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

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

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

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

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