Space of modular forms of level 2175, weight 2, and trivial character

• Label: 2175.2.a
• Level: $2175$
• Weight: $2$
• Character orbit: 2175.a
• Rep. character: $\chi_{2175}(1,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $30$
• Sturm bound: $600$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(600\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2175))\).

	Total	New	Old
Modular forms	312	88	224
Cusp forms	289	88	201
Eisenstein series	23	0	23

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

\(3\)	\(5\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	\(16\)
\(+\)	\(-\)	\(+\)	\(-\)	\(14\)
\(+\)	\(-\)	\(-\)	\(+\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(14\)
\(-\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)	\(15\)
Plus space			\(+\)	\(29\)
Minus space			\(-\)	\(59\)

Trace form

88 * q - 2 * q^3 + 90 * q^4 + 4 * q^6 + 12 * q^8 + 88 * q^9 + 8 * q^11 + 2 * q^12 - 12 * q^13 + 36 * q^14 + 118 * q^16 + 12 * q^19 - 12 * q^21 + 30 * q^22 + 12 * q^23 + 6 * q^24 + 44 * q^26 - 2 * q^27 + 10 * q^28 + 4 * q^31 + 24 * q^32 - 2 * q^34 + 90 * q^36 - 8 * q^37 - 16 * q^38 - 8 * q^39 + 8 * q^41 + 14 * q^42 + 20 * q^44 + 44 * q^46 + 24 * q^47 + 2 * q^48 + 96 * q^49 + 16 * q^51 - 34 * q^52 - 4 * q^53 + 4 * q^54 + 96 * q^56 - 12 * q^57 - 2 * q^58 + 40 * q^59 + 28 * q^61 + 4 * q^62 + 132 * q^64 + 16 * q^66 + 40 * q^67 + 20 * q^68 + 32 * q^69 + 84 * q^71 + 12 * q^72 + 4 * q^74 + 56 * q^76 + 56 * q^77 + 10 * q^78 - 8 * q^79 + 88 * q^81 + 28 * q^82 + 56 * q^83 - 64 * q^84 + 84 * q^86 - 6 * q^87 + 76 * q^88 + 32 * q^89 + 4 * q^91 + 4 * q^92 + 4 * q^93 - 74 * q^94 - 24 * q^96 - 16 * q^97 + 44 * q^98 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2175))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	29
2175.2.a.a	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓				435.2.c.a	$1$	$0$	\(-2\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{7}+\cdots\)
2175.2.a.b	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓				435.2.a.d	$1$	$2$	\(-1\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}-4q^{7}+3q^{8}+\cdots\)
2175.2.a.c	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓				435.2.c.b	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+2q^{7}+3q^{8}+\cdots\)
2175.2.a.d	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓				435.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-2q^{7}+q^{9}+3q^{11}+\cdots\)
2175.2.a.e	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓	✓			2175.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{7}+q^{9}-2q^{11}+2q^{12}+\cdots\)
2175.2.a.f	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓	✓			2175.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{7}+q^{9}-2q^{11}-2q^{12}+\cdots\)
2175.2.a.g	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓				435.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+2q^{7}+q^{9}+q^{11}-2q^{12}+\cdots\)
2175.2.a.h	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓				435.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}+4q^{7}-3q^{8}+\cdots\)
2175.2.a.i	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓				435.2.c.b	$1$	$0$	\(1\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-2q^{7}-3q^{8}+\cdots\)
2175.2.a.j	$2175$	$2$	2175.a	1.a	$1$	$1$	$1$	$17.367$	\(\Q\)	None	✓				435.2.c.a	$1$	$1$	\(2\)	\(-1\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+2q^{7}+\cdots\)
2175.2.a.k	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{2}) \)	None	✓				435.2.c.c	$1$	$1$	\(-2\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}-\beta q^{7}+3q^{8}+\cdots\)
2175.2.a.l	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{5}) \)	None	✓				87.2.a.a	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
2175.2.a.m	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{17}) \)	None	✓				435.2.a.h	$1$	$0$	\(-1\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+\beta q^{6}+(-2+\cdots)q^{7}+\cdots\)
2175.2.a.n	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{2}) \)	None	✓	✓			2175.2.a.n	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}-\beta q^{6}+q^{7}-2\beta q^{8}+\cdots\)
2175.2.a.o	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{5}) \)	None	✓				435.2.a.g	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+3q^{4}+\beta q^{6}-2q^{7}+\cdots\)
2175.2.a.p	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{2}) \)	None	✓	✓			2175.2.a.n	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+\beta q^{6}-q^{7}-2\beta q^{8}+\cdots\)
2175.2.a.q	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{5}) \)	None	✓				435.2.a.e	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
2175.2.a.r	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{21}) \)	None	✓				435.2.a.f	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(3+\beta )q^{4}-\beta q^{6}-q^{7}+\cdots\)
2175.2.a.s	$2175$	$2$	2175.a	1.a	$1$	$2$	$2$	$17.367$	\(\Q(\sqrt{2}) \)	None	✓				435.2.c.c	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}+\beta q^{7}-3q^{8}+\cdots\)
2175.2.a.t	$2175$	$2$	2175.a	1.a	$1$	$3$	$3$	$17.367$	3.3.229.1	None	✓		✓		87.2.a.b	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+q^{3}+(2+\beta _{1})q^{4}+\cdots\)
2175.2.a.u	$2175$	$2$	2175.a	1.a	$1$	$3$	$3$	$17.367$	3.3.469.1	None	✓				435.2.a.i	$1$	$0$	\(-1\)	\(3\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
2175.2.a.v	$2175$	$2$	2175.a	1.a	$1$	$4$	$4$	$17.367$	4.4.2225.1	None	✓				435.2.a.j	$1$	$0$	\(3\)	\(4\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.2.a.w	$2175$	$2$	2175.a	1.a	$1$	$5$	$5$	$17.367$	5.5.246832.1	None	✓		✓		435.2.c.e	$1$	$1$	\(-3\)	\(-5\)	\(0\)	\(-8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
2175.2.a.x	$2175$	$2$	2175.a	1.a	$1$	$5$	$5$	$17.367$	5.5.331312.1	None	✓		✓		435.2.c.d	$1$	$1$	\(-2\)	\(5\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+q^{3}+(1+\beta _{1}-\beta _{3}-\beta _{4})q^{4}+\cdots\)
2175.2.a.y	$2175$	$2$	2175.a	1.a	$1$	$5$	$5$	$17.367$	5.5.331312.1	None	✓				435.2.c.d	$1$	$0$	\(2\)	\(-5\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}-q^{3}+(1+\beta _{1}-\beta _{3}-\beta _{4})q^{4}+\cdots\)
2175.2.a.z	$2175$	$2$	2175.a	1.a	$1$	$5$	$5$	$17.367$	5.5.246832.1	None	✓				435.2.c.e	$1$	$0$	\(3\)	\(5\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2175.2.a.ba	$2175$	$2$	2175.a	1.a	$1$	$7$	$7$	$17.367$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		2175.2.a.ba	$1$	$0$	\(-2\)	\(-7\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
2175.2.a.bb	$2175$	$2$	2175.a	1.a	$1$	$7$	$7$	$17.367$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		2175.2.a.ba	$1$	$0$	\(2\)	\(7\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
2175.2.a.bc	$2175$	$2$	2175.a	1.a	$1$	$8$	$8$	$17.367$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		2175.2.a.bc	$1$	$0$	\(-2\)	\(-8\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
2175.2.a.bd	$2175$	$2$	2175.a	1.a	$1$	$8$	$8$	$17.367$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		2175.2.a.bc	$1$	$0$	\(2\)	\(8\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2175))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2175)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(725))\)\(^{\oplus 2}\)