# Properties

 Label 56.2.p Level 56 Weight 2 Character orbit p Rep. character $$\chi_{56}(37,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 12 Newform subspaces 1 Sturm bound 16 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ = $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 56.p (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$56$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$16$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(56, [\chi])$$.

Total New Old
Modular forms 20 20 0
Cusp forms 12 12 0
Eisenstein series 8 8 0

## Trace form

 $$12q - 2q^{2} - 8q^{6} - 4q^{7} + 4q^{8} + O(q^{10})$$ $$12q - 2q^{2} - 8q^{6} - 4q^{7} + 4q^{8} - 8q^{10} - 2q^{12} - 16q^{14} - 20q^{15} + 8q^{16} - 2q^{17} + 6q^{18} + 8q^{20} + 12q^{22} + 2q^{23} + 18q^{24} - 4q^{25} - 2q^{26} + 26q^{28} + 14q^{30} + 10q^{31} - 12q^{32} - 14q^{33} + 32q^{34} - 32q^{36} + 18q^{38} + 4q^{39} + 10q^{40} - 8q^{41} + 22q^{42} - 30q^{44} - 4q^{46} + 30q^{47} - 56q^{48} - 12q^{49} - 16q^{50} - 32q^{52} + 2q^{54} + 4q^{55} - 40q^{56} - 4q^{57} - 22q^{58} + 6q^{60} - 28q^{62} + 44q^{63} + 24q^{64} + 8q^{65} - 38q^{66} + 4q^{68} - 48q^{70} + 32q^{71} + 20q^{72} - 10q^{73} + 18q^{74} + 52q^{76} + 52q^{78} - 22q^{79} + 36q^{80} + 22q^{81} - 26q^{82} + 56q^{84} + 40q^{86} - 20q^{87} - 14q^{88} - 10q^{89} + 52q^{90} - 20q^{92} + 42q^{94} - 34q^{95} + 16q^{96} + 40q^{97} + 52q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(56, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
56.2.p.a $$12$$ $$0.447$$ 12.0.$$\cdots$$.1 None $$-2$$ $$0$$ $$0$$ $$-4$$ $$q+(-\beta _{1}+\beta _{5})q^{2}+\beta _{6}q^{3}-\beta _{3}q^{4}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} - 4 T^{4} - 4 T^{5} - 4 T^{6} - 8 T^{7} - 16 T^{8} + 32 T^{10} + 64 T^{11} + 64 T^{12}$$
$3$ $$1 + 9 T^{2} + 35 T^{4} + 92 T^{6} + 253 T^{8} + 683 T^{10} + 1774 T^{12} + 6147 T^{14} + 20493 T^{16} + 67068 T^{18} + 229635 T^{20} + 531441 T^{22} + 531441 T^{24}$$
$5$ $$1 + 17 T^{2} + 147 T^{4} + 788 T^{6} + 2793 T^{8} + 5371 T^{10} + 7494 T^{12} + 134275 T^{14} + 1745625 T^{16} + 12312500 T^{18} + 57421875 T^{20} + 166015625 T^{22} + 244140625 T^{24}$$
$7$ $$( 1 + 2 T + 5 T^{2} + 32 T^{3} + 35 T^{4} + 98 T^{5} + 343 T^{6} )^{2}$$
$11$ $$1 + 29 T^{2} + 327 T^{4} + 2696 T^{6} + 24957 T^{8} - 44693 T^{10} - 3466650 T^{12} - 5407853 T^{14} + 365395437 T^{16} + 4776128456 T^{18} + 70095354087 T^{20} + 752185313429 T^{22} + 3138428376721 T^{24}$$
$13$ $$( 1 - 46 T^{2} + 1191 T^{4} - 18804 T^{6} + 201279 T^{8} - 1313806 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 + T - 33 T^{2} + 8 T^{3} + 565 T^{4} - 425 T^{5} - 9946 T^{6} - 7225 T^{7} + 163285 T^{8} + 39304 T^{9} - 2756193 T^{10} + 1419857 T^{11} + 24137569 T^{12} )^{2}$$
$19$ $$1 + 53 T^{2} + 1215 T^{4} + 18608 T^{6} + 219741 T^{8} - 1587413 T^{10} - 104375754 T^{12} - 573056093 T^{14} + 28636866861 T^{16} + 875429753648 T^{18} + 20635029094815 T^{20} + 324946511663453 T^{22} + 2213314919066161 T^{24}$$
$23$ $$( 1 - T - 61 T^{2} + 12 T^{3} + 2381 T^{4} + 29 T^{5} - 63238 T^{6} + 667 T^{7} + 1259549 T^{8} + 146004 T^{9} - 17070301 T^{10} - 6436343 T^{11} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 - 102 T^{2} + 4567 T^{4} - 141988 T^{6} + 3840847 T^{8} - 72142662 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 - 5 T - 71 T^{2} + 126 T^{3} + 4725 T^{4} - 4133 T^{5} - 158330 T^{6} - 128123 T^{7} + 4540725 T^{8} + 3753666 T^{9} - 65569991 T^{10} - 143145755 T^{11} + 887503681 T^{12} )^{2}$$
$37$ $$1 + 161 T^{2} + 13515 T^{4} + 840428 T^{6} + 43791513 T^{8} + 2009438227 T^{10} + 80476513014 T^{12} + 2750920932763 T^{14} + 82072345795593 T^{16} + 2156308314463052 T^{18} + 47471159819742315 T^{20} + 774182083959273689 T^{22} + 6582952005840035281 T^{24}$$
$41$ $$( 1 + 2 T + 83 T^{2} + 80 T^{3} + 3403 T^{4} + 3362 T^{5} + 68921 T^{6} )^{4}$$
$43$ $$( 1 - 94 T^{2} + 3543 T^{4} - 112068 T^{6} + 6551007 T^{8} - 321367294 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 - 15 T + 57 T^{2} - 78 T^{3} + 2013 T^{4} - 1383 T^{5} - 125066 T^{6} - 65001 T^{7} + 4446717 T^{8} - 8098194 T^{9} + 278141817 T^{10} - 3440175105 T^{11} + 10779215329 T^{12} )^{2}$$
$53$ $$1 + 161 T^{2} + 12171 T^{4} + 564332 T^{6} + 17660601 T^{8} + 173268691 T^{10} - 11132063562 T^{12} + 486711753019 T^{14} + 139350636639081 T^{16} + 12508058244650828 T^{18} + 757762691996674731 T^{20} + 28156882728847600889 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$1 + 177 T^{2} + 16739 T^{4} + 834316 T^{6} + 7372221 T^{8} - 3081613133 T^{10} - 272281952338 T^{12} - 10727095315973 T^{14} + 89331863228781 T^{16} + 35191894105224556 T^{18} + 2457794695058729219 T^{20} + 90467665334213527977 T^{22} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 + 73 T^{2} - 3941 T^{4} - 121236 T^{6} + 22314201 T^{8} - 326849717 T^{10} - 131320604138 T^{12} - 1216207796957 T^{14} + 308958879088041 T^{16} - 6246124106030196 T^{18} - 755518520522284421 T^{20} + 52074032551390429873 T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 361 T^{2} + 73723 T^{4} + 10542804 T^{6} + 1164287325 T^{8} + 103688059123 T^{10} + 7623484120510 T^{12} + 465455697403147 T^{14} + 23461694764841325 T^{16} + 953684993364861876 T^{18} + 29936527392508244443 T^{20} +$$$$65\!\cdots\!89$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$( 1 - 8 T + 157 T^{2} - 704 T^{3} + 11147 T^{4} - 40328 T^{5} + 357911 T^{6} )^{4}$$
$73$ $$( 1 + 5 T - 101 T^{2} + 52 T^{3} + 5233 T^{4} - 28921 T^{5} - 430618 T^{6} - 2111233 T^{7} + 27886657 T^{8} + 20228884 T^{9} - 2868222341 T^{10} + 10365357965 T^{11} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 + 11 T - 137 T^{2} - 656 T^{3} + 26965 T^{4} + 70973 T^{5} - 1975630 T^{6} + 5606867 T^{7} + 168288565 T^{8} - 323433584 T^{9} - 5336161097 T^{10} + 33847620389 T^{11} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 - 446 T^{2} + 86503 T^{4} - 9357636 T^{6} + 595919167 T^{8} - 21166411166 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 + 5 T - 133 T^{2} - 1452 T^{3} + 5297 T^{4} + 76103 T^{5} + 246758 T^{6} + 6773167 T^{7} + 41957537 T^{8} - 1023614988 T^{9} - 8344718053 T^{10} + 27920297245 T^{11} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 - 10 T + 255 T^{2} - 1968 T^{3} + 24735 T^{4} - 94090 T^{5} + 912673 T^{6} )^{4}$$