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Properties

Label 19.6.a
Level $19$
Weight $6$
Character orbit 19.a
Rep. character $\chi_{19}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $4$
Sturm bound $10$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 19.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(10\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(19))\).

Total New Old
Modular forms 10 8 2
Cusp forms 8 8 0
Eisenstein series 2 0 2

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

\(19\)Dim
\(+\)\(3\)
\(-\)\(5\)

Trace form

\( 8 q - 6 q^{2} + 2 q^{3} + 122 q^{4} - 13 q^{5} + 106 q^{6} - 37 q^{7} - 504 q^{8} + 712 q^{9} + O(q^{10}) \) \( 8 q - 6 q^{2} + 2 q^{3} + 122 q^{4} - 13 q^{5} + 106 q^{6} - 37 q^{7} - 504 q^{8} + 712 q^{9} + 368 q^{10} - 401 q^{11} - 360 q^{12} - 1014 q^{13} + 1228 q^{14} - 2566 q^{15} + 890 q^{16} + 2453 q^{17} - 4974 q^{18} + 722 q^{19} - 4568 q^{20} + 3286 q^{21} + 1648 q^{22} + 4768 q^{23} + 1902 q^{24} + 8911 q^{25} + 2138 q^{26} - 5092 q^{27} - 10674 q^{28} + 1520 q^{29} + 22316 q^{30} + 11324 q^{31} - 6144 q^{32} - 12994 q^{33} - 37036 q^{34} - 507 q^{35} - 788 q^{36} - 844 q^{37} + 4332 q^{38} - 16320 q^{39} + 21900 q^{40} - 12712 q^{41} - 45574 q^{42} + 36739 q^{43} + 10976 q^{44} + 2003 q^{45} - 908 q^{46} + 35505 q^{47} + 40080 q^{48} + 12535 q^{49} + 20054 q^{50} - 75506 q^{51} - 13060 q^{52} - 50462 q^{53} + 138310 q^{54} - 37683 q^{55} + 66564 q^{56} + 6498 q^{57} + 24962 q^{58} - 2186 q^{59} - 249524 q^{60} - 123553 q^{61} + 159748 q^{62} - 53493 q^{63} - 103918 q^{64} + 132744 q^{65} + 162092 q^{66} + 49600 q^{67} + 226978 q^{68} + 93336 q^{69} - 180756 q^{70} - 80058 q^{71} - 335160 q^{72} + 86233 q^{73} - 237200 q^{74} + 90676 q^{75} + 28880 q^{76} + 24835 q^{77} + 130900 q^{78} - 307768 q^{79} - 366860 q^{80} + 439324 q^{81} + 31628 q^{82} - 116560 q^{83} + 268452 q^{84} + 68709 q^{85} + 40956 q^{86} + 311484 q^{87} + 438504 q^{88} - 230842 q^{89} - 238876 q^{90} - 405932 q^{91} + 463610 q^{92} - 331008 q^{93} + 14328 q^{94} + 108661 q^{95} - 241378 q^{96} + 243218 q^{97} - 508862 q^{98} - 432977 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(19))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 19
19.6.a.a 19.a 1.a $1$ $3.047$ \(\Q\) None \(-6\) \(4\) \(54\) \(248\) $-$ $\mathrm{SU}(2)$ \(q-6q^{2}+4q^{3}+4q^{4}+54q^{5}-24q^{6}+\cdots\)
19.6.a.b 19.a 1.a $1$ $3.047$ \(\Q\) None \(-2\) \(-1\) \(-24\) \(-167\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}-28q^{4}-24q^{5}+2q^{6}+\cdots\)
19.6.a.c 19.a 1.a $2$ $3.047$ \(\Q(\sqrt{177}) \) None \(-7\) \(-7\) \(-133\) \(72\) $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{2}+(-5+3\beta )q^{3}+(21+\cdots)q^{4}+\cdots\)
19.6.a.d 19.a 1.a $4$ $3.047$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(9\) \(6\) \(90\) \(-190\) $-$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1}+\beta _{2})q^{2}+(3+3\beta _{2})q^{3}+(23+\cdots)q^{4}+\cdots\)