Space of modular forms of level 49, weight 24, and trivial character

• Label: 49.24.a
• Level: $49$
• Weight: $24$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $8$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{24}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	112	81	31
Cusp forms	104	76	28
Eisenstein series	8	5	3

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

\(7\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(57\)	\(40\)	\(17\)		\(53\)	\(38\)	\(15\)		\(4\)	\(2\)	\(2\)
\(-\)		\(55\)	\(41\)	\(14\)		\(51\)	\(38\)	\(13\)		\(4\)	\(3\)	\(1\)

Trace form

76 * q + 3016 * q^2 - 14812 * q^3 + 308876940 * q^4 - 70357686 * q^5 - 977238878 * q^6 + 60328022868 * q^8 + 2275482394984 * q^9 - 516370708644 * q^10 + 609308381344 * q^11 + 5250626436874 * q^12 - 16892448134702 * q^13 + 64242362913752 * q^15 + 1320438430895412 * q^16 + 196881509864214 * q^17 - 648527892746308 * q^18 + 369104034325012 * q^19 - 2045914432541448 * q^20 + 4445088272861972 * q^22 + 14023914681277472 * q^23 - 21779464333029858 * q^24 + 142825101280206676 * q^25 + 70791262105466784 * q^26 - 39878062991555320 * q^27 - 12995413618115308 * q^29 - 462585822045925764 * q^30 - 53113370563869056 * q^31 + 49133230132756940 * q^32 - 561549957516375184 * q^33 + 690416966607895086 * q^34 + 10479309478139316124 * q^36 - 1209089277385234568 * q^37 - 2437561909084819650 * q^38 + 4832887103513270200 * q^39 - 2094683559943245600 * q^40 + 3971990631558638766 * q^41 + 7841084268148333144 * q^43 - 3988896472446726616 * q^44 - 53678722161806953582 * q^45 + 28067526279924307348 * q^46 + 14595090121907093904 * q^47 + 18708077177935510498 * q^48 + 133386264522204388756 * q^50 - 133372014577883406112 * q^51 - 320302556374686356996 * q^52 + 274212098877948976988 * q^53 - 215777178530681773796 * q^54 + 506652358782199671048 * q^55 + 193017806814638184276 * q^57 - 1216198093520751943184 * q^58 - 479354853592301755236 * q^59 + 209021829615157593328 * q^60 - 655363780789877326622 * q^61 - 1361091067871880065532 * q^62 + 5314170048837834772108 * q^64 + 4085626193653434539676 * q^65 + 2306828466354279594160 * q^66 - 2263003025470744010568 * q^67 + 2968715291861146719402 * q^68 - 368673610851365213184 * q^69 - 3640397279706871365656 * q^71 - 3351902125044209870604 * q^72 + 4486366362324123661678 * q^73 - 1479329062767658617420 * q^74 + 3638206101514274554012 * q^75 + 6371309178929012062822 * q^76 - 48150425843863310407440 * q^78 + 22601828070658059360248 * q^79 + 42697473452481464433264 * q^80 + 50413630462804082740444 * q^81 - 6302208047936872561602 * q^82 - 16491004102812431453772 * q^83 - 12063102280455705093984 * q^85 + 71649574184861698238568 * q^86 - 70212627350190299918440 * q^87 - 100904305472998229731368 * q^88 - 52980497818563397275522 * q^89 + 454826140908197644012 * q^90 + 222912153524527277491384 * q^92 - 55048959303292161821212 * q^93 - 17242762920550975759476 * q^94 + 177179577944182861396488 * q^95 + 54463542621991161002414 * q^96 + 124429755946559974476454 * q^97 + 194726820285924545389048 * q^99

Decomposition of \(S_{24}^{\mathrm{new}}(\Gamma_0(49))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7
49.24.a.a	$49$	$24$	49.a	1.a	$1$	$1$	$1$	$164.250$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	✓			49.24.a.a	$2$	$1$	\(-5197\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-5197q^{2}+18620201q^{4}-53173588821q^{8}+\cdots\)
49.24.a.b	$49$	$24$	49.a	1.a	$1$	$2$	$2$	$164.250$	\(\Q(\sqrt{144169}) \)	None	✓				1.24.a.a	$1$	$1$	\(1080\)	\(-339480\)	\(-73069020\)	\(0\)		$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(540-\beta )q^{2}+(-169740-48\beta )q^{3}+\cdots\)
49.24.a.c	$49$	$24$	49.a	1.a	$1$	$5$	$5$	$164.250$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				7.24.a.a	$1$	$1$	\(-1014\)	\(454020\)	\(74149194\)	\(0\)		$-$	$-$	$2^{9}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-203-\beta _{1})q^{2}+(90801-2^{4}\beta _{1}+\cdots)q^{3}+\cdots\)
49.24.a.d	$49$	$24$	49.a	1.a	$1$	$6$	$6$	$164.250$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				7.24.a.b	$1$	$1$	\(-2115\)	\(-129352\)	\(-71437860\)	\(0\)		$-$	$-$	$2^{11}\cdot 3^{4}\cdot 5\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(-352-\beta _{1})q^{2}+(-21591+65\beta _{1}+\cdots)q^{3}+\cdots\)
49.24.a.e	$49$	$24$	49.a	1.a	$1$	$10$	$10$	$164.250$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			49.24.a.e	$2$	$1$	\(6164\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{37}\cdot 3^{13}\cdot 5^{5}\cdot 7^{10}\cdot 11$	$\mathrm{SU}(2)$	\(q+(616+\beta _{1})q^{2}+\beta _{5}q^{3}+(3372918+\cdots)q^{4}+\cdots\)
49.24.a.f	$49$	$24$	49.a	1.a	$1$	$14$	$14$	$164.250$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓		✓		7.24.c.a	$1$	$1$	\(-966\)	\(-177148\)	\(-74771022\)	\(0\)		$-$	$-$	$2^{49}\cdot 3^{13}\cdot 5^{3}\cdot 7^{23}$	$\mathrm{SU}(2)$	\(q+(-69+\beta _{1})q^{2}+(-12653-9\beta _{1}+\cdots)q^{3}+\cdots\)
49.24.a.g	$49$	$24$	49.a	1.a	$1$	$14$	$14$	$164.250$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				7.24.c.a	$1$	$0$	\(-966\)	\(177148\)	\(74771022\)	\(0\)		$+$	$+$	$2^{49}\cdot 3^{13}\cdot 5^{3}\cdot 7^{23}$	$\mathrm{SU}(2)$	\(q+(-69+\beta _{1})q^{2}+(12653+9\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
49.24.a.h	$49$	$24$	49.a	1.a	$1$	$24$	$24$	$164.250$		None	✓	✓	✓		49.24.a.h	$2$	$0$	\(6030\)	\(0\)	\(0\)	\(0\)		$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{24}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{24}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{24}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{24}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)