Space of modular forms of level 245, weight 4, and trivial character

• Label: 245.4.a
• Level: $245$
• Weight: $4$
• Character orbit: 245.a
• Rep. character: $\chi_{245}(1,\cdot)$
• Character field: $\Q$
• Dimension: $41$
• Newform subspaces: $16$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$245 = 5 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	245.a (trivial)
Character field:			$\Q$
Newform subspaces:			$16$
Sturm bound:			$112$
Trace bound:			$3$
Distinguishing $T_p$:			$2$, $3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(245))$.

	Total	New	Old
Modular forms	92	41	51
Cusp forms	76	41	35
Eisenstein series	16	0	16

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

$5$	$7$	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$26$	$11$	$15$		$22$	$11$	$11$		$4$	$0$	$4$
$+$	$-$	$-$		$21$	$9$	$12$		$17$	$9$	$8$		$4$	$0$	$4$
$-$	$+$	$-$		$22$	$9$	$13$		$18$	$9$	$9$		$4$	$0$	$4$
$-$	$-$	$+$		$23$	$12$	$11$		$19$	$12$	$7$		$4$	$0$	$4$
Plus space		$+$		$49$	$23$	$26$		$41$	$23$	$18$		$8$	$0$	$8$
Minus space		$-$		$43$	$18$	$25$		$35$	$18$	$17$		$8$	$0$	$8$

Trace form

41 * q + 2 * q^3 + 148 * q^4 + 5 * q^5 - 36 * q^8 + 341 * q^9 + 40 * q^10 + 28 * q^11 + 188 * q^12 + 22 * q^13 - 50 * q^15 + 772 * q^16 + 170 * q^17 + 356 * q^18 - 344 * q^19 + 40 * q^20 - 32 * q^22 - 238 * q^23 - 44 * q^24 + 1025 * q^25 + 156 * q^26 - 76 * q^27 - 110 * q^29 - 240 * q^30 + 156 * q^31 - 1228 * q^32 - 360 * q^33 - 360 * q^34 + 1932 * q^36 - 1342 * q^37 + 500 * q^38 - 372 * q^39 + 240 * q^40 - 458 * q^41 - 694 * q^43 + 996 * q^44 - 275 * q^45 - 1148 * q^46 + 1058 * q^47 + 2028 * q^48 + 1308 * q^51 - 2072 * q^52 - 102 * q^53 - 788 * q^54 + 520 * q^55 + 2672 * q^57 - 2084 * q^58 + 1472 * q^59 + 260 * q^60 - 762 * q^61 + 552 * q^62 + 2540 * q^64 - 870 * q^65 - 1092 * q^66 + 2254 * q^67 - 2692 * q^68 - 204 * q^69 - 180 * q^71 + 1768 * q^72 - 190 * q^73 + 508 * q^74 + 50 * q^75 - 4084 * q^76 - 2136 * q^78 + 3880 * q^79 + 1120 * q^80 + 3733 * q^81 + 508 * q^82 + 2202 * q^83 + 30 * q^85 - 4408 * q^86 + 1684 * q^87 - 4784 * q^88 + 994 * q^89 + 2780 * q^90 - 6548 * q^92 - 2352 * q^93 + 444 * q^94 + 20 * q^95 + 6396 * q^96 - 4254 * q^97 - 7332 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(245))$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	7
245.4.a.a	$245$	$4$	245.a	1.a	$1$	$1$	$1$	$14.455$	$\Q$	None	✓				5.4.a.a	$1$	$0$	$-4$	$-2$	$5$	$0$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-4q^{2}-2q^{3}+8q^{4}+5q^{5}+8q^{6}+\cdots$
245.4.a.b	$245$	$4$	245.a	1.a	$1$	$1$	$1$	$14.455$	$\Q$	None	✓	✓			245.4.a.b	$1$	$0$	$1$	$-6$	$5$	$0$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-6q^{3}-7q^{4}+5q^{5}-6q^{6}+\cdots$
245.4.a.c	$245$	$4$	245.a	1.a	$1$	$1$	$1$	$14.455$	$\Q$	None	✓	✓			245.4.a.b	$1$	$1$	$1$	$6$	$-5$	$0$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+6q^{3}-7q^{4}-5q^{5}+6q^{6}+\cdots$
245.4.a.d	$245$	$4$	245.a	1.a	$1$	$1$	$1$	$14.455$	$\Q$	None	✓				35.4.a.a	$1$	$0$	$1$	$8$	$5$	$0$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+8q^{3}-7q^{4}+5q^{5}+8q^{6}+\cdots$
245.4.a.e	$245$	$4$	245.a	1.a	$1$	$1$	$1$	$14.455$	$\Q$	None	✓				35.4.e.a	$1$	$1$	$3$	$-2$	$5$	$0$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+3q^{2}-2q^{3}+q^{4}+5q^{5}-6q^{6}+\cdots$
245.4.a.f	$245$	$4$	245.a	1.a	$1$	$1$	$1$	$14.455$	$\Q$	None	✓				35.4.e.a	$1$	$1$	$3$	$2$	$-5$	$0$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+3q^{2}+2q^{3}+q^{4}-5q^{5}+6q^{6}+\cdots$
245.4.a.g	$245$	$4$	245.a	1.a	$1$	$2$	$2$	$14.455$	$\Q(\sqrt{2})$	None	✓				35.4.e.b	$1$	$1$	$-6$	$-2$	$10$	$0$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-3+\beta )q^{2}+(-1-3\beta )q^{3}+(3-6\beta )q^{4}+\cdots$
245.4.a.h	$245$	$4$	245.a	1.a	$1$	$2$	$2$	$14.455$	$\Q(\sqrt{2})$	None	✓				35.4.e.b	$1$	$1$	$-6$	$2$	$-10$	$0$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-3+\beta )q^{2}+(1+3\beta )q^{3}+(3-6\beta )q^{4}+\cdots$
245.4.a.i	$245$	$4$	245.a	1.a	$1$	$2$	$2$	$14.455$	$\Q(\sqrt{11})$	None	✓	✓			245.4.a.i	$1$	$1$	$2$	$-10$	$-10$	$0$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}-5q^{3}+(4+2\beta )q^{4}-5q^{5}+\cdots$
245.4.a.j	$245$	$4$	245.a	1.a	$1$	$2$	$2$	$14.455$	$\Q(\sqrt{11})$	None	✓	✓			245.4.a.i	$1$	$0$	$2$	$10$	$10$	$0$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+5q^{3}+(4+2\beta )q^{4}+5q^{5}+\cdots$
245.4.a.k	$245$	$4$	245.a	1.a	$1$	$2$	$2$	$14.455$	$\Q(\sqrt{2})$	None	✓				35.4.a.b	$1$	$0$	$8$	$-2$	$10$	$0$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(4+\beta )q^{2}+(-1+4\beta )q^{3}+(10+8\beta )q^{4}+\cdots$
245.4.a.l	$245$	$4$	245.a	1.a	$1$	$3$	$3$	$14.455$	3.3.14360.1	None	✓		✓		35.4.a.c	$1$	$1$	$-3$	$-2$	$-15$	$0$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\cdots$
245.4.a.m	$245$	$4$	245.a	1.a	$1$	$5$	$5$	$14.455$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓				35.4.e.c	$1$	$0$	$1$	$-8$	$-25$	$0$		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(-2+\beta _{2})q^{3}+(7+\beta _{3})q^{4}+\cdots$
245.4.a.n	$245$	$4$	245.a	1.a	$1$	$5$	$5$	$14.455$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓		✓		35.4.e.c	$1$	$0$	$1$	$8$	$25$	$0$		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(7+\beta _{3})q^{4}+\cdots$
245.4.a.o	$245$	$4$	245.a	1.a	$1$	$6$	$6$	$14.455$	6.6.1163891200.1	None	✓	✓	✓		245.4.a.o	$1$	$1$	$-2$	$-16$	$30$	$0$		$-$	$+$	$-$	$2\cdot 7$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(-3+\beta _{5})q^{3}+(2+3\beta _{1}+\cdots)q^{4}+\cdots$
245.4.a.p	$245$	$4$	245.a	1.a	$1$	$6$	$6$	$14.455$	6.6.1163891200.1	None	✓	✓	✓		245.4.a.o	$1$	$0$	$-2$	$16$	$-30$	$0$		$+$	$+$	$+$	$2\cdot 7$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(3-\beta _{5})q^{3}+(2+3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(245))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(245)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(7))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(35))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(49))$$^{\oplus 2}$