Space of modular forms of level 75, weight 18, and trivial character

• Label: 75.18.a
• Level: $75$
• Weight: $18$
• Character orbit: 75.a
• Rep. character: $\chi_{75}(1,\cdot)$
• Character field: $\Q$
• Dimension: $53$
• Newform subspaces: $12$
• Sturm bound: $180$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(180\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(75))\).

	Total	New	Old
Modular forms	176	53	123
Cusp forms	164	53	111
Eisenstein series	12	0	12

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(43\)	\(12\)	\(31\)		\(40\)	\(12\)	\(28\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(45\)	\(14\)	\(31\)		\(42\)	\(14\)	\(28\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(45\)	\(14\)	\(31\)		\(42\)	\(14\)	\(28\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(43\)	\(13\)	\(30\)		\(40\)	\(13\)	\(27\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(86\)	\(25\)	\(61\)		\(80\)	\(25\)	\(55\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(90\)	\(28\)	\(62\)		\(84\)	\(28\)	\(56\)		\(6\)	\(0\)	\(6\)

Trace form

53 * q + 286 * q^2 + 6561 * q^3 + 3342400 * q^4 - 2558790 * q^6 - 1083720 * q^7 + 187753692 * q^8 + 2281476213 * q^9 - 542763216 * q^11 + 834847884 * q^12 + 6550855782 * q^13 + 977230044 * q^14 + 138044494644 * q^16 + 24725878138 * q^17 + 12311362206 * q^18 - 81170613702 * q^19 + 211467971538 * q^21 - 464497896204 * q^22 - 989320551120 * q^23 - 392539879836 * q^24 - 1471621644504 * q^26 + 282429536481 * q^27 - 55262829900 * q^28 + 4773930941526 * q^29 - 9673901279558 * q^31 + 13191284901260 * q^32 + 16188261278796 * q^33 - 41654776149820 * q^34 + 143879360270400 * q^36 - 17848511890050 * q^37 - 125293650377216 * q^38 - 24372092584872 * q^39 + 260225753676582 * q^41 + 128643609330180 * q^42 + 15454441879116 * q^43 - 295009624218288 * q^44 - 390542252507168 * q^46 + 654034443355912 * q^47 - 310044022672896 * q^48 + 447296503668083 * q^49 + 250916041735434 * q^51 + 2574260197395288 * q^52 + 977430855936310 * q^53 - 110147519227590 * q^54 - 2433549274655400 * q^56 + 2350224840324204 * q^57 - 4508178366394872 * q^58 - 5303372078559288 * q^59 + 556695540451588 * q^61 + 14297885748645240 * q^62 - 46650592482120 * q^63 - 13546203895474164 * q^64 - 15916425375672 * q^66 + 9045170871240948 * q^67 - 12054651575795968 * q^68 - 6259576744831716 * q^69 + 593900449765176 * q^71 + 8082180796243932 * q^72 + 3675642918560250 * q^73 - 1423200332015436 * q^74 - 14732129251780048 * q^76 - 3392237604490560 * q^77 + 3400663726204812 * q^78 - 36470138625005400 * q^79 + 98210070009147573 * q^81 - 53233806178256580 * q^82 + 22530496278136548 * q^83 + 69035446331453016 * q^84 + 44199743484515364 * q^86 + 30844620036341478 * q^87 + 5681772158905332 * q^88 + 153152840078569398 * q^89 + 120486499223910442 * q^91 - 460118593723336320 * q^92 - 52139329393996440 * q^93 + 440633834780028848 * q^94 - 185144722774560612 * q^96 - 148550647784559582 * q^97 + 625684792783536158 * q^98 - 23364176728214736 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(75))\) 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
75.18.a.a	$75$	$18$	75.a	1.a	$1$	$1$	$1$	$137.417$	\(\Q\)	None	✓				3.18.a.a	$1$	$0$	\(-204\)	\(6561\)	\(0\)	\(20846560\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-204q^{2}+3^{8}q^{3}-89456q^{4}-1338444q^{6}+\cdots\)
75.18.a.b	$75$	$18$	75.a	1.a	$1$	$2$	$2$	$137.417$	\(\Q(\sqrt{14569}) \)	None	✓				3.18.a.b	$1$	$1$	\(-594\)	\(-13122\)	\(0\)	\(-24471568\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-297-\beta )q^{2}-3^{8}q^{3}+(88258+\cdots)q^{4}+\cdots\)
75.18.a.c	$75$	$18$	75.a	1.a	$1$	$2$	$2$	$137.417$	\(\Q(\sqrt{849}) \)	None	✓				15.18.a.a	$1$	$1$	\(356\)	\(-13122\)	\(0\)	\(20754552\)		$+$	$+$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(178-\beta )q^{2}-3^{8}q^{3}+(175688-356\beta )q^{4}+\cdots\)
75.18.a.d	$75$	$18$	75.a	1.a	$1$	$3$	$3$	$137.417$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				15.18.a.c	$1$	$1$	\(253\)	\(-19683\)	\(0\)	\(4332484\)		$+$	$+$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(84+\beta _{1})q^{2}-3^{8}q^{3}+(-2408+137\beta _{1}+\cdots)q^{4}+\cdots\)
75.18.a.e	$75$	$18$	75.a	1.a	$1$	$3$	$3$	$137.417$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				15.18.a.b	$1$	$0$	\(442\)	\(19683\)	\(0\)	\(-4962644\)		$-$	$+$	$-$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(147-\beta _{1})q^{2}+3^{8}q^{3}+(99318-14^{2}\beta _{1}+\cdots)q^{4}+\cdots\)
75.18.a.f	$75$	$18$	75.a	1.a	$1$	$4$	$4$	$137.417$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				15.18.a.d	$1$	$0$	\(33\)	\(26244\)	\(0\)	\(-17583104\)		$-$	$+$	$-$	$2^{6}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(8+\beta _{1})q^{2}+3^{8}q^{3}+(109856-65\beta _{1}+\cdots)q^{4}+\cdots\)
75.18.a.g	$75$	$18$	75.a	1.a	$1$	$5$	$5$	$137.417$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		75.18.a.g	$1$	$1$	\(-204\)	\(-32805\)	\(0\)	\(2169469\)		$+$	$+$	$+$	$2^{12}\cdot 3^{3}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+(-41-\beta _{1})q^{2}-3^{8}q^{3}+(52431+\cdots)q^{4}+\cdots\)
75.18.a.h	$75$	$18$	75.a	1.a	$1$	$5$	$5$	$137.417$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			75.18.a.g	$1$	$1$	\(204\)	\(32805\)	\(0\)	\(-2169469\)		$-$	$-$	$+$	$2^{12}\cdot 3^{3}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q+(41+\beta _{1})q^{2}+3^{8}q^{3}+(52431-21\beta _{1}+\cdots)q^{4}+\cdots\)
75.18.a.i	$75$	$18$	75.a	1.a	$1$	$6$	$6$	$137.417$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		75.18.a.i	$1$	$0$	\(-338\)	\(39366\)	\(0\)	\(20999794\)		$-$	$+$	$-$	$2^{10}\cdot 3^{5}\cdot 5^{7}$	$\mathrm{SU}(2)$	\(q+(-56-\beta _{1})q^{2}+3^{8}q^{3}+(65542+\cdots)q^{4}+\cdots\)
75.18.a.j	$75$	$18$	75.a	1.a	$1$	$6$	$6$	$137.417$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			75.18.a.i	$1$	$0$	\(338\)	\(-39366\)	\(0\)	\(-20999794\)		$+$	$-$	$-$	$2^{10}\cdot 3^{5}\cdot 5^{7}$	$\mathrm{SU}(2)$	\(q+(56+\beta _{1})q^{2}-3^{8}q^{3}+(65542-3\beta _{1}+\cdots)q^{4}+\cdots\)
75.18.a.k	$75$	$18$	75.a	1.a	$1$	$8$	$8$	$137.417$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		15.18.b.a	$1$	$1$	\(-189\)	\(52488\)	\(0\)	\(-1557468\)		$-$	$-$	$+$	$2^{15}\cdot 3^{10}\cdot 5^{14}$	$\mathrm{SU}(2)$	\(q+(-24+\beta _{1})q^{2}+3^{8}q^{3}+(53927+\cdots)q^{4}+\cdots\)
75.18.a.l	$75$	$18$	75.a	1.a	$1$	$8$	$8$	$137.417$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		15.18.b.a	$1$	$0$	\(189\)	\(-52488\)	\(0\)	\(1557468\)		$+$	$-$	$-$	$2^{15}\cdot 3^{10}\cdot 5^{14}$	$\mathrm{SU}(2)$	\(q+(24-\beta _{1})q^{2}-3^{8}q^{3}+(53927-21\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(75)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)