Space of modular forms of level 900, weight 2, and Character 900.k

• Label: 900.2.k
• Level: $900$
• Weight: $2$
• Character orbit: 900.k
• Rep. character: $\chi_{900}(307,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $86$
• Newform subspaces: $14$
• Sturm bound: $360$
• Trace bound: $26$

Defining parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.k (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(360\)
Trace bound:			\(26\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(900, [\chi])\).

	Total	New	Old
Modular forms	408	94	314
Cusp forms	312	86	226
Eisenstein series	96	8	88

Trace form

86 * q - 2 * q^2 - 8 * q^8 + 2 * q^13 + 20 * q^16 - 14 * q^17 + 16 * q^22 + 52 * q^26 + 32 * q^28 + 28 * q^32 + 6 * q^37 + 16 * q^38 + 24 * q^41 + 16 * q^46 - 52 * q^52 + 22 * q^53 - 56 * q^56 - 24 * q^58 + 56 * q^61 - 56 * q^62 - 28 * q^68 + 38 * q^73 + 36 * q^76 + 48 * q^77 + 32 * q^82 + 80 * q^86 - 32 * q^88 + 56 * q^92 + 54 * q^97 + 38 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(900, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
900.2.k.a	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-5}) \)					100.2.e.a	$4$	$1$	\(-2\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(i-1)q^{2}-2 i q^{4}+(3 i-3)q^{7}+\cdots\)
900.2.k.b	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					180.2.k.a	$4$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-i-1)q^{2}+2 i q^{4}+(-2 i+2)q^{8}+\cdots\)
900.2.k.c	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					20.2.e.a	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-i-1)q^{2}+2 i q^{4}+(-2 i+2)q^{8}+\cdots\)
900.2.k.d	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					180.2.k.a	$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(i+1)q^{2}+2 i q^{4}+(2 i-2)q^{8}+\cdots\)
900.2.k.e	$900$	$2$	900.k	20.e	$4$	$2$	$1$	$7.187$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-5}) \)					100.2.e.a	$4$	$0$	\(2\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-i+1)q^{2}-2 i q^{4}+(-3 i+3)q^{7}+\cdots\)
900.2.k.f	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	\(\Q(\zeta_{24})\)	None		✓			900.2.k.f	$8$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{3}-\beta_1)q^{2}+(\beta_{4}+\beta_1)q^{4}+(-\beta_{7}+\beta_{6}+\cdots+\beta_{2})q^{7}+\cdots\)
900.2.k.g	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	\(\Q(\zeta_{24})\)	None					300.2.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{4} q^{2}+(-\beta_{6}+\beta_{2}-2)q^{4}+\cdots\)
900.2.k.h	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	8.0.157351936.1	None					300.2.j.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+(\beta _{4}+\beta _{6})q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
900.2.k.i	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	8.0.3317760000.5	\(\Q(\sqrt{-15}) \)		✓			900.2.k.i	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q+\beta _{5}q^{2}+(\beta _{2}-\beta _{4})q^{4}+(\beta _{1}-\beta _{3})q^{8}+\cdots\)
900.2.k.j	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	8.0.3317760000.5	None					100.2.e.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{5}q^{2}+(-\beta _{2}+\beta _{3})q^{4}+(\beta _{4}+2\beta _{6}+\cdots)q^{8}+\cdots\)
900.2.k.k	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	8.0.157351936.1	None					180.2.k.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{7}q^{2}+(-2\beta _{3}+\beta _{4})q^{4}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
900.2.k.l	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	\(\Q(\zeta_{24})\)	None					300.2.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{7}+\beta_1)q^{2}+(-\beta_{6}+\beta_{2})q^{4}+\cdots\)
900.2.k.m	$900$	$2$	900.k	20.e	$4$	$8$	$4$	$7.187$	\(\Q(\zeta_{24})\)	None		✓			900.2.k.f	$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_1+1)q^{2}+(-\beta_{4}+2\beta_{3}-\beta_1)q^{4}+\cdots\)
900.2.k.n	$900$	$2$	900.k	20.e	$4$	$12$	$6$	$7.187$	12.0.\(\cdots\).1	None			✓		60.2.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{2}-\beta _{2}q^{4}+(\beta _{4}+\beta _{7})q^{7}+(-1+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(900, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(900, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 2}\)