Properties

Label 4032.2.k
Level 4032
Weight 2
Character orbit k
Rep. character \(\chi_{4032}(3905,\cdot)\)
Character field \(\Q\)
Dimension 64
Newform subspaces 8
Sturm bound 1536
Trace bound 67

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Defining parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.k (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(1536\)
Trace bound: \(67\)
Distinguishing \(T_p\): \(5\), \(43\), \(67\)

Dimensions

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

Total New Old
Modular forms 816 64 752
Cusp forms 720 64 656
Eisenstein series 96 0 96

Trace form

\( 64q + O(q^{10}) \) \( 64q + 64q^{25} - 32q^{37} + O(q^{100}) \)

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.k.a \(4\) \(32.196\) \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(0\) \(-4\) \(q-\beta _{3}q^{5}+(-1+\beta _{1})q^{7}-\beta _{2}q^{11}+\cdots\)
4032.2.k.b \(4\) \(32.196\) \(\Q(\sqrt{-2}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+(\beta _{1}+\beta _{3})q^{23}+\cdots\)
4032.2.k.c \(4\) \(32.196\) \(\Q(\sqrt{-2}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{7}+(\beta _{1}-\beta _{3})q^{11}+(-\beta _{1}-\beta _{3})q^{23}+\cdots\)
4032.2.k.d \(4\) \(32.196\) \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(0\) \(4\) \(q-\beta _{3}q^{5}+(1-\beta _{1})q^{7}+\beta _{2}q^{11}-2\beta _{1}q^{13}+\cdots\)
4032.2.k.e \(8\) \(32.196\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(-8\) \(q-\zeta_{16}^{7}q^{5}+(-1-\zeta_{16}^{6})q^{7}+(\zeta_{16}+\cdots)q^{11}+\cdots\)
4032.2.k.f \(8\) \(32.196\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(8\) \(q-\zeta_{16}^{7}q^{5}+(1+\zeta_{16}^{4})q^{7}+(\zeta_{16}+\zeta_{16}^{2}+\cdots)q^{11}+\cdots\)
4032.2.k.g \(16\) \(32.196\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{5}+\beta _{6}q^{7}-\beta _{3}q^{11}+\beta _{13}q^{13}+\cdots\)
4032.2.k.h \(16\) \(32.196\) 16.0.\(\cdots\).7 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+\beta _{3}q^{7}+\beta _{4}q^{11}+\beta _{9}q^{13}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(4032, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(672, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1344, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2016, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 5 T^{2} )^{4} \))(\( ( 1 + 5 T^{2} )^{4} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 22 T^{4} + 100 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 + 4 T^{2} + 22 T^{4} + 100 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 + 12 T^{2} + 78 T^{4} + 300 T^{6} + 625 T^{8} )^{4} \))(\( ( 1 + 8 T^{2} + 38 T^{4} + 200 T^{6} + 625 T^{8} )^{4} \))
$7$ (\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 10 T^{2} + 28 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 4 T + 10 T^{2} - 28 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 70 T^{4} + 196 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 - 4 T^{2} - 10 T^{4} - 196 T^{6} + 2401 T^{8} )^{2} \))
$11$ (\( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \))(\( 1 - 206 T^{4} + 14641 T^{8} \))(\( 1 - 206 T^{4} + 14641 T^{8} \))(\( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 24 T^{2} + 314 T^{4} - 2904 T^{6} + 14641 T^{8} )^{4} \))(\( ( 1 - 28 T^{2} + 410 T^{4} - 3388 T^{6} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 - 2 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 2 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 310 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 310 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 36 T^{2} + 630 T^{4} - 6084 T^{6} + 28561 T^{8} )^{4} \))(\( ( 1 - 12 T^{2} + 262 T^{4} - 2028 T^{6} + 28561 T^{8} )^{4} \))
$17$ (\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 12 T^{2} + 582 T^{4} - 3468 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 12 T^{2} + 582 T^{4} - 3468 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 382 T^{4} + 8092 T^{6} + 83521 T^{8} )^{4} \))(\( ( 1 + 56 T^{2} + 1334 T^{4} + 16184 T^{6} + 83521 T^{8} )^{4} \))
$19$ (\( ( 1 - 14 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 14 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 44 T^{2} + 1078 T^{4} - 15884 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 44 T^{2} + 1078 T^{4} - 15884 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 12 T^{2} - 42 T^{4} - 4332 T^{6} + 130321 T^{8} )^{4} \))(\( ( 1 - 36 T^{2} + 934 T^{4} - 12996 T^{6} + 130321 T^{8} )^{4} \))
$23$ (\( ( 1 - 28 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 734 T^{4} + 279841 T^{8} \))(\( 1 - 734 T^{4} + 279841 T^{8} \))(\( ( 1 - 28 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 48 T^{2} + 1346 T^{4} - 25392 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 48 T^{2} + 1346 T^{4} - 25392 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 40 T^{2} + 810 T^{4} - 21160 T^{6} + 279841 T^{8} )^{4} \))(\( ( 1 - 76 T^{2} + 2474 T^{4} - 40204 T^{6} + 279841 T^{8} )^{4} \))
$29$ (\( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \))(\( 1 + 1234 T^{4} + 707281 T^{8} \))(\( 1 + 1234 T^{4} + 707281 T^{8} \))(\( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 80 T^{2} + 3154 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3154 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{8} \))(\( ( 1 + 1234 T^{4} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 + 4 T^{2} - 122 T^{4} + 3844 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 4 T^{2} - 122 T^{4} + 3844 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 60 T^{2} + 2694 T^{4} - 57660 T^{6} + 923521 T^{8} )^{4} \))(\( ( 1 - 44 T^{2} + 1958 T^{4} - 42284 T^{6} + 923521 T^{8} )^{4} \))
$37$ (\( ( 1 - 8 T + 37 T^{2} )^{4} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 8 T + 37 T^{2} )^{4} \))(\( ( 1 + 4 T + 37 T^{2} )^{8} \))(\( ( 1 + 4 T + 37 T^{2} )^{8} \))(\( ( 1 + 66 T^{2} + 1369 T^{4} )^{8} \))(\( ( 1 + 4 T + 50 T^{2} + 148 T^{3} + 1369 T^{4} )^{8} \))
$41$ (\( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 84 T^{2} + 3558 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 84 T^{2} + 3558 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 3166 T^{4} + 47068 T^{6} + 2825761 T^{8} )^{4} \))(\( ( 1 + 88 T^{2} + 3926 T^{4} + 147928 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 - 2 T + 43 T^{2} )^{4} \))(\( ( 1 + 58 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{4} \))(\( ( 1 - 4 T + 18 T^{2} - 172 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 + 4 T + 18 T^{2} + 172 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 + 20 T^{2} + 3510 T^{4} + 36980 T^{6} + 3418801 T^{8} )^{4} \))(\( ( 1 + 140 T^{2} + 8486 T^{4} + 258860 T^{6} + 3418801 T^{8} )^{4} \))
$47$ (\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 60 T^{2} + 2118 T^{4} + 132540 T^{6} + 4879681 T^{8} )^{4} \))(\( ( 1 - 20 T^{2} + 4070 T^{4} - 44180 T^{6} + 4879681 T^{8} )^{4} \))
$53$ (\( ( 1 + 56 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 5582 T^{4} + 7890481 T^{8} \))(\( 1 - 5582 T^{4} + 7890481 T^{8} \))(\( ( 1 + 56 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 176 T^{2} + 13234 T^{4} - 494384 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 176 T^{2} + 13234 T^{4} - 494384 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 176 T^{2} + 13234 T^{4} - 494384 T^{6} + 7890481 T^{8} )^{4} \))(\( ( 1 - 88 T^{2} + 2809 T^{4} )^{8} \))
$59$ (\( ( 1 - 74 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 - 74 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 59 T^{2} )^{8} \))(\( ( 1 + 59 T^{2} )^{8} \))(\( ( 1 + 108 T^{2} + 9366 T^{4} + 375948 T^{6} + 12117361 T^{8} )^{4} \))(\( ( 1 + 76 T^{2} + 6614 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{4} \))
$61$ (\( ( 1 - 26 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 26 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} )^{8} \))(\( ( 1 - 61 T^{2} )^{8} \))(\( ( 1 - 84 T^{2} + 9078 T^{4} - 312564 T^{6} + 13845841 T^{8} )^{4} \))(\( ( 1 - 36 T^{2} + 7318 T^{4} - 133956 T^{6} + 13845841 T^{8} )^{4} \))
$67$ (\( ( 1 + 8 T + 67 T^{2} )^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{4} \))(\( ( 1 - 4 T + 67 T^{2} )^{4} \))(\( ( 1 - 8 T + 67 T^{2} )^{4} \))(\( ( 1 + 8 T + 118 T^{2} + 536 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 - 8 T + 118 T^{2} - 536 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 + 60 T^{2} - 490 T^{4} + 269340 T^{6} + 20151121 T^{8} )^{4} \))(\( ( 1 + 62 T^{2} + 4489 T^{4} )^{8} \))
$71$ (\( ( 1 - 124 T^{2} + 5041 T^{4} )^{2} \))(\( 1 + 2914 T^{4} + 25411681 T^{8} \))(\( 1 + 2914 T^{4} + 25411681 T^{8} \))(\( ( 1 - 124 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 208 T^{2} + 20610 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 208 T^{2} + 20610 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 40 T^{2} + 10410 T^{4} - 201640 T^{6} + 25411681 T^{8} )^{4} \))(\( ( 1 - 252 T^{2} + 25706 T^{4} - 1270332 T^{6} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 - 122 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 122 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 148 T^{2} + 13542 T^{4} - 788692 T^{6} + 28398241 T^{8} )^{4} \))(\( ( 1 - 12 T^{2} + 5206 T^{4} - 63948 T^{6} + 28398241 T^{8} )^{4} \))
$79$ (\( ( 1 + 4 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 4 T + 79 T^{2} )^{4} \))(\( ( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 16 T + 190 T^{2} + 1264 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 - 20 T^{2} + 6310 T^{4} - 124820 T^{6} + 38950081 T^{8} )^{4} \))(\( ( 1 + 86 T^{2} + 6241 T^{4} )^{8} \))
$83$ (\( ( 1 + 118 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 118 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 12 T^{2} + 13302 T^{4} + 82668 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 12 T^{2} + 13302 T^{4} + 82668 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 76 T^{2} - 266 T^{4} + 523564 T^{6} + 47458321 T^{8} )^{4} \))(\( ( 1 + 124 T^{2} + 17174 T^{4} + 854236 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 276 T^{2} + 34854 T^{4} + 2186196 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 276 T^{2} + 34854 T^{4} + 2186196 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} - 3170 T^{4} + 221788 T^{6} + 62742241 T^{8} )^{4} \))(\( ( 1 + 248 T^{2} + 31190 T^{4} + 1964408 T^{6} + 62742241 T^{8} )^{4} \))
$97$ (\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 228 T^{2} + 31686 T^{4} - 2145252 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 228 T^{2} + 31686 T^{4} - 2145252 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 308 T^{2} + 42502 T^{4} - 2897972 T^{6} + 88529281 T^{8} )^{4} \))(\( ( 1 - 172 T^{2} + 17142 T^{4} - 1618348 T^{6} + 88529281 T^{8} )^{4} \))
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