Space of modular forms of level 7, weight 8, and trivial character

• Label: 7.8.a
• Level: $7$
• Weight: $8$
• Character orbit: 7.a
• Rep. character: $\chi_{7}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $5$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	7.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(5\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(7))\).

	Total	New	Old
Modular forms	5	3	2
Cusp forms	3	3	0
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.

\(7\)	Dim
\(+\)	\(2\)
\(-\)	\(1\)

Trace form

3 * q - 9 * q^2 + 52 * q^3 + 89 * q^4 + 246 * q^5 - 754 * q^6 - 343 * q^7 + 135 * q^8 + 1351 * q^9 - 4316 * q^10 - 2724 * q^11 + 14966 * q^12 - 2618 * q^13 - 1029 * q^14 + 27688 * q^15 - 31807 * q^16 - 15474 * q^17 - 81433 * q^18 + 88180 * q^19 + 50568 * q^20 - 46648 * q^21 + 79312 * q^22 - 97848 * q^23 - 82590 * q^24 - 129619 * q^25 + 408576 * q^26 + 150040 * q^27 - 93639 * q^28 + 62250 * q^29 - 403408 * q^30 - 241504 * q^31 - 55689 * q^32 + 267184 * q^33 - 53982 * q^34 - 142002 * q^35 + 443393 * q^36 + 231026 * q^37 - 10350 * q^38 - 427448 * q^39 - 163680 * q^40 + 224406 * q^41 + 431494 * q^42 - 652548 * q^43 + 619128 * q^44 + 1141342 * q^45 + 536736 * q^46 - 1608144 * q^47 - 1700578 * q^48 + 352947 * q^49 - 913011 * q^50 + 797736 * q^51 - 441084 * q^52 + 2799642 * q^53 - 3415420 * q^54 + 435272 * q^55 + 859215 * q^56 - 1525640 * q^57 + 3776230 * q^58 + 2139420 * q^59 + 2899904 * q^60 - 3853394 * q^61 - 1398948 * q^62 - 753571 * q^63 - 762831 * q^64 - 2963772 * q^65 - 325168 * q^66 + 2080756 * q^67 + 1572438 * q^68 + 3565872 * q^69 + 1826132 * q^70 - 5995704 * q^71 + 1073775 * q^72 + 2480222 * q^73 - 11488242 * q^74 + 3087548 * q^75 - 3237990 * q^76 - 2885316 * q^77 + 16500344 * q^78 - 8077680 * q^79 - 5520624 * q^80 - 264797 * q^81 + 12216722 * q^82 + 12355812 * q^83 - 2482634 * q^84 + 2330604 * q^85 + 4187256 * q^86 - 8891240 * q^87 - 10690440 * q^88 + 15330750 * q^89 - 19124332 * q^90 - 2636298 * q^91 + 7864176 * q^92 - 22560816 * q^93 - 11211372 * q^94 - 2948280 * q^95 + 26043346 * q^96 + 20639486 * q^97 - 1058841 * q^98 - 4554068 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\) 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
7.8.a.a	$7$	$8$	7.a	1.a	$1$	$1$	$1$	$2.187$	\(\Q\)	None	✓	✓	✓	✓	7.8.a.a	$1$	$1$	\(-6\)	\(-42\)	\(-84\)	\(343\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}-42q^{3}-92q^{4}-84q^{5}+252q^{6}+\cdots\)
7.8.a.b	$7$	$8$	7.a	1.a	$1$	$2$	$2$	$2.187$	\(\Q(\sqrt{865}) \)	None	✓	✓	✓	✓	7.8.a.b	$1$	$0$	\(-3\)	\(94\)	\(330\)	\(-686\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(46+2\beta )q^{3}+(89+3\beta )q^{4}+\cdots\)