Properties

Label 225.6.a
Level $225$
Weight $6$
Character orbit 225.a
Rep. character $\chi_{225}(1,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $22$
Sturm bound $180$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(180\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(225))\).

Total New Old
Modular forms 162 41 121
Cusp forms 138 38 100
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.
\(+\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(11\)
\(-\)\(-\)\(+\)\(11\)
Plus space\(+\)\(18\)
Minus space\(-\)\(20\)

Trace form

\( 38 q + 578 q^{4} - 80 q^{7} - 180 q^{8} + O(q^{10}) \) \( 38 q + 578 q^{4} - 80 q^{7} - 180 q^{8} + 324 q^{11} + 100 q^{13} - 1896 q^{14} + 8330 q^{16} + 1860 q^{17} + 4036 q^{19} + 2640 q^{22} + 2400 q^{23} - 13524 q^{26} - 9920 q^{28} + 8916 q^{29} - 164 q^{31} - 20460 q^{32} + 10650 q^{34} + 12340 q^{37} + 61560 q^{38} - 22428 q^{41} - 11240 q^{43} + 14862 q^{44} + 39948 q^{46} - 31920 q^{47} + 48330 q^{49} + 4840 q^{52} + 77340 q^{53} - 118260 q^{56} - 114000 q^{58} - 19788 q^{59} + 25264 q^{61} - 51000 q^{62} + 149786 q^{64} - 18200 q^{67} + 291240 q^{68} - 109584 q^{71} - 177380 q^{73} + 179124 q^{74} + 265450 q^{76} + 24000 q^{77} + 79336 q^{79} + 210000 q^{82} + 120120 q^{83} - 297756 q^{86} + 87360 q^{88} - 210492 q^{89} + 216188 q^{91} - 256320 q^{92} + 343512 q^{94} - 35540 q^{97} + 367440 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\) 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
225.6.a.a 225.a 1.a $1$ $36.086$ \(\Q\) None \(-6\) \(0\) \(0\) \(40\) $-$ $+$ $\mathrm{SU}(2)$ \(q-6q^{2}+4q^{4}+40q^{7}+168q^{8}+564q^{11}+\cdots\)
225.6.a.b 225.a 1.a $1$ $36.086$ \(\Q\) None \(-4\) \(0\) \(0\) \(-225\) $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-2^{4}q^{4}-15^{2}q^{7}+192q^{8}+\cdots\)
225.6.a.c 225.a 1.a $1$ $36.086$ \(\Q\) None \(-2\) \(0\) \(0\) \(132\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-28q^{4}+132q^{7}+120q^{8}+\cdots\)
225.6.a.d 225.a 1.a $1$ $36.086$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-25\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2^{5}q^{4}-5^{2}q^{7}-775q^{13}+2^{10}q^{16}+\cdots\)
225.6.a.e 225.a 1.a $1$ $36.086$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(25\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2^{5}q^{4}+5^{2}q^{7}+775q^{13}+2^{10}q^{16}+\cdots\)
225.6.a.f 225.a 1.a $1$ $36.086$ \(\Q\) None \(2\) \(0\) \(0\) \(-192\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-28q^{4}-192q^{7}-120q^{8}+\cdots\)
225.6.a.g 225.a 1.a $1$ $36.086$ \(\Q\) None \(4\) \(0\) \(0\) \(225\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-2^{4}q^{4}+15^{2}q^{7}-192q^{8}+\cdots\)
225.6.a.h 225.a 1.a $1$ $36.086$ \(\Q\) None \(7\) \(0\) \(0\) \(-12\) $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{2}+17q^{4}-12q^{7}-105q^{8}+\cdots\)
225.6.a.i 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{89}) \) None \(-9\) \(0\) \(0\) \(108\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{2}+(6+9\beta )q^{4}+(42+24\beta )q^{7}+\cdots\)
225.6.a.j 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{31}) \) None \(-6\) \(0\) \(0\) \(102\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta )q^{2}+(8-6\beta )q^{4}+(51-24\beta )q^{7}+\cdots\)
225.6.a.k 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{145}) \) None \(-5\) \(0\) \(0\) \(-80\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}+(8+5\beta )q^{4}+(-30+\cdots)q^{7}+\cdots\)
225.6.a.l 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{241}) \) None \(-5\) \(0\) \(0\) \(200\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}+(2^{5}+5\beta )q^{4}+(102+\cdots)q^{7}+\cdots\)
225.6.a.m 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{409}) \) None \(-1\) \(0\) \(0\) \(112\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(70+\beta )q^{4}+(2^{6}-2^{4}\beta )q^{7}+\cdots\)
225.6.a.n 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{11}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+12q^{4}+9\beta q^{7}-20\beta q^{8}+\cdots\)
225.6.a.o 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{70}) \) None \(0\) \(0\) \(0\) \(-90\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+38q^{4}-45q^{7}+6\beta q^{8}-40\beta q^{11}+\cdots\)
225.6.a.p 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{70}) \) None \(0\) \(0\) \(0\) \(90\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+38q^{4}+45q^{7}+6\beta q^{8}+40\beta q^{11}+\cdots\)
225.6.a.q 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-\beta q^{2}+93q^{4}-61\beta q^{8}+4649q^{16}+\cdots\)
225.6.a.r 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{145}) \) None \(5\) \(0\) \(0\) \(-80\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{2}+(13-5\beta )q^{4}+(-50+20\beta )q^{7}+\cdots\)
225.6.a.s 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{241}) \) None \(5\) \(0\) \(0\) \(-200\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{2}+(37-5\beta )q^{4}+(-98-4\beta )q^{7}+\cdots\)
225.6.a.t 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{31}) \) None \(6\) \(0\) \(0\) \(-102\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{2}+(8+6\beta )q^{4}+(-51-24\beta )q^{7}+\cdots\)
225.6.a.u 225.a 1.a $2$ $36.086$ \(\Q(\sqrt{89}) \) None \(9\) \(0\) \(0\) \(-108\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(5-\beta )q^{2}+(15-9\beta )q^{4}+(-66+24\beta )q^{7}+\cdots\)
225.6.a.v 225.a 1.a $4$ $36.086$ \(\Q(\sqrt{5}, \sqrt{14})\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-12q^{4}-\beta _{2}q^{7}+44\beta _{1}q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

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