Properties

Label 8400.2.a
Level $8400$
Weight $2$
Character orbit 8400.a
Rep. character $\chi_{8400}(1,\cdot)$
Character field $\Q$
Dimension $114$
Newform subspaces $90$
Sturm bound $3840$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 90 \)
Sturm bound: \(3840\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8400))\).

Total New Old
Modular forms 1992 114 1878
Cusp forms 1849 114 1735
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.

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(7\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(7\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(51\)
Minus space\(-\)\(63\)

Trace form

\( 114q + 2q^{7} + 114q^{9} + O(q^{10}) \) \( 114q + 2q^{7} + 114q^{9} + 4q^{11} - 4q^{13} + 4q^{17} - 8q^{19} + 4q^{23} - 12q^{29} - 8q^{31} - 12q^{37} - 8q^{39} + 4q^{41} - 24q^{43} - 24q^{47} + 114q^{49} + 4q^{51} - 12q^{53} - 24q^{59} + 12q^{61} + 2q^{63} + 16q^{67} + 16q^{69} - 20q^{71} + 20q^{73} - 8q^{77} - 48q^{79} + 114q^{81} - 64q^{83} + 24q^{87} + 36q^{89} + 12q^{91} - 8q^{93} + 20q^{97} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\) 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
8400.2.a.a \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{7}+q^{9}-6q^{11}-q^{13}+3q^{17}+\cdots\)
8400.2.a.b \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.c \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}-2q^{11}-4q^{13}+\cdots\)
8400.2.a.d \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}-2q^{11}-q^{13}+3q^{17}+\cdots\)
8400.2.a.e \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
8400.2.a.f \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}-2q^{13}-2q^{17}+\cdots\)
8400.2.a.g \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}+q^{11}-4q^{13}-2q^{17}+\cdots\)
8400.2.a.h \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}+q^{11}+2q^{13}-6q^{19}+\cdots\)
8400.2.a.i \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{7}+q^{9}+3q^{11}-4q^{13}+\cdots\)
8400.2.a.j \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{7}+q^{9}+3q^{11}+2q^{13}+\cdots\)
8400.2.a.k \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
8400.2.a.l \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
8400.2.a.m \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
8400.2.a.n \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}-5q^{11}-2q^{13}+\cdots\)
8400.2.a.o \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
8400.2.a.p \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.q \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.r \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}-3q^{11}-2q^{17}+\cdots\)
8400.2.a.s \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
8400.2.a.t \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}-2q^{11}+q^{13}-q^{17}+\cdots\)
8400.2.a.u \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
8400.2.a.v \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
8400.2.a.w \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
8400.2.a.x \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}-q^{11}-4q^{13}-2q^{17}+\cdots\)
8400.2.a.y \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}-6q^{13}+2q^{17}+\cdots\)
8400.2.a.z \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+2q^{13}-6q^{17}+\cdots\)
8400.2.a.ba \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}+q^{11}-2q^{13}+8q^{17}+\cdots\)
8400.2.a.bb \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+2q^{11}-7q^{13}+\cdots\)
8400.2.a.bc \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+2q^{11}-4q^{13}+\cdots\)
8400.2.a.bd \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
8400.2.a.be \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+2q^{11}+5q^{13}+\cdots\)
8400.2.a.bf \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+3q^{11}+2q^{13}+\cdots\)
8400.2.a.bg \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
8400.2.a.bh \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
8400.2.a.bi \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}+5q^{11}+4q^{13}+\cdots\)
8400.2.a.bj \(1\) \(67.074\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
8400.2.a.bk \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}-5q^{11}+2q^{13}+\cdots\)
8400.2.a.bl \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.bm \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.bn \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
8400.2.a.bo \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}-3q^{11}+2q^{17}+\cdots\)
8400.2.a.bp \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)
8400.2.a.bq \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}-2q^{11}-4q^{13}+\cdots\)
8400.2.a.br \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
8400.2.a.bs \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
8400.2.a.bt \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}-2q^{11}-q^{13}+q^{17}+\cdots\)
8400.2.a.bu \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}-2q^{11}+3q^{13}+\cdots\)
8400.2.a.bv \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}-q^{11}+4q^{13}+2q^{17}+\cdots\)
8400.2.a.bw \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}-6q^{13}+2q^{17}+\cdots\)
8400.2.a.bx \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}+2q^{13}-6q^{17}+\cdots\)
8400.2.a.by \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}+q^{11}+2q^{13}-8q^{17}+\cdots\)
8400.2.a.bz \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+2q^{11}-5q^{13}+\cdots\)
8400.2.a.ca \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
8400.2.a.cb \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+2q^{11}+7q^{13}+\cdots\)
8400.2.a.cc \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
8400.2.a.cd \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
8400.2.a.ce \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
8400.2.a.cf \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
8400.2.a.cg \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-q^{7}+q^{9}+5q^{11}-4q^{13}+\cdots\)
8400.2.a.ch \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+6q^{11}+2q^{13}+\cdots\)
8400.2.a.ci \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}-6q^{11}+q^{13}-3q^{17}+\cdots\)
8400.2.a.cj \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}-6q^{11}+4q^{13}+\cdots\)
8400.2.a.ck \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}-2q^{11}+q^{13}-3q^{17}+\cdots\)
8400.2.a.cl \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}-2q^{13}-2q^{17}+\cdots\)
8400.2.a.cm \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}-2q^{13}+6q^{17}+\cdots\)
8400.2.a.cn \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}-2q^{13}+6q^{17}+\cdots\)
8400.2.a.co \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}+6q^{13}-2q^{17}+\cdots\)
8400.2.a.cp \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}+q^{11}-2q^{13}-6q^{19}+\cdots\)
8400.2.a.cq \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}+q^{11}+4q^{13}+2q^{17}+\cdots\)
8400.2.a.cr \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
8400.2.a.cs \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
8400.2.a.ct \(1\) \(67.074\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
8400.2.a.cu \(2\) \(67.074\) \(\Q(\sqrt{17}) \) None \(0\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{7}+q^{9}+(-1-\beta )q^{11}+(2+\cdots)q^{13}+\cdots\)
8400.2.a.cv \(2\) \(67.074\) \(\Q(\sqrt{73}) \) None \(0\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}-\beta q^{11}+(1+\beta )q^{13}+\cdots\)
8400.2.a.cw \(2\) \(67.074\) \(\Q(\sqrt{13}) \) None \(0\) \(-2\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{7}+q^{9}+3q^{11}+(1+\beta )q^{13}+\cdots\)
8400.2.a.cx \(2\) \(67.074\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+(-2-\beta )q^{11}-\beta q^{13}+\cdots\)
8400.2.a.cy \(2\) \(67.074\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+(1+2\beta )q^{11}+(3+\cdots)q^{13}+\cdots\)
8400.2.a.cz \(2\) \(67.074\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(2\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}+2q^{11}-\beta q^{13}+\cdots\)
8400.2.a.da \(2\) \(67.074\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+(1+2\beta )q^{11}+(-3+\cdots)q^{13}+\cdots\)
8400.2.a.db \(2\) \(67.074\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+2q^{11}-\beta q^{13}+\cdots\)
8400.2.a.dc \(2\) \(67.074\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(0\) \(2\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}+(-1-\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
8400.2.a.dd \(2\) \(67.074\) \(\Q(\sqrt{73}) \) None \(0\) \(2\) \(0\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}-\beta q^{11}+(-1-\beta )q^{13}+\cdots\)
8400.2.a.de \(2\) \(67.074\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(2\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}+\beta q^{11}-2q^{13}+\cdots\)
8400.2.a.df \(2\) \(67.074\) \(\Q(\sqrt{13}) \) None \(0\) \(2\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{7}+q^{9}+3q^{11}+(-1+\beta )q^{13}+\cdots\)
8400.2.a.dg \(3\) \(67.074\) 3.3.148.1 None \(0\) \(-3\) \(0\) \(-3\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{7}+q^{9}-2q^{11}+(2-\beta _{2})q^{13}+\cdots\)
8400.2.a.dh \(3\) \(67.074\) 3.3.148.1 None \(0\) \(-3\) \(0\) \(-3\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{7}+q^{9}+(-1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
8400.2.a.di \(3\) \(67.074\) 3.3.148.1 None \(0\) \(-3\) \(0\) \(-3\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{7}+q^{9}+(1+\beta _{1}-\beta _{2})q^{11}+\cdots\)
8400.2.a.dj \(3\) \(67.074\) 3.3.148.1 None \(0\) \(3\) \(0\) \(3\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}-2q^{11}+(-2+\beta _{2})q^{13}+\cdots\)
8400.2.a.dk \(3\) \(67.074\) 3.3.148.1 None \(0\) \(3\) \(0\) \(3\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}+(-1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
8400.2.a.dl \(3\) \(67.074\) 3.3.148.1 None \(0\) \(3\) \(0\) \(3\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}+(1+\beta _{1}-\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8400)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(840))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1680))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4200))\)\(^{\oplus 2}\)