Properties

Label 400.6.a
Level $400$
Weight $6$
Character orbit 400.a
Rep. character $\chi_{400}(1,\cdot)$
Character field $\Q$
Dimension $46$
Newform subspaces $27$
Sturm bound $360$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 27 \)
Sturm bound: \(360\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(400))\).

Total New Old
Modular forms 318 49 269
Cusp forms 282 46 236
Eisenstein series 36 3 33

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

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(11\)
\(+\)\(-\)\(-\)\(13\)
\(-\)\(+\)\(-\)\(11\)
\(-\)\(-\)\(+\)\(11\)
Plus space\(+\)\(22\)
Minus space\(-\)\(24\)

Trace form

\( 46 q - 10 q^{3} + 110 q^{7} + 3462 q^{9} + O(q^{10}) \) \( 46 q - 10 q^{3} + 110 q^{7} + 3462 q^{9} + 64 q^{11} - 60 q^{13} - 400 q^{17} + 644 q^{19} - 580 q^{21} + 90 q^{23} - 7180 q^{27} - 196 q^{29} + 2868 q^{31} + 5920 q^{33} + 620 q^{37} - 28540 q^{39} - 2540 q^{41} + 20630 q^{43} + 41950 q^{47} + 97098 q^{49} - 22904 q^{51} + 11060 q^{53} + 10040 q^{57} + 38968 q^{59} + 40648 q^{61} + 4190 q^{63} - 64350 q^{67} + 9284 q^{69} - 37300 q^{71} + 43520 q^{73} - 16880 q^{77} - 151360 q^{79} + 307602 q^{81} - 94130 q^{83} - 23580 q^{87} + 58760 q^{89} + 266308 q^{91} - 163560 q^{93} + 32760 q^{97} - 93644 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
400.6.a.a $1$ $64.154$ \(\Q\) None \(0\) \(-26\) \(0\) \(-22\) $-$ $+$ \(q-26q^{3}-22q^{7}+433q^{9}+768q^{11}+\cdots\)
400.6.a.b $1$ $64.154$ \(\Q\) None \(0\) \(-18\) \(0\) \(242\) $+$ $+$ \(q-18q^{3}+242q^{7}+3^{4}q^{9}-656q^{11}+\cdots\)
400.6.a.c $1$ $64.154$ \(\Q\) None \(0\) \(-14\) \(0\) \(-158\) $-$ $-$ \(q-14q^{3}-158q^{7}-47q^{9}+148q^{11}+\cdots\)
400.6.a.d $1$ $64.154$ \(\Q\) None \(0\) \(-12\) \(0\) \(-88\) $-$ $+$ \(q-12q^{3}-88q^{7}-99q^{9}-540q^{11}+\cdots\)
400.6.a.e $1$ $64.154$ \(\Q\) None \(0\) \(-11\) \(0\) \(-142\) $-$ $+$ \(q-11q^{3}-142q^{7}-122q^{9}-777q^{11}+\cdots\)
400.6.a.f $1$ $64.154$ \(\Q\) None \(0\) \(-8\) \(0\) \(-108\) $+$ $+$ \(q-8q^{3}-108q^{7}-179q^{9}+604q^{11}+\cdots\)
400.6.a.g $1$ $64.154$ \(\Q\) None \(0\) \(-4\) \(0\) \(192\) $-$ $+$ \(q-4q^{3}+192q^{7}-227q^{9}+148q^{11}+\cdots\)
400.6.a.h $1$ $64.154$ \(\Q\) None \(0\) \(-2\) \(0\) \(-62\) $+$ $+$ \(q-2q^{3}-62q^{7}-239q^{9}+12^{2}q^{11}+\cdots\)
400.6.a.i $1$ $64.154$ \(\Q\) None \(0\) \(6\) \(0\) \(-118\) $-$ $+$ \(q+6q^{3}-118q^{7}-207q^{9}-192q^{11}+\cdots\)
400.6.a.j $1$ $64.154$ \(\Q\) None \(0\) \(11\) \(0\) \(142\) $-$ $-$ \(q+11q^{3}+142q^{7}-122q^{9}-777q^{11}+\cdots\)
400.6.a.k $1$ $64.154$ \(\Q\) None \(0\) \(14\) \(0\) \(158\) $-$ $-$ \(q+14q^{3}+158q^{7}-47q^{9}+148q^{11}+\cdots\)
400.6.a.l $1$ $64.154$ \(\Q\) None \(0\) \(20\) \(0\) \(-24\) $+$ $+$ \(q+20q^{3}-24q^{7}+157q^{9}-124q^{11}+\cdots\)
400.6.a.m $1$ $64.154$ \(\Q\) None \(0\) \(22\) \(0\) \(218\) $-$ $+$ \(q+22q^{3}+218q^{7}+241q^{9}+480q^{11}+\cdots\)
400.6.a.n $1$ $64.154$ \(\Q\) None \(0\) \(24\) \(0\) \(-172\) $-$ $+$ \(q+24q^{3}-172q^{7}+333q^{9}-132q^{11}+\cdots\)
400.6.a.o $2$ $64.154$ \(\Q(\sqrt{241}) \) None \(0\) \(-20\) \(0\) \(-200\) $-$ $-$ \(q+(-10-\beta )q^{3}+(-10^{2}-2\beta )q^{7}+\cdots\)
400.6.a.p $2$ $64.154$ \(\Q(\sqrt{409}) \) None \(0\) \(-20\) \(0\) \(40\) $-$ $+$ \(q+(-10-\beta )q^{3}+(20+6\beta )q^{7}+(266+\cdots)q^{9}+\cdots\)
400.6.a.q $2$ $64.154$ \(\Q(\sqrt{129}) \) None \(0\) \(-12\) \(0\) \(52\) $+$ $+$ \(q+(-6-\beta )q^{3}+(26-3\beta )q^{7}+(309+\cdots)q^{9}+\cdots\)
400.6.a.r $2$ $64.154$ \(\Q(\sqrt{241}) \) None \(0\) \(-8\) \(0\) \(8\) $+$ $+$ \(q+(-4-\beta )q^{3}+(4+2\beta )q^{7}+(14+8\beta )q^{9}+\cdots\)
400.6.a.s $2$ $64.154$ \(\Q(\sqrt{31}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ \(q+\beta q^{3}-11\beta q^{7}-119q^{9}+10^{2}q^{11}+\cdots\)
400.6.a.t $2$ $64.154$ \(\Q(\sqrt{11}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ \(q+3\beta q^{3}-9\beta q^{7}+153q^{9}-252q^{11}+\cdots\)
400.6.a.u $2$ $64.154$ \(\Q(\sqrt{241}) \) None \(0\) \(8\) \(0\) \(-8\) $+$ $-$ \(q+(4+\beta )q^{3}+(-4-2\beta )q^{7}+(14+8\beta )q^{9}+\cdots\)
400.6.a.v $2$ $64.154$ \(\Q(\sqrt{409}) \) None \(0\) \(20\) \(0\) \(-40\) $-$ $-$ \(q+(10-\beta )q^{3}+(-20+6\beta )q^{7}+(266+\cdots)q^{9}+\cdots\)
400.6.a.w $2$ $64.154$ \(\Q(\sqrt{241}) \) None \(0\) \(20\) \(0\) \(200\) $-$ $+$ \(q+(10-\beta )q^{3}+(10^{2}-2\beta )q^{7}+(98-20\beta )q^{9}+\cdots\)
400.6.a.x $3$ $64.154$ 3.3.47217.1 None \(0\) \(-1\) \(0\) \(70\) $+$ $-$ \(q-\beta _{1}q^{3}+(24-\beta _{1}-\beta _{2})q^{7}+(50+5\beta _{1}+\cdots)q^{9}+\cdots\)
400.6.a.y $3$ $64.154$ 3.3.47217.1 None \(0\) \(1\) \(0\) \(-70\) $+$ $+$ \(q+\beta _{1}q^{3}+(-24+\beta _{1}+\beta _{2})q^{7}+(50+\cdots)q^{9}+\cdots\)
400.6.a.z $4$ $64.154$ 4.4.1595208.1 None \(0\) \(-4\) \(0\) \(-148\) $+$ $-$ \(q+(-1-\beta _{1})q^{3}+(-37-\beta _{1}+\beta _{3})q^{7}+\cdots\)
400.6.a.ba $4$ $64.154$ 4.4.1595208.1 None \(0\) \(4\) \(0\) \(148\) $+$ $-$ \(q+(1+\beta _{1})q^{3}+(37+\beta _{1}-\beta _{3})q^{7}+(5^{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(400)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)