# Properties

 Label 112.2.j Level 112 Weight 2 Character orbit j Rep. character $$\chi_{112}(27,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 28 Newform subspaces 4 Sturm bound 32 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ = $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 112.j (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$112$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$32$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(112, [\chi])$$.

Total New Old
Modular forms 36 36 0
Cusp forms 28 28 0
Eisenstein series 8 8 0

## Trace form

 $$28q - 4q^{2} - 2q^{4} - 4q^{7} - 16q^{8} + O(q^{10})$$ $$28q - 4q^{2} - 2q^{4} - 4q^{7} - 16q^{8} - 8q^{11} + 2q^{14} - 14q^{16} - 6q^{18} + 4q^{21} + 22q^{22} - 32q^{28} - 12q^{29} + 24q^{30} - 24q^{32} - 4q^{35} + 32q^{36} - 12q^{37} - 8q^{39} - 40q^{42} + 58q^{44} + 8q^{46} - 4q^{49} + 30q^{50} - 32q^{51} - 12q^{53} - 22q^{56} + 66q^{58} + 24q^{60} + 46q^{64} - 8q^{65} - 48q^{67} + 8q^{70} + 56q^{71} + 46q^{72} - 2q^{74} - 16q^{77} - 88q^{78} - 4q^{81} + 80q^{84} + 16q^{85} - 30q^{86} + 30q^{88} + 20q^{91} - 88q^{92} - 16q^{93} + 60q^{98} + 64q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(112, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
112.2.j.a $$4$$ $$0.894$$ $$\Q(i, \sqrt{6})$$ None $$-4$$ $$0$$ $$0$$ $$-4$$ $$q+(-1-\beta _{1})q^{2}+2\beta _{1}q^{4}+(-\beta _{2}-\beta _{3})q^{5}+\cdots$$
112.2.j.b $$4$$ $$0.894$$ $$\Q(i, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{7}+\cdots$$
112.2.j.c $$4$$ $$0.894$$ $$\Q(\zeta_{12})$$ None $$4$$ $$0$$ $$0$$ $$-8$$ $$q+(1+\zeta_{12})q^{2}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots$$
112.2.j.d $$16$$ $$0.894$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-4$$ $$0$$ $$0$$ $$8$$ $$q-\beta _{9}q^{2}+\beta _{15}q^{3}+(-1-\beta _{6}-\beta _{7}+\cdots)q^{4}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T + 2 T^{2} )^{2}$$)($$1 - 3 T^{2} + 4 T^{4}$$)($$( 1 - 2 T + 2 T^{2} )^{2}$$)($$( 1 + 2 T + 4 T^{2} + 8 T^{3} + 10 T^{4} + 16 T^{5} + 16 T^{6} + 16 T^{7} + 16 T^{8} )^{2}$$)
$3$ ($$( 1 + 9 T^{4} )^{2}$$)($$( 1 + 9 T^{4} )^{2}$$)($$( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} )^{2}$$)($$1 - 32 T^{4} + 644 T^{8} - 8864 T^{12} + 92358 T^{16} - 717984 T^{20} + 4225284 T^{24} - 17006112 T^{28} + 43046721 T^{32}$$)
$5$ ($$( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} )( 1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4} )$$)($$( 1 + 25 T^{4} )^{2}$$)($$1 - 34 T^{4} + 625 T^{8}$$)($$1 - 188 T^{8} - 1856 T^{12} + 721606 T^{16} - 1160000 T^{20} - 73437500 T^{24} + 152587890625 T^{32}$$)
$7$ ($$( 1 + 2 T + 7 T^{2} )^{2}$$)($$( 1 - 7 T^{2} )^{2}$$)($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 - 4 T + 12 T^{2} - 36 T^{3} + 86 T^{4} - 252 T^{5} + 588 T^{6} - 1372 T^{7} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 4 T + 11 T^{2} )^{2}( 1 - 6 T^{2} + 121 T^{4} )$$)($$( 1 - 6 T + 18 T^{2} - 66 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4} )^{8}$$)
$13$ ($$1 - 142 T^{4} + 28561 T^{8}$$)($$( 1 + 169 T^{4} )^{2}$$)($$1 + 62 T^{4} + 28561 T^{8}$$)($$1 + 256 T^{4} + 62916 T^{8} + 4935360 T^{12} + 1067752518 T^{16} + 140958816960 T^{20} + 51322514042436 T^{24} + 5964309791355136 T^{28} + 665416609183179841 T^{32}$$)
$17$ ($$( 1 - 10 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 17 T^{2} )^{4}$$)($$( 1 - 17 T^{2} )^{4}$$)($$( 1 - 48 T^{2} + 1372 T^{4} - 33104 T^{6} + 651526 T^{8} - 9567056 T^{10} + 114590812 T^{12} - 1158603312 T^{14} + 6975757441 T^{16} )^{2}$$)
$19$ ($$1 - 622 T^{4} + 130321 T^{8}$$)($$( 1 + 361 T^{4} )^{2}$$)($$1 - 466 T^{4} + 130321 T^{8}$$)($$1 + 1120 T^{4} + 377476 T^{8} - 66979872 T^{12} - 74664260666 T^{16} - 8728883898912 T^{20} + 6410887442464516 T^{24} + 2478912709354100320 T^{28} +$$$$28\!\cdots\!81$$$$T^{32}$$)
$23$ ($$( 1 - 4 T + 23 T^{2} )^{4}$$)($$( 1 + 8 T + 23 T^{2} )^{4}$$)($$( 1 - 4 T + 23 T^{2} )^{4}$$)($$( 1 + 76 T^{2} - 16 T^{3} + 2442 T^{4} - 368 T^{5} + 40204 T^{6} + 279841 T^{8} )^{4}$$)
$29$ ($$( 1 - 4 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}( 1 - 54 T^{2} + 841 T^{4} )$$)($$( 1 + 2 T + 2 T^{2} + 58 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 128 T^{3} + 124 T^{4} - 5504 T^{5} + 8192 T^{6} - 56576 T^{7} - 904026 T^{8} - 1640704 T^{9} + 6889472 T^{10} - 134237056 T^{11} + 87702844 T^{12} + 2625427072 T^{13} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 + 38 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 31 T^{2} )^{4}$$)($$( 1 + 14 T^{2} + 961 T^{4} )^{2}$$)($$( 1 + 128 T^{2} + 8764 T^{4} + 408192 T^{6} + 14409414 T^{8} + 392272512 T^{10} + 8093738044 T^{12} + 113600471168 T^{14} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 2 T + 37 T^{2} )^{2}$$)($$( 1 + 6 T + 37 T^{2} )^{2}( 1 - 38 T^{2} + 1369 T^{4} )$$)($$( 1 - 2 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 64 T^{3} + 2076 T^{4} + 5312 T^{5} + 2048 T^{6} + 110208 T^{7} + 2291366 T^{8} + 4077696 T^{9} + 2803712 T^{10} + 269068736 T^{11} + 3890758236 T^{12} - 4438013248 T^{13} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 + 58 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 70 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 112 T^{2} + 5596 T^{4} + 139408 T^{6} + 2824390 T^{8} + 234344848 T^{10} + 15812958556 T^{12} + 532011674992 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 12 T + 43 T^{2} )^{2}( 1 + 58 T^{2} + 1849 T^{4} )$$)($$( 1 + 2 T + 2 T^{2} + 86 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 64 T^{3} + 2884 T^{4} - 6208 T^{5} + 2048 T^{6} - 149632 T^{7} + 4212966 T^{8} - 6434176 T^{9} + 3786752 T^{10} - 493579456 T^{11} + 9859822084 T^{12} + 9408540352 T^{13} + 11688200277601 T^{16} )^{2}$$)
$47$ ($$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 46 T^{2} + 2209 T^{4} )^{2}$$)($$( 1 + 64 T^{2} + 6460 T^{4} + 424256 T^{6} + 18837830 T^{8} + 937181504 T^{10} + 31522739260 T^{12} + 689869781056 T^{14} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 + 10 T + 53 T^{2} )^{2}( 1 - 6 T^{2} + 2809 T^{4} )$$)($$( 1 - 6 T + 18 T^{2} - 318 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 128 T^{3} - 2788 T^{4} + 6528 T^{5} + 8192 T^{6} + 165120 T^{7} + 1283878 T^{8} + 8751360 T^{9} + 23011328 T^{10} + 971869056 T^{11} - 21998661028 T^{12} - 53529023104 T^{13} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$1 - 2062 T^{4} + 12117361 T^{8}$$)($$( 1 + 3481 T^{4} )^{2}$$)($$1 - 5938 T^{4} + 12117361 T^{8}$$)($$1 + 13536 T^{4} + 105796996 T^{8} + 565374993760 T^{12} + 2271869649992518 T^{16} + 6850852899762667360 T^{20} +$$$$15\!\cdots\!16$$$$T^{24} +$$$$24\!\cdots\!16$$$$T^{28} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$1 + 4658 T^{4} + 13845841 T^{8}$$)($$( 1 + 3721 T^{4} )^{2}$$)($$1 - 2818 T^{4} + 13845841 T^{8}$$)($$1 - 5056 T^{4} + 23160132 T^{8} - 86679311488 T^{12} + 177647957367366 T^{16} - 1200147964852321408 T^{20} +$$$$44\!\cdots\!92$$$$T^{24} -$$$$13\!\cdots\!76$$$$T^{28} +$$$$36\!\cdots\!61$$$$T^{32}$$)
$67$ ($$( 1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 - 4 T + 67 T^{2} )^{2}( 1 - 118 T^{2} + 4489 T^{4} )$$)($$( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 24 T + 288 T^{2} + 2920 T^{3} + 33092 T^{4} + 360168 T^{5} + 3376736 T^{6} + 30121176 T^{7} + 258152998 T^{8} + 2018118792 T^{9} + 15158167904 T^{10} + 108325208184 T^{11} + 666840896132 T^{12} + 3942365312440 T^{13} + 26052014064672 T^{14} + 145457078527752 T^{15} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 - 2 T + 71 T^{2} )^{4}$$)($$( 1 + 114 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 4 T + 71 T^{2} )^{4}$$)($$( 1 - 8 T + 140 T^{2} - 552 T^{3} + 8070 T^{4} - 39192 T^{5} + 705740 T^{6} - 2863288 T^{7} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 + 50 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + 73 T^{2} )^{4}$$)($$( 1 + 38 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + 488 T^{2} + 109788 T^{4} + 14861528 T^{6} + 1322555910 T^{8} + 79197082712 T^{10} + 3117786082908 T^{12} + 73851102429032 T^{14} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 - 142 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 94 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 154 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 248 T^{2} + 44188 T^{4} - 5247432 T^{6} + 480379078 T^{8} - 32749223112 T^{10} + 1721126179228 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 + 6889 T^{4} )^{2}$$)($$( 1 + 6889 T^{4} )^{2}$$)($$1 + 11822 T^{4} + 47458321 T^{8}$$)($$1 - 8864 T^{4} + 76814212 T^{8} - 447540754464 T^{12} + 1619408186640838 T^{16} - 21239532785934694944 T^{20} +$$$$17\!\cdots\!92$$$$T^{24} -$$$$94\!\cdots\!04$$$$T^{28} +$$$$50\!\cdots\!81$$$$T^{32}$$)
$89$ ($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 166 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 200 T^{2} + 34908 T^{4} + 4599288 T^{6} + 427472262 T^{8} + 36430960248 T^{10} + 2190206148828 T^{12} + 99396258192200 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 97 T^{2} )^{4}$$)($$( 1 - 464 T^{2} + 108060 T^{4} - 16830512 T^{6} + 1903290630 T^{8} - 158358287408 T^{10} + 9566474104860 T^{12} - 386499010287056 T^{14} + 7837433594376961 T^{16} )^{2}$$)