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Properties

Label 1575.4.a
Level $1575$
Weight $4$
Character orbit 1575.a
Rep. character $\chi_{1575}(1,\cdot)$
Character field $\Q$
Dimension $143$
Newform subspaces $47$
Sturm bound $960$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 47 \)
Sturm bound: \(960\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1575))\).

Total New Old
Modular forms 744 143 601
Cusp forms 696 143 553
Eisenstein series 48 0 48

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

\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(16\)
\(+\)\(+\)\(-\)$-$\(10\)
\(+\)\(-\)\(+\)$-$\(14\)
\(+\)\(-\)\(-\)$+$\(18\)
\(-\)\(+\)\(+\)$-$\(19\)
\(-\)\(+\)\(-\)$+$\(22\)
\(-\)\(-\)\(+\)$+$\(23\)
\(-\)\(-\)\(-\)$-$\(21\)
Plus space\(+\)\(79\)
Minus space\(-\)\(64\)

Trace form

\( 143 q - q^{2} + 569 q^{4} - 7 q^{7} + 3 q^{8} + O(q^{10}) \) \( 143 q - q^{2} + 569 q^{4} - 7 q^{7} + 3 q^{8} - 108 q^{11} - 128 q^{13} + 7 q^{14} + 2469 q^{16} - 86 q^{17} - 110 q^{19} + 256 q^{22} + 412 q^{23} - 1036 q^{26} - 35 q^{28} - 410 q^{29} + 324 q^{31} - 597 q^{32} + 362 q^{34} + 86 q^{37} + 722 q^{38} - 274 q^{41} + 240 q^{43} - 870 q^{44} + 1406 q^{46} + 604 q^{47} + 7007 q^{49} - 836 q^{52} + 394 q^{53} - 147 q^{56} - 658 q^{58} + 562 q^{59} + 296 q^{61} - 3516 q^{62} + 9031 q^{64} + 156 q^{67} - 490 q^{68} + 2248 q^{71} - 1026 q^{73} + 6316 q^{74} - 738 q^{76} + 896 q^{77} + 4260 q^{79} + 6650 q^{82} + 422 q^{83} + 4222 q^{86} + 4204 q^{88} + 1530 q^{89} + 196 q^{91} + 1368 q^{92} + 10024 q^{94} - 1270 q^{97} - 49 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\) 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 7
1575.4.a.a 1575.a 1.a $1$ $92.928$ \(\Q\) None \(-3\) \(0\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6q^{11}+\cdots\)
1575.4.a.b 1575.a 1.a $1$ $92.928$ \(\Q\) None \(-3\) \(0\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6^{2}q^{11}+\cdots\)
1575.4.a.c 1575.a 1.a $1$ $92.928$ \(\Q\) None \(-3\) \(0\) \(0\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+60q^{11}+\cdots\)
1575.4.a.d 1575.a 1.a $1$ $92.928$ \(\Q\) None \(-2\) \(0\) \(0\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{4}+7q^{7}+24q^{8}+21q^{11}+\cdots\)
1575.4.a.e 1575.a 1.a $1$ $92.928$ \(\Q\) None \(-1\) \(0\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-7q^{4}+7q^{7}+15q^{8}+8q^{11}+\cdots\)
1575.4.a.f 1575.a 1.a $1$ $92.928$ \(\Q\) None \(0\) \(0\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{4}-7q^{7}-42q^{11}-20q^{13}+\cdots\)
1575.4.a.g 1575.a 1.a $1$ $92.928$ \(\Q\) None \(1\) \(0\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}-7q^{7}-15q^{8}-12q^{11}+\cdots\)
1575.4.a.h 1575.a 1.a $1$ $92.928$ \(\Q\) None \(2\) \(0\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-4q^{4}-7q^{7}-24q^{8}+21q^{11}+\cdots\)
1575.4.a.i 1575.a 1.a $1$ $92.928$ \(\Q\) None \(3\) \(0\) \(0\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}-7q^{7}-21q^{8}-60q^{11}+\cdots\)
1575.4.a.j 1575.a 1.a $1$ $92.928$ \(\Q\) None \(3\) \(0\) \(0\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}+7q^{7}-21q^{8}+6q^{11}+\cdots\)
1575.4.a.k 1575.a 1.a $1$ $92.928$ \(\Q\) None \(4\) \(0\) \(0\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+8q^{4}+7q^{7}-62q^{11}+62q^{13}+\cdots\)
1575.4.a.l 1575.a 1.a $1$ $92.928$ \(\Q\) None \(5\) \(0\) \(0\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{2}+17q^{4}-7q^{7}+45q^{8}-12q^{11}+\cdots\)
1575.4.a.m 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{17}) \) None \(-7\) \(0\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{2}+(5+7\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.n 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{5}) \) None \(-4\) \(0\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}+(1+4\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.o 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{17}) \) None \(-3\) \(0\) \(0\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(-3+3\beta )q^{4}-7q^{7}+\cdots\)
1575.4.a.p 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{57}) \) None \(-3\) \(0\) \(0\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(7+3\beta )q^{4}-7q^{7}+\cdots\)
1575.4.a.q 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.r 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(0\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(-4+\cdots)q^{8}+\cdots\)
1575.4.a.s 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{41}) \) None \(-1\) \(0\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2+\beta )q^{4}+7q^{7}+(-10+5\beta )q^{8}+\cdots\)
1575.4.a.t 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+11q^{4}+7q^{7}+3\beta q^{8}+10\beta q^{11}+\cdots\)
1575.4.a.u 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(0\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(4-11\beta )q^{8}+\cdots\)
1575.4.a.v 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{41}) \) None \(1\) \(0\) \(0\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{4}-7q^{7}+(10-5\beta )q^{8}+\cdots\)
1575.4.a.w 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{65}) \) None \(1\) \(0\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(8+\beta )q^{4}+7q^{7}+(2^{4}+\beta )q^{8}+\cdots\)
1575.4.a.x 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{17}) \) None \(3\) \(0\) \(0\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.y 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{41}) \) None \(3\) \(0\) \(0\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(3+3\beta )q^{4}-7q^{7}+(5^{2}+\cdots)q^{8}+\cdots\)
1575.4.a.z 1575.a 1.a $2$ $92.928$ \(\Q(\sqrt{2}) \) None \(8\) \(0\) \(0\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{2}+(10+8\beta )q^{4}+7q^{7}+(24+\cdots)q^{8}+\cdots\)
1575.4.a.ba 1575.a 1.a $3$ $92.928$ 3.3.14360.1 None \(-3\) \(0\) \(0\) \(-21\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bb 1575.a 1.a $3$ $92.928$ 3.3.22952.1 None \(-2\) \(0\) \(0\) \(-21\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bc 1575.a 1.a $3$ $92.928$ 3.3.2292.1 None \(-1\) \(0\) \(0\) \(-21\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots\)
1575.4.a.bd 1575.a 1.a $3$ $92.928$ 3.3.2292.1 None \(1\) \(0\) \(0\) \(21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots\)
1575.4.a.be 1575.a 1.a $3$ $92.928$ 3.3.22952.1 None \(2\) \(0\) \(0\) \(-21\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots\)
1575.4.a.bf 1575.a 1.a $4$ $92.928$ 4.4.26729725.1 None \(-6\) \(0\) \(0\) \(-28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bg 1575.a 1.a $4$ $92.928$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-4\) \(0\) \(0\) \(28\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1575.4.a.bh 1575.a 1.a $4$ $92.928$ 4.4.3030748.1 None \(0\) \(0\) \(0\) \(-28\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}-7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bi 1575.a 1.a $4$ $92.928$ 4.4.3030748.1 None \(0\) \(0\) \(0\) \(28\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}+7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bj 1575.a 1.a $4$ $92.928$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(-28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}-7q^{7}+(2+4\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bk 1575.a 1.a $4$ $92.928$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(28\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+7q^{7}+(-2+\cdots)q^{8}+\cdots\)
1575.4.a.bl 1575.a 1.a $4$ $92.928$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(4\) \(0\) \(0\) \(-28\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}-7q^{7}+\cdots\)
1575.4.a.bm 1575.a 1.a $4$ $92.928$ 4.4.26729725.1 None \(6\) \(0\) \(0\) \(28\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots\)
1575.4.a.bn 1575.a 1.a $5$ $92.928$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-4\) \(0\) \(0\) \(35\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1575.4.a.bo 1575.a 1.a $5$ $92.928$ 5.5.78066700.1 None \(-1\) \(0\) \(0\) \(35\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}+7q^{7}+\cdots\)
1575.4.a.bp 1575.a 1.a $5$ $92.928$ 5.5.78066700.1 None \(1\) \(0\) \(0\) \(-35\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}-7q^{7}+\cdots\)
1575.4.a.bq 1575.a 1.a $5$ $92.928$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(0\) \(0\) \(-35\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1575.4.a.br 1575.a 1.a $8$ $92.928$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-56\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bs 1575.a 1.a $8$ $92.928$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(56\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bt 1575.a 1.a $10$ $92.928$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(-70\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bu 1575.a 1.a $10$ $92.928$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(70\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1575))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1575)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)