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Properties

Label 150.12.a
Level $150$
Weight $12$
Character orbit 150.a
Rep. character $\chi_{150}(1,\cdot)$
Character field $\Q$
Dimension $35$
Newform subspaces $21$
Sturm bound $360$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 21 \)
Sturm bound: \(360\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(150))\).

Total New Old
Modular forms 342 35 307
Cusp forms 318 35 283
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
\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(5\)
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(5\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(19\)
Minus space\(-\)\(16\)

Trace form

\( 35 q + 32 q^{2} + 243 q^{3} + 35840 q^{4} - 7776 q^{6} - 66564 q^{7} + 32768 q^{8} + 2066715 q^{9} + O(q^{10}) \) \( 35 q + 32 q^{2} + 243 q^{3} + 35840 q^{4} - 7776 q^{6} - 66564 q^{7} + 32768 q^{8} + 2066715 q^{9} + 1576964 q^{11} + 248832 q^{12} - 535218 q^{13} - 4739328 q^{14} + 36700160 q^{16} + 12946866 q^{17} + 1889568 q^{18} - 8629828 q^{19} - 53040096 q^{21} + 26517504 q^{22} - 44563248 q^{23} - 7962624 q^{24} + 479552 q^{26} + 14348907 q^{27} - 68161536 q^{28} - 516520182 q^{29} + 223708848 q^{31} + 33554432 q^{32} + 55304856 q^{33} + 626468032 q^{34} + 2116316160 q^{36} + 982129566 q^{37} + 254711680 q^{38} - 1639121994 q^{39} + 687879086 q^{41} - 37666944 q^{42} - 801672828 q^{43} + 1614811136 q^{44} - 2461759232 q^{46} - 6561777744 q^{47} + 254803968 q^{48} + 6291308019 q^{49} + 1952650314 q^{51} - 548063232 q^{52} + 9616568142 q^{53} - 459165024 q^{54} - 4853071872 q^{56} + 6330961620 q^{57} + 2313007680 q^{58} + 10002122500 q^{59} - 22104036206 q^{61} - 4954789376 q^{62} - 3930537636 q^{63} + 37580963840 q^{64} - 704474496 q^{66} + 28415926836 q^{67} + 13257590784 q^{68} - 45832612968 q^{69} + 42551520744 q^{71} + 1934917632 q^{72} - 51675989298 q^{73} - 44505117120 q^{74} - 8836943872 q^{76} + 75411643392 q^{77} + 19438926912 q^{78} - 42762858304 q^{79} + 122037454035 q^{81} - 3821958336 q^{82} - 69372722508 q^{83} - 54313058304 q^{84} - 129838339200 q^{86} - 7974596610 q^{87} + 27153924096 q^{88} + 24562221086 q^{89} + 180733483840 q^{91} - 45632765952 q^{92} - 26276126544 q^{93} + 69121301504 q^{94} - 8153726976 q^{96} - 79953135954 q^{97} + 342959344416 q^{98} + 93118147236 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(150))\) 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
150.12.a.a 150.a 1.a $1$ $115.251$ \(\Q\) None \(-32\) \(-243\) \(0\) \(-32936\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.b 150.a 1.a $1$ $115.251$ \(\Q\) None \(-32\) \(-243\) \(0\) \(-10556\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.c 150.a 1.a $1$ $115.251$ \(\Q\) None \(-32\) \(243\) \(0\) \(-29348\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.d 150.a 1.a $1$ $115.251$ \(\Q\) None \(-32\) \(243\) \(0\) \(5152\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.e 150.a 1.a $1$ $115.251$ \(\Q\) None \(32\) \(-243\) \(0\) \(-56672\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.f 150.a 1.a $1$ $115.251$ \(\Q\) None \(32\) \(-243\) \(0\) \(50008\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.g 150.a 1.a $1$ $115.251$ \(\Q\) None \(32\) \(243\) \(0\) \(-72464\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.h 150.a 1.a $1$ $115.251$ \(\Q\) None \(32\) \(243\) \(0\) \(22876\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.i 150.a 1.a $1$ $115.251$ \(\Q\) None \(32\) \(243\) \(0\) \(57376\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.j 150.a 1.a $2$ $115.251$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-64\) \(-486\) \(0\) \(24058\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.k 150.a 1.a $2$ $115.251$ \(\Q(\sqrt{1129}) \) None \(-64\) \(-486\) \(0\) \(27738\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.l 150.a 1.a $2$ $115.251$ \(\Q(\sqrt{499}) \) None \(-64\) \(-486\) \(0\) \(65438\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.m 150.a 1.a $2$ $115.251$ \(\Q(\sqrt{94291}) \) None \(-64\) \(486\) \(0\) \(21394\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.n 150.a 1.a $2$ $115.251$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-64\) \(486\) \(0\) \(37214\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.o 150.a 1.a $2$ $115.251$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(64\) \(-486\) \(0\) \(-37214\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.p 150.a 1.a $2$ $115.251$ \(\Q(\sqrt{94291}) \) None \(64\) \(-486\) \(0\) \(-21394\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.q 150.a 1.a $2$ $115.251$ \(\Q(\sqrt{499}) \) None \(64\) \(486\) \(0\) \(-65438\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.r 150.a 1.a $2$ $115.251$ \(\Q(\sqrt{1129}) \) None \(64\) \(486\) \(0\) \(-27738\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.s 150.a 1.a $2$ $115.251$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(64\) \(486\) \(0\) \(-24058\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.t 150.a 1.a $3$ $115.251$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-96\) \(729\) \(0\) \(-67384\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.u 150.a 1.a $3$ $115.251$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(96\) \(-729\) \(0\) \(67384\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(150)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)