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Properties

Label 525.4.d
Level $525$
Weight $4$
Character orbit 525.d
Rep. character $\chi_{525}(274,\cdot)$
Character field $\Q$
Dimension $56$
Newform subspaces $15$
Sturm bound $320$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(320\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(525, [\chi])\).

Total New Old
Modular forms 252 56 196
Cusp forms 228 56 172
Eisenstein series 24 0 24

Trace form

\( 56 q - 216 q^{4} + 24 q^{6} - 504 q^{9} + O(q^{10}) \) \( 56 q - 216 q^{4} + 24 q^{6} - 504 q^{9} + 1016 q^{16} + 96 q^{19} + 84 q^{21} - 648 q^{24} + 184 q^{26} + 560 q^{29} + 96 q^{31} + 160 q^{34} + 1944 q^{36} - 576 q^{39} - 2112 q^{41} - 1580 q^{44} + 2044 q^{46} - 2744 q^{49} + 2304 q^{51} - 216 q^{54} + 84 q^{56} + 112 q^{59} - 4208 q^{61} - 5988 q^{64} - 1296 q^{66} + 2304 q^{69} - 1616 q^{71} + 8708 q^{74} - 8952 q^{76} - 6904 q^{79} + 4536 q^{81} - 1008 q^{84} - 1460 q^{86} + 4288 q^{89} + 1456 q^{91} + 1216 q^{94} + 7968 q^{96} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(525, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
525.4.d.a 525.d 5.b $2$ $30.976$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+5iq^{2}+3iq^{3}-17q^{4}-15q^{6}+\cdots\)
525.4.d.b 525.d 5.b $2$ $30.976$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}+3iq^{3}-8q^{4}-12q^{6}-7iq^{7}+\cdots\)
525.4.d.c 525.d 5.b $2$ $30.976$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{2}-3iq^{3}-q^{4}+9q^{6}-7iq^{7}+\cdots\)
525.4.d.d 525.d 5.b $2$ $30.976$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{2}-3iq^{3}-q^{4}+9q^{6}-7iq^{7}+\cdots\)
525.4.d.e 525.d 5.b $2$ $30.976$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+3iq^{3}+4q^{4}-6q^{6}+7iq^{7}+\cdots\)
525.4.d.f 525.d 5.b $2$ $30.976$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-3iq^{3}+8q^{4}-7iq^{7}-9q^{9}+42q^{11}+\cdots\)
525.4.d.g 525.d 5.b $4$ $30.976$ \(\Q(i, \sqrt{57})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}-\beta _{2})q^{2}-3\beta _{2}q^{3}+(-10+3\beta _{3})q^{4}+\cdots\)
525.4.d.h 525.d 5.b $4$ $30.976$ \(\Q(i, \sqrt{65})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+3\beta _{2}q^{3}+(-9+\beta _{3})q^{4}+\cdots\)
525.4.d.i 525.d 5.b $4$ $30.976$ \(\Q(i, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+4\beta _{2})q^{2}-3\beta _{2}q^{3}+(-5-7\beta _{3})q^{4}+\cdots\)
525.4.d.j 525.d 5.b $4$ $30.976$ \(\Q(i, \sqrt{41})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-6+3\beta _{3})q^{4}+\cdots\)
525.4.d.k 525.d 5.b $4$ $30.976$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2\beta _{1}+\beta _{2})q^{2}-3\beta _{1}q^{3}+(-1+\cdots)q^{4}+\cdots\)
525.4.d.l 525.d 5.b $4$ $30.976$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{8}+\zeta_{8}^{2})q^{2}-3\zeta_{8}q^{3}+(-1-2\zeta_{8}^{3})q^{4}+\cdots\)
525.4.d.m 525.d 5.b $4$ $30.976$ \(\Q(i, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}-\beta _{2})q^{2}-3\beta _{2}q^{3}+3\beta _{3}q^{4}+\cdots\)
525.4.d.n 525.d 5.b $8$ $30.976$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-3\beta _{3}q^{3}+(-4+\beta _{2})q^{4}+\cdots\)
525.4.d.o 525.d 5.b $8$ $30.976$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{4})q^{2}-3\beta _{4}q^{3}+(-5+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(525, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)