Space of modular forms of level 49, weight 22, and trivial character

• Label: 49.22.a
• Level: $49$
• Weight: $22$
• Character orbit: 49.a
• Rep. character: $\chi_{49}(1,\cdot)$
• Character field: $\Q$
• Dimension: $69$
• Newform subspaces: $8$
• Sturm bound: $102$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(102\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(49))\).

	Total	New	Old
Modular forms	102	74	28
Cusp forms	94	69	25
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
\(+\)		\(50\)	\(35\)	\(15\)		\(46\)	\(33\)	\(13\)		\(4\)	\(2\)	\(2\)
\(-\)		\(52\)	\(39\)	\(13\)		\(48\)	\(36\)	\(12\)		\(4\)	\(3\)	\(1\)

Trace form

69 * q + 288 * q^2 - 128842 * q^3 + 68796140 * q^4 + 41640914 * q^5 - 57397886 * q^6 - 6326056668 * q^8 + 198780124867 * q^9 - 38036595604 * q^10 + 22346871674 * q^11 - 465998656214 * q^12 + 254283852998 * q^13 - 777783445750 * q^15 + 63688572427572 * q^16 + 2483071056396 * q^17 + 30697464728180 * q^18 + 13415723563298 * q^19 + 225488051615848 * q^20 - 272796343028028 * q^22 - 377409266701662 * q^23 - 166593058901682 * q^24 + 4956861206855553 * q^25 + 697088818177696 * q^26 - 3219173916471100 * q^27 - 3063043625462270 * q^29 + 14411245755170988 * q^30 + 8790568781116276 * q^31 - 5801120849031508 * q^32 + 14576486478194816 * q^33 - 1367743002110562 * q^34 + 30544510070393116 * q^36 + 39037880926076180 * q^37 + 202986820597254190 * q^38 - 80288059479880700 * q^39 + 143841217819931184 * q^40 - 8963531927668892 * q^41 - 301040838544717700 * q^43 + 188043750573718872 * q^44 + 1430155808841222890 * q^45 - 529241139654628556 * q^46 + 912252609604559916 * q^47 - 2697167689781413982 * q^48 + 175812809150646764 * q^50 + 2959642068676244402 * q^51 - 194286838693817604 * q^52 - 2823697892252236656 * q^53 - 2626336074996853604 * q^54 + 568804826451770776 * q^55 - 4981039943015243082 * q^57 - 5345937811930847696 * q^58 - 1208095872867738414 * q^59 + 15116231207662864384 * q^60 - 560942239522196350 * q^61 + 22046242285381503252 * q^62 + 120949572424887993548 * q^64 + 4293532657346501268 * q^65 - 114570303320241509216 * q^66 + 28634340347289436698 * q^67 + 22520606424157123002 * q^68 - 50260567954512185760 * q^69 - 21422328180050613664 * q^71 - 3261564781434553452 * q^72 + 17919746668505420368 * q^73 - 76711991875949438828 * q^74 - 602719016400413150 * q^75 + 168296942370576272022 * q^76 + 70623843862870583040 * q^78 - 47346203742046316358 * q^79 + 275757509990677004416 * q^80 + 497103600861438627517 * q^81 - 203591541535398480338 * q^82 - 228323000393364620982 * q^83 - 260468633940798287518 * q^85 - 286122576596686611640 * q^86 + 1123854421404845115020 * q^87 + 788496585315109090344 * q^88 + 938419369137921644792 * q^89 - 292380858026228411972 * q^90 - 1336502469896420871576 * q^92 + 270710738783316581186 * q^93 + 168155190652975397340 * q^94 - 1032114481900198545706 * q^95 + 1432591235788044923486 * q^96 - 330756189337511158004 * q^97 + 4326731834103672995440 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(49))\) 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
49.22.a.a	$49$	$22$	49.a	1.a	$1$	$1$	$1$	$136.944$	\(\Q\)	None	✓				1.22.a.a	$1$	$0$	\(-288\)	\(128844\)	\(-21640950\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-288q^{2}+128844q^{3}-2014208q^{4}+\cdots\)
49.22.a.b	$49$	$22$	49.a	1.a	$1$	$1$	$1$	$136.944$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	✓			49.22.a.b	$2$	$0$	\(2795\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+2795q^{2}+5714873q^{4}+10111530195q^{8}+\cdots\)
49.22.a.c	$49$	$22$	49.a	1.a	$1$	$5$	$5$	$136.944$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				7.22.a.a	$1$	$0$	\(-2278\)	\(5810\)	\(60216716\)	\(0\)		$-$	$-$	$2^{9}\cdot 3^{2}\cdot 5\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-456+\beta _{1})q^{2}+(1155+18\beta _{1}+\cdots)q^{3}+\cdots\)
49.22.a.d	$49$	$22$	49.a	1.a	$1$	$6$	$6$	$136.944$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				7.22.a.b	$1$	$0$	\(2565\)	\(-263496\)	\(3065148\)	\(0\)		$-$	$-$	$2^{8}\cdot 3^{4}\cdot 5\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(428-\beta _{1})q^{2}+(-43924+15\beta _{1}+\cdots)q^{3}+\cdots\)
49.22.a.e	$49$	$22$	49.a	1.a	$1$	$10$	$10$	$136.944$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			49.22.a.e	$2$	$0$	\(-460\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{32}\cdot 3^{13}\cdot 5\cdot 7^{10}$	$\mathrm{SU}(2)$	\(q+(-46+\beta _{2})q^{2}-\beta _{1}q^{3}+(524612+\cdots)q^{4}+\cdots\)
49.22.a.f	$49$	$22$	49.a	1.a	$1$	$13$	$13$	$136.944$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓				7.22.c.a	$1$	$1$	\(-286\)	\(-118097\)	\(-19296893\)	\(0\)		$+$	$+$	$2^{43}\cdot 3^{9}\cdot 5^{3}\cdot 7^{19}$	$\mathrm{SU}(2)$	\(q+(-22-\beta _{1})q^{2}+(-9084+6\beta _{1}+\cdots)q^{3}+\cdots\)
49.22.a.g	$49$	$22$	49.a	1.a	$1$	$13$	$13$	$136.944$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		7.22.c.a	$1$	$0$	\(-286\)	\(118097\)	\(19296893\)	\(0\)		$-$	$-$	$2^{43}\cdot 3^{9}\cdot 5^{3}\cdot 7^{19}$	$\mathrm{SU}(2)$	\(q+(-22-\beta _{1})q^{2}+(9084-6\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)
49.22.a.h	$49$	$22$	49.a	1.a	$1$	$20$	$20$	$136.944$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	✓	✓		49.22.a.h	$2$	$1$	\(-1474\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{44}\cdot 3^{8}\cdot 5^{4}\cdot 7^{50}$	$\mathrm{SU}(2)$	\(q+(-74+\beta _{1})q^{2}-\beta _{3}q^{3}+(1023984+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)