# 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

# Related objects

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