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Properties

Label 350.6
Level 350
Weight 6
Dimension 5300
Nonzero newspaces 12
Sturm bound 43200
Trace bound 4

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(43200\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(350))\).

Total New Old
Modular forms 18336 5300 13036
Cusp forms 17664 5300 12364
Eisenstein series 672 0 672

Trace form

\( 5300 q - 16 q^{2} + 34 q^{3} + 32 q^{4} + 170 q^{5} - 104 q^{6} - 392 q^{7} - 256 q^{8} + 1394 q^{9} + O(q^{10}) \) \( 5300 q - 16 q^{2} + 34 q^{3} + 32 q^{4} + 170 q^{5} - 104 q^{6} - 392 q^{7} - 256 q^{8} + 1394 q^{9} - 360 q^{10} - 3404 q^{11} + 544 q^{12} + 4106 q^{13} + 3016 q^{14} + 1640 q^{15} + 4608 q^{16} - 7196 q^{17} + 2120 q^{18} + 16318 q^{19} - 1280 q^{20} + 10910 q^{21} - 2800 q^{22} - 26416 q^{23} - 15744 q^{24} - 77334 q^{25} - 4312 q^{26} + 64804 q^{27} + 34752 q^{28} + 132656 q^{29} + 80448 q^{30} + 44316 q^{31} + 6144 q^{32} - 102324 q^{33} - 69672 q^{34} - 141572 q^{35} - 31200 q^{36} - 166286 q^{37} - 58760 q^{38} + 52344 q^{39} - 640 q^{40} + 254624 q^{41} + 282776 q^{42} + 339012 q^{43} + 44096 q^{44} - 27502 q^{45} - 57040 q^{46} - 250060 q^{47} + 8704 q^{48} - 44540 q^{49} - 22600 q^{50} - 93700 q^{51} - 14944 q^{52} - 315354 q^{53} - 218208 q^{54} - 86064 q^{55} - 51840 q^{56} + 405556 q^{57} + 120912 q^{58} + 463646 q^{59} - 225280 q^{60} + 349594 q^{61} + 343312 q^{62} + 652016 q^{63} + 57344 q^{64} + 577738 q^{65} + 124960 q^{66} - 128496 q^{67} + 177984 q^{68} - 280704 q^{69} - 47360 q^{70} - 733960 q^{71} - 225280 q^{72} - 1302024 q^{73} - 1104528 q^{74} - 1671736 q^{75} - 152480 q^{76} + 3776 q^{77} + 113120 q^{78} + 591216 q^{79} + 43520 q^{80} + 1225430 q^{81} + 609040 q^{82} + 923310 q^{83} + 140832 q^{84} + 1309922 q^{85} - 597264 q^{86} - 104204 q^{87} - 84480 q^{88} + 1356646 q^{89} + 126744 q^{90} + 1836430 q^{91} + 214784 q^{92} + 2492356 q^{93} + 1212128 q^{94} + 680728 q^{95} + 133120 q^{96} - 624120 q^{97} - 1264 q^{98} - 3029740 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(350))\)

We only show spaces with even 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
350.6.a \(\chi_{350}(1, \cdot)\) 350.6.a.a 1 1
350.6.a.b 1
350.6.a.c 1
350.6.a.d 1
350.6.a.e 1
350.6.a.f 1
350.6.a.g 1
350.6.a.h 1
350.6.a.i 1
350.6.a.j 1
350.6.a.k 1
350.6.a.l 1
350.6.a.m 1
350.6.a.n 1
350.6.a.o 2
350.6.a.p 2
350.6.a.q 2
350.6.a.r 2
350.6.a.s 2
350.6.a.t 2
350.6.a.u 3
350.6.a.v 3
350.6.a.w 3
350.6.a.x 3
350.6.a.y 5
350.6.a.z 5
350.6.c \(\chi_{350}(99, \cdot)\) 350.6.c.a 2 1
350.6.c.b 2
350.6.c.c 2
350.6.c.d 2
350.6.c.e 2
350.6.c.f 2
350.6.c.g 2
350.6.c.h 2
350.6.c.i 4
350.6.c.j 4
350.6.c.k 4
350.6.c.l 4
350.6.c.m 6
350.6.c.n 6
350.6.e \(\chi_{350}(51, \cdot)\) n/a 128 2
350.6.g \(\chi_{350}(293, \cdot)\) n/a 120 2
350.6.h \(\chi_{350}(71, \cdot)\) n/a 296 4
350.6.j \(\chi_{350}(149, \cdot)\) n/a 120 2
350.6.m \(\chi_{350}(29, \cdot)\) n/a 304 4
350.6.o \(\chi_{350}(143, \cdot)\) n/a 240 4
350.6.q \(\chi_{350}(11, \cdot)\) n/a 800 8
350.6.r \(\chi_{350}(13, \cdot)\) n/a 800 8
350.6.u \(\chi_{350}(9, \cdot)\) n/a 800 8
350.6.x \(\chi_{350}(3, \cdot)\) n/a 1600 16

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(350))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(350)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(175))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(350))\)\(^{\oplus 1}\)