Space of modular forms of level 1936, weight 2, and trivial character

• Label: 1936.2.a
• Level: $1936$
• Weight: $2$
• Character orbit: 1936.a
• Rep. character: $\chi_{1936}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $29$
• Sturm bound: $528$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(528\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1936))\).

	Total	New	Old
Modular forms	300	59	241
Cusp forms	229	50	179
Eisenstein series	71	9	62

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

\(2\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(12\)
\(+\)	\(-\)	\(-\)	\(15\)
\(-\)	\(+\)	\(-\)	\(13\)
\(-\)	\(-\)	\(+\)	\(10\)
Plus space		\(+\)	\(22\)
Minus space		\(-\)	\(28\)

Trace form

50 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 + 40 * q^9 + 2 * q^13 + 4 * q^15 - 2 * q^17 + 4 * q^19 + 8 * q^21 + 16 * q^23 + 28 * q^25 + 4 * q^27 + 10 * q^29 - 2 * q^31 + 12 * q^35 + 10 * q^37 + 8 * q^39 - 2 * q^41 - 16 * q^43 + 8 * q^45 + 22 * q^47 + 12 * q^49 + 28 * q^51 + 2 * q^53 + 8 * q^57 - 6 * q^59 - 6 * q^61 - 32 * q^63 + 4 * q^65 + 12 * q^67 - 8 * q^69 + 2 * q^71 - 18 * q^73 + 8 * q^75 - 20 * q^79 + 18 * q^81 + 8 * q^83 - 4 * q^85 + 40 * q^87 - 28 * q^89 - 56 * q^91 - 24 * q^93 - 48 * q^95 - 10 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1936))\) 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	11
1936.2.a.a	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				121.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-2q^{7}+q^{9}-q^{13}-2q^{15}+\cdots\)
1936.2.a.b	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				121.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+2q^{7}+q^{9}+q^{13}-2q^{15}+\cdots\)
1936.2.a.c	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				44.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+2q^{7}-2q^{9}+4q^{13}+\cdots\)
1936.2.a.d	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				968.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}-2q^{9}+4q^{13}+\cdots\)
1936.2.a.e	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				968.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+4q^{7}-2q^{9}-4q^{13}+\cdots\)
1936.2.a.f	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				968.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-4q^{7}-3q^{9}+3q^{13}+3q^{17}+\cdots\)
1936.2.a.g	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				968.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+4q^{7}-3q^{9}-3q^{13}-3q^{17}+\cdots\)
1936.2.a.h	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	\(\Q(\sqrt{-11}) \)	✓				121.2.a.b	$2$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+q^{3}-3q^{5}-2q^{9}-3q^{15}+9q^{23}+\cdots\)
1936.2.a.i	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				11.2.a.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}-2q^{9}-4q^{13}+\cdots\)
1936.2.a.j	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				242.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{5}-2q^{7}+q^{9}+5q^{13}+\cdots\)
1936.2.a.k	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				242.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{5}+2q^{7}+q^{9}-5q^{13}+\cdots\)
1936.2.a.l	$1936$	$2$	1936.a	1.a	$1$	$1$	$1$	$15.459$	\(\Q\)	None	✓				88.2.a.a	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-3q^{5}-2q^{7}+6q^{9}-9q^{15}+\cdots\)
1936.2.a.m	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{3}) \)	None	✓				484.2.a.e	$2$	$0$	\(0\)	\(-4\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{5}+2\beta q^{7}+q^{9}+3\beta q^{13}+\cdots\)
1936.2.a.n	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{5}) \)	None	✓				22.2.c.a	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(2-2\beta )q^{5}-2q^{7}+\cdots\)
1936.2.a.o	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{5}) \)	None	✓				22.2.c.a	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(2-2\beta )q^{5}+2q^{7}+\cdots\)
1936.2.a.p	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{3}) \)	None	✓				968.2.a.k	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-2+\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
1936.2.a.q	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{3}) \)	None	✓				968.2.a.k	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-2+\beta )q^{5}+(1-\beta )q^{7}+\cdots\)
1936.2.a.r	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{17}) \)	None	✓				88.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(2-\beta )q^{5}-2\beta q^{7}+(1+\beta )q^{9}+\cdots\)
1936.2.a.s	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{33}) \)	\(\Q(\sqrt{-11}) \)	✓				484.2.a.d	$2$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-\beta q^{3}+(2-\beta )q^{5}+(5+\beta )q^{9}+(8-\beta )q^{15}+\cdots\)
1936.2.a.t	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{5}) \)	None	✓				88.2.i.a	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}-\beta q^{7}+(-2+\cdots)q^{9}+\cdots\)
1936.2.a.u	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{5}) \)	None	✓				88.2.i.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}+\beta q^{7}+(-2+\cdots)q^{9}+\cdots\)
1936.2.a.v	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{3}) \)	None	✓				242.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+\beta q^{5}+(-3+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
1936.2.a.w	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{5}) \)	None	✓				968.2.a.f	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+\beta q^{5}+(-1-\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
1936.2.a.x	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{5}) \)	None	✓				968.2.a.f	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+\beta q^{5}+(1+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
1936.2.a.y	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{3}) \)	None	✓				242.2.a.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+\beta q^{5}+(3-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
1936.2.a.z	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{5}) \)	None	✓				44.2.e.a	$1$	$1$	\(0\)	\(3\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-\beta q^{5}+(1-3\beta )q^{7}+(-1+\cdots)q^{9}+\cdots\)
1936.2.a.ba	$1936$	$2$	1936.a	1.a	$1$	$2$	$2$	$15.459$	\(\Q(\sqrt{5}) \)	None	✓				44.2.e.a	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-\beta q^{5}+(-1+3\beta )q^{7}+\cdots\)
1936.2.a.bb	$1936$	$2$	1936.a	1.a	$1$	$4$	$4$	$15.459$	4.4.5225.1	None	✓		✓		88.2.i.b	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-1+\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
1936.2.a.bc	$1936$	$2$	1936.a	1.a	$1$	$4$	$4$	$15.459$	4.4.5225.1	None	✓		✓		88.2.i.b	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-1+\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1936))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1936)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(484))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(968))\)\(^{\oplus 2}\)