# Properties

 Label 128.4.a Level 128 Weight 4 Character orbit a Rep. character $$\chi_{128}(1,\cdot)$$ Character field $$\Q$$ Dimension 12 Newform subspaces 8 Sturm bound 64 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 128.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$64$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(128))$$.

Total New Old
Modular forms 56 12 44
Cusp forms 40 12 28
Eisenstein series 16 0 16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$Dim.
$$+$$$$7$$
$$-$$$$5$$

## Trace form

 $$12q + 108q^{9} + O(q^{10})$$ $$12q + 108q^{9} + 104q^{17} + 212q^{25} + 464q^{33} - 392q^{41} - 852q^{49} - 1360q^{57} - 2512q^{65} - 24q^{73} + 1596q^{81} + 1960q^{89} + 4328q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(128))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2
128.4.a.a $$1$$ $$7.552$$ $$\Q$$ None $$0$$ $$-2$$ $$-6$$ $$20$$ $$-$$ $$q-2q^{3}-6q^{5}+20q^{7}-23q^{9}-14q^{11}+\cdots$$
128.4.a.b $$1$$ $$7.552$$ $$\Q$$ None $$0$$ $$-2$$ $$6$$ $$-20$$ $$-$$ $$q-2q^{3}+6q^{5}-20q^{7}-23q^{9}-14q^{11}+\cdots$$
128.4.a.c $$1$$ $$7.552$$ $$\Q$$ None $$0$$ $$2$$ $$-6$$ $$-20$$ $$-$$ $$q+2q^{3}-6q^{5}-20q^{7}-23q^{9}+14q^{11}+\cdots$$
128.4.a.d $$1$$ $$7.552$$ $$\Q$$ None $$0$$ $$2$$ $$6$$ $$20$$ $$+$$ $$q+2q^{3}+6q^{5}+20q^{7}-23q^{9}+14q^{11}+\cdots$$
128.4.a.e $$2$$ $$7.552$$ $$\Q(\sqrt{3})$$ None $$0$$ $$-4$$ $$-4$$ $$-8$$ $$-$$ $$q+(-2+\beta )q^{3}+(-2-2\beta )q^{5}+(-4+\cdots)q^{7}+\cdots$$
128.4.a.f $$2$$ $$7.552$$ $$\Q(\sqrt{3})$$ None $$0$$ $$-4$$ $$4$$ $$8$$ $$+$$ $$q+(-2+\beta )q^{3}+(2+2\beta )q^{5}+(4+2\beta )q^{7}+\cdots$$
128.4.a.g $$2$$ $$7.552$$ $$\Q(\sqrt{3})$$ None $$0$$ $$4$$ $$-4$$ $$8$$ $$+$$ $$q+(2+\beta )q^{3}+(-2+2\beta )q^{5}+(4-2\beta )q^{7}+\cdots$$
128.4.a.h $$2$$ $$7.552$$ $$\Q(\sqrt{3})$$ None $$0$$ $$4$$ $$4$$ $$-8$$ $$+$$ $$q+(2+\beta )q^{3}+(2-2\beta )q^{5}+(-4+2\beta )q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(128))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(128)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(64))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 2 T + 27 T^{2}$$)($$1 + 2 T + 27 T^{2}$$)($$1 - 2 T + 27 T^{2}$$)($$1 - 2 T + 27 T^{2}$$)($$1 + 4 T + 10 T^{2} + 108 T^{3} + 729 T^{4}$$)($$1 + 4 T + 10 T^{2} + 108 T^{3} + 729 T^{4}$$)($$1 - 4 T + 10 T^{2} - 108 T^{3} + 729 T^{4}$$)($$1 - 4 T + 10 T^{2} - 108 T^{3} + 729 T^{4}$$)
$5$ ($$1 + 6 T + 125 T^{2}$$)($$1 - 6 T + 125 T^{2}$$)($$1 + 6 T + 125 T^{2}$$)($$1 - 6 T + 125 T^{2}$$)($$1 + 4 T + 62 T^{2} + 500 T^{3} + 15625 T^{4}$$)($$1 - 4 T + 62 T^{2} - 500 T^{3} + 15625 T^{4}$$)($$1 + 4 T + 62 T^{2} + 500 T^{3} + 15625 T^{4}$$)($$1 - 4 T + 62 T^{2} - 500 T^{3} + 15625 T^{4}$$)
$7$ ($$1 - 20 T + 343 T^{2}$$)($$1 + 20 T + 343 T^{2}$$)($$1 + 20 T + 343 T^{2}$$)($$1 - 20 T + 343 T^{2}$$)($$1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4}$$)($$1 - 8 T + 510 T^{2} - 2744 T^{3} + 117649 T^{4}$$)($$1 - 8 T + 510 T^{2} - 2744 T^{3} + 117649 T^{4}$$)($$1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4}$$)
$11$ ($$1 + 14 T + 1331 T^{2}$$)($$1 + 14 T + 1331 T^{2}$$)($$1 - 14 T + 1331 T^{2}$$)($$1 - 14 T + 1331 T^{2}$$)($$1 + 92 T + 4730 T^{2} + 122452 T^{3} + 1771561 T^{4}$$)($$1 + 92 T + 4730 T^{2} + 122452 T^{3} + 1771561 T^{4}$$)($$1 - 92 T + 4730 T^{2} - 122452 T^{3} + 1771561 T^{4}$$)($$1 - 92 T + 4730 T^{2} - 122452 T^{3} + 1771561 T^{4}$$)
$13$ ($$1 + 54 T + 2197 T^{2}$$)($$1 - 54 T + 2197 T^{2}$$)($$1 + 54 T + 2197 T^{2}$$)($$1 - 54 T + 2197 T^{2}$$)($$1 + 100 T + 5166 T^{2} + 219700 T^{3} + 4826809 T^{4}$$)($$1 - 100 T + 5166 T^{2} - 219700 T^{3} + 4826809 T^{4}$$)($$1 + 100 T + 5166 T^{2} + 219700 T^{3} + 4826809 T^{4}$$)($$1 - 100 T + 5166 T^{2} - 219700 T^{3} + 4826809 T^{4}$$)
$17$ ($$1 + 66 T + 4913 T^{2}$$)($$1 + 66 T + 4913 T^{2}$$)($$1 + 66 T + 4913 T^{2}$$)($$1 + 66 T + 4913 T^{2}$$)($$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$)($$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$)($$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$)($$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$)
$19$ ($$1 + 162 T + 6859 T^{2}$$)($$1 + 162 T + 6859 T^{2}$$)($$1 - 162 T + 6859 T^{2}$$)($$1 - 162 T + 6859 T^{2}$$)($$1 + 4 T + 11370 T^{2} + 27436 T^{3} + 47045881 T^{4}$$)($$1 + 4 T + 11370 T^{2} + 27436 T^{3} + 47045881 T^{4}$$)($$1 - 4 T + 11370 T^{2} - 27436 T^{3} + 47045881 T^{4}$$)($$1 - 4 T + 11370 T^{2} - 27436 T^{3} + 47045881 T^{4}$$)
$23$ ($$1 - 172 T + 12167 T^{2}$$)($$1 + 172 T + 12167 T^{2}$$)($$1 + 172 T + 12167 T^{2}$$)($$1 - 172 T + 12167 T^{2}$$)($$1 - 8 T + 8798 T^{2} - 97336 T^{3} + 148035889 T^{4}$$)($$1 + 8 T + 8798 T^{2} + 97336 T^{3} + 148035889 T^{4}$$)($$1 + 8 T + 8798 T^{2} + 97336 T^{3} + 148035889 T^{4}$$)($$1 - 8 T + 8798 T^{2} - 97336 T^{3} + 148035889 T^{4}$$)
$29$ ($$1 - 2 T + 24389 T^{2}$$)($$1 + 2 T + 24389 T^{2}$$)($$1 - 2 T + 24389 T^{2}$$)($$1 + 2 T + 24389 T^{2}$$)($$1 + 84 T + 45742 T^{2} + 2048676 T^{3} + 594823321 T^{4}$$)($$1 - 84 T + 45742 T^{2} - 2048676 T^{3} + 594823321 T^{4}$$)($$1 + 84 T + 45742 T^{2} + 2048676 T^{3} + 594823321 T^{4}$$)($$1 - 84 T + 45742 T^{2} - 2048676 T^{3} + 594823321 T^{4}$$)
$31$ ($$1 + 128 T + 29791 T^{2}$$)($$1 - 128 T + 29791 T^{2}$$)($$1 - 128 T + 29791 T^{2}$$)($$1 + 128 T + 29791 T^{2}$$)($$1 + 384 T + 84158 T^{2} + 11439744 T^{3} + 887503681 T^{4}$$)($$1 - 384 T + 84158 T^{2} - 11439744 T^{3} + 887503681 T^{4}$$)($$1 - 384 T + 84158 T^{2} - 11439744 T^{3} + 887503681 T^{4}$$)($$1 + 384 T + 84158 T^{2} + 11439744 T^{3} + 887503681 T^{4}$$)
$37$ ($$1 + 158 T + 50653 T^{2}$$)($$1 - 158 T + 50653 T^{2}$$)($$1 + 158 T + 50653 T^{2}$$)($$1 - 158 T + 50653 T^{2}$$)($$1 - 172 T + 99294 T^{2} - 8712316 T^{3} + 2565726409 T^{4}$$)($$1 + 172 T + 99294 T^{2} + 8712316 T^{3} + 2565726409 T^{4}$$)($$1 - 172 T + 99294 T^{2} - 8712316 T^{3} + 2565726409 T^{4}$$)($$1 + 172 T + 99294 T^{2} + 8712316 T^{3} + 2565726409 T^{4}$$)
$41$ ($$1 - 202 T + 68921 T^{2}$$)($$1 - 202 T + 68921 T^{2}$$)($$1 - 202 T + 68921 T^{2}$$)($$1 - 202 T + 68921 T^{2}$$)($$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$)($$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$)($$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$)($$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$)
$43$ ($$1 - 298 T + 79507 T^{2}$$)($$1 - 298 T + 79507 T^{2}$$)($$1 + 298 T + 79507 T^{2}$$)($$1 + 298 T + 79507 T^{2}$$)($$1 + 300 T + 180314 T^{2} + 23852100 T^{3} + 6321363049 T^{4}$$)($$1 + 300 T + 180314 T^{2} + 23852100 T^{3} + 6321363049 T^{4}$$)($$1 - 300 T + 180314 T^{2} - 23852100 T^{3} + 6321363049 T^{4}$$)($$1 - 300 T + 180314 T^{2} - 23852100 T^{3} + 6321363049 T^{4}$$)
$47$ ($$1 + 408 T + 103823 T^{2}$$)($$1 - 408 T + 103823 T^{2}$$)($$1 - 408 T + 103823 T^{2}$$)($$1 + 408 T + 103823 T^{2}$$)($$1 + 16 T + 114782 T^{2} + 1661168 T^{3} + 10779215329 T^{4}$$)($$1 - 16 T + 114782 T^{2} - 1661168 T^{3} + 10779215329 T^{4}$$)($$1 - 16 T + 114782 T^{2} - 1661168 T^{3} + 10779215329 T^{4}$$)($$1 + 16 T + 114782 T^{2} + 1661168 T^{3} + 10779215329 T^{4}$$)
$53$ ($$1 - 690 T + 148877 T^{2}$$)($$1 + 690 T + 148877 T^{2}$$)($$1 - 690 T + 148877 T^{2}$$)($$1 + 690 T + 148877 T^{2}$$)($$1 - 12 T + 288382 T^{2} - 1786524 T^{3} + 22164361129 T^{4}$$)($$1 + 12 T + 288382 T^{2} + 1786524 T^{3} + 22164361129 T^{4}$$)($$1 - 12 T + 288382 T^{2} - 1786524 T^{3} + 22164361129 T^{4}$$)($$1 + 12 T + 288382 T^{2} + 1786524 T^{3} + 22164361129 T^{4}$$)
$59$ ($$1 - 322 T + 205379 T^{2}$$)($$1 - 322 T + 205379 T^{2}$$)($$1 + 322 T + 205379 T^{2}$$)($$1 + 322 T + 205379 T^{2}$$)($$1 - 644 T + 462170 T^{2} - 132264076 T^{3} + 42180533641 T^{4}$$)($$1 - 644 T + 462170 T^{2} - 132264076 T^{3} + 42180533641 T^{4}$$)($$1 + 644 T + 462170 T^{2} + 132264076 T^{3} + 42180533641 T^{4}$$)($$1 + 644 T + 462170 T^{2} + 132264076 T^{3} + 42180533641 T^{4}$$)
$61$ ($$1 - 298 T + 226981 T^{2}$$)($$1 + 298 T + 226981 T^{2}$$)($$1 - 298 T + 226981 T^{2}$$)($$1 + 298 T + 226981 T^{2}$$)($$1 + 292 T + 240078 T^{2} + 66278452 T^{3} + 51520374361 T^{4}$$)($$1 - 292 T + 240078 T^{2} - 66278452 T^{3} + 51520374361 T^{4}$$)($$1 + 292 T + 240078 T^{2} + 66278452 T^{3} + 51520374361 T^{4}$$)($$1 - 292 T + 240078 T^{2} - 66278452 T^{3} + 51520374361 T^{4}$$)
$67$ ($$1 + 202 T + 300763 T^{2}$$)($$1 + 202 T + 300763 T^{2}$$)($$1 - 202 T + 300763 T^{2}$$)($$1 - 202 T + 300763 T^{2}$$)($$1 - 172 T + 278250 T^{2} - 51731236 T^{3} + 90458382169 T^{4}$$)($$1 - 172 T + 278250 T^{2} - 51731236 T^{3} + 90458382169 T^{4}$$)($$1 + 172 T + 278250 T^{2} + 51731236 T^{3} + 90458382169 T^{4}$$)($$1 + 172 T + 278250 T^{2} + 51731236 T^{3} + 90458382169 T^{4}$$)
$71$ ($$1 + 700 T + 357911 T^{2}$$)($$1 - 700 T + 357911 T^{2}$$)($$1 - 700 T + 357911 T^{2}$$)($$1 + 700 T + 357911 T^{2}$$)($$1 - 408 T + 672766 T^{2} - 146027688 T^{3} + 128100283921 T^{4}$$)($$1 + 408 T + 672766 T^{2} + 146027688 T^{3} + 128100283921 T^{4}$$)($$1 + 408 T + 672766 T^{2} + 146027688 T^{3} + 128100283921 T^{4}$$)($$1 - 408 T + 672766 T^{2} - 146027688 T^{3} + 128100283921 T^{4}$$)
$73$ ($$1 + 418 T + 389017 T^{2}$$)($$1 + 418 T + 389017 T^{2}$$)($$1 + 418 T + 389017 T^{2}$$)($$1 + 418 T + 389017 T^{2}$$)($$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$)($$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$)($$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$)($$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$)
$79$ ($$1 - 744 T + 493039 T^{2}$$)($$1 + 744 T + 493039 T^{2}$$)($$1 + 744 T + 493039 T^{2}$$)($$1 - 744 T + 493039 T^{2}$$)($$1 + 400 T + 963870 T^{2} + 197215600 T^{3} + 243087455521 T^{4}$$)($$1 - 400 T + 963870 T^{2} - 197215600 T^{3} + 243087455521 T^{4}$$)($$1 - 400 T + 963870 T^{2} - 197215600 T^{3} + 243087455521 T^{4}$$)($$1 + 400 T + 963870 T^{2} + 197215600 T^{3} + 243087455521 T^{4}$$)
$83$ ($$1 - 678 T + 571787 T^{2}$$)($$1 - 678 T + 571787 T^{2}$$)($$1 + 678 T + 571787 T^{2}$$)($$1 + 678 T + 571787 T^{2}$$)($$1 + 948 T + 1360138 T^{2} + 542054076 T^{3} + 326940373369 T^{4}$$)($$1 + 948 T + 1360138 T^{2} + 542054076 T^{3} + 326940373369 T^{4}$$)($$1 - 948 T + 1360138 T^{2} - 542054076 T^{3} + 326940373369 T^{4}$$)($$1 - 948 T + 1360138 T^{2} - 542054076 T^{3} + 326940373369 T^{4}$$)
$89$ ($$1 + 82 T + 704969 T^{2}$$)($$1 + 82 T + 704969 T^{2}$$)($$1 + 82 T + 704969 T^{2}$$)($$1 + 82 T + 704969 T^{2}$$)($$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$)($$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$)($$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$)($$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$)
$97$ ($$1 + 1122 T + 912673 T^{2}$$)($$1 + 1122 T + 912673 T^{2}$$)($$1 + 1122 T + 912673 T^{2}$$)($$1 + 1122 T + 912673 T^{2}$$)($$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$)($$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$)($$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$)($$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$)