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Properties

Label 300.3.l
Level $300$
Weight $3$
Character orbit 300.l
Rep. character $\chi_{300}(107,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $136$
Newform subspaces $8$
Sturm bound $180$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 60 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 8 \)
Sturm bound: \(180\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(7\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(300, [\chi])\).

Total New Old
Modular forms 264 152 112
Cusp forms 216 136 80
Eisenstein series 48 16 32

Trace form

\( 136 q - 4 q^{6} + O(q^{10}) \) \( 136 q - 4 q^{6} + 20 q^{12} + 8 q^{13} + 48 q^{16} + 24 q^{18} + 24 q^{21} + 76 q^{22} + 84 q^{28} + 40 q^{33} - 164 q^{36} + 40 q^{37} - 236 q^{42} - 504 q^{46} - 196 q^{48} - 304 q^{52} + 72 q^{57} - 180 q^{58} - 144 q^{61} + 660 q^{66} + 600 q^{72} - 104 q^{73} + 1064 q^{76} + 408 q^{78} - 168 q^{81} + 720 q^{82} + 580 q^{88} - 368 q^{93} - 1780 q^{96} - 72 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(300, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
300.3.l.a 300.l 60.l $4$ $8.174$ \(\Q(i, \sqrt{14})\) None \(-8\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-2q^{2}+(1+\beta _{2}+\beta _{3})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
300.3.l.b 300.l 60.l $4$ $8.174$ \(\Q(i, \sqrt{14})\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-2\beta _{2}q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}-4q^{4}+\cdots\)
300.3.l.c 300.l 60.l $4$ $8.174$ \(\Q(i, \sqrt{14})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-2\beta _{2}q^{2}+(1+\beta _{1}-\beta _{2})q^{3}-4q^{4}+\cdots\)
300.3.l.d 300.l 60.l $4$ $8.174$ \(\Q(i, \sqrt{14})\) None \(8\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+2q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+4q^{4}+\cdots\)
300.3.l.e 300.l 60.l $8$ $8.174$ 8.0.3317760000.4 \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{2}+3\beta _{5}q^{3}+(4\beta _{4}+\beta _{6})q^{4}+\cdots\)
300.3.l.f 300.l 60.l $8$ $8.174$ 8.0.40960000.1 \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-\beta _{6}q^{2}+(-\beta _{3}+\beta _{7})q^{3}-4\beta _{2}q^{4}+\cdots\)
300.3.l.g 300.l 60.l $40$ $8.174$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
300.3.l.h 300.l 60.l $64$ $8.174$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{3}^{\mathrm{old}}(300, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(300, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)