Properties

Label 7350.2.a
Level $7350$
Weight $2$
Character orbit 7350.a
Rep. character $\chi_{7350}(1,\cdot)$
Character field $\Q$
Dimension $131$
Newform subspaces $99$
Sturm bound $3360$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 99 \)
Sturm bound: \(3360\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\), \(23\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7350))\).

Total New Old
Modular forms 1776 131 1645
Cusp forms 1585 131 1454
Eisenstein series 191 0 191

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

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(10\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(7\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(9\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(7\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(8\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(10\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(10\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(58\)
Minus space\(-\)\(73\)

Trace form

\( 131q + q^{2} + q^{3} + 131q^{4} - q^{6} + q^{8} + 131q^{9} + O(q^{10}) \) \( 131q + q^{2} + q^{3} + 131q^{4} - q^{6} + q^{8} + 131q^{9} - 12q^{11} + q^{12} + 6q^{13} + 131q^{16} - 6q^{17} + q^{18} + 4q^{19} - 4q^{22} - 24q^{23} - q^{24} - 22q^{26} + q^{27} - 30q^{29} - 8q^{31} + q^{32} - 10q^{34} + 131q^{36} - 42q^{37} - 12q^{38} - 10q^{39} - 50q^{41} - 44q^{43} - 12q^{44} - 16q^{47} + q^{48} - 6q^{51} + 6q^{52} + 38q^{53} - q^{54} - 20q^{57} + 30q^{58} - 4q^{59} - 14q^{61} - 8q^{62} + 131q^{64} - 4q^{66} - 36q^{67} - 6q^{68} - 8q^{69} - 16q^{71} + q^{72} + 6q^{73} + 10q^{74} + 4q^{76} - 6q^{78} - 20q^{79} + 131q^{81} + 10q^{82} + 28q^{83} - 12q^{86} - 22q^{87} - 4q^{88} + 14q^{89} - 24q^{92} - 16q^{93} - 16q^{94} - q^{96} + 6q^{97} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 7
7350.2.a.a \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.b \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.c \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.d \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.e \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.f \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.g \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.h \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.i \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.j \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.k \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.l \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.m \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.n \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.o \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.p \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.q \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.r \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.s \(1\) \(58.690\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.t \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.u \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.v \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.w \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.x \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.y \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.z \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.ba \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bb \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bc \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bd \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.be \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bf \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bg \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bh \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bi \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bj \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bk \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bl \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bm \(1\) \(58.690\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.bn \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bo \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bp \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bq \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.br \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bs \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bt \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bu \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bv \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bw \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bx \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.by \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.bz \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.ca \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cb \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cc \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cd \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.ce \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cf \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cg \(1\) \(58.690\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.ch \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.ci \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cj \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.ck \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cl \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cm \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cn \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.co \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cp \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cq \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cr \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cs \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.ct \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cu \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cv \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cw \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cx \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cy \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.cz \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.da \(1\) \(58.690\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.db \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.dc \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.dd \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.de \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.df \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.dg \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.dh \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.di \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.dj \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.dk \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.dl \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.dm \(2\) \(58.690\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.dn \(3\) \(58.690\) 3.3.2700.1 None \(-3\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.do \(3\) \(58.690\) 3.3.2700.1 None \(-3\) \(3\) \(0\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.dp \(3\) \(58.690\) 3.3.2700.1 None \(3\) \(-3\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.dq \(3\) \(58.690\) 3.3.2700.1 None \(3\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.dr \(4\) \(58.690\) 4.4.10304.1 None \(-4\) \(-4\) \(0\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.ds \(4\) \(58.690\) 4.4.10304.1 None \(-4\) \(4\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
7350.2.a.dt \(4\) \(58.690\) 4.4.10304.1 None \(4\) \(-4\) \(0\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
7350.2.a.du \(4\) \(58.690\) 4.4.10304.1 None \(4\) \(4\) \(0\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7350))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7350)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(735))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1470))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\)\(^{\oplus 2}\)