Newform orbit 37.11.d.a

• Label: 37.11.d.a
• Level: $37$
• Weight: $11$
• Character orbit: 37.d
• Analytic conductor: $23.508$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	37.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.5082183489\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 12 q^{2} + 11094 q^{5} + 2046 q^{6} - 4 q^{7} - 20280 q^{8} - 964620 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
6.1	−44.7438	+	44.7438i			69.9886i		−	2980.01i	3419.76	+	3419.76i	−3131.55	−	3131.55i	−14682.2			87519.2	+	87519.2i	54150.6			−306026.
6.2	−39.3471	+	39.3471i		−	165.913i		−	2072.39i	−2246.50	−	2246.50i	6528.21	+	6528.21i	12584.4			41250.9	+	41250.9i	31521.7			176787.
6.3	−38.1461	+	38.1461i			393.049i		−	1886.25i	−1487.12	−	1487.12i	−14993.3	−	14993.3i	−8790.45			32891.6	+	32891.6i	−95438.7			113456.
6.4	−36.6981	+	36.6981i		−	471.964i		−	1669.50i	255.153	+	255.153i	17320.2	+	17320.2i	−23325.3			23688.8	+	23688.8i	−163701.			−18727.3
6.5	−34.6099	+	34.6099i			261.549i		−	1371.69i	987.224	+	987.224i	−9052.19	−	9052.19i	23552.7			12033.6	+	12033.6i	−9358.87			−68335.4
6.6	−29.4663	+	29.4663i		−	91.0754i		−	712.523i	1067.16	+	1067.16i	2683.65	+	2683.65i	−14396.8			−9178.08	−	9178.08i	50754.3			−62890.5
6.7	−28.7783	+	28.7783i		−	308.352i		−	632.385i	4127.67	+	4127.67i	8873.85	+	8873.85i	28375.7			−11270.0	−	11270.0i	−36031.8			−237575.
6.8	−26.8365	+	26.8365i			144.991i		−	416.394i	−3621.85	−	3621.85i	−3891.06	−	3891.06i	−22976.9			−16306.0	−	16306.0i	38026.5			194395.
6.9	−21.6828	+	21.6828i			236.801i			83.7140i	1066.61	+	1066.61i	−5134.51	−	5134.51i	10682.7			−24018.3	−	24018.3i	2974.12			−46254.1
6.10	−15.9821	+	15.9821i			314.319i			513.146i	3627.33	+	3627.33i	−5023.47	−	5023.47i	−25310.7			−24566.8	−	24566.8i	−39747.3			−115945.
6.11	−14.5575	+	14.5575i		−	414.474i			600.159i	−2714.51	−	2714.51i	6033.70	+	6033.70i	11282.5			−23643.7	−	23643.7i	−112740.			79032.8
6.12	−13.2079	+	13.2079i		−	208.898i			675.105i	−342.436	−	342.436i	2759.10	+	2759.10i	−3090.02			−22441.5	−	22441.5i	15410.7			9045.69
6.13	−10.9958	+	10.9958i			127.791i			782.185i	−3825.90	−	3825.90i	−1405.16	−	1405.16i	26128.7			−19860.5	−	19860.5i	42718.5			84137.7
6.14	−4.55945	+	4.55945i			467.484i			982.423i	−1353.22	−	1353.22i	−2131.47	−	2131.47i	4509.07			−9148.19	−	9148.19i	−159492.			12339.9
6.15	1.07520	−	1.07520i		−	33.9150i			1021.69i	2177.49	+	2177.49i	−36.4653	−	36.4653i	6473.29			2199.52	+	2199.52i	57898.8			4682.46
6.16	2.81409	−	2.81409i			90.9323i			1008.16i	−2198.37	−	2198.37i	255.892	+	255.892i	−24301.2			5718.68	+	5718.68i	50780.3			−12372.8
6.17	3.53840	−	3.53840i		−	354.328i			998.959i	3636.07	+	3636.07i	−1253.76	−	1253.76i	−25459.9			7158.05	+	7158.05i	−66499.6			25731.8
6.18	4.21658	−	4.21658i			126.107i			988.441i	1887.87	+	1887.87i	531.741	+	531.741i	15808.2			8485.62	+	8485.62i	43146.0			15920.7
6.19	13.7409	−	13.7409i		−	378.325i			646.374i	945.559	+	945.559i	−5198.53	−	5198.53i	24801.2			22952.5	+	22952.5i	−84080.6			25985.7
6.20	14.9926	−	14.9926i			263.976i			574.445i	−850.730	−	850.730i	3957.69	+	3957.69i	−15042.9			23964.8	+	23964.8i	−10634.5			−25509.3

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.d	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	37.11.d.a	✓	60
37.d	odd	4	1	inner	37.11.d.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.11.d.a	✓	60	1.a	even	1	1	trivial
37.11.d.a	✓	60	37.d	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(37, [\chi])\).