Newform orbit 63.6.i.a

• Label: 63.6.i.a
• Level: $63$
• Weight: $6$
• Character orbit: 63.i
• 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.i (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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 3 q^{3} - 1154 q^{4} - 3 q^{5} + 96 q^{6} - 30 q^{7} + 489 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1		−	10.9868i	6.07666	−	14.3553i	−88.7089			−10.8523	−	18.7967i	−157.718	−	66.7628i	50.5087	−	119.398i			623.047i	−169.148	−	174.464i	−206.515	+	119.231i
5.2		−	10.2891i	11.9653	+	9.99158i	−73.8652			41.5228	+	71.9196i	102.804	−	123.112i	−128.780	+	14.9271i			430.754i	43.3366	+	239.104i	739.986	−	427.231i
5.3		−	10.1846i	−12.3585	+	9.50098i	−71.7270			19.2556	+	33.3516i	96.7641	+	125.867i	124.028	+	37.7381i			404.606i	62.4628	−	234.835i	339.675	−	196.111i
5.4		−	9.00493i	−9.57147	−	12.3039i	−49.0887			9.25446	+	16.0292i	−110.796	+	86.1904i	−36.2660	+	124.466i			153.883i	−59.7739	+	235.534i	144.342	−	83.3357i
5.5		−	8.93539i	−15.4544	+	2.03988i	−47.8413			−44.7261	−	77.4678i	18.2271	+	138.091i	−111.739	−	65.7371i			141.548i	234.678	−	63.0502i	−692.205	+	399.645i
5.6		−	8.84415i	15.5553	−	1.01693i	−46.2191			−40.6414	−	70.3929i	−8.99385	−	137.573i	−36.9707	+	124.258i			125.755i	240.932	−	31.6371i	−622.566	+	359.439i
5.7		−	8.78035i	6.59181	+	14.1261i	−45.0945			−22.6857	−	39.2928i	124.032	−	57.8784i	78.2643	−	103.352i			114.974i	−156.096	+	186.234i	−345.004	+	199.188i
5.8		−	6.84620i	14.4986	−	5.72635i	−14.8704			28.3002	+	49.0174i	−39.2037	−	99.2602i	125.210	+	33.6063i		−	117.272i	177.418	−	166.048i	335.583	−	193.749i
5.9		−	6.53003i	5.54388	−	14.5693i	−10.6412			26.4452	+	45.8044i	−95.1381	−	36.2017i	−106.124	−	74.4630i		−	139.473i	−181.531	−	161.541i	299.104	−	172.688i
5.10		−	6.29189i	−7.03689	+	13.9098i	−7.58787			19.0177	+	32.9397i	87.5189	+	44.2753i	−124.098	−	37.5069i		−	153.598i	−143.964	−	195.763i	207.253	−	119.658i
5.11		−	5.53719i	−14.8489	−	4.74460i	1.33949			39.7402	+	68.8320i	−26.2718	+	82.2210i	64.2142	−	112.621i		−	184.607i	197.977	+	140.904i	381.136	−	220.049i
5.12		−	5.28457i	0.760712	+	15.5699i	4.07335			−18.3459	−	31.7760i	82.2801	−	4.02003i	43.8244	+	122.010i		−	190.632i	−241.843	+	23.6884i	−167.922	+	96.9499i
5.13		−	5.10258i	−6.99587	−	13.9305i	5.96367			−37.1817	−	64.4006i	−71.0813	+	35.6970i	111.893	−	65.4751i		−	193.713i	−145.116	+	194.911i	−328.609	+	189.723i
5.14		−	3.19435i	8.04498	−	13.3521i	21.7961			−32.5888	−	56.4455i	−42.6513	−	25.6985i	−122.855	+	41.3954i		−	171.844i	−113.557	−	214.835i	−180.307	+	104.100i
5.15		−	2.88308i	14.7732	+	4.97529i	23.6878			−10.7979	−	18.7025i	14.3441	−	42.5922i	−20.3787	−	128.030i		−	160.553i	193.493	+	147.001i	−53.9207	+	31.1311i
5.16		−	2.72832i	−15.0723	+	3.97815i	24.5563			−18.6475	−	32.2984i	10.8537	+	41.1221i	73.1597	+	107.026i		−	154.304i	211.349	−	119.920i	−88.1204	+	50.8763i
5.17		−	1.65180i	12.2585	+	9.62963i	29.2716			29.8897	+	51.7705i	15.9062	−	20.2485i	−44.0986	+	121.911i		−	101.208i	57.5403	+	236.089i	85.5146	−	49.3719i
5.18		−	0.279751i	−15.2895	−	3.03836i	31.9217			9.68296	+	16.7714i	−0.849983	+	4.27725i	−129.532	+	5.33078i		−	17.8822i	224.537	+	92.9099i	4.69181	−	2.70882i
5.19		−	0.0780188i	−1.61775	+	15.5043i	31.9939			46.1545	+	79.9420i	1.20963	+	0.126215i	103.353	−	78.2630i		−	4.99273i	−237.766	−	50.1642i	6.23698	−	3.60092i
5.20		−	0.0365780i	−2.02604	−	15.4562i	31.9987			31.2888	+	54.1937i	−0.565358	+	0.0741087i	39.4223	+	123.503i		−	2.34094i	−234.790	+	62.6300i	1.98230	−	1.14448i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.i	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.i.a	✓	76
3.b	odd	2	1		189.6.i.a		76
7.d	odd	6	1		63.6.s.a	yes	76
9.c	even	3	1		189.6.s.a		76
9.d	odd	6	1		63.6.s.a	yes	76
21.g	even	6	1		189.6.s.a		76
63.i	even	6	1	inner	63.6.i.a	✓	76
63.t	odd	6	1		189.6.i.a		76

    

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

Hecke kernels

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