Newform orbit 639.2.i.a

• Label: 639.2.i.a
• Level: $639$
• Weight: $2$
• Character orbit: 639.i
• Analytic conductor: $5.102$
• Analytic rank: $0$
• Dimension: $28$
• CM discriminant: -71
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	639.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.10244068916\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
212.1	−2.44899	+	1.41392i	−1.45586	+	0.938338i	2.99835	−	5.19330i	3.04872	+	1.76018i	2.23864	−	4.35645i		0				11.3021i	1.23904	−	2.73217i	−9.95503
212.2	−2.42904	+	1.40241i	−0.674301	−	1.59541i	2.93349	−	5.08095i	−1.65029	−	0.952795i	3.87531	+	2.92966i		0				10.8461i	−2.09063	+	2.15157i	5.34482
212.3	−1.76143	+	1.01696i	1.29974	+	1.14484i	1.06843	−	1.85058i	3.78317	+	2.18421i	−3.45368	−	0.694769i		0				0.278373i	0.378674	+	2.97601i	−8.88506
212.4	−1.56577	+	0.903997i	0.904553	−	1.47709i	0.634422	−	1.09885i	−3.00710	−	1.73615i	−0.0810390	+	3.13049i		0			−	1.32193i	−1.36357	−	2.67221i	6.27789
212.5	−1.48807	+	0.859135i	1.71881	−	0.213740i	0.476227	−	0.824849i	1.65029	+	0.952795i	−2.37407	+	1.79475i		0			−	1.79997i	2.90863	−	0.734759i	−3.27432
212.6	−1.26753	+	0.731808i	−0.0846957	+	1.72998i	0.0710852	−	0.123123i	−3.04872	−	1.76018i	−1.15866	−	2.25478i		0			−	2.71915i	−2.98565	−	0.293043i	5.15245
212.7	0.232567	−	0.134273i	−1.66776	−	0.467529i	−0.963942	+	1.66960i	−0.0333787	−	0.0192712i	−0.450642	+	0.115202i		0				1.05481i	2.56283	+	1.55945i	−0.0103504
212.8	0.496503	−	0.286656i	−0.174090	+	1.72328i	−0.835656	+	1.44740i	−1.71044	−	0.987520i	0.407553	+	0.905518i		0				2.10481i	−2.93939	−	0.600012i	−1.13232
212.9	0.593398	−	0.342598i	−1.64133	−	0.553191i	−0.765253	+	1.32546i	−3.78317	−	2.18421i	−1.16349	+	0.234056i		0				2.41909i	2.38796	+	1.81594i	−2.99323
212.10	0.848457	−	0.489857i	0.826918	−	1.52191i	−0.520080	+	0.900806i	3.00710	+	1.73615i	−0.0439133	−	1.69635i		0				2.97849i	−1.63241	−	2.51699i	3.40186
212.11	2.05144	−	1.18440i	1.70545	−	0.302384i	1.80560	−	3.12740i	−3.76831	−	2.17564i	3.14049	−	2.64026i		0			−	3.81663i	2.81713	−	1.03140i	−10.3073
212.12	2.18490	−	1.26145i	−0.590853	−	1.62816i	2.18252	−	3.78024i	3.76831	+	2.17564i	−3.34479	−	2.81202i		0			−	5.96678i	−2.30179	+	1.92400i	10.9778
212.13	2.22802	−	1.28635i	1.23877	+	1.21056i	2.30938	−	3.99997i	0.0333787	+	0.0192712i	4.31720	+	1.10365i		0			−	6.73730i	0.0691059	+	2.99920i	0.0991580
212.14	2.32554	−	1.34265i	−1.40536	+	1.01241i	2.60541	−	4.51271i	1.71044	+	0.987520i	−1.90891	+	4.24129i		0			−	8.62203i	0.950067	−	2.84559i	5.30357
425.1	−2.44899	−	1.41392i	−1.45586	−	0.938338i	2.99835	+	5.19330i	3.04872	−	1.76018i	2.23864	+	4.35645i		0			−	11.3021i	1.23904	+	2.73217i	−9.95503
425.2	−2.42904	−	1.40241i	−0.674301	+	1.59541i	2.93349	+	5.08095i	−1.65029	+	0.952795i	3.87531	−	2.92966i		0			−	10.8461i	−2.09063	−	2.15157i	5.34482
425.3	−1.76143	−	1.01696i	1.29974	−	1.14484i	1.06843	+	1.85058i	3.78317	−	2.18421i	−3.45368	+	0.694769i		0			−	0.278373i	0.378674	−	2.97601i	−8.88506
425.4	−1.56577	−	0.903997i	0.904553	+	1.47709i	0.634422	+	1.09885i	−3.00710	+	1.73615i	−0.0810390	−	3.13049i		0				1.32193i	−1.36357	+	2.67221i	6.27789
425.5	−1.48807	−	0.859135i	1.71881	+	0.213740i	0.476227	+	0.824849i	1.65029	−	0.952795i	−2.37407	−	1.79475i		0				1.79997i	2.90863	+	0.734759i	−3.27432
425.6	−1.26753	−	0.731808i	−0.0846957	−	1.72998i	0.0710852	+	0.123123i	−3.04872	+	1.76018i	−1.15866	+	2.25478i		0				2.71915i	−2.98565	+	0.293043i	5.15245

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.b	odd	2	1	CM by \(\Q(\sqrt{-71}) \)
9.d	odd	6	1	inner
639.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	639.2.i.a	✓	28
9.d	odd	6	1	inner	639.2.i.a	✓	28
71.b	odd	2	1	CM	639.2.i.a	✓	28
639.i	even	6	1	inner	639.2.i.a	✓	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
639.2.i.a	✓	28	1.a	even	1	1	trivial
639.2.i.a	✓	28	9.d	odd	6	1	inner
639.2.i.a	✓	28	71.b	odd	2	1	CM
639.2.i.a	✓	28	639.i	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^28 - 28*T2^26 + 476*T2^24 - 5264*T2^22 - 63*T2^21 + 43120*T2^20 + 1764*T2^19 - 260288*T2^18 - 21168*T2^17 + 1204224*T2^16 + 162288*T2^15 - 4125340*T2^14 - 691488*T2^13 + 10588718*T2^12 + 1778112*T2^11 - 18684904*T2^10 - 592704*T2^9 + 24223024*T2^8 - 5551623*T2^7 - 19502000*T2^6 + 11339874*T2^5 + 7648704*T2^4 - 11063808*T2^3 + 5130496*T2^2 - 1104264*T2 + 97969