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

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

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\) 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
225.4.a.a 225.a 1.a $1$ $13.275$ \(\Q\) None \(-5\) \(0\) \(0\) \(30\) $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}+17q^{4}+30q^{7}-45q^{8}+50q^{11}+\cdots\)
225.4.a.b 225.a 1.a $1$ $13.275$ \(\Q\) None \(-4\) \(0\) \(0\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+8q^{4}-6q^{7}-2^{5}q^{11}+38q^{13}+\cdots\)
225.4.a.c 225.a 1.a $1$ $13.275$ \(\Q\) None \(-1\) \(0\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}+6q^{7}+15q^{8}+43q^{11}+\cdots\)
225.4.a.d 225.a 1.a $1$ $13.275$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-20\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}-20q^{7}+70q^{13}+2^{6}q^{16}+\cdots\)
225.4.a.e 225.a 1.a $1$ $13.275$ \(\Q\) None \(1\) \(0\) \(0\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}-6q^{7}-15q^{8}+43q^{11}+\cdots\)
225.4.a.f 225.a 1.a $1$ $13.275$ \(\Q\) None \(1\) \(0\) \(0\) \(24\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}+24q^{7}-15q^{8}-52q^{11}+\cdots\)
225.4.a.g 225.a 1.a $1$ $13.275$ \(\Q\) None \(3\) \(0\) \(0\) \(-20\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}-20q^{7}-21q^{8}+24q^{11}+\cdots\)
225.4.a.h 225.a 1.a $1$ $13.275$ \(\Q\) None \(5\) \(0\) \(0\) \(30\) $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}+17q^{4}+30q^{7}+45q^{8}-50q^{11}+\cdots\)
225.4.a.i 225.a 1.a $2$ $13.275$ \(\Q(\sqrt{41}) \) None \(-3\) \(0\) \(0\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(3+3\beta )q^{4}+(6-6\beta )q^{7}+\cdots\)
225.4.a.j 225.a 1.a $2$ $13.275$ \(\Q(\sqrt{19}) \) None \(-2\) \(0\) \(0\) \(-26\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(12-2\beta )q^{4}+(-13+\cdots)q^{7}+\cdots\)
225.4.a.k 225.a 1.a $2$ $13.275$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-\beta q^{2}-3q^{4}+11\beta q^{8}-31q^{16}+\cdots\)
225.4.a.l 225.a 1.a $2$ $13.275$ \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(-30\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+2q^{4}-15q^{7}-6\beta q^{8}-20\beta q^{11}+\cdots\)
225.4.a.m 225.a 1.a $2$ $13.275$ \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(30\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+2q^{4}+15q^{7}-6\beta q^{8}+20\beta q^{11}+\cdots\)
225.4.a.n 225.a 1.a $2$ $13.275$ \(\Q(\sqrt{19}) \) None \(2\) \(0\) \(0\) \(26\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(12+2\beta )q^{4}+(13-4\beta )q^{7}+\cdots\)
225.4.a.o 225.a 1.a $2$ $13.275$ \(\Q(\sqrt{41}) \) None \(3\) \(0\) \(0\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(3+3\beta )q^{4}+(-6+6\beta )q^{7}+\cdots\)

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

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