Newform orbit 63.6.g.a

• Label: 63.6.g.a
• Level: $63$
• Weight: $6$
• Character orbit: 63.g
• 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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1041806482\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + q^{2} - q^{3} - 575 q^{4} - 202 q^{5} - 116 q^{6} + 28 q^{7} - 72 q^{8} + 299 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−5.45858	+	9.45455i	6.02055	+	14.3789i	−43.5923	−	75.5041i	−81.8996			−168.810	−	21.5669i	116.686	−	56.4925i	602.459			−170.506	+	173.138i	447.056	−	774.324i
4.2	−5.19274	+	8.99409i	−14.3357	+	6.12265i	−37.9291	−	65.6951i	78.9178			19.3741	−	160.730i	78.2959	+	103.328i	455.487			168.026	−	175.545i	−409.800	+	709.794i
4.3	−4.90850	+	8.50178i	−9.78720	−	12.1330i	−32.1868	−	55.7492i	−0.906572			151.193	−	23.6535i	−27.1127	−	126.775i	317.812			−51.4216	+	237.497i	4.44991	−	7.70748i
4.4	−4.86454	+	8.42562i	13.8209	−	7.20991i	−31.3274	−	54.2607i	−33.1194			−6.48422	+	151.523i	−126.153	+	29.8716i	298.243			139.034	−	199.295i	161.111	−	279.052i
4.5	−4.57252	+	7.91984i	10.9649	+	11.0802i	−25.8159	−	44.7144i	83.7927			−137.891	+	36.1755i	−129.140	+	11.3996i	179.533			−2.54290	+	242.987i	−383.144	+	663.624i
4.6	−4.40700	+	7.63314i	3.38767	−	15.2159i	−22.8433	−	39.5657i	11.3542			101.216	+	92.9150i	85.1554	+	97.7525i	120.633			−220.047	−	103.093i	−50.0378	+	86.6681i
4.7	−4.16732	+	7.21801i	−15.3204	+	2.87825i	−18.7331	−	32.4467i	−101.394			43.0699	−	122.578i	−84.9872	+	97.8988i	45.5588			226.431	−	88.1920i	422.542	−	731.864i
4.8	−3.53515	+	6.12306i	15.2582	+	3.19159i	−8.99455	−	15.5790i	32.9744			−73.4824	+	82.1443i	129.546	−	4.97106i	−99.0612			222.628	+	97.3960i	−116.569	+	201.904i
4.9	−3.47017	+	6.01050i	−8.53482	+	13.0444i	−8.08411	−	14.0021i	11.3731			−48.7863	−	96.5649i	−20.3156	−	128.040i	−109.878			−97.3138	−	222.663i	−39.4664	+	68.3578i
4.10	−2.80913	+	4.86555i	0.935686	+	15.5604i	0.217629	+	0.376944i	−19.0252			−78.3381	−	39.1584i	30.5840	+	125.983i	−182.229			−241.249	+	29.1192i	53.4442	−	92.5680i
4.11	−2.66412	+	4.61439i	14.7925	−	4.91762i	1.80491	+	3.12620i	−76.4115			−16.7171	+	81.3594i	27.6739	−	126.654i	−189.738			194.634	−	145.487i	203.570	−	352.593i
4.12	−2.45815	+	4.25764i	−13.1905	−	8.30722i	3.91502	+	6.78102i	42.3225			67.7934	−	35.7401i	−115.362	+	59.1492i	−195.816			104.980	+	219.153i	−104.035	+	180.194i
4.13	−2.27973	+	3.94860i	6.66259	−	14.0929i	5.60571	+	9.70937i	107.131			40.4584	+	58.4358i	−22.2750	−	127.714i	−197.020			−154.220	−	187.790i	−244.230	+	423.019i
4.14	−2.18387	+	3.78258i	−7.05203	−	13.9021i	6.46140	+	11.1915i	−87.8756			67.9866	+	3.68562i	129.631	−	1.69202i	−196.211			−143.538	+	196.076i	191.909	−	332.396i
4.15	−1.46936	+	2.54501i	−15.5712	−	0.733322i	11.6820	+	20.2337i	35.4936			24.7460	−	38.5513i	128.007	−	20.5232i	−162.699			241.924	+	22.8374i	−52.1529	+	90.3315i
4.16	−1.34728	+	2.33355i	10.7103	+	11.3265i	12.3697	+	21.4249i	−54.9827			−40.8607	+	9.73323i	−126.245	−	29.4838i	−152.887			−13.5776	+	242.620i	74.0768	−	128.305i
4.17	−0.992584	+	1.71921i	5.06511	−	14.7426i	14.0296	+	24.2999i	−33.8405			20.3180	+	23.3413i	−94.9899	+	88.2265i	−119.227			−191.689	−	149.346i	33.5895	−	58.1787i
4.18	−0.487864	+	0.845006i	15.4052	−	2.38330i	15.5240	+	26.8883i	41.3255			−5.50174	+	14.1802i	3.17928	+	129.603i	−61.5177			231.640	−	73.4303i	−20.1612	+	34.9203i
4.19	0.372053	−	0.644415i	−7.56275	+	13.6310i	15.7232	+	27.2333i	97.3099			5.97029	+	9.94501i	−107.248	+	72.8345i	47.2108			−128.610	−	206.176i	36.2044	−	62.7079i
4.20	0.482480	−	0.835680i	−14.7836	+	4.94422i	15.5344	+	26.9064i	−67.7724			−3.00101	+	14.7398i	−77.6500	−	103.815i	60.8589			194.109	−	146.187i	−32.6989	+	56.6361i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.g	even	3	1	inner

Twists

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

    

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

Hecke kernels

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