Properties

Label 4.38.a
Level 4
Weight 38
Character orbit 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

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

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 272163492 T + 660985375731284745 T^{2} + \)\(83\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!35\)\( T^{4} + \)\(55\!\cdots\!48\)\( T^{5} + \)\(91\!\cdots\!47\)\( T^{6} \)
$5$ \( 1 - 3641045116194 T + \)\(10\!\cdots\!75\)\( T^{2} - \)\(68\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!75\)\( T^{4} - \)\(19\!\cdots\!50\)\( T^{5} + \)\(38\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 1514184507161592 T + \)\(35\!\cdots\!97\)\( T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + \)\(66\!\cdots\!79\)\( T^{4} - \)\(52\!\cdots\!08\)\( T^{5} + \)\(63\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + 11609057084515238700 T + \)\(34\!\cdots\!13\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(39\!\cdots\!11\)\( T^{6} \)
$13$ \( 1 + 70576684907789880678 T + \)\(10\!\cdots\!95\)\( T^{2} - \)\(90\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} + \)\(19\!\cdots\!42\)\( T^{5} + \)\(44\!\cdots\!37\)\( T^{6} \)
$17$ \( 1 - \)\(21\!\cdots\!94\)\( T + \)\(72\!\cdots\!75\)\( T^{2} - \)\(14\!\cdots\!12\)\( T^{3} + \)\(24\!\cdots\!75\)\( T^{4} - \)\(24\!\cdots\!26\)\( T^{5} + \)\(38\!\cdots\!33\)\( T^{6} \)
$19$ \( 1 - \)\(15\!\cdots\!08\)\( T + \)\(45\!\cdots\!05\)\( T^{2} - \)\(61\!\cdots\!80\)\( T^{3} + \)\(93\!\cdots\!95\)\( T^{4} - \)\(67\!\cdots\!68\)\( T^{5} + \)\(87\!\cdots\!19\)\( T^{6} \)
$23$ \( 1 - \)\(86\!\cdots\!16\)\( T + \)\(51\!\cdots\!93\)\( T^{2} - \)\(75\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!79\)\( T^{4} - \)\(50\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!27\)\( T^{6} \)
$29$ \( 1 - \)\(10\!\cdots\!98\)\( T + \)\(23\!\cdots\!95\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(29\!\cdots\!55\)\( T^{4} - \)\(16\!\cdots\!38\)\( T^{5} + \)\(21\!\cdots\!29\)\( T^{6} \)
$31$ \( 1 - \)\(21\!\cdots\!08\)\( T + \)\(44\!\cdots\!21\)\( T^{2} + \)\(12\!\cdots\!68\)\( T^{3} + \)\(67\!\cdots\!31\)\( T^{4} - \)\(50\!\cdots\!68\)\( T^{5} + \)\(34\!\cdots\!31\)\( T^{6} \)
$37$ \( 1 - \)\(31\!\cdots\!02\)\( T + \)\(59\!\cdots\!07\)\( T^{2} - \)\(71\!\cdots\!80\)\( T^{3} + \)\(63\!\cdots\!19\)\( T^{4} - \)\(34\!\cdots\!78\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \)
$41$ \( 1 - \)\(26\!\cdots\!78\)\( T + \)\(35\!\cdots\!71\)\( T^{2} - \)\(29\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!51\)\( T^{4} - \)\(58\!\cdots\!58\)\( T^{5} + \)\(10\!\cdots\!41\)\( T^{6} \)
$43$ \( 1 - \)\(54\!\cdots\!80\)\( T + \)\(17\!\cdots\!29\)\( T^{2} - \)\(34\!\cdots\!80\)\( T^{3} + \)\(48\!\cdots\!47\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{5} + \)\(20\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - \)\(41\!\cdots\!84\)\( T + \)\(76\!\cdots\!05\)\( T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(56\!\cdots\!35\)\( T^{4} - \)\(22\!\cdots\!96\)\( T^{5} + \)\(40\!\cdots\!03\)\( T^{6} \)
$53$ \( 1 + \)\(15\!\cdots\!06\)\( T + \)\(82\!\cdots\!23\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(51\!\cdots\!99\)\( T^{4} + \)\(62\!\cdots\!14\)\( T^{5} + \)\(24\!\cdots\!97\)\( T^{6} \)
$59$ \( 1 + \)\(54\!\cdots\!44\)\( T + \)\(10\!\cdots\!69\)\( T^{2} + \)\(33\!\cdots\!64\)\( T^{3} + \)\(33\!\cdots\!11\)\( T^{4} + \)\(59\!\cdots\!84\)\( T^{5} + \)\(36\!\cdots\!59\)\( T^{6} \)
$61$ \( 1 - \)\(11\!\cdots\!86\)\( T + \)\(19\!\cdots\!95\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!95\)\( T^{4} - \)\(15\!\cdots\!26\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} \)
$67$ \( 1 - \)\(16\!\cdots\!92\)\( T + \)\(14\!\cdots\!77\)\( T^{2} - \)\(85\!\cdots\!00\)\( T^{3} + \)\(51\!\cdots\!79\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{5} + \)\(49\!\cdots\!83\)\( T^{6} \)
$71$ \( 1 - \)\(10\!\cdots\!04\)\( T + \)\(81\!\cdots\!45\)\( T^{2} - \)\(40\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!95\)\( T^{4} - \)\(10\!\cdots\!24\)\( T^{5} + \)\(30\!\cdots\!71\)\( T^{6} \)
$73$ \( 1 + \)\(85\!\cdots\!58\)\( T + \)\(16\!\cdots\!75\)\( T^{2} + \)\(20\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!75\)\( T^{4} + \)\(65\!\cdots\!22\)\( T^{5} + \)\(67\!\cdots\!77\)\( T^{6} \)
$79$ \( 1 + \)\(26\!\cdots\!36\)\( T + \)\(65\!\cdots\!09\)\( T^{2} + \)\(89\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} + \)\(70\!\cdots\!16\)\( T^{5} + \)\(43\!\cdots\!79\)\( T^{6} \)
$83$ \( 1 + \)\(68\!\cdots\!04\)\( T + \)\(25\!\cdots\!53\)\( T^{2} + \)\(88\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!19\)\( T^{4} + \)\(70\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!67\)\( T^{6} \)
$89$ \( 1 - \)\(24\!\cdots\!42\)\( T + \)\(60\!\cdots\!75\)\( T^{2} - \)\(72\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!75\)\( T^{4} - \)\(44\!\cdots\!22\)\( T^{5} + \)\(24\!\cdots\!89\)\( T^{6} \)
$97$ \( 1 + \)\(46\!\cdots\!98\)\( T + \)\(54\!\cdots\!87\)\( T^{2} + \)\(96\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!19\)\( T^{4} + \)\(49\!\cdots\!62\)\( T^{5} + \)\(34\!\cdots\!53\)\( T^{6} \)
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