Space of modular forms of level 414, weight 4, and trivial character

• Label: 414.4.a
• Level: $414$
• Weight: $4$
• Character orbit: 414.a
• Rep. character: $\chi_{414}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $14$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(414))\).

	Total	New	Old
Modular forms	224	28	196
Cusp forms	208	28	180
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.

\(2\)	\(3\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(17\)
Minus space			\(-\)	\(11\)

Trace form

28 * q - 4 * q^2 + 112 * q^4 - 10 * q^5 + 40 * q^7 - 16 * q^8 + 4 * q^10 + 34 * q^11 + 4 * q^13 + 40 * q^14 + 448 * q^16 - 88 * q^17 - 98 * q^19 - 40 * q^20 + 148 * q^22 + 616 * q^25 - 120 * q^26 + 160 * q^28 + 208 * q^29 + 452 * q^31 - 64 * q^32 - 256 * q^34 + 352 * q^35 - 310 * q^37 - 60 * q^38 + 16 * q^40 - 192 * q^41 - 214 * q^43 + 136 * q^44 + 92 * q^46 - 156 * q^47 + 1876 * q^49 - 420 * q^50 + 16 * q^52 + 1810 * q^53 + 424 * q^55 + 160 * q^56 + 512 * q^58 - 216 * q^59 + 1538 * q^61 + 128 * q^62 + 1792 * q^64 + 2840 * q^65 + 1250 * q^67 - 352 * q^68 + 544 * q^70 + 2436 * q^71 + 1944 * q^73 + 220 * q^74 - 392 * q^76 - 1048 * q^77 + 1980 * q^79 - 160 * q^80 - 352 * q^82 + 742 * q^83 + 2620 * q^85 + 1404 * q^86 + 592 * q^88 + 2136 * q^89 - 644 * q^91 - 816 * q^94 - 2128 * q^95 - 1824 * q^97 + 540 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(414))\) 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	3	23
414.4.a.a	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	\(\Q\)	None	✓				138.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(-34\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+2q^{5}-34q^{7}-8q^{8}+\cdots\)
414.4.a.b	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	\(\Q\)	None	✓				46.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(20\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+20q^{5}+2q^{7}-8q^{8}+\cdots\)
414.4.a.c	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	\(\Q\)	None	✓				138.4.a.b	$1$	$1$	\(2\)	\(0\)	\(-2\)	\(-32\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-2q^{5}-2^{5}q^{7}+8q^{8}+\cdots\)
414.4.a.d	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	\(\Q\)	None	✓				46.4.a.a	$1$	$0$	\(2\)	\(0\)	\(10\)	\(-12\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+10q^{5}-12q^{7}+8q^{8}+\cdots\)
414.4.a.e	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	\(\Q\)	None	✓				138.4.a.a	$1$	$0$	\(2\)	\(0\)	\(10\)	\(32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+10q^{5}+2^{5}q^{7}+8q^{8}+\cdots\)
414.4.a.f	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	\(\Q(\sqrt{73}) \)	None	✓				46.4.a.d	$1$	$1$	\(-4\)	\(0\)	\(-10\)	\(12\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-4-2\beta )q^{5}+(4+4\beta )q^{7}+\cdots\)
414.4.a.g	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	\(\Q(\sqrt{2}) \)	None	✓				138.4.a.e	$1$	$1$	\(-4\)	\(0\)	\(-8\)	\(12\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-4+\beta )q^{5}+(6+\beta )q^{7}+\cdots\)
414.4.a.h	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	414.4.a.h	$1$	$1$	\(-4\)	\(0\)	\(10\)	\(-10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(5+3\beta )q^{5}+(-5+5\beta )q^{7}+\cdots\)
414.4.a.i	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	414.4.a.h	$1$	$1$	\(4\)	\(0\)	\(-10\)	\(-10\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-5+3\beta )q^{5}+(-5+\cdots)q^{7}+\cdots\)
414.4.a.j	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	\(\Q(\sqrt{41}) \)	None	✓		✓		46.4.a.c	$1$	$1$	\(4\)	\(0\)	\(-10\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-4-2\beta )q^{5}+(2+2\beta )q^{7}+\cdots\)
414.4.a.k	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	\(\Q(\sqrt{277}) \)	None	✓		✓		138.4.a.d	$1$	$0$	\(4\)	\(0\)	\(-2\)	\(28\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-1-\beta )q^{5}+14q^{7}+\cdots\)
414.4.a.l	$414$	$4$	414.a	1.a	$1$	$3$	$3$	$24.427$	3.3.16372.1	None	✓		✓		138.4.a.f	$1$	$0$	\(-6\)	\(0\)	\(-20\)	\(10\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-7-\beta _{2})q^{5}+(3+\beta _{1}+\cdots)q^{7}+\cdots\)
414.4.a.m	$414$	$4$	414.a	1.a	$1$	$4$	$4$	$24.427$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	414.4.a.m	$1$	$0$	\(-8\)	\(0\)	\(0\)	\(18\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-\beta _{3}q^{5}+(5+\beta _{2})q^{7}+\cdots\)
414.4.a.n	$414$	$4$	414.a	1.a	$1$	$4$	$4$	$24.427$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	414.4.a.m	$1$	$0$	\(8\)	\(0\)	\(0\)	\(18\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+\beta _{3}q^{5}+(5+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(414))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(414)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 2}\)