Properties

Label 4032.2.h
Level 4032
Weight 2
Character orbit h
Rep. character \(\chi_{4032}(575,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 8
Sturm bound 1536
Trace bound 25

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Defining parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(1536\)
Trace bound: \(25\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4032, [\chi])\).

Total New Old
Modular forms 816 48 768
Cusp forms 720 48 672
Eisenstein series 96 0 96

Trace form

\( 48q + O(q^{10}) \) \( 48q - 48q^{25} - 48q^{49} - 64q^{61} + 64q^{85} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4032, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4032.2.h.a \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-2\zeta_{8}^{2}q^{5}+\zeta_{8}q^{7}-\zeta_{8}^{3}q^{11}-6q^{13}+\cdots\)
4032.2.h.b \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{2}q^{5}+\zeta_{8}q^{7}+2\zeta_{8}^{3}q^{11}+5\zeta_{8}^{2}q^{17}+\cdots\)
4032.2.h.c \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-2\zeta_{8}^{2}q^{5}+\zeta_{8}q^{7}+\zeta_{8}^{3}q^{11}+2q^{13}+\cdots\)
4032.2.h.d \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{7}+3\zeta_{8}^{3}q^{11}+2q^{13}+2\zeta_{8}^{2}q^{17}+\cdots\)
4032.2.h.e \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{2}q^{5}+\zeta_{8}q^{7}+4\zeta_{8}^{3}q^{11}+4q^{13}+\cdots\)
4032.2.h.f \(8\) \(32.196\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{24}^{2}+\zeta_{24}^{7})q^{5}+\zeta_{24}q^{7}+(\zeta_{24}^{3}+\cdots)q^{11}+\cdots\)
4032.2.h.g \(8\) \(32.196\) 8.0.5473632256.1 None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{5}+\beta _{6})q^{5}-\beta _{1}q^{7}+(-\beta _{4}-2\beta _{7})q^{11}+\cdots\)
4032.2.h.h \(12\) \(32.196\) 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{5}+\beta _{3}q^{7}+(-\beta _{5}-\beta _{6})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4032, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4032, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(672, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1344, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2016, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} )^{4} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 6 T^{4} - 100 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 2 T^{2} + 34 T^{4} - 50 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 12 T^{2} + 95 T^{4} - 568 T^{6} + 2375 T^{8} - 7500 T^{10} + 15625 T^{12} )^{2} \))
$7$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{6} \))
$11$ (\( ( 1 + 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 114 T^{4} + 1936 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 18 T^{2} + 170 T^{4} + 2178 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 38 T^{2} + 715 T^{4} + 9068 T^{6} + 86515 T^{8} + 556358 T^{10} + 1771561 T^{12} )^{2} \))
$13$ (\( ( 1 + 6 T + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 - 4 T + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 13 T^{2} )^{8} \))(\( ( 1 - 2 T + 10 T^{2} - 26 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 + 11 T^{2} + 16 T^{3} + 143 T^{4} + 2197 T^{6} )^{4} \))
$17$ (\( ( 1 - 6 T + 17 T^{2} )^{2}( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 16 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 52 T^{2} + 1206 T^{4} - 15028 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 26 T^{2} + 322 T^{4} - 7514 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 68 T^{2} + 2167 T^{4} - 44168 T^{6} + 626263 T^{8} - 5679428 T^{10} + 24137569 T^{12} )^{2} \))
$19$ (\( ( 1 + 26 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 44 T^{2} + 1014 T^{4} - 15884 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 50 T^{2} + 1639 T^{4} - 35804 T^{6} + 591679 T^{8} - 6516050 T^{10} + 47045881 T^{12} )^{2} \))
$23$ (\( ( 1 + 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 64 T^{2} + 1890 T^{4} + 33856 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 66 T^{2} + 1994 T^{4} + 34914 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 2 T^{2} + 979 T^{4} + 3772 T^{6} + 517891 T^{8} - 559682 T^{10} + 148035889 T^{12} )^{2} \))
$29$ (\( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 64 T^{2} + 2514 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 32 T^{2} + 850 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{6} \))
$31$ (\( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 28 T^{2} + 390 T^{4} - 26908 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 20 T^{2} + 934 T^{4} + 19220 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 58 T^{2} + 3727 T^{4} - 113644 T^{6} + 3581647 T^{8} - 53564218 T^{10} + 887503681 T^{12} )^{2} \))
$37$ (\( ( 1 + 8 T + 37 T^{2} )^{4} \))(\( ( 1 + 2 T + 37 T^{2} )^{4} \))(\( ( 1 + 37 T^{2} )^{4} \))(\( ( 1 + 8 T + 37 T^{2} )^{4} \))(\( ( 1 + 2 T + 37 T^{2} )^{4} \))(\( ( 1 - 8 T + 42 T^{2} - 296 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 - 14 T + 106 T^{2} - 518 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 + 2 T + 47 T^{2} + 276 T^{3} + 1739 T^{4} + 2738 T^{5} + 50653 T^{6} )^{4} \))
$41$ (\( ( 1 - 10 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 64 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 52 T^{2} + 3606 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 66 T^{2} + 3074 T^{4} - 110946 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 132 T^{2} + 10823 T^{4} - 527752 T^{6} + 18193463 T^{8} - 373000452 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( ( 1 + 58 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 - 98 T^{2} + 4567 T^{4} - 172988 T^{6} + 8444383 T^{8} - 335042498 T^{10} + 6321363049 T^{12} )^{2} \))
$47$ (\( ( 1 + 86 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 86 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 86 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 22 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 + 20 T^{2} - 2282 T^{4} + 44180 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 10 T^{2} + 4591 T^{4} + 70732 T^{6} + 10141519 T^{8} + 48796810 T^{10} + 10779215329 T^{12} )^{2} \))
$53$ (\( ( 1 - 104 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 88 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )^{4} \))(\( ( 1 - 128 T^{2} + 8626 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 184 T^{2} + 16171 T^{4} - 975856 T^{6} + 45424339 T^{8} - 1451848504 T^{10} + 22164361129 T^{12} )^{2} \))
$59$ (\( ( 1 - 82 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 86 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 110 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 110 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 110 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 132 T^{2} + 8870 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 146 T^{2} + 7799 T^{4} + 306396 T^{6} + 27148319 T^{8} + 1769134706 T^{10} + 42180533641 T^{12} )^{2} \))
$61$ (\( ( 1 + 2 T + 61 T^{2} )^{4} \))(\( ( 1 + 14 T + 61 T^{2} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )^{4} \))(\( ( 1 - 2 T + 61 T^{2} )^{4} \))(\( ( 1 + 6 T + 61 T^{2} )^{4} \))(\( ( 1 + 110 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 2 T + 61 T^{2} )^{8} \))(\( ( 1 + 14 T + 187 T^{2} + 1700 T^{3} + 11407 T^{4} + 52094 T^{5} + 226981 T^{6} )^{4} \))
$67$ (\( ( 1 + 62 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 + 44 T^{2} + 2550 T^{4} + 197516 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 232 T^{2} + 22366 T^{4} - 1041448 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 186 T^{2} + 13175 T^{4} - 680684 T^{6} + 59142575 T^{8} - 3748108506 T^{10} + 90458382169 T^{12} )^{2} \))
$71$ (\( ( 1 - 20 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 92 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 160 T^{2} + 14754 T^{4} + 806560 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 130 T^{2} + 14154 T^{4} + 655330 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 158 T^{2} + 5683 T^{4} - 107140 T^{6} + 28648003 T^{8} + 4015045598 T^{10} + 128100283921 T^{12} )^{2} \))
$73$ (\( ( 1 - 2 T + 73 T^{2} )^{4} \))(\( ( 1 + 4 T + 73 T^{2} )^{4} \))(\( ( 1 + 6 T + 73 T^{2} )^{4} \))(\( ( 1 - 14 T + 73 T^{2} )^{4} \))(\( ( 1 + 8 T + 73 T^{2} )^{4} \))(\( ( 1 - 8 T + 150 T^{2} - 584 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 + 6 T + 138 T^{2} + 438 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 + 47 T^{2} - 352 T^{3} + 3431 T^{4} + 389017 T^{6} )^{4} \))
$79$ (\( ( 1 - 58 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 154 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 50 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 8 T^{2} + 11886 T^{4} - 49928 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 162 T^{2} + 19359 T^{4} - 2006332 T^{6} + 120819519 T^{8} - 6309913122 T^{10} + 243087455521 T^{12} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 38 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 116 T^{2} + 10230 T^{4} - 799124 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 44 T^{2} + 9910 T^{4} + 303116 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 370 T^{2} + 65191 T^{4} + 6834652 T^{6} + 449100799 T^{8} + 17559578770 T^{10} + 326940373369 T^{12} )^{2} \))
$89$ (\( ( 1 + 22 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 80 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 106 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 80 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 340 T^{2} + 44694 T^{4} - 2693140 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 338 T^{2} + 44386 T^{4} - 2677298 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 308 T^{2} + 49895 T^{4} - 5241384 T^{6} + 395218295 T^{8} - 19324610228 T^{10} + 496981290961 T^{12} )^{2} \))
$97$ (\( ( 1 + 6 T + 97 T^{2} )^{4} \))(\( ( 1 - 12 T + 97 T^{2} )^{4} \))(\( ( 1 + 14 T + 97 T^{2} )^{4} \))(\( ( 1 - 14 T + 97 T^{2} )^{4} \))(\( ( 1 + 16 T + 97 T^{2} )^{4} \))(\( ( 1 - 16 T + 246 T^{2} - 1552 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 + 6 T + 186 T^{2} + 582 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 + 135 T^{2} + 128 T^{3} + 13095 T^{4} + 912673 T^{6} )^{4} \))
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