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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
525.4.a.a \(1\) \(30.976\) \(\Q\) None \(-5\) \(3\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q-5q^{2}+3q^{3}+17q^{4}-15q^{6}-7q^{7}+\cdots\)
525.4.a.b \(1\) \(30.976\) \(\Q\) None \(-4\) \(3\) \(0\) \(7\) \(-\) \(+\) \(-\) \(q-4q^{2}+3q^{3}+8q^{4}-12q^{6}+7q^{7}+\cdots\)
525.4.a.c \(1\) \(30.976\) \(\Q\) None \(-3\) \(-3\) \(0\) \(7\) \(+\) \(-\) \(-\) \(q-3q^{2}-3q^{3}+q^{4}+9q^{6}+7q^{7}+\cdots\)
525.4.a.d \(1\) \(30.976\) \(\Q\) None \(-2\) \(3\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q-2q^{2}+3q^{3}-4q^{4}-6q^{6}-7q^{7}+\cdots\)
525.4.a.e \(1\) \(30.976\) \(\Q\) None \(0\) \(3\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q+3q^{3}-8q^{4}-7q^{7}+9q^{9}+42q^{11}+\cdots\)
525.4.a.f \(1\) \(30.976\) \(\Q\) None \(2\) \(-3\) \(0\) \(7\) \(+\) \(-\) \(-\) \(q+2q^{2}-3q^{3}-4q^{4}-6q^{6}+7q^{7}+\cdots\)
525.4.a.g \(1\) \(30.976\) \(\Q\) None \(3\) \(3\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q+3q^{2}+3q^{3}+q^{4}+9q^{6}-7q^{7}+\cdots\)
525.4.a.h \(1\) \(30.976\) \(\Q\) None \(3\) \(3\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q+3q^{2}+3q^{3}+q^{4}+9q^{6}-7q^{7}+\cdots\)
525.4.a.i \(2\) \(30.976\) \(\Q(\sqrt{41}) \) None \(-3\) \(-6\) \(0\) \(-14\) \(+\) \(+\) \(+\) \(q+(-1-\beta )q^{2}-3q^{3}+(3+3\beta )q^{4}+\cdots\)
525.4.a.j \(2\) \(30.976\) \(\Q(\sqrt{17}) \) None \(-3\) \(6\) \(0\) \(14\) \(-\) \(-\) \(-\) \(q+(-1-\beta )q^{2}+3q^{3}+(-3+3\beta )q^{4}+\cdots\)
525.4.a.k \(2\) \(30.976\) \(\Q(\sqrt{65}) \) None \(-1\) \(-6\) \(0\) \(14\) \(+\) \(+\) \(-\) \(q-\beta q^{2}-3q^{3}+(8+\beta )q^{4}+3\beta q^{6}+\cdots\)
525.4.a.l \(2\) \(30.976\) \(\Q(\sqrt{2}) \) None \(2\) \(6\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+(1+\beta )q^{2}+3q^{3}+(1+2\beta )q^{4}+(3+\cdots)q^{6}+\cdots\)
525.4.a.m \(2\) \(30.976\) \(\Q(\sqrt{17}) \) None \(3\) \(-6\) \(0\) \(-14\) \(+\) \(+\) \(+\) \(q+(1+\beta )q^{2}-3q^{3}+(-3+3\beta )q^{4}+\cdots\)
525.4.a.n \(2\) \(30.976\) \(\Q(\sqrt{57}) \) None \(3\) \(-6\) \(0\) \(-14\) \(+\) \(+\) \(+\) \(q+(1+\beta )q^{2}-3q^{3}+(7+3\beta )q^{4}+(-3+\cdots)q^{6}+\cdots\)
525.4.a.o \(2\) \(30.976\) \(\Q(\sqrt{5}) \) None \(4\) \(-6\) \(0\) \(14\) \(+\) \(+\) \(-\) \(q+(2+\beta )q^{2}-3q^{3}+(1+4\beta )q^{4}+(-6+\cdots)q^{6}+\cdots\)
525.4.a.p \(2\) \(30.976\) \(\Q(\sqrt{17}) \) None \(7\) \(6\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+(4-\beta )q^{2}+3q^{3}+(12-7\beta )q^{4}+(12+\cdots)q^{6}+\cdots\)
525.4.a.q \(3\) \(30.976\) 3.3.2292.1 None \(-1\) \(9\) \(0\) \(21\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}+3q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
525.4.a.r \(3\) \(30.976\) 3.3.2292.1 None \(1\) \(-9\) \(0\) \(-21\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}-3q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
525.4.a.s \(4\) \(30.976\) 4.4.26729725.1 None \(-6\) \(-12\) \(0\) \(28\) \(+\) \(+\) \(-\) \(q+(-1-\beta _{1})q^{2}-3q^{3}+(3+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.4.a.t \(4\) \(30.976\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(0\) \(-28\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
525.4.a.u \(4\) \(30.976\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(12\) \(0\) \(28\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
525.4.a.v \(4\) \(30.976\) 4.4.26729725.1 None \(6\) \(12\) \(0\) \(-28\) \(-\) \(-\) \(+\) \(q+(1+\beta _{1})q^{2}+3q^{3}+(3+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
525.4.a.w \(5\) \(30.976\) 5.5.78066700.1 None \(-1\) \(15\) \(0\) \(-35\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{2}+3q^{3}+(5-\beta _{1}-\beta _{3})q^{4}+\cdots\)
525.4.a.x \(5\) \(30.976\) 5.5.78066700.1 None \(1\) \(-15\) \(0\) \(35\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-3q^{3}+(5-\beta _{1}-\beta _{3})q^{4}+\cdots\)

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

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