Properties

Label 900.6.a
Level $900$
Weight $6$
Character orbit 900.a
Rep. character $\chi_{900}(1,\cdot)$
Character field $\Q$
Dimension $39$
Newform subspaces $24$
Sturm bound $1080$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 900.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(1080\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(900))\).

Total New Old
Modular forms 936 39 897
Cusp forms 864 39 825
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.
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(13\)
Plus space\(+\)\(18\)
Minus space\(-\)\(21\)

Trace form

\( 39 q - 46 q^{7} + O(q^{10}) \) \( 39 q - 46 q^{7} - 96 q^{11} + 398 q^{13} - 900 q^{17} + 2628 q^{19} - 1050 q^{23} - 3192 q^{29} - 588 q^{31} - 11230 q^{37} - 17268 q^{41} - 11698 q^{43} - 11550 q^{47} + 69975 q^{49} + 3120 q^{53} + 46980 q^{59} + 8418 q^{61} + 51506 q^{67} - 9084 q^{71} + 110 q^{73} - 89760 q^{77} - 19548 q^{79} - 7710 q^{83} + 68088 q^{89} + 110352 q^{91} + 177086 q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(900))\) 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
900.6.a.a $1$ $144.345$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-236\) $-$ $+$ $+$ \(q-236q^{7}-1202q^{13}-1432q^{19}+\cdots\)
900.6.a.b $1$ $144.345$ \(\Q\) None \(0\) \(0\) \(0\) \(-218\) $-$ $-$ $+$ \(q-218q^{7}+480q^{11}+622q^{13}+186q^{17}+\cdots\)
900.6.a.c $1$ $144.345$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-211\) $-$ $+$ $-$ \(q-211q^{7}-427q^{13}+3143q^{19}+\cdots\)
900.6.a.d $1$ $144.345$ \(\Q\) None \(0\) \(0\) \(0\) \(-91\) $-$ $-$ $-$ \(q-91q^{7}+174q^{11}+785q^{13}-1794q^{17}+\cdots\)
900.6.a.e $1$ $144.345$ \(\Q\) None \(0\) \(0\) \(0\) \(-56\) $-$ $-$ $+$ \(q-56q^{7}-156q^{11}-350q^{13}+786q^{17}+\cdots\)
900.6.a.f $1$ $144.345$ \(\Q\) None \(0\) \(0\) \(0\) \(-44\) $-$ $-$ $+$ \(q-44q^{7}-6^{3}q^{11}-770q^{13}+534q^{17}+\cdots\)
900.6.a.g $1$ $144.345$ \(\Q\) None \(0\) \(0\) \(0\) \(16\) $-$ $-$ $+$ \(q+2^{4}q^{7}+564q^{11}+370q^{13}-1086q^{17}+\cdots\)
900.6.a.h $1$ $144.345$ \(\Q\) None \(0\) \(0\) \(0\) \(88\) $-$ $-$ $+$ \(q+88q^{7}-540q^{11}+418q^{13}+594q^{17}+\cdots\)
900.6.a.i $1$ $144.345$ \(\Q\) None \(0\) \(0\) \(0\) \(91\) $-$ $-$ $+$ \(q+91q^{7}+174q^{11}-785q^{13}+1794q^{17}+\cdots\)
900.6.a.j $1$ $144.345$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(211\) $-$ $+$ $+$ \(q+211q^{7}+427q^{13}+3143q^{19}+\cdots\)
900.6.a.k $1$ $144.345$ \(\Q\) None \(0\) \(0\) \(0\) \(244\) $-$ $-$ $+$ \(q+244q^{7}+12^{2}q^{11}-50q^{13}-1914q^{17}+\cdots\)
900.6.a.l $2$ $144.345$ \(\Q(\sqrt{94}) \) None \(0\) \(0\) \(0\) \(-142\) $-$ $+$ $+$ \(q-71q^{7}+\beta q^{11}-137q^{13}+\beta q^{17}+\cdots\)
900.6.a.m $2$ $144.345$ \(\Q(\sqrt{409}) \) None \(0\) \(0\) \(0\) \(-40\) $-$ $-$ $+$ \(q+(-20-2\beta )q^{7}+(30-5\beta )q^{11}+(460+\cdots)q^{13}+\cdots\)
900.6.a.n $2$ $144.345$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(-22\) $-$ $-$ $+$ \(q+(-11+\beta )q^{7}+(-186-3\beta )q^{11}+\cdots\)
900.6.a.o $2$ $144.345$ \(\Q(\sqrt{61}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ \(q-8\beta q^{7}-80q^{11}+24\beta q^{13}+35\beta q^{17}+\cdots\)
900.6.a.p $2$ $144.345$ \(\Q(\sqrt{61}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ \(q-8\beta q^{7}+80q^{11}+24\beta q^{13}-35\beta q^{17}+\cdots\)
900.6.a.q $2$ $144.345$ \(\Q(\sqrt{31}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $-$ \(q+11\beta q^{7}+10^{2}q^{11}+66\beta q^{13}+88\beta q^{17}+\cdots\)
900.6.a.r $2$ $144.345$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(22\) $-$ $-$ $-$ \(q+(11+\beta )q^{7}+(-186+3\beta )q^{11}+(5+\cdots)q^{13}+\cdots\)
900.6.a.s $2$ $144.345$ \(\Q(\sqrt{409}) \) None \(0\) \(0\) \(0\) \(40\) $-$ $-$ $-$ \(q+(20+2\beta )q^{7}+(30-5\beta )q^{11}+(-460+\cdots)q^{13}+\cdots\)
900.6.a.t $2$ $144.345$ \(\Q(\sqrt{241}) \) None \(0\) \(0\) \(0\) \(80\) $-$ $+$ $+$ \(q+(40-\beta )q^{7}+(-120-5\beta )q^{11}+(340+\cdots)q^{13}+\cdots\)
900.6.a.u $2$ $144.345$ \(\Q(\sqrt{241}) \) None \(0\) \(0\) \(0\) \(80\) $-$ $+$ $+$ \(q+(40-\beta )q^{7}+(120+5\beta )q^{11}+(340+\cdots)q^{13}+\cdots\)
900.6.a.v $2$ $144.345$ \(\Q(\sqrt{94}) \) None \(0\) \(0\) \(0\) \(142\) $-$ $+$ $-$ \(q+71q^{7}-\beta q^{11}+137q^{13}+\beta q^{17}+\cdots\)
900.6.a.w $3$ $144.345$ 3.3.535753.1 None \(0\) \(0\) \(0\) \(-88\) $-$ $-$ $-$ \(q+(-29+\beta _{2})q^{7}+(-48+\beta _{1}+4\beta _{2})q^{11}+\cdots\)
900.6.a.x $3$ $144.345$ 3.3.535753.1 None \(0\) \(0\) \(0\) \(88\) $-$ $-$ $-$ \(q+(29-\beta _{2})q^{7}+(-48+\beta _{1}+4\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(900))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(900)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 18}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 18}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)