# Properties

 Label 400.8.a Level $400$ Weight $8$ Character orbit 400.a Rep. character $\chi_{400}(1,\cdot)$ Character field $\Q$ Dimension $65$ Newform subspaces $38$ Sturm bound $480$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial) Character field: $$\Q$$ Newform subspaces: $$38$$ Sturm bound: $$480$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_0(400))$$.

Total New Old
Modular forms 438 68 370
Cusp forms 402 65 337
Eisenstein series 36 3 33

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

$$2$$$$5$$FrickeDim
$$+$$$$+$$$$+$$$$16$$
$$+$$$$-$$$$-$$$$17$$
$$-$$$$+$$$$-$$$$15$$
$$-$$$$-$$$$+$$$$17$$
Plus space$$+$$$$33$$
Minus space$$-$$$$32$$

## Trace form

 $$65 q + 26 q^{3} + 662 q^{7} + 46317 q^{9} + O(q^{10})$$ $$65 q + 26 q^{3} + 662 q^{7} + 46317 q^{9} - 3388 q^{11} + 3278 q^{13} - 11258 q^{17} - 60448 q^{19} + 3356 q^{21} + 7282 q^{23} + 12188 q^{27} - 68378 q^{29} + 8340 q^{31} - 166544 q^{33} - 104322 q^{37} + 268748 q^{39} - 133454 q^{41} - 142462 q^{43} - 1589618 q^{47} + 6831653 q^{49} + 1514416 q^{51} - 392442 q^{53} + 725144 q^{57} - 5391876 q^{59} - 5518 q^{61} + 4296854 q^{63} + 4606486 q^{67} + 2445284 q^{69} - 3556636 q^{71} + 1901070 q^{73} + 7780240 q^{77} + 9605360 q^{79} + 27940077 q^{81} - 3397918 q^{83} - 19863444 q^{87} + 10608630 q^{89} - 18655020 q^{91} + 9069672 q^{93} + 8946158 q^{97} - 46333544 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_0(400))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
400.8.a.a $1$ $124.954$ $$\Q$$ None $$0$$ $$-87$$ $$0$$ $$-1366$$ $-$ $+$ $$q-87q^{3}-1366q^{7}+5382q^{9}+1083q^{11}+\cdots$$
400.8.a.b $1$ $124.954$ $$\Q$$ None $$0$$ $$-84$$ $$0$$ $$-456$$ $+$ $+$ $$q-84q^{3}-456q^{7}+4869q^{9}+2524q^{11}+\cdots$$
400.8.a.c $1$ $124.954$ $$\Q$$ None $$0$$ $$-69$$ $$0$$ $$174$$ $+$ $+$ $$q-69q^{3}+174q^{7}+2574q^{9}-7111q^{11}+\cdots$$
400.8.a.d $1$ $124.954$ $$\Q$$ None $$0$$ $$-57$$ $$0$$ $$1174$$ $-$ $-$ $$q-57q^{3}+1174q^{7}+1062q^{9}+7563q^{11}+\cdots$$
400.8.a.e $1$ $124.954$ $$\Q$$ None $$0$$ $$-48$$ $$0$$ $$-1644$$ $-$ $+$ $$q-48q^{3}-1644q^{7}+117q^{9}-172q^{11}+\cdots$$
400.8.a.f $1$ $124.954$ $$\Q$$ None $$0$$ $$-43$$ $$0$$ $$-974$$ $-$ $-$ $$q-43q^{3}-974q^{7}-338q^{9}-87q^{11}+\cdots$$
400.8.a.g $1$ $124.954$ $$\Q$$ None $$0$$ $$-36$$ $$0$$ $$776$$ $+$ $+$ $$q-6^{2}q^{3}+776q^{7}-891q^{9}+124q^{11}+\cdots$$
400.8.a.h $1$ $124.954$ $$\Q$$ None $$0$$ $$-34$$ $$0$$ $$-106$$ $+$ $-$ $$q-34q^{3}-106q^{7}-1031q^{9}+1324q^{11}+\cdots$$
400.8.a.i $1$ $124.954$ $$\Q$$ None $$0$$ $$-9$$ $$0$$ $$694$$ $+$ $-$ $$q-9q^{3}+694q^{7}-2106q^{9}-4901q^{11}+\cdots$$
400.8.a.j $1$ $124.954$ $$\Q$$ None $$0$$ $$-6$$ $$0$$ $$-706$$ $-$ $+$ $$q-6q^{3}-706q^{7}-2151q^{9}+3840q^{11}+\cdots$$
400.8.a.k $1$ $124.954$ $$\Q$$ None $$0$$ $$9$$ $$0$$ $$-694$$ $+$ $+$ $$q+9q^{3}-694q^{7}-2106q^{9}-4901q^{11}+\cdots$$
400.8.a.l $1$ $124.954$ $$\Q$$ None $$0$$ $$12$$ $$0$$ $$1016$$ $-$ $+$ $$q+12q^{3}+1016q^{7}-2043q^{9}-1092q^{11}+\cdots$$
400.8.a.m $1$ $124.954$ $$\Q$$ None $$0$$ $$28$$ $$0$$ $$104$$ $-$ $+$ $$q+28q^{3}+104q^{7}-1403q^{9}+5148q^{11}+\cdots$$
400.8.a.n $1$ $124.954$ $$\Q$$ None $$0$$ $$34$$ $$0$$ $$106$$ $+$ $-$ $$q+34q^{3}+106q^{7}-1031q^{9}+1324q^{11}+\cdots$$
400.8.a.o $1$ $124.954$ $$\Q$$ None $$0$$ $$43$$ $$0$$ $$974$$ $-$ $+$ $$q+43q^{3}+974q^{7}-338q^{9}-87q^{11}+\cdots$$
400.8.a.p $1$ $124.954$ $$\Q$$ None $$0$$ $$44$$ $$0$$ $$-1224$$ $+$ $+$ $$q+44q^{3}-1224q^{7}-251q^{9}+3164q^{11}+\cdots$$
400.8.a.q $1$ $124.954$ $$\Q$$ None $$0$$ $$57$$ $$0$$ $$-1174$$ $-$ $+$ $$q+57q^{3}-1174q^{7}+1062q^{9}+7563q^{11}+\cdots$$
400.8.a.r $1$ $124.954$ $$\Q$$ None $$0$$ $$69$$ $$0$$ $$-174$$ $+$ $-$ $$q+69q^{3}-174q^{7}+2574q^{9}-7111q^{11}+\cdots$$
400.8.a.s $1$ $124.954$ $$\Q$$ None $$0$$ $$87$$ $$0$$ $$1366$$ $-$ $-$ $$q+87q^{3}+1366q^{7}+5382q^{9}+1083q^{11}+\cdots$$
400.8.a.t $2$ $124.954$ $$\Q(\sqrt{31})$$ None $$0$$ $$-56$$ $$0$$ $$-408$$ $-$ $-$ $$q+(-28+\beta )q^{3}+(-204-7\beta )q^{7}+\cdots$$
400.8.a.u $2$ $124.954$ $$\Q(\sqrt{2521})$$ None $$0$$ $$-40$$ $$0$$ $$-280$$ $-$ $-$ $$q+(-20-\beta )q^{3}+(-140-6\beta )q^{7}+\cdots$$
400.8.a.v $2$ $124.954$ $$\Q(\sqrt{649})$$ None $$0$$ $$-40$$ $$0$$ $$600$$ $-$ $+$ $$q+(-20-\beta )q^{3}+(300+42\beta )q^{7}+(-1138+\cdots)q^{9}+\cdots$$
400.8.a.w $2$ $124.954$ $$\Q(\sqrt{1129})$$ None $$0$$ $$-20$$ $$0$$ $$1660$$ $-$ $+$ $$q+(-10-\beta )q^{3}+(830+9\beta )q^{7}+(2429+\cdots)q^{9}+\cdots$$
400.8.a.x $2$ $124.954$ $$\Q(\sqrt{3889})$$ None $$0$$ $$-4$$ $$0$$ $$-1144$$ $+$ $-$ $$q+(-2-\beta )q^{3}+(-572-2\beta )q^{7}+(1706+\cdots)q^{9}+\cdots$$
400.8.a.y $2$ $124.954$ $$\Q(\sqrt{29})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $-$ $$q-3\beta q^{3}-39\beta q^{7}-1143q^{9}+6828q^{11}+\cdots$$
400.8.a.z $2$ $124.954$ $$\Q(\sqrt{46})$$ None $$0$$ $$4$$ $$0$$ $$844$$ $+$ $+$ $$q+(2+\beta )q^{3}+(422+17\beta )q^{7}+(761+\cdots)q^{9}+\cdots$$
400.8.a.ba $2$ $124.954$ $$\Q(\sqrt{3889})$$ None $$0$$ $$4$$ $$0$$ $$1144$$ $+$ $+$ $$q+(2+\beta )q^{3}+(572+2\beta )q^{7}+(1706+\cdots)q^{9}+\cdots$$
400.8.a.bb $2$ $124.954$ $$\Q(\sqrt{19})$$ None $$0$$ $$20$$ $$0$$ $$-100$$ $-$ $+$ $$q+(10+\beta )q^{3}+(-50-7\beta )q^{7}+(2777+\cdots)q^{9}+\cdots$$
400.8.a.bc $2$ $124.954$ $$\Q(\sqrt{6})$$ None $$0$$ $$36$$ $$0$$ $$-404$$ $+$ $+$ $$q+(18+\beta )q^{3}+(-202-63\beta )q^{7}+(-1479+\cdots)q^{9}+\cdots$$
400.8.a.bd $2$ $124.954$ $$\Q(\sqrt{649})$$ None $$0$$ $$40$$ $$0$$ $$-600$$ $-$ $-$ $$q+(20-\beta )q^{3}+(-300+42\beta )q^{7}+(-1138+\cdots)q^{9}+\cdots$$
400.8.a.be $2$ $124.954$ $$\Q(\sqrt{2521})$$ None $$0$$ $$40$$ $$0$$ $$280$$ $-$ $+$ $$q+(20-\beta )q^{3}+(140-6\beta )q^{7}+(734+\cdots)q^{9}+\cdots$$
400.8.a.bf $2$ $124.954$ $$\Q(\sqrt{31})$$ None $$0$$ $$56$$ $$0$$ $$408$$ $-$ $-$ $$q+(28+\beta )q^{3}+(204-7\beta )q^{7}+(1697+\cdots)q^{9}+\cdots$$
400.8.a.bg $2$ $124.954$ $$\Q(\sqrt{601})$$ None $$0$$ $$76$$ $$0$$ $$796$$ $+$ $+$ $$q+(38-\beta )q^{3}+(398-7\beta )q^{7}+(1661+\cdots)q^{9}+\cdots$$
400.8.a.bh $3$ $124.954$ 3.3.40101.1 None $$0$$ $$-97$$ $$0$$ $$1630$$ $+$ $-$ $$q+(-2^{5}-\beta _{1})q^{3}+(543+2\beta _{1}+\beta _{2})q^{7}+\cdots$$
400.8.a.bi $3$ $124.954$ 3.3.40101.1 None $$0$$ $$97$$ $$0$$ $$-1630$$ $+$ $+$ $$q+(2^{5}+\beta _{1})q^{3}+(-543-2\beta _{1}-\beta _{2})q^{7}+\cdots$$
400.8.a.bj $4$ $124.954$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$0$$ $$-32$$ $$0$$ $$-1632$$ $+$ $-$ $$q+(-8-\beta _{1})q^{3}+(-408+3\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots$$
400.8.a.bk $4$ $124.954$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $-$ $$q+\beta _{1}q^{3}+(6\beta _{1}+\beta _{2})q^{7}+(509+\beta _{3})q^{9}+\cdots$$
400.8.a.bl $4$ $124.954$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$0$$ $$32$$ $$0$$ $$1632$$ $+$ $-$ $$q+(8+\beta _{1})q^{3}+(408-3\beta _{1}+2\beta _{2}-\beta _{3})q^{7}+\cdots$$

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_0(400))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_0(400)) \simeq$$ $$S_{8}^{\mathrm{new}}(\Gamma_0(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 9}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 10}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(25))$$$$^{\oplus 5}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(200))$$$$^{\oplus 2}$$