Properties

Label 825.4.a
Level $825$
Weight $4$
Character orbit 825.a
Rep. character $\chi_{825}(1,\cdot)$
Character field $\Q$
Dimension $94$
Newform subspaces $30$
Sturm bound $480$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(480\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(825))\).

Total New Old
Modular forms 372 94 278
Cusp forms 348 94 254
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.
\(+\)\(+\)\(+\)\(+\)\(12\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(12\)
\(+\)\(-\)\(-\)\(+\)\(14\)
\(-\)\(+\)\(+\)\(-\)\(11\)
\(-\)\(+\)\(-\)\(+\)\(14\)
\(-\)\(-\)\(+\)\(+\)\(12\)
\(-\)\(-\)\(-\)\(-\)\(10\)
Plus space\(+\)\(52\)
Minus space\(-\)\(42\)

Trace form

\( 94 q + 4 q^{2} + 356 q^{4} + 12 q^{6} - 32 q^{7} - 36 q^{8} + 846 q^{9} + O(q^{10}) \) \( 94 q + 4 q^{2} + 356 q^{4} + 12 q^{6} - 32 q^{7} - 36 q^{8} + 846 q^{9} - 24 q^{12} + 172 q^{13} + 28 q^{14} + 1388 q^{16} - 24 q^{17} + 36 q^{18} + 32 q^{19} - 36 q^{21} + 44 q^{22} - 108 q^{23} + 324 q^{24} + 1532 q^{26} + 216 q^{28} + 480 q^{29} - 480 q^{31} - 84 q^{32} + 66 q^{33} + 368 q^{34} + 3204 q^{36} - 236 q^{37} + 592 q^{38} - 504 q^{39} + 560 q^{41} - 468 q^{42} + 584 q^{43} + 616 q^{44} - 296 q^{46} + 172 q^{47} + 240 q^{48} + 4526 q^{49} + 864 q^{51} - 232 q^{52} + 776 q^{53} + 108 q^{54} + 252 q^{56} + 84 q^{57} + 1248 q^{58} + 40 q^{59} - 5460 q^{61} + 2096 q^{62} - 288 q^{63} + 7972 q^{64} - 264 q^{66} + 1344 q^{67} + 136 q^{68} - 1140 q^{69} - 396 q^{71} - 324 q^{72} - 3740 q^{73} - 5776 q^{74} - 248 q^{76} - 176 q^{77} - 852 q^{78} + 3824 q^{79} + 7614 q^{81} - 2040 q^{82} + 1912 q^{83} + 2400 q^{84} - 2552 q^{86} - 504 q^{87} + 924 q^{88} - 6700 q^{89} - 3960 q^{91} + 2460 q^{92} - 1824 q^{93} + 9576 q^{94} + 10044 q^{96} + 812 q^{97} + 6316 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\) 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
825.4.a.a 825.a 1.a $1$ $48.677$ \(\Q\) None \(-5\) \(-3\) \(0\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}-3q^{3}+17q^{4}+15q^{6}-3q^{7}+\cdots\)
825.4.a.b 825.a 1.a $1$ $48.677$ \(\Q\) None \(-4\) \(3\) \(0\) \(21\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+3q^{3}+8q^{4}-12q^{6}+21q^{7}+\cdots\)
825.4.a.c 825.a 1.a $1$ $48.677$ \(\Q\) None \(-3\) \(-3\) \(0\) \(-7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}-3q^{3}+q^{4}+9q^{6}-7q^{7}+\cdots\)
825.4.a.d 825.a 1.a $1$ $48.677$ \(\Q\) None \(-1\) \(-3\) \(0\) \(-36\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-7q^{4}+3q^{6}-6^{2}q^{7}+\cdots\)
825.4.a.e 825.a 1.a $1$ $48.677$ \(\Q\) None \(0\) \(3\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-8q^{4}-2q^{7}+9q^{9}-11q^{11}+\cdots\)
825.4.a.f 825.a 1.a $1$ $48.677$ \(\Q\) None \(1\) \(3\) \(0\) \(26\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}-7q^{4}+3q^{6}+26q^{7}+\cdots\)
825.4.a.g 825.a 1.a $1$ $48.677$ \(\Q\) None \(3\) \(3\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+3q^{3}+q^{4}+9q^{6}+7q^{7}+\cdots\)
825.4.a.h 825.a 1.a $1$ $48.677$ \(\Q\) None \(4\) \(-3\) \(0\) \(-21\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-3q^{3}+8q^{4}-12q^{6}-21q^{7}+\cdots\)
825.4.a.i 825.a 1.a $1$ $48.677$ \(\Q\) None \(5\) \(-3\) \(0\) \(32\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}-3q^{3}+17q^{4}-15q^{6}+2^{5}q^{7}+\cdots\)
825.4.a.j 825.a 1.a $1$ $48.677$ \(\Q\) None \(5\) \(3\) \(0\) \(3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}+3q^{3}+17q^{4}+15q^{6}+3q^{7}+\cdots\)
825.4.a.k 825.a 1.a $2$ $48.677$ \(\Q(\sqrt{33}) \) None \(-1\) \(-6\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-3q^{3}+\beta q^{4}+3\beta q^{6}+(-2+\cdots)q^{7}+\cdots\)
825.4.a.l 825.a 1.a $2$ $48.677$ \(\Q(\sqrt{97}) \) None \(-1\) \(6\) \(0\) \(-24\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{3}+(2^{4}+\beta )q^{4}-3\beta q^{6}+\cdots\)
825.4.a.m 825.a 1.a $2$ $48.677$ \(\Q(\sqrt{17}) \) None \(1\) \(-6\) \(0\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-3q^{3}+(-4+\beta )q^{4}-3\beta q^{6}+\cdots\)
825.4.a.n 825.a 1.a $3$ $48.677$ 3.3.788.1 None \(-1\) \(9\) \(0\) \(16\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+3q^{3}+(-2-\beta _{1})q^{4}-3\beta _{2}q^{6}+\cdots\)
825.4.a.o 825.a 1.a $3$ $48.677$ 3.3.3368.1 None \(-1\) \(9\) \(0\) \(-16\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(3+2\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
825.4.a.p 825.a 1.a $3$ $48.677$ 3.3.1957.1 None \(-1\) \(9\) \(0\) \(-6\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+2\beta _{2})q^{4}+\cdots\)
825.4.a.q 825.a 1.a $3$ $48.677$ 3.3.3368.1 None \(1\) \(-9\) \(0\) \(16\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(3+2\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
825.4.a.r 825.a 1.a $3$ $48.677$ 3.3.47528.1 None \(2\) \(-9\) \(0\) \(-10\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(10+\beta _{2})q^{4}+\cdots\)
825.4.a.s 825.a 1.a $3$ $48.677$ 3.3.23612.1 None \(4\) \(9\) \(0\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+3q^{3}+(7+\beta _{1}+\beta _{2})q^{4}+\cdots\)
825.4.a.t 825.a 1.a $4$ $48.677$ 4.4.1540841.1 None \(-4\) \(-12\) \(0\) \(-34\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(7-\beta _{1}+\beta _{3})q^{4}+\cdots\)
825.4.a.u 825.a 1.a $4$ $48.677$ 4.4.91035289.2 None \(-2\) \(12\) \(0\) \(-11\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+3q^{3}+(-4-\beta _{3})q^{4}+3\beta _{3}q^{6}+\cdots\)
825.4.a.v 825.a 1.a $4$ $48.677$ 4.4.91035289.2 None \(2\) \(-12\) \(0\) \(11\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{2}-3q^{3}+(-4-\beta _{3})q^{4}+3\beta _{3}q^{6}+\cdots\)
825.4.a.w 825.a 1.a $5$ $48.677$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-4\) \(-15\) \(0\) \(38\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(9-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.4.a.x 825.a 1.a $5$ $48.677$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-1\) \(15\) \(0\) \(-18\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(1+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
825.4.a.y 825.a 1.a $5$ $48.677$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(1\) \(-15\) \(0\) \(18\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(1+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
825.4.a.z 825.a 1.a $5$ $48.677$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(15\) \(0\) \(-38\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(9-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.4.a.ba 825.a 1.a $7$ $48.677$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-5\) \(21\) \(0\) \(-34\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.4.a.bb 825.a 1.a $7$ $48.677$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-3\) \(-21\) \(0\) \(-50\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(6+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
825.4.a.bc 825.a 1.a $7$ $48.677$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(3\) \(21\) \(0\) \(50\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
825.4.a.bd 825.a 1.a $7$ $48.677$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(5\) \(-21\) \(0\) \(34\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(825))\) into lower level spaces

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