Newform orbit 37.6.c.a

• Label: 37.6.c.a
• Level: $37$
• Weight: $6$
• Character orbit: 37.c
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 4 q^{2} + 16 q^{3} - 230 q^{4} - 51 q^{5} + 60 q^{6} + 50 q^{7} + 24 q^{8} - 1085 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
10.1	−5.29373	+	9.16901i	−7.75625	−	13.4342i	−40.0472	−	69.3637i	14.2918	+	24.7542i	164.238			10.7449	+	18.6107i	509.197			1.18106	−	2.04566i	−302.628
10.2	−5.15115	+	8.92206i	13.8137	+	23.9261i	−37.0688	−	64.2050i	−33.8746	−	58.6725i	−284.626			−29.2828	−	50.7192i	434.114			−260.137	+	450.571i	697.972
10.3	−3.72321	+	6.44878i	5.98171	+	10.3606i	−11.7245	−	20.3075i	37.3727	+	64.7314i	−89.0845			68.5781	+	118.781i	−63.6740			49.9383	−	86.4957i	−556.584
10.4	−3.51823	+	6.09376i	−0.728242	−	1.26135i	−8.75591	−	15.1657i	−24.4115	−	42.2819i	10.2485			−37.5928	−	65.1127i	−101.945			120.439	−	208.607i	343.541
10.5	−2.35757	+	4.08344i	−14.3898	−	24.9239i	4.88369	+	8.45880i	−24.5682	−	42.5534i	135.700			109.024	+	188.835i	−196.939			−292.634	+	506.856i	231.686
10.6	−2.03969	+	3.53285i	−10.2415	−	17.7388i	7.67933	+	13.3010i	36.0654	+	62.4672i	83.5580			−123.244	−	213.465i	−193.194			−88.2770	+	152.900i	−294.249
10.7	−1.17581	+	2.03656i	6.67175	+	11.5558i	13.2349	+	22.9236i	−7.00965	−	12.1411i	−31.3789			52.0933	+	90.2282i	−137.499			32.4755	−	56.2492i	32.9681
10.8	−0.480658	+	0.832525i	12.5533	+	21.7430i	15.5379	+	26.9125i	14.2475	+	24.6775i	−24.1355			−96.5783	−	167.278i	−60.6359			−193.673	+	335.451i	−27.3928
10.9	0.684514	−	1.18561i	−2.47268	−	4.28281i	15.0629	+	26.0897i	−49.2630	−	85.3260i	−6.77035			−21.8515	−	37.8479i	85.0519			109.272	−	189.264i	−134.885
10.10	1.06983	−	1.85300i	−4.46391	−	7.73172i	13.7109	+	23.7480i	22.0008	+	38.1065i	−19.1024			29.8851	+	51.7625i	127.142			81.6470	−	141.417i	94.1481
10.11	2.93501	−	5.08359i	12.1402	+	21.0274i	−1.22857	−	2.12795i	−29.0869	−	50.3799i	142.526			97.1808	+	168.322i	173.417			−173.269	+	300.110i	−341.481
10.12	3.64161	−	6.30745i	−11.9080	−	20.6253i	−10.5226	−	18.2257i	−4.92186	−	8.52492i	−173.457			−8.42954	−	14.6004i	79.7857			−162.101	+	280.767i	−71.6940
10.13	3.71247	−	6.43018i	6.84687	+	11.8591i	−11.5648	−	20.0308i	46.2802	+	80.1596i	101.675			3.53945	+	6.13051i	65.8619			27.7406	−	48.0482i	687.254
10.14	4.11131	−	7.12099i	3.90426	+	6.76238i	−17.8057	−	30.8404i	−21.1612	−	36.6522i	64.2065			−110.755	−	191.833i	−29.6949			91.0134	−	157.640i	−348.000
10.15	5.58532	−	9.67406i	−1.95144	−	3.38000i	−46.3916	−	80.3526i	−1.46163	−	2.53162i	−43.5978			81.6887	+	141.489i	−678.988			113.884	−	197.252i	−32.6548
26.1	−5.29373	−	9.16901i	−7.75625	+	13.4342i	−40.0472	+	69.3637i	14.2918	−	24.7542i	164.238			10.7449	−	18.6107i	509.197			1.18106	+	2.04566i	−302.628
26.2	−5.15115	−	8.92206i	13.8137	−	23.9261i	−37.0688	+	64.2050i	−33.8746	+	58.6725i	−284.626			−29.2828	+	50.7192i	434.114			−260.137	−	450.571i	697.972
26.3	−3.72321	−	6.44878i	5.98171	−	10.3606i	−11.7245	+	20.3075i	37.3727	−	64.7314i	−89.0845			68.5781	−	118.781i	−63.6740			49.9383	+	86.4957i	−556.584
26.4	−3.51823	−	6.09376i	−0.728242	+	1.26135i	−8.75591	+	15.1657i	−24.4115	+	42.2819i	10.2485			−37.5928	+	65.1127i	−101.945			120.439	+	208.607i	343.541
26.5	−2.35757	−	4.08344i	−14.3898	+	24.9239i	4.88369	−	8.45880i	−24.5682	+	42.5534i	135.700			109.024	−	188.835i	−196.939			−292.634	−	506.856i	231.686

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	37.6.c.a	✓	30
37.c	even	3	1	inner	37.6.c.a	✓	30

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.6.c.a	✓	30	1.a	even	1	1	trivial
37.6.c.a	✓	30	37.c	even	3	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).