Properties

Label 245.2.a
Level 245
Weight 2
Character orbit a
Rep. character \(\chi_{245}(1,\cdot)\)
Character field \(\Q\)
Dimension 13
Newform subspaces 8
Sturm bound 56
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(56\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(245))\).

Total New Old
Modular forms 36 13 23
Cusp forms 21 13 8
Eisenstein series 15 0 15

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

\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(9\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(245))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 7
245.2.a.a \(1\) \(1.956\) \(\Q\) None \(-2\) \(-3\) \(1\) \(0\) \(-\) \(-\) \(q-2q^{2}-3q^{3}+2q^{4}+q^{5}+6q^{6}+\cdots\)
245.2.a.b \(1\) \(1.956\) \(\Q\) None \(-2\) \(3\) \(-1\) \(0\) \(+\) \(-\) \(q-2q^{2}+3q^{3}+2q^{4}-q^{5}-6q^{6}+\cdots\)
245.2.a.c \(1\) \(1.956\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(-\) \(-\) \(q-q^{3}-2q^{4}+q^{5}-2q^{9}-3q^{11}+\cdots\)
245.2.a.d \(2\) \(1.956\) \(\Q(\sqrt{17}) \) None \(-1\) \(1\) \(-2\) \(0\) \(+\) \(-\) \(q-\beta q^{2}+(1-\beta )q^{3}+(2+\beta )q^{4}-q^{5}+\cdots\)
245.2.a.e \(2\) \(1.956\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) \(+\) \(+\) \(q+\beta q^{2}+(-1-\beta )q^{3}-q^{5}+(-2-\beta )q^{6}+\cdots\)
245.2.a.f \(2\) \(1.956\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) \(-\) \(+\) \(q+\beta q^{2}+(1+\beta )q^{3}+q^{5}+(2+\beta )q^{6}+\cdots\)
245.2.a.g \(2\) \(1.956\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(-2\) \(0\) \(+\) \(-\) \(q+(1+\beta )q^{2}+(-1+\beta )q^{3}+(1+2\beta )q^{4}+\cdots\)
245.2.a.h \(2\) \(1.956\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(2\) \(0\) \(-\) \(+\) \(q+(1+\beta )q^{2}+(1-\beta )q^{3}+(1+2\beta )q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(245))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(245)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))
$5$ (\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$7$ 1
$11$ (\( 1 - T + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 + 3 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 + 5 T + 28 T^{2} + 65 T^{3} + 169 T^{4} \))(\( 1 + 6 T + 33 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 33 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 5 T + 36 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 - 2 T + 17 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 2 T + 17 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 30 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 30 T^{2} - 68 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} \))(\( ( 1 + 6 T + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} )^{2} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 12 T + 80 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 - 12 T + 80 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 45 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 45 T^{2} + 46 T^{3} + 529 T^{4} \))
$29$ (\( 1 + T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 35 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 35 T^{2} + 174 T^{3} + 841 T^{4} \))(\( ( 1 + T + 29 T^{2} )^{2} \))(\( ( 1 + T + 29 T^{2} )^{2} \))
$31$ (\( 1 + 6 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 + 12 T + 80 T^{2} + 372 T^{3} + 961 T^{4} \))(\( 1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} )^{2} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 + 2 T + 66 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 - 10 T + 99 T^{2} - 410 T^{3} + 1681 T^{4} \))(\( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 6 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 10 T + 94 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( 1 - 10 T + 109 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 - 10 T + 109 T^{2} - 430 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 - 5 T + 62 T^{2} - 235 T^{3} + 2209 T^{4} \))(\( 1 - 6 T + 85 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T + 85 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( ( 1 - 2 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 2 T + 90 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 88 T^{2} + 2809 T^{4} \))(\( 1 + 88 T^{2} + 2809 T^{4} \))(\( 1 + 8 T + 114 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 114 T^{2} + 424 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( 1 + 4 T + 104 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 - 4 T + 104 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 - 8 T + 62 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 62 T^{2} + 472 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 6 T - 22 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 114 T^{2} + 3721 T^{4} \))(\( 1 + 114 T^{2} + 3721 T^{4} \))(\( 1 - 6 T + 59 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 + 6 T + 59 T^{2} + 366 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 14 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 70 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 132 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 132 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 22 T + 253 T^{2} - 1474 T^{3} + 4489 T^{4} \))(\( 1 - 22 T + 253 T^{2} - 1474 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( 1 + 12 T + 170 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( 1 + 12 T + 170 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 8 T + 94 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 + 74 T^{2} + 5329 T^{4} \))(\( 1 + 74 T^{2} + 5329 T^{4} \))(\( 1 + 4 T + 142 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 4 T + 142 T^{2} - 292 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 + 14 T + 135 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( 1 + 14 T + 135 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( 1 - 24 T + 294 T^{2} - 1896 T^{3} + 6241 T^{4} \))(\( 1 - 24 T + 294 T^{2} - 1896 T^{3} + 6241 T^{4} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 + 2 T + 5 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 5 T^{2} - 166 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 12 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 6 T + 170 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( ( 1 - 8 T + 89 T^{2} )^{2} \))(\( ( 1 + 8 T + 89 T^{2} )^{2} \))(\( 1 - 6 T + 155 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T + 155 T^{2} + 534 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 15 T + 97 T^{2} \))(\( 1 + 15 T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 - 9 T + 108 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( 1 + 18 T + 257 T^{2} + 1746 T^{3} + 9409 T^{4} \))(\( 1 - 18 T + 257 T^{2} - 1746 T^{3} + 9409 T^{4} \))(\( 1 + 12 T + 198 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 - 12 T + 198 T^{2} - 1164 T^{3} + 9409 T^{4} \))
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