Space of modular forms of level 650, weight 2, and trivial character

• Label: 650.2.a
• Level: $650$
• Weight: $2$
• Character orbit: 650.a
• Rep. character: $\chi_{650}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $15$
• Sturm bound: $210$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	650.a (trivial)
Character field:			$\Q$
Newform subspaces:			$15$
Sturm bound:			$210$
Trace bound:			$7$
Distinguishing $T_p$:			$3$, $7$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(650))$.

	Total	New	Old
Modular forms	116	19	97
Cusp forms	93	19	74
Eisenstein series	23	0	23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$2$	$5$	$13$	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$11$	$3$	$8$		$9$	$3$	$6$		$2$	$0$	$2$
$+$	$+$	$-$	$-$		$17$	$3$	$14$		$14$	$3$	$11$		$3$	$0$	$3$
$+$	$-$	$+$	$-$		$16$	$3$	$13$		$13$	$3$	$10$		$3$	$0$	$3$
$+$	$-$	$-$	$+$		$13$	$1$	$12$		$10$	$1$	$9$		$3$	$0$	$3$
$-$	$+$	$+$	$-$		$16$	$3$	$13$		$13$	$3$	$10$		$3$	$0$	$3$
$-$	$+$	$-$	$+$		$13$	$0$	$13$		$10$	$0$	$10$		$3$	$0$	$3$
$-$	$-$	$+$	$+$		$15$	$1$	$14$		$12$	$1$	$11$		$3$	$0$	$3$
$-$	$-$	$-$	$-$		$15$	$5$	$10$		$12$	$5$	$7$		$3$	$0$	$3$
Plus space			$+$		$52$	$5$	$47$		$41$	$5$	$36$		$11$	$0$	$11$
Minus space			$-$		$64$	$14$	$50$		$52$	$14$	$38$		$12$	$0$	$12$

Trace form

19 * q - q^2 + 2 * q^3 + 19 * q^4 + 4 * q^6 + 8 * q^7 - q^8 + 25 * q^9 + 4 * q^11 + 2 * q^12 - q^13 + 2 * q^14 + 19 * q^16 + 8 * q^17 - 5 * q^18 - 8 * q^19 + 12 * q^21 + 4 * q^22 - 4 * q^23 + 4 * q^24 + 3 * q^26 + 14 * q^27 + 8 * q^28 + 10 * q^29 + 16 * q^31 - q^32 - 20 * q^33 - 10 * q^34 + 25 * q^36 - 2 * q^37 - 2 * q^41 + 18 * q^42 + 14 * q^43 + 4 * q^44 + 20 * q^46 + 2 * q^48 + 53 * q^49 - 50 * q^51 - q^52 - 26 * q^53 - 8 * q^54 + 2 * q^56 + 8 * q^57 - 2 * q^58 - 36 * q^59 + 18 * q^61 + 4 * q^62 + 19 * q^64 - 12 * q^66 + 8 * q^68 - 68 * q^69 - 48 * q^71 - 5 * q^72 - 2 * q^73 + 4 * q^74 - 8 * q^76 - 24 * q^77 - 2 * q^78 + 28 * q^79 - 29 * q^81 - 26 * q^82 - 24 * q^83 + 12 * q^84 - 16 * q^87 + 4 * q^88 - 58 * q^89 + 10 * q^91 - 4 * q^92 + 32 * q^93 + 2 * q^94 + 4 * q^96 - 6 * q^97 + 7 * q^98 - 52 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(650))$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	13
650.2.a.a	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓	✓			650.2.a.a	$1$	$1$	$-1$	$-3$	$0$	$0$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-3q^{3}+q^{4}+3q^{6}-q^{8}+6q^{9}+\cdots$
650.2.a.b	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓	✓	✓	✓	650.2.a.b	$1$	$1$	$-1$	$-2$	$0$	$-1$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{7}-q^{8}+\cdots$
650.2.a.c	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓				130.2.a.c	$1$	$0$	$-1$	$-2$	$0$	$4$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}+4q^{7}-q^{8}+\cdots$
650.2.a.d	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓				130.2.a.b	$1$	$1$	$-1$	$0$	$0$	$0$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}-3q^{9}-q^{13}+q^{16}+\cdots$
650.2.a.e	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓	✓			650.2.a.e	$1$	$0$	$-1$	$1$	$0$	$4$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+4q^{7}-q^{8}+\cdots$
650.2.a.f	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓	✓			650.2.a.f	$1$	$1$	$-1$	$2$	$0$	$-5$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-5q^{7}-q^{8}+\cdots$
650.2.a.g	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓				26.2.a.b	$1$	$0$	$-1$	$3$	$0$	$-1$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{7}-q^{8}+\cdots$
650.2.a.h	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓	✓			650.2.a.f	$1$	$0$	$1$	$-2$	$0$	$5$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+5q^{7}+q^{8}+\cdots$
650.2.a.i	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓	✓	✓	✓	650.2.a.e	$1$	$1$	$1$	$-1$	$0$	$-4$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots$
650.2.a.j	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓				26.2.a.a	$1$	$0$	$1$	$-1$	$0$	$1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots$
650.2.a.k	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓	✓			650.2.a.b	$1$	$0$	$1$	$2$	$0$	$1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots$
650.2.a.l	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓				130.2.a.a	$1$	$0$	$1$	$2$	$0$	$4$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}+4q^{7}+q^{8}+\cdots$
650.2.a.m	$650$	$2$	650.a	1.a	$1$	$1$	$1$	$5.190$	$\Q$	None	✓	✓			650.2.a.a	$1$	$0$	$1$	$3$	$0$	$0$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{8}+6q^{9}+\cdots$
650.2.a.n	$650$	$2$	650.a	1.a	$1$	$3$	$3$	$5.190$	3.3.940.1	None	✓		✓	✓	130.2.b.a	$1$	$0$	$-3$	$0$	$0$	$2$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
650.2.a.o	$650$	$2$	650.a	1.a	$1$	$3$	$3$	$5.190$	3.3.940.1	None	✓		✓		130.2.b.a	$1$	$0$	$3$	$0$	$0$	$-2$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(650))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(650)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(50))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(65))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(130))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(325))$$^{\oplus 2}$