Space of modular forms of level 338, weight 4, and trivial character

• Label: 338.4.a
• Level: $338$
• Weight: $4$
• Character orbit: 338.a
• Rep. character: $\chi_{338}(1,\cdot)$
• Character field: $\Q$
• Dimension: $39$
• Newform subspaces: $15$
• Sturm bound: $182$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$338 = 2 \cdot 13^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	338.a (trivial)
Character field:			$\Q$
Newform subspaces:			$15$
Sturm bound:			$182$
Trace bound:			$5$
Distinguishing $T_p$:			$3$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(338))$.

	Total	New	Old
Modular forms	151	39	112
Cusp forms	123	39	84
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
$+$	$+$	$+$	$11$
$+$	$-$	$-$	$9$
$-$	$+$	$-$	$7$
$-$	$-$	$+$	$12$
Plus space		$+$	$23$
Minus space		$-$	$16$

Trace form

39 * q - 2 * q^2 - 6 * q^3 + 156 * q^4 - 10 * q^5 - 4 * q^7 - 8 * q^8 + 433 * q^9 + 24 * q^10 + 84 * q^11 - 24 * q^12 + 88 * q^14 + 56 * q^15 + 624 * q^16 + 6 * q^17 + 38 * q^18 - 168 * q^19 - 40 * q^20 - 172 * q^21 - 12 * q^22 + 44 * q^23 + 755 * q^25 + 372 * q^27 - 16 * q^28 - 116 * q^29 + 328 * q^30 - 212 * q^31 - 32 * q^32 + 308 * q^33 - 196 * q^34 + 684 * q^35 + 1732 * q^36 + 198 * q^37 - 100 * q^38 + 96 * q^40 + 358 * q^41 - 248 * q^42 - 642 * q^43 + 336 * q^44 + 442 * q^45 - 296 * q^46 + 372 * q^47 - 96 * q^48 + 1295 * q^49 - 734 * q^50 - 88 * q^51 - 968 * q^53 - 72 * q^54 - 240 * q^55 + 352 * q^56 - 112 * q^57 - 884 * q^58 - 48 * q^59 + 224 * q^60 + 316 * q^61 - 16 * q^62 - 348 * q^63 + 2496 * q^64 + 656 * q^66 + 1036 * q^67 + 24 * q^68 - 260 * q^69 + 2328 * q^70 + 1228 * q^71 + 152 * q^72 + 798 * q^73 - 1064 * q^74 - 162 * q^75 - 672 * q^76 + 2436 * q^77 - 1952 * q^79 - 160 * q^80 + 4015 * q^81 + 2228 * q^82 - 948 * q^83 - 688 * q^84 + 2072 * q^85 + 512 * q^86 + 1604 * q^87 - 48 * q^88 - 1562 * q^89 - 984 * q^90 + 176 * q^92 - 1056 * q^93 + 240 * q^94 - 3112 * q^95 + 2794 * q^97 - 1842 * q^98 - 1160 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(338))$ 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
338.4.a.a	$338$	$4$	338.a	1.a	$1$	$1$	$1$	$19.943$	$\Q$	None	✓				26.4.c.a	$1$	$0$	$-2$	$-3$	$2$	$-5$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-3q^{3}+4q^{4}+2q^{5}+6q^{6}+\cdots$
338.4.a.b	$338$	$4$	338.a	1.a	$1$	$1$	$1$	$19.943$	$\Q$	None	✓				26.4.a.b	$1$	$0$	$-2$	$-1$	$-17$	$35$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-q^{3}+4q^{4}-17q^{5}+2q^{6}+\cdots$
338.4.a.c	$338$	$4$	338.a	1.a	$1$	$1$	$1$	$19.943$	$\Q$	None	✓				26.4.a.c	$1$	$0$	$-2$	$4$	$18$	$-20$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+4q^{3}+4q^{4}+18q^{5}-8q^{6}+\cdots$
338.4.a.d	$338$	$4$	338.a	1.a	$1$	$1$	$1$	$19.943$	$\Q$	None	✓				26.4.c.a	$1$	$1$	$2$	$-3$	$-2$	$5$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-3q^{3}+4q^{4}-2q^{5}-6q^{6}+\cdots$
338.4.a.e	$338$	$4$	338.a	1.a	$1$	$1$	$1$	$19.943$	$\Q$	None	✓				26.4.a.a	$1$	$1$	$2$	$3$	$-11$	$-19$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+3q^{3}+4q^{4}-11q^{5}+6q^{6}+\cdots$
338.4.a.f	$338$	$4$	338.a	1.a	$1$	$2$	$2$	$19.943$	$\Q(\sqrt{217})$	None	✓				26.4.b.a	$1$	$1$	$-4$	$-3$	$-19$	$-7$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(-1-\beta )q^{3}+4q^{4}+(-9+\cdots)q^{5}+\cdots$
338.4.a.g	$338$	$4$	338.a	1.a	$1$	$2$	$2$	$19.943$	$\Q(\sqrt{217})$	None	✓				26.4.c.b	$1$	$0$	$-4$	$-3$	$7$	$45$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(-1-\beta )q^{3}+4q^{4}+(4-\beta )q^{5}+\cdots$
338.4.a.h	$338$	$4$	338.a	1.a	$1$	$2$	$2$	$19.943$	$\Q(\sqrt{217})$	None	✓				26.4.c.b	$1$	$1$	$4$	$-3$	$-7$	$-45$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-1-\beta )q^{3}+4q^{4}+(-4+\cdots)q^{5}+\cdots$
338.4.a.i	$338$	$4$	338.a	1.a	$1$	$2$	$2$	$19.943$	$\Q(\sqrt{217})$	None	✓				26.4.b.a	$1$	$0$	$4$	$-3$	$19$	$7$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-1-\beta )q^{3}+4q^{4}+(9+\beta )q^{5}+\cdots$
338.4.a.j	$338$	$4$	338.a	1.a	$1$	$3$	$3$	$19.943$	$\Q(\zeta_{14})^+$	None	✓	✓			338.4.a.j	$1$	$1$	$-6$	$-12$	$12$	$-27$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(-5-3\beta _{2})q^{3}+4q^{4}+(1+\cdots)q^{5}+\cdots$
338.4.a.k	$338$	$4$	338.a	1.a	$1$	$3$	$3$	$19.943$	$\Q(\zeta_{14})^+$	None	✓	✓	✓		338.4.a.j	$1$	$1$	$6$	$-12$	$-12$	$27$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-5-3\beta _{2})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots$
338.4.a.l	$338$	$4$	338.a	1.a	$1$	$4$	$4$	$19.943$	4.4.1859472.2	None	✓		✓		26.4.e.a	$1$	$1$	$-8$	$6$	$4$	$-20$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots$
338.4.a.m	$338$	$4$	338.a	1.a	$1$	$4$	$4$	$19.943$	4.4.1859472.2	None	✓				26.4.e.a	$1$	$0$	$8$	$6$	$-4$	$20$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots$
338.4.a.n	$338$	$4$	338.a	1.a	$1$	$6$	$6$	$19.943$	6.6.6681389953.1	None	✓	✓	✓		338.4.a.n	$1$	$0$	$-12$	$9$	$-18$	$-25$		$+$	$+$	$+$	$13^{2}$	$\mathrm{SU}(2)$	$q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(-3+\beta _{1}+\cdots)q^{5}+\cdots$
338.4.a.o	$338$	$4$	338.a	1.a	$1$	$6$	$6$	$19.943$	6.6.6681389953.1	None	✓	✓	✓		338.4.a.n	$1$	$0$	$12$	$9$	$18$	$25$		$-$	$-$	$+$	$13^{2}$	$\mathrm{SU}(2)$	$q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(3-\beta _{1}+\cdots)q^{5}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(338))$ into lower level spaces

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