Properties

Label 350.2.e
Level 350
Weight 2
Character orbit e
Rep. character \(\chi_{350}(51,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 24
Newforms 12
Sturm bound 120
Trace bound 7

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newforms: \( 12 \)
Sturm bound: \(120\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(350, [\chi])\).

Total New Old
Modular forms 144 24 120
Cusp forms 96 24 72
Eisenstein series 48 0 48

Trace form

\(24q \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(24q \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{18} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 8q^{22} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 8q^{26} \) \(\mathstrut -\mathstrut 24q^{27} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 20q^{31} \) \(\mathstrut -\mathstrut 24q^{33} \) \(\mathstrut -\mathstrut 16q^{34} \) \(\mathstrut +\mathstrut 16q^{36} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 24q^{41} \) \(\mathstrut -\mathstrut 4q^{42} \) \(\mathstrut +\mathstrut 40q^{43} \) \(\mathstrut -\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 20q^{47} \) \(\mathstrut -\mathstrut 12q^{51} \) \(\mathstrut +\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 12q^{54} \) \(\mathstrut -\mathstrut 4q^{56} \) \(\mathstrut +\mathstrut 24q^{57} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut +\mathstrut 48q^{62} \) \(\mathstrut -\mathstrut 36q^{63} \) \(\mathstrut +\mathstrut 24q^{64} \) \(\mathstrut -\mathstrut 4q^{68} \) \(\mathstrut +\mathstrut 48q^{69} \) \(\mathstrut -\mathstrut 24q^{71} \) \(\mathstrut +\mathstrut 8q^{72} \) \(\mathstrut -\mathstrut 28q^{73} \) \(\mathstrut +\mathstrut 20q^{74} \) \(\mathstrut -\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 8q^{77} \) \(\mathstrut +\mathstrut 16q^{78} \) \(\mathstrut +\mathstrut 12q^{79} \) \(\mathstrut -\mathstrut 20q^{81} \) \(\mathstrut -\mathstrut 16q^{82} \) \(\mathstrut +\mathstrut 32q^{83} \) \(\mathstrut +\mathstrut 28q^{84} \) \(\mathstrut -\mathstrut 8q^{86} \) \(\mathstrut +\mathstrut 24q^{87} \) \(\mathstrut +\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut 8q^{89} \) \(\mathstrut +\mathstrut 28q^{91} \) \(\mathstrut -\mathstrut 24q^{92} \) \(\mathstrut -\mathstrut 12q^{93} \) \(\mathstrut +\mathstrut 4q^{94} \) \(\mathstrut -\mathstrut 64q^{97} \) \(\mathstrut -\mathstrut 24q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(350, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
350.2.e.a \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-2\) \(0\) \(-5\) \(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.b \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-2\) \(0\) \(4\) \(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.c \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
350.2.e.d \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(1\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(2-3\zeta_{6})q^{7}+\cdots\)
350.2.e.e \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(1\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.f \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(0\) \(5\) \(q-\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.g \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(0\) \(-5\) \(q+\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.h \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(-2\) \(0\) \(4\) \(q+\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.i \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-2+3\zeta_{6})q^{7}+\cdots\)
350.2.e.j \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(3-2\zeta_{6})q^{7}+\cdots\)
350.2.e.k \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(2\) \(0\) \(5\) \(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
350.2.e.l \(2\) \(2.795\) \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(0\) \(-1\) \(q+\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)