Space of modular forms of level 45, weight 18, and trivial character

• Label: 45.18.a
• Level: $45$
• Weight: $18$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $8$
• Sturm bound: $108$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(108\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(45))\).

	Total	New	Old
Modular forms	106	29	77
Cusp forms	98	29	69
Eisenstein series	8	0	8

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

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(9\)
\(-\)	\(-\)	\(+\)	\(8\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(15\)

Trace form

29 * q + 286 * q^2 + 2031680 * q^4 - 390625 * q^5 + 20354440 * q^7 + 204595932 * q^8 + 419531250 * q^10 - 1022680364 * q^11 + 5451754558 * q^13 - 16276829388 * q^14 + 150546980420 * q^16 - 72245108282 * q^17 - 79490828492 * q^19 - 174392187500 * q^20 + 92131753224 * q^22 - 1485217461480 * q^23 + 4425048828125 * q^25 - 1352199009344 * q^26 + 7021765562560 * q^28 + 246838789610 * q^29 - 3144397452560 * q^31 + 38235370464140 * q^32 + 14984419420668 * q^34 - 8265000000000 * q^35 - 13846437287690 * q^37 + 19128452034784 * q^38 + 84682214062500 * q^40 + 5340275153966 * q^41 + 96648722194804 * q^43 - 395371878847708 * q^44 + 61441329298464 * q^46 - 362932458800048 * q^47 + 1799555550570525 * q^49 + 43640136718750 * q^50 + 1725912541683712 * q^52 + 1034837105877850 * q^53 - 440715595312500 * q^55 + 383240643656220 * q^56 + 1190652655775532 * q^58 + 2692306791986260 * q^59 - 1150572475494002 * q^61 + 11170117726853400 * q^62 + 14198317833591308 * q^64 - 152551967968750 * q^65 + 2850899027703292 * q^67 - 12509097103936768 * q^68 + 2762615015625000 * q^70 - 22932197266145224 * q^71 + 7254994713931810 * q^73 + 27113869215040712 * q^74 - 72511114104996536 * q^76 - 44472995664575280 * q^77 + 1233030392407552 * q^79 - 14160877400000000 * q^80 + 24226736259246180 * q^82 - 57563515358178492 * q^83 + 20870214063281250 * q^85 + 138551273721727456 * q^86 - 192906685844826912 * q^88 + 79925409803423670 * q^89 - 7119329368588096 * q^91 - 4756879689516480 * q^92 - 118619612107468152 * q^94 - 61575465564062500 * q^95 - 14491656651760118 * q^97 + 399976137086259518 * q^98

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(45))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
45.18.a.a	$45$	$18$	45.a	1.a	$1$	$2$	$2$	$82.450$	\(\Q(\sqrt{39}) \)	None	✓				5.18.a.a	$1$	$1$	\(-680\)	\(0\)	\(781250\)	\(-22820700\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-340+\beta )q^{2}+(35072-680\beta )q^{4}+\cdots\)
45.18.a.b	$45$	$18$	45.a	1.a	$1$	$2$	$2$	$82.450$	\(\Q(\sqrt{849}) \)	None	✓				15.18.a.a	$1$	$0$	\(356\)	\(0\)	\(-781250\)	\(-20754552\)		$-$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(178-\beta )q^{2}+(175688-356\beta )q^{4}+\cdots\)
45.18.a.c	$45$	$18$	45.a	1.a	$1$	$3$	$3$	$82.450$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				5.18.a.b	$1$	$0$	\(-118\)	\(0\)	\(-1171875\)	\(2139308\)		$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-39+\beta _{1})q^{2}+(5574-14^{2}\beta _{1}+\cdots)q^{4}+\cdots\)
45.18.a.d	$45$	$18$	45.a	1.a	$1$	$3$	$3$	$82.450$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				15.18.a.c	$1$	$1$	\(253\)	\(0\)	\(1171875\)	\(-4332484\)		$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(84+\beta _{1})q^{2}+(-2408+137\beta _{1}+\cdots)q^{4}+\cdots\)
45.18.a.e	$45$	$18$	45.a	1.a	$1$	$3$	$3$	$82.450$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				15.18.a.b	$1$	$1$	\(442\)	\(0\)	\(1171875\)	\(4962644\)		$-$	$-$	$+$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(147-\beta _{1})q^{2}+(99318-14^{2}\beta _{1}+\cdots)q^{4}+\cdots\)
45.18.a.f	$45$	$18$	45.a	1.a	$1$	$4$	$4$	$82.450$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		15.18.a.d	$1$	$0$	\(33\)	\(0\)	\(-1562500\)	\(17583104\)		$-$	$+$	$-$	$2^{6}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(8+\beta _{1})q^{2}+(109856-65\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
45.18.a.g	$45$	$18$	45.a	1.a	$1$	$6$	$6$	$82.450$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	45.18.a.g	$1$	$1$	\(-665\)	\(0\)	\(-2343750\)	\(21788560\)		$+$	$+$	$+$	$2^{11}\cdot 3^{12}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(-111+\beta _{1})q^{2}+(71921-112\beta _{1}+\cdots)q^{4}+\cdots\)
45.18.a.h	$45$	$18$	45.a	1.a	$1$	$6$	$6$	$82.450$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	45.18.a.g	$1$	$0$	\(665\)	\(0\)	\(2343750\)	\(21788560\)		$+$	$-$	$-$	$2^{11}\cdot 3^{12}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(111-\beta _{1})q^{2}+(71921-112\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(45)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)