Space of modular forms of level 1400, weight 2, and trivial character

• Label: 1400.2.a
• Level: $1400$
• Weight: $2$
• Character orbit: 1400.a
• Rep. character: $\chi_{1400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $20$
• Sturm bound: $480$
• Trace bound: $13$

Defining parameters

Level:	$N$	$=$	$1400 = 2^{3} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1400.a (trivial)
Character field:			$\Q$
Newform subspaces:			$20$
Sturm bound:			$480$
Trace bound:			$13$
Distinguishing $T_p$:			$3$, $11$, $13$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(1400))$.

	Total	New	Old
Modular forms	264	28	236
Cusp forms	217	28	189
Eisenstein series	47	0	47

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

$2$	$5$	$7$	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$27$	$4$	$23$		$22$	$4$	$18$		$5$	$0$	$5$
$+$	$+$	$-$	$-$		$39$	$5$	$34$		$33$	$5$	$28$		$6$	$0$	$6$
$+$	$-$	$+$	$-$		$37$	$4$	$33$		$31$	$4$	$27$		$6$	$0$	$6$
$+$	$-$	$-$	$+$		$29$	$2$	$27$		$23$	$2$	$21$		$6$	$0$	$6$
$-$	$+$	$+$	$-$		$33$	$3$	$30$		$27$	$3$	$24$		$6$	$0$	$6$
$-$	$+$	$-$	$+$		$33$	$2$	$31$		$27$	$2$	$25$		$6$	$0$	$6$
$-$	$-$	$+$	$+$		$35$	$3$	$32$		$29$	$3$	$26$		$6$	$0$	$6$
$-$	$-$	$-$	$-$		$31$	$5$	$26$		$25$	$5$	$20$		$6$	$0$	$6$
Plus space			$+$		$124$	$11$	$113$		$101$	$11$	$90$		$23$	$0$	$23$
Minus space			$-$		$140$	$17$	$123$		$116$	$17$	$99$		$24$	$0$	$24$

Trace form

28 * q + 2 * q^3 + 20 * q^9 + 8 * q^11 - 2 * q^13 - 4 * q^17 + 6 * q^19 - 2 * q^21 + 20 * q^27 + 12 * q^29 - 4 * q^31 - 24 * q^33 + 12 * q^37 + 16 * q^41 + 4 * q^43 - 4 * q^47 + 28 * q^49 + 56 * q^51 + 24 * q^53 + 12 * q^57 + 18 * q^59 + 38 * q^61 - 4 * q^63 + 32 * q^67 + 8 * q^69 - 8 * q^71 - 8 * q^73 + 4 * q^77 - 16 * q^79 - 26 * q^83 + 12 * q^87 + 4 * q^89 + 10 * q^91 + 32 * q^93 - 36 * q^97 + 44 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(1400))$ 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	7
1400.2.a.a	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓				56.2.a.b	$1$	$0$	$0$	$-2$	$0$	$-1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}-q^{7}+q^{9}+2q^{17}-2q^{19}+\cdots$
1400.2.a.b	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓	✓			1400.2.a.b	$1$	$1$	$0$	$-2$	$0$	$-1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}-q^{7}+q^{9}+5q^{11}-8q^{17}+\cdots$
1400.2.a.c	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓	✓			1400.2.a.c	$1$	$1$	$0$	$-2$	$0$	$1$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}+q^{7}+q^{9}+q^{11}-4q^{13}+\cdots$
1400.2.a.d	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓				280.2.g.a	$1$	$1$	$0$	$-1$	$0$	$1$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+q^{7}-2q^{9}-q^{11}+q^{13}+3q^{17}+\cdots$
1400.2.a.e	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓	✓			1400.2.a.e	$1$	$0$	$0$	$-1$	$0$	$1$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+q^{7}-2q^{9}-q^{11}+6q^{13}+\cdots$
1400.2.a.f	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓	✓			1400.2.a.f	$1$	$0$	$0$	$0$	$0$	$-1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{7}-3q^{9}+q^{11}+2q^{13}+4q^{17}+\cdots$
1400.2.a.g	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓				56.2.a.a	$1$	$0$	$0$	$0$	$0$	$1$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{7}-3q^{9}-4q^{11}-2q^{13}+6q^{17}+\cdots$
1400.2.a.h	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓	✓			1400.2.a.f	$1$	$1$	$0$	$0$	$0$	$1$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{7}-3q^{9}+q^{11}-2q^{13}-4q^{17}+\cdots$
1400.2.a.i	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓	✓			1400.2.a.e	$1$	$1$	$0$	$1$	$0$	$-1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-q^{7}-2q^{9}-q^{11}-6q^{13}+\cdots$
1400.2.a.j	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓				280.2.g.a	$1$	$1$	$0$	$1$	$0$	$-1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-q^{7}-2q^{9}-q^{11}-q^{13}-3q^{17}+\cdots$
1400.2.a.k	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓				280.2.a.b	$1$	$1$	$0$	$1$	$0$	$1$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{3}+q^{7}-2q^{9}-5q^{11}-q^{13}+\cdots$
1400.2.a.l	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓	✓			1400.2.a.c	$1$	$0$	$0$	$2$	$0$	$-1$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-q^{7}+q^{9}+q^{11}+4q^{13}+\cdots$
1400.2.a.m	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓	✓			1400.2.a.b	$1$	$0$	$0$	$2$	$0$	$1$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}+q^{7}+q^{9}+5q^{11}+8q^{17}+\cdots$
1400.2.a.n	$1400$	$2$	1400.a	1.a	$1$	$1$	$1$	$11.179$	$\Q$	None	✓				280.2.a.a	$1$	$0$	$0$	$3$	$0$	$-1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+3q^{3}-q^{7}+6q^{9}-5q^{11}+5q^{13}+\cdots$
1400.2.a.o	$1400$	$2$	1400.a	1.a	$1$	$2$	$2$	$11.179$	$\Q(\sqrt{17})$	None	✓	✓			1400.2.a.o	$1$	$1$	$0$	$-1$	$0$	$-2$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{3}-q^{7}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots$
1400.2.a.p	$1400$	$2$	1400.a	1.a	$1$	$2$	$2$	$11.179$	$\Q(\sqrt{17})$	None	✓				280.2.a.d	$1$	$1$	$0$	$-1$	$0$	$-2$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{3}-q^{7}+(1+\beta )q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots$
1400.2.a.q	$1400$	$2$	1400.a	1.a	$1$	$2$	$2$	$11.179$	$\Q(\sqrt{17})$	None	✓	✓			1400.2.a.o	$1$	$0$	$0$	$1$	$0$	$2$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+q^{7}+(1+\beta )q^{9}+(-3+2\beta )q^{11}+\cdots$
1400.2.a.r	$1400$	$2$	1400.a	1.a	$1$	$2$	$2$	$11.179$	$\Q(\sqrt{33})$	None	✓		✓		280.2.a.c	$1$	$0$	$0$	$1$	$0$	$2$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+q^{7}+(5+\beta )q^{9}+(4-\beta )q^{11}+\cdots$
1400.2.a.s	$1400$	$2$	1400.a	1.a	$1$	$3$	$3$	$11.179$	3.3.568.1	None	✓		✓		280.2.g.b	$1$	$0$	$0$	$-1$	$0$	$3$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots$
1400.2.a.t	$1400$	$2$	1400.a	1.a	$1$	$3$	$3$	$11.179$	3.3.568.1	None	✓		✓		280.2.g.b	$1$	$0$	$0$	$1$	$0$	$-3$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}-q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(1400))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(1400)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))$$^{\oplus 9}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(20))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(35))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(40))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(50))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(56))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(70))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(100))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(140))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(175))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(200))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(280))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(350))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(700))$$^{\oplus 2}$