Properties

Label 450.4.a
Level $450$
Weight $4$
Character orbit 450.a
Rep. character $\chi_{450}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $22$
Sturm bound $360$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(360\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(450))\).

Total New Old
Modular forms 294 24 270
Cusp forms 246 24 222
Eisenstein series 48 0 48

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.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(13\)
Minus space\(-\)\(11\)

Trace form

\( 24 q + 96 q^{4} - 36 q^{7} + O(q^{10}) \) \( 24 q + 96 q^{4} - 36 q^{7} - 78 q^{11} - 84 q^{13} - 8 q^{14} + 384 q^{16} - 132 q^{17} + 110 q^{19} + 372 q^{23} - 416 q^{26} - 144 q^{28} + 360 q^{29} + 148 q^{31} + 188 q^{34} + 588 q^{37} + 192 q^{38} - 138 q^{41} + 492 q^{43} - 312 q^{44} + 536 q^{46} - 444 q^{47} + 672 q^{49} - 336 q^{52} + 564 q^{53} - 32 q^{56} + 336 q^{58} + 960 q^{59} - 1272 q^{61} + 1488 q^{62} + 1536 q^{64} - 732 q^{67} - 528 q^{68} + 3312 q^{71} - 1788 q^{73} - 728 q^{74} + 440 q^{76} - 1968 q^{77} + 1340 q^{79} - 96 q^{82} - 1572 q^{83} + 304 q^{86} + 3690 q^{89} - 992 q^{91} + 1488 q^{92} + 1888 q^{94} - 828 q^{97} - 2496 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(450))\) 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
450.4.a.a $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(-34\) $+$ $-$ $-$ \(q-2q^{2}+4q^{4}-34q^{7}-8q^{8}-3^{3}q^{11}+\cdots\)
450.4.a.b $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(-32\) $+$ $-$ $+$ \(q-2q^{2}+4q^{4}-2^{5}q^{7}-8q^{8}+60q^{11}+\cdots\)
450.4.a.c $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(-14\) $+$ $+$ $+$ \(q-2q^{2}+4q^{4}-14q^{7}-8q^{8}-6q^{11}+\cdots\)
450.4.a.d $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(-11\) $+$ $+$ $+$ \(q-2q^{2}+4q^{4}-11q^{7}-8q^{8}+6^{2}q^{11}+\cdots\)
450.4.a.e $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(-2\) $+$ $-$ $-$ \(q-2q^{2}+4q^{4}-2q^{7}-8q^{8}-70q^{11}+\cdots\)
450.4.a.f $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(1\) $+$ $-$ $+$ \(q-2q^{2}+4q^{4}+q^{7}-8q^{8}-42q^{11}+\cdots\)
450.4.a.g $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(11\) $+$ $+$ $-$ \(q-2q^{2}+4q^{4}+11q^{7}-8q^{8}-6^{2}q^{11}+\cdots\)
450.4.a.h $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(16\) $+$ $-$ $+$ \(q-2q^{2}+4q^{4}+2^{4}q^{7}-8q^{8}-12q^{11}+\cdots\)
450.4.a.i $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(23\) $+$ $-$ $-$ \(q-2q^{2}+4q^{4}+23q^{7}-8q^{8}+30q^{11}+\cdots\)
450.4.a.j $1$ $26.551$ \(\Q\) None \(-2\) \(0\) \(0\) \(26\) $+$ $-$ $-$ \(q-2q^{2}+4q^{4}+26q^{7}-8q^{8}+28q^{11}+\cdots\)
450.4.a.k $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(-26\) $-$ $-$ $-$ \(q+2q^{2}+4q^{4}-26q^{7}+8q^{8}+28q^{11}+\cdots\)
450.4.a.l $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(-23\) $-$ $-$ $+$ \(q+2q^{2}+4q^{4}-23q^{7}+8q^{8}+30q^{11}+\cdots\)
450.4.a.m $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(-14\) $-$ $+$ $+$ \(q+2q^{2}+4q^{4}-14q^{7}+8q^{8}+6q^{11}+\cdots\)
450.4.a.n $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(-11\) $-$ $+$ $+$ \(q+2q^{2}+4q^{4}-11q^{7}+8q^{8}-6^{2}q^{11}+\cdots\)
450.4.a.o $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ \(q+2q^{2}+4q^{4}-q^{7}+8q^{8}-42q^{11}+\cdots\)
450.4.a.p $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(2\) $-$ $-$ $-$ \(q+2q^{2}+4q^{4}+2q^{7}+8q^{8}-70q^{11}+\cdots\)
450.4.a.q $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(4\) $-$ $-$ $+$ \(q+2q^{2}+4q^{4}+4q^{7}+8q^{8}-12q^{11}+\cdots\)
450.4.a.r $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(4\) $-$ $-$ $+$ \(q+2q^{2}+4q^{4}+4q^{7}+8q^{8}+48q^{11}+\cdots\)
450.4.a.s $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(11\) $-$ $+$ $-$ \(q+2q^{2}+4q^{4}+11q^{7}+8q^{8}+6^{2}q^{11}+\cdots\)
450.4.a.t $1$ $26.551$ \(\Q\) None \(2\) \(0\) \(0\) \(34\) $-$ $-$ $+$ \(q+2q^{2}+4q^{4}+34q^{7}+8q^{8}-3^{3}q^{11}+\cdots\)
450.4.a.u $2$ $26.551$ \(\Q(\sqrt{31}) \) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $-$ \(q-2q^{2}+4q^{4}+\beta q^{7}-8q^{8}-\beta q^{11}+\cdots\)
450.4.a.v $2$ $26.551$ \(\Q(\sqrt{31}) \) None \(4\) \(0\) \(0\) \(0\) $-$ $+$ $-$ \(q+2q^{2}+4q^{4}+\beta q^{7}+8q^{8}+\beta q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(450))\) into lower level spaces

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