Space of modular forms of level 4508, weight 2, and trivial character

• Label: 4508.2.a
• Level: $4508$
• Weight: $2$
• Character orbit: 4508.a
• Rep. character: $\chi_{4508}(1,\cdot)$
• Character field: $\Q$
• Dimension: $74$
• Newform subspaces: $15$
• Sturm bound: $1344$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 4508 = 2^{2} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4508.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(1344\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4508))\).

	Total	New	Old
Modular forms	696	74	622
Cusp forms	649	74	575
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.

\(2\)	\(7\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(76\)	\(0\)	\(76\)		\(69\)	\(0\)	\(69\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(100\)	\(0\)	\(100\)		\(92\)	\(0\)	\(92\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(100\)	\(0\)	\(100\)		\(92\)	\(0\)	\(92\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(76\)	\(0\)	\(76\)		\(68\)	\(0\)	\(68\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(80\)	\(17\)	\(63\)		\(76\)	\(17\)	\(59\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(92\)	\(17\)	\(75\)		\(88\)	\(17\)	\(71\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(92\)	\(20\)	\(72\)		\(88\)	\(20\)	\(68\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(80\)	\(20\)	\(60\)		\(76\)	\(20\)	\(56\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(336\)	\(37\)	\(299\)		\(313\)	\(37\)	\(276\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(360\)	\(37\)	\(323\)		\(336\)	\(37\)	\(299\)		\(24\)	\(0\)	\(24\)

Trace form

74 * q - 2 * q^3 - 2 * q^5 + 64 * q^9 - 6 * q^11 - 6 * q^13 - 6 * q^15 - 6 * q^17 - 12 * q^19 + 66 * q^25 - 2 * q^27 - 14 * q^29 + 6 * q^31 - 18 * q^33 + 14 * q^37 + 14 * q^39 - 6 * q^41 + 28 * q^43 - 12 * q^45 - 2 * q^47 + 14 * q^51 - 36 * q^53 + 24 * q^55 + 48 * q^57 - 12 * q^59 - 4 * q^61 - 50 * q^65 + 2 * q^67 + 4 * q^69 + 14 * q^71 + 6 * q^73 + 10 * q^75 + 28 * q^79 + 50 * q^81 + 18 * q^83 - 60 * q^85 + 30 * q^87 - 12 * q^89 + 18 * q^93 + 12 * q^95 - 34 * q^97 - 116 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4508))\) 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	7	23
4508.2.a.a	$4508$	$2$	4508.a	1.a	$1$	$1$	$1$	$35.997$	\(\Q\)	None	✓				92.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+q^{13}+6q^{17}-2q^{19}+\cdots\)
4508.2.a.b	$4508$	$2$	4508.a	1.a	$1$	$1$	$1$	$35.997$	\(\Q\)	None	✓				644.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-2q^{9}-2q^{11}+q^{13}+\cdots\)
4508.2.a.c	$4508$	$2$	4508.a	1.a	$1$	$1$	$1$	$35.997$	\(\Q\)	None	✓				644.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-2q^{11}+3q^{13}-q^{23}+\cdots\)
4508.2.a.d	$4508$	$2$	4508.a	1.a	$1$	$1$	$1$	$35.997$	\(\Q\)	None	✓				92.2.a.a	$1$	$0$	\(0\)	\(3\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}+6q^{9}+2q^{11}+5q^{13}+\cdots\)
4508.2.a.e	$4508$	$2$	4508.a	1.a	$1$	$4$	$4$	$35.997$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4508.2.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{2})q^{9}-2\beta _{2}q^{11}+\cdots\)
4508.2.a.f	$4508$	$2$	4508.a	1.a	$1$	$5$	$5$	$35.997$	5.5.6963152.1	None	✓				644.2.a.d	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
4508.2.a.g	$4508$	$2$	4508.a	1.a	$1$	$5$	$5$	$35.997$	5.5.8580816.1	None	✓				644.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{4})q^{5}+(2+\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
4508.2.a.h	$4508$	$2$	4508.a	1.a	$1$	$6$	$6$	$35.997$	6.6.217653248.1	None	✓	✓			4508.2.a.h	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{5})q^{5}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
4508.2.a.i	$4508$	$2$	4508.a	1.a	$1$	$6$	$6$	$35.997$	6.6.524361728.1	None	✓	✓			4508.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{5}+(1+\beta _{2})q^{9}+\cdots\)
4508.2.a.j	$4508$	$2$	4508.a	1.a	$1$	$6$	$6$	$35.997$	6.6.73156608.1	None	✓	✓			4508.2.a.j	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{5})q^{5}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
4508.2.a.k	$4508$	$2$	4508.a	1.a	$1$	$7$	$7$	$35.997$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		644.2.i.b	$1$	$1$	\(0\)	\(-5\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+\beta _{5}q^{5}+(1-\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
4508.2.a.l	$4508$	$2$	4508.a	1.a	$1$	$7$	$7$	$35.997$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		644.2.i.a	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}-\beta _{5}q^{5}+(1+\beta _{1}-\beta _{4}+\beta _{5}+\cdots)q^{9}+\cdots\)
4508.2.a.m	$4508$	$2$	4508.a	1.a	$1$	$7$	$7$	$35.997$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				644.2.i.a	$1$	$0$	\(0\)	\(3\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+\beta _{5}q^{5}+(1+\beta _{1}-\beta _{4}+\beta _{5}+\cdots)q^{9}+\cdots\)
4508.2.a.n	$4508$	$2$	4508.a	1.a	$1$	$7$	$7$	$35.997$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		644.2.i.b	$1$	$0$	\(0\)	\(5\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-\beta _{5}q^{5}+(1-\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
4508.2.a.o	$4508$	$2$	4508.a	1.a	$1$	$10$	$10$	$35.997$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		4508.2.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{4}q^{5}+\beta _{2}q^{9}+(\beta _{5}+\beta _{6}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4508))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4508)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(322))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(644))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1127))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2254))\)\(^{\oplus 2}\)