Space of modular forms of level 3400, weight 2, and Character 3400.e

• Label: 3400.2.e
• Level: $3400$
• Weight: $2$
• Character orbit: 3400.e
• Rep. character: $\chi_{3400}(2449,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $15$
• Sturm bound: $1080$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3400.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(1080\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3400, [\chi])\).

	Total	New	Old
Modular forms	564	72	492
Cusp forms	516	72	444
Eisenstein series	48	0	48

Trace form

72 * q - 64 * q^9 - 20 * q^11 + 8 * q^19 + 16 * q^21 - 36 * q^29 + 8 * q^31 - 16 * q^39 + 32 * q^41 - 48 * q^49 - 12 * q^51 - 12 * q^61 + 40 * q^69 + 16 * q^71 - 8 * q^79 - 8 * q^81 - 40 * q^89 - 96 * q^91 + 36 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(3400, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3400.2.e.a	$3400$	$2$	3400.e	5.b	$2$	$2$	$2$	$27.149$	\(\Q(\sqrt{-1}) \)	None					136.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-2 i q^{7}-q^{9}-6 q^{11}+\cdots\)
3400.2.e.b	$3400$	$2$	3400.e	5.b	$2$	$2$	$2$	$27.149$	\(\Q(\sqrt{-1}) \)	None					680.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-2 i q^{7}-q^{9}-2 q^{11}+\cdots\)
3400.2.e.c	$3400$	$2$	3400.e	5.b	$2$	$2$	$2$	$27.149$	\(\Q(\sqrt{-1}) \)	None					136.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-q^{9}+2 q^{11}-6 i q^{13}+\cdots\)
3400.2.e.d	$3400$	$2$	3400.e	5.b	$2$	$2$	$2$	$27.149$	\(\Q(\sqrt{-1}) \)	None					680.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 i q^{7}+2 q^{9}+4 q^{11}+\cdots\)
3400.2.e.e	$3400$	$2$	3400.e	5.b	$2$	$2$	$2$	$27.149$	\(\Q(\sqrt{-1}) \)	None					680.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 q^{9}-2 i q^{13}-i q^{17}+4 q^{19}+\cdots\)
3400.2.e.f	$3400$	$2$	3400.e	5.b	$2$	$4$	$4$	$27.149$	\(\Q(i, \sqrt{5})\)	None					136.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
3400.2.e.g	$3400$	$2$	3400.e	5.b	$2$	$4$	$4$	$27.149$	\(\Q(\zeta_{12})\)	None					680.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_{2}+\beta_1)q^{3}+(-\beta_{2}-3\beta_1)q^{7}+\cdots\)
3400.2.e.h	$3400$	$2$	3400.e	5.b	$2$	$4$	$4$	$27.149$	\(\Q(\zeta_{8})\)	None					680.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{2} q^{3}+\beta_{2} q^{7}+q^{9}+(3\beta_{3}-2)q^{11}+\cdots\)
3400.2.e.i	$3400$	$2$	3400.e	5.b	$2$	$6$	$6$	$27.149$	6.0.3534400.1	None					680.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{4}+\beta _{5})q^{3}+(-\beta _{2}-\beta _{4})q^{7}+(-3+\cdots)q^{9}+\cdots\)
3400.2.e.j	$3400$	$2$	3400.e	5.b	$2$	$6$	$6$	$27.149$	6.0.16516096.2	None					680.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-2\beta _{3}-\beta _{5})q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots\)
3400.2.e.k	$3400$	$2$	3400.e	5.b	$2$	$6$	$6$	$27.149$	6.0.3356224.1	None					680.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(\beta _{4}-\beta _{5})q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots\)
3400.2.e.l	$3400$	$2$	3400.e	5.b	$2$	$6$	$6$	$27.149$	6.0.5089536.1	None		✓			3400.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(\beta _{2}+\beta _{4}-\beta _{5})q^{7}+(-1+\cdots)q^{9}+\cdots\)
3400.2.e.m	$3400$	$2$	3400.e	5.b	$2$	$6$	$6$	$27.149$	6.0.350464.1	None		✓			3400.2.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(-\beta _{3}+\beta _{5})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3400.2.e.n	$3400$	$2$	3400.e	5.b	$2$	$10$	$10$	$27.149$	10.0.\(\cdots\).1	None		✓			3400.2.a.r	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{2}+\beta _{9})q^{3}+(\beta _{2}+\beta _{4}-\beta _{5}-\beta _{9})q^{7}+\cdots\)
3400.2.e.o	$3400$	$2$	3400.e	5.b	$2$	$10$	$10$	$27.149$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			3400.2.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{4}-\beta _{8})q^{7}+(-2-\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3400, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(340, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(425, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(680, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(850, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1700, [\chi])\)\(^{\oplus 2}\)