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

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

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\) into newform subspaces

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

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

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