Space of modular forms of level 3400, weight 2, and trivial character

• Label: 3400.2.a
• Level: $3400$
• Weight: $2$
• Character orbit: 3400.a
• Rep. character: $\chi_{3400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $25$
• Sturm bound: $1080$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(1080\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3400))\).

	Total	New	Old
Modular forms	564	76	488
Cusp forms	517	76	441
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.

\(2\)	\(5\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(66\)	\(7\)	\(59\)		\(61\)	\(7\)	\(54\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(72\)	\(11\)	\(61\)		\(66\)	\(11\)	\(55\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(73\)	\(12\)	\(61\)		\(67\)	\(12\)	\(55\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(71\)	\(8\)	\(63\)		\(65\)	\(8\)	\(57\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(75\)	\(8\)	\(67\)		\(69\)	\(8\)	\(61\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(69\)	\(10\)	\(59\)		\(63\)	\(10\)	\(53\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(68\)	\(10\)	\(58\)		\(62\)	\(10\)	\(52\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(70\)	\(10\)	\(60\)		\(64\)	\(10\)	\(54\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(274\)	\(35\)	\(239\)		\(251\)	\(35\)	\(216\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(290\)	\(41\)	\(249\)		\(266\)	\(41\)	\(225\)		\(24\)	\(0\)	\(24\)

Trace form

76 * q - 2 * q^3 - 8 * q^7 + 72 * q^9 + 2 * q^11 - 4 * q^13 + 2 * q^17 - 8 * q^19 + 8 * q^21 - 4 * q^23 + 4 * q^27 + 6 * q^29 - 16 * q^31 + 4 * q^33 - 2 * q^37 - 12 * q^39 + 4 * q^41 - 20 * q^43 + 8 * q^47 + 84 * q^49 + 6 * q^51 + 20 * q^53 + 8 * q^57 - 4 * q^59 - 14 * q^61 - 24 * q^63 + 12 * q^67 - 8 * q^69 + 12 * q^71 + 24 * q^73 - 32 * q^77 - 60 * q^79 + 88 * q^81 - 12 * q^83 + 32 * q^87 + 16 * q^89 - 72 * q^91 - 8 * q^93 - 12 * q^97 - 22 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3400))\) 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	17
3400.2.a.a	$3400$	$2$	3400.a	1.a	$1$	$1$	$1$	$27.149$	\(\Q\)	None	✓				680.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
3400.2.a.b	$3400$	$2$	3400.a	1.a	$1$	$1$	$1$	$27.149$	\(\Q\)	None	✓				136.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+2q^{11}+6q^{13}+q^{17}+\cdots\)
3400.2.a.c	$3400$	$2$	3400.a	1.a	$1$	$1$	$1$	$27.149$	\(\Q\)	None	✓				680.2.e.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}+2q^{11}-5q^{13}+\cdots\)
3400.2.a.d	$3400$	$2$	3400.a	1.a	$1$	$1$	$1$	$27.149$	\(\Q\)	None	✓				680.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+2q^{13}-q^{17}-4q^{19}+8q^{23}+\cdots\)
3400.2.a.e	$3400$	$2$	3400.a	1.a	$1$	$1$	$1$	$27.149$	\(\Q\)	None	✓				680.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}+4q^{11}+q^{13}+\cdots\)
3400.2.a.f	$3400$	$2$	3400.a	1.a	$1$	$1$	$1$	$27.149$	\(\Q\)	None	✓				680.2.e.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}+2q^{11}+5q^{13}+\cdots\)
3400.2.a.g	$3400$	$2$	3400.a	1.a	$1$	$1$	$1$	$27.149$	\(\Q\)	None	✓				136.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
3400.2.a.h	$3400$	$2$	3400.a	1.a	$1$	$2$	$2$	$27.149$	\(\Q(\sqrt{2}) \)	None	✓				680.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}-q^{9}+(-2+3\beta )q^{11}+\cdots\)
3400.2.a.i	$3400$	$2$	3400.a	1.a	$1$	$2$	$2$	$27.149$	\(\Q(\sqrt{5}) \)	None	✓				136.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
3400.2.a.j	$3400$	$2$	3400.a	1.a	$1$	$2$	$2$	$27.149$	\(\Q(\sqrt{3}) \)	None	✓				680.2.a.d	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(3-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
3400.2.a.k	$3400$	$2$	3400.a	1.a	$1$	$3$	$3$	$27.149$	3.3.940.1	None	✓				680.2.a.h	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2})q^{7}+(3+\cdots)q^{9}+\cdots\)
3400.2.a.l	$3400$	$2$	3400.a	1.a	$1$	$3$	$3$	$27.149$	3.3.1016.1	None	✓				680.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-2-\beta _{2})q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3400.2.a.m	$3400$	$2$	3400.a	1.a	$1$	$3$	$3$	$27.149$	3.3.564.1	None	✓	✓	✓		3400.2.a.m	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3400.2.a.n	$3400$	$2$	3400.a	1.a	$1$	$3$	$3$	$27.149$	3.3.229.1	None	✓				680.2.a.f	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-\beta _{1}+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3400.2.a.o	$3400$	$2$	3400.a	1.a	$1$	$3$	$3$	$27.149$	3.3.148.1	None	✓	✓			3400.2.a.o	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{2})q^{7}+(-1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3400.2.a.p	$3400$	$2$	3400.a	1.a	$1$	$3$	$3$	$27.149$	3.3.148.1	None	✓	✓			3400.2.a.o	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{2})q^{7}+(-1+\beta _{1}+\cdots)q^{9}+\cdots\)
3400.2.a.q	$3400$	$2$	3400.a	1.a	$1$	$3$	$3$	$27.149$	3.3.564.1	None	✓	✓			3400.2.a.m	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3400.2.a.r	$3400$	$2$	3400.a	1.a	$1$	$5$	$5$	$27.149$	5.5.1981136.1	None	✓	✓	✓		3400.2.a.r	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1}+\beta _{3}-\beta _{4})q^{7}+\cdots\)
3400.2.a.s	$3400$	$2$	3400.a	1.a	$1$	$5$	$5$	$27.149$	5.5.4153680.1	None	✓	✓			3400.2.a.s	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{3})q^{7}+(2+\beta _{2}-\beta _{3}+\cdots)q^{9}+\cdots\)
3400.2.a.t	$3400$	$2$	3400.a	1.a	$1$	$5$	$5$	$27.149$	5.5.563792.1	None	✓				680.2.e.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{3}-\beta _{4})q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
3400.2.a.u	$3400$	$2$	3400.a	1.a	$1$	$5$	$5$	$27.149$	5.5.563792.1	None	✓		✓		680.2.e.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3}+\beta _{4})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
3400.2.a.v	$3400$	$2$	3400.a	1.a	$1$	$5$	$5$	$27.149$	5.5.4153680.1	None	✓	✓	✓		3400.2.a.s	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{3})q^{7}+(2+\beta _{2}-\beta _{3}+\cdots)q^{9}+\cdots\)
3400.2.a.w	$3400$	$2$	3400.a	1.a	$1$	$5$	$5$	$27.149$	5.5.1981136.1	None	✓	✓			3400.2.a.r	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3}+\beta _{4})q^{7}+\cdots\)
3400.2.a.x	$3400$	$2$	3400.a	1.a	$1$	$6$	$6$	$27.149$	6.6.358395264.1	None	✓		✓		680.2.e.c	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{4}q^{7}+(3-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
3400.2.a.y	$3400$	$2$	3400.a	1.a	$1$	$6$	$6$	$27.149$	6.6.358395264.1	None	✓		✓		680.2.e.c	$1$	$0$	\(0\)	\(4\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+\beta _{4}q^{7}+(3-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3400)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(340))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(425))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(680))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(850))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1700))\)\(^{\oplus 2}\)