Properties

Label 9075.2.a
Level $9075$
Weight $2$
Character orbit 9075.a
Rep. character $\chi_{9075}(1,\cdot)$
Character field $\Q$
Dimension $345$
Newform subspaces $105$
Sturm bound $2640$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 105 \)
Sturm bound: \(2640\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(2\), \(7\), \(13\), \(17\), \(19\), \(23\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9075))\).

Total New Old
Modular forms 1392 345 1047
Cusp forms 1249 345 904
Eisenstein series 143 0 143

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.
\(+\)\(+\)\(+\)\(+\)\(36\)
\(+\)\(+\)\(-\)\(-\)\(45\)
\(+\)\(-\)\(+\)\(-\)\(46\)
\(+\)\(-\)\(-\)\(+\)\(45\)
\(-\)\(+\)\(+\)\(-\)\(48\)
\(-\)\(+\)\(-\)\(+\)\(35\)
\(-\)\(-\)\(+\)\(+\)\(38\)
\(-\)\(-\)\(-\)\(-\)\(52\)
Plus space\(+\)\(154\)
Minus space\(-\)\(191\)

Trace form

\( 345q - 3q^{2} + q^{3} + 341q^{4} - q^{6} + 4q^{7} - 15q^{8} + 345q^{9} + O(q^{10}) \) \( 345q - 3q^{2} + q^{3} + 341q^{4} - q^{6} + 4q^{7} - 15q^{8} + 345q^{9} + 7q^{12} - 2q^{13} + 345q^{16} - 10q^{17} - 3q^{18} + 14q^{19} + 10q^{21} - 4q^{23} - 9q^{24} + 42q^{26} + q^{27} + 4q^{28} - 6q^{29} - 10q^{31} - 23q^{32} - 10q^{34} + 341q^{36} + 2q^{37} - 4q^{38} + 16q^{39} + 26q^{41} - 16q^{42} - 16q^{46} - 8q^{47} - q^{48} + 363q^{49} + 6q^{51} - 54q^{52} - 30q^{53} - q^{54} - 12q^{56} + 16q^{57} + 6q^{58} + 4q^{59} + 8q^{61} + 32q^{62} + 4q^{63} + 341q^{64} + 24q^{67} + 50q^{68} - 8q^{69} + 12q^{71} - 15q^{72} - 14q^{73} - 22q^{74} + 44q^{76} - 6q^{78} + 4q^{79} + 345q^{81} + 14q^{82} + 28q^{83} + 24q^{84} + 4q^{86} + 2q^{87} - 26q^{89} + 70q^{91} + 8q^{92} + 44q^{93} + 64q^{94} - 9q^{96} + 38q^{97} + 77q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 11
9075.2.a.a \(1\) \(72.464\) \(\Q\) None \(-2\) \(-1\) \(0\) \(3\) \(+\) \(-\) \(-\) \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+3q^{7}+\cdots\)
9075.2.a.b \(1\) \(72.464\) \(\Q\) None \(-2\) \(1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+q^{7}+q^{9}+\cdots\)
9075.2.a.c \(1\) \(72.464\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}-q^{4}+q^{6}-4q^{7}+3q^{8}+\cdots\)
9075.2.a.d \(1\) \(72.464\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}-q^{4}+q^{6}-2q^{7}+3q^{8}+\cdots\)
9075.2.a.e \(1\) \(72.464\) \(\Q\) None \(-1\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}-q^{4}+q^{6}+q^{7}+3q^{8}+\cdots\)
9075.2.a.f \(1\) \(72.464\) \(\Q\) None \(-1\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}-q^{4}-q^{6}-4q^{7}+3q^{8}+\cdots\)
9075.2.a.g \(1\) \(72.464\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
9075.2.a.h \(1\) \(72.464\) \(\Q\) None \(-1\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}-q^{4}-q^{6}+q^{7}+3q^{8}+\cdots\)
9075.2.a.i \(1\) \(72.464\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{4}-q^{7}+q^{9}+2q^{12}-q^{13}+\cdots\)
9075.2.a.j \(1\) \(72.464\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{4}-q^{7}+q^{9}+2q^{12}+2q^{13}+\cdots\)
9075.2.a.k \(1\) \(72.464\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{4}+q^{7}+q^{9}+2q^{12}-2q^{13}+\cdots\)
9075.2.a.l \(1\) \(72.464\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}-2q^{4}+q^{7}+q^{9}-2q^{12}+q^{13}+\cdots\)
9075.2.a.m \(1\) \(72.464\) \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}-q^{4}-q^{6}-q^{7}-3q^{8}+\cdots\)
9075.2.a.n \(1\) \(72.464\) \(\Q\) None \(1\) \(-1\) \(0\) \(2\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}-q^{4}-q^{6}+2q^{7}-3q^{8}+\cdots\)
9075.2.a.o \(1\) \(72.464\) \(\Q\) None \(1\) \(-1\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}-q^{4}-q^{6}+4q^{7}-3q^{8}+\cdots\)
9075.2.a.p \(1\) \(72.464\) \(\Q\) None \(1\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-q^{7}-3q^{8}+\cdots\)
9075.2.a.q \(1\) \(72.464\) \(\Q\) None \(1\) \(1\) \(0\) \(4\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}-q^{4}+q^{6}+4q^{7}-3q^{8}+\cdots\)
9075.2.a.r \(1\) \(72.464\) \(\Q\) None \(1\) \(1\) \(0\) \(4\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}+q^{6}+4q^{7}-3q^{8}+\cdots\)
9075.2.a.s \(1\) \(72.464\) \(\Q\) None \(2\) \(1\) \(0\) \(-3\) \(-\) \(+\) \(-\) \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}-3q^{7}+\cdots\)
9075.2.a.t \(1\) \(72.464\) \(\Q\) None \(2\) \(1\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}-q^{7}+q^{9}+\cdots\)
9075.2.a.u \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-3\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{6}+\cdots\)
9075.2.a.v \(2\) \(72.464\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
9075.2.a.w \(2\) \(72.464\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
9075.2.a.x \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(-6\) \(+\) \(+\) \(+\) \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
9075.2.a.y \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(-6\) \(+\) \(-\) \(+\) \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
9075.2.a.z \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(-6\) \(+\) \(-\) \(+\) \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
9075.2.a.ba \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
9075.2.a.bb \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-1\) \(2\) \(0\) \(-6\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
9075.2.a.bc \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-1\) \(2\) \(0\) \(-6\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
9075.2.a.bd \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-1\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
9075.2.a.be \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(-1\) \(2\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
9075.2.a.bf \(2\) \(72.464\) \(\Q(\sqrt{15}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-q^{3}-2q^{4}+\beta q^{7}+q^{9}+2q^{12}+\cdots\)
9075.2.a.bg \(2\) \(72.464\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+\beta q^{2}-q^{3}+q^{4}-\beta q^{6}-2\beta q^{7}+\cdots\)
9075.2.a.bh \(2\) \(72.464\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(4\) \(+\) \(+\) \(-\) \(q+\beta q^{2}-q^{3}+q^{4}-\beta q^{6}+2q^{7}-\beta q^{8}+\cdots\)
9075.2.a.bi \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-\beta q^{2}-q^{3}+3q^{4}+\beta q^{6}+2\beta q^{7}+\cdots\)
9075.2.a.bj \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-\beta q^{2}-q^{3}+3q^{4}+\beta q^{6}-\beta q^{8}+\cdots\)
9075.2.a.bk \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q-\beta q^{2}-q^{3}+3q^{4}+\beta q^{6}-\beta q^{7}+\cdots\)
9075.2.a.bl \(2\) \(72.464\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{3}-2q^{4}+\beta q^{7}+q^{9}-2q^{12}+\cdots\)
9075.2.a.bm \(2\) \(72.464\) \(\Q(\sqrt{15}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+q^{3}-2q^{4}+\beta q^{7}+q^{9}-2q^{12}+\cdots\)
9075.2.a.bn \(2\) \(72.464\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+q^{3}+q^{4}+\beta q^{6}-2\beta q^{7}+\cdots\)
9075.2.a.bo \(2\) \(72.464\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+q^{3}+q^{4}+\beta q^{6}-2\beta q^{7}+\cdots\)
9075.2.a.bp \(2\) \(72.464\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+q^{3}+q^{4}+\beta q^{6}-\beta q^{8}+q^{9}+\cdots\)
9075.2.a.bq \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q-\beta q^{2}+q^{3}+3q^{4}-\beta q^{6}-\beta q^{8}+\cdots\)
9075.2.a.br \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-\beta q^{2}+q^{3}+3q^{4}-\beta q^{6}-\beta q^{7}+\cdots\)
9075.2.a.bs \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
9075.2.a.bt \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(0\) \(6\) \(+\) \(-\) \(-\) \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
9075.2.a.bu \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(0\) \(6\) \(+\) \(-\) \(-\) \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
9075.2.a.bv \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(0\) \(6\) \(+\) \(+\) \(-\) \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
9075.2.a.bw \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
9075.2.a.bx \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
9075.2.a.by \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(6\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
9075.2.a.bz \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(6\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
9075.2.a.ca \(2\) \(72.464\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
9075.2.a.cb \(2\) \(72.464\) \(\Q(\sqrt{5}) \) None \(3\) \(2\) \(0\) \(2\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1+\beta )q^{6}+\cdots\)
9075.2.a.cc \(3\) \(72.464\) 3.3.148.1 None \(-3\) \(-3\) \(0\) \(-8\) \(+\) \(-\) \(-\) \(q+(-1-\beta _{2})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
9075.2.a.cd \(3\) \(72.464\) 3.3.568.1 None \(-2\) \(3\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+q^{3}+(3+\beta _{2})q^{4}+\cdots\)
9075.2.a.ce \(3\) \(72.464\) 3.3.469.1 None \(-1\) \(-3\) \(0\) \(-3\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.cf \(3\) \(72.464\) 3.3.148.1 None \(-1\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.cg \(3\) \(72.464\) 3.3.148.1 None \(-1\) \(3\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.ch \(3\) \(72.464\) 3.3.148.1 None \(1\) \(-3\) \(0\) \(4\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.ci \(3\) \(72.464\) 3.3.469.1 None \(1\) \(-3\) \(0\) \(3\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.cj \(3\) \(72.464\) 3.3.568.1 None \(2\) \(-3\) \(0\) \(-3\) \(+\) \(+\) \(-\) \(q+(1-\beta _{1})q^{2}-q^{3}+(3+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
9075.2.a.ck \(3\) \(72.464\) 3.3.148.1 None \(3\) \(3\) \(0\) \(8\) \(-\) \(-\) \(-\) \(q+(1+\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
9075.2.a.cl \(4\) \(72.464\) 4.4.725.1 None \(-5\) \(-4\) \(0\) \(2\) \(+\) \(+\) \(+\) \(q+(-2+\beta _{1}+\beta _{2})q^{2}-q^{3}+(3-\beta _{1}+\cdots)q^{4}+\cdots\)
9075.2.a.cm \(4\) \(72.464\) 4.4.725.1 None \(-3\) \(-4\) \(0\) \(-6\) \(+\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{2}-q^{3}+(-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
9075.2.a.cn \(4\) \(72.464\) 4.4.5725.1 None \(-1\) \(-4\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q+(\beta _{1}-\beta _{2})q^{2}-q^{3}+(3-\beta _{2}-\beta _{3})q^{4}+\cdots\)
9075.2.a.co \(4\) \(72.464\) \(\Q(\zeta_{15})^+\) None \(-1\) \(4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+(\beta _{2}+\beta _{3})q^{2}+q^{3}+\beta _{1}q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots\)
9075.2.a.cp \(4\) \(72.464\) 4.4.5725.1 None \(-1\) \(4\) \(0\) \(-4\) \(-\) \(+\) \(+\) \(q+(\beta _{1}-\beta _{2})q^{2}+q^{3}+(3-\beta _{2}-\beta _{3})q^{4}+\cdots\)
9075.2.a.cq \(4\) \(72.464\) 4.4.5725.1 None \(-1\) \(4\) \(0\) \(8\) \(-\) \(+\) \(+\) \(q+(\beta _{1}-\beta _{2})q^{2}+q^{3}+(3-\beta _{2}-\beta _{3})q^{4}+\cdots\)
9075.2.a.cr \(4\) \(72.464\) \(\Q(\zeta_{24})^+\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+\beta _{2}q^{2}-q^{3}-\beta _{2}q^{6}-\beta _{1}q^{7}-2\beta _{2}q^{8}+\cdots\)
9075.2.a.cs \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}-2\beta _{3}q^{7}+\cdots\)
9075.2.a.ct \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}+\beta _{1}q^{7}+\cdots\)
9075.2.a.cu \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+\beta _{3}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{3}q^{6}-2\beta _{1}q^{7}+\cdots\)
9075.2.a.cv \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{11})\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{3})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.cw \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{11})\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{3})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.cx \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{5})\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+(\beta _{1}+\beta _{2})q^{2}-q^{3}+(2+\beta _{3})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
9075.2.a.cy \(4\) \(72.464\) \(\Q(\zeta_{24})^+\) None \(0\) \(4\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta _{2}q^{2}+q^{3}+\beta _{2}q^{6}-\beta _{1}q^{7}-2\beta _{2}q^{8}+\cdots\)
9075.2.a.cz \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(4\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta _{3}q^{2}+q^{3}-\beta _{2}q^{4}+\beta _{3}q^{6}-2\beta _{3}q^{7}+\cdots\)
9075.2.a.da \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(4\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta _{3}q^{2}+q^{3}-\beta _{2}q^{4}+\beta _{3}q^{6}+\beta _{1}q^{7}+\cdots\)
9075.2.a.db \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{11})\) None \(0\) \(4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{3})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.dc \(4\) \(72.464\) 4.4.8112.1 None \(0\) \(4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-\beta _{2}q^{2}+q^{3}+(2-\beta _{3})q^{4}-\beta _{2}q^{6}+\cdots\)
9075.2.a.dd \(4\) \(72.464\) \(\Q(\sqrt{3}, \sqrt{5})\) None \(0\) \(4\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+(\beta _{1}+\beta _{2})q^{2}+q^{3}+(2+\beta _{3})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
9075.2.a.de \(4\) \(72.464\) 4.4.5725.1 None \(1\) \(-4\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q+(-\beta _{1}+\beta _{2})q^{2}-q^{3}+(3-\beta _{2}-\beta _{3})q^{4}+\cdots\)
9075.2.a.df \(4\) \(72.464\) \(\Q(\zeta_{15})^+\) None \(1\) \(4\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+(-\beta _{2}-\beta _{3})q^{2}+q^{3}+\beta _{1}q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
9075.2.a.dg \(4\) \(72.464\) 4.4.5725.1 None \(1\) \(4\) \(0\) \(-8\) \(-\) \(+\) \(-\) \(q+(-\beta _{1}+\beta _{2})q^{2}+q^{3}+(3-\beta _{2}-\beta _{3})q^{4}+\cdots\)
9075.2.a.dh \(4\) \(72.464\) 4.4.5725.1 None \(1\) \(4\) \(0\) \(4\) \(-\) \(+\) \(-\) \(q+(-\beta _{1}+\beta _{2})q^{2}+q^{3}+(3-\beta _{2}-\beta _{3})q^{4}+\cdots\)
9075.2.a.di \(4\) \(72.464\) 4.4.725.1 None \(3\) \(-4\) \(0\) \(6\) \(+\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}-q^{3}+(-\beta _{1}+\beta _{2})q^{4}+\cdots\)
9075.2.a.dj \(4\) \(72.464\) 4.4.725.1 None \(5\) \(-4\) \(0\) \(-2\) \(+\) \(+\) \(-\) \(q+(2-\beta _{1}-\beta _{2})q^{2}-q^{3}+(3-\beta _{1}-2\beta _{2}+\cdots)q^{4}+\cdots\)
9075.2.a.dk \(5\) \(72.464\) 5.5.9444552.1 None \(-1\) \(-5\) \(0\) \(4\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.dl \(5\) \(72.464\) 5.5.9444552.1 None \(-1\) \(5\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.dm \(5\) \(72.464\) 5.5.9444552.1 None \(1\) \(-5\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.dn \(5\) \(72.464\) 5.5.9444552.1 None \(1\) \(5\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.do \(6\) \(72.464\) 6.6.860280160.1 None \(-1\) \(-6\) \(0\) \(4\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.dp \(6\) \(72.464\) 6.6.860280160.1 None \(-1\) \(6\) \(0\) \(4\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.dq \(6\) \(72.464\) 6.6.437199552.1 None \(0\) \(6\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.dr \(6\) \(72.464\) 6.6.860280160.1 None \(1\) \(-6\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.ds \(6\) \(72.464\) 6.6.860280160.1 None \(1\) \(6\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.dt \(8\) \(72.464\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(-8\) \(0\) \(-8\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.du \(8\) \(72.464\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(8\) \(0\) \(-8\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.dv \(8\) \(72.464\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(-8\) \(0\) \(8\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.dw \(8\) \(72.464\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(8\) \(0\) \(8\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.dx \(12\) \(72.464\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-4\) \(-12\) \(0\) \(-8\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
9075.2.a.dy \(12\) \(72.464\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-4\) \(12\) \(0\) \(-8\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.dz \(12\) \(72.464\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(4\) \(-12\) \(0\) \(8\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
9075.2.a.ea \(12\) \(72.464\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(4\) \(12\) \(0\) \(8\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9075))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9075)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(605))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(825))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\)\(^{\oplus 2}\)