Space of modular forms of level 170, weight 2, and trivial character

• Label: 170.2.a
• Level: $170$
• Weight: $2$
• Character orbit: 170.a
• Rep. character: $\chi_{170}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $6$
• Sturm bound: $54$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$170 = 2 \cdot 5 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	170.a (trivial)
Character field:			$\Q$
Newform subspaces:			$6$
Sturm bound:			$54$
Trace bound:			$5$
Distinguishing $T_p$:			$3$, $7$, $13$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(170))$.

	Total	New	Old
Modular forms	30	7	23
Cusp forms	23	7	16
Eisenstein series	7	0	7

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

$2$	$5$	$17$	Fricke	Dim
$+$	$+$	$-$	$-$	$2$
$+$	$-$	$+$	$-$	$1$
$+$	$-$	$-$	$+$	$1$
$-$	$+$	$+$	$-$	$1$
$-$	$-$	$-$	$-$	$2$
Plus space			$+$	$1$
Minus space			$-$	$6$

Trace form

7 * q - q^2 + 7 * q^4 + q^5 + 8 * q^7 - q^8 + 7 * q^9 + q^10 - 8 * q^11 + 2 * q^13 - 4 * q^15 + 7 * q^16 + 3 * q^17 - 5 * q^18 + q^20 - 8 * q^21 - 8 * q^22 - 16 * q^23 + 7 * q^25 + 6 * q^26 + 8 * q^28 - 2 * q^29 - 8 * q^31 - q^32 - 16 * q^33 - q^34 - 4 * q^35 + 7 * q^36 - 2 * q^37 - 4 * q^38 - 8 * q^39 + q^40 - 18 * q^41 - 24 * q^42 - 4 * q^43 - 8 * q^44 - 3 * q^45 + 7 * q^49 - q^50 - 4 * q^51 + 2 * q^52 - 22 * q^53 - 24 * q^54 - 12 * q^55 + 16 * q^57 - 2 * q^58 + 8 * q^59 - 4 * q^60 + 22 * q^61 + 24 * q^63 + 7 * q^64 + 6 * q^65 + 24 * q^66 + 20 * q^67 + 3 * q^68 + 16 * q^69 + 4 * q^70 + 8 * q^71 - 5 * q^72 + 38 * q^73 + 6 * q^74 - 16 * q^78 - 8 * q^79 + q^80 - 25 * q^81 + 14 * q^82 + 12 * q^83 - 8 * q^84 + q^85 - 4 * q^86 + 40 * q^87 - 8 * q^88 + 2 * q^89 + 13 * q^90 + 40 * q^91 - 16 * q^92 + 8 * q^93 + 12 * q^94 - 20 * q^95 - 10 * q^97 + 31 * q^98 - 32 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(170))$ 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	17
170.2.a.a	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	$\Q$	None	✓	✓			170.2.a.a	$1$	$0$	$-1$	$-2$	$-1$	$2$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}+2q^{7}+\cdots$
170.2.a.b	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	$\Q$	None	✓	✓	✓	✓	170.2.a.b	$1$	$1$	$-1$	$-2$	$1$	$-2$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-2q^{7}+\cdots$
170.2.a.c	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	$\Q$	None	✓	✓	✓	✓	170.2.a.c	$1$	$0$	$-1$	$1$	$1$	$2$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots$
170.2.a.d	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	$\Q$	None	✓	✓			170.2.a.d	$1$	$0$	$-1$	$3$	$-1$	$2$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}+2q^{7}+\cdots$
170.2.a.e	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	$\Q$	None	✓	✓	✓	✓	170.2.a.e	$1$	$0$	$1$	$1$	$-1$	$2$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots$
170.2.a.f	$170$	$2$	170.a	1.a	$1$	$2$	$2$	$1.357$	$\Q(\sqrt{17})$	None	✓	✓	✓	✓	170.2.a.f	$1$	$0$	$2$	$-1$	$2$	$2$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta q^{3}+q^{4}+q^{5}-\beta q^{6}+2\beta q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(170))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(170)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(34))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(85))$$^{\oplus 2}$