Newform orbit 63.6.s.a

• Label: 63.6.s.a
• Level: $63$
• Weight: $6$
• Character orbit: 63.s
• Analytic conductor: $10.104$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1041806482\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 3 q^{2} - 3 q^{3} + 577 q^{4} - 6 q^{5} - 96 q^{6} - 30 q^{7} - 81 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−9.58962	+	5.53657i	−0.877361	+	15.5637i	45.3072	−	78.4743i	21.5696			−77.7562	−	154.108i	61.1717	+	114.302i			649.045i	−241.460	−	27.3101i	−206.844	+	119.422i
47.2	−9.06532	+	5.23386i	7.40328	−	13.7183i	38.7866	−	67.1804i	85.8214			4.68654	+	163.108i	39.9810	−	123.323i			477.048i	−133.383	−	203.121i	−777.998	+	449.178i
47.3	−8.69257	+	5.01866i	−15.4559	−	2.02881i	34.3739	−	59.5373i	−49.6832			144.533	−	59.9322i	56.4374	−	116.713i			368.849i	234.768	+	62.7140i	431.875	−	249.343i
47.4	−8.37859	+	4.83738i	9.11067	−	12.6489i	30.8005	−	53.3480i	−104.481			−15.1468	+	150.052i	−115.003	+	59.8436i			286.382i	−76.9914	−	230.481i	875.403	−	505.414i
47.5	−7.50494	+	4.33298i	14.2230	+	6.38014i	21.5494	−	37.3246i	15.7794			−134.388	+	13.7454i	−117.294	−	55.2192i			96.1815i	161.588	+	181.489i	−118.424	+	68.3718i
47.6	−7.40857	+	4.27734i	−9.98088	−	11.9742i	20.5913	−	35.6651i	38.9005			125.162	+	46.0202i	−32.6607	+	125.460i			78.5539i	−43.7639	+	239.027i	−288.197	+	166.391i
47.7	−7.18813	+	4.15007i	15.2650	+	3.15899i	18.4461	−	31.9497i	−13.4718			−122.837	+	40.6437i	121.599	+	44.9527i			40.6067i	223.042	+	96.4440i	96.8369	−	55.9088i
47.8	−6.85527	+	3.95789i	−11.6841	+	10.3190i	15.3298	−	26.5520i	−25.8967			39.2562	−	116.984i	−120.641	+	47.4627i		−	10.6099i	30.0365	−	241.136i	177.529	−	102.497i
47.9	−5.95180	+	3.43627i	−2.44206	+	15.3960i	7.61597	−	13.1912i	82.4113			−38.3702	−	100.025i	−27.2714	−	126.741i		−	115.239i	−231.073	−	75.1959i	−490.496	+	283.188i
47.10	−5.22444	+	3.01633i	1.37473	+	15.5277i	2.19649	−	3.80443i	−106.554			−54.0189	−	76.9769i	99.6671	−	82.9064i		−	166.544i	−239.220	+	42.6928i	556.683	−	321.401i
47.11	−4.52842	+	2.61449i	−15.1040	+	3.85599i	−2.32894	+	4.03383i	57.3397			58.3159	−	56.9508i	128.059	+	20.1980i		−	191.683i	213.263	−	116.482i	−259.658	+	149.914i
47.12	−4.44864	+	2.56843i	6.98225	−	13.9373i	−2.80638	+	4.86080i	−22.3974			4.73533	+	79.9354i	125.358	+	33.0507i		−	193.211i	−145.496	−	194.627i	99.6378	−	57.5259i
47.13	−4.43807	+	2.56232i	−5.73757	−	14.4941i	−2.86900	+	4.96925i	−12.0147			62.6024	+	49.6246i	−44.5801	−	121.736i		−	193.394i	−177.161	+	166.322i	53.3223	−	30.7856i
47.14	−3.42420	+	1.97696i	13.4136	−	7.94196i	−8.18323	+	14.1738i	96.1527			−30.2300	+	53.7131i	−67.3980	+	110.745i		−	191.237i	116.850	−	213.061i	−329.246	+	190.090i
47.15	−2.69688	+	1.55704i	8.07896	+	13.3316i	−11.1512	+	19.3145i	−14.7675			−42.5458	−	23.3743i	−76.3428	+	104.780i		−	169.103i	−112.461	+	215.410i	39.8260	−	22.9936i
47.16	−1.69409	+	0.978081i	−13.0902	−	8.46437i	−14.0867	+	24.3989i	−99.5152			30.4548	+	1.53606i	54.4830	+	117.638i		−	117.709i	99.7087	+	221.601i	168.587	−	97.3339i
47.17	−1.45110	+	0.837793i	14.6627	−	5.29211i	−14.5962	+	25.2814i	−41.1571			−16.8433	+	19.9637i	−56.9736	−	116.452i		−	102.533i	186.987	−	155.193i	59.7231	−	34.4811i
47.18	−1.22697	+	0.708392i	−14.9068	+	4.55938i	−14.9964	+	25.9745i	−39.2064			15.0604	−	16.1541i	−128.843	−	14.3719i		−	87.8303i	201.424	−	135.931i	48.1052	−	27.7735i
47.19	0.0316775	−	0.0182890i	12.3725	+	9.48272i	−15.9993	+	27.7117i	62.5776			0.565358	+	0.0741087i	87.2452	−	95.8920i			2.34094i	63.1559	+	234.649i	1.98230	−	1.14448i
47.20	0.0675663	−	0.0390094i	−14.2360	−	6.35113i	−15.9970	+	27.7075i	92.3091			−1.20963	+	0.126215i	−119.454	−	50.3751i			4.99273i	162.326	+	180.829i	6.23698	−	3.60092i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	63.6.s.a	yes	76
3.b	odd	2	1		189.6.s.a		76
7.d	odd	6	1		63.6.i.a	✓	76
9.c	even	3	1		189.6.i.a		76
9.d	odd	6	1		63.6.i.a	✓	76
21.g	even	6	1		189.6.i.a		76
63.k	odd	6	1		189.6.s.a		76
63.s	even	6	1	inner	63.6.s.a	yes	76

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.6.i.a	✓	76	7.d	odd	6	1
63.6.i.a	✓	76	9.d	odd	6	1
63.6.s.a	yes	76	1.a	even	1	1	trivial
63.6.s.a	yes	76	63.s	even	6	1	inner
189.6.i.a		76	9.c	even	3	1
189.6.i.a		76	21.g	even	6	1
189.6.s.a		76	3.b	odd	2	1
189.6.s.a		76	63.k	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(63, [\chi])\).