# Properties

 Label 693.2.c Level 693 Weight 2 Character orbit c Rep. character $$\chi_{693}(307,\cdot)$$ Character field $$\Q$$ Dimension 38 Newform subspaces 5 Sturm bound 192 Trace bound 4

# Related objects

## Defining parameters

 Level: $$N$$ = $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 693.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$77$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$192$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(693, [\chi])$$.

Total New Old
Modular forms 104 42 62
Cusp forms 88 38 50
Eisenstein series 16 4 12

## Trace form

 $$38q - 42q^{4} + O(q^{10})$$ $$38q - 42q^{4} - 8q^{11} - 6q^{14} + 46q^{16} - 6q^{22} + 4q^{23} - 54q^{25} + 16q^{37} + 52q^{44} + 34q^{49} - 44q^{53} - 26q^{56} + 16q^{58} - 34q^{64} - 12q^{67} - 68q^{70} + 44q^{71} - 2q^{77} - 28q^{86} + 82q^{88} - 24q^{91} + 48q^{92} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(693, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
693.2.c.a $$2$$ $$5.534$$ $$\Q(\sqrt{-7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta q^{2}-5q^{4}+\beta q^{7}+3\beta q^{8}+(-2+\cdots)q^{11}+\cdots$$
693.2.c.b $$4$$ $$5.534$$ $$\Q(\sqrt{-2}, \sqrt{-5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}-\beta _{3}q^{5}+(\beta _{1}-2\beta _{2})q^{7}-2\beta _{2}q^{8}+\cdots$$
693.2.c.c $$4$$ $$5.534$$ $$\Q(i, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+q^{4}-\beta _{3}q^{7}+3\beta _{1}q^{8}+(2\beta _{1}+\cdots)q^{11}+\cdots$$
693.2.c.d $$12$$ $$5.534$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ $$\Q(\sqrt{-231})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{2}+(-2+\beta _{1})q^{4}+\beta _{4}q^{5}+\beta _{6}q^{7}+\cdots$$
693.2.c.e $$16$$ $$5.534$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{6}q^{2}+(-1+\beta _{4})q^{4}-\beta _{1}q^{5}+(\beta _{5}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(693, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(693, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(77, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(231, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - T + 2 T^{2} )( 1 + T + 2 T^{2} )$$)($$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$)($$( 1 - 3 T^{2} + 4 T^{4} )^{2}$$)($$( 1 - 5 T^{6} + 64 T^{12} )^{2}$$)($$( 1 - 5 T^{2} + 16 T^{4} - 45 T^{6} + 101 T^{8} - 180 T^{10} + 256 T^{12} - 320 T^{14} + 256 T^{16} )^{2}$$)
$3$ ()()()()()
$5$ ($$( 1 - 5 T^{2} )^{2}$$)($$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 5 T^{2} )^{4}$$)($$( 1 + 58 T^{6} + 15625 T^{12} )^{2}$$)($$( 1 - 14 T^{2} + 121 T^{4} - 774 T^{6} + 4196 T^{8} - 19350 T^{10} + 75625 T^{12} - 218750 T^{14} + 390625 T^{16} )^{2}$$)
$7$ ($$1 + 7 T^{2}$$)($$1 + 4 T^{2} + 49 T^{4}$$)($$( 1 + 7 T^{2} )^{2}$$)($$( 1 - 7 T^{2} )^{6}$$)($$1 - 4 T^{4} - 314 T^{8} - 9604 T^{12} + 5764801 T^{16}$$)
$11$ ($$1 + 4 T + 11 T^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{2}$$)($$1 - 6 T^{2} + 121 T^{4}$$)($$( 1 + 11 T^{2} )^{6}$$)($$( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} )^{4}$$)
$13$ ($$( 1 + 13 T^{2} )^{2}$$)($$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$)($$( 1 + 13 T^{2} )^{4}$$)($$( 1 - 1094 T^{6} + 4826809 T^{12} )^{2}$$)($$( 1 + 70 T^{2} + 2381 T^{4} + 51950 T^{6} + 796636 T^{8} + 8779550 T^{10} + 68003741 T^{12} + 337876630 T^{14} + 815730721 T^{16} )^{2}$$)
$17$ ($$( 1 + 17 T^{2} )^{2}$$)($$( 1 + 17 T^{2} )^{4}$$)($$( 1 + 17 T^{2} )^{4}$$)($$( 1 + 17 T^{2} )^{12}$$)($$( 1 + 20 T^{2} + 1076 T^{4} + 16300 T^{6} + 457366 T^{8} + 4710700 T^{10} + 89868596 T^{12} + 482751380 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 - 13190 T^{6} + 47045881 T^{12} )^{2}$$)($$( 1 + 102 T^{2} + 5093 T^{4} + 161814 T^{6} + 3616780 T^{8} + 58414854 T^{10} + 663724853 T^{12} + 4798679862 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$( 1 - 8 T + 23 T^{2} )^{2}$$)($$( 1 - 3 T + 23 T^{2} )^{4}$$)($$( 1 + 18 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 23 T^{2} )^{12}$$)($$( 1 + 6 T + 48 T^{2} + 270 T^{3} + 1086 T^{4} + 6210 T^{5} + 25392 T^{6} + 73002 T^{7} + 279841 T^{8} )^{4}$$)
$29$ ($$( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$)($$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 54 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 3722 T^{6} + 594823321 T^{12} )^{2}$$)($$( 1 - 158 T^{2} + 12293 T^{4} - 609286 T^{6} + 20954380 T^{8} - 512409526 T^{10} + 8694605333 T^{12} - 93982084718 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 - 31 T^{2} )^{2}$$)($$( 1 - 17 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 31 T^{2} )^{12}$$)($$( 1 - 108 T^{2} + 7028 T^{4} - 324436 T^{6} + 11297430 T^{8} - 311782996 T^{10} + 6490505588 T^{12} - 95850397548 T^{14} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 + 6 T + 37 T^{2} )^{2}$$)($$( 1 + T + 37 T^{2} )^{4}$$)($$( 1 - 6 T + 37 T^{2} )^{4}$$)($$( 1 - 446 T^{3} + 50653 T^{6} )^{4}$$)($$( 1 - 2 T + 97 T^{2} - 250 T^{3} + 4636 T^{4} - 9250 T^{5} + 132793 T^{6} - 101306 T^{7} + 1874161 T^{8} )^{4}$$)
$41$ ($$( 1 + 41 T^{2} )^{2}$$)($$( 1 - 8 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 41 T^{2} )^{12}$$)($$( 1 + 192 T^{2} + 16508 T^{4} + 888384 T^{6} + 38165830 T^{8} + 1493373504 T^{10} + 46647662588 T^{12} + 912020014272 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} )$$)($$( 1 - 68 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 12 T + 43 T^{2} )^{2}( 1 + 12 T + 43 T^{2} )^{2}$$)($$( 1 - 43 T^{2} )^{12}$$)($$( 1 - 140 T^{2} + 13796 T^{4} - 867860 T^{6} + 43988886 T^{8} - 1604673140 T^{10} + 47165778596 T^{12} - 884990826860 T^{14} + 11688200277601 T^{16} )^{2}$$)
$47$ ($$( 1 - 47 T^{2} )^{2}$$)($$( 1 - 74 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 - 47 T^{2} )^{4}$$)($$( 1 - 176846 T^{6} + 10779215329 T^{12} )^{2}$$)($$( 1 - 230 T^{2} + 28521 T^{4} - 2266390 T^{6} + 126631796 T^{8} - 5006455510 T^{10} + 139173381801 T^{12} - 2479219525670 T^{14} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 + 10 T + 53 T^{2} )^{2}$$)($$( 1 + 53 T^{2} )^{4}$$)($$( 1 - 6 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 + 53 T^{2} )^{12}$$)($$( 1 + 6 T + 188 T^{2} + 850 T^{3} + 14486 T^{4} + 45050 T^{5} + 528092 T^{6} + 893262 T^{7} + 7890481 T^{8} )^{4}$$)
$59$ ($$( 1 - 59 T^{2} )^{2}$$)($$( 1 - 113 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 59 T^{2} )^{4}$$)($$( 1 + 185530 T^{6} + 42180533641 T^{12} )^{2}$$)($$( 1 - 278 T^{2} + 40073 T^{4} - 3829126 T^{6} + 263019940 T^{8} - 13329187606 T^{10} + 485579007353 T^{12} - 11726188352198 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 61 T^{2} )^{2}$$)($$( 1 + 112 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 61 T^{2} )^{4}$$)($$( 1 + 10 T^{2} + 3721 T^{4} )^{6}$$)($$( 1 + 312 T^{2} + 48988 T^{4} + 5009224 T^{6} + 360566630 T^{8} + 18639322504 T^{10} + 678280058908 T^{12} + 16074356800632 T^{14} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$( 1 + 4 T + 67 T^{2} )^{2}$$)($$( 1 - 11 T + 67 T^{2} )^{4}$$)($$( 1 + 4 T + 67 T^{2} )^{4}$$)($$( 1 - 1076 T^{3} + 300763 T^{6} )^{4}$$)($$( 1 + 8 T + 207 T^{2} + 1600 T^{3} + 18936 T^{4} + 107200 T^{5} + 929223 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{4}$$)
$71$ ($$( 1 - 16 T + 71 T^{2} )^{2}$$)($$( 1 + 9 T + 71 T^{2} )^{4}$$)($$( 1 + 114 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 71 T^{2} )^{12}$$)($$( 1 - 12 T + 248 T^{2} - 1804 T^{3} + 23150 T^{4} - 128084 T^{5} + 1250168 T^{6} - 4294932 T^{7} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 + 73 T^{2} )^{2}$$)($$( 1 + 136 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + 73 T^{2} )^{4}$$)($$( 1 + 733234 T^{6} + 151334226289 T^{12} )^{2}$$)($$( 1 - 10 T^{2} + 18381 T^{4} - 178370 T^{6} + 140146796 T^{8} - 950533730 T^{10} + 521988067821 T^{12} - 1513342262890 T^{14} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} )$$)($$( 1 - 86 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 79 T^{2} )^{12}$$)($$( 1 - 168 T^{2} + 16988 T^{4} - 1063576 T^{6} + 82499910 T^{8} - 6637777816 T^{10} + 661683976028 T^{12} - 40838692527528 T^{14} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 + 83 T^{2} )^{2}$$)($$( 1 + 76 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 83 T^{2} )^{4}$$)($$( 1 + 83 T^{2} )^{12}$$)($$( 1 + 180 T^{2} + 21956 T^{4} + 2097260 T^{6} + 156627606 T^{8} + 14448024140 T^{10} + 1041994895876 T^{12} + 58849267206420 T^{14} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 - 89 T^{2} )^{2}$$)($$( 1 - 173 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 - 89 T^{2} )^{4}$$)($$( 1 + 130 T^{2} + 7921 T^{4} )^{6}$$)($$( 1 - 528 T^{2} + 133148 T^{4} - 20911856 T^{6} + 2232438150 T^{8} - 165642811376 T^{10} + 8354003904668 T^{12} - 262406121627408 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 - 97 T^{2} )^{2}$$)($$( 1 - 149 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 97 T^{2} )^{12}$$)($$( 1 - 60 T^{2} + 15076 T^{4} - 1727620 T^{6} + 110798006 T^{8} - 16255176580 T^{10} + 1334667440356 T^{12} - 49978320295740 T^{14} + 7837433594376961 T^{16} )^{2}$$)