Space of modular forms of level 7, weight 22, and trivial character

• Label: 7.22.a
• Level: $7$
• Weight: $22$
• Character orbit: 7.a
• Rep. character: $\chi_{7}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $2$
• Sturm bound: $14$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	7.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(14\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(7))\).

	Total	New	Old
Modular forms	15	11	4
Cusp forms	13	11	2
Eisenstein series	2	0	2

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
\(+\)		\(7\)	\(5\)	\(2\)		\(6\)	\(5\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)		\(8\)	\(6\)	\(2\)		\(7\)	\(6\)	\(1\)		\(1\)	\(0\)	\(1\)

Trace form

11 * q + 287 * q^2 + 257686 * q^3 + 15809161 * q^4 - 63281864 * q^5 + 20290814 * q^6 + 282475249 * q^7 + 4103699055 * q^8 + 50006802667 * q^9 + 44269189204 * q^10 + 53191743976 * q^11 + 206480040662 * q^12 - 173662063204 * q^13 + 1368027630907 * q^14 + 4381319269888 * q^15 + 8438757071329 * q^16 - 5535353986398 * q^17 - 34196618583049 * q^18 - 5494935211558 * q^19 - 181898676998248 * q^20 + 76072279407194 * q^21 + 311547897098824 * q^22 + 347358696320496 * q^23 + 319153590439602 * q^24 + 1535576027170153 * q^25 - 720307893638368 * q^26 + 2662576846532020 * q^27 - 1110800867088367 * q^28 + 4849545159076318 * q^29 - 14927779067899888 * q^30 - 10691109957426708 * q^31 + 11868609655602127 * q^32 - 26781225254493488 * q^33 + 2246800485951138 * q^34 + 16143903401540432 * q^35 + 172457990753826761 * q^36 - 74778332232037098 * q^37 - 205268007642555310 * q^38 + 100019441561005288 * q^39 - 169465653843227184 * q^40 + 29586335072215250 * q^41 + 203283409549385158 * q^42 + 533480505501994576 * q^43 - 652483865050083520 * q^44 - 1563040398841319240 * q^45 + 901386648804565008 * q^46 - 1059213113920171548 * q^47 + 3197480240340600926 * q^48 + 877714929273732011 * q^49 + 969710831904750349 * q^50 - 1618025303650537692 * q^51 + 31897784716424452 * q^52 - 3354384351079501614 * q^53 + 2786636031139308644 * q^54 + 1481132629859277824 * q^55 + 1794848943117392463 * q^56 - 1128176350909108220 * q^57 + 7101472695097825870 * q^58 + 7183978615610091234 * q^59 - 28877066152405052512 * q^60 - 5629674914955952912 * q^61 - 21498886426604098836 * q^62 - 5866451136856878463 * q^63 - 28059757735216506847 * q^64 + 9485762772220105840 * q^65 + 118085268087815526752 * q^66 - 6035357847576403948 * q^67 - 16372673728283654586 * q^68 + 40746026409083183424 * q^69 - 37181714941933516444 * q^70 + 28065118041679869312 * q^71 + 13153529348221037055 * q^72 + 25365012842597517126 * q^73 - 103059192765738253722 * q^74 - 493284906537314350 * q^75 - 184251057634957793942 * q^76 + 48827180725908055688 * q^77 + 1026861031589297912 * q^78 + 33657865163468253400 * q^79 - 359791216574016831616 * q^80 + 504760583855001142519 * q^81 + 197652174229769129234 * q^82 + 179411145690403571826 * q^83 + 187180021529307607654 * q^84 + 313537119570937305072 * q^85 - 57189673024170868496 * q^86 - 1671832042458458239460 * q^87 + 990289623305993605680 * q^88 - 434115879238077635762 * q^89 + 330651619946256160772 * q^90 + 152227986404659350220 * q^91 - 890380388725635019008 * q^92 - 984408211886598553608 * q^93 - 125830565410079247324 * q^94 + 1253026376960724092992 * q^95 - 1896623874184902767198 * q^96 - 477518868817518026478 * q^97 + 22900380427414644287 * q^98 - 484685185597480418648 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(7))\) 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
7.22.a.a	$7$	$22$	7.a	1.a	$1$	$5$	$5$	$19.563$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	7.22.a.a	$1$	$1$	\(-2278\)	\(-5810\)	\(-60216716\)	\(-1412376245\)		$+$	$+$	$2^{9}\cdot 3^{2}\cdot 5\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-456+\beta _{1})q^{2}+(-1155-18\beta _{1}+\cdots)q^{3}+\cdots\)
7.22.a.b	$7$	$22$	7.a	1.a	$1$	$6$	$6$	$19.563$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	7.22.a.b	$1$	$0$	\(2565\)	\(263496\)	\(-3065148\)	\(1694851494\)		$-$	$-$	$2^{8}\cdot 3^{4}\cdot 5\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(428-\beta _{1})q^{2}+(43924-15\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(7))\) into lower level spaces

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