Properties

Label 201.2.a
Level 201
Weight 2
Character orbit a
Rep. character \(\chi_{201}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 5
Sturm bound 45
Trace bound 2

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Defining parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 201.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(45\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(201))\).

Total New Old
Modular forms 24 11 13
Cusp forms 21 11 10
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(67\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(3\)
Minus space\(-\)\(8\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(201))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 67
201.2.a.a \(1\) \(1.605\) \(\Q\) None \(-2\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{9}-6q^{11}+\cdots\)
201.2.a.b \(1\) \(1.605\) \(\Q\) None \(-1\) \(1\) \(-1\) \(-5\) \(-\) \(-\) \(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}-5q^{7}+\cdots\)
201.2.a.c \(1\) \(1.605\) \(\Q\) None \(1\) \(-1\) \(-3\) \(-3\) \(+\) \(+\) \(q+q^{2}-q^{3}-q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
201.2.a.d \(3\) \(1.605\) 3.3.148.1 None \(3\) \(-3\) \(1\) \(1\) \(+\) \(-\) \(q+(1+\beta _{2})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
201.2.a.e \(5\) \(1.605\) 5.5.1025428.1 None \(0\) \(5\) \(-3\) \(7\) \(-\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(201))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(201)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 3 T + 5 T^{2} - 7 T^{3} + 10 T^{4} - 12 T^{5} + 8 T^{6} \))(\( 1 + 2 T^{2} + 5 T^{4} + 2 T^{5} + 10 T^{6} + 16 T^{8} + 32 T^{10} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{5} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - T + 12 T^{2} - 9 T^{3} + 60 T^{4} - 25 T^{5} + 125 T^{6} \))(\( 1 + 3 T + 16 T^{2} + 41 T^{3} + 125 T^{4} + 276 T^{5} + 625 T^{6} + 1025 T^{7} + 2000 T^{8} + 1875 T^{9} + 3125 T^{10} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - T + 16 T^{2} - 13 T^{3} + 112 T^{4} - 49 T^{5} + 343 T^{6} \))(\( 1 - 7 T + 38 T^{2} - 133 T^{3} + 425 T^{4} - 1112 T^{5} + 2975 T^{6} - 6517 T^{7} + 13034 T^{8} - 16807 T^{9} + 16807 T^{10} \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 10 T + 57 T^{2} - 216 T^{3} + 627 T^{4} - 1210 T^{5} + 1331 T^{6} \))(\( 1 + 35 T^{2} - 4 T^{3} + 606 T^{4} - 120 T^{5} + 6666 T^{6} - 484 T^{7} + 46585 T^{8} + 161051 T^{10} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 8 T + 51 T^{2} + 212 T^{3} + 663 T^{4} + 1352 T^{5} + 2197 T^{6} \))(\( 1 - 10 T + 85 T^{2} - 484 T^{3} + 2382 T^{4} - 9236 T^{5} + 30966 T^{6} - 81796 T^{7} + 186745 T^{8} - 285610 T^{9} + 371293 T^{10} \))
$17$ (\( 1 + 7 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 23 T^{2} + 52 T^{3} + 391 T^{4} + 4913 T^{6} \))(\( 1 + 5 T + 39 T^{2} + 244 T^{3} + 1180 T^{4} + 4838 T^{5} + 20060 T^{6} + 70516 T^{7} + 191607 T^{8} + 417605 T^{9} + 1419857 T^{10} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 13 T^{2} + 56 T^{3} + 247 T^{4} + 722 T^{5} + 6859 T^{6} \))(\( 1 - 5 T + 49 T^{2} - 132 T^{3} + 808 T^{4} - 1422 T^{5} + 15352 T^{6} - 47652 T^{7} + 336091 T^{8} - 651605 T^{9} + 2476099 T^{10} \))
$23$ (\( 1 + T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 7 T + 23 T^{2} \))(\( 1 - 3 T + 38 T^{2} - 43 T^{3} + 874 T^{4} - 1587 T^{5} + 12167 T^{6} \))(\( 1 + 2 T + 101 T^{2} + 192 T^{3} + 4335 T^{4} + 6712 T^{5} + 99705 T^{6} + 101568 T^{7} + 1228867 T^{8} + 559682 T^{9} + 6436343 T^{10} \))
$29$ (\( 1 - T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 4 T + 39 T^{2} - 168 T^{3} + 1131 T^{4} - 3364 T^{5} + 24389 T^{6} \))(\( 1 - 3 T + 47 T^{2} - 124 T^{3} + 1932 T^{4} - 4194 T^{5} + 56028 T^{6} - 104284 T^{7} + 1146283 T^{8} - 2121843 T^{9} + 20511149 T^{10} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 - 11 T + 80 T^{2} - 387 T^{3} + 2480 T^{4} - 10571 T^{5} + 29791 T^{6} \))(\( 1 - 9 T + 154 T^{2} - 943 T^{3} + 9185 T^{4} - 41200 T^{5} + 284735 T^{6} - 906223 T^{7} + 4587814 T^{8} - 8311689 T^{9} + 28629151 T^{10} \))
$37$ (\( 1 - 3 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 + 9 T + 98 T^{2} + 497 T^{3} + 3626 T^{4} + 12321 T^{5} + 50653 T^{6} \))(\( 1 - 8 T + 117 T^{2} - 746 T^{3} + 6797 T^{4} - 34118 T^{5} + 251489 T^{6} - 1021274 T^{7} + 5926401 T^{8} - 14993288 T^{9} + 69343957 T^{10} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - T + 62 T^{2} - 179 T^{3} + 2542 T^{4} - 1681 T^{5} + 68921 T^{6} \))(\( 1 + 7 T + 190 T^{2} + 1019 T^{3} + 14951 T^{4} + 60056 T^{5} + 612991 T^{6} + 1712939 T^{7} + 13094990 T^{8} + 19780327 T^{9} + 115856201 T^{10} \))
$43$ (\( 1 + 6 T + 43 T^{2} \))(\( 1 - 7 T + 43 T^{2} \))(\( 1 - 9 T + 43 T^{2} \))(\( ( 1 + T + 43 T^{2} )^{3} \))(\( 1 - T + 124 T^{2} + 33 T^{3} + 8725 T^{4} + 480 T^{5} + 375175 T^{6} + 61017 T^{7} + 9858868 T^{8} - 3418801 T^{9} + 147008443 T^{10} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 18 T + 201 T^{2} - 1640 T^{3} + 9447 T^{4} - 39762 T^{5} + 103823 T^{6} \))(\( 1 + 5 T + 189 T^{2} + 692 T^{3} + 15424 T^{4} + 42974 T^{5} + 724928 T^{6} + 1528628 T^{7} + 19622547 T^{8} + 24398405 T^{9} + 229345007 T^{10} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( 1 + 5 T + 53 T^{2} \))(\( 1 - T + 53 T^{2} \))(\( 1 - 7 T + 82 T^{2} - 873 T^{3} + 4346 T^{4} - 19663 T^{5} + 148877 T^{6} \))(\( 1 + 15 T + 168 T^{2} + 1247 T^{3} + 8491 T^{4} + 46324 T^{5} + 450023 T^{6} + 3502823 T^{7} + 25011336 T^{8} + 118357215 T^{9} + 418195493 T^{10} \))
$59$ (\( 1 - 3 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 - 15 T + 152 T^{2} - 1145 T^{3} + 8968 T^{4} - 52215 T^{5} + 205379 T^{6} \))(\( 1 + 6 T + 191 T^{2} + 1132 T^{3} + 18867 T^{4} + 91308 T^{5} + 1113153 T^{6} + 3940492 T^{7} + 39227389 T^{8} + 72704166 T^{9} + 714924299 T^{10} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 2 T + 107 T^{2} + 360 T^{3} + 6527 T^{4} + 7442 T^{5} + 226981 T^{6} \))(\( 1 - 6 T + 209 T^{2} - 420 T^{3} + 16170 T^{4} - 2732 T^{5} + 986370 T^{6} - 1562820 T^{7} + 47439029 T^{8} - 83075046 T^{9} + 844596301 T^{10} \))
$67$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{5} \))
$71$ (\( 1 + 16 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 18 T + 281 T^{2} - 2456 T^{3} + 19951 T^{4} - 90738 T^{5} + 357911 T^{6} \))(\( 1 - 22 T + 375 T^{2} - 4100 T^{3} + 42078 T^{4} - 349772 T^{5} + 2987538 T^{6} - 20668100 T^{7} + 134216625 T^{8} - 559056982 T^{9} + 1804229351 T^{10} \))
$73$ (\( 1 + 7 T + 73 T^{2} \))(\( 1 + 13 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 + 19 T + 302 T^{2} + 2871 T^{3} + 22046 T^{4} + 101251 T^{5} + 389017 T^{6} \))(\( 1 + 81 T^{2} + 534 T^{3} + 11057 T^{4} - 874 T^{5} + 807161 T^{6} + 2845686 T^{7} + 31510377 T^{8} + 2073071593 T^{10} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 28 T + 485 T^{2} - 5112 T^{3} + 38315 T^{4} - 174748 T^{5} + 493039 T^{6} \))(\( 1 - 28 T + 371 T^{2} - 2912 T^{3} + 17042 T^{4} - 111624 T^{5} + 1346318 T^{6} - 18173792 T^{7} + 182917469 T^{8} - 1090602268 T^{9} + 3077056399 T^{10} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - T + 83 T^{2} \))(\( 1 - 5 T + 83 T^{2} \))(\( 1 + 7 T + 228 T^{2} + 1137 T^{3} + 18924 T^{4} + 48223 T^{5} + 571787 T^{6} \))(\( 1 - 9 T + 186 T^{2} - 169 T^{3} + 5585 T^{4} + 99852 T^{5} + 463555 T^{6} - 1164241 T^{7} + 106352382 T^{8} - 427124889 T^{9} + 3939040643 T^{10} \))
$89$ (\( 1 + 15 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 6 T + 119 T^{2} + 1184 T^{3} + 10591 T^{4} + 47526 T^{5} + 704969 T^{6} \))(\( 1 + 11 T + 365 T^{2} + 3632 T^{3} + 59750 T^{4} + 469970 T^{5} + 5317750 T^{6} + 28769072 T^{7} + 257313685 T^{8} + 690164651 T^{9} + 5584059449 T^{10} \))
$97$ (\( 1 - 4 T + 97 T^{2} \))(\( 1 + 12 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 + 8 T + 51 T^{2} + 620 T^{3} + 4947 T^{4} + 75272 T^{5} + 912673 T^{6} \))(\( 1 + 14 T + 309 T^{2} + 1468 T^{3} + 20994 T^{4} - 15492 T^{5} + 2036418 T^{6} + 13812412 T^{7} + 282015957 T^{8} + 1239409934 T^{9} + 8587340257 T^{10} \))
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