Properties

Label 4.38.a
Level 4
Weight 38
Character orbit a
Rep. character \(\chi_{4}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 1
Sturm bound 19
Trace bound 0

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Defining parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 4.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(19\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(4))\).

Total New Old
Modular forms 20 3 17
Cusp forms 17 3 14
Eisenstein series 3 0 3

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

\(2\)Dim.
\(-\)\(3\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 272163492q^{3} \) \(\mathstrut +\mathstrut 3641045116194q^{5} \) \(\mathstrut +\mathstrut 1514184507161592q^{7} \) \(\mathstrut +\mathstrut 102953932588056663q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 272163492q^{3} \) \(\mathstrut +\mathstrut 3641045116194q^{5} \) \(\mathstrut +\mathstrut 1514184507161592q^{7} \) \(\mathstrut +\mathstrut 102953932588056663q^{9} \) \(\mathstrut -\mathstrut 11609057084515238700q^{11} \) \(\mathstrut -\mathstrut 70576684907789880678q^{13} \) \(\mathstrut -\mathstrut 2933969461740641207448q^{15} \) \(\mathstrut +\mathstrut 21449865005452482229494q^{17} \) \(\mathstrut +\mathstrut 158301069447345491378508q^{19} \) \(\mathstrut -\mathstrut 1151851369077659765142432q^{21} \) \(\mathstrut +\mathstrut 866923505554430216912616q^{23} \) \(\mathstrut +\mathstrut 21645722139716923137352461q^{25} \) \(\mathstrut +\mathstrut 147832683778944657999655704q^{27} \) \(\mathstrut +\mathstrut 1010653042580699333097500298q^{29} \) \(\mathstrut +\mathstrut 2180097844230541775332759008q^{31} \) \(\mathstrut +\mathstrut 31048498538042201781617598480q^{33} \) \(\mathstrut +\mathstrut 100605979662426959198661319248q^{35} \) \(\mathstrut +\mathstrut 311298722814240927017686969602q^{37} \) \(\mathstrut +\mathstrut 979152159930324950801131779528q^{39} \) \(\mathstrut +\mathstrut 2635940958803787572844371014878q^{41} \) \(\mathstrut +\mathstrut 5419069191259593947069584358580q^{43} \) \(\mathstrut +\mathstrut 8099313551230663632839319909594q^{45} \) \(\mathstrut +\mathstrut 4156587745521390586896893605584q^{47} \) \(\mathstrut -\mathstrut 13495549806538130861979895910709q^{49} \) \(\mathstrut -\mathstrut 89987135087681316961287710640264q^{51} \) \(\mathstrut -\mathstrut 159108065600661790773554978344206q^{53} \) \(\mathstrut -\mathstrut 501378639325706351372809856249160q^{55} \) \(\mathstrut -\mathstrut 698546758941918182526174576638352q^{57} \) \(\mathstrut -\mathstrut 540908935574431553277616956343644q^{59} \) \(\mathstrut +\mathstrut 1178713661089330885859771001735786q^{61} \) \(\mathstrut +\mathstrut 3403110131746397253746662721587992q^{63} \) \(\mathstrut +\mathstrut 6554640894858129925078606186828668q^{65} \) \(\mathstrut +\mathstrut 16889218571987157419703623848158492q^{67} \) \(\mathstrut +\mathstrut 17143397087023799436715277896360224q^{69} \) \(\mathstrut +\mathstrut 10309319797367225088925816018968504q^{71} \) \(\mathstrut -\mathstrut 8520196523581770817793340839438658q^{73} \) \(\mathstrut -\mathstrut 84054788877364659786396368345168412q^{75} \) \(\mathstrut -\mathstrut 208861611906418376677492203082464480q^{77} \) \(\mathstrut -\mathstrut 264775084715781163780628803947993936q^{79} \) \(\mathstrut -\mathstrut 370309441007212745944873457987660853q^{81} \) \(\mathstrut -\mathstrut 68956651634275475176080339084470004q^{83} \) \(\mathstrut +\mathstrut 522499848715075310236425753489988836q^{85} \) \(\mathstrut +\mathstrut 911140443393561025920716736778215048q^{87} \) \(\mathstrut +\mathstrut 2493391276449702933740991498959962542q^{89} \) \(\mathstrut +\mathstrut 2807109998185271068204514151779860752q^{91} \) \(\mathstrut +\mathstrut 5409735857664632612107028249216969088q^{93} \) \(\mathstrut +\mathstrut 3240594516611672218958902717719033864q^{95} \) \(\mathstrut -\mathstrut 4697277125747917937176391731430481498q^{97} \) \(\mathstrut -\mathstrut 14740636216794395787136495035497749500q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(4))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
4.38.a.a \(3\) \(34.686\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(-272163492\) \(36\!\cdots\!94\) \(15\!\cdots\!92\) \(-\) \(q+(-90721164-\beta _{1})q^{3}+(1213681705398+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_0(4))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_0(4)) \cong \) \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)