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Properties

Label 150.8.a
Level $150$
Weight $8$
Character orbit 150.a
Rep. character $\chi_{150}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $19$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(150))\).

Total New Old
Modular forms 222 21 201
Cusp forms 198 21 177
Eisenstein series 24 0 24

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
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(3\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(12\)
Minus space\(-\)\(9\)

Trace form

\( 21 q - 8 q^{2} + 27 q^{3} + 1344 q^{4} - 216 q^{6} + 1348 q^{7} - 512 q^{8} + 15309 q^{9} + O(q^{10}) \) \( 21 q - 8 q^{2} + 27 q^{3} + 1344 q^{4} - 216 q^{6} + 1348 q^{7} - 512 q^{8} + 15309 q^{9} - 2060 q^{11} + 1728 q^{12} - 17366 q^{13} + 5376 q^{14} + 86016 q^{16} - 16122 q^{17} - 5832 q^{18} + 2124 q^{19} - 106704 q^{21} + 1920 q^{23} - 13824 q^{24} - 47312 q^{26} + 19683 q^{27} + 86272 q^{28} - 98034 q^{29} + 118728 q^{31} - 32768 q^{32} - 136080 q^{33} - 567984 q^{34} + 979776 q^{36} + 537946 q^{37} + 826208 q^{38} + 674838 q^{39} + 1404754 q^{41} - 14688 q^{42} - 1167308 q^{43} - 131840 q^{44} - 1606080 q^{46} + 3079680 q^{47} + 110592 q^{48} + 3120621 q^{49} - 902934 q^{51} - 1111424 q^{52} - 3839982 q^{53} - 157464 q^{54} + 344064 q^{56} - 2257092 q^{57} + 3839472 q^{58} - 2657980 q^{59} + 12091590 q^{61} + 324416 q^{62} + 982692 q^{63} + 5505024 q^{64} - 3762720 q^{66} - 1355372 q^{67} - 1031808 q^{68} + 2733480 q^{69} + 7138200 q^{71} - 373248 q^{72} + 4353154 q^{73} + 14209392 q^{74} + 135936 q^{76} + 13088640 q^{77} + 5271696 q^{78} + 5532552 q^{79} + 11160261 q^{81} + 6771888 q^{82} - 24926076 q^{83} - 6829056 q^{84} + 14617824 q^{86} + 442422 q^{87} - 3005198 q^{89} - 26731248 q^{91} + 122880 q^{92} + 19031976 q^{93} - 7228800 q^{94} - 884736 q^{96} + 1936306 q^{97} - 32555208 q^{98} - 1501740 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(150))\) 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
150.8.a.a 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(-27\) \(0\) \(-1604\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-1604q^{7}+\cdots\)
150.8.a.b 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(-27\) \(0\) \(-349\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-349q^{7}+\cdots\)
150.8.a.c 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(-27\) \(0\) \(391\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+391q^{7}+\cdots\)
150.8.a.d 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(-27\) \(0\) \(1126\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+1126q^{7}+\cdots\)
150.8.a.e 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(-27\) \(0\) \(1576\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+1576q^{7}+\cdots\)
150.8.a.f 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(27\) \(0\) \(-1427\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}-1427q^{7}+\cdots\)
150.8.a.g 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(27\) \(0\) \(-512\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}-2^{9}q^{7}+\cdots\)
150.8.a.h 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(27\) \(0\) \(713\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+713q^{7}+\cdots\)
150.8.a.i 150.a 1.a $1$ $46.858$ \(\Q\) None \(-8\) \(27\) \(0\) \(988\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+988q^{7}+\cdots\)
150.8.a.j 150.a 1.a $1$ $46.858$ \(\Q\) None \(8\) \(-27\) \(0\) \(-713\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}-713q^{7}+\cdots\)
150.8.a.k 150.a 1.a $1$ $46.858$ \(\Q\) None \(8\) \(-27\) \(0\) \(232\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+232q^{7}+\cdots\)
150.8.a.l 150.a 1.a $1$ $46.858$ \(\Q\) None \(8\) \(-27\) \(0\) \(1427\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+1427q^{7}+\cdots\)
150.8.a.m 150.a 1.a $1$ $46.858$ \(\Q\) None \(8\) \(27\) \(0\) \(-1126\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-1126q^{7}+\cdots\)
150.8.a.n 150.a 1.a $1$ $46.858$ \(\Q\) None \(8\) \(27\) \(0\) \(-416\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-416q^{7}+\cdots\)
150.8.a.o 150.a 1.a $1$ $46.858$ \(\Q\) None \(8\) \(27\) \(0\) \(-391\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-391q^{7}+\cdots\)
150.8.a.p 150.a 1.a $1$ $46.858$ \(\Q\) None \(8\) \(27\) \(0\) \(349\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+349q^{7}+\cdots\)
150.8.a.q 150.a 1.a $1$ $46.858$ \(\Q\) None \(8\) \(27\) \(0\) \(1084\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+1084q^{7}+\cdots\)
150.8.a.r 150.a 1.a $2$ $46.858$ \(\Q(\sqrt{2641}) \) None \(-16\) \(54\) \(0\) \(-564\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+(-282+\cdots)q^{7}+\cdots\)
150.8.a.s 150.a 1.a $2$ $46.858$ \(\Q(\sqrt{2641}) \) None \(16\) \(-54\) \(0\) \(564\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+(282+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(150)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)