Space of modular forms of level 49, weight 20, and trivial character

• Label: 49.20.a
• Level: $49$
• Weight: $20$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $63$
• Newform subspaces: $8$
• Sturm bound: $93$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(93\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	92	68	24
Cusp forms	84	63	21
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
\(+\)		\(47\)	\(34\)	\(13\)		\(43\)	\(32\)	\(11\)		\(4\)	\(2\)	\(2\)
\(-\)		\(45\)	\(34\)	\(11\)		\(41\)	\(31\)	\(10\)		\(4\)	\(3\)	\(1\)

Trace form

63 * q - 456 * q^2 + 11288 * q^3 + 16665036 * q^4 - 4621576 * q^5 + 9984994 * q^6 + 35620212 * q^8 + 25659749107 * q^9 - 936451204 * q^10 - 4775590852 * q^11 + 3579921226 * q^12 - 27252155952 * q^13 - 57026512408 * q^15 + 4825263732532 * q^16 - 665125685052 * q^17 - 1061634118804 * q^18 - 77746959144 * q^19 - 388241020328 * q^20 - 2721934956428 * q^22 - 19890815531616 * q^23 - 29614050341634 * q^24 + 241992356826741 * q^25 - 18530418284000 * q^26 + 77252961639824 * q^27 + 115848121505158 * q^29 + 117432753174876 * q^30 - 506727819201040 * q^31 + 26964347900300 * q^32 + 95777853088112 * q^33 + 448906286497422 * q^34 + 9185938052192284 * q^36 - 1450785207443706 * q^37 - 3575448053365730 * q^38 + 60636958834144 * q^39 - 1809558646433280 * q^40 + 1009727471849236 * q^41 + 11100656419431572 * q^43 - 5098031928971256 * q^44 - 4282939471559272 * q^45 + 2406914414364372 * q^46 - 5912881223743824 * q^47 - 9890000485425566 * q^48 - 93229271479023484 * q^50 + 38757211777708808 * q^51 + 61838578121254396 * q^52 + 34725181468584330 * q^53 + 50134336900706332 * q^54 - 24947524231708832 * q^55 + 35485152827630760 * q^57 + 23536217839455984 * q^58 + 89780664933810792 * q^59 + 50577704066381968 * q^60 + 14676790747251912 * q^61 + 15816162235400868 * q^62 + 692938298190576140 * q^64 - 685309033328654784 * q^65 + 39403194201250576 * q^66 + 58993903136243372 * q^67 - 347874069486555318 * q^68 - 137526039339464736 * q^69 + 1045836705831219344 * q^71 + 540805462408955700 * q^72 - 1327907837901643204 * q^73 + 3167264278652096500 * q^74 - 1274612278059871928 * q^75 + 725829945628816966 * q^76 + 3062383767051137424 * q^78 - 3044893199816313504 * q^79 + 111191835928620304 * q^80 + 8824922595211438807 * q^81 - 2674939090934251106 * q^82 + 360054595421047128 * q^83 + 698016058625871176 * q^85 + 6967961769604823720 * q^86 + 4606011031196631344 * q^87 - 7196417785111979016 * q^88 - 5828513845627034836 * q^89 - 6976634315563944308 * q^90 - 9826044971509388136 * q^92 + 824084932545300152 * q^93 + 16525896427974934956 * q^94 + 11759896615847602568 * q^95 - 25706091416339209330 * q^96 - 13804782798602046076 * q^97 - 55503700856908827404 * q^99

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(49))\) 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
49.20.a.a	$49$	$20$	49.a	1.a	$1$	$1$	$1$	$112.120$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	✓			49.20.a.a	$2$	$1$	\(-797\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-797q^{2}+110921q^{4}+329453499q^{8}+\cdots\)
49.20.a.b	$49$	$20$	49.a	1.a	$1$	$1$	$1$	$112.120$	\(\Q\)	None	✓				1.20.a.a	$1$	$1$	\(456\)	\(-50652\)	\(2377410\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+456q^{2}-50652q^{3}-316352q^{4}+\cdots\)
49.20.a.c	$49$	$20$	49.a	1.a	$1$	$4$	$4$	$112.120$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				7.20.a.a	$1$	$1$	\(-342\)	\(29526\)	\(2486610\)	\(0\)		$-$	$-$	$2^{5}\cdot 3^{2}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-86+\beta _{1})q^{2}+(7378+8\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
49.20.a.d	$49$	$20$	49.a	1.a	$1$	$5$	$5$	$112.120$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				7.20.a.b	$1$	$1$	\(-115\)	\(32414\)	\(-9485596\)	\(0\)		$-$	$-$	$2^{6}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-23-\beta _{1})q^{2}+(6483-4\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)
49.20.a.e	$49$	$20$	49.a	1.a	$1$	$8$	$8$	$112.120$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			49.20.a.e	$2$	$1$	\(-684\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{23}\cdot 3^{6}\cdot 5^{3}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+(-86-\beta _{1})q^{2}+\beta _{2}q^{3}+(305916+\cdots)q^{4}+\cdots\)
49.20.a.f	$49$	$20$	49.a	1.a	$1$	$12$	$12$	$112.120$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		7.20.c.a	$1$	$1$	\(458\)	\(-19684\)	\(-4737292\)	\(0\)		$-$	$-$	$2^{38}\cdot 3^{10}\cdot 5^{3}\cdot 7^{16}$	$\mathrm{SU}(2)$	\(q+(38+\beta _{1})q^{2}+(-1640-3\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)
49.20.a.g	$49$	$20$	49.a	1.a	$1$	$12$	$12$	$112.120$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				7.20.c.a	$1$	$0$	\(458\)	\(19684\)	\(4737292\)	\(0\)		$+$	$+$	$2^{38}\cdot 3^{10}\cdot 5^{3}\cdot 7^{16}$	$\mathrm{SU}(2)$	\(q+(38+\beta _{1})q^{2}+(1640+3\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
49.20.a.h	$49$	$20$	49.a	1.a	$1$	$20$	$20$	$112.120$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	✓	✓		49.20.a.h	$2$	$0$	\(110\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{43}\cdot 3^{12}\cdot 5^{4}\cdot 7^{50}$	$\mathrm{SU}(2)$	\(q+(6-\beta _{1})q^{2}-\beta _{3}q^{3}+(324794-26\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)