Space of modular forms of level 3800, weight 2, and Character 3800.d

• Label: 3800.2.d
• Level: $3800$
• Weight: $2$
• Character orbit: 3800.d
• Rep. character: $\chi_{3800}(3649,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $17$
• Sturm bound: $1200$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(1200\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3800, [\chi])\).

	Total	New	Old
Modular forms	624	82	542
Cusp forms	576	82	494
Eisenstein series	48	0	48

Trace form

82 * q - 78 * q^9 - 12 * q^11 + 6 * q^19 + 16 * q^21 - 28 * q^29 - 8 * q^31 - 32 * q^39 - 4 * q^41 - 42 * q^49 + 72 * q^51 - 32 * q^61 + 8 * q^69 + 48 * q^71 - 16 * q^79 + 98 * q^81 - 12 * q^89 + 32 * q^91 - 52 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(3800, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3800.2.d.a	$3800$	$2$	3800.d	5.b	$2$	$2$	$2$	$30.343$	\(\Q(\sqrt{-1}) \)	None					760.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+i q^{7}-6 q^{9}+4 q^{11}+\cdots\)
3800.2.d.b	$3800$	$2$	3800.d	5.b	$2$	$2$	$2$	$30.343$	\(\Q(\sqrt{-1}) \)	None					760.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-q^{9}-4 q^{11}-2\beta q^{13}-\beta q^{17}+\cdots\)
3800.2.d.c	$3800$	$2$	3800.d	5.b	$2$	$2$	$2$	$30.343$	\(\Q(\sqrt{-1}) \)	None					760.2.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-2\beta q^{7}-q^{9}-4 q^{11}-3\beta q^{17}+\cdots\)
3800.2.d.d	$3800$	$2$	3800.d	5.b	$2$	$2$	$2$	$30.343$	\(\Q(\sqrt{-1}) \)	None					152.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-3 i q^{7}-q^{9}-3 q^{11}+\cdots\)
3800.2.d.e	$3800$	$2$	3800.d	5.b	$2$	$2$	$2$	$30.343$	\(\Q(\sqrt{-1}) \)	None					760.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+2\beta q^{7}-q^{9}+4 q^{11}+2\beta q^{13}+\cdots\)
3800.2.d.f	$3800$	$2$	3800.d	5.b	$2$	$2$	$2$	$30.343$	\(\Q(\sqrt{-1}) \)	None					152.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-3 i q^{7}+2 q^{9}+2 q^{11}+\cdots\)
3800.2.d.g	$3800$	$2$	3800.d	5.b	$2$	$2$	$2$	$30.343$	\(\Q(\sqrt{-1}) \)	None					760.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 q^{9}-4 q^{11}-3\beta q^{13}+3\beta q^{17}+\cdots\)
3800.2.d.h	$3800$	$2$	3800.d	5.b	$2$	$4$	$4$	$30.343$	\(\Q(\zeta_{12})\)	None					760.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{2} q^{3}+(\beta_{3}-1)q^{9}+2 q^{11}+\beta_1 q^{13}+\cdots\)
3800.2.d.i	$3800$	$2$	3800.d	5.b	$2$	$4$	$4$	$30.343$	\(\Q(\zeta_{8})\)	None					760.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{2} q^{3}+2\beta_{2} q^{7}+q^{9}+(-\beta_{3}+2)q^{11}+\cdots\)
3800.2.d.j	$3800$	$2$	3800.d	5.b	$2$	$6$	$6$	$30.343$	6.0.59105344.1	None					152.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(\beta _{1}+\beta _{2}-\beta _{4})q^{7}+(-3+\cdots)q^{9}+\cdots\)
3800.2.d.k	$3800$	$2$	3800.d	5.b	$2$	$6$	$6$	$30.343$	6.0.5161984.1	None					760.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(\beta _{3}-\beta _{5})q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots\)
3800.2.d.l	$3800$	$2$	3800.d	5.b	$2$	$6$	$6$	$30.343$	6.0.3356224.1	None					760.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+\beta _{5}q^{7}+(-1+\beta _{1})q^{9}+(\beta _{2}+\cdots)q^{13}+\cdots\)
3800.2.d.m	$3800$	$2$	3800.d	5.b	$2$	$6$	$6$	$30.343$	6.0.4227136.2	None		✓			3800.2.a.u	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+\beta _{3}q^{7}+(-\beta _{2}-\beta _{5})q^{9}+\cdots\)
3800.2.d.n	$3800$	$2$	3800.d	5.b	$2$	$6$	$6$	$30.343$	6.0.399424.1	None					760.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(\beta _{4}-\beta _{5})q^{7}-\beta _{1}q^{9}+2\beta _{1}q^{11}+\cdots\)
3800.2.d.o	$3800$	$2$	3800.d	5.b	$2$	$6$	$6$	$30.343$	6.0.419904.1	None		✓			3800.2.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+2\beta _{5})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3800.2.d.p	$3800$	$2$	3800.d	5.b	$2$	$12$	$12$	$30.343$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			3800.2.a.bb	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{9}q^{7}+(-1+\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
3800.2.d.q	$3800$	$2$	3800.d	5.b	$2$	$12$	$12$	$30.343$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			3800.2.a.ba	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{8})q^{7}+(-1-\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3800, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3800, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(95, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(190, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(380, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(475, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(760, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(950, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1900, [\chi])\)\(^{\oplus 2}\)