# Properties

 Label 2001.2.a.l Level $2001$ Weight $2$ Character orbit 2001.a Self dual yes Analytic conductor $15.978$ Analytic rank $0$ Dimension $11$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.9780654445$$ Analytic rank: $$0$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ Defining polynomial: $$x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} - 93 x^{2} - 369 x - 108$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{10}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} - 93 x^{2} - 369 x - 108$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu - 1$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{10} + 5 \nu^{9} + 9 \nu^{8} - 63 \nu^{7} - 13 \nu^{6} + 251 \nu^{5} - 27 \nu^{4} - 360 \nu^{3} - 4 \nu^{2} + 165 \nu + 48$$$$)/6$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{10} + 7 \nu^{9} + 21 \nu^{8} - 87 \nu^{7} - 59 \nu^{6} + 343 \nu^{5} + 63 \nu^{4} - 498 \nu^{3} - 170 \nu^{2} + 261 \nu + 153$$$$)/9$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{10} - \nu^{9} + 24 \nu^{8} + 6 \nu^{7} - 187 \nu^{6} + 23 \nu^{5} + 567 \nu^{4} - 150 \nu^{3} - 598 \nu^{2} + 117 \nu + 162$$$$)/9$$ $$\beta_{7}$$ $$=$$ $$\nu^{8} - 2 \nu^{7} - 12 \nu^{6} + 22 \nu^{5} + 44 \nu^{4} - 68 \nu^{3} - 57 \nu^{2} + 52 \nu + 31$$ $$\beta_{8}$$ $$=$$ $$($$$$-4 \nu^{10} + 14 \nu^{9} + 42 \nu^{8} - 174 \nu^{7} - 109 \nu^{6} + 686 \nu^{5} + 18 \nu^{4} - 996 \nu^{3} - 7 \nu^{2} + 504 \nu + 135$$$$)/9$$ $$\beta_{9}$$ $$=$$ $$($$$$2 \nu^{10} - 7 \nu^{9} - 30 \nu^{8} + 105 \nu^{7} + 167 \nu^{6} - 541 \nu^{5} - 450 \nu^{4} + 1101 \nu^{3} + 620 \nu^{2} - 693 \nu - 369$$$$)/9$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{10} + 5 \nu^{9} + 15 \nu^{8} - 75 \nu^{7} - 91 \nu^{6} + 389 \nu^{5} + 303 \nu^{4} - 822 \nu^{3} - 538 \nu^{2} + 591 \nu + 354$$$$)/6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{5} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 22$$ $$\nu^{5}$$ $$=$$ $$\beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 29 \beta_{1} + 10$$ $$\nu^{6}$$ $$=$$ $$12 \beta_{9} + \beta_{8} + 12 \beta_{7} + 10 \beta_{5} + 12 \beta_{3} + 47 \beta_{2} + 14 \beta_{1} + 135$$ $$\nu^{7}$$ $$=$$ $$12 \beta_{10} + 16 \beta_{9} + 12 \beta_{8} + 3 \beta_{7} + \beta_{6} - 10 \beta_{5} - 10 \beta_{4} + 82 \beta_{3} + 25 \beta_{2} + 184 \beta_{1} + 89$$ $$\nu^{8}$$ $$=$$ $$2 \beta_{10} + 110 \beta_{9} + 14 \beta_{8} + 107 \beta_{7} + 2 \beta_{6} + 78 \beta_{5} + 2 \beta_{4} + 112 \beta_{3} + 319 \beta_{2} + 142 \beta_{1} + 875$$ $$\nu^{9}$$ $$=$$ $$105 \beta_{10} + 172 \beta_{9} + 106 \beta_{8} + 53 \beta_{7} + 15 \beta_{6} - 73 \beta_{5} - 71 \beta_{4} + 629 \beta_{3} + 240 \beta_{2} + 1229 \beta_{1} + 752$$ $$\nu^{10}$$ $$=$$ $$38 \beta_{10} + 910 \beta_{9} + 138 \beta_{8} + 856 \beta_{7} + 30 \beta_{6} + 559 \beta_{5} + 36 \beta_{4} + 954 \beta_{3} + 2194 \beta_{2} + 1266 \beta_{1} + 5861$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.51124 −1.96728 −1.94502 −1.05971 −0.661934 −0.467085 1.17662 1.76484 2.24285 2.70316 2.72479
−2.51124 −1.00000 4.30634 2.07440 2.51124 −0.329384 −5.79178 1.00000 −5.20933
1.2 −1.96728 −1.00000 1.87020 −3.90206 1.96728 −0.839519 0.255353 1.00000 7.67646
1.3 −1.94502 −1.00000 1.78310 0.890641 1.94502 −3.69089 0.421884 1.00000 −1.73231
1.4 −1.05971 −1.00000 −0.877023 −1.30384 1.05971 0.720797 3.04880 1.00000 1.38169
1.5 −0.661934 −1.00000 −1.56184 −1.16115 0.661934 4.80000 2.35770 1.00000 0.768607
1.6 −0.467085 −1.00000 −1.78183 −0.0105419 0.467085 −1.85912 1.76644 1.00000 0.00492395
1.7 1.17662 −1.00000 −0.615555 2.85352 −1.17662 3.62966 −3.07753 1.00000 3.35752
1.8 1.76484 −1.00000 1.11465 −2.54815 −1.76484 −4.97592 −1.56250 1.00000 −4.49707
1.9 2.24285 −1.00000 3.03039 3.33222 −2.24285 0.336371 2.31101 1.00000 7.47368
1.10 2.70316 −1.00000 5.30708 2.79988 −2.70316 0.880738 8.93955 1.00000 7.56852
1.11 2.72479 −1.00000 5.42450 −1.02492 −2.72479 4.32726 9.33107 1.00000 −2.79269
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$23$$ $$-1$$
$$29$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2001.2.a.l 11
3.b odd 2 1 6003.2.a.m 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.l 11 1.a even 1 1 trivial
6003.2.a.m 11 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2001))$$:

 $$T_{2}^{11} - \cdots$$ $$T_{5}^{11} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-108 - 369 T - 93 T^{2} + 634 T^{3} + 285 T^{4} - 408 T^{5} - 152 T^{6} + 124 T^{7} + 30 T^{8} - 18 T^{9} - 2 T^{10} + T^{11}$$
$3$ $$( 1 + T )^{11}$$
$5$ $$-8 - 764 T - 470 T^{2} + 1499 T^{3} + 896 T^{4} - 862 T^{5} - 387 T^{6} + 233 T^{7} + 53 T^{8} - 27 T^{9} - 2 T^{10} + T^{11}$$
$7$ $$152 - 148 T - 1696 T^{2} + 1668 T^{3} + 2999 T^{4} - 2453 T^{5} - 1342 T^{6} + 599 T^{7} + 124 T^{8} - 45 T^{9} - 3 T^{10} + T^{11}$$
$11$ $$-5372 + 11145 T + 38739 T^{2} - 34305 T^{3} - 24255 T^{4} + 27781 T^{5} - 5422 T^{6} - 1416 T^{7} + 545 T^{8} - 14 T^{9} - 11 T^{10} + T^{11}$$
$13$ $$-5683 - 25917 T + 46260 T^{2} + 16133 T^{3} - 23745 T^{4} - 4924 T^{5} + 4327 T^{6} + 814 T^{7} - 300 T^{8} - 59 T^{9} + 5 T^{10} + T^{11}$$
$17$ $$-14592 - 14272 T + 166480 T^{2} - 34139 T^{3} - 124414 T^{4} + 65783 T^{5} - 1436 T^{6} - 5261 T^{7} + 1033 T^{8} - 15 T^{10} + T^{11}$$
$19$ $$-49408 + 888768 T - 2441296 T^{2} + 1555540 T^{3} - 26768 T^{4} - 163231 T^{5} + 15022 T^{6} + 6635 T^{7} - 564 T^{8} - 129 T^{9} + 6 T^{10} + T^{11}$$
$23$ $$( -1 + T )^{11}$$
$29$ $$( 1 + T )^{11}$$
$31$ $$756944 - 1271500 T - 1024830 T^{2} + 2690635 T^{3} - 1166407 T^{4} - 24880 T^{5} + 103632 T^{6} - 16667 T^{7} - 744 T^{8} + 399 T^{9} - 35 T^{10} + T^{11}$$
$37$ $$759412 - 4285026 T + 1706907 T^{2} + 3978338 T^{3} + 1217937 T^{4} - 149144 T^{5} - 142013 T^{6} - 26369 T^{7} - 1153 T^{8} + 210 T^{9} + 28 T^{10} + T^{11}$$
$41$ $$-5506272 + 1794292 T + 18641848 T^{2} - 19928025 T^{3} + 4926400 T^{4} + 284928 T^{5} - 217167 T^{6} + 11359 T^{7} + 2661 T^{8} - 223 T^{9} - 10 T^{10} + T^{11}$$
$43$ $$7091488 - 21192896 T + 10203752 T^{2} + 6291148 T^{3} - 1694580 T^{4} - 538505 T^{5} + 73524 T^{6} + 17249 T^{7} - 1159 T^{8} - 224 T^{9} + 6 T^{10} + T^{11}$$
$47$ $$-958100256 + 1889856696 T - 518914924 T^{2} - 57882414 T^{3} + 28528161 T^{4} - 172270 T^{5} - 558298 T^{6} + 23031 T^{7} + 4758 T^{8} - 274 T^{9} - 15 T^{10} + T^{11}$$
$53$ $$-80314464 - 121829440 T + 103796304 T^{2} + 10456620 T^{3} - 10758516 T^{4} - 1055028 T^{5} + 300829 T^{6} + 33441 T^{7} - 2664 T^{8} - 331 T^{9} + 7 T^{10} + T^{11}$$
$59$ $$88926336 + 38952576 T - 75760704 T^{2} - 46432000 T^{3} - 4148272 T^{4} + 2040868 T^{5} + 396208 T^{6} - 8635 T^{7} - 5672 T^{8} - 190 T^{9} + 20 T^{10} + T^{11}$$
$61$ $$-40052104 + 11083000 T + 32870768 T^{2} - 2516827 T^{3} - 5611830 T^{4} - 751 T^{5} + 236307 T^{6} + 4939 T^{7} - 3725 T^{8} - 132 T^{9} + 20 T^{10} + T^{11}$$
$67$ $$-26096464 - 99183273 T - 70762696 T^{2} + 27467238 T^{3} + 19078496 T^{4} + 1733774 T^{5} - 502721 T^{6} - 108095 T^{7} - 5170 T^{8} + 335 T^{9} + 39 T^{10} + T^{11}$$
$71$ $$66188592 + 19279308 T - 68607985 T^{2} + 16492659 T^{3} + 10974181 T^{4} - 6443998 T^{5} + 1301874 T^{6} - 105310 T^{7} - 1141 T^{8} + 780 T^{9} - 49 T^{10} + T^{11}$$
$73$ $$-42266597504 - 9422277184 T + 3162939872 T^{2} + 629782120 T^{3} - 70425376 T^{4} - 14132492 T^{5} + 626147 T^{6} + 137562 T^{7} - 2349 T^{8} - 607 T^{9} + 3 T^{10} + T^{11}$$
$79$ $$-360776992 + 220695456 T + 87218808 T^{2} - 60239820 T^{3} - 4012248 T^{4} + 4406159 T^{5} - 110903 T^{6} - 87149 T^{7} + 6197 T^{8} + 320 T^{9} - 41 T^{10} + T^{11}$$
$83$ $$-15116544 - 131639904 T - 262335024 T^{2} + 32542560 T^{3} + 22633992 T^{4} - 1934766 T^{5} - 602883 T^{6} + 45171 T^{7} + 5490 T^{8} - 411 T^{9} - 13 T^{10} + T^{11}$$
$89$ $$18922248 - 104668812 T + 83043198 T^{2} - 9447395 T^{3} - 7289204 T^{4} + 1848545 T^{5} + 60779 T^{6} - 53965 T^{7} + 3873 T^{8} + 222 T^{9} - 34 T^{10} + T^{11}$$
$97$ $$-580896 - 599376 T + 1364024 T^{2} + 1130236 T^{3} - 813224 T^{4} - 447994 T^{5} + 73245 T^{6} + 31853 T^{7} - 2551 T^{8} - 340 T^{9} + 11 T^{10} + T^{11}$$