Space of modular forms of level 1058, weight 4, and trivial character

• Label: 1058.4.a
• Level: $1058$
• Weight: $4$
• Character orbit: 1058.a
• Rep. character: $\chi_{1058}(1,\cdot)$
• Character field: $\Q$
• Dimension: $126$
• Newform subspaces: $23$
• Sturm bound: $552$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1058 = 2 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1058.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(552\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1058))\).

	Total	New	Old
Modular forms	438	126	312
Cusp forms	390	126	264
Eisenstein series	48	0	48

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

\(2\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(33\)
\(+\)	\(-\)	\(-\)	\(30\)
\(-\)	\(+\)	\(-\)	\(27\)
\(-\)	\(-\)	\(+\)	\(36\)
Plus space		\(+\)	\(69\)
Minus space		\(-\)	\(57\)

Trace form

126 * q + 8 * q^3 + 504 * q^4 + 10 * q^5 + 8 * q^6 - 8 * q^7 + 1042 * q^9 + 20 * q^10 + 34 * q^11 + 32 * q^12 + 112 * q^13 - 40 * q^14 - 4 * q^15 + 2016 * q^16 - 28 * q^17 + 128 * q^18 - 38 * q^19 + 40 * q^20 + 260 * q^21 + 188 * q^22 + 32 * q^24 + 3222 * q^25 - 192 * q^26 + 404 * q^27 - 32 * q^28 + 388 * q^29 - 480 * q^30 + 244 * q^31 - 948 * q^33 + 520 * q^34 - 504 * q^35 + 4168 * q^36 - 294 * q^37 + 436 * q^38 + 444 * q^39 + 80 * q^40 - 516 * q^41 + 64 * q^42 + 366 * q^43 + 136 * q^44 + 326 * q^45 - 1192 * q^47 + 128 * q^48 + 7494 * q^49 - 592 * q^50 + 988 * q^51 + 448 * q^52 - 690 * q^53 + 128 * q^54 - 1356 * q^55 - 160 * q^56 + 1312 * q^57 - 296 * q^58 - 1384 * q^59 - 16 * q^60 + 642 * q^61 + 840 * q^62 - 420 * q^63 + 8064 * q^64 + 2256 * q^65 + 96 * q^66 - 426 * q^67 - 112 * q^68 + 368 * q^70 + 800 * q^71 + 512 * q^72 + 1436 * q^73 - 220 * q^74 + 5616 * q^75 - 152 * q^76 + 872 * q^77 - 304 * q^78 - 1948 * q^79 + 160 * q^80 + 7134 * q^81 - 1264 * q^82 - 2146 * q^83 + 1040 * q^84 - 2028 * q^85 - 1380 * q^86 + 3204 * q^87 + 752 * q^88 + 1044 * q^89 + 4804 * q^90 + 1956 * q^91 - 3804 * q^93 - 1056 * q^94 - 2828 * q^95 + 128 * q^96 + 428 * q^97 - 832 * q^98 - 2586 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\) 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	23
1058.4.a.a	$1058$	$4$	1058.a	1.a	$1$	$1$	$1$	$62.424$	\(\Q\)	None	✓				46.4.a.a	$1$	$1$	\(-2\)	\(-1\)	\(10\)	\(12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+4q^{4}+10q^{5}+2q^{6}+\cdots\)
1058.4.a.b	$1058$	$4$	1058.a	1.a	$1$	$1$	$1$	$62.424$	\(\Q\)	None	✓				46.4.a.b	$1$	$0$	\(2\)	\(-9\)	\(20\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-9q^{3}+4q^{4}+20q^{5}-18q^{6}+\cdots\)
1058.4.a.c	$1058$	$4$	1058.a	1.a	$1$	$1$	$1$	$62.424$	\(\Q\)	None	✓	✓			1058.4.a.c	$1$	$0$	\(2\)	\(7\)	\(-18\)	\(30\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+7q^{3}+4q^{4}-18q^{5}+14q^{6}+\cdots\)
1058.4.a.d	$1058$	$4$	1058.a	1.a	$1$	$1$	$1$	$62.424$	\(\Q\)	None	✓	✓			1058.4.a.c	$1$	$0$	\(2\)	\(7\)	\(18\)	\(-30\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+7q^{3}+4q^{4}+18q^{5}+14q^{6}+\cdots\)
1058.4.a.e	$1058$	$4$	1058.a	1.a	$1$	$2$	$2$	$62.424$	\(\Q(\sqrt{3}) \)	None	✓	✓			1058.4.a.e	$2$	$1$	\(-4\)	\(-10\)	\(0\)	\(0\)		$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}-5q^{3}+4q^{4}-\beta q^{5}+10q^{6}+\cdots\)
1058.4.a.f	$1058$	$4$	1058.a	1.a	$1$	$2$	$2$	$62.424$	\(\Q(\sqrt{41}) \)	None	✓				46.4.a.c	$1$	$1$	\(-4\)	\(-1\)	\(-10\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1-3\beta )q^{3}+4q^{4}+(-4-2\beta )q^{5}+\cdots\)
1058.4.a.g	$1058$	$4$	1058.a	1.a	$1$	$2$	$2$	$62.424$	\(\Q(\sqrt{3}) \)	None	✓	✓			1058.4.a.g	$1$	$1$	\(-4\)	\(2\)	\(-12\)	\(18\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1+3\beta )q^{3}+4q^{4}+(-6-\beta )q^{5}+\cdots\)
1058.4.a.h	$1058$	$4$	1058.a	1.a	$1$	$2$	$2$	$62.424$	\(\Q(\sqrt{3}) \)	None	✓	✓			1058.4.a.g	$1$	$1$	\(-4\)	\(2\)	\(12\)	\(-18\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1+3\beta )q^{3}+4q^{4}+(6+\beta )q^{5}+\cdots\)
1058.4.a.i	$1058$	$4$	1058.a	1.a	$1$	$2$	$2$	$62.424$	\(\Q(\sqrt{26}) \)	None	✓	✓			1058.4.a.i	$2$	$0$	\(4\)	\(-8\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{3}+4q^{4}+\beta q^{5}-8q^{6}+\cdots\)
1058.4.a.j	$1058$	$4$	1058.a	1.a	$1$	$2$	$2$	$62.424$	\(\Q(\sqrt{73}) \)	None	✓				46.4.a.d	$1$	$0$	\(4\)	\(3\)	\(-10\)	\(-12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1+\beta )q^{3}+4q^{4}+(-6+2\beta )q^{5}+\cdots\)
1058.4.a.k	$1058$	$4$	1058.a	1.a	$1$	$3$	$3$	$62.424$	3.3.6584.1	None	✓	✓			1058.4.a.k	$1$	$1$	\(-6\)	\(-2\)	\(-9\)	\(30\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(-3+\cdots)q^{5}+\cdots\)
1058.4.a.l	$1058$	$4$	1058.a	1.a	$1$	$3$	$3$	$62.424$	3.3.6584.1	None	✓	✓			1058.4.a.k	$1$	$1$	\(-6\)	\(-2\)	\(9\)	\(-30\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(3+2\beta _{1}+\cdots)q^{5}+\cdots\)
1058.4.a.m	$1058$	$4$	1058.a	1.a	$1$	$3$	$3$	$62.424$	3.3.123880.1	None	✓	✓			1058.4.a.m	$1$	$0$	\(6\)	\(9\)	\(-4\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(3+\beta _{1})q^{3}+4q^{4}+(-1+\beta _{2})q^{5}+\cdots\)
1058.4.a.n	$1058$	$4$	1058.a	1.a	$1$	$3$	$3$	$62.424$	3.3.123880.1	None	✓	✓			1058.4.a.m	$1$	$0$	\(6\)	\(9\)	\(4\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(3+\beta _{1})q^{3}+4q^{4}+(1-\beta _{2})q^{5}+\cdots\)
1058.4.a.o	$1058$	$4$	1058.a	1.a	$1$	$4$	$4$	$62.424$	\(\Q(\sqrt{2}, \sqrt{13})\)	None	✓	✓			1058.4.a.o	$2$	$1$	\(8\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+\beta _{3}q^{3}+4q^{4}+(-5\beta _{1}-6\beta _{2}+\cdots)q^{5}+\cdots\)
1058.4.a.p	$1058$	$4$	1058.a	1.a	$1$	$6$	$6$	$62.424$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			1058.4.a.p	$2$	$0$	\(-12\)	\(12\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+(2+\beta _{2})q^{3}+4q^{4}+(4\beta _{3}+\beta _{5})q^{5}+\cdots\)
1058.4.a.q	$1058$	$4$	1058.a	1.a	$1$	$8$	$8$	$62.424$	8.8.\(\cdots\).1	None	✓	✓			1058.4.a.q	$2$	$1$	\(16\)	\(-24\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-3+\beta _{1}-\beta _{4})q^{3}+4q^{4}+\cdots\)
1058.4.a.r	$1058$	$4$	1058.a	1.a	$1$	$8$	$8$	$62.424$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			1058.4.a.r	$2$	$0$	\(16\)	\(6\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1-\beta _{3})q^{3}+4q^{4}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
1058.4.a.s	$1058$	$4$	1058.a	1.a	$1$	$12$	$12$	$62.424$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			1058.4.a.s	$2$	$0$	\(-24\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-\beta _{1}+\beta _{2})q^{3}+4q^{4}+(\beta _{7}+\cdots)q^{5}+\cdots\)
1058.4.a.t	$1058$	$4$	1058.a	1.a	$1$	$15$	$15$	$62.424$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓		✓		46.4.c.b	$1$	$1$	\(-30\)	\(1\)	\(0\)	\(-80\)		$+$	$-$	$-$	$23^{4}$	$\mathrm{SU}(2)$	\(q-2q^{2}+\beta _{4}q^{3}+4q^{4}-\beta _{7}q^{5}-2\beta _{4}q^{6}+\cdots\)
1058.4.a.u	$1058$	$4$	1058.a	1.a	$1$	$15$	$15$	$62.424$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓		✓		46.4.c.b	$1$	$0$	\(-30\)	\(1\)	\(0\)	\(80\)		$+$	$+$	$+$	$23^{4}$	$\mathrm{SU}(2)$	\(q-2q^{2}+\beta _{4}q^{3}+4q^{4}+\beta _{7}q^{5}-2\beta _{4}q^{6}+\cdots\)
1058.4.a.v	$1058$	$4$	1058.a	1.a	$1$	$15$	$15$	$62.424$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓		✓		46.4.c.a	$1$	$1$	\(30\)	\(3\)	\(-50\)	\(-84\)		$-$	$+$	$-$	$23^{4}$	$\mathrm{SU}(2)$	\(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+(-3+\beta _{10}+\cdots)q^{5}+\cdots\)
1058.4.a.w	$1058$	$4$	1058.a	1.a	$1$	$15$	$15$	$62.424$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓		✓		46.4.c.a	$1$	$0$	\(30\)	\(3\)	\(50\)	\(84\)		$-$	$-$	$+$	$23^{4}$	$\mathrm{SU}(2)$	\(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+(3-\beta _{10})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1058))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1058)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(529))\)\(^{\oplus 2}\)