Properties

Label 338.4
Level 338
Weight 4
Dimension 3507
Nonzero newspaces 8
Sturm bound 28392
Trace bound 1

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Defining parameters

Level: \( N \) = \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 8 \)
Sturm bound: \(28392\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(338))\).

Total New Old
Modular forms 10875 3507 7368
Cusp forms 10419 3507 6912
Eisenstein series 456 0 456

Trace form

\( 3507 q - 144 q^{7} - 48 q^{8} + O(q^{10}) \) \( 3507 q - 144 q^{7} - 48 q^{8} + 180 q^{10} + 240 q^{11} + 96 q^{12} + 288 q^{13} + 240 q^{14} + 288 q^{15} - 474 q^{17} - 540 q^{18} - 1368 q^{19} - 264 q^{20} - 240 q^{21} + 456 q^{23} + 750 q^{25} + 1656 q^{27} - 576 q^{28} - 2106 q^{29} - 1632 q^{30} - 1128 q^{31} - 912 q^{33} + 864 q^{34} + 1560 q^{35} + 1536 q^{36} + 1374 q^{37} + 2736 q^{38} + 1404 q^{39} + 2718 q^{41} + 1872 q^{42} + 336 q^{43} + 384 q^{44} - 210 q^{45} - 864 q^{46} - 1704 q^{47} - 3672 q^{49} - 5388 q^{50} - 8880 q^{51} - 852 q^{52} - 2088 q^{53} - 3312 q^{54} - 192 q^{56} + 1128 q^{57} + 1188 q^{58} + 2328 q^{59} + 2112 q^{60} + 1326 q^{61} + 4176 q^{62} + 1440 q^{63} + 384 q^{64} - 165 q^{65} + 5952 q^{66} + 2472 q^{67} + 312 q^{68} + 7752 q^{69} + 2448 q^{70} + 4416 q^{71} + 384 q^{72} + 3168 q^{73} - 5772 q^{74} - 5400 q^{75} - 5472 q^{76} - 5856 q^{77} - 5832 q^{78} - 7440 q^{79} - 1056 q^{80} - 3744 q^{81} + 276 q^{82} + 8088 q^{83} + 2688 q^{84} + 9222 q^{85} + 3360 q^{86} + 9984 q^{87} + 1920 q^{89} + 10800 q^{90} + 2160 q^{91} + 4224 q^{92} - 3336 q^{93} + 9360 q^{94} - 4272 q^{95} - 72 q^{97} - 9216 q^{98} - 11088 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(338))\)

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
338.4.a \(\chi_{338}(1, \cdot)\) 338.4.a.a 1 1
338.4.a.b 1
338.4.a.c 1
338.4.a.d 1
338.4.a.e 1
338.4.a.f 2
338.4.a.g 2
338.4.a.h 2
338.4.a.i 2
338.4.a.j 3
338.4.a.k 3
338.4.a.l 4
338.4.a.m 4
338.4.a.n 6
338.4.a.o 6
338.4.b \(\chi_{338}(337, \cdot)\) 338.4.b.a 2 1
338.4.b.b 2
338.4.b.c 2
338.4.b.d 2
338.4.b.e 4
338.4.b.f 6
338.4.b.g 8
338.4.b.h 12
338.4.c \(\chi_{338}(191, \cdot)\) 338.4.c.a 2 2
338.4.c.b 2
338.4.c.c 2
338.4.c.d 2
338.4.c.e 2
338.4.c.f 2
338.4.c.g 2
338.4.c.h 4
338.4.c.i 4
338.4.c.j 4
338.4.c.k 6
338.4.c.l 6
338.4.c.m 8
338.4.c.n 8
338.4.c.o 12
338.4.c.p 12
338.4.e \(\chi_{338}(23, \cdot)\) 338.4.e.a 4 2
338.4.e.b 4
338.4.e.c 4
338.4.e.d 4
338.4.e.e 8
338.4.e.f 8
338.4.e.g 8
338.4.e.h 12
338.4.e.i 24
338.4.g \(\chi_{338}(27, \cdot)\) n/a 540 12
338.4.h \(\chi_{338}(25, \cdot)\) n/a 552 12
338.4.i \(\chi_{338}(3, \cdot)\) n/a 1080 24
338.4.k \(\chi_{338}(17, \cdot)\) n/a 1104 24

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(338))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(338)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(169))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(338))\)\(^{\oplus 1}\)