Properties

Label 1200.4.a
Level $1200$
Weight $4$
Character orbit 1200.a
Rep. character $\chi_{1200}(1,\cdot)$
Character field $\Q$
Dimension $57$
Newform subspaces $47$
Sturm bound $960$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 47 \)
Sturm bound: \(960\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1200))\).

Total New Old
Modular forms 756 57 699
Cusp forms 684 57 627
Eisenstein series 72 0 72

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

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(8\)
\(+\)\(+\)\(-\)\(-\)\(7\)
\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(9\)
\(-\)\(+\)\(+\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(31\)
Minus space\(-\)\(26\)

Trace form

\( 57q - 3q^{3} + 32q^{7} + 513q^{9} + O(q^{10}) \) \( 57q - 3q^{3} + 32q^{7} + 513q^{9} - 20q^{11} + 46q^{13} + 26q^{17} + 192q^{19} - 80q^{23} - 27q^{27} - 58q^{29} - 84q^{31} + 12q^{33} - 74q^{37} + 54q^{39} - 414q^{41} - 836q^{43} - 168q^{47} + 2713q^{49} + 678q^{51} + 1054q^{53} - 84q^{57} - 628q^{59} + 734q^{61} + 288q^{63} + 1140q^{67} + 264q^{69} - 1512q^{71} + 970q^{73} + 768q^{77} - 880q^{79} + 4617q^{81} - 3644q^{83} - 162q^{87} + 1426q^{89} + 2468q^{91} - 456q^{93} + 802q^{97} - 180q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\) 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
1200.4.a.a \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-28\) \(-\) \(+\) \(+\) \(q-3q^{3}-28q^{7}+9q^{9}+24q^{11}+70q^{13}+\cdots\)
1200.4.a.b \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-16\) \(-\) \(+\) \(+\) \(q-3q^{3}-2^{4}q^{7}+9q^{9}-12q^{11}-38q^{13}+\cdots\)
1200.4.a.c \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-16\) \(+\) \(+\) \(+\) \(q-3q^{3}-2^{4}q^{7}+9q^{9}+28q^{11}+26q^{13}+\cdots\)
1200.4.a.d \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-13\) \(-\) \(+\) \(+\) \(q-3q^{3}-13q^{7}+9q^{9}-6q^{11}-5q^{13}+\cdots\)
1200.4.a.e \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-10\) \(+\) \(+\) \(-\) \(q-3q^{3}-10q^{7}+9q^{9}+14q^{11}-82q^{13}+\cdots\)
1200.4.a.f \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-10\) \(+\) \(+\) \(-\) \(q-3q^{3}-10q^{7}+9q^{9}+46q^{11}-34q^{13}+\cdots\)
1200.4.a.g \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-5\) \(+\) \(+\) \(-\) \(q-3q^{3}-5q^{7}+9q^{9}-14q^{11}+q^{13}+\cdots\)
1200.4.a.h \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(q-3q^{3}-2q^{7}+9q^{9}-70q^{11}-54q^{13}+\cdots\)
1200.4.a.i \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-3q^{3}-q^{7}+9q^{9}-42q^{11}+67q^{13}+\cdots\)
1200.4.a.j \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-3q^{3}+9q^{9}-4q^{11}-54q^{13}-114q^{17}+\cdots\)
1200.4.a.k \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(4\) \(+\) \(+\) \(+\) \(q-3q^{3}+4q^{7}+9q^{9}-72q^{11}+6q^{13}+\cdots\)
1200.4.a.l \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(4\) \(+\) \(+\) \(-\) \(q-3q^{3}+4q^{7}+9q^{9}+28q^{11}+2^{4}q^{13}+\cdots\)
1200.4.a.m \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(7\) \(-\) \(+\) \(-\) \(q-3q^{3}+7q^{7}+9q^{9}+54q^{11}-55q^{13}+\cdots\)
1200.4.a.n \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(19\) \(+\) \(+\) \(-\) \(q-3q^{3}+19q^{7}+9q^{9}-22q^{11}+q^{13}+\cdots\)
1200.4.a.o \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(20\) \(-\) \(+\) \(+\) \(q-3q^{3}+20q^{7}+9q^{9}+24q^{11}-74q^{13}+\cdots\)
1200.4.a.p \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(20\) \(+\) \(+\) \(+\) \(q-3q^{3}+20q^{7}+9q^{9}+56q^{11}+86q^{13}+\cdots\)
1200.4.a.q \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(22\) \(-\) \(+\) \(-\) \(q-3q^{3}+22q^{7}+9q^{9}+14q^{11}+30q^{13}+\cdots\)
1200.4.a.r \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(23\) \(-\) \(+\) \(+\) \(q-3q^{3}+23q^{7}+9q^{9}+30q^{11}-29q^{13}+\cdots\)
1200.4.a.s \(1\) \(70.802\) \(\Q\) None \(0\) \(-3\) \(0\) \(32\) \(-\) \(+\) \(+\) \(q-3q^{3}+2^{5}q^{7}+9q^{9}-6^{2}q^{11}+10q^{13}+\cdots\)
1200.4.a.t \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(-24\) \(-\) \(-\) \(+\) \(q+3q^{3}-24q^{7}+9q^{9}-52q^{11}-22q^{13}+\cdots\)
1200.4.a.u \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(-24\) \(+\) \(-\) \(+\) \(q+3q^{3}-24q^{7}+9q^{9}+28q^{11}+74q^{13}+\cdots\)
1200.4.a.v \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(-23\) \(-\) \(-\) \(-\) \(q+3q^{3}-23q^{7}+9q^{9}+30q^{11}+29q^{13}+\cdots\)
1200.4.a.w \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(-22\) \(-\) \(-\) \(-\) \(q+3q^{3}-22q^{7}+9q^{9}+14q^{11}-30q^{13}+\cdots\)
1200.4.a.x \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(-19\) \(+\) \(-\) \(+\) \(q+3q^{3}-19q^{7}+9q^{9}-22q^{11}-q^{13}+\cdots\)
1200.4.a.y \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}-7q^{7}+9q^{9}+54q^{11}+55q^{13}+\cdots\)
1200.4.a.z \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(-4\) \(+\) \(-\) \(-\) \(q+3q^{3}-4q^{7}+9q^{9}+28q^{11}-2^{4}q^{13}+\cdots\)
1200.4.a.ba \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(q+3q^{3}-4q^{7}+9q^{9}+48q^{11}-2q^{13}+\cdots\)
1200.4.a.bb \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+3q^{3}+q^{7}+9q^{9}-42q^{11}-67q^{13}+\cdots\)
1200.4.a.bc \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+3q^{3}+2q^{7}+9q^{9}-70q^{11}+54q^{13}+\cdots\)
1200.4.a.bd \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(5\) \(+\) \(-\) \(+\) \(q+3q^{3}+5q^{7}+9q^{9}-14q^{11}-q^{13}+\cdots\)
1200.4.a.be \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(8\) \(-\) \(-\) \(+\) \(q+3q^{3}+8q^{7}+9q^{9}-6^{2}q^{11}+10q^{13}+\cdots\)
1200.4.a.bf \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(8\) \(+\) \(-\) \(+\) \(q+3q^{3}+8q^{7}+9q^{9}-20q^{11}-22q^{13}+\cdots\)
1200.4.a.bg \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(10\) \(+\) \(-\) \(-\) \(q+3q^{3}+10q^{7}+9q^{9}+14q^{11}+82q^{13}+\cdots\)
1200.4.a.bh \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(10\) \(+\) \(-\) \(-\) \(q+3q^{3}+10q^{7}+9q^{9}+46q^{11}+34q^{13}+\cdots\)
1200.4.a.bi \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(13\) \(-\) \(-\) \(-\) \(q+3q^{3}+13q^{7}+9q^{9}-6q^{11}+5q^{13}+\cdots\)
1200.4.a.bj \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(20\) \(+\) \(-\) \(+\) \(q+3q^{3}+20q^{7}+9q^{9}-2^{4}q^{11}-58q^{13}+\cdots\)
1200.4.a.bk \(1\) \(70.802\) \(\Q\) None \(0\) \(3\) \(0\) \(32\) \(-\) \(-\) \(+\) \(q+3q^{3}+2^{5}q^{7}+9q^{9}+60q^{11}+34q^{13}+\cdots\)
1200.4.a.bl \(2\) \(70.802\) \(\Q(\sqrt{19}) \) None \(0\) \(-6\) \(0\) \(-26\) \(-\) \(+\) \(-\) \(q-3q^{3}+(-13+\beta )q^{7}+9q^{9}+(-14+\cdots)q^{11}+\cdots\)
1200.4.a.bm \(2\) \(70.802\) \(\Q(\sqrt{181}) \) None \(0\) \(-6\) \(0\) \(-6\) \(+\) \(+\) \(+\) \(q-3q^{3}+(-3-\beta )q^{7}+9q^{9}+(-4+\cdots)q^{11}+\cdots\)
1200.4.a.bn \(2\) \(70.802\) \(\Q(\sqrt{41}) \) None \(0\) \(-6\) \(0\) \(-6\) \(-\) \(+\) \(-\) \(q-3q^{3}+(-3-3\beta )q^{7}+9q^{9}+(21+\cdots)q^{11}+\cdots\)
1200.4.a.bo \(2\) \(70.802\) \(\Q(\sqrt{129}) \) None \(0\) \(-6\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q-3q^{3}+(1+3\beta )q^{7}+9q^{9}+(-37+\cdots)q^{11}+\cdots\)
1200.4.a.bp \(2\) \(70.802\) \(\Q(\sqrt{109}) \) None \(0\) \(-6\) \(0\) \(2\) \(+\) \(+\) \(+\) \(q-3q^{3}+(1+\beta )q^{7}+9q^{9}+(8-3\beta )q^{11}+\cdots\)
1200.4.a.bq \(2\) \(70.802\) \(\Q(\sqrt{129}) \) None \(0\) \(6\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(q+3q^{3}+(-1-3\beta )q^{7}+9q^{9}+(-37+\cdots)q^{11}+\cdots\)
1200.4.a.br \(2\) \(70.802\) \(\Q(\sqrt{109}) \) None \(0\) \(6\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(q+3q^{3}+(-1-\beta )q^{7}+9q^{9}+(8-3\beta )q^{11}+\cdots\)
1200.4.a.bs \(2\) \(70.802\) \(\Q(\sqrt{181}) \) None \(0\) \(6\) \(0\) \(6\) \(+\) \(-\) \(-\) \(q+3q^{3}+(3+\beta )q^{7}+9q^{9}+(-4-\beta )q^{11}+\cdots\)
1200.4.a.bt \(2\) \(70.802\) \(\Q(\sqrt{41}) \) None \(0\) \(6\) \(0\) \(6\) \(-\) \(-\) \(-\) \(q+3q^{3}+(3+3\beta )q^{7}+9q^{9}+(21-3\beta )q^{11}+\cdots\)
1200.4.a.bu \(2\) \(70.802\) \(\Q(\sqrt{19}) \) None \(0\) \(6\) \(0\) \(26\) \(-\) \(-\) \(+\) \(q+3q^{3}+(13+\beta )q^{7}+9q^{9}+(-14+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1200))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1200)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)