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Properties

Label 522.4.a
Level $522$
Weight $4$
Character orbit 522.a
Rep. character $\chi_{522}(1,\cdot)$
Character field $\Q$
Dimension $35$
Newform subspaces $17$
Sturm bound $360$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 522.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(360\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(522))\).

Total New Old
Modular forms 278 35 243
Cusp forms 262 35 227
Eisenstein series 16 0 16

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

\(2\)\(3\)\(29\)FrickeDim
\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(5\)
\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(-\)$+$\(4\)
\(-\)\(-\)\(+\)$+$\(6\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(19\)
Minus space\(-\)\(16\)

Trace form

\( 35 q - 2 q^{2} + 140 q^{4} + 32 q^{5} + 44 q^{7} - 8 q^{8} + O(q^{10}) \) \( 35 q - 2 q^{2} + 140 q^{4} + 32 q^{5} + 44 q^{7} - 8 q^{8} + 20 q^{10} + 40 q^{11} + 20 q^{13} - 64 q^{14} + 560 q^{16} - 190 q^{17} - 128 q^{19} + 128 q^{20} - 108 q^{22} + 260 q^{23} + 867 q^{25} - 228 q^{26} + 176 q^{28} - 87 q^{29} + 200 q^{31} - 32 q^{32} - 180 q^{34} - 1228 q^{35} - 330 q^{37} + 336 q^{38} + 80 q^{40} - 762 q^{41} - 8 q^{43} + 160 q^{44} + 264 q^{46} + 456 q^{47} + 1387 q^{49} + 450 q^{50} + 80 q^{52} + 1000 q^{53} - 900 q^{55} - 256 q^{56} + 58 q^{58} + 1108 q^{59} + 1922 q^{61} + 1604 q^{62} + 2240 q^{64} + 658 q^{65} - 1276 q^{67} - 760 q^{68} - 224 q^{70} - 144 q^{71} - 102 q^{73} + 316 q^{74} - 512 q^{76} + 1576 q^{77} + 1272 q^{79} + 512 q^{80} - 1460 q^{82} - 244 q^{83} + 1560 q^{85} + 236 q^{86} - 432 q^{88} + 3310 q^{89} + 324 q^{91} + 1040 q^{92} + 772 q^{94} + 2376 q^{95} - 3198 q^{97} + 174 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(522))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 29
522.4.a.a 522.a 1.a $1$ $30.799$ \(\Q\) None \(-2\) \(0\) \(10\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+10q^{5}+7q^{7}-8q^{8}+\cdots\)
522.4.a.b 522.a 1.a $1$ $30.799$ \(\Q\) None \(-2\) \(0\) \(15\) \(-18\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+15q^{5}-18q^{7}-8q^{8}+\cdots\)
522.4.a.c 522.a 1.a $1$ $30.799$ \(\Q\) None \(-2\) \(0\) \(18\) \(-29\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+18q^{5}-29q^{7}-8q^{8}+\cdots\)
522.4.a.d 522.a 1.a $1$ $30.799$ \(\Q\) None \(2\) \(0\) \(-21\) \(19\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-21q^{5}+19q^{7}+8q^{8}+\cdots\)
522.4.a.e 522.a 1.a $1$ $30.799$ \(\Q\) None \(2\) \(0\) \(-5\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-2q^{7}+8q^{8}+\cdots\)
522.4.a.f 522.a 1.a $1$ $30.799$ \(\Q\) None \(2\) \(0\) \(0\) \(-17\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-17q^{7}+8q^{8}+23q^{11}+\cdots\)
522.4.a.g 522.a 1.a $1$ $30.799$ \(\Q\) None \(2\) \(0\) \(8\) \(19\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+8q^{5}+19q^{7}+8q^{8}+\cdots\)
522.4.a.h 522.a 1.a $1$ $30.799$ \(\Q\) None \(2\) \(0\) \(14\) \(-21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+14q^{5}-21q^{7}+8q^{8}+\cdots\)
522.4.a.i 522.a 1.a $2$ $30.799$ \(\Q(\sqrt{457}) \) None \(-4\) \(0\) \(5\) \(14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(3-\beta )q^{5}+7q^{7}-8q^{8}+\cdots\)
522.4.a.j 522.a 1.a $2$ $30.799$ \(\Q(\sqrt{6}) \) None \(4\) \(0\) \(10\) \(-16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(5+6\beta )q^{5}+(-8+8\beta )q^{7}+\cdots\)
522.4.a.k 522.a 1.a $3$ $30.799$ 3.3.19816.1 None \(-6\) \(0\) \(-20\) \(24\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-6-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
522.4.a.l 522.a 1.a $3$ $30.799$ 3.3.717105.1 None \(-6\) \(0\) \(-7\) \(34\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-2-\beta _{1})q^{5}+(12+\cdots)q^{7}+\cdots\)
522.4.a.m 522.a 1.a $3$ $30.799$ 3.3.7404.1 None \(-6\) \(0\) \(5\) \(-25\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(2-\beta _{2})q^{5}+(-7+2\beta _{1}+\cdots)q^{7}+\cdots\)
522.4.a.n 522.a 1.a $3$ $30.799$ 3.3.7404.1 None \(6\) \(0\) \(-5\) \(-25\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-2+\beta _{2})q^{5}+(-7+\cdots)q^{7}+\cdots\)
522.4.a.o 522.a 1.a $3$ $30.799$ 3.3.973681.1 None \(6\) \(0\) \(5\) \(18\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(2+\beta _{2})q^{5}+(6-\beta _{1}+\cdots)q^{7}+\cdots\)
522.4.a.p 522.a 1.a $4$ $30.799$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(0\) \(-15\) \(31\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-4+\beta _{1})q^{5}+(8+\beta _{2}+\cdots)q^{7}+\cdots\)
522.4.a.q 522.a 1.a $4$ $30.799$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(0\) \(15\) \(31\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(4-\beta _{1})q^{5}+(8+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(522))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(522)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 2}\)