# Properties

 Label 201.2.e.c Level 201 Weight 2 Character orbit 201.e Analytic conductor 1.605 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.3665654523963.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 8 x^{8} + 21 x^{6} - 5 x^{5} + 26 x^{4} + 4 x^{3} + 13 x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-1127 \nu^{9} + 7711 \nu^{8} - 25937 \nu^{7} + 48871 \nu^{6} - 53276 \nu^{5} + 58183 \nu^{4} - 144911 \nu^{3} + 39957 \nu^{2} - 9814 \nu - 47489$$$$)/44248$$ $$\beta_{3}$$ $$=$$ $$($$$$1899 \nu^{9} - 6083 \nu^{8} + 20461 \nu^{7} - 17723 \nu^{6} + 42028 \nu^{5} - 45899 \nu^{4} + 89131 \nu^{3} - 31521 \nu^{2} + 7742 \nu - 12403$$$$)/44248$$ $$\beta_{4}$$ $$=$$ $$($$$$-3473 \nu^{9} + 2993 \nu^{8} - 22135 \nu^{7} - 25035 \nu^{6} - 95096 \nu^{5} - 47811 \nu^{4} - 116057 \nu^{3} - 66953 \nu^{2} - 90294 \nu - 6827$$$$)/22124$$ $$\beta_{5}$$ $$=$$ $$($$$$12403 \nu^{9} - 22907 \nu^{8} + 93141 \nu^{7} + 20461 \nu^{6} + 242740 \nu^{5} - 19987 \nu^{4} + 276579 \nu^{3} + 138743 \nu^{2} + 129718 \nu + 14781$$$$)/44248$$ $$\beta_{6}$$ $$=$$ $$($$$$-14781 \nu^{9} + 41965 \nu^{8} - 141155 \nu^{7} + 93141 \nu^{6} - 289940 \nu^{5} + 316645 \nu^{4} - 404293 \nu^{3} + 217455 \nu^{2} - 53410 \nu + 129813$$$$)/44248$$ $$\beta_{7}$$ $$=$$ $$($$$$3911 \nu^{9} - 7629 \nu^{8} + 31695 \nu^{7} - 1369 \nu^{6} + 89918 \nu^{5} - 22367 \nu^{4} + 104757 \nu^{3} + 1699 \nu^{2} + 52952 \nu - 12251$$$$)/11062$$ $$\beta_{8}$$ $$=$$ $$($$$$2133 \nu^{9} - 2953 \nu^{8} + 14961 \nu^{7} + 9085 \nu^{6} + 49566 \nu^{5} + 14424 \nu^{4} + 58697 \nu^{3} + 31963 \nu^{2} + 49012 \nu + 3317$$$$)/5531$$ $$\beta_{9}$$ $$=$$ $$($$$$-9205 \nu^{9} + 14061 \nu^{8} - 61375 \nu^{7} - 41255 \nu^{6} - 165532 \nu^{5} - 38715 \nu^{4} - 146785 \nu^{3} - 130277 \nu^{2} - 56110 \nu + 12005$$$$)/22124$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 7 \beta_{3} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{9} + 7 \beta_{8} - 2 \beta_{5} + 8 \beta_{4} + 2 \beta_{2} - 18 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{9} + 18 \beta_{8} - 20 \beta_{7} - 18 \beta_{6} - \beta_{5} - 53 \beta_{3} - 53 \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$-61 \beta_{7} - 53 \beta_{6} - 61 \beta_{4} - 145 \beta_{3} - 20 \beta_{2} + 7$$ $$\nu^{7}$$ $$=$$ $$-61 \beta_{9} - 145 \beta_{8} + 11 \beta_{5} - 165 \beta_{4} - 61 \beta_{2} + 411 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-165 \beta_{9} - 411 \beta_{8} + 472 \beta_{7} + 411 \beta_{6} + 40 \beta_{5} + 1143 \beta_{3} + 1143 \beta_{1} - 40$$ $$\nu^{9}$$ $$=$$ $$1308 \beta_{7} + 1143 \beta_{6} + 1308 \beta_{4} + 3209 \beta_{3} + 472 \beta_{2} - 95$$

## Character values

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

 $$n$$ $$68$$ $$136$$ $$\chi(n)$$ $$1$$ $$-\beta_{5}$$

## 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
 0.586579 − 1.01598i −0.725148 + 1.25599i 1.39901 − 2.42316i 0.133354 − 0.230976i −0.393795 + 0.682072i 0.586579 + 1.01598i −0.725148 − 1.25599i 1.39901 + 2.42316i 0.133354 + 0.230976i −0.393795 − 0.682072i
−1.32873 2.30142i 1.00000 −2.53104 + 4.38389i 2.48430 −1.32873 2.30142i 0.160379 0.277784i 8.13733 1.00000 −3.30096 5.71743i
37.2 −0.453162 0.784899i 1.00000 0.589289 1.02068i 3.35662 −0.453162 0.784899i −0.380391 + 0.658856i −2.88082 1.00000 −1.52109 2.63461i
37.3 −0.0732181 0.126817i 1.00000 0.989278 1.71348i −1.65158 −0.0732181 0.126817i 1.22031 2.11364i −0.582605 1.00000 0.120926 + 0.209450i
37.4 0.538290 + 0.932346i 1.00000 0.420487 0.728305i −0.343289 0.538290 + 0.932346i −1.74135 + 3.01611i 3.05854 1.00000 −0.184789 0.320064i
37.5 1.31682 + 2.28079i 1.00000 −2.46802 + 4.27473i −0.846046 1.31682 + 2.28079i 0.241054 0.417518i −7.73244 1.00000 −1.11409 1.92966i
163.1 −1.32873 + 2.30142i 1.00000 −2.53104 4.38389i 2.48430 −1.32873 + 2.30142i 0.160379 + 0.277784i 8.13733 1.00000 −3.30096 + 5.71743i
163.2 −0.453162 + 0.784899i 1.00000 0.589289 + 1.02068i 3.35662 −0.453162 + 0.784899i −0.380391 0.658856i −2.88082 1.00000 −1.52109 + 2.63461i
163.3 −0.0732181 + 0.126817i 1.00000 0.989278 + 1.71348i −1.65158 −0.0732181 + 0.126817i 1.22031 + 2.11364i −0.582605 1.00000 0.120926 0.209450i
163.4 0.538290 0.932346i 1.00000 0.420487 + 0.728305i −0.343289 0.538290 0.932346i −1.74135 3.01611i 3.05854 1.00000 −0.184789 + 0.320064i
163.5 1.31682 2.28079i 1.00000 −2.46802 4.27473i −0.846046 1.31682 2.28079i 0.241054 + 0.417518i −7.73244 1.00000 −1.11409 + 1.92966i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.e.c 10
3.b odd 2 1 603.2.g.f 10
67.c even 3 1 inner 201.2.e.c 10
201.g odd 6 1 603.2.g.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.e.c 10 1.a even 1 1 trivial
201.2.e.c 10 67.c even 3 1 inner
603.2.g.f 10 3.b odd 2 1
603.2.g.f 10 201.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 5 T^{4} + T^{5} - 2 T^{6} - 4 T^{7} - 11 T^{8} - T^{9} + 31 T^{10} - 2 T^{11} - 44 T^{12} - 32 T^{13} - 32 T^{14} + 32 T^{15} + 320 T^{16} - 512 T^{18} + 1024 T^{20}$$
$3$ $$( 1 - T )^{10}$$
$5$ $$( 1 - 3 T + 19 T^{2} - 49 T^{3} + 176 T^{4} - 336 T^{5} + 880 T^{6} - 1225 T^{7} + 2375 T^{8} - 1875 T^{9} + 3125 T^{10} )^{2}$$
$7$ $$1 + T - 25 T^{2} - 30 T^{3} + 343 T^{4} + 411 T^{5} - 3169 T^{6} - 3189 T^{7} + 23137 T^{8} + 9593 T^{9} - 159214 T^{10} + 67151 T^{11} + 1133713 T^{12} - 1093827 T^{13} - 7608769 T^{14} + 6907677 T^{15} + 40353607 T^{16} - 24706290 T^{17} - 144120025 T^{18} + 40353607 T^{19} + 282475249 T^{20}$$
$11$ $$1 - 6 T - 2 T^{2} - 32 T^{3} + 475 T^{4} + 152 T^{5} - 668 T^{6} - 21446 T^{7} + 9413 T^{8} - 22120 T^{9} + 881362 T^{10} - 243320 T^{11} + 1138973 T^{12} - 28544626 T^{13} - 9780188 T^{14} + 24479752 T^{15} + 841491475 T^{16} - 623589472 T^{17} - 428717762 T^{18} - 14147686146 T^{19} + 25937424601 T^{20}$$
$13$ $$1 + T - 39 T^{2} - 4 T^{3} + 873 T^{4} - 297 T^{5} - 9619 T^{6} + 9753 T^{7} + 44855 T^{8} - 59517 T^{9} + 141714 T^{10} - 773721 T^{11} + 7580495 T^{12} + 21427341 T^{13} - 274728259 T^{14} - 110274021 T^{15} + 4213804257 T^{16} - 250994068 T^{17} - 31813498119 T^{18} + 10604499373 T^{19} + 137858491849 T^{20}$$
$17$ $$1 - 8 T - 5 T^{2} + 38 T^{3} + 942 T^{4} - 1960 T^{5} - 3632 T^{6} - 31122 T^{7} - 17598 T^{8} + 154944 T^{9} + 3171184 T^{10} + 2634048 T^{11} - 5085822 T^{12} - 152902386 T^{13} - 303348272 T^{14} - 2782919720 T^{15} + 22737589998 T^{16} + 15592869574 T^{17} - 34878787205 T^{18} - 948703011976 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 + 5 T + 9 T^{2} + 20 T^{3} - 543 T^{4} - 2347 T^{5} - 3137 T^{6} - 25261 T^{7} + 112571 T^{8} + 1056217 T^{9} + 2867534 T^{10} + 20068123 T^{11} + 40638131 T^{12} - 173265199 T^{13} - 408816977 T^{14} - 5811404353 T^{15} - 25545913383 T^{16} + 17877434780 T^{17} + 152852067369 T^{18} + 1613438488895 T^{19} + 6131066257801 T^{20}$$
$23$ $$1 + 7 T - 28 T^{2} - 169 T^{3} + 490 T^{4} - 2635 T^{5} - 47402 T^{6} - 68213 T^{7} + 1115861 T^{8} + 3074918 T^{9} - 10160692 T^{10} + 70723114 T^{11} + 590290469 T^{12} - 829947571 T^{13} - 13265023082 T^{14} - 16959763805 T^{15} + 72537585610 T^{16} - 575415500543 T^{17} - 2192707587868 T^{18} + 12608068630241 T^{19} + 41426511213649 T^{20}$$
$29$ $$1 + 12 T + 49 T^{2} - 698 T^{3} - 8162 T^{4} - 29314 T^{5} + 242314 T^{6} + 2648902 T^{7} + 8527334 T^{8} - 51892474 T^{9} - 519226304 T^{10} - 1504881746 T^{11} + 7171487894 T^{12} + 64604070878 T^{13} + 171384088234 T^{14} - 601263821786 T^{15} - 4854947946002 T^{16} - 12040413663682 T^{17} + 24512074235089 T^{18} + 174085751710428 T^{19} + 420707233300201 T^{20}$$
$31$ $$1 + 12 T + 26 T^{2} - 608 T^{3} - 5365 T^{4} - 10912 T^{5} + 130200 T^{6} + 1149340 T^{7} + 3094221 T^{8} - 19496512 T^{9} - 205293462 T^{10} - 604391872 T^{11} + 2973546381 T^{12} + 34239987940 T^{13} + 120242434200 T^{14} - 312401295712 T^{15} - 4761457248565 T^{16} - 16727669379488 T^{17} + 22175166973466 T^{18} + 317275465928052 T^{19} + 819628286980801 T^{20}$$
$37$ $$1 + 17 T + 79 T^{2} - 250 T^{3} - 3783 T^{4} - 21237 T^{5} - 177053 T^{6} - 967703 T^{7} + 1810739 T^{8} + 55321381 T^{9} + 416535186 T^{10} + 2046891097 T^{11} + 2478901691 T^{12} - 49017060059 T^{13} - 331825827533 T^{14} - 1472657614809 T^{15} - 9706143005247 T^{16} - 23732969283250 T^{17} + 277485876859759 T^{18} + 2209349576516309 T^{19} + 4808584372417849 T^{20}$$
$41$ $$1 - 13 T - 3 T^{2} + 528 T^{3} - 423 T^{4} + 633 T^{5} - 145243 T^{6} + 325119 T^{7} + 5344351 T^{8} - 15495599 T^{9} - 86063806 T^{10} - 635319559 T^{11} + 8983854031 T^{12} + 22407526599 T^{13} - 410422004923 T^{14} + 73336975233 T^{15} - 2009294093943 T^{16} + 102830256609168 T^{17} - 23954775687363 T^{18} - 4255965147121493 T^{19} + 13422659310152401 T^{20}$$
$43$ $$( 1 + 2 T + 136 T^{2} + 213 T^{3} + 9751 T^{4} + 12954 T^{5} + 419293 T^{6} + 393837 T^{7} + 10812952 T^{8} + 6837602 T^{9} + 147008443 T^{10} )^{2}$$
$47$ $$1 + 25 T + 229 T^{2} + 1600 T^{3} + 17233 T^{4} + 107717 T^{5} + 1759 T^{6} - 2747789 T^{7} - 27929365 T^{8} - 392253539 T^{9} - 3590510690 T^{10} - 18435916333 T^{11} - 61695967285 T^{12} - 285283697347 T^{13} + 8583358879 T^{14} + 24704356119019 T^{15} + 185758217764657 T^{16} + 810596992740800 T^{17} + 5452784645543269 T^{18} + 27978261827569175 T^{19} + 52599132235830049 T^{20}$$
$53$ $$( 1 + 6 T + 156 T^{2} + 881 T^{3} + 13099 T^{4} + 61642 T^{5} + 694247 T^{6} + 2474729 T^{7} + 23224812 T^{8} + 47342886 T^{9} + 418195493 T^{10} )^{2}$$
$59$ $$( 1 + 6 T + 170 T^{2} + 709 T^{3} + 16557 T^{4} + 62562 T^{5} + 976863 T^{6} + 2468029 T^{7} + 34914430 T^{8} + 72704166 T^{9} + 714924299 T^{10} )^{2}$$
$61$ $$1 - 9 T - 143 T^{2} + 720 T^{3} + 17731 T^{4} - 41977 T^{5} - 1063593 T^{6} + 191383 T^{7} + 37041985 T^{8} + 39709251 T^{9} - 1285232122 T^{10} + 2422264311 T^{11} + 137833226185 T^{12} + 43440304723 T^{13} - 14726339566713 T^{14} - 35453618927077 T^{15} + 913507757794891 T^{16} + 2262774841935120 T^{17} - 27414145758611183 T^{18} - 105247314835507269 T^{19} + 713342911662882601 T^{20}$$
$67$ $$1 - 2 T + 108 T^{2} - 663 T^{3} - 609 T^{4} - 70902 T^{5} - 40803 T^{6} - 2976207 T^{7} + 32482404 T^{8} - 40302242 T^{9} + 1350125107 T^{10}$$
$71$ $$1 - 29 T + 172 T^{2} + 227 T^{3} + 42010 T^{4} - 655063 T^{5} + 551678 T^{6} - 7889593 T^{7} + 675212805 T^{8} - 3579881474 T^{9} - 7395575892 T^{10} - 254171584654 T^{11} + 3403747750005 T^{12} - 2823772120223 T^{13} + 14019065350718 T^{14} - 1181883891354113 T^{15} + 5381492927521210 T^{16} + 2064592275954757 T^{17} + 111069607374270892 T^{18} - 1329606520835021899 T^{19} + 3255243551009881201 T^{20}$$
$73$ $$1 - 12 T - 243 T^{2} + 2166 T^{3} + 50962 T^{4} - 296558 T^{5} - 6805814 T^{6} + 19816122 T^{7} + 747409258 T^{8} - 754426666 T^{9} - 59977062832 T^{10} - 55073146618 T^{11} + 3982943935882 T^{12} + 7708808332074 T^{13} - 193273146173174 T^{14} - 614785965476894 T^{15} + 7712294840140018 T^{16} + 23928665192364102 T^{17} - 195969802330261683 T^{18} - 706459040499214956 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$1 + T - 163 T^{2} - 2432 T^{3} + 17745 T^{4} + 386997 T^{5} + 1447855 T^{6} - 43350169 T^{7} - 351131257 T^{8} + 1211710577 T^{9} + 44054595094 T^{10} + 95725135583 T^{11} - 2191410174937 T^{12} - 21373323973591 T^{13} + 56394069526255 T^{14} + 1190811595243803 T^{15} + 4313586898220145 T^{16} - 46703906654338688 T^{17} - 247288736014769443 T^{18} + 119851595982618319 T^{19} + 9468276082626847201 T^{20}$$
$83$ $$1 + 6 T - 333 T^{2} - 1472 T^{3} + 66934 T^{4} + 204696 T^{5} - 9747600 T^{6} - 15450810 T^{7} + 1133101468 T^{8} + 537045640 T^{9} - 104602238660 T^{10} + 44574788120 T^{11} + 7805936013052 T^{12} - 8834572297470 T^{13} - 462604729779600 T^{14} + 806305863459528 T^{15} + 21883426951080646 T^{16} - 39944267056730944 T^{17} - 750013313302300653 T^{18} + 1121641531605242418 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$( 1 + 2 T + 38 T^{2} - 847 T^{3} + 4877 T^{4} - 52366 T^{5} + 434053 T^{6} - 6709087 T^{7} + 26788822 T^{8} + 125484482 T^{9} + 5584059449 T^{10} )^{2}$$
$97$ $$1 - 11 T - 377 T^{2} + 2738 T^{3} + 105319 T^{4} - 490863 T^{5} - 19253567 T^{6} + 44453801 T^{7} + 2768038153 T^{8} - 2124828485 T^{9} - 299323562154 T^{10} - 206108363045 T^{11} + 26044470981577 T^{12} + 40571783920073 T^{13} - 1704504443195327 T^{14} - 4215207600571791 T^{15} + 87727778587117351 T^{16} + 221225702901073394 T^{17} - 2954712465080114297 T^{18} - 8362541645200217387 T^{19} + 73742412689492826049 T^{20}$$