Properties

Label 4032.2.v
Level 4032
Weight 2
Character orbit v
Rep. character \(\chi_{4032}(1583,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 96
Newform subspaces 5
Sturm bound 1536
Trace bound 13

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Defining parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.v (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 48 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 5 \)
Sturm bound: \(1536\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

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

Total New Old
Modular forms 1600 96 1504
Cusp forms 1472 96 1376
Eisenstein series 128 0 128

Trace form

\( 96q + O(q^{10}) \) \( 96q + 32q^{19} - 64q^{43} + 96q^{49} - 128q^{55} + 64q^{61} - 16q^{67} + 64q^{85} + O(q^{100}) \)

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.v.a \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-4\) \(q+(\zeta_{8}^{2}+\zeta_{8}^{3})q^{5}-q^{7}+(-\zeta_{8}^{2}+\zeta_{8}^{3})q^{11}+\cdots\)
4032.2.v.b \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-4\) \(q+(\zeta_{8}^{2}+\zeta_{8}^{3})q^{5}-q^{7}+(2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{11}+\cdots\)
4032.2.v.c \(12\) \(32.196\) 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(12\) \(q+(\beta _{3}-\beta _{5})q^{5}+q^{7}+(\beta _{1}+\beta _{3}+\beta _{5}+\cdots)q^{11}+\cdots\)
4032.2.v.d \(36\) \(32.196\) None \(0\) \(0\) \(0\) \(36\)
4032.2.v.e \(40\) \(32.196\) None \(0\) \(0\) \(0\) \(-40\)

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

\( S_{2}^{\mathrm{old}}(4032, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1344, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))
$3$ (\( \))(\( \))(\( \))
$5$ (\( ( 1 - 8 T^{2} + 25 T^{4} )( 1 + 8 T^{2} + 25 T^{4} ) \))(\( ( 1 - 8 T^{2} + 25 T^{4} )( 1 + 8 T^{2} + 25 T^{4} ) \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{3}( 1 + 8 T^{2} + 25 T^{4} )^{3} \))
$7$ (\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{12} \))
$11$ (\( 1 + 82 T^{4} + 14641 T^{8} \))(\( 1 - 206 T^{4} + 14641 T^{8} \))(\( 1 + 22 T^{4} + 14047 T^{8} + 1317556 T^{12} + 205662127 T^{16} + 4715895382 T^{20} + 3138428376721 T^{24} \))
$13$ (\( ( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 88 T^{3} + 315 T^{4} - 1728 T^{5} + 7616 T^{6} - 22464 T^{7} + 53235 T^{8} - 193336 T^{9} + 913952 T^{10} - 2970344 T^{11} + 4826809 T^{12} )^{2} \))
$17$ (\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 6 T + 17 T^{2} )^{2}( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} + 223 T^{4} - 3076 T^{6} + 64447 T^{8} + 167042 T^{10} + 24137569 T^{12} )^{2} \))
$19$ (\( ( 1 + 361 T^{4} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} + 152 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 361 T^{4} )^{6} \))
$23$ (\( ( 1 + 4 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 108 T^{2} + 5363 T^{4} - 156760 T^{6} + 2837027 T^{8} - 30222828 T^{10} + 148035889 T^{12} )^{2} \))
$29$ (\( 1 + 1234 T^{4} + 707281 T^{8} \))(\( 1 - 1646 T^{4} + 707281 T^{8} \))(\( 1 - 3530 T^{4} + 5536831 T^{8} - 5509074188 T^{12} + 3916095366511 T^{16} - 1765869837752330 T^{20} + 353814783205469041 T^{24} \))
$31$ (\( ( 1 + 2 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 106 T^{2} + 5071 T^{4} - 169228 T^{6} + 4873231 T^{8} - 97893226 T^{10} + 887503681 T^{12} )^{2} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 74 T^{3} + 1369 T^{4} )^{6} \))
$41$ (\( ( 1 + 74 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 118 T^{2} + 4879 T^{4} + 140692 T^{6} + 8201599 T^{8} + 333439798 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( ( 1 - 6 T + 18 T^{2} - 258 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 14 T + 98 T^{2} + 602 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 10 T + 50 T^{2} - 398 T^{3} - 1785 T^{4} + 35220 T^{5} - 183748 T^{6} + 1514460 T^{7} - 3300465 T^{8} - 31643786 T^{9} + 170940050 T^{10} - 1470084430 T^{11} + 6321363049 T^{12} )^{2} \))
$47$ (\( ( 1 + 62 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 154 T^{2} + 13423 T^{4} + 756268 T^{6} + 29651407 T^{8} + 751470874 T^{10} + 10779215329 T^{12} )^{2} \))
$53$ (\( ( 1 + 2809 T^{4} )^{2} \))(\( 1 - 5582 T^{4} + 7890481 T^{8} \))(\( 1 + 790 T^{4} - 2418209 T^{8} + 5899921588 T^{12} - 19080832168529 T^{16} + 49185155424975190 T^{20} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 + 3442 T^{4} + 12117361 T^{8} \))(\( 1 - 4046 T^{4} + 12117361 T^{8} \))(\( 1 - 6314 T^{4} + 48612511 T^{8} - 159936163532 T^{12} + 589055344903471 T^{16} - 927087383033682794 T^{20} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( ( 1 + 4 T + 8 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 488 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 16 T + 128 T^{2} + 1440 T^{3} + 18779 T^{4} + 143024 T^{5} + 921472 T^{6} + 8724464 T^{7} + 69876659 T^{8} + 326852640 T^{9} + 1772267648 T^{10} + 13513540816 T^{11} + 51520374361 T^{12} )^{2} \))
$67$ (\( ( 1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 22 T + 242 T^{2} + 2850 T^{3} + 35511 T^{4} + 315092 T^{5} + 2399612 T^{6} + 21111164 T^{7} + 159408879 T^{8} + 857174550 T^{9} + 4876571282 T^{10} + 29702752354 T^{11} + 90458382169 T^{12} )^{2} \))
$71$ (\( ( 1 - 44 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 92 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 188 T^{2} + 25939 T^{4} - 2091512 T^{6} + 130758499 T^{8} - 4777396028 T^{10} + 128100283921 T^{12} )^{2} \))
$73$ (\( ( 1 + 50 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 16 T + 73 T^{2} )^{2}( 1 + 16 T + 73 T^{2} )^{2} \))(\( ( 1 - 282 T^{2} + 42047 T^{4} - 3793004 T^{6} + 224068463 T^{8} - 8008303962 T^{10} + 151334226289 T^{12} )^{2} \))
$79$ (\( ( 1 + 98 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 154 T^{2} + 1711 T^{4} + 731348 T^{6} + 10678351 T^{8} - 5998312474 T^{10} + 243087455521 T^{12} )^{2} \))
$83$ (\( 1 + 8722 T^{4} + 47458321 T^{8} \))(\( 1 + 8722 T^{4} + 47458321 T^{8} \))(\( 1 - 12810 T^{4} + 46034687 T^{8} + 36989567092 T^{12} + 2184728952780527 T^{16} - 28851863493701115210 T^{20} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( ( 1 + 50 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 110 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 46 T^{2} + 3343 T^{4} + 916708 T^{6} + 26479903 T^{8} + 2886143086 T^{10} + 496981290961 T^{12} )^{2} \))
$97$ (\( ( 1 + 6 T + 97 T^{2} )^{4} \))(\( ( 1 + 10 T + 97 T^{2} )^{4} \))(\( ( 1 + 14 T - 33 T^{2} - 1948 T^{3} - 3201 T^{4} + 131726 T^{5} + 912673 T^{6} )^{4} \))
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