Properties

Label 825.6.a
Level $825$
Weight $6$
Character orbit 825.a
Rep. character $\chi_{825}(1,\cdot)$
Character field $\Q$
Dimension $160$
Newform subspaces $25$
Sturm bound $720$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 25 \)
Sturm bound: \(720\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(825))\).

Total New Old
Modular forms 612 160 452
Cusp forms 588 160 428
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.

\(3\)\(5\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(18\)
\(+\)\(+\)\(-\)\(-\)\(21\)
\(+\)\(-\)\(+\)\(-\)\(22\)
\(+\)\(-\)\(-\)\(+\)\(20\)
\(-\)\(+\)\(+\)\(-\)\(19\)
\(-\)\(+\)\(-\)\(+\)\(16\)
\(-\)\(-\)\(+\)\(+\)\(21\)
\(-\)\(-\)\(-\)\(-\)\(23\)
Plus space\(+\)\(75\)
Minus space\(-\)\(85\)

Trace form

\( 160 q - 8 q^{2} - 18 q^{3} + 2644 q^{4} - 180 q^{6} - 36 q^{7} + 108 q^{8} + 12960 q^{9} + O(q^{10}) \) \( 160 q - 8 q^{2} - 18 q^{3} + 2644 q^{4} - 180 q^{6} - 36 q^{7} + 108 q^{8} + 12960 q^{9} - 864 q^{12} - 544 q^{13} - 1508 q^{14} + 45132 q^{16} + 2700 q^{17} - 648 q^{18} - 6856 q^{19} + 2664 q^{21} - 968 q^{22} + 4896 q^{23} - 540 q^{24} - 22324 q^{26} - 1458 q^{27} + 11240 q^{28} + 26148 q^{29} + 18556 q^{31} + 16428 q^{32} - 2178 q^{33} + 46080 q^{34} + 214164 q^{36} + 7872 q^{37} + 59320 q^{38} - 4896 q^{39} + 32636 q^{41} + 13068 q^{42} - 50236 q^{43} + 4840 q^{44} + 67416 q^{46} - 62360 q^{47} - 77040 q^{48} + 371972 q^{49} + 23328 q^{51} - 33704 q^{52} + 94088 q^{53} - 14580 q^{54} - 69732 q^{56} + 32220 q^{57} - 40704 q^{58} - 72608 q^{59} + 56964 q^{61} - 156976 q^{62} - 2916 q^{63} + 865796 q^{64} + 17424 q^{66} - 60648 q^{67} + 200344 q^{68} + 21096 q^{69} - 60456 q^{71} + 8748 q^{72} + 162488 q^{73} + 179720 q^{74} + 67192 q^{76} + 29524 q^{77} - 61740 q^{78} - 56420 q^{79} + 1049760 q^{81} + 20568 q^{82} + 98992 q^{83} + 84240 q^{84} + 202096 q^{86} + 95904 q^{87} - 161172 q^{88} - 454528 q^{89} + 102588 q^{91} - 116724 q^{92} - 236376 q^{93} - 879944 q^{94} - 491940 q^{96} + 380896 q^{97} + 233152 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\) 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 11
825.6.a.a 825.a 1.a $1$ $132.317$ \(\Q\) None \(-1\) \(-9\) \(0\) \(26\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-9q^{3}-31q^{4}+9q^{6}+26q^{7}+\cdots\)
825.6.a.b 825.a 1.a $1$ $132.317$ \(\Q\) None \(2\) \(9\) \(0\) \(-148\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+9q^{3}-28q^{4}+18q^{6}-148q^{7}+\cdots\)
825.6.a.c 825.a 1.a $2$ $132.317$ \(\Q(\sqrt{33}) \) None \(-13\) \(-18\) \(0\) \(-146\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-6-\beta )q^{2}-9q^{3}+(12+13\beta )q^{4}+\cdots\)
825.6.a.d 825.a 1.a $2$ $132.317$ \(\Q(\sqrt{313}) \) None \(-1\) \(18\) \(0\) \(18\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+9q^{3}+(46+\beta )q^{4}-9\beta q^{6}+\cdots\)
825.6.a.e 825.a 1.a $2$ $132.317$ \(\Q(\sqrt{177}) \) None \(5\) \(18\) \(0\) \(286\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{2}+9q^{3}+(21-5\beta )q^{4}+(3^{3}+\cdots)q^{6}+\cdots\)
825.6.a.f 825.a 1.a $3$ $132.317$ 3.3.307532.1 None \(-7\) \(27\) \(0\) \(-92\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{2}+9q^{3}+(24+2\beta _{1}+\cdots)q^{4}+\cdots\)
825.6.a.g 825.a 1.a $3$ $132.317$ 3.3.18257.1 None \(-2\) \(27\) \(0\) \(68\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+9q^{3}+(-14-\beta _{1}+\cdots)q^{4}+\cdots\)
825.6.a.h 825.a 1.a $3$ $132.317$ 3.3.788.1 None \(-2\) \(27\) \(0\) \(-152\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+9q^{3}+(-8-4\beta _{1}+\cdots)q^{4}+\cdots\)
825.6.a.i 825.a 1.a $3$ $132.317$ 3.3.3368.1 None \(2\) \(-27\) \(0\) \(232\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}-9q^{3}+(8+4\beta _{2})q^{4}+\cdots\)
825.6.a.j 825.a 1.a $3$ $132.317$ 3.3.34253.1 None \(7\) \(-27\) \(0\) \(172\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{2}-9q^{3}+(7+4\beta _{1}+\beta _{2})q^{4}+\cdots\)
825.6.a.k 825.a 1.a $5$ $132.317$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(1\) \(45\) \(0\) \(-116\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+9q^{3}+(5^{2}+\beta _{1}+\beta _{3})q^{4}+\cdots\)
825.6.a.l 825.a 1.a $5$ $132.317$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(2\) \(-45\) \(0\) \(-184\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-9q^{3}+(2^{4}+\beta _{2})q^{4}-9\beta _{1}q^{6}+\cdots\)
825.6.a.m 825.a 1.a $7$ $132.317$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-9\) \(63\) \(0\) \(-65\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+9q^{3}+(13+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.6.a.n 825.a 1.a $7$ $132.317$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(-63\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-9q^{3}+(28+\beta _{2})q^{4}+9\beta _{1}q^{6}+\cdots\)
825.6.a.o 825.a 1.a $7$ $132.317$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(9\) \(-63\) \(0\) \(65\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}-9q^{3}+(13+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.6.a.p 825.a 1.a $8$ $132.317$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-9\) \(72\) \(0\) \(66\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+9q^{3}+(13+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.6.a.q 825.a 1.a $8$ $132.317$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(9\) \(-72\) \(0\) \(-66\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}-9q^{3}+(13+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.6.a.r 825.a 1.a $9$ $132.317$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-1\) \(81\) \(0\) \(57\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+9q^{3}+(19+\beta _{2})q^{4}-9\beta _{1}q^{6}+\cdots\)
825.6.a.s 825.a 1.a $9$ $132.317$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(1\) \(-81\) \(0\) \(-57\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-9q^{3}+(19+\beta _{2})q^{4}-9\beta _{1}q^{6}+\cdots\)
825.6.a.t 825.a 1.a $10$ $132.317$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-1\) \(90\) \(0\) \(188\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+9q^{3}+(22+\beta _{2})q^{4}-9\beta _{1}q^{6}+\cdots\)
825.6.a.u 825.a 1.a $10$ $132.317$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(1\) \(-90\) \(0\) \(-188\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-9q^{3}+(22+\beta _{2})q^{4}-9\beta _{1}q^{6}+\cdots\)
825.6.a.v 825.a 1.a $13$ $132.317$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(-13\) \(-117\) \(0\) \(-304\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-9q^{3}+(2^{4}-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.6.a.w 825.a 1.a $13$ $132.317$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(-3\) \(117\) \(0\) \(-284\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+9q^{3}+(17+\beta _{1}+\beta _{2})q^{4}+\cdots\)
825.6.a.x 825.a 1.a $13$ $132.317$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(3\) \(-117\) \(0\) \(284\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-9q^{3}+(17+\beta _{1}+\beta _{2})q^{4}+\cdots\)
825.6.a.y 825.a 1.a $13$ $132.317$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(13\) \(117\) \(0\) \(304\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+9q^{3}+(2^{4}-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(825))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(825)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 2}\)