Properties

Label 38.10.a
Level $38$
Weight $10$
Character orbit 38.a
Rep. character $\chi_{38}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $5$
Sturm bound $50$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(50\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(38))\).

Total New Old
Modular forms 47 13 34
Cusp forms 43 13 30
Eisenstein series 4 0 4

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

\(2\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(5\)
Minus space\(-\)\(8\)

Trace form

\( 13q - 16q^{2} + 296q^{3} + 3328q^{4} - 2308q^{5} + 1952q^{6} + 5944q^{7} - 4096q^{8} + 75239q^{9} + O(q^{10}) \) \( 13q - 16q^{2} + 296q^{3} + 3328q^{4} - 2308q^{5} + 1952q^{6} + 5944q^{7} - 4096q^{8} + 75239q^{9} - 7840q^{10} + 12554q^{11} + 75776q^{12} + 43522q^{13} + 127424q^{14} - 179392q^{15} + 851968q^{16} - 642222q^{17} - 58064q^{18} - 130321q^{19} - 590848q^{20} + 1565376q^{21} + 1745856q^{22} + 3201686q^{23} + 499712q^{24} + 4285001q^{25} - 8220992q^{26} + 8524160q^{27} + 1521664q^{28} + 7376494q^{29} + 11703488q^{30} + 17897460q^{31} - 1048576q^{32} - 46155212q^{33} + 23043808q^{34} + 7874562q^{35} + 19261184q^{36} - 4660078q^{37} - 6255408q^{38} + 101912994q^{39} - 2007040q^{40} - 16660946q^{41} - 24989216q^{42} - 43079262q^{43} + 3213824q^{44} - 101525240q^{45} - 15514560q^{46} + 52264094q^{47} + 19398656q^{48} + 37038761q^{49} - 90002160q^{50} - 23820580q^{51} + 11141632q^{52} + 96950450q^{53} - 45272032q^{54} - 132700946q^{55} + 32620544q^{56} - 21112002q^{57} + 235968832q^{58} - 358398768q^{59} - 45924352q^{60} - 229861552q^{61} - 166077504q^{62} - 595784674q^{63} + 218103808q^{64} + 78516376q^{65} + 336624960q^{66} - 222504220q^{67} - 164408832q^{68} - 191261588q^{69} + 166765632q^{70} + 57336224q^{71} - 14864384q^{72} - 438580250q^{73} - 657503904q^{74} + 1236573180q^{75} - 33362176q^{76} + 713884098q^{77} - 14188480q^{78} - 1270558956q^{79} - 151257088q^{80} + 1574745341q^{81} - 225638176q^{82} + 1130392872q^{83} + 400736256q^{84} + 59139426q^{85} - 219498240q^{86} - 1166432318q^{87} + 446939136q^{88} + 1297806654q^{89} + 1927922080q^{90} + 2158403728q^{91} + 819631616q^{92} + 500285756q^{93} - 37863552q^{94} - 653168852q^{95} + 127926272q^{96} + 2433642478q^{97} - 727871248q^{98} + 1414521614q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(38))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 19
38.10.a.a \(1\) \(19.571\) \(\Q\) None \(16\) \(-119\) \(-684\) \(9149\) \(-\) \(-\) \(q+2^{4}q^{2}-119q^{3}+2^{8}q^{4}-684q^{5}+\cdots\)
38.10.a.b \(1\) \(19.571\) \(\Q\) None \(16\) \(102\) \(-1581\) \(-4865\) \(-\) \(-\) \(q+2^{4}q^{2}+102q^{3}+2^{8}q^{4}-1581q^{5}+\cdots\)
38.10.a.c \(3\) \(19.571\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-48\) \(3\) \(486\) \(-13317\) \(+\) \(+\) \(q-2^{4}q^{2}+(1+\beta _{1})q^{3}+2^{8}q^{4}+(162+\cdots)q^{5}+\cdots\)
38.10.a.d \(4\) \(19.571\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-64\) \(84\) \(-1395\) \(12307\) \(+\) \(-\) \(q-2^{4}q^{2}+(21+\beta _{1})q^{3}+2^{8}q^{4}+(-350+\cdots)q^{5}+\cdots\)
38.10.a.e \(4\) \(19.571\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(64\) \(226\) \(866\) \(2670\) \(-\) \(+\) \(q+2^{4}q^{2}+(57+\beta _{2})q^{3}+2^{8}q^{4}+(218+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(38))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(38)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)