Space of modular forms of level 3800, weight 2, and trivial character

• Label: 3800.2.a
• Level: $3800$
• Weight: $2$
• Character orbit: 3800.a
• Rep. character: $\chi_{3800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $85$
• Newform subspaces: $31$
• Sturm bound: $1200$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 31 \)
Sturm bound:			\(1200\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3800))\).

	Total	New	Old
Modular forms	624	85	539
Cusp forms	577	85	492
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.

\(2\)	\(5\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(9\)
\(+\)	\(+\)	\(-\)	\(-\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)	\(11\)
\(+\)	\(-\)	\(-\)	\(+\)	\(11\)
\(-\)	\(+\)	\(+\)	\(-\)	\(13\)
\(-\)	\(+\)	\(-\)	\(+\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)	\(14\)
Plus space			\(+\)	\(35\)
Minus space			\(-\)	\(50\)

Trace form

85 * q - 4 * q^7 + 79 * q^9 + 6 * q^11 - 2 * q^13 + 6 * q^17 + 3 * q^19 + 16 * q^21 + 6 * q^23 + 30 * q^29 - 12 * q^31 + 20 * q^33 + 14 * q^37 + 22 * q^39 + 14 * q^41 + 2 * q^43 + 6 * q^47 + 81 * q^49 + 20 * q^51 + 14 * q^53 + 2 * q^57 - 8 * q^59 + 16 * q^61 + 6 * q^63 - 4 * q^67 - 4 * q^69 + 40 * q^71 + 18 * q^73 - 14 * q^77 - 36 * q^79 + 101 * q^81 - 20 * q^83 - 18 * q^87 + 30 * q^89 + 12 * q^93 - 6 * q^97 - 14 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\) 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	19
3800.2.a.a	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				760.2.a.e	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{7}+6q^{9}+4q^{11}-q^{13}+\cdots\)
3800.2.a.b	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				760.2.a.d	$1$	$2$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}-4q^{11}-6q^{17}+\cdots\)
3800.2.a.c	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				760.2.d.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}+4q^{11}+8q^{17}+\cdots\)
3800.2.a.d	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				152.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}-2q^{9}+2q^{11}-q^{13}+\cdots\)
3800.2.a.e	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				760.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-4q^{11}+6q^{13}+6q^{17}-q^{19}+\cdots\)
3800.2.a.f	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				760.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
3800.2.a.g	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				760.2.d.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}+4q^{11}-8q^{17}+\cdots\)
3800.2.a.h	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				760.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-4q^{11}-4q^{13}+2q^{17}+\cdots\)
3800.2.a.i	$3800$	$2$	3800.a	1.a	$1$	$1$	$1$	$30.343$	\(\Q\)	None	✓				152.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{7}+q^{9}-3q^{11}+4q^{13}+\cdots\)
3800.2.a.j	$3800$	$2$	3800.a	1.a	$1$	$2$	$2$	$30.343$	\(\Q(\sqrt{3}) \)	None	✓				760.2.a.g	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1-2\beta )q^{9}+2q^{11}+\cdots\)
3800.2.a.k	$3800$	$2$	3800.a	1.a	$1$	$2$	$2$	$30.343$	\(\Q(\sqrt{5}) \)	None	✓				760.2.d.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+2\beta q^{7}+(3+2\beta )q^{9}+\cdots\)
3800.2.a.l	$3800$	$2$	3800.a	1.a	$1$	$2$	$2$	$30.343$	\(\Q(\sqrt{2}) \)	None	✓				760.2.d.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1-\beta )q^{7}-2\beta q^{9}+\cdots\)
3800.2.a.m	$3800$	$2$	3800.a	1.a	$1$	$2$	$2$	$30.343$	\(\Q(\sqrt{2}) \)	None	✓				760.2.d.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-2+2\beta )q^{7}-q^{9}+(2-2\beta )q^{11}+\cdots\)
3800.2.a.n	$3800$	$2$	3800.a	1.a	$1$	$2$	$2$	$30.343$	\(\Q(\sqrt{2}) \)	None	✓				760.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{7}-q^{9}+(2-2\beta )q^{11}+\cdots\)
3800.2.a.o	$3800$	$2$	3800.a	1.a	$1$	$2$	$2$	$30.343$	\(\Q(\sqrt{2}) \)	None	✓				760.2.d.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2+2\beta )q^{7}-q^{9}+(2+2\beta )q^{11}+\cdots\)
3800.2.a.p	$3800$	$2$	3800.a	1.a	$1$	$2$	$2$	$30.343$	\(\Q(\sqrt{2}) \)	None	✓				760.2.d.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1-\beta )q^{7}+2\beta q^{9}+\cdots\)
3800.2.a.q	$3800$	$2$	3800.a	1.a	$1$	$2$	$2$	$30.343$	\(\Q(\sqrt{5}) \)	None	✓				760.2.d.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2\beta q^{7}+(3+2\beta )q^{9}+(-1+\cdots)q^{13}+\cdots\)
3800.2.a.r	$3800$	$2$	3800.a	1.a	$1$	$3$	$3$	$30.343$	3.3.961.1	None	✓				152.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-2+\beta _{1}-\beta _{2})q^{7}+(5-\beta _{1}+\cdots)q^{9}+\cdots\)
3800.2.a.s	$3800$	$2$	3800.a	1.a	$1$	$3$	$3$	$30.343$	\(\Q(\zeta_{18})^+\)	None	✓	✓			3800.2.a.s	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}-2\beta _{2})q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3800.2.a.t	$3800$	$2$	3800.a	1.a	$1$	$3$	$3$	$30.343$	\(\Q(\zeta_{18})^+\)	None	✓	✓	✓		3800.2.a.s	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}+2\beta _{2})q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3800.2.a.u	$3800$	$2$	3800.a	1.a	$1$	$3$	$3$	$30.343$	3.3.257.1	None	✓	✓			3800.2.a.u	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{2}q^{7}+(\beta _{1}-\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
3800.2.a.v	$3800$	$2$	3800.a	1.a	$1$	$3$	$3$	$30.343$	3.3.257.1	None	✓	✓	✓		3800.2.a.u	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{2}q^{7}+(\beta _{1}-\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
3800.2.a.w	$3800$	$2$	3800.a	1.a	$1$	$3$	$3$	$30.343$	3.3.316.1	None	✓				760.2.a.i	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-2\beta _{1}+\beta _{2})q^{7}+\beta _{2}q^{9}+\cdots\)
3800.2.a.x	$3800$	$2$	3800.a	1.a	$1$	$3$	$3$	$30.343$	3.3.229.1	None	✓				760.2.a.j	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-\beta _{1}q^{7}+(1-\beta _{1})q^{9}+(-2+\cdots)q^{13}+\cdots\)
3800.2.a.y	$3800$	$2$	3800.a	1.a	$1$	$3$	$3$	$30.343$	3.3.568.1	None	✓				760.2.a.h	$1$	$0$	\(0\)	\(1\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+\beta _{2})q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
3800.2.a.z	$3800$	$2$	3800.a	1.a	$1$	$6$	$6$	$30.343$	6.6.253565184.1	None	✓		✓		760.2.d.e	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{1}-\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
3800.2.a.ba	$3800$	$2$	3800.a	1.a	$1$	$6$	$6$	$30.343$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		3800.2.a.ba	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{4})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
3800.2.a.bb	$3800$	$2$	3800.a	1.a	$1$	$6$	$6$	$30.343$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			3800.2.a.bb	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3800.2.a.bc	$3800$	$2$	3800.a	1.a	$1$	$6$	$6$	$30.343$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		3800.2.a.ba	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{4})q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
3800.2.a.bd	$3800$	$2$	3800.a	1.a	$1$	$6$	$6$	$30.343$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		3800.2.a.bb	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3800.2.a.be	$3800$	$2$	3800.a	1.a	$1$	$6$	$6$	$30.343$	6.6.253565184.1	None	✓				760.2.d.e	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3800)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(760))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(950))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\)\(^{\oplus 2}\)