Properties

Label 225.4.a
Level $225$
Weight $4$
Character orbit 225.a
Rep. character $\chi_{225}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $15$
Sturm bound $120$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(120\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(225))\).

Total New Old
Modular forms 102 25 77
Cusp forms 78 22 56
Eisenstein series 24 3 21

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

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(12\)
Minus space\(-\)\(10\)

Trace form

\( 22q + 82q^{4} + 38q^{7} - 36q^{8} + O(q^{10}) \) \( 22q + 82q^{4} + 38q^{7} - 36q^{8} + 54q^{11} + 52q^{13} - 48q^{14} + 154q^{16} + 66q^{17} + 14q^{19} - 352q^{22} - 366q^{23} + 924q^{26} + 944q^{28} + 30q^{29} - 292q^{31} + 372q^{32} - 42q^{34} - 192q^{37} - 792q^{38} - 36q^{41} - 386q^{43} + 582q^{44} - 684q^{46} - 282q^{47} + 42q^{49} + 504q^{52} + 114q^{53} + 540q^{56} + 304q^{58} + 60q^{59} + 1184q^{61} + 744q^{62} + 2506q^{64} + 1394q^{67} + 360q^{68} - 2196q^{71} + 976q^{73} - 3588q^{74} - 622q^{76} - 1536q^{77} - 3184q^{79} - 5024q^{82} + 1650q^{83} - 1116q^{86} - 4224q^{88} + 180q^{89} - 164q^{91} + 432q^{92} - 4392q^{94} + 3624q^{97} + 1632q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
225.4.a.a \(1\) \(13.275\) \(\Q\) None \(-5\) \(0\) \(0\) \(30\) \(+\) \(+\) \(q-5q^{2}+17q^{4}+30q^{7}-45q^{8}+50q^{11}+\cdots\)
225.4.a.b \(1\) \(13.275\) \(\Q\) None \(-4\) \(0\) \(0\) \(-6\) \(-\) \(+\) \(q-4q^{2}+8q^{4}-6q^{7}-2^{5}q^{11}+38q^{13}+\cdots\)
225.4.a.c \(1\) \(13.275\) \(\Q\) None \(-1\) \(0\) \(0\) \(6\) \(-\) \(+\) \(q-q^{2}-7q^{4}+6q^{7}+15q^{8}+43q^{11}+\cdots\)
225.4.a.d \(1\) \(13.275\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-20\) \(+\) \(+\) \(q-8q^{4}-20q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
225.4.a.e \(1\) \(13.275\) \(\Q\) None \(1\) \(0\) \(0\) \(-6\) \(-\) \(-\) \(q+q^{2}-7q^{4}-6q^{7}-15q^{8}+43q^{11}+\cdots\)
225.4.a.f \(1\) \(13.275\) \(\Q\) None \(1\) \(0\) \(0\) \(24\) \(-\) \(+\) \(q+q^{2}-7q^{4}+24q^{7}-15q^{8}-52q^{11}+\cdots\)
225.4.a.g \(1\) \(13.275\) \(\Q\) None \(3\) \(0\) \(0\) \(-20\) \(-\) \(+\) \(q+3q^{2}+q^{4}-20q^{7}-21q^{8}+24q^{11}+\cdots\)
225.4.a.h \(1\) \(13.275\) \(\Q\) None \(5\) \(0\) \(0\) \(30\) \(+\) \(+\) \(q+5q^{2}+17q^{4}+30q^{7}+45q^{8}-50q^{11}+\cdots\)
225.4.a.i \(2\) \(13.275\) \(\Q(\sqrt{41}) \) None \(-3\) \(0\) \(0\) \(6\) \(-\) \(-\) \(q+(-1-\beta )q^{2}+(3+3\beta )q^{4}+(6-6\beta )q^{7}+\cdots\)
225.4.a.j \(2\) \(13.275\) \(\Q(\sqrt{19}) \) None \(-2\) \(0\) \(0\) \(-26\) \(-\) \(+\) \(q+(-1+\beta )q^{2}+(12-2\beta )q^{4}+(-13+\cdots)q^{7}+\cdots\)
225.4.a.k \(2\) \(13.275\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-\beta q^{2}-3q^{4}+11\beta q^{8}-31q^{16}+\cdots\)
225.4.a.l \(2\) \(13.275\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(-30\) \(+\) \(-\) \(q+\beta q^{2}+2q^{4}-15q^{7}-6\beta q^{8}-20\beta q^{11}+\cdots\)
225.4.a.m \(2\) \(13.275\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(30\) \(+\) \(+\) \(q+\beta q^{2}+2q^{4}+15q^{7}-6\beta q^{8}+20\beta q^{11}+\cdots\)
225.4.a.n \(2\) \(13.275\) \(\Q(\sqrt{19}) \) None \(2\) \(0\) \(0\) \(26\) \(-\) \(-\) \(q+(1+\beta )q^{2}+(12+2\beta )q^{4}+(13-4\beta )q^{7}+\cdots\)
225.4.a.o \(2\) \(13.275\) \(\Q(\sqrt{41}) \) None \(3\) \(0\) \(0\) \(-6\) \(-\) \(-\) \(q+(1+\beta )q^{2}+(3+3\beta )q^{4}+(-6+6\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(225)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)