Space of modular forms of level 49, weight 16, and Character 49.c

• Label: 49.16.c
• Level: $49$
• Weight: $16$
• Character orbit: 49.c
• Rep. character: $\chi_{49}(18,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $96$
• Newform subspaces: $10$
• Sturm bound: $74$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	49.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(74\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(49, [\chi])\).

	Total	New	Old
Modular forms	148	104	44
Cusp forms	132	96	36
Eisenstein series	16	8	8

Trace form

96 * q + 179 * q^2 + 4375 * q^3 - 704713 * q^4 + 234087 * q^5 + 40964 * q^6 - 7837338 * q^8 - 225493405 * q^9 + 22514534 * q^10 - 54200275 * q^11 + 288834308 * q^12 - 513884980 * q^13 + 3236700562 * q^15 - 11500111173 * q^16 - 972374613 * q^17 + 12968999857 * q^18 + 3111766217 * q^19 - 53399044104 * q^20 + 14884988988 * q^22 - 56887099097 * q^23 + 52758967800 * q^24 - 151215767133 * q^25 + 99659957460 * q^26 - 284144943530 * q^27 + 241712289640 * q^29 + 195654455670 * q^30 + 74937071755 * q^31 - 463665224579 * q^32 - 194333417915 * q^33 + 2143875710340 * q^34 + 2696368740650 * q^36 - 21466637629 * q^37 + 817486402194 * q^38 - 196485738190 * q^39 - 1785313599048 * q^40 + 644745625524 * q^41 + 4778113549240 * q^43 + 5550656063284 * q^44 + 137659135474 * q^45 - 2502482903474 * q^46 + 3161157736821 * q^47 - 13371673883680 * q^48 + 35447795176034 * q^50 - 5074456102217 * q^51 + 18449613599928 * q^52 + 20927678004895 * q^53 + 42184117534538 * q^54 - 91307835023930 * q^55 - 46468813561194 * q^57 - 52168281890402 * q^58 + 61004320427463 * q^59 - 65262488389588 * q^60 - 11174039585989 * q^61 - 162863764600908 * q^62 + 551105866802042 * q^64 + 243887079672558 * q^65 + 251023951343858 * q^66 - 240326224586955 * q^67 - 47805418774092 * q^68 + 43835892593682 * q^69 - 228008208875248 * q^71 + 1001448153005133 * q^72 + 372680541702511 * q^73 - 423929720137764 * q^74 + 53908157287628 * q^75 - 1273783253096760 * q^76 + 1710084510385944 * q^78 + 569139340456215 * q^79 + 1781313856759248 * q^80 - 1535265150652708 * q^81 + 487753126538860 * q^82 - 2181968918091144 * q^83 + 2317905681916514 * q^85 + 1118691485584968 * q^86 + 2581959840409390 * q^87 - 866473331154648 * q^88 + 983172302764239 * q^89 - 8798853346789576 * q^90 + 4502749887413912 * q^92 + 2852695948989595 * q^93 + 5442401319513690 * q^94 - 1821600247662375 * q^95 + 1944756092404384 * q^96 - 5279284314998540 * q^97 + 6130137690209732 * q^99

Decomposition of \(S_{16}^{\mathrm{new}}(49, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
49.16.c.a	$49$	$16$	49.c	7.c	$3$	$2$	$1$	$69.920$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-7}) \)		✓			49.16.a.b	$4$	$0$	\(-275\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-275\zeta_{6}q^{2}+(-42857+42857\zeta_{6})q^{4}+\cdots\)
49.16.c.b	$49$	$16$	49.c	7.c	$3$	$2$	$1$	$69.920$	\(\Q(\sqrt{-3}) \)	None					1.16.a.a	$2$	$0$	\(-216\)	\(-3348\)	\(52110\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-6^{3}\zeta_{6}q^{2}+(-3348+3348\zeta_{6})q^{3}+\cdots\)
49.16.c.c	$49$	$16$	49.c	7.c	$3$	$2$	$1$	$69.920$	\(\Q(\sqrt{-3}) \)	None					1.16.a.a	$2$	$0$	\(-216\)	\(3348\)	\(-52110\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-6^{3}\zeta_{6}q^{2}+(3348-3348\zeta_{6})q^{3}+\cdots\)
49.16.c.d	$49$	$16$	49.c	7.c	$3$	$6$	$3$	$69.920$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					7.16.a.a	$2$	$0$	\(438\)	\(-1860\)	\(-111414\)	\(0\)	$2^{4}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(146-146\beta _{1}-\beta _{2}-\beta _{4})q^{2}+(-620\beta _{1}+\cdots)q^{3}+\cdots\)
49.16.c.e	$49$	$16$	49.c	7.c	$3$	$6$	$3$	$69.920$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					7.16.a.a	$2$	$0$	\(438\)	\(1860\)	\(111414\)	\(0\)	$2^{4}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(146-146\beta _{1}-\beta _{2}-\beta _{4})q^{2}+(620\beta _{1}+\cdots)q^{3}+\cdots\)
49.16.c.f	$49$	$16$	49.c	7.c	$3$	$8$	$4$	$69.920$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					7.16.a.b	$2$	$0$	\(-93\)	\(-8554\)	\(-7770\)	\(0\)	$2^{14}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-23\beta _{1}+\beta _{3}+\beta _{4})q^{2}+(-2138+\cdots)q^{3}+\cdots\)
49.16.c.g	$49$	$16$	49.c	7.c	$3$	$8$	$4$	$69.920$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					7.16.a.b	$2$	$0$	\(-93\)	\(8554\)	\(7770\)	\(0\)	$2^{14}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-23\beta _{1}+\beta _{3}+\beta _{4})q^{2}+(2138-2138\beta _{1}+\cdots)q^{3}+\cdots\)
49.16.c.h	$49$	$16$	49.c	7.c	$3$	$12$	$6$	$69.920$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			49.16.a.e	$4$	$0$	\(364\)	\(0\)	\(0\)	\(0\)	$2^{32}\cdot 3^{14}\cdot 5^{2}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-61\beta _{1}+\beta _{2}+\beta _{3})q^{2}+\beta _{5}q^{3}+\cdots\)
49.16.c.i	$49$	$16$	49.c	7.c	$3$	$18$	$9$	$69.920$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None					7.16.c.a	$2$	$0$	\(-90\)	\(4375\)	\(234087\)	\(0\)	$2^{40}\cdot 3^{10}\cdot 5^{6}\cdot 7^{16}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-10+10\beta _{1}+\beta _{2}-\beta _{3})q^{2}+(486\beta _{1}+\cdots)q^{3}+\cdots\)
49.16.c.j	$49$	$16$	49.c	7.c	$3$	$32$	$16$	$69.920$		None		✓	✓		49.16.a.h	$4$	$0$	\(-78\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{16}^{\mathrm{old}}(49, [\chi])\) into lower level spaces

\( S_{16}^{\mathrm{old}}(49, [\chi]) \simeq \) \(S_{16}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)