Properties

Label 1521.4.a
Level $1521$
Weight $4$
Character orbit 1521.a
Rep. character $\chi_{1521}(1,\cdot)$
Character field $\Q$
Dimension $188$
Newform subspaces $40$
Sturm bound $728$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 40 \)
Sturm bound: \(728\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1521))\).

Total New Old
Modular forms 574 199 375
Cusp forms 518 188 330
Eisenstein series 56 11 45

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

\(3\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(41\)
\(+\)\(-\)\(-\)\(36\)
\(-\)\(+\)\(-\)\(54\)
\(-\)\(-\)\(+\)\(57\)
Plus space\(+\)\(98\)
Minus space\(-\)\(90\)

Trace form

\( 188 q + 718 q^{4} + 6 q^{5} - 22 q^{7} + O(q^{10}) \) \( 188 q + 718 q^{4} + 6 q^{5} - 22 q^{7} + 90 q^{10} - 42 q^{11} - 86 q^{14} + 2658 q^{16} + 34 q^{17} + 106 q^{19} - 192 q^{20} - 64 q^{22} - 80 q^{23} + 3958 q^{25} - 728 q^{28} + 356 q^{29} + 422 q^{31} - 180 q^{32} + 184 q^{34} + 388 q^{35} - 188 q^{37} + 1146 q^{38} + 1502 q^{40} + 402 q^{41} + 6 q^{43} - 600 q^{44} + 1996 q^{46} + 282 q^{47} + 8648 q^{49} - 144 q^{50} - 980 q^{53} - 404 q^{55} - 784 q^{56} + 1172 q^{58} - 1734 q^{59} - 1906 q^{61} - 638 q^{62} + 6636 q^{64} - 1298 q^{67} + 3120 q^{68} + 3388 q^{70} - 1098 q^{71} + 680 q^{73} + 3364 q^{74} + 2112 q^{76} + 2852 q^{77} + 624 q^{79} - 1392 q^{80} - 498 q^{82} - 1422 q^{83} + 712 q^{85} + 5004 q^{86} - 768 q^{88} - 534 q^{89} - 2052 q^{92} - 5110 q^{94} - 3852 q^{95} - 428 q^{97} - 5424 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\) 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 13
1521.4.a.a 1521.a 1.a $1$ $89.742$ \(\Q\) None \(-5\) \(0\) \(-7\) \(13\) $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}+17q^{4}-7q^{5}+13q^{7}-45q^{8}+\cdots\)
1521.4.a.b 1521.a 1.a $1$ $89.742$ \(\Q\) None \(-4\) \(0\) \(-17\) \(20\) $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+8q^{4}-17q^{5}+20q^{7}+68q^{10}+\cdots\)
1521.4.a.c 1521.a 1.a $1$ $89.742$ \(\Q\) None \(-3\) \(0\) \(9\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}+9q^{5}+2q^{7}+21q^{8}+\cdots\)
1521.4.a.d 1521.a 1.a $1$ $89.742$ \(\Q\) None \(-3\) \(0\) \(9\) \(15\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}+9q^{5}+15q^{7}+21q^{8}+\cdots\)
1521.4.a.e 1521.a 1.a $1$ $89.742$ \(\Q\) None \(-1\) \(0\) \(7\) \(10\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}+7q^{5}+10q^{7}+15q^{8}+\cdots\)
1521.4.a.f 1521.a 1.a $1$ $89.742$ \(\Q\) None \(0\) \(0\) \(-12\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{4}-12q^{5}-2q^{7}-6^{2}q^{11}+2^{6}q^{16}+\cdots\)
1521.4.a.g 1521.a 1.a $1$ $89.742$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-20\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}-20q^{7}+2^{6}q^{16}-56q^{19}+\cdots\)
1521.4.a.h 1521.a 1.a $1$ $89.742$ \(\Q\) None \(1\) \(0\) \(-7\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}-7q^{5}-10q^{7}-15q^{8}+\cdots\)
1521.4.a.i 1521.a 1.a $1$ $89.742$ \(\Q\) None \(3\) \(0\) \(-9\) \(-15\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}-9q^{5}-15q^{7}-21q^{8}+\cdots\)
1521.4.a.j 1521.a 1.a $1$ $89.742$ \(\Q\) None \(3\) \(0\) \(-9\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}-9q^{5}-2q^{7}-21q^{8}+\cdots\)
1521.4.a.k 1521.a 1.a $1$ $89.742$ \(\Q\) None \(4\) \(0\) \(17\) \(-20\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+8q^{4}+17q^{5}-20q^{7}+68q^{10}+\cdots\)
1521.4.a.l 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{17}) \) None \(-5\) \(0\) \(-15\) \(-15\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}+5\beta q^{4}+(-10+5\beta )q^{5}+\cdots\)
1521.4.a.m 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{4}-3\beta q^{5}+6\beta q^{7}+30\beta q^{11}+\cdots\)
1521.4.a.n 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-8q^{4}+\beta q^{7}+2^{6}q^{16}-5\beta q^{19}+\cdots\)
1521.4.a.o 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-5q^{4}+\beta q^{5}+8\beta q^{7}-13\beta q^{8}+\cdots\)
1521.4.a.p 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(44\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{4}-4\beta q^{5}+22q^{7}-9\beta q^{8}+\cdots\)
1521.4.a.q 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2\beta q^{2}+4q^{4}+8\beta q^{5}+13\beta q^{7}+\cdots\)
1521.4.a.r 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(-3\) \(9\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-4+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
1521.4.a.s 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{14}) \) None \(2\) \(0\) \(24\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(7+2\beta )q^{4}+(12-2\beta )q^{5}+\cdots\)
1521.4.a.t 1521.a 1.a $2$ $89.742$ \(\Q(\sqrt{17}) \) None \(5\) \(0\) \(15\) \(15\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{2}+(5-5\beta )q^{4}+(5+5\beta )q^{5}+\cdots\)
1521.4.a.u 1521.a 1.a $3$ $89.742$ 3.3.3144.1 None \(2\) \(0\) \(4\) \(-30\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(2-2\beta _{2})q^{5}+\cdots\)
1521.4.a.v 1521.a 1.a $4$ $89.742$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2\) \(0\) \(6\) \(-14\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(5+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1521.4.a.w 1521.a 1.a $4$ $89.742$ 4.4.1362828.1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{3})q^{4}+(2\beta _{1}+\beta _{2})q^{5}+\cdots\)
1521.4.a.x 1521.a 1.a $4$ $89.742$ 4.4.5054412.1 None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(7+\beta _{3})q^{4}-\beta _{2}q^{5}+6\beta _{1}q^{7}+\cdots\)
1521.4.a.y 1521.a 1.a $4$ $89.742$ 4.4.8112.1 \(\Q(\sqrt{-39}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-\beta _{1}q^{2}+(8-\beta _{2})q^{4}+(2\beta _{1}-\beta _{3})q^{5}+\cdots\)
1521.4.a.z 1521.a 1.a $4$ $89.742$ \(\Q(\sqrt{3}, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+9q^{4}+(3\beta _{1}-2\beta _{3})q^{5}+(11\beta _{1}+\cdots)q^{7}+\cdots\)
1521.4.a.ba 1521.a 1.a $4$ $89.742$ 4.4.1520092.1 None \(0\) \(0\) \(0\) \(-36\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(14+\beta _{2})q^{4}-\beta _{3}q^{5}+(-8+\cdots)q^{7}+\cdots\)
1521.4.a.bb 1521.a 1.a $4$ $89.742$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(2\) \(0\) \(-6\) \(14\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1521.4.a.bc 1521.a 1.a $8$ $89.742$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-22\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+\beta _{3}q^{5}+(-3+\cdots)q^{7}+\cdots\)
1521.4.a.bd 1521.a 1.a $8$ $89.742$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(22\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+\beta _{3}q^{5}+(3-\beta _{5}+\cdots)q^{7}+\cdots\)
1521.4.a.be 1521.a 1.a $9$ $89.742$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-8\) \(0\) \(-41\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(3-2\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
1521.4.a.bf 1521.a 1.a $9$ $89.742$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-6\) \(0\) \(-33\) \(83\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{3})q^{2}+(5+\beta _{3}+\beta _{5})q^{4}+\cdots\)
1521.4.a.bg 1521.a 1.a $9$ $89.742$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-5\) \(0\) \(-30\) \(38\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(5+\beta _{5}+\beta _{6}+\beta _{8})q^{4}+\cdots\)
1521.4.a.bh 1521.a 1.a $9$ $89.742$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(5\) \(0\) \(30\) \(-38\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(5+\beta _{5}+\beta _{6}+\beta _{8})q^{4}+\cdots\)
1521.4.a.bi 1521.a 1.a $9$ $89.742$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(6\) \(0\) \(33\) \(-83\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{3})q^{2}+(5+\beta _{3}+\beta _{5})q^{4}+(3+\cdots)q^{5}+\cdots\)
1521.4.a.bj 1521.a 1.a $9$ $89.742$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(8\) \(0\) \(41\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(3-2\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
1521.4.a.bk 1521.a 1.a $10$ $89.742$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(6+\beta _{4})q^{4}+(\beta _{1}+\beta _{2}-\beta _{8}+\cdots)q^{5}+\cdots\)
1521.4.a.bl 1521.a 1.a $12$ $89.742$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{6}q^{2}+(3+\beta _{2})q^{4}+(-\beta _{6}+\beta _{8}+\cdots)q^{5}+\cdots\)
1521.4.a.bm 1521.a 1.a $18$ $89.742$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(0\) \(0\) \(0\) \(-94\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(\beta _{1}-\beta _{14})q^{5}+\cdots\)
1521.4.a.bn 1521.a 1.a $18$ $89.742$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(0\) \(0\) \(0\) \(94\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(\beta _{1}-\beta _{14})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1521))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1521)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 2}\)