Space of modular forms of level 338, weight 8, and trivial character

• Label: 338.8.a
• Level: $338$
• Weight: $8$
• Character orbit: 338.a
• Rep. character: $\chi_{338}(1,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $18$
• Sturm bound: $364$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(364\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(338))\).

	Total	New	Old
Modular forms	333	90	243
Cusp forms	305	90	215
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
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(86\)	\(23\)	\(63\)		\(79\)	\(23\)	\(56\)		\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(-\)		\(81\)	\(22\)	\(59\)		\(74\)	\(22\)	\(52\)		\(7\)	\(0\)	\(7\)
\(-\)	\(+\)	\(-\)		\(82\)	\(20\)	\(62\)		\(75\)	\(20\)	\(55\)		\(7\)	\(0\)	\(7\)
\(-\)	\(-\)	\(+\)		\(84\)	\(25\)	\(59\)		\(77\)	\(25\)	\(52\)		\(7\)	\(0\)	\(7\)
Plus space		\(+\)		\(170\)	\(48\)	\(122\)		\(156\)	\(48\)	\(108\)		\(14\)	\(0\)	\(14\)
Minus space		\(-\)		\(163\)	\(42\)	\(121\)		\(149\)	\(42\)	\(107\)		\(14\)	\(0\)	\(14\)

Trace form

90 * q + 66 * q^3 + 5760 * q^4 - 320 * q^5 - 96 * q^6 + 1120 * q^7 + 63676 * q^9 - 2096 * q^10 - 11116 * q^11 + 4224 * q^12 - 9760 * q^14 - 30064 * q^15 + 368640 * q^16 - 10832 * q^17 - 9664 * q^18 + 50992 * q^19 - 20480 * q^20 - 79780 * q^21 + 20592 * q^22 + 209280 * q^23 - 6144 * q^24 + 1431200 * q^25 + 343416 * q^27 + 71680 * q^28 - 19838 * q^29 + 189088 * q^30 + 45776 * q^31 + 474044 * q^33 - 330752 * q^34 - 195516 * q^35 + 4075264 * q^36 + 344128 * q^37 + 379152 * q^38 - 134144 * q^40 - 377844 * q^41 + 513952 * q^42 - 220414 * q^43 - 711424 * q^44 - 1747148 * q^45 - 1030368 * q^46 - 1251232 * q^47 + 270336 * q^48 + 12972494 * q^49 + 2081216 * q^50 - 4329556 * q^51 + 3890646 * q^53 + 1379232 * q^54 - 1042600 * q^55 - 624640 * q^56 - 3943408 * q^57 - 1853088 * q^58 - 55592 * q^59 - 1924096 * q^60 + 6374662 * q^61 + 3755648 * q^62 + 12519096 * q^63 + 23592960 * q^64 + 9948224 * q^66 - 2560804 * q^67 - 693248 * q^68 - 367412 * q^69 + 1030048 * q^70 - 3572536 * q^71 - 618496 * q^72 - 16453276 * q^73 + 1062480 * q^74 + 26922138 * q^75 + 3263488 * q^76 - 3534804 * q^77 + 16740636 * q^79 - 1310720 * q^80 + 35155210 * q^81 - 4394208 * q^82 - 17016516 * q^83 - 5105920 * q^84 - 16280588 * q^85 - 5213792 * q^86 + 9286940 * q^87 + 1317888 * q^88 + 36560556 * q^89 - 27409584 * q^90 + 13393920 * q^92 - 2367312 * q^93 - 8512384 * q^94 + 3679668 * q^95 - 393216 * q^96 - 34547408 * q^97 - 4300800 * q^98 - 61078784 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(338))\) 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
338.8.a.a	$338$	$8$	338.a	1.a	$1$	$1$	$1$	$105.586$	\(\Q\)	None	✓				26.8.a.b	$1$	$0$	\(-8\)	\(-87\)	\(-321\)	\(181\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-87q^{3}+2^{6}q^{4}-321q^{5}+\cdots\)
338.8.a.b	$338$	$8$	338.a	1.a	$1$	$1$	$1$	$105.586$	\(\Q\)	None	✓				26.8.a.c	$1$	$0$	\(-8\)	\(-27\)	\(245\)	\(587\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+245q^{5}+\cdots\)
338.8.a.c	$338$	$8$	338.a	1.a	$1$	$1$	$1$	$105.586$	\(\Q\)	None	✓				26.8.a.a	$1$	$1$	\(8\)	\(-39\)	\(-385\)	\(293\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-39q^{3}+2^{6}q^{4}-385q^{5}+\cdots\)
338.8.a.d	$338$	$8$	338.a	1.a	$1$	$1$	$1$	$105.586$	\(\Q\)	None	✓				2.8.a.a	$1$	$1$	\(8\)	\(12\)	\(210\)	\(-1016\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+12q^{3}+2^{6}q^{4}+210q^{5}+\cdots\)
338.8.a.e	$338$	$8$	338.a	1.a	$1$	$2$	$2$	$105.586$	\(\Q(\sqrt{2305}) \)	None	✓				26.8.a.e	$1$	$0$	\(-16\)	\(87\)	\(-215\)	\(-705\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(44-\beta )q^{3}+2^{6}q^{4}+(-110+\cdots)q^{5}+\cdots\)
338.8.a.f	$338$	$8$	338.a	1.a	$1$	$2$	$2$	$105.586$	\(\Q(\sqrt{105}) \)	None	✓				26.8.a.d	$1$	$1$	\(16\)	\(-12\)	\(146\)	\(1780\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-6-7\beta )q^{3}+2^{6}q^{4}+(73+\cdots)q^{5}+\cdots\)
338.8.a.g	$338$	$8$	338.a	1.a	$1$	$3$	$3$	$105.586$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				26.8.c.a	$1$	$0$	\(-24\)	\(0\)	\(333\)	\(1160\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-8q^{2}-\beta _{1}q^{3}+2^{6}q^{4}+(110+2\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.a.h	$338$	$8$	338.a	1.a	$1$	$3$	$3$	$105.586$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				26.8.c.a	$1$	$1$	\(24\)	\(0\)	\(-333\)	\(-1160\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}-\beta _{1}q^{3}+2^{6}q^{4}+(-110-2\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.a.i	$338$	$8$	338.a	1.a	$1$	$4$	$4$	$105.586$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				26.8.c.b	$1$	$0$	\(-32\)	\(0\)	\(278\)	\(548\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+\beta _{1}q^{3}+2^{6}q^{4}+(70-\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.a.j	$338$	$8$	338.a	1.a	$1$	$4$	$4$	$105.586$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				26.8.c.b	$1$	$1$	\(32\)	\(0\)	\(-278\)	\(-548\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+8q^{2}+\beta _{1}q^{3}+2^{6}q^{4}+(-70+\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.a.k	$338$	$8$	338.a	1.a	$1$	$5$	$5$	$105.586$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				26.8.b.a	$1$	$1$	\(-40\)	\(27\)	\(-71\)	\(-237\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-8q^{2}+(5+\beta _{1})q^{3}+2^{6}q^{4}+(-13+\cdots)q^{5}+\cdots\)
338.8.a.l	$338$	$8$	338.a	1.a	$1$	$5$	$5$	$105.586$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				26.8.b.a	$1$	$0$	\(40\)	\(27\)	\(71\)	\(237\)		$-$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}+(5+\beta _{1})q^{3}+2^{6}q^{4}+(13+2\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.a.m	$338$	$8$	338.a	1.a	$1$	$8$	$8$	$105.586$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				26.8.e.a	$1$	$1$	\(-64\)	\(0\)	\(-304\)	\(-2160\)		$+$	$-$	$-$	$2^{6}\cdot 13^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+\beta _{1}q^{3}+2^{6}q^{4}+(-38-\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.a.n	$338$	$8$	338.a	1.a	$1$	$8$	$8$	$105.586$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				26.8.e.a	$1$	$0$	\(64\)	\(0\)	\(304\)	\(2160\)		$-$	$-$	$+$	$2^{6}\cdot 13^{3}$	$\mathrm{SU}(2)$	\(q+8q^{2}+\beta _{1}q^{3}+2^{6}q^{4}+(38+\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.a.o	$338$	$8$	338.a	1.a	$1$	$9$	$9$	$105.586$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		338.8.a.o	$1$	$1$	\(-72\)	\(-69\)	\(318\)	\(1432\)		$+$	$-$	$-$	$2^{3}\cdot 7\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-8-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+\cdots\)
338.8.a.p	$338$	$8$	338.a	1.a	$1$	$9$	$9$	$105.586$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		338.8.a.o	$1$	$1$	\(72\)	\(-69\)	\(-318\)	\(-1432\)		$-$	$+$	$-$	$2^{3}\cdot 7\cdot 13^{4}$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-8-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+\cdots\)
338.8.a.q	$338$	$8$	338.a	1.a	$1$	$12$	$12$	$105.586$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		338.8.a.q	$1$	$0$	\(-96\)	\(108\)	\(-292\)	\(364\)		$+$	$+$	$+$	$2^{6}\cdot 13^{8}$	$\mathrm{SU}(2)$	\(q-8q^{2}+(9-\beta _{1})q^{3}+2^{6}q^{4}+(-24+\cdots)q^{5}+\cdots\)
338.8.a.r	$338$	$8$	338.a	1.a	$1$	$12$	$12$	$105.586$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		338.8.a.q	$1$	$0$	\(96\)	\(108\)	\(292\)	\(-364\)		$-$	$-$	$+$	$2^{6}\cdot 13^{8}$	$\mathrm{SU}(2)$	\(q+8q^{2}+(9-\beta _{1})q^{3}+2^{6}q^{4}+(24+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(338))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(338)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)