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Properties

Label 75.16.a
Level $75$
Weight $16$
Character orbit 75.a
Rep. character $\chi_{75}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $12$
Sturm bound $160$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(160\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(75))\).

Total New Old
Modular forms 156 48 108
Cusp forms 144 48 96
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(5\)FrickeDim
\(+\)\(+\)$+$\(12\)
\(+\)\(-\)$-$\(12\)
\(-\)\(+\)$-$\(10\)
\(-\)\(-\)$+$\(14\)
Plus space\(+\)\(26\)
Minus space\(-\)\(22\)

Trace form

\( 48 q - 50 q^{2} + 726896 q^{4} - 354294 q^{6} - 4058260 q^{7} - 3276180 q^{8} + 229582512 q^{9} + O(q^{10}) \) \( 48 q - 50 q^{2} + 726896 q^{4} - 354294 q^{6} - 4058260 q^{7} - 3276180 q^{8} + 229582512 q^{9} + 167119632 q^{11} + 180077580 q^{12} - 530982860 q^{13} - 849482196 q^{14} + 11262788020 q^{16} - 2352919100 q^{17} - 239148450 q^{18} + 5002155164 q^{19} + 4418955972 q^{21} - 9848735500 q^{22} + 52008734160 q^{23} - 1407386988 q^{24} + 147238192056 q^{26} - 301634512780 q^{28} - 54689043552 q^{29} + 259300712748 q^{31} - 955331623540 q^{32} - 56996937900 q^{33} + 2696194011812 q^{34} + 3476721034224 q^{36} - 1956026687940 q^{37} + 437259124480 q^{38} - 287693665164 q^{39} + 3696743539488 q^{41} - 3693457606860 q^{42} + 2314176851440 q^{43} + 8628550897968 q^{44} + 14727235874848 q^{46} - 11095654943000 q^{47} + 8900911540800 q^{48} + 59751178922300 q^{49} - 7201189942416 q^{51} - 52839833011080 q^{52} + 22413604171180 q^{53} - 1694577218886 q^{54} - 107960445502920 q^{56} - 11493099655200 q^{57} + 56775511710760 q^{58} - 6062540832864 q^{59} - 37476021915060 q^{61} - 1168619570040 q^{62} - 19410531773940 q^{63} + 441738595805164 q^{64} + 35326205495208 q^{66} + 94346908492760 q^{67} - 143287628985280 q^{68} + 15530062992264 q^{69} - 4976823871824 q^{71} - 15669867378420 q^{72} + 99447919824400 q^{73} - 70904742244476 q^{74} - 44392276669136 q^{76} + 792858769260480 q^{77} + 299552484450780 q^{78} + 184902892810280 q^{79} + 1098086037838128 q^{81} + 243902751043900 q^{82} - 1105597254652200 q^{83} + 409146257942136 q^{84} - 54661045261788 q^{86} - 451104387558660 q^{87} - 630876517035660 q^{88} + 367258710623664 q^{89} - 479718407587372 q^{91} + 3569937228127200 q^{92} - 64261961018640 q^{93} - 199303067805904 q^{94} + 1932884016536412 q^{96} - 1889570860207320 q^{97} + 2542262216683310 q^{98} + 799328019147408 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(75))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
75.16.a.a 75.a 1.a $1$ $107.020$ \(\Q\) None \(72\) \(-2187\) \(0\) \(2149000\) $+$ $+$ $\mathrm{SU}(2)$ \(q+72q^{2}-3^{7}q^{3}-27584q^{4}-54^{3}q^{6}+\cdots\)
75.16.a.b 75.a 1.a $1$ $107.020$ \(\Q\) None \(234\) \(2187\) \(0\) \(1373344\) $-$ $+$ $\mathrm{SU}(2)$ \(q+234q^{2}+3^{7}q^{3}+21988q^{4}+511758q^{6}+\cdots\)
75.16.a.c 75.a 1.a $2$ $107.020$ \(\Q(\sqrt{79}) \) None \(-208\) \(4374\) \(0\) \(-195776\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-104+\beta )q^{2}+3^{7}q^{3}+(-12^{3}+\cdots)q^{4}+\cdots\)
75.16.a.d 75.a 1.a $2$ $107.020$ \(\Q(\sqrt{5641}) \) None \(158\) \(-4374\) \(0\) \(395136\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(79-\beta )q^{2}-3^{7}q^{3}+(24242-158\beta )q^{4}+\cdots\)
75.16.a.e 75.a 1.a $3$ $107.020$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-191\) \(-6561\) \(0\) \(-2816048\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2^{6}+\beta _{1})q^{2}-3^{7}q^{3}+(17533+\cdots)q^{4}+\cdots\)
75.16.a.f 75.a 1.a $3$ $107.020$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-115\) \(6561\) \(0\) \(-4963916\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-38-\beta _{1})q^{2}+3^{7}q^{3}+(45742+\cdots)q^{4}+\cdots\)
75.16.a.g 75.a 1.a $4$ $107.020$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-56\) \(8748\) \(0\) \(-971852\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-14+\beta _{1})q^{2}+3^{7}q^{3}+(-1298+\cdots)q^{4}+\cdots\)
75.16.a.h 75.a 1.a $4$ $107.020$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(56\) \(-8748\) \(0\) \(971852\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(14-\beta _{1})q^{2}-3^{7}q^{3}+(-1298+41\beta _{1}+\cdots)q^{4}+\cdots\)
75.16.a.i 75.a 1.a $6$ $107.020$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-234\) \(13122\) \(0\) \(2590222\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-39+\beta _{1})q^{2}+3^{7}q^{3}+(15519+\cdots)q^{4}+\cdots\)
75.16.a.j 75.a 1.a $6$ $107.020$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(234\) \(-13122\) \(0\) \(-2590222\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(39-\beta _{1})q^{2}-3^{7}q^{3}+(15519-2^{6}\beta _{1}+\cdots)q^{4}+\cdots\)
75.16.a.k 75.a 1.a $8$ $107.020$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-273\) \(-17496\) \(0\) \(-1149126\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-34-\beta _{1})q^{2}-3^{7}q^{3}+(20114+\cdots)q^{4}+\cdots\)
75.16.a.l 75.a 1.a $8$ $107.020$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(273\) \(17496\) \(0\) \(1149126\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(34+\beta _{1})q^{2}+3^{7}q^{3}+(20114+33\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(75)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)