Properties

Label 402.2.a
Level 402
Weight 2
Character orbit a
Rep. character \(\chi_{402}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 7
Sturm bound 136
Trace bound 5

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Defining parameters

Level: \( N \) = \( 402 = 2 \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 402.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(136\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(402))\).

Total New Old
Modular forms 72 11 61
Cusp forms 65 11 54
Eisenstein series 7 0 7

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

\(2\)\(3\)\(67\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(9\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(402))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 67
402.2.a.a \(1\) \(3.210\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-3\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-3q^{7}+\cdots\)
402.2.a.b \(1\) \(3.210\) \(\Q\) None \(-1\) \(1\) \(-3\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}-q^{7}+\cdots\)
402.2.a.c \(1\) \(3.210\) \(\Q\) None \(-1\) \(1\) \(2\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-q^{8}+\cdots\)
402.2.a.d \(1\) \(3.210\) \(\Q\) None \(1\) \(-1\) \(2\) \(2\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+2q^{7}+\cdots\)
402.2.a.e \(2\) \(3.210\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(6\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+2\beta q^{5}+q^{6}+(3+\cdots)q^{7}+\cdots\)
402.2.a.f \(2\) \(3.210\) \(\Q(\sqrt{41}) \) None \(2\) \(-2\) \(1\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+\beta q^{5}-q^{6}-\beta q^{7}+\cdots\)
402.2.a.g \(3\) \(3.210\) 3.3.316.1 None \(3\) \(3\) \(3\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+(1-\beta _{2})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(402))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(402)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))
$5$ (\( 1 - T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))(\( 1 - T - 5 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 11 T^{2} - 26 T^{3} + 55 T^{4} - 75 T^{5} + 125 T^{6} \))
$7$ (\( 1 + 3 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 6 T + 20 T^{2} - 42 T^{3} + 49 T^{4} \))(\( 1 + T + 4 T^{2} + 7 T^{3} + 49 T^{4} \))(\( 1 - T + 17 T^{2} - 12 T^{3} + 119 T^{4} - 49 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))(\( 1 + 5 T^{2} + 16 T^{3} + 55 T^{4} + 1331 T^{6} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 2 T + 24 T^{2} + 26 T^{3} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( 1 + 6 T + 17 T^{2} + 20 T^{3} + 221 T^{4} + 1014 T^{5} + 2197 T^{6} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 22 T^{2} + 289 T^{4} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( 1 + 2 T + 23 T^{2} + 76 T^{3} + 391 T^{4} + 578 T^{5} + 4913 T^{6} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( 1 + 2 T - 2 T^{2} + 38 T^{3} + 361 T^{4} \))(\( ( 1 + 2 T + 19 T^{2} )^{3} \))
$23$ (\( 1 + 3 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 14 T + 92 T^{2} - 322 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 56 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 - 3 T + 33 T^{2} - 84 T^{3} + 759 T^{4} - 1587 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 2 T - 16 T^{2} + 58 T^{3} + 841 T^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( 1 - 6 T + 65 T^{2} - 212 T^{3} + 1885 T^{4} - 5046 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 9 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 10 T + 60 T^{2} - 310 T^{3} + 961 T^{4} \))(\( 1 + 7 T + 64 T^{2} + 217 T^{3} + 961 T^{4} \))(\( 1 + 13 T + 145 T^{2} + 868 T^{3} + 4495 T^{4} + 12493 T^{5} + 29791 T^{6} \))
$37$ (\( 1 + 3 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( 1 - 15 T + 120 T^{2} - 555 T^{3} + 1369 T^{4} \))(\( 1 + 7 T - 25 T^{2} - 510 T^{3} - 925 T^{4} + 9583 T^{5} + 50653 T^{6} \))
$41$ (\( 1 - 3 T + 41 T^{2} \))(\( 1 - 3 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 4 T + 74 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 + T + 72 T^{2} + 41 T^{3} + 1681 T^{4} \))(\( 1 - 5 T + 39 T^{2} - 66 T^{3} + 1599 T^{4} - 8405 T^{5} + 68921 T^{6} \))
$43$ (\( 1 + 7 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 + 11 T + 106 T^{2} + 473 T^{3} + 1849 T^{4} \))(\( 1 + T + 25 T^{2} - 266 T^{3} + 1075 T^{4} + 1849 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 2 T + 20 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 + 10 T + 78 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 + 10 T + 95 T^{2} + 812 T^{3} + 4465 T^{4} + 22090 T^{5} + 103823 T^{6} \))
$53$ (\( 1 + 3 T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 8 T + 110 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 3 T + 16 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 15 T + 227 T^{2} - 1682 T^{3} + 12031 T^{4} - 42135 T^{5} + 148877 T^{6} \))
$59$ (\( 1 - 3 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 12 T + 142 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 7 T + 38 T^{2} + 413 T^{3} + 3481 T^{4} \))(\( 1 - 3 T + 117 T^{2} - 346 T^{3} + 6903 T^{4} - 10443 T^{5} + 205379 T^{6} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T + 128 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 14 T + 130 T^{2} + 854 T^{3} + 3721 T^{4} \))(\( 1 + 125 T^{2} - 124 T^{3} + 7625 T^{4} + 226981 T^{6} \))
$67$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))
$71$ (\( 1 - 4 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 - 6 T + 124 T^{2} - 426 T^{3} + 5041 T^{4} \))(\( 1 + 14 T + 150 T^{2} + 994 T^{3} + 5041 T^{4} \))(\( 1 - 6 T + 167 T^{2} - 620 T^{3} + 11857 T^{4} - 30246 T^{5} + 357911 T^{6} \))
$73$ (\( 1 - 11 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( 1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4} \))(\( 1 + T + 163 T^{2} - 26 T^{3} + 11899 T^{4} + 5329 T^{5} + 389017 T^{6} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 6 T + 140 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 6 T + 126 T^{2} + 474 T^{3} + 6241 T^{4} \))(\( 1 + 26 T + 367 T^{2} + 3644 T^{3} + 28993 T^{4} + 162266 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 9 T + 83 T^{2} \))(\( 1 - 15 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 16 T + 182 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 130 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 + 3 T + 65 T^{2} - 46 T^{3} + 5395 T^{4} + 20667 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - 16 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( 1 + 18 T + 218 T^{2} + 1602 T^{3} + 7921 T^{4} \))(\( 1 - 16 T + 287 T^{2} - 2480 T^{3} + 25543 T^{4} - 126736 T^{5} + 704969 T^{6} \))
$97$ (\( 1 + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 86 T^{2} + 9409 T^{4} \))(\( 1 + 14 T + 202 T^{2} + 1358 T^{3} + 9409 T^{4} \))(\( 1 - 4 T + 199 T^{2} - 280 T^{3} + 19303 T^{4} - 37636 T^{5} + 912673 T^{6} \))
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