LMFDB - Space of modular forms of level 338, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	338.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 8 $
Sturm bound:			$91$
Trace bound:			$3$
Distinguishing $T_p$:			$3$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(338))$.

	Total	New	Old
Modular forms	59	12	47
Cusp forms	32	12	20
Eisenstein series	27	0	27

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

$2$	$13$	Fricke	Dim.
$+$	$+$	$+$	$2$
$+$	$-$	$-$	$4$
$-$	$+$	$-$	$5$
$-$	$-$	$+$	$1$
Plus space		$+$	$3$
Minus space		$-$	$9$

Trace form

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

$ 12q + 2q^{3} + 12q^{4} + 4q^{5} + 4q^{6} + 10q^{9} - 2q^{10} - 4q^{11} + 2q^{12} - 4q^{14} + 12q^{16} + 4q^{17} - 8q^{18} - 8q^{19} + 4q^{20} + 4q^{21} + 6q^{22} + 4q^{24} + 10q^{25} + 8q^{27} - 14q^{29} - 12q^{30} - 12q^{33} - 12q^{35} + 10q^{36} + 4q^{37} - 6q^{38} - 2q^{40} - 4q^{42} + 2q^{43} - 4q^{44} + 4q^{46} - 16q^{47} + 2q^{48} + 16q^{49} + 8q^{50} - 20q^{51} - 18q^{53} + 4q^{54} + 8q^{55} - 4q^{56} + 16q^{57} + 4q^{58} + 16q^{59} - 10q^{61} - 16q^{62} - 8q^{63} + 12q^{64} - 8q^{66} - 12q^{67} + 4q^{68} - 28q^{69} + 4q^{70} + 8q^{71} - 8q^{72} + 8q^{73} - 18q^{74} - 38q^{75} - 8q^{76} - 12q^{77} - 20q^{79} + 4q^{80} - 20q^{81} - 4q^{82} - 12q^{83} + 4q^{84} - 12q^{85} + 4q^{86} - 20q^{87} + 6q^{88} - 2q^{90} + 16q^{93} - 16q^{94} - 12q^{95} + 4q^{96} - 4q^{97} + 24q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(338))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs		$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	2	13	$q$-expansion
338.2.a.a	$1$	$2.699$	$\Q$	None	$-1$	$-3$	$1$	$-1$	$+$	$+$	$q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots$
338.2.a.b	$1$	$2.699$	$\Q$	None	$-1$	$-1$	$3$	$3$	$+$	$-$	$q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots$
338.2.a.c	$1$	$2.699$	$\Q$	None	$-1$	$0$	$1$	$-4$	$+$	$+$	$q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots$
338.2.a.d	$1$	$2.699$	$\Q$	None	$1$	$-1$	$-3$	$-3$	$-$	$-$	$q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots$
338.2.a.e	$1$	$2.699$	$\Q$	None	$1$	$0$	$-1$	$4$	$-$	$+$	$q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots$
338.2.a.f	$1$	$2.699$	$\Q$	None	$1$	$1$	$3$	$1$	$-$	$+$	$q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots$
338.2.a.g	$3$	$2.699$	$\Q(\zeta_{14})^+$	None	$-3$	$3$	$-2$	$4$	$+$	$-$	$q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots$
338.2.a.h	$3$	$2.699$	$\Q(\zeta_{14})^+$	None	$3$	$3$	$2$	$-4$	$-$	$+$	$q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(338))$ into lower level spaces

$ S_{2}^{\mathrm{old}}(\Gamma_0(338)) \cong $ $S_{2}^{\mathrm{new}}(\Gamma_0(26))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(169))$$^{\oplus 2}$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(91\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

\(2\)	\(13\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(9\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more

Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(338))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(338))\) into lower level spaces

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	338.2.a

Level	$338$
Weight	$2$
Character orbit	338.a
Rep. character	$\chi_{338}(1,\cdot)$
Character field	$\Q$
Dimension	$12$
Newform subspaces	$8$
Sturm bound	$91$
Trace bound	$3$

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs		$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	2	13	$q$-expansion
338.2.a.a	\(1\)	\(2.699\)	\(\Q\)	None	\(-1\)	\(-3\)	\(1\)	\(-1\)	\(+\)	\(+\)	\(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
338.2.a.b	\(1\)	\(2.699\)	\(\Q\)	None	\(-1\)	\(-1\)	\(3\)	\(3\)	\(+\)	\(-\)	\(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
338.2.a.c	\(1\)	\(2.699\)	\(\Q\)	None	\(-1\)	\(0\)	\(1\)	\(-4\)	\(+\)	\(+\)	\(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots\)
338.2.a.d	\(1\)	\(2.699\)	\(\Q\)	None	\(1\)	\(-1\)	\(-3\)	\(-3\)	\(-\)	\(-\)	\(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
338.2.a.e	\(1\)	\(2.699\)	\(\Q\)	None	\(1\)	\(0\)	\(-1\)	\(4\)	\(-\)	\(+\)	\(q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots\)
338.2.a.f	\(1\)	\(2.699\)	\(\Q\)	None	\(1\)	\(1\)	\(3\)	\(1\)	\(-\)	\(+\)	\(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots\)
338.2.a.g	\(3\)	\(2.699\)	\(\Q(\zeta_{14})^+\)	None	\(-3\)	\(3\)	\(-2\)	\(4\)	\(+\)	\(-\)	\(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots\)
338.2.a.h	\(3\)	\(2.699\)	\(\Q(\zeta_{14})^+\)	None	\(3\)	\(3\)	\(2\)	\(-4\)	\(-\)	\(+\)	\(q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots\)