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

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

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
1575.4.a.a \(1\) \(92.928\) \(\Q\) None \(-3\) \(0\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6q^{11}+\cdots\)
1575.4.a.b \(1\) \(92.928\) \(\Q\) None \(-3\) \(0\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6^{2}q^{11}+\cdots\)
1575.4.a.c \(1\) \(92.928\) \(\Q\) None \(-3\) \(0\) \(0\) \(-7\) \(+\) \(+\) \(+\) \(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+60q^{11}+\cdots\)
1575.4.a.d \(1\) \(92.928\) \(\Q\) None \(-2\) \(0\) \(0\) \(7\) \(-\) \(-\) \(-\) \(q-2q^{2}-4q^{4}+7q^{7}+24q^{8}+21q^{11}+\cdots\)
1575.4.a.e \(1\) \(92.928\) \(\Q\) None \(-1\) \(0\) \(0\) \(7\) \(-\) \(+\) \(-\) \(q-q^{2}-7q^{4}+7q^{7}+15q^{8}+8q^{11}+\cdots\)
1575.4.a.f \(1\) \(92.928\) \(\Q\) None \(0\) \(0\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q-8q^{4}-7q^{7}-42q^{11}-20q^{13}+\cdots\)
1575.4.a.g \(1\) \(92.928\) \(\Q\) None \(1\) \(0\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q+q^{2}-7q^{4}-7q^{7}-15q^{8}-12q^{11}+\cdots\)
1575.4.a.h \(1\) \(92.928\) \(\Q\) None \(2\) \(0\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q+2q^{2}-4q^{4}-7q^{7}-24q^{8}+21q^{11}+\cdots\)
1575.4.a.i \(1\) \(92.928\) \(\Q\) None \(3\) \(0\) \(0\) \(-7\) \(+\) \(+\) \(+\) \(q+3q^{2}+q^{4}-7q^{7}-21q^{8}-60q^{11}+\cdots\)
1575.4.a.j \(1\) \(92.928\) \(\Q\) None \(3\) \(0\) \(0\) \(7\) \(-\) \(-\) \(-\) \(q+3q^{2}+q^{4}+7q^{7}-21q^{8}+6q^{11}+\cdots\)
1575.4.a.k \(1\) \(92.928\) \(\Q\) None \(4\) \(0\) \(0\) \(7\) \(-\) \(+\) \(-\) \(q+4q^{2}+8q^{4}+7q^{7}-62q^{11}+62q^{13}+\cdots\)
1575.4.a.l \(1\) \(92.928\) \(\Q\) None \(5\) \(0\) \(0\) \(-7\) \(-\) \(+\) \(+\) \(q+5q^{2}+17q^{4}-7q^{7}+45q^{8}-12q^{11}+\cdots\)
1575.4.a.m \(2\) \(92.928\) \(\Q(\sqrt{17}) \) None \(-7\) \(0\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+(-3-\beta )q^{2}+(5+7\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.n \(2\) \(92.928\) \(\Q(\sqrt{5}) \) None \(-4\) \(0\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+(-2-\beta )q^{2}+(1+4\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.o \(2\) \(92.928\) \(\Q(\sqrt{17}) \) None \(-3\) \(0\) \(0\) \(-14\) \(-\) \(+\) \(+\) \(q+(-1-\beta )q^{2}+(-3+3\beta )q^{4}-7q^{7}+\cdots\)
1575.4.a.p \(2\) \(92.928\) \(\Q(\sqrt{57}) \) None \(-3\) \(0\) \(0\) \(-14\) \(-\) \(+\) \(+\) \(q+(-1-\beta )q^{2}+(7+3\beta )q^{4}-7q^{7}+\cdots\)
1575.4.a.q \(2\) \(92.928\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.r \(2\) \(92.928\) \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(0\) \(14\) \(+\) \(+\) \(-\) \(q-\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(-4+\cdots)q^{8}+\cdots\)
1575.4.a.s \(2\) \(92.928\) \(\Q(\sqrt{41}) \) None \(-1\) \(0\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+(2+\beta )q^{4}+7q^{7}+(-10+5\beta )q^{8}+\cdots\)
1575.4.a.t \(2\) \(92.928\) \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(14\) \(+\) \(+\) \(-\) \(q+\beta q^{2}+11q^{4}+7q^{7}+3\beta q^{8}+10\beta q^{11}+\cdots\)
1575.4.a.u \(2\) \(92.928\) \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(0\) \(14\) \(+\) \(+\) \(-\) \(q+\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(4-11\beta )q^{8}+\cdots\)
1575.4.a.v \(2\) \(92.928\) \(\Q(\sqrt{41}) \) None \(1\) \(0\) \(0\) \(-14\) \(-\) \(-\) \(+\) \(q+\beta q^{2}+(2+\beta )q^{4}-7q^{7}+(10-5\beta )q^{8}+\cdots\)
1575.4.a.w \(2\) \(92.928\) \(\Q(\sqrt{65}) \) None \(1\) \(0\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+\beta q^{2}+(8+\beta )q^{4}+7q^{7}+(2^{4}+\beta )q^{8}+\cdots\)
1575.4.a.x \(2\) \(92.928\) \(\Q(\sqrt{17}) \) None \(3\) \(0\) \(0\) \(14\) \(-\) \(-\) \(-\) \(q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.y \(2\) \(92.928\) \(\Q(\sqrt{41}) \) None \(3\) \(0\) \(0\) \(-14\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{2}+(3+3\beta )q^{4}-7q^{7}+(5^{2}+\cdots)q^{8}+\cdots\)
1575.4.a.z \(2\) \(92.928\) \(\Q(\sqrt{2}) \) None \(8\) \(0\) \(0\) \(14\) \(-\) \(+\) \(-\) \(q+(4+\beta )q^{2}+(10+8\beta )q^{4}+7q^{7}+(24+\cdots)q^{8}+\cdots\)
1575.4.a.ba \(3\) \(92.928\) 3.3.14360.1 None \(-3\) \(0\) \(0\) \(-21\) \(-\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bb \(3\) \(92.928\) 3.3.22952.1 None \(-2\) \(0\) \(0\) \(-21\) \(+\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bc \(3\) \(92.928\) 3.3.2292.1 None \(-1\) \(0\) \(0\) \(-21\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots\)
1575.4.a.bd \(3\) \(92.928\) 3.3.2292.1 None \(1\) \(0\) \(0\) \(21\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots\)
1575.4.a.be \(3\) \(92.928\) 3.3.22952.1 None \(2\) \(0\) \(0\) \(-21\) \(+\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots\)
1575.4.a.bf \(4\) \(92.928\) 4.4.26729725.1 None \(-6\) \(0\) \(0\) \(-28\) \(-\) \(-\) \(+\) \(q+(-1-\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bg \(4\) \(92.928\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-4\) \(0\) \(0\) \(28\) \(-\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1575.4.a.bh \(4\) \(92.928\) 4.4.3030748.1 None \(0\) \(0\) \(0\) \(-28\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}-7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bi \(4\) \(92.928\) 4.4.3030748.1 None \(0\) \(0\) \(0\) \(28\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}+7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bj \(4\) \(92.928\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(-28\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}-7q^{7}+(2+4\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bk \(4\) \(92.928\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(28\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+7q^{7}+(-2+\cdots)q^{8}+\cdots\)
1575.4.a.bl \(4\) \(92.928\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(4\) \(0\) \(0\) \(-28\) \(-\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}-7q^{7}+\cdots\)
1575.4.a.bm \(4\) \(92.928\) 4.4.26729725.1 None \(6\) \(0\) \(0\) \(28\) \(-\) \(+\) \(-\) \(q+(1+\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots\)
1575.4.a.bn \(5\) \(92.928\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-4\) \(0\) \(0\) \(35\) \(-\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1575.4.a.bo \(5\) \(92.928\) 5.5.78066700.1 None \(-1\) \(0\) \(0\) \(35\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}+7q^{7}+\cdots\)
1575.4.a.bp \(5\) \(92.928\) 5.5.78066700.1 None \(1\) \(0\) \(0\) \(-35\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}-7q^{7}+\cdots\)
1575.4.a.bq \(5\) \(92.928\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(0\) \(0\) \(-35\) \(-\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1575.4.a.br \(8\) \(92.928\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-56\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bs \(8\) \(92.928\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(56\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bt \(10\) \(92.928\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(-70\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bu \(10\) \(92.928\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(70\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)

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

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