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Properties

Label 392.3
Level 392
Weight 3
Dimension 4981
Nonzero newspaces 12
Newform subspaces 52
Sturm bound 28224
Trace bound 3

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Defining parameters

Level: \( N \) = \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 52 \)
Sturm bound: \(28224\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(392))\).

Total New Old
Modular forms 9768 5175 4593
Cusp forms 9048 4981 4067
Eisenstein series 720 194 526

Trace form

\( 4981 q - 32 q^{2} - 32 q^{3} - 26 q^{4} - 26 q^{6} - 36 q^{7} - 62 q^{8} - 113 q^{9} + O(q^{10}) \) \( 4981 q - 32 q^{2} - 32 q^{3} - 26 q^{4} - 26 q^{6} - 36 q^{7} - 62 q^{8} - 113 q^{9} - 30 q^{10} - 40 q^{11} - 38 q^{12} - 36 q^{14} - 6 q^{15} - 14 q^{16} - 10 q^{17} + 88 q^{18} + 104 q^{19} + 222 q^{20} + 84 q^{21} + 218 q^{22} + 114 q^{23} + 202 q^{24} + 37 q^{25} + 30 q^{26} + 10 q^{27} - 84 q^{28} - 318 q^{30} - 342 q^{31} - 362 q^{32} - 496 q^{33} - 538 q^{34} - 288 q^{35} - 614 q^{36} - 216 q^{37} - 358 q^{38} - 138 q^{39} - 330 q^{40} - 106 q^{41} - 252 q^{42} + 296 q^{43} - 502 q^{44} + 552 q^{45} - 570 q^{46} + 762 q^{47} - 902 q^{48} + 132 q^{49} - 566 q^{50} + 806 q^{51} - 354 q^{52} + 360 q^{53} - 206 q^{54} - 42 q^{55} + 6 q^{56} - 472 q^{57} + 174 q^{58} - 736 q^{59} + 918 q^{60} - 504 q^{61} + 726 q^{62} - 660 q^{63} + 766 q^{64} - 564 q^{65} + 1778 q^{66} - 736 q^{67} + 1658 q^{68} + 948 q^{70} + 474 q^{71} + 2230 q^{72} + 110 q^{73} + 1482 q^{74} + 772 q^{75} + 842 q^{76} + 360 q^{77} + 786 q^{78} + 786 q^{79} + 654 q^{80} + 1609 q^{81} + 206 q^{82} + 1316 q^{83} - 144 q^{84} + 672 q^{85} - 706 q^{86} + 846 q^{87} - 862 q^{88} + 350 q^{89} - 2058 q^{90} - 234 q^{91} - 1794 q^{92} + 216 q^{93} - 2838 q^{94} - 2238 q^{95} - 3950 q^{96} + 1190 q^{97} - 1572 q^{98} - 2902 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(392))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
392.3.c \(\chi_{392}(97, \cdot)\) 392.3.c.a 4 1
392.3.c.b 8
392.3.c.c 8
392.3.d \(\chi_{392}(295, \cdot)\) None 0 1
392.3.g \(\chi_{392}(99, \cdot)\) 392.3.g.a 1 1
392.3.g.b 2
392.3.g.c 2
392.3.g.d 2
392.3.g.e 2
392.3.g.f 2
392.3.g.g 2
392.3.g.h 4
392.3.g.i 6
392.3.g.j 6
392.3.g.k 6
392.3.g.l 6
392.3.g.m 8
392.3.g.n 8
392.3.g.o 20
392.3.h \(\chi_{392}(293, \cdot)\) 392.3.h.a 28 1
392.3.h.b 48
392.3.j \(\chi_{392}(117, \cdot)\) 392.3.j.a 4 2
392.3.j.b 4
392.3.j.c 4
392.3.j.d 16
392.3.j.e 28
392.3.j.f 96
392.3.k \(\chi_{392}(67, \cdot)\) 392.3.k.a 2 2
392.3.k.b 2
392.3.k.c 2
392.3.k.d 2
392.3.k.e 4
392.3.k.f 4
392.3.k.g 4
392.3.k.h 4
392.3.k.i 8
392.3.k.j 8
392.3.k.k 12
392.3.k.l 12
392.3.k.m 16
392.3.k.n 16
392.3.k.o 16
392.3.k.p 40
392.3.n \(\chi_{392}(79, \cdot)\) None 0 2
392.3.o \(\chi_{392}(129, \cdot)\) 392.3.o.a 8 2
392.3.o.b 8
392.3.o.c 8
392.3.o.d 16
392.3.r \(\chi_{392}(13, \cdot)\) 392.3.r.a 660 6
392.3.s \(\chi_{392}(43, \cdot)\) 392.3.s.a 660 6
392.3.v \(\chi_{392}(15, \cdot)\) None 0 6
392.3.w \(\chi_{392}(41, \cdot)\) 392.3.w.a 168 6
392.3.ba \(\chi_{392}(17, \cdot)\) 392.3.ba.a 336 12
392.3.bb \(\chi_{392}(23, \cdot)\) None 0 12
392.3.be \(\chi_{392}(11, \cdot)\) 392.3.be.a 1320 12
392.3.bf \(\chi_{392}(5, \cdot)\) 392.3.bf.a 1320 12

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(392))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(392)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 2}\)