Properties

 Label 600.4.f Level $600$ Weight $4$ Character orbit 600.f Rep. character $\chi_{600}(49,\cdot)$ Character field $\Q$ Dimension $26$ Newform subspaces $11$ Sturm bound $480$ Trace bound $19$

Related objects

Defining parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 600.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$480$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(600, [\chi])$$.

Total New Old
Modular forms 384 26 358
Cusp forms 336 26 310
Eisenstein series 48 0 48

Trace form

 $$26 q - 234 q^{9} + O(q^{10})$$ $$26 q - 234 q^{9} + 56 q^{11} - 92 q^{19} - 84 q^{21} + 556 q^{29} + 492 q^{31} - 24 q^{39} - 1012 q^{41} + 42 q^{49} + 612 q^{51} + 440 q^{59} - 136 q^{61} + 552 q^{69} + 1536 q^{71} - 592 q^{79} + 2106 q^{81} - 5340 q^{89} + 1484 q^{91} - 504 q^{99} + O(q^{100})$$

Decomposition of $$S_{4}^{\mathrm{new}}(600, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
600.4.f.a $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+20iq^{7}-9q^{9}-56q^{11}+\cdots$$
600.4.f.b $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+24iq^{7}-9q^{9}-28q^{11}+\cdots$$
600.4.f.c $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+2^{4}iq^{7}-9q^{9}-28q^{11}+\cdots$$
600.4.f.d $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}-9q^{9}+4q^{11}-54iq^{13}+\cdots$$
600.4.f.e $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+5iq^{7}-9q^{9}+14q^{11}+\cdots$$
600.4.f.f $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+20iq^{7}-9q^{9}+2^{4}q^{11}+\cdots$$
600.4.f.g $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{3}+8iq^{7}-9q^{9}+20q^{11}+\cdots$$
600.4.f.h $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+19iq^{7}-9q^{9}+22q^{11}+\cdots$$
600.4.f.i $2$ $35.401$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+4iq^{7}-9q^{9}+72q^{11}+\cdots$$
600.4.f.j $4$ $35.401$ $$\Q(i, \sqrt{109})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{7}-9q^{9}+(-8+\cdots)q^{11}+\cdots$$
600.4.f.k $4$ $35.401$ $$\Q(i, \sqrt{181})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{3}+(-3\beta _{1}+\beta _{2})q^{7}-9q^{9}+\cdots$$

Decomposition of $$S_{4}^{\mathrm{old}}(600, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(600, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$