# 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

# Related objects

## 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

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