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Properties

Label 525.4.a
Level $525$
Weight $4$
Character orbit 525.a
Rep. character $\chi_{525}(1,\cdot)$
Character field $\Q$
Dimension $56$
Newform subspaces $24$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(320\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(525))\).

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

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

\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(7\)
\(+\)\(-\)\(-\)$+$\(7\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(9\)
\(-\)\(-\)\(+\)$+$\(9\)
\(-\)\(-\)\(-\)$-$\(5\)
Plus space\(+\)\(31\)
Minus space\(-\)\(25\)

Trace form

\( 56 q + 6 q^{2} + 194 q^{4} + 12 q^{6} + 14 q^{7} + 30 q^{8} + 504 q^{9} + O(q^{10}) \) \( 56 q + 6 q^{2} + 194 q^{4} + 12 q^{6} + 14 q^{7} + 30 q^{8} + 504 q^{9} + 20 q^{11} - 24 q^{12} + 112 q^{13} - 70 q^{14} + 858 q^{16} - 56 q^{17} + 54 q^{18} + 40 q^{19} - 42 q^{21} - 152 q^{22} - 212 q^{23} + 324 q^{24} + 868 q^{26} + 70 q^{28} + 920 q^{29} - 72 q^{31} + 1086 q^{32} + 288 q^{33} + 644 q^{34} + 1746 q^{36} + 152 q^{37} - 712 q^{38} - 336 q^{39} + 176 q^{41} + 84 q^{42} + 240 q^{43} + 716 q^{44} + 2228 q^{46} - 824 q^{47} + 432 q^{48} + 2744 q^{49} + 276 q^{51} + 1636 q^{52} - 480 q^{53} + 108 q^{54} - 378 q^{56} + 192 q^{57} - 268 q^{58} - 952 q^{59} - 3832 q^{61} + 2520 q^{62} + 126 q^{63} + 3790 q^{64} + 1896 q^{66} - 168 q^{67} + 2596 q^{68} + 336 q^{69} - 1676 q^{71} + 270 q^{72} - 2520 q^{73} - 9280 q^{74} + 3672 q^{76} - 1176 q^{77} + 1464 q^{78} + 336 q^{79} + 4536 q^{81} + 1604 q^{82} - 4208 q^{83} - 504 q^{84} - 9340 q^{86} + 792 q^{87} - 4952 q^{88} - 5120 q^{89} - 1148 q^{91} - 1568 q^{92} - 816 q^{93} + 9904 q^{94} + 7764 q^{96} + 4520 q^{97} + 294 q^{98} + 180 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 7
525.4.a.a 525.a 1.a $1$ $30.976$ \(\Q\) None \(-5\) \(3\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}+3q^{3}+17q^{4}-15q^{6}-7q^{7}+\cdots\)
525.4.a.b 525.a 1.a $1$ $30.976$ \(\Q\) None \(-4\) \(3\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+3q^{3}+8q^{4}-12q^{6}+7q^{7}+\cdots\)
525.4.a.c 525.a 1.a $1$ $30.976$ \(\Q\) None \(-3\) \(-3\) \(0\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}-3q^{3}+q^{4}+9q^{6}+7q^{7}+\cdots\)
525.4.a.d 525.a 1.a $1$ $30.976$ \(\Q\) None \(-2\) \(3\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}-4q^{4}-6q^{6}-7q^{7}+\cdots\)
525.4.a.e 525.a 1.a $1$ $30.976$ \(\Q\) None \(0\) \(3\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-8q^{4}-7q^{7}+9q^{9}+42q^{11}+\cdots\)
525.4.a.f 525.a 1.a $1$ $30.976$ \(\Q\) None \(2\) \(-3\) \(0\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}-4q^{4}-6q^{6}+7q^{7}+\cdots\)
525.4.a.g 525.a 1.a $1$ $30.976$ \(\Q\) None \(3\) \(3\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}+9q^{6}-7q^{7}+\cdots\)
525.4.a.h 525.a 1.a $1$ $30.976$ \(\Q\) None \(3\) \(3\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}+9q^{6}-7q^{7}+\cdots\)
525.4.a.i 525.a 1.a $2$ $30.976$ \(\Q(\sqrt{41}) \) None \(-3\) \(-6\) \(0\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}-3q^{3}+(3+3\beta )q^{4}+\cdots\)
525.4.a.j 525.a 1.a $2$ $30.976$ \(\Q(\sqrt{17}) \) None \(-3\) \(6\) \(0\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3q^{3}+(-3+3\beta )q^{4}+\cdots\)
525.4.a.k 525.a 1.a $2$ $30.976$ \(\Q(\sqrt{65}) \) None \(-1\) \(-6\) \(0\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-3q^{3}+(8+\beta )q^{4}+3\beta q^{6}+\cdots\)
525.4.a.l 525.a 1.a $2$ $30.976$ \(\Q(\sqrt{2}) \) None \(2\) \(6\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3q^{3}+(1+2\beta )q^{4}+(3+\cdots)q^{6}+\cdots\)
525.4.a.m 525.a 1.a $2$ $30.976$ \(\Q(\sqrt{17}) \) None \(3\) \(-6\) \(0\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(-3+3\beta )q^{4}+\cdots\)
525.4.a.n 525.a 1.a $2$ $30.976$ \(\Q(\sqrt{57}) \) None \(3\) \(-6\) \(0\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(7+3\beta )q^{4}+(-3+\cdots)q^{6}+\cdots\)
525.4.a.o 525.a 1.a $2$ $30.976$ \(\Q(\sqrt{5}) \) None \(4\) \(-6\) \(0\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{2}-3q^{3}+(1+4\beta )q^{4}+(-6+\cdots)q^{6}+\cdots\)
525.4.a.p 525.a 1.a $2$ $30.976$ \(\Q(\sqrt{17}) \) None \(7\) \(6\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(4-\beta )q^{2}+3q^{3}+(12-7\beta )q^{4}+(12+\cdots)q^{6}+\cdots\)
525.4.a.q 525.a 1.a $3$ $30.976$ 3.3.2292.1 None \(-1\) \(9\) \(0\) \(21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
525.4.a.r 525.a 1.a $3$ $30.976$ 3.3.2292.1 None \(1\) \(-9\) \(0\) \(-21\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
525.4.a.s 525.a 1.a $4$ $30.976$ 4.4.26729725.1 None \(-6\) \(-12\) \(0\) \(28\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}-3q^{3}+(3+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.4.a.t 525.a 1.a $4$ $30.976$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(0\) \(-28\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
525.4.a.u 525.a 1.a $4$ $30.976$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(12\) \(0\) \(28\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
525.4.a.v 525.a 1.a $4$ $30.976$ 4.4.26729725.1 None \(6\) \(12\) \(0\) \(-28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+3q^{3}+(3+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.4.a.w 525.a 1.a $5$ $30.976$ 5.5.78066700.1 None \(-1\) \(15\) \(0\) \(-35\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(5-\beta _{1}-\beta _{3})q^{4}+\cdots\)
525.4.a.x 525.a 1.a $5$ $30.976$ 5.5.78066700.1 None \(1\) \(-15\) \(0\) \(35\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(5-\beta _{1}-\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(525))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(525)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)