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Properties

Label 150.16.a
Level $150$
Weight $16$
Character orbit 150.a
Rep. character $\chi_{150}(1,\cdot)$
Character field $\Q$
Dimension $47$
Newform subspaces $25$
Sturm bound $480$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 25 \)
Sturm bound: \(480\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(150))\).

Total New Old
Modular forms 462 47 415
Cusp forms 438 47 391
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
\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(-\)$-$\(6\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(6\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(7\)
\(-\)\(-\)\(+\)$+$\(6\)
\(-\)\(-\)\(-\)$-$\(5\)
Plus space\(+\)\(25\)
Minus space\(-\)\(22\)

Trace form

\( 47 q - 128 q^{2} - 2187 q^{3} + 770048 q^{4} - 279936 q^{6} + 3997996 q^{7} - 2097152 q^{8} + 224799543 q^{9} + O(q^{10}) \) \( 47 q - 128 q^{2} - 2187 q^{3} + 770048 q^{4} - 279936 q^{6} + 3997996 q^{7} - 2097152 q^{8} + 224799543 q^{9} - 156310124 q^{11} - 35831808 q^{12} + 739645342 q^{13} + 17571840 q^{14} + 12616466432 q^{16} + 352378386 q^{17} - 612220032 q^{18} + 1754471736 q^{19} - 1151840412 q^{21} + 16833183744 q^{22} - 15543105888 q^{23} - 4586471424 q^{24} - 109221084416 q^{26} - 10460353203 q^{27} + 65503166464 q^{28} + 163773506874 q^{29} - 44483983372 q^{31} - 34359738368 q^{32} - 160328165184 q^{33} - 636537998080 q^{34} + 3683115712512 q^{36} + 2873309791726 q^{37} - 867367897600 q^{38} + 1010603099070 q^{39} - 4744241600498 q^{41} + 526412929536 q^{42} - 3855099858068 q^{43} - 2560985071616 q^{44} + 2200615150592 q^{46} + 3753378637536 q^{47} - 587068342272 q^{48} + 15558206513043 q^{49} + 6187073550318 q^{51} + 12118349283328 q^{52} - 25629535870938 q^{53} - 1338925209984 q^{54} + 287897026560 q^{56} - 5870482087500 q^{57} - 18492959842560 q^{58} - 39868718385580 q^{59} - 26300485384514 q^{61} - 26867001957376 q^{62} + 19122290930124 q^{63} + 206708186021888 q^{64} - 34766854193664 q^{66} + 32936107146796 q^{67} + 5773367476224 q^{68} + 54420580306368 q^{69} - 51059435570424 q^{71} - 10030613004288 q^{72} - 14920904984138 q^{73} + 123595262863104 q^{74} + 28745264922624 q^{76} - 1133839458139008 q^{77} - 14933392512768 q^{78} + 282020663716248 q^{79} + 1075209245383167 q^{81} + 174829154421504 q^{82} + 569940577494492 q^{83} - 18871753310208 q^{84} - 306794256486912 q^{86} + 341891957491290 q^{87} + 275794882461696 q^{88} + 193057622889358 q^{89} + 68623659177804 q^{91} - 254658246868992 q^{92} - 406353339759384 q^{93} + 481797003790336 q^{94} - 75144747810816 q^{96} + 1811270559819046 q^{97} + 1328971506750336 q^{98} - 747626477478156 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(150))\) 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
150.16.a.a 150.a 1.a $1$ $214.040$ \(\Q\) None \(-128\) \(-2187\) \(0\) \(-762104\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.b 150.a 1.a $1$ $214.040$ \(\Q\) None \(-128\) \(-2187\) \(0\) \(-511994\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.c 150.a 1.a $1$ $214.040$ \(\Q\) None \(-128\) \(-2187\) \(0\) \(3067456\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.d 150.a 1.a $1$ $214.040$ \(\Q\) None \(-128\) \(2187\) \(0\) \(-2412032\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.e 150.a 1.a $1$ $214.040$ \(\Q\) None \(-128\) \(2187\) \(0\) \(265468\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.f 150.a 1.a $1$ $214.040$ \(\Q\) None \(-128\) \(2187\) \(0\) \(3034528\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.g 150.a 1.a $1$ $214.040$ \(\Q\) None \(128\) \(-2187\) \(0\) \(-918428\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.h 150.a 1.a $1$ $214.040$ \(\Q\) None \(128\) \(2187\) \(0\) \(-2025056\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.i 150.a 1.a $1$ $214.040$ \(\Q\) None \(128\) \(2187\) \(0\) \(511994\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.j 150.a 1.a $1$ $214.040$ \(\Q\) None \(128\) \(2187\) \(0\) \(1107904\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.k 150.a 1.a $1$ $214.040$ \(\Q\) None \(128\) \(2187\) \(0\) \(3785404\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.l 150.a 1.a $2$ $214.040$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-256\) \(-4374\) \(0\) \(-2234368\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.m 150.a 1.a $2$ $214.040$ \(\Q(\sqrt{59119}) \) None \(-256\) \(-4374\) \(0\) \(1317302\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.n 150.a 1.a $2$ $214.040$ \(\Q(\sqrt{5569}) \) None \(-256\) \(-4374\) \(0\) \(1391922\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.o 150.a 1.a $2$ $214.040$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-256\) \(4374\) \(0\) \(-840854\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.p 150.a 1.a $2$ $214.040$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(256\) \(-4374\) \(0\) \(840854\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.q 150.a 1.a $2$ $214.040$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(256\) \(-4374\) \(0\) \(1089224\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.r 150.a 1.a $2$ $214.040$ \(\Q(\sqrt{5569}) \) None \(256\) \(4374\) \(0\) \(-1391922\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.s 150.a 1.a $2$ $214.040$ \(\Q(\sqrt{59119}) \) None \(256\) \(4374\) \(0\) \(-1317302\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.t 150.a 1.a $3$ $214.040$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-384\) \(-6561\) \(0\) \(-701247\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.u 150.a 1.a $3$ $214.040$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-384\) \(6561\) \(0\) \(87519\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.v 150.a 1.a $3$ $214.040$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(384\) \(-6561\) \(0\) \(-87519\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.w 150.a 1.a $3$ $214.040$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(384\) \(6561\) \(0\) \(701247\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}+6^{7}q^{6}+\cdots\)
150.16.a.x 150.a 1.a $4$ $214.040$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-512\) \(8748\) \(0\) \(228762\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)
150.16.a.y 150.a 1.a $4$ $214.040$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(512\) \(-8748\) \(0\) \(-228762\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}-6^{7}q^{6}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(150)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)