Properties

Label 525.2.d
Level $525$
Weight $2$
Character orbit 525.d
Rep. character $\chi_{525}(274,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $5$
Sturm bound $160$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(160\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(525, [\chi])\).

Total New Old
Modular forms 92 16 76
Cusp forms 68 16 52
Eisenstein series 24 0 24

Trace form

\( 16q - 20q^{4} + 8q^{6} - 16q^{9} + O(q^{10}) \) \( 16q - 20q^{4} + 8q^{6} - 16q^{9} + 12q^{16} - 8q^{19} - 4q^{21} - 24q^{24} - 64q^{26} + 40q^{29} + 32q^{31} - 8q^{34} + 20q^{36} - 48q^{41} + 68q^{44} - 4q^{46} - 16q^{49} - 8q^{54} - 12q^{56} + 8q^{59} + 24q^{64} - 24q^{69} - 16q^{71} - 44q^{74} + 40q^{76} + 16q^{81} + 12q^{84} + 20q^{86} + 56q^{89} - 16q^{91} - 16q^{94} + 16q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(525, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
525.2.d.a \(2\) \(4.192\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}+iq^{3}+q^{4}-q^{6}+iq^{7}+3iq^{8}+\cdots\)
525.2.d.b \(2\) \(4.192\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-iq^{3}+q^{4}+q^{6}+iq^{7}+3iq^{8}+\cdots\)
525.2.d.c \(4\) \(4.192\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}-\beta _{1}q^{3}-3q^{4}+\beta _{3}q^{6}-\beta _{1}q^{7}+\cdots\)
525.2.d.d \(4\) \(4.192\) \(\Q(i, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-2+\beta _{3})q^{4}+(1+\cdots)q^{6}+\cdots\)
525.2.d.e \(4\) \(4.192\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{3})q^{2}+\beta _{3}q^{3}+3\beta _{2}q^{4}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(525, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 3 T^{2} + 4 T^{4} \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( ( 1 + T^{2} + 4 T^{4} )^{2} \))(\( 1 - T^{2} + 5 T^{4} - 4 T^{6} + 16 T^{8} \))(\( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \))
$3$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))
$5$ 1
$7$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))
$11$ (\( ( 1 - 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{4} \))(\( ( 1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( ( 1 - 6 T^{2} + 169 T^{4} )^{2} \))(\( 1 - 24 T^{2} + 430 T^{4} - 4056 T^{6} + 28561 T^{8} \))(\( 1 - 24 T^{2} + 302 T^{4} - 4056 T^{6} + 28561 T^{8} \))
$17$ (\( 1 + 2 T^{2} + 289 T^{4} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))(\( 1 - 40 T^{2} + 926 T^{4} - 11560 T^{6} + 83521 T^{8} \))(\( 1 + 24 T^{2} + 542 T^{4} + 6936 T^{6} + 83521 T^{8} \))
$19$ (\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 6 T + 34 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 14 T^{2} + 899 T^{4} + 7406 T^{6} + 279841 T^{8} \))(\( ( 1 - 21 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{4} \))(\( ( 1 - 8 T + 61 T^{2} - 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} - 116 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 42 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( 1 - 38 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 20 T^{2} + 1558 T^{4} + 27380 T^{6} + 1874161 T^{8} \))(\( 1 - 26 T^{2} + 1035 T^{4} - 35594 T^{6} + 1874161 T^{8} \))(\( 1 - 106 T^{2} + 5467 T^{4} - 145114 T^{6} + 1874161 T^{8} \))
$41$ (\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 8 T + 41 T^{2} )^{4} \))
$43$ (\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 6 T^{2} + 1849 T^{4} )^{2} \))(\( 1 - 74 T^{2} + 3195 T^{4} - 136826 T^{6} + 3418801 T^{8} \))(\( 1 - 90 T^{2} + 5003 T^{4} - 166410 T^{6} + 3418801 T^{8} \))
$47$ (\( ( 1 - 47 T^{2} )^{2} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 + 4 T^{2} - 698 T^{4} + 8836 T^{6} + 4879681 T^{8} \))(\( 1 - 64 T^{2} + 2894 T^{4} - 141376 T^{6} + 4879681 T^{8} \))(\( 1 - 128 T^{2} + 8014 T^{4} - 282752 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 - 44 T^{2} + 982 T^{4} - 123596 T^{6} + 7890481 T^{8} \))(\( 1 - 76 T^{2} + 3734 T^{4} - 213484 T^{6} + 7890481 T^{8} \))(\( 1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 14 T + 154 T^{2} - 826 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 6 T + 122 T^{2} - 354 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{4} \))(\( ( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( ( 1 - 118 T^{2} + 4489 T^{4} )^{2} \))(\( 1 + 46 T^{2} + 2019 T^{4} + 206494 T^{6} + 20151121 T^{8} \))(\( 1 - 146 T^{2} + 11427 T^{4} - 655394 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 3 T + 71 T^{2} )^{4} \))(\( ( 1 + 10 T + 147 T^{2} + 710 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 124 T^{2} + 9382 T^{4} - 660796 T^{6} + 28398241 T^{8} \))(\( 1 - 248 T^{2} + 25566 T^{4} - 1321592 T^{6} + 28398241 T^{8} \))(\( 1 - 280 T^{2} + 30238 T^{4} - 1492120 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 - 16 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 8 T + 57 T^{2} - 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T + 169 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 44 T^{2} - 6218 T^{4} - 303116 T^{6} + 47458321 T^{8} \))(\( 1 - 196 T^{2} + 20054 T^{4} - 1350244 T^{6} + 47458321 T^{8} \))(\( 1 - 164 T^{2} + 15382 T^{4} - 1129796 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 - 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{4} \))(\( ( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 2 T + 134 T^{2} + 178 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} ) \))(\( ( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} ) \))(\( 1 - 316 T^{2} + 42502 T^{4} - 2973244 T^{6} + 88529281 T^{8} \))(\( 1 - 156 T^{2} + 11590 T^{4} - 1467804 T^{6} + 88529281 T^{8} \))(\( 1 - 220 T^{2} + 25798 T^{4} - 2069980 T^{6} + 88529281 T^{8} \))
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