LMFDB - Space of modular forms of level 39, weight 2, and Character 39.e

Defining parameters

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	39.e (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 13 $
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$ 2 $
Sturm bound:			$9$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(39, [\chi])$.

	Total	New	Old
Modular forms	14	6	8
Cusp forms	6	6	0
Eisenstein series	8	0	8

Trace form

$ 6 q - 2 q^{2} - q^{3} - 4 q^{4} - 8 q^{5} + q^{7} + 12 q^{8} - 3 q^{9} + O(q^{10}) $

\( 6 q - 2 q^{2} - q^{3} - 4 q^{4} - 8 q^{5} + q^{7} + 12 q^{8} - 3 q^{9} - 6 q^{10} + 6 q^{11} + 12 q^{12} - 5 q^{13} - 16 q^{14} + 2 q^{15} - 2 q^{16} + 6 q^{17} + 4 q^{18} - 2 q^{20} - 10 q^{21} + 4 q^{22} + 2 q^{23} - 12 q^{24} - 2 q^{25} + 30 q^{26} + 2 q^{27} + 18 q^{28} - 8 q^{30} + 10 q^{31} - 14 q^{32} + 2 q^{33} + 4 q^{34} + 6 q^{35} - 4 q^{36} - 12 q^{37} - 40 q^{38} - 6 q^{39} - 4 q^{40} - 10 q^{41} + 12 q^{42} - 11 q^{43} - 16 q^{44} + 4 q^{45} + 4 q^{46} + 12 q^{47} - 4 q^{48} + 4 q^{49} + 28 q^{50} + 16 q^{51} - 30 q^{52} + 4 q^{53} - 8 q^{55} + 28 q^{56} + 24 q^{57} + 26 q^{58} + 14 q^{59} - 17 q^{61} + 4 q^{62} + q^{63} + 28 q^{64} + 4 q^{65} - 3 q^{67} - 18 q^{68} - 10 q^{69} - 8 q^{70} - 34 q^{71} - 6 q^{72} - 2 q^{73} + 2 q^{74} - 7 q^{75} - 4 q^{76} + 4 q^{77} - 28 q^{78} + 22 q^{79} - 22 q^{80} - 3 q^{81} - 18 q^{82} - 48 q^{83} + 14 q^{84} - 14 q^{85} - 2 q^{87} + 12 q^{88} - 4 q^{89} + 12 q^{90} + 31 q^{91} + 32 q^{92} + 3 q^{93} + 28 q^{94} + 20 q^{95} + 8 q^{96} + 15 q^{97} - 22 q^{98} - 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(39, [\chi])$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				$q$-expansion
Label	Dim.	$A$	Field	CM	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	$q$-expansion
39.2.e.a	$2$	$0.311$	$\Q(\sqrt{-3}) $	None	$-1$	$1$	$-2$	$-2$	$q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$
39.2.e.b	$4$	$0.311$	$\Q(\sqrt{-3}, \sqrt{17})$	None	$-1$	$-2$	$-6$	$3$	$q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots$

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Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	39.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(9\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(39, [\chi])\) into newform subspaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	39.2.e

Level	$39$
Weight	$2$
Character orbit	39.e
Rep. character	$\chi_{39}(16,\cdot)$
Character field	$\Q(\zeta_{3})$
Dimension	$6$
Newform subspaces	$2$
Sturm bound	$9$
Trace bound	$1$

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	$q$-expansion
39.2.e.a	$2$	$0.311$	\(\Q(\sqrt{-3}) \)	None	\(-1\)	\(1\)	\(-2\)	\(-2\)	\(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
39.2.e.b	$4$	$0.311$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None	\(-1\)	\(-2\)	\(-6\)	\(3\)	\(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)