# Properties

 Label 37.8.a Level $37$ Weight $8$ Character orbit 37.a Rep. character $\chi_{37}(1,\cdot)$ Character field $\Q$ Dimension $21$ Newform subspaces $2$ Sturm bound $25$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 37.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$25$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_0(37))$$.

Total New Old
Modular forms 23 21 2
Cusp forms 21 21 0
Eisenstein series 2 0 2

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

$$37$$Dim
$$+$$$$11$$
$$-$$$$10$$

## Trace form

 $$21 q - 8 q^{2} + 26 q^{3} + 1396 q^{4} - 248 q^{5} - 258 q^{6} + 1742 q^{7} + 60 q^{8} + 16007 q^{9} + O(q^{10})$$ $$21 q - 8 q^{2} + 26 q^{3} + 1396 q^{4} - 248 q^{5} - 258 q^{6} + 1742 q^{7} + 60 q^{8} + 16007 q^{9} - 4034 q^{10} + 1090 q^{11} + 8310 q^{12} - 4596 q^{13} - 47158 q^{14} - 30042 q^{15} + 110868 q^{16} - 18612 q^{17} + 7746 q^{18} + 49406 q^{19} - 140584 q^{20} + 32482 q^{21} - 151318 q^{22} - 40744 q^{23} + 95088 q^{24} + 273787 q^{25} + 1294 q^{26} + 359282 q^{27} + 435428 q^{28} - 122406 q^{29} - 1002100 q^{30} - 461808 q^{31} + 563192 q^{32} - 396262 q^{33} - 390976 q^{34} - 268342 q^{35} + 1467574 q^{36} - 50653 q^{37} + 1080540 q^{38} - 264526 q^{39} - 270086 q^{40} + 1339728 q^{41} - 398570 q^{42} - 1966658 q^{43} - 727066 q^{44} - 2566066 q^{45} + 866570 q^{46} + 734994 q^{47} + 3862210 q^{48} + 2068199 q^{49} - 6194872 q^{50} + 618534 q^{51} - 4051336 q^{52} - 2864624 q^{53} + 7445238 q^{54} + 686906 q^{55} - 7953768 q^{56} - 3054846 q^{57} + 4291174 q^{58} - 1295920 q^{59} - 5738032 q^{60} + 3638938 q^{61} + 3868418 q^{62} + 4046420 q^{63} + 8389684 q^{64} + 2555560 q^{65} + 11536406 q^{66} - 4784216 q^{67} - 18588 q^{68} + 1958704 q^{69} + 2511040 q^{70} - 360678 q^{71} + 5271960 q^{72} - 12759148 q^{73} - 2026120 q^{74} + 9607954 q^{75} + 6244140 q^{76} + 23391266 q^{77} - 31320222 q^{78} - 5189462 q^{79} - 12065104 q^{80} + 2203637 q^{81} + 10253218 q^{82} + 12873414 q^{83} - 32820544 q^{84} + 25165332 q^{85} - 13459684 q^{86} - 18438288 q^{87} - 44255800 q^{88} - 9020624 q^{89} + 18065504 q^{90} + 11639102 q^{91} - 34314620 q^{92} - 5266552 q^{93} + 71670 q^{94} + 11847816 q^{95} + 22813492 q^{96} + 680570 q^{97} + 7917594 q^{98} - 31076936 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_0(37))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 37
37.8.a.a $10$ $11.558$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-24$$ $$-95$$ $$-624$$ $$-501$$ $-$ $$q+(-2-\beta _{1})q^{2}+(-10+\beta _{1}-\beta _{7}+\cdots)q^{3}+\cdots$$
37.8.a.b $11$ $11.558$ $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ None $$16$$ $$121$$ $$376$$ $$2243$$ $+$ $$q+(1+\beta _{1})q^{2}+(11+\beta _{2})q^{3}+(71+2\beta _{1}+\cdots)q^{4}+\cdots$$