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

• 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: $5$

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	Dim
\(+\)	\(+\)	\(+\)	\(28\)
\(+\)	\(-\)	\(-\)	\(30\)
\(-\)	\(+\)	\(-\)	\(31\)
\(-\)	\(-\)	\(+\)	\(27\)
Plus space		\(+\)	\(55\)
Minus space		\(-\)	\(61\)

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 - 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	13
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

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