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Properties

Label 4.38.a
Level $4$
Weight $38$
Character orbit 4.a
Rep. character $\chi_{4}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $1$
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: \(\Q\)
Newform subspaces: \( 1 \)
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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(4))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
4.38.a.a 4.a 1.a $3$ $34.686$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(-272163492\) \(36\!\cdots\!94\) \(15\!\cdots\!92\) $-$ $\mathrm{SU}(2)$ \(q+(-90721164-\beta _{1})q^{3}+(1213681705398+\cdots)q^{5}+\cdots\)

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

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