LMFDB - Space of modular forms of level 338, weight 10, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	338.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 18 $
Sturm bound:			$455$
Trace bound:			$5$
Distinguishing $T_p$:			$3$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{10}(\Gamma_0(338))$.

	Total	New	Old
Modular forms	423	116	307
Cusp forms	395	116	279
Eisenstein series	28	0	28

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	Total			Cusp			Eisenstein
$2$	$13$	Fricke	All	New	Old	All	New	Old	All	New	Old
$+$	$+$	$+$	$103$	$28$	$75$	$96$	$28$	$68$	$7$	$0$	$7$
$+$	$-$	$-$	$108$	$30$	$78$	$101$	$30$	$71$	$7$	$0$	$7$
$-$	$+$	$-$	$107$	$31$	$76$	$100$	$31$	$69$	$7$	$0$	$7$
$-$	$-$	$+$	$105$	$27$	$78$	$98$	$27$	$71$	$7$	$0$	$7$
Plus space		$+$	$208$	$55$	$153$	$194$	$55$	$139$	$14$	$0$	$14$
Minus space		$-$	$215$	$61$	$154$	$201$	$61$	$140$	$14$	$0$	$14$

Trace form

$ 116 q + 6 q^{3} + 29696 q^{4} + 2428 q^{5} - 2496 q^{6} - 1180 q^{7} + 742654 q^{9} + 37344 q^{10} + 50144 q^{11} + 1536 q^{12} - 112192 q^{14} + 26864 q^{15} + 7602176 q^{16} - 1133024 q^{17} + 292736 q^{18}+ \cdots + 6184680436 q^{99}+O(q^{100}) $

116 * q + 6 * q^3 + 29696 * q^4 + 2428 * q^5 - 2496 * q^6 - 1180 * q^7 + 742654 * q^9 + 37344 * q^10 + 50144 * q^11 + 1536 * q^12 - 112192 * q^14 + 26864 * q^15 + 7602176 * q^16 - 1133024 * q^17 + 292736 * q^18 - 1003748 * q^19 + 621568 * q^20 + 1318796 * q^21 - 2584992 * q^22 - 4348236 * q^23 - 638976 * q^24 + 43709058 * q^25 + 3565020 * q^27 - 302080 * q^28 + 13295158 * q^29 + 1154368 * q^30 - 12553764 * q^31 + 9944252 * q^33 + 19081216 * q^34 - 2528052 * q^35 + 190119424 * q^36 + 14144908 * q^37 - 5710944 * q^38 + 9560064 * q^40 + 62310264 * q^41 - 15161408 * q^42 + 12363346 * q^43 + 12836864 * q^44 - 16074512 * q^45 + 36739136 * q^46 - 11195932 * q^47 + 393216 * q^48 + 632511124 * q^49 - 19040128 * q^50 + 206075264 * q^51 - 136724670 * q^53 - 177181632 * q^54 + 125124600 * q^55 - 28721152 * q^56 - 183224560 * q^57 + 99567680 * q^58 + 54241756 * q^59 + 6877184 * q^60 + 25019766 * q^61 + 67342592 * q^62 - 842734212 * q^63 + 1946157056 * q^64 + 262039424 * q^66 - 312664320 * q^67 - 290054144 * q^68 + 357645940 * q^69 + 332467776 * q^70 + 296235812 * q^71 + 74940416 * q^72 + 557073280 * q^73 - 123161376 * q^74 + 118852002 * q^75 - 256959488 * q^76 + 1033459644 * q^77 - 1178812176 * q^79 + 159121408 * q^80 + 5741961652 * q^81 - 313225280 * q^82 + 1193508480 * q^83 + 337611776 * q^84 + 325342444 * q^85 + 1737090112 * q^86 - 716486260 * q^87 - 661757952 * q^88 - 1435566888 * q^89 + 2336204256 * q^90 - 1113148416 * q^92 + 1752294000 * q^93 + 1059316992 * q^94 + 1569077040 * q^95 - 163577856 * q^96 + 313419948 * q^97 - 926367744 * q^98 + 6184680436 * q^99

Decomposition of $S_{10}^{\mathrm{new}}(\Gamma_0(338))$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	2	13	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
338.10.a.a	$338$	$10$	338.a	1.a	$1$	$1$	$1$	$174.082$	$\Q$	None	✓				2.10.a.a	$1$	$1$	$-16$	$-156$	$-870$	$952$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}-156q^{3}+2^{8}q^{4}-870q^{5}+\cdots$
338.10.a.b	$338$	$10$	338.a	1.a	$1$	$1$	$1$	$174.082$	$\Q$	None	✓				26.10.a.c	$1$	$1$	$-16$	$75$	$1979$	$10115$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+75q^{3}+2^{8}q^{4}+1979q^{5}+\cdots$
338.10.a.c	$338$	$10$	338.a	1.a	$1$	$1$	$1$	$174.082$	$\Q$	None	✓				26.10.a.a	$1$	$0$	$16$	$-273$	$-1015$	$-3955$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}-273q^{3}+2^{8}q^{4}-1015q^{5}+\cdots$
338.10.a.d	$338$	$10$	338.a	1.a	$1$	$1$	$1$	$174.082$	$\Q$	None	✓				26.10.a.b	$1$	$0$	$16$	$192$	$1310$	$5810$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+192q^{3}+2^{8}q^{4}+1310q^{5}+\cdots$
338.10.a.e	$338$	$10$	338.a	1.a	$1$	$3$	$3$	$174.082$	3.3.2119705.1	None	✓				26.10.a.e	$1$	$1$	$-48$	$156$	$1272$	$-17058$	$+$	$+$	$+$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(52-\beta _{1}-\beta _{2})q^{3}+2^{8}q^{4}+\cdots$
338.10.a.f	$338$	$10$	338.a	1.a	$1$	$3$	$3$	$174.082$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓				26.10.a.d	$1$	$0$	$48$	$0$	$-248$	$2956$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}-\beta _{2}q^{3}+2^{8}q^{4}+(-79+11\beta _{1}+\cdots)q^{5}+\cdots$
338.10.a.g	$338$	$10$	338.a	1.a	$1$	$5$	$5$	$174.082$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				26.10.b.a	$1$	$0$	$-80$	$81$	$-1213$	$8715$	$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(2^{4}+\beta _{1})q^{3}+2^{8}q^{4}+(-3^{5}+\cdots)q^{5}+\cdots$
338.10.a.h	$338$	$10$	338.a	1.a	$1$	$5$	$5$	$174.082$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				26.10.c.a	$1$	$1$	$-80$	$81$	$-914$	$-3323$	$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(2^{4}+\beta _{1})q^{3}+2^{8}q^{4}+(-183+\cdots)q^{5}+\cdots$
338.10.a.i	$338$	$10$	338.a	1.a	$1$	$5$	$5$	$174.082$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				26.10.c.a	$1$	$0$	$80$	$81$	$914$	$3323$	$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+(2^{4}+\beta _{1})q^{3}+2^{8}q^{4}+(183+\cdots)q^{5}+\cdots$
338.10.a.j	$338$	$10$	338.a	1.a	$1$	$5$	$5$	$174.082$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				26.10.b.a	$1$	$1$	$80$	$81$	$1213$	$-8715$	$-$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+(2^{4}+\beta _{1})q^{3}+2^{8}q^{4}+(3^{5}+\cdots)q^{5}+\cdots$
338.10.a.k	$338$	$10$	338.a	1.a	$1$	$6$	$6$	$174.082$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓				26.10.c.b	$1$	$1$	$-96$	$81$	$-1693$	$-473$	$+$	$+$	$+$	$2^{6}\cdot 3^{4}\cdot 13$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(14-\beta _{1})q^{3}+2^{8}q^{4}+(-281+\cdots)q^{5}+\cdots$
338.10.a.l	$338$	$10$	338.a	1.a	$1$	$6$	$6$	$174.082$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓				26.10.c.b	$1$	$0$	$96$	$81$	$1693$	$473$	$-$	$+$	$-$	$2^{6}\cdot 3^{4}\cdot 13$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+(14-\beta _{1})q^{3}+2^{8}q^{4}+(281+\cdots)q^{5}+\cdots$
338.10.a.m	$338$	$10$	338.a	1.a	$1$	$10$	$10$	$174.082$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓				26.10.e.a	$1$	$0$	$-160$	$-162$	$3088$	$11376$	$+$	$-$	$-$	$2^{10}\cdot 3\cdot 13^{6}$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(-2^{4}-\beta _{1})q^{3}+2^{8}q^{4}+\cdots$
338.10.a.n	$338$	$10$	338.a	1.a	$1$	$10$	$10$	$174.082$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓				26.10.e.a	$1$	$1$	$160$	$-162$	$-3088$	$-11376$	$-$	$-$	$+$	$2^{10}\cdot 3\cdot 13^{6}$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+(-2^{4}-\beta _{1})q^{3}+2^{8}q^{4}+\cdots$
338.10.a.o	$338$	$10$	338.a	1.a	$1$	$12$	$12$	$174.082$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	✓	✓		338.10.a.o	$1$	$1$	$-192$	$-399$	$562$	$-3161$	$+$	$+$	$+$	$2^{9}\cdot 3^{3}\cdot 7\cdot 13^{8}$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(-33+\beta _{1}-\beta _{2})q^{3}+2^{8}q^{4}+\cdots$
338.10.a.p	$338$	$10$	338.a	1.a	$1$	$12$	$12$	$174.082$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	✓	✓		338.10.a.o	$1$	$1$	$192$	$-399$	$-562$	$3161$	$-$	$-$	$+$	$2^{9}\cdot 3^{3}\cdot 7\cdot 13^{8}$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+(-33+\beta _{1}-\beta _{2})q^{3}+2^{8}q^{4}+\cdots$
338.10.a.q	$338$	$10$	338.a	1.a	$1$	$15$	$15$	$174.082$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	None	✓	✓	✓		338.10.a.q	$1$	$0$	$-240$	$324$	$-2164$	$-4227$	$+$	$-$	$-$	$2^{12}\cdot 3^{3}\cdot 13^{13}$	$\mathrm{SU}(2)$	$q-2^{4}q^{2}+(22+\beta _{1})q^{3}+2^{8}q^{4}+(-145+\cdots)q^{5}+\cdots$
338.10.a.r	$338$	$10$	338.a	1.a	$1$	$15$	$15$	$174.082$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	None	✓	✓	✓		338.10.a.q	$1$	$0$	$240$	$324$	$2164$	$4227$	$-$	$+$	$-$	$2^{12}\cdot 3^{3}\cdot 13^{13}$	$\mathrm{SU}(2)$	$q+2^{4}q^{2}+(22+\beta _{1})q^{3}+2^{8}q^{4}+(145+\cdots)q^{5}+\cdots$

Decomposition of $S_{10}^{\mathrm{old}}(\Gamma_0(338))$ into lower level spaces

$ S_{10}^{\mathrm{old}}(\Gamma_0(338)) \simeq $ $S_{10}^{\mathrm{new}}(\Gamma_0(2))$$^{\oplus 3}$$\oplus$$S_{10}^{\mathrm{new}}(\Gamma_0(13))$$^{\oplus 4}$$\oplus$$S_{10}^{\mathrm{new}}(\Gamma_0(26))$$^{\oplus 2}$$\oplus$$S_{10}^{\mathrm{new}}(\Gamma_0(169))$$^{\oplus 2}$

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Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(455\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

\(2\)	\(13\)	Fricke	Total			Cusp			Eisenstein
\(2\)	\(13\)	Fricke	All	New	Old	All	New	Old	All	New	Old
\(+\)	\(+\)	\(+\)	\(103\)	\(28\)	\(75\)	\(96\)	\(28\)	\(68\)	\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(-\)	\(108\)	\(30\)	\(78\)	\(101\)	\(30\)	\(71\)	\(7\)	\(0\)	\(7\)
\(-\)	\(+\)	\(-\)	\(107\)	\(31\)	\(76\)	\(100\)	\(31\)	\(69\)	\(7\)	\(0\)	\(7\)
\(-\)	\(-\)	\(+\)	\(105\)	\(27\)	\(78\)	\(98\)	\(27\)	\(71\)	\(7\)	\(0\)	\(7\)
Plus space		\(+\)	\(208\)	\(55\)	\(153\)	\(194\)	\(55\)	\(139\)	\(14\)	\(0\)	\(14\)
Minus space		\(-\)	\(215\)	\(61\)	\(154\)	\(201\)	\(61\)	\(140\)	\(14\)	\(0\)	\(14\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(338))\) into newform subspaces

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(338))\) 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	338.10.a

Level	$338$
Weight	$10$
Character orbit	338.a
Rep. character	$\chi_{338}(1,\cdot)$
Character field	$\Q$
Dimension	$116$
Newform subspaces	$18$
Sturm bound	$455$
Trace bound	$5$