# 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

# Related objects

## Newspace parameters

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

## 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: $$2^{4}$$ Twist minimal: yes 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.b odd 2 1 inner
201.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.d.a 20
3.b odd 2 1 inner 201.2.d.a 20
67.b odd 2 1 inner 201.2.d.a 20
201.d even 2 1 inner 201.2.d.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.d.a 20 1.a even 1 1 trivial
201.2.d.a 20 3.b odd 2 1 inner
201.2.d.a 20 67.b odd 2 1 inner
201.2.d.a 20 201.d even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 7 T^{2} + 26 T^{4} + 71 T^{6} + 173 T^{8} + 372 T^{10} + 692 T^{12} + 1136 T^{14} + 1664 T^{16} + 1792 T^{18} + 1024 T^{20} )^{2}$$
$3$ $$1 + 7 T^{2} + 32 T^{4} + 128 T^{6} + 423 T^{8} + 1186 T^{10} + 3807 T^{12} + 10368 T^{14} + 23328 T^{16} + 45927 T^{18} + 59049 T^{20}$$
$5$ $$( 1 + 26 T^{2} + 367 T^{4} + 3506 T^{6} + 25161 T^{8} + 141088 T^{10} + 629025 T^{12} + 2191250 T^{14} + 5734375 T^{16} + 10156250 T^{18} + 9765625 T^{20} )^{2}$$
$7$ $$( 1 - 23 T^{2} + 283 T^{4} - 2507 T^{6} + 19167 T^{8} - 139372 T^{10} + 939183 T^{12} - 6019307 T^{14} + 33294667 T^{16} - 132590423 T^{18} + 282475249 T^{20} )^{2}$$
$11$ $$( 1 + 58 T^{2} + 1862 T^{4} + 40628 T^{6} + 660125 T^{8} + 8230836 T^{10} + 79875125 T^{12} + 594834548 T^{14} + 3298646582 T^{16} + 12432815098 T^{18} + 25937424601 T^{20} )^{2}$$
$13$ $$( 1 - 57 T^{2} + 1668 T^{4} - 35816 T^{6} + 625183 T^{8} - 8962430 T^{10} + 105655927 T^{12} - 1022940776 T^{14} + 8051117412 T^{16} - 46496651097 T^{18} + 137858491849 T^{20} )^{2}$$
$17$ $$( 1 - 69 T^{2} + 2897 T^{4} - 84448 T^{6} + 1943438 T^{8} - 36126838 T^{10} + 561653582 T^{12} - 7053181408 T^{14} + 69926537393 T^{16} - 481327263429 T^{18} + 2015993900449 T^{20} )^{2}$$
$19$ $$( 1 - 6 T + 77 T^{2} - 399 T^{3} + 2694 T^{4} - 10816 T^{5} + 51186 T^{6} - 144039 T^{7} + 528143 T^{8} - 781926 T^{9} + 2476099 T^{10} )^{4}$$
$23$ $$( 1 - 135 T^{2} + 9753 T^{4} - 466864 T^{6} + 16291345 T^{8} - 428543479 T^{10} + 8618121505 T^{12} - 130647688624 T^{14} + 1443794025417 T^{16} - 10571983012935 T^{18} + 41426511213649 T^{20} )^{2}$$
$29$ $$( 1 - 155 T^{2} + 11412 T^{4} - 541365 T^{6} + 19456967 T^{8} - 596118168 T^{10} + 16363309247 T^{12} - 382897178565 T^{14} + 6788123739252 T^{16} - 77538194008955 T^{18} + 420707233300201 T^{20} )^{2}$$
$31$ $$( 1 - 128 T^{2} + 9279 T^{4} - 470100 T^{6} + 18750365 T^{8} - 628047028 T^{10} + 18019100765 T^{12} - 434147222100 T^{14} + 8235146655999 T^{16} - 109170052792448 T^{18} + 819628286980801 T^{20} )^{2}$$
$37$ $$( 1 - 6 T + 126 T^{2} - 349 T^{3} + 5773 T^{4} - 8281 T^{5} + 213601 T^{6} - 477781 T^{7} + 6382278 T^{8} - 11244966 T^{9} + 69343957 T^{10} )^{4}$$
$41$ $$( 1 + 273 T^{2} + 36843 T^{4} + 3219131 T^{6} + 201840205 T^{8} + 9492902282 T^{10} + 339293384605 T^{12} + 9096494833691 T^{14} + 175008090551163 T^{16} + 2179884587550033 T^{18} + 13422659310152401 T^{20} )^{2}$$
$43$ $$( 1 - 193 T^{2} + 18277 T^{4} - 1220287 T^{6} + 67258147 T^{8} - 3150400198 T^{10} + 124360313803 T^{12} - 4171918415887 T^{14} + 115535552446573 T^{16} - 2255822653576993 T^{18} + 21611482313284249 T^{20} )^{2}$$
$47$ $$( 1 - 168 T^{2} + 14919 T^{4} - 890619 T^{6} + 44275280 T^{8} - 2059302358 T^{10} + 97804093520 T^{12} - 4345936612539 T^{14} + 160815113493351 T^{16} - 4000296159175848 T^{18} + 52599132235830049 T^{20} )^{2}$$
$53$ $$( 1 + 308 T^{2} + 47691 T^{4} + 4933406 T^{6} + 378892457 T^{8} + 22619448350 T^{10} + 1064308911713 T^{12} + 38926946308286 T^{14} + 1057040546603139 T^{16} + 19175984646699188 T^{18} + 174887470365513049 T^{20} )^{2}$$
$59$ $$( 1 - 125 T^{2} + 15350 T^{4} - 875993 T^{6} + 60034785 T^{8} - 2515766660 T^{10} + 208981086585 T^{12} - 10614723414473 T^{14} + 647471191389350 T^{16} - 18353804700540125 T^{18} + 511116753300641401 T^{20} )^{2}$$
$61$ $$( 1 - 295 T^{2} + 47942 T^{4} - 5519398 T^{6} + 481865517 T^{8} - 32973713998 T^{10} + 1793021588757 T^{12} - 76420707123718 T^{14} + 2469989787615062 T^{16} - 56553657334197895 T^{18} + 713342911662882601 T^{20} )^{2}$$
$67$ $$( 1 - 16 T + 95 T^{2} - 280 T^{3} - 2006 T^{4} + 39184 T^{5} - 134402 T^{6} - 1256920 T^{7} + 28572485 T^{8} - 322417936 T^{9} + 1350125107 T^{10} )^{2}$$
$71$ $$( 1 - 460 T^{2} + 100726 T^{4} - 14076066 T^{6} + 1429238317 T^{8} - 113368312844 T^{10} + 7204790355997 T^{12} - 357696498926946 T^{14} + 12903029198226646 T^{16} - 297046624373050060 T^{18} + 3255243551009881201 T^{20} )^{2}$$
$73$ $$( 1 - T + 206 T^{2} - 103 T^{3} + 20115 T^{4} - 6904 T^{5} + 1468395 T^{6} - 548887 T^{7} + 80137502 T^{8} - 28398241 T^{9} + 2073071593 T^{10} )^{4}$$
$79$ $$( 1 - 411 T^{2} + 63834 T^{4} - 3197614 T^{6} - 304842923 T^{8} + 50995950578 T^{10} - 1902524682443 T^{12} - 124547324306734 T^{14} + 15517244635727514 T^{16} - 623531720871596571 T^{18} + 9468276082626847201 T^{20} )^{2}$$
$83$ $$( 1 - 365 T^{2} + 75403 T^{4} - 11235039 T^{6} + 1291616557 T^{8} - 118995779466 T^{10} + 8897946461173 T^{12} - 533196087309519 T^{14} + 24652284973142707 T^{16} - 822086664730749965 T^{18} + 15516041187205853449 T^{20} )^{2}$$
$89$ $$( 1 - 703 T^{2} + 234596 T^{4} - 48864245 T^{6} + 7035486623 T^{8} - 731920278252 T^{10} + 55728089540783 T^{12} - 3065852236073045 T^{14} + 116589822934286756 T^{16} - 2767421930408562943 T^{18} + 31181719929966183601 T^{20} )^{2}$$
$97$ $$( 1 - 684 T^{2} + 228862 T^{4} - 49086542 T^{6} + 7453302405 T^{8} - 836001815156 T^{10} + 70128122328645 T^{12} - 4345596270036302 T^{14} + 190635638992060798 T^{16} - 5360804578553841324 T^{18} + 73742412689492826049 T^{20} )^{2}$$