Properties

Label 32.6.a
Level 32
Weight 6
Character orbit a
Rep. character \(\chi_{32}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 4
Sturm bound 24
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 32.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(32))\).

Total New Old
Modular forms 24 5 19
Cusp forms 16 5 11
Eisenstein series 8 0 8

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.
\(+\)\(2\)
\(-\)\(3\)

Trace form

\( 5q + 38q^{5} + 449q^{9} + O(q^{10}) \) \( 5q + 38q^{5} + 449q^{9} + 110q^{13} + 1610q^{17} - 5888q^{21} - 4277q^{25} + 3678q^{29} + 13184q^{33} - 7258q^{37} - 30766q^{41} + 63214q^{45} + 57789q^{49} - 76490q^{53} - 132992q^{57} + 108510q^{61} + 113476q^{65} - 226560q^{69} - 120318q^{73} + 195328q^{77} + 270029q^{81} - 131124q^{85} - 174702q^{89} + 152576q^{93} + 359642q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(32))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
32.6.a.a \(1\) \(5.132\) \(\Q\) None \(0\) \(-8\) \(14\) \(-208\) \(+\) \(q-8q^{3}+14q^{5}-208q^{7}-179q^{9}+\cdots\)
32.6.a.b \(1\) \(5.132\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-82\) \(0\) \(+\) \(q-82q^{5}-3^{5}q^{9}-1194q^{13}+2242q^{17}+\cdots\)
32.6.a.c \(1\) \(5.132\) \(\Q\) None \(0\) \(8\) \(14\) \(208\) \(-\) \(q+8q^{3}+14q^{5}+208q^{7}-179q^{9}+\cdots\)
32.6.a.d \(2\) \(5.132\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(92\) \(0\) \(-\) \(q+\beta q^{3}+46q^{5}-6\beta q^{7}+525q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(32))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(32)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 8 T + 243 T^{2} \))(\( 1 + 243 T^{2} \))(\( 1 - 8 T + 243 T^{2} \))(\( 1 - 282 T^{2} + 59049 T^{4} \))
$5$ (\( 1 - 14 T + 3125 T^{2} \))(\( 1 + 82 T + 3125 T^{2} \))(\( 1 - 14 T + 3125 T^{2} \))(\( ( 1 - 46 T + 3125 T^{2} )^{2} \))
$7$ (\( 1 + 208 T + 16807 T^{2} \))(\( 1 + 16807 T^{2} \))(\( 1 - 208 T + 16807 T^{2} \))(\( 1 + 5966 T^{2} + 282475249 T^{4} \))
$11$ (\( 1 + 536 T + 161051 T^{2} \))(\( 1 + 161051 T^{2} \))(\( 1 - 536 T + 161051 T^{2} \))(\( 1 + 315190 T^{2} + 25937424601 T^{4} \))
$13$ (\( 1 - 694 T + 371293 T^{2} \))(\( 1 + 1194 T + 371293 T^{2} \))(\( 1 - 694 T + 371293 T^{2} \))(\( ( 1 + 42 T + 371293 T^{2} )^{2} \))
$17$ (\( 1 + 1278 T + 1419857 T^{2} \))(\( 1 - 2242 T + 1419857 T^{2} \))(\( 1 + 1278 T + 1419857 T^{2} \))(\( ( 1 - 962 T + 1419857 T^{2} )^{2} \))
$19$ (\( 1 - 1112 T + 2476099 T^{2} \))(\( 1 + 2476099 T^{2} \))(\( 1 + 1112 T + 2476099 T^{2} \))(\( 1 + 632198 T^{2} + 6131066257801 T^{4} \))
$23$ (\( 1 - 3216 T + 6436343 T^{2} \))(\( 1 + 6436343 T^{2} \))(\( 1 + 3216 T + 6436343 T^{2} \))(\( 1 + 2891758 T^{2} + 41426511213649 T^{4} \))
$29$ (\( 1 - 2918 T + 20511149 T^{2} \))(\( 1 - 2950 T + 20511149 T^{2} \))(\( 1 - 2918 T + 20511149 T^{2} \))(\( ( 1 + 2554 T + 20511149 T^{2} )^{2} \))
$31$ (\( 1 + 2624 T + 28629151 T^{2} \))(\( 1 + 28629151 T^{2} \))(\( 1 - 2624 T + 28629151 T^{2} \))(\( 1 + 53276990 T^{2} + 819628286980801 T^{4} \))
$37$ (\( 1 + 9458 T + 69343957 T^{2} \))(\( 1 + 12242 T + 69343957 T^{2} \))(\( 1 + 9458 T + 69343957 T^{2} \))(\( ( 1 - 11950 T + 69343957 T^{2} )^{2} \))
$41$ (\( 1 - 170 T + 115856201 T^{2} \))(\( 1 + 20950 T + 115856201 T^{2} \))(\( 1 - 170 T + 115856201 T^{2} \))(\( ( 1 + 5078 T + 115856201 T^{2} )^{2} \))
$43$ (\( 1 + 19928 T + 147008443 T^{2} \))(\( 1 + 147008443 T^{2} \))(\( 1 - 19928 T + 147008443 T^{2} \))(\( 1 + 136416374 T^{2} + 21611482313284249 T^{4} \))
$47$ (\( 1 - 32 T + 229345007 T^{2} \))(\( 1 + 229345007 T^{2} \))(\( 1 + 32 T + 229345007 T^{2} \))(\( 1 + 307289566 T^{2} + 52599132235830049 T^{4} \))
$53$ (\( 1 + 22178 T + 418195493 T^{2} \))(\( 1 - 7294 T + 418195493 T^{2} \))(\( 1 + 22178 T + 418195493 T^{2} \))(\( ( 1 + 19714 T + 418195493 T^{2} )^{2} \))
$59$ (\( 1 - 41480 T + 714924299 T^{2} \))(\( 1 + 714924299 T^{2} \))(\( 1 + 41480 T + 714924299 T^{2} \))(\( 1 + 1350713110 T^{2} + 511116753300641401 T^{4} \))
$61$ (\( 1 - 15462 T + 844596301 T^{2} \))(\( 1 - 18950 T + 844596301 T^{2} \))(\( 1 - 15462 T + 844596301 T^{2} \))(\( ( 1 - 29318 T + 844596301 T^{2} )^{2} \))
$67$ (\( 1 + 20744 T + 1350125107 T^{2} \))(\( 1 + 1350125107 T^{2} \))(\( 1 - 20744 T + 1350125107 T^{2} \))(\( 1 + 2415413606 T^{2} + 1822837804551761449 T^{4} \))
$71$ (\( 1 - 28592 T + 1804229351 T^{2} \))(\( 1 + 1804229351 T^{2} \))(\( 1 + 28592 T + 1804229351 T^{2} \))(\( 1 - 2948789810 T^{2} + 3255243551009881201 T^{4} \))
$73$ (\( 1 + 53670 T + 2073071593 T^{2} \))(\( 1 + 88806 T + 2073071593 T^{2} \))(\( 1 + 53670 T + 2073071593 T^{2} \))(\( ( 1 - 37914 T + 2073071593 T^{2} )^{2} \))
$79$ (\( 1 + 69152 T + 3077056399 T^{2} \))(\( 1 + 3077056399 T^{2} \))(\( 1 - 69152 T + 3077056399 T^{2} \))(\( 1 - 1729880290 T^{2} + 9468276082626847201 T^{4} \))
$83$ (\( 1 + 37800 T + 3939040643 T^{2} \))(\( 1 + 3939040643 T^{2} \))(\( 1 - 37800 T + 3939040643 T^{2} \))(\( 1 + 6331666438 T^{2} + 15516041187205853449 T^{4} \))
$89$ (\( 1 + 126806 T + 5584059449 T^{2} \))(\( 1 - 51050 T + 5584059449 T^{2} \))(\( 1 + 126806 T + 5584059449 T^{2} \))(\( ( 1 - 13930 T + 5584059449 T^{2} )^{2} \))
$97$ (\( 1 - 62290 T + 8587340257 T^{2} \))(\( 1 + 92142 T + 8587340257 T^{2} \))(\( 1 - 62290 T + 8587340257 T^{2} \))(\( ( 1 - 163602 T + 8587340257 T^{2} )^{2} \))
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