Properties

Label 450.6.a
Level $450$
Weight $6$
Character orbit 450.a
Rep. character $\chi_{450}(1,\cdot)$
Character field $\Q$
Dimension $40$
Newform subspaces $32$
Sturm bound $540$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 32 \)
Sturm bound: \(540\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(450))\).

Total New Old
Modular forms 474 40 434
Cusp forms 426 40 386
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
\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(6\)
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(4\)
\(-\)\(-\)\(+\)$+$\(5\)
\(-\)\(-\)\(-\)$-$\(7\)
Plus space\(+\)\(19\)
Minus space\(-\)\(21\)

Trace form

\( 40 q + 640 q^{4} + 40 q^{7} + O(q^{10}) \) \( 40 q + 640 q^{4} + 40 q^{7} - 162 q^{11} - 140 q^{13} + 880 q^{14} + 10240 q^{16} - 2640 q^{17} - 4246 q^{19} - 2280 q^{23} + 6592 q^{26} + 640 q^{28} - 5652 q^{29} + 10772 q^{31} - 19704 q^{34} - 2420 q^{37} - 19200 q^{38} + 33666 q^{41} + 12040 q^{43} - 2592 q^{44} - 40656 q^{46} + 37200 q^{47} + 162336 q^{49} - 2240 q^{52} - 77640 q^{53} + 14080 q^{56} + 21600 q^{58} + 1200 q^{59} - 50536 q^{61} - 17760 q^{62} + 163840 q^{64} + 1000 q^{67} - 42240 q^{68} + 37608 q^{71} + 141940 q^{73} + 57328 q^{74} - 67936 q^{76} + 79440 q^{77} - 57940 q^{79} + 8640 q^{82} - 180480 q^{83} + 279904 q^{86} + 198606 q^{89} - 452000 q^{91} - 36480 q^{92} - 50112 q^{94} - 99740 q^{97} - 44160 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\) 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
450.6.a.a 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(-233\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-233q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.b 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(-164\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-164q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.c 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(-158\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-158q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.d 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(-98\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-98q^{7}-2^{6}q^{8}+354q^{11}+\cdots\)
450.6.a.e 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(-79\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-79q^{7}-2^{6}q^{8}-150q^{11}+\cdots\)
450.6.a.f 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-4q^{7}-2^{6}q^{8}+500q^{11}+\cdots\)
450.6.a.g 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+q^{7}-2^{6}q^{8}+210q^{11}+\cdots\)
450.6.a.h 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(22\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+22q^{7}-2^{6}q^{8}+768q^{11}+\cdots\)
450.6.a.i 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(47\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+47q^{7}-2^{6}q^{8}-222q^{11}+\cdots\)
450.6.a.j 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(142\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+142q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.k 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(148\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+148q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.l 450.a 1.a $1$ $72.173$ \(\Q\) None \(-4\) \(0\) \(0\) \(172\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+172q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.m 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(-176\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-176q^{7}+2^{6}q^{8}+\cdots\)
450.6.a.n 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(-142\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-142q^{7}+2^{6}q^{8}+\cdots\)
450.6.a.o 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(-98\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-98q^{7}+2^{6}q^{8}-354q^{11}+\cdots\)
450.6.a.p 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(-47\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-47q^{7}+2^{6}q^{8}-222q^{11}+\cdots\)
450.6.a.q 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(-32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-2^{5}q^{7}+2^{6}q^{8}-12q^{11}+\cdots\)
450.6.a.r 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-q^{7}+2^{6}q^{8}+210q^{11}+\cdots\)
450.6.a.s 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+4q^{7}+2^{6}q^{8}+500q^{11}+\cdots\)
450.6.a.t 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(79\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+79q^{7}+2^{6}q^{8}-150q^{11}+\cdots\)
450.6.a.u 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(118\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+118q^{7}+2^{6}q^{8}+\cdots\)
450.6.a.v 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(148\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+148q^{7}+2^{6}q^{8}+\cdots\)
450.6.a.w 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(158\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+158q^{7}+2^{6}q^{8}+\cdots\)
450.6.a.x 450.a 1.a $1$ $72.173$ \(\Q\) None \(4\) \(0\) \(0\) \(233\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+233q^{7}+2^{6}q^{8}+\cdots\)
450.6.a.y 450.a 1.a $2$ $72.173$ \(\Q(\sqrt{4081}) \) None \(-8\) \(0\) \(0\) \(-100\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(-50-\beta )q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.z 450.a 1.a $2$ $72.173$ \(\Q(\sqrt{19}) \) None \(-8\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+\beta q^{7}-2^{6}q^{8}+37\beta q^{11}+\cdots\)
450.6.a.ba 450.a 1.a $2$ $72.173$ \(\Q(\sqrt{4081}) \) None \(-8\) \(0\) \(0\) \(100\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(50-\beta )q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.bb 450.a 1.a $2$ $72.173$ \(\Q(\sqrt{1249}) \) None \(-8\) \(0\) \(0\) \(114\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(57-\beta )q^{7}-2^{6}q^{8}+\cdots\)
450.6.a.bc 450.a 1.a $2$ $72.173$ \(\Q(\sqrt{1249}) \) None \(8\) \(0\) \(0\) \(-114\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(-57-\beta )q^{7}+2^{6}q^{8}+\cdots\)
450.6.a.bd 450.a 1.a $2$ $72.173$ \(\Q(\sqrt{4081}) \) None \(8\) \(0\) \(0\) \(-100\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(-50-\beta )q^{7}+2^{6}q^{8}+\cdots\)
450.6.a.be 450.a 1.a $2$ $72.173$ \(\Q(\sqrt{19}) \) None \(8\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+\beta q^{7}+2^{6}q^{8}-37\beta q^{11}+\cdots\)
450.6.a.bf 450.a 1.a $2$ $72.173$ \(\Q(\sqrt{4081}) \) None \(8\) \(0\) \(0\) \(100\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(50-\beta )q^{7}+2^{6}q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(450))\) into lower level spaces

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