Space of modular forms of level 42, weight 12, and Character 42.e

• Label: 42.12.e
• Level: $42$
• Weight: $12$
• Character orbit: 42.e
• Rep. character: $\chi_{42}(25,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $28$
• Newform subspaces: $4$
• Sturm bound: $96$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	42.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(96\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(42, [\chi])\).

	Total	New	Old
Modular forms	184	28	156
Cusp forms	168	28	140
Eisenstein series	16	0	16

Trace form

28 * q - 14336 * q^4 - 10124 * q^5 + 31104 * q^6 - 72418 * q^7 - 826686 * q^9 - 318272 * q^10 - 1638040 * q^11 + 1978584 * q^13 - 335360 * q^14 + 4555764 * q^15 - 14680064 * q^16 + 1559804 * q^17 - 5097220 * q^19 + 20733952 * q^20 + 29958012 * q^21 - 51113344 * q^22 - 20646748 * q^23 - 15925248 * q^24 + 65193944 * q^25 - 67420544 * q^26 + 126420992 * q^28 + 276341656 * q^29 - 115675776 * q^30 - 371468682 * q^31 - 67370778 * q^33 + 844796672 * q^34 + 150672680 * q^35 + 1693052928 * q^36 - 1351393444 * q^37 - 891388032 * q^38 + 251218260 * q^39 - 325910528 * q^40 + 5451177312 * q^41 + 24960960 * q^42 - 3530671024 * q^43 - 1677352960 * q^44 - 597812076 * q^45 + 3449698688 * q^46 - 3475255764 * q^47 + 10706104834 * q^49 - 5634512384 * q^50 - 2024725572 * q^51 - 1013035008 * q^52 + 9444322068 * q^53 - 918330048 * q^54 + 13463802380 * q^55 + 7451312128 * q^56 + 11007159336 * q^57 - 1514752448 * q^58 - 26063526872 * q^59 - 2332551168 * q^60 - 20964535920 * q^61 + 24035449600 * q^62 - 3013860960 * q^63 + 30064771072 * q^64 - 40538303292 * q^65 - 10018660608 * q^66 + 21217199788 * q^67 + 1597239296 * q^68 + 10716278616 * q^69 + 1230037184 * q^70 - 42719023432 * q^71 - 43156310732 * q^73 + 3430345344 * q^74 - 14813488008 * q^75 + 10439106560 * q^76 + 9063313036 * q^77 + 27329156352 * q^78 + 57995267574 * q^79 - 10615783424 * q^80 - 48814981614 * q^81 - 927854592 * q^82 - 238907176288 * q^83 - 22924394496 * q^84 + 213787952168 * q^85 + 30817020288 * q^86 - 20620471158 * q^87 + 26170032128 * q^88 + 187906684896 * q^89 + 37587286656 * q^90 - 358553662500 * q^91 + 42284539904 * q^92 + 124846567812 * q^93 - 3636794112 * q^94 - 401515267132 * q^95 - 16307453952 * q^96 - 242632624124 * q^97 + 36348195072 * q^98 + 193449247920 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(42, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
42.12.e.a	$42$	$12$	42.e	7.c	$3$	$6$	$3$	$32.270$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓			42.12.e.a	$2$	$0$	\(-96\)	\(729\)	\(1045\)	\(45731\)	$2^{8}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2^{5}-2^{5}\beta _{2})q^{2}-3^{5}\beta _{2}q^{3}+2^{10}\beta _{2}q^{4}+\cdots\)
42.12.e.b	$42$	$12$	42.e	7.c	$3$	$6$	$3$	$32.270$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓			42.12.e.b	$2$	$0$	\(96\)	\(-729\)	\(1331\)	\(-83545\)	$2^{4}\cdot 3\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2^{5}+2^{5}\beta _{1})q^{2}+3^{5}\beta _{1}q^{3}+2^{10}\beta _{1}q^{4}+\cdots\)
42.12.e.c	$42$	$12$	42.e	7.c	$3$	$8$	$4$	$32.270$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			42.12.e.c	$2$	$0$	\(-128\)	\(-972\)	\(-11080\)	\(31758\)	$2^{6}\cdot 3^{3}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-2^{5}\beta _{1}q^{2}+(-3^{5}+3^{5}\beta _{1})q^{3}+(-2^{10}+\cdots)q^{4}+\cdots\)
42.12.e.d	$42$	$12$	42.e	7.c	$3$	$8$	$4$	$32.270$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			42.12.e.d	$2$	$0$	\(128\)	\(972\)	\(-1420\)	\(-66362\)	$2^{8}\cdot 3^{3}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+2^{5}\beta _{1}q^{2}+(3^{5}-3^{5}\beta _{1})q^{3}+(-2^{10}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(42, [\chi]) \simeq \) \(S_{12}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)