Properties

Label 354.2.a
Level 354
Weight 2
Character orbit a
Rep. character \(\chi_{354}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 8
Sturm bound 120
Trace bound 5

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Defining parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 354.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(120\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(354))\).

Total New Old
Modular forms 64 11 53
Cusp forms 57 11 46
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.

\(2\)\(3\)\(59\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(9\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(354))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 59
354.2.a.a \(1\) \(2.827\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
354.2.a.b \(1\) \(2.827\) \(\Q\) None \(-1\) \(-1\) \(2\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-q^{8}+\cdots\)
354.2.a.c \(1\) \(2.827\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
354.2.a.d \(1\) \(2.827\) \(\Q\) None \(1\) \(-1\) \(-4\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}-q^{7}+\cdots\)
354.2.a.e \(1\) \(2.827\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
354.2.a.f \(1\) \(2.827\) \(\Q\) None \(1\) \(-1\) \(4\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+4q^{5}-q^{6}+q^{8}+\cdots\)
354.2.a.g \(2\) \(2.827\) \(\Q(\sqrt{11}) \) None \(-2\) \(2\) \(2\) \(8\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+(1+\beta )q^{5}-q^{6}+\cdots\)
354.2.a.h \(3\) \(2.827\) 3.3.316.1 None \(3\) \(3\) \(2\) \(-1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(354))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(354)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 2 T - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 9 T^{2} - 12 T^{3} + 45 T^{4} - 50 T^{5} + 125 T^{6} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( 1 + T + 5 T^{2} + 30 T^{3} + 35 T^{4} + 49 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 5 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( 1 - T + T^{2} + 54 T^{3} + 11 T^{4} - 121 T^{5} + 1331 T^{6} \))
$13$ (\( 1 - T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 2 T + 16 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 - T + 35 T^{2} - 24 T^{3} + 455 T^{4} - 169 T^{5} + 2197 T^{6} \))
$17$ (\( 1 - T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 7 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 10 T^{2} + 289 T^{4} \))(\( 1 + 3 T + 47 T^{2} + 98 T^{3} + 799 T^{4} + 867 T^{5} + 4913 T^{6} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( 1 + 29 T^{2} - 16 T^{3} + 551 T^{4} + 6859 T^{6} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 6 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 10 T + 85 T^{2} + 444 T^{3} + 1955 T^{4} + 5290 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 10 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 10 T + 72 T^{2} + 290 T^{3} + 841 T^{4} \))(\( 1 + 8 T + 93 T^{2} + 420 T^{3} + 2697 T^{4} + 6728 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 2 T + 52 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 2 T + 79 T^{2} + 92 T^{3} + 2449 T^{4} + 1922 T^{5} + 29791 T^{6} \))
$37$ (\( 1 - 9 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 2 T + 64 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 9 T + 35 T^{2} - 40 T^{3} + 1295 T^{4} - 12321 T^{5} + 50653 T^{6} \))
$41$ (\( 1 + 5 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 3 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 + T + 31 T^{2} + 246 T^{3} + 1271 T^{4} + 1681 T^{5} + 68921 T^{6} \))
$43$ (\( 1 - 3 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( 1 + 7 T + 113 T^{2} + 474 T^{3} + 4859 T^{4} + 12943 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 4 T + 54 T^{2} - 188 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 101 T^{2} + 784 T^{3} + 4747 T^{4} + 17672 T^{5} + 103823 T^{6} \))
$53$ (\( 1 - 4 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 8 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 14 T + 144 T^{2} + 742 T^{3} + 2809 T^{4} \))(\( 1 + 18 T + 209 T^{2} + 1652 T^{3} + 11077 T^{4} + 50562 T^{5} + 148877 T^{6} \))
$59$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 2 T + 112 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 29 T^{2} - 724 T^{3} + 1769 T^{4} + 226981 T^{6} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 4 T + 94 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 161 T^{2} - 1104 T^{3} + 10787 T^{4} - 35912 T^{5} + 300763 T^{6} \))
$71$ (\( 1 - T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 - 6 T + 140 T^{2} - 426 T^{3} + 5041 T^{4} \))(\( 1 + 9 T + 201 T^{2} + 1172 T^{3} + 14271 T^{4} + 45369 T^{5} + 357911 T^{6} \))
$73$ (\( 1 + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 16 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 4 T - 26 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 8 T - T^{2} + 768 T^{3} - 73 T^{4} - 42632 T^{5} + 389017 T^{6} \))
$79$ (\( 1 + 3 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 5 T + 79 T^{2} \))(\( 1 - 5 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( 1 + 19 T + 301 T^{2} + 2746 T^{3} + 23779 T^{4} + 118579 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 7 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 - T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( 1 + 7 T + 209 T^{2} + 890 T^{3} + 17347 T^{4} + 48223 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - 16 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 - 4 T + 138 T^{2} - 356 T^{3} + 7921 T^{4} \))(\( 1 + 3 T^{2} + 736 T^{3} + 267 T^{4} + 704969 T^{6} \))
$97$ (\( 1 + 12 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 4 T + 97 T^{2} \))(\( 1 + 4 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 24 T + 294 T^{2} + 2328 T^{3} + 9409 T^{4} \))(\( 1 - 28 T + 535 T^{2} - 6088 T^{3} + 51895 T^{4} - 263452 T^{5} + 912673 T^{6} \))
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