 \nu^{6} - 408321 \nu^{4} - 828873 \nu^{2} - 1397493 \)\()/104976\)
\(\beta_{16}\)\(=\)\((\)\( 236 \nu^{18} + 1013 \nu^{16} + 4861 \nu^{14} + 18589 \nu^{12} + 49545 \nu^{10} + 153860 \nu^{8} + 513117 \nu^{6} + 1090989 \nu^{4} + 2633877 \nu^{2} + 4258089 \)\()/209952\)
\(\beta_{17}\)\(=\)\((\)\( 553 \nu^{19} + 1162 \nu^{17} + 6914 \nu^{15} + 28970 \nu^{13} + 68769 \nu^{11} + 226153 \nu^{9} + 779778 \nu^{7} + 996138 \nu^{5} + 3838914 \nu^{3} + 6147657 \nu \)\()/1259712\)
\(\beta_{18}\)\(=\)\((\)\( -377 \nu^{19} - 1226 \nu^{17} - 5818 \nu^{15} - 21994 \nu^{13} - 55881 \nu^{11} - 171641 \nu^{9} - 632034 \nu^{7} - 1192482 \nu^{5} - 2975778 \nu^{3} - 4730481 \nu \)\()/629856\)
\(\beta_{19}\)\(=\)\((\)\( -676 \nu^{19} - 2815 \nu^{17} - 13559 \nu^{15} - 51671 \nu^{13} - 135099 \nu^{11} - 423628 \nu^{9} - 1417527 \nu^{7} - 2941191 \nu^{5} - 7155135 \nu^{3} - 11934459 \nu \)\()/629856\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{19} - \beta_{18} + \beta_{17} + \beta_{10} + \beta_{7}\)
\(\nu^{4}\)\(=\)\(-\beta_{16} - 2 \beta_{15} - \beta_{12} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 1\)
\(\nu^{5}\)\(=\)\(-2 \beta_{18} - 2 \beta_{17} - \beta_{14} + \beta_{13} + 3 \beta_{11} - \beta_{10} - 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{16} + 3 \beta_{15} + 5 \beta_{12} + \beta_{9} - 6 \beta_{8} - 2 \beta_{6} + 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 5\)
\(\nu^{7}\)\(=\)\(-\beta_{19} - \beta_{18} - 5 \beta_{17} + 9 \beta_{14} - 6 \beta_{13} - 5 \beta_{11} - \beta_{10} + 9 \beta_{7} + 2 \beta_{5} - 6 \beta_{1}\)
\(\nu^{8}\)\(=\)\(16 \beta_{16} + 23 \beta_{15} - 6 \beta_{12} - 3 \beta_{9} + 19 \beta_{8} - 2 \beta_{6} - \beta_{4} + 3 \beta_{3} - 12 \beta_{2} + 11\)
\(\nu^{9}\)\(=\)\(8 \beta_{18} + 20 \beta_{17} + 16 \beta_{14} - 25 \beta_{13} + 37 \beta_{11} + 14 \beta_{10} - 31 \beta_{7} + 21 \beta_{5} + 30 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-16 \beta_{16} - \beta_{15} + 6 \beta_{12} - 5 \beta_{9} + 15 \beta_{8} - 65 \beta_{6} - 39 \beta_{4} - 53 \beta_{3} + 2 \beta_{2} + 41\)
\(\nu^{11}\)\(=\)\(-28 \beta_{19} + 82 \beta_{18} - 50 \beta_{14} + 35 \beta_{13} - 105 \beta_{11} - 52 \beta_{10} + 7 \beta_{7} - 228 \beta_{5} - 61 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-24 \beta_{16} - 87 \beta_{15} - 94 \beta_{12} + 39 \beta_{9} + 37 \beta_{8} + 181 \beta_{6} - 99 \beta_{4} - 27 \beta_{3} + 43 \beta_{2} - 541\)
\(\nu^{13}\)\(=\)\(-25 \beta_{19} + 61 \beta_{18} + 11 \beta_{17} - 108 \beta_{14} + 225 \beta_{13} - 25 \beta_{11} - 71 \beta_{10} - 18 \beta_{7} + 272 \beta_{5} - 494 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-119 \beta_{16} - 309 \beta_{15} + 143 \beta_{12} - 71 \beta_{9} - 471 \beta_{8} + 133 \beta_{6} + 82 \beta_{4} + 299 \beta_{3} - 361 \beta_{2} + 1371\)
\(\nu^{15}\)\(=\)\(-612 \beta_{19} + 340 \beta_{18} - 742 \beta_{17} - 369 \beta_{14} + 156 \beta_{13} - 214 \beta_{11} - 371 \beta_{10} + 300 \beta_{7} + 342 \beta_{5} + 943 \beta_{1}\)
\(\nu^{16}\)\(=\)\(231 \beta_{16} + 860 \beta_{15} + 925 \beta_{12} - 28 \beta_{9} + \beta_{8} + 441 \beta_{6} + 187 \beta_{4} - 227 \beta_{3} + 1782 \beta_{2} + 423\)
\(\nu^{17}\)\(=\)\(1673 \beta_{19} - 1231 \beta_{18} + 1745 \beta_{17} + 991 \beta_{14} - 1075 \beta_{13} - 1030 \beta_{11} + 2415 \beta_{10} + 1072 \beta_{7} + 3510 \beta_{5} + 3908 \beta_{1}\)
\(\nu^{18}\)\(=\)\(-384 \beta_{16} - 2536 \beta_{15} - 3108 \beta_{12} - 658 \beta_{9} + 4292 \beta_{8} + 409 \beta_{6} + 826 \beta_{4} + 1216 \beta_{3} + 1614 \beta_{2} + 5391\)
\(\nu^{19}\)\(=\)\(3396 \beta_{19} - 6374 \beta_{18} + 2844 \beta_{17} - 3026 \beta_{14} + 1052 \beta_{13} + 5722 \beta_{11} + 294 \beta_{10} - 7482 \beta_{7} - 5353 \beta_{5} + 3942 \beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
200.1
−0.362922 + 1.69360i
−0.362922 1.69360i
1.16595 + 1.28084i
1.16595 1.28084i
−1.24906 + 1.19993i
−1.24906 1.19993i
1.58783 + 0.691953i
1.58783 0.691953i
−0.421281 + 1.68004i
−0.421281 1.68004i
0.421281 + 1.68004i
0.421281 1.68004i
−1.58783 + 0.691953i
−1.58783 0.691953i
1.24906 + 1.19993i
1.24906 1.19993i
−1.16595 + 1.28084i
−1.16595 1.28084i
0.362922 + 1.69360i
0.362922 1.69360i
−2.50456 −0.362922 1.69360i 4.27284 2.28042 0.908961 + 4.24173i 3.79080i −5.69246 −2.73658 + 1.22929i −5.71146
200.2 −2.50456 −0.362922 + 1.69360i 4.27284 2.28042 0.908961 4.24173i 3.79080i −5.69246 −2.73658 1.22929i −5.71146
200.3 −2.09324 1.16595 1.28084i 2.38166 −1.47666 −2.44062 + 2.68111i 2.11750i −0.798907 −0.281099 2.98680i 3.09102
200.4 −2.09324 1.16595 + 1.28084i 2.38166 −1.47666 −2.44062 2.68111i 2.11750i −0.798907 −0.281099 + 2.98680i 3.09102
200.5 −1.18817 −1.24906 1.19993i −0.588256 −2.08509 1.48410 + 1.42572i 0.284155i 3.07528 0.120318 + 2.99759i 2.47744
200.6 −1.18817 −1.24906 + 1.19993i −0.588256 −2.08509 1.48410 1.42572i 0.284155i 3.07528 0.120318 2.99759i 2.47744
200.7 −0.758634 1.58783 0.691953i −1.42447 1.15744 −1.20458 + 0.524939i 4.81799i 2.59792 2.04240 2.19741i −0.878075
200.8 −0.758634 1.58783 + 0.691953i −1.42447 1.15744 −1.20458 0.524939i 4.81799i 2.59792 2.04240 + 2.19741i −0.878075
200.9 −0.598526 −0.421281 1.68004i −1.64177 3.30634 0.252148 + 1.00555i 2.20280i 2.17969 −2.64504 + 1.41554i −1.97893
200.10 −0.598526 −0.421281 + 1.68004i −1.64177 3.30634 0.252148 1.00555i 2.20280i 2.17969 −2.64504 1.41554i −1.97893
200.11 0.598526 0.421281 1.68004i −1.64177 −3.30634 0.252148 1.00555i 2.20280i −2.17969 −2.64504 1.41554i −1.97893
200.12 0.598526 0.421281 + 1.68004i −1.64177 −3.30634 0.252148 + 1.00555i 2.20280i −2.17969 −2.64504 + 1.41554i −1.97893
200.13 0.758634 −1.58783 0.691953i −1.42447 −1.15744 −1.20458 0.524939i 4.81799i −2.59792 2.04240 + 2.19741i −0.878075
200.14 0.758634 −1.58783 + 0.691953i −1.42447 −1.15744 −1.20458 + 0.524939i 4.81799i −2.59792 2.04240 2.19741i −0.878075
200.15 1.18817 1.24906 1.19993i −0.588256 2.08509 1.48410 1.42572i 0.284155i −3.07528 0.120318 2.99759i 2.47744
200.16 1.18817 1.24906 + 1.19993i −0.588256 2.08509 1.48410 + 1.42572i 0.284155i −3.07528 0.120318 + 2.99759i 2.47744
200.17 2.09324 −1.16595 1.28084i 2.38166 1.47666 −2.44062 2.68111i 2.11750i 0.798907 −0.281099 + 2.98680i 3.09102
200.18 2.09324 −1.16595 + 1.28084i 2.38166 1.47666 −2.44062 + 2.68111i 2.11750i 0.798907 −0.281099 2.98680i 3.09102
200.19 2.50456 0.362922 1.69360i 4.27284 −2.28042 0.908961 4.24173i 3.79080i 5.69246 −2.73658 1.22929i −5.71146
200.20 2.50456 0.362922 + 1.69360i 4.27284 −2.28042 0.908961 + 4.24173i 3.79080i 5.69246 −2.73658 + 1.22929i −5.71146
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 200.20
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
67.b Odd 1 yes
201.d Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(201, [\chi])\).