Properties

Label 4032.2.j
Level 4032
Weight 2
Character orbit j
Rep. character \(\chi_{4032}(2591,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 6
Sturm bound 1536
Trace bound 43

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Defining parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.j (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(1536\)
Trace bound: \(43\)
Distinguishing \(T_p\): \(5\), \(19\), \(43\)

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

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

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.j.a \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\zeta_{8}^{3}q^{5}-\zeta_{8}q^{7}-\zeta_{8}^{2}q^{11}-2\zeta_{8}q^{13}+\cdots\)
4032.2.j.b \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}q^{7}-\zeta_{8}^{2}q^{11}-6\zeta_{8}q^{13}-\zeta_{8}^{3}q^{23}+\cdots\)
4032.2.j.c \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}q^{7}+\zeta_{8}^{2}q^{11}+6\zeta_{8}q^{13}-\zeta_{8}^{3}q^{23}+\cdots\)
4032.2.j.d \(4\) \(32.196\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\zeta_{8}^{3}q^{5}-\zeta_{8}q^{7}-\zeta_{8}^{2}q^{11}+2\zeta_{8}q^{13}+\cdots\)
4032.2.j.e \(16\) \(32.196\) 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{13}q^{5}+\beta _{1}q^{7}+\beta _{9}q^{11}-\beta _{5}q^{13}+\cdots\)
4032.2.j.f \(16\) \(32.196\) 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{12}q^{5}+\beta _{8}q^{7}+(-\beta _{11}-\beta _{13}+\cdots)q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(4032, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(672, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1344, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2016, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 5 T^{2} )^{4} \))(\( ( 1 + 5 T^{2} )^{4} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 54 T^{4} + 250 T^{6} + 625 T^{8} )^{4} \))(\( ( 1 + 2 T^{2} + 6 T^{4} + 50 T^{6} + 625 T^{8} )^{4} \))
$7$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{8} \))(\( ( 1 + T^{2} )^{8} \))
$11$ (\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 510 T^{4} - 4114 T^{6} + 14641 T^{8} )^{4} \))(\( ( 1 - 2 T^{2} - 162 T^{4} - 242 T^{6} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 510 T^{4} - 5408 T^{6} + 28561 T^{8} )^{4} \))(\( ( 1 - 16 T^{2} + 222 T^{4} - 2704 T^{6} + 28561 T^{8} )^{4} \))
$17$ (\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 174 T^{4} - 6358 T^{6} + 83521 T^{8} )^{4} \))(\( ( 1 - 14 T^{2} + 222 T^{4} - 4046 T^{6} + 83521 T^{8} )^{4} \))
$19$ (\( ( 1 + 4 T + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 - 4 T + 19 T^{2} )^{4} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{8} \))(\( ( 1 + 26 T^{2} + 361 T^{4} )^{8} \))
$23$ (\( ( 1 + 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 28 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 28 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 2262 T^{4} + 37030 T^{6} + 279841 T^{8} )^{4} \))(\( ( 1 + 38 T^{2} + 1014 T^{4} + 20102 T^{6} + 279841 T^{8} )^{4} \))
$29$ (\( ( 1 + 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 354 T^{4} - 6728 T^{6} + 707281 T^{8} )^{4} \))(\( ( 1 + 52 T^{2} + 841 T^{4} )^{8} \))
$31$ (\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 + 22 T^{2} + 961 T^{4} )^{8} \))(\( ( 1 - 58 T^{2} + 961 T^{4} )^{8} \))
$37$ (\( ( 1 + 26 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 80 T^{2} + 3582 T^{4} - 109520 T^{6} + 1874161 T^{8} )^{4} \))(\( ( 1 - 112 T^{2} + 5694 T^{4} - 153328 T^{6} + 1874161 T^{8} )^{4} \))
$41$ (\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 3342 T^{4} + 43706 T^{6} + 2825761 T^{8} )^{4} \))(\( ( 1 - 110 T^{2} + 5982 T^{4} - 184910 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 + 2 T + 43 T^{2} )^{4} \))(\( ( 1 + 6 T + 43 T^{2} )^{4} \))(\( ( 1 - 6 T + 43 T^{2} )^{4} \))(\( ( 1 - 2 T + 43 T^{2} )^{4} \))(\( ( 1 + 20 T^{2} - 1578 T^{4} + 36980 T^{6} + 3418801 T^{8} )^{4} \))(\( ( 1 + 26 T^{2} + 1849 T^{4} )^{8} \))
$47$ (\( ( 1 + 86 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 86 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 2209 T^{4} )^{8} \))(\( ( 1 + 47 T^{2} )^{16} \))
$53$ (\( ( 1 + 104 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 104 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 136 T^{2} + 8898 T^{4} + 382024 T^{6} + 7890481 T^{8} )^{4} \))(\( ( 1 + 104 T^{2} + 5442 T^{4} + 292136 T^{6} + 7890481 T^{8} )^{4} \))
$59$ (\( ( 1 - 46 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} )^{16} \))(\( ( 1 - 22 T^{2} + 3481 T^{4} )^{8} \))
$61$ (\( ( 1 - 86 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 86 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 86 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 86 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{8}( 1 + 14 T + 61 T^{2} )^{8} \))(\( ( 1 - 100 T^{2} + 7062 T^{4} - 372100 T^{6} + 13845841 T^{8} )^{4} \))
$67$ (\( ( 1 + 4 T + 67 T^{2} )^{4} \))(\( ( 1 - 12 T + 67 T^{2} )^{4} \))(\( ( 1 + 12 T + 67 T^{2} )^{4} \))(\( ( 1 - 4 T + 67 T^{2} )^{4} \))(\( ( 1 + 200 T^{2} + 18222 T^{4} + 897800 T^{6} + 20151121 T^{8} )^{4} \))(\( ( 1 + 232 T^{2} + 22254 T^{4} + 1041448 T^{6} + 20151121 T^{8} )^{4} \))
$71$ (\( ( 1 - 20 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 124 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 124 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 254 T^{2} + 26022 T^{4} + 1280414 T^{6} + 25411681 T^{8} )^{4} \))(\( ( 1 + 158 T^{2} + 12678 T^{4} + 796478 T^{6} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )^{4} \))(\( ( 1 + 2 T + 73 T^{2} )^{4} \))(\( ( 1 + 2 T + 73 T^{2} )^{4} \))(\( ( 1 - 10 T + 73 T^{2} )^{4} \))(\( ( 1 + 10 T + 150 T^{2} + 730 T^{3} + 5329 T^{4} )^{8} \))(\( ( 1 - 2 T + 102 T^{2} - 146 T^{3} + 5329 T^{4} )^{8} \))
$79$ (\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 58 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 58 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 80 T^{2} + 7278 T^{4} + 499280 T^{6} + 38950081 T^{8} )^{4} \))(\( ( 1 - 176 T^{2} + 15726 T^{4} - 1098416 T^{6} + 38950081 T^{8} )^{4} \))
$83$ (\( ( 1 - 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 122 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 122 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 5382 T^{4} - 27556 T^{6} + 47458321 T^{8} )^{4} \))(\( ( 1 - 164 T^{2} + 14022 T^{4} - 1129796 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 - 50 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 106 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 106 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 166 T^{2} + 22542 T^{4} - 1314886 T^{6} + 62742241 T^{8} )^{4} \))(\( ( 1 - 302 T^{2} + 38238 T^{4} - 2392142 T^{6} + 62742241 T^{8} )^{4} \))
$97$ (\( ( 1 - 2 T + 97 T^{2} )^{4} \))(\( ( 1 + 10 T + 97 T^{2} )^{4} \))(\( ( 1 + 10 T + 97 T^{2} )^{4} \))(\( ( 1 - 2 T + 97 T^{2} )^{4} \))(\( ( 1 + 14 T + 222 T^{2} + 1358 T^{3} + 9409 T^{4} )^{8} \))(\( ( 1 - 22 T + 270 T^{2} - 2134 T^{3} + 9409 T^{4} )^{8} \))
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