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.

\(7\)	Dim
\(+\)	\(33\)
\(-\)	\(36\)

Trace form

\( 69 q + 288 q^{2} - 128842 q^{3} + 68796140 q^{4} + 41640914 q^{5} - 57397886 q^{6} - 6326056668 q^{8} + 198780124867 q^{9} + O(q^{10}) \)

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

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

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