Space of modular forms of level 4600, weight 2, and trivial character

• Label: 4600.2.a
• Level: $4600$
• Weight: $2$
• Character orbit: 4600.a
• Rep. character: $\chi_{4600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $104$
• Newform subspaces: $37$
• Sturm bound: $1440$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 37 \)
Sturm bound:			\(1440\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4600))\).

	Total	New	Old
Modular forms	744	104	640
Cusp forms	697	104	593
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.

\(2\)	\(5\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(81\)	\(13\)	\(68\)		\(76\)	\(13\)	\(63\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(105\)	\(14\)	\(91\)		\(99\)	\(14\)	\(85\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(103\)	\(14\)	\(89\)		\(97\)	\(14\)	\(83\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(83\)	\(12\)	\(71\)		\(77\)	\(12\)	\(65\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(87\)	\(12\)	\(75\)		\(81\)	\(12\)	\(69\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(99\)	\(11\)	\(88\)		\(93\)	\(11\)	\(82\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(101\)	\(13\)	\(88\)		\(95\)	\(13\)	\(82\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(85\)	\(15\)	\(70\)		\(79\)	\(15\)	\(64\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(364\)	\(49\)	\(315\)		\(341\)	\(49\)	\(292\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(380\)	\(55\)	\(325\)		\(356\)	\(55\)	\(301\)		\(24\)	\(0\)	\(24\)

Trace form

104 * q - 8 * q^7 + 96 * q^9 + 2 * q^11 - 4 * q^13 + 2 * q^19 - 4 * q^21 - 12 * q^27 + 16 * q^29 - 20 * q^31 - 12 * q^33 + 10 * q^37 + 20 * q^39 + 4 * q^41 - 10 * q^43 - 4 * q^47 + 112 * q^49 + 40 * q^51 - 18 * q^53 - 32 * q^57 + 48 * q^59 + 14 * q^61 + 20 * q^63 + 6 * q^67 - 4 * q^69 + 28 * q^71 - 16 * q^73 - 8 * q^77 - 68 * q^79 + 88 * q^81 + 14 * q^83 + 20 * q^87 + 12 * q^89 - 60 * q^91 + 4 * q^93 - 16 * q^97 + 74 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4600))\) 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	23
4600.2.a.a	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				184.2.a.d	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{7}+6q^{9}+5q^{13}+6q^{17}+\cdots\)
4600.2.a.b	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				920.2.e.a	$1$	$2$	\(0\)	\(-2\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-3q^{7}+q^{9}-6q^{13}-7q^{17}+\cdots\)
4600.2.a.c	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓	✓			4600.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}-5q^{11}+q^{13}+\cdots\)
4600.2.a.d	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓	✓			4600.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{7}+q^{9}+5q^{11}+5q^{13}+\cdots\)
4600.2.a.e	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓	✓			4600.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
4600.2.a.f	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				920.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}-q^{13}+4q^{17}+\cdots\)
4600.2.a.g	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				184.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-3q^{9}+6q^{11}+2q^{13}-6q^{17}+\cdots\)
4600.2.a.h	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				920.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-3q^{9}-6q^{11}+2q^{13}+3q^{17}+\cdots\)
4600.2.a.i	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				184.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}-4q^{11}+5q^{13}+\cdots\)
4600.2.a.j	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				920.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+2q^{11}+5q^{13}+4q^{17}+\cdots\)
4600.2.a.k	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				184.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}-2q^{11}-7q^{13}+\cdots\)
4600.2.a.l	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓	✓			4600.2.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
4600.2.a.m	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓	✓			4600.2.a.d	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{7}+q^{9}+5q^{11}-5q^{13}+\cdots\)
4600.2.a.n	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓	✓			4600.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{7}+q^{9}-5q^{11}-q^{13}+\cdots\)
4600.2.a.o	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				920.2.e.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{7}+q^{9}+6q^{13}+7q^{17}+\cdots\)
4600.2.a.p	$4600$	$2$	4600.a	1.a	$1$	$1$	$1$	$36.731$	\(\Q\)	None	✓				920.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{7}+6q^{9}-q^{13}+6q^{21}+\cdots\)
4600.2.a.q	$4600$	$2$	4600.a	1.a	$1$	$2$	$2$	$36.731$	\(\Q(\sqrt{5}) \)	None	✓	✓			4600.2.a.q	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(2+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
4600.2.a.r	$4600$	$2$	4600.a	1.a	$1$	$2$	$2$	$36.731$	\(\Q(\sqrt{17}) \)	None	✓				920.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-2\beta q^{7}+(1+\beta )q^{9}-4q^{11}+\cdots\)
4600.2.a.s	$4600$	$2$	4600.a	1.a	$1$	$2$	$2$	$36.731$	\(\Q(\sqrt{17}) \)	None	✓				184.2.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+\beta )q^{9}+2\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
4600.2.a.t	$4600$	$2$	4600.a	1.a	$1$	$2$	$2$	$36.731$	\(\Q(\sqrt{17}) \)	None	✓				920.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{7}+(1+\beta )q^{9}-2q^{11}+\cdots\)
4600.2.a.u	$4600$	$2$	4600.a	1.a	$1$	$2$	$2$	$36.731$	\(\Q(\sqrt{5}) \)	None	✓	✓			4600.2.a.q	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
4600.2.a.v	$4600$	$2$	4600.a	1.a	$1$	$3$	$3$	$36.731$	3.3.229.1	None	✓				920.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-2+\beta _{2})q^{7}+(1+\cdots)q^{9}+\cdots\)
4600.2.a.w	$4600$	$2$	4600.a	1.a	$1$	$3$	$3$	$36.731$	\(\Q(\zeta_{14})^+\)	None	✓	✓			4600.2.a.w	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-2-2\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots\)
4600.2.a.x	$4600$	$2$	4600.a	1.a	$1$	$3$	$3$	$36.731$	3.3.2597.1	None	✓				920.2.a.h	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1})q^{7}+(3+\beta _{2})q^{9}+\cdots\)
4600.2.a.y	$4600$	$2$	4600.a	1.a	$1$	$3$	$3$	$36.731$	3.3.621.1	None	✓				920.2.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{1})q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
4600.2.a.z	$4600$	$2$	4600.a	1.a	$1$	$3$	$3$	$36.731$	\(\Q(\zeta_{14})^+\)	None	✓	✓			4600.2.a.w	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2-2\beta _{1})q^{7}+(-2\beta _{1}+\cdots)q^{9}+\cdots\)
4600.2.a.ba	$4600$	$2$	4600.a	1.a	$1$	$4$	$4$	$36.731$	4.4.15529.1	None	✓	✓			4600.2.a.ba	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(-\beta _{1}+\beta _{2})q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
4600.2.a.bb	$4600$	$2$	4600.a	1.a	$1$	$4$	$4$	$36.731$	4.4.15529.1	None	✓	✓	✓		4600.2.a.ba	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(\beta _{1}-\beta _{2})q^{7}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
4600.2.a.bc	$4600$	$2$	4600.a	1.a	$1$	$5$	$5$	$36.731$	5.5.791953.1	None	✓	✓			4600.2.a.bc	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{2}+\beta _{4})q^{7}+\cdots\)
4600.2.a.bd	$4600$	$2$	4600.a	1.a	$1$	$5$	$5$	$36.731$	5.5.521397.1	None	✓	✓			4600.2.a.bd	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
4600.2.a.be	$4600$	$2$	4600.a	1.a	$1$	$5$	$5$	$36.731$	5.5.13955077.1	None	✓				920.2.a.j	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{3}+\beta _{4})q^{7}+(2+\beta _{2}+\beta _{4})q^{9}+\cdots\)
4600.2.a.bf	$4600$	$2$	4600.a	1.a	$1$	$5$	$5$	$36.731$	5.5.521397.1	None	✓	✓	✓		4600.2.a.bd	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
4600.2.a.bg	$4600$	$2$	4600.a	1.a	$1$	$5$	$5$	$36.731$	5.5.791953.1	None	✓	✓			4600.2.a.bc	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{2}-\beta _{4})q^{7}+\cdots\)
4600.2.a.bh	$4600$	$2$	4600.a	1.a	$1$	$7$	$7$	$36.731$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		920.2.e.b	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{3})q^{7}+(\beta _{2}+\beta _{5}+\cdots)q^{9}+\cdots\)
4600.2.a.bi	$4600$	$2$	4600.a	1.a	$1$	$7$	$7$	$36.731$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		920.2.e.b	$1$	$1$	\(0\)	\(3\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{3})q^{7}+(\beta _{2}+\beta _{5}+\beta _{6})q^{9}+\cdots\)
4600.2.a.bj	$4600$	$2$	4600.a	1.a	$1$	$8$	$8$	$36.731$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		920.2.e.c	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-7\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{7})q^{7}+(2+\beta _{3}+\beta _{5}+\cdots)q^{9}+\cdots\)
4600.2.a.bk	$4600$	$2$	4600.a	1.a	$1$	$8$	$8$	$36.731$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		920.2.e.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(7\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{7})q^{7}+(2+\beta _{3}+\beta _{5}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4600)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(575))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(920))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2300))\)\(^{\oplus 2}\)