Space of modular forms of level 38, weight 6, and trivial character

• Label: 38.6.a
• Level: $38$
• Weight: $6$
• Character orbit: 38.a
• Rep. character: $\chi_{38}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $4$
• Sturm bound: $30$
• Trace bound: $2$

Defining parameters

Level:	$N$	$=$	$38 = 2 \cdot 19$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	38.a (trivial)
Character field:			$\Q$
Newform subspaces:			$4$
Sturm bound:			$30$
Trace bound:			$2$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{6}(\Gamma_0(38))$.

	Total	New	Old
Modular forms	27	7	20
Cusp forms	23	7	16
Eisenstein series	4	0	4

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

$2$	$19$	Fricke	Dim
$+$	$+$	$+$	$1$
$+$	$-$	$-$	$2$
$-$	$+$	$-$	$3$
$-$	$-$	$+$	$1$
Plus space		$+$	$2$
Minus space		$-$	$5$

Trace form

7 * q + 4 * q^2 - 4 * q^3 + 112 * q^4 + 22 * q^5 + 8 * q^6 + 194 * q^7 + 64 * q^8 + 221 * q^9 + 200 * q^10 + 320 * q^11 - 64 * q^12 + 1338 * q^13 + 80 * q^14 + 1868 * q^15 + 1792 * q^16 - 1316 * q^17 + 628 * q^18 - 361 * q^19 + 352 * q^20 - 2940 * q^21 - 1424 * q^22 - 3430 * q^23 + 128 * q^24 - 5899 * q^25 - 2144 * q^26 + 3464 * q^27 + 3104 * q^28 - 5566 * q^29 - 7792 * q^30 - 11164 * q^31 + 1024 * q^32 - 6800 * q^33 + 3080 * q^34 - 7788 * q^35 + 3536 * q^36 + 10038 * q^37 - 4332 * q^38 + 5046 * q^39 + 3200 * q^40 + 25722 * q^41 + 8632 * q^42 - 18208 * q^43 + 5120 * q^44 + 10810 * q^45 + 688 * q^46 - 2056 * q^47 - 1024 * q^48 - 2563 * q^49 - 16260 * q^50 + 47816 * q^51 + 21408 * q^52 + 53370 * q^53 - 22072 * q^54 + 60424 * q^55 + 1280 * q^56 - 6498 * q^57 - 25376 * q^58 - 31028 * q^59 + 29888 * q^60 + 102938 * q^61 - 46192 * q^62 + 22580 * q^63 + 28672 * q^64 - 108904 * q^65 - 56400 * q^66 - 29516 * q^67 - 21056 * q^68 - 121196 * q^69 + 65232 * q^70 + 5892 * q^71 + 10048 * q^72 - 115600 * q^73 + 20104 * q^74 - 136620 * q^75 - 5776 * q^76 - 7592 * q^77 - 2800 * q^78 + 202780 * q^79 + 5632 * q^80 - 350161 * q^81 - 39768 * q^82 + 79616 * q^83 - 47040 * q^84 - 101964 * q^85 - 83968 * q^86 - 214610 * q^87 - 22784 * q^88 + 202750 * q^89 + 85720 * q^90 + 481736 * q^91 - 54880 * q^92 + 289700 * q^93 + 61312 * q^94 - 72922 * q^95 + 2048 * q^96 + 16390 * q^97 + 243076 * q^98 + 324236 * q^99

Decomposition of $S_{6}^{\mathrm{new}}(\Gamma_0(38))$ 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	19
38.6.a.a	$38$	$6$	38.a	1.a	$1$	$1$	$1$	$6.095$	$\Q$	None	✓	✓	✓	✓	38.6.a.a	$1$	$1$	$-4$	$-6$	$31$	$-27$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-4q^{2}-6q^{3}+2^{4}q^{4}+31q^{5}+24q^{6}+\cdots$
38.6.a.b	$38$	$6$	38.a	1.a	$1$	$1$	$1$	$6.095$	$\Q$	None	✓	✓	✓	✓	38.6.a.b	$1$	$1$	$4$	$-14$	$-45$	$-121$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+4q^{2}-14q^{3}+2^{4}q^{4}-45q^{5}-56q^{6}+\cdots$
38.6.a.c	$38$	$6$	38.a	1.a	$1$	$2$	$2$	$6.095$	$\Q(\sqrt{1441})$	None	✓	✓	✓	✓	38.6.a.c	$1$	$0$	$-8$	$3$	$-45$	$114$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-4q^{2}+(2-\beta )q^{3}+2^{4}q^{4}+(-21+\cdots)q^{5}+\cdots$
38.6.a.d	$38$	$6$	38.a	1.a	$1$	$3$	$3$	$6.095$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓	✓	38.6.a.d	$1$	$0$	$12$	$13$	$81$	$228$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+4q^{2}+(4+\beta _{1})q^{3}+2^{4}q^{4}+(3^{3}-\beta _{1}+\cdots)q^{5}+\cdots$

Decomposition of $S_{6}^{\mathrm{old}}(\Gamma_0(38))$ into lower level spaces

$S_{6}^{\mathrm{old}}(\Gamma_0(38)) \simeq$ $S_{6}^{\mathrm{new}}(\Gamma_0(19))$$^{\oplus 2}$