Newform orbit 864.2.y.d

• Label: 864.2.y.d
• Level: $864$
• Weight: $2$
• Character orbit: 864.y
• Analytic conductor: $6.899$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.y (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89907473464\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
97.1		0		−1.71652	+	0.231448i		0		−3.16823	−	1.15314i		0		0.327751	−	1.85877i		0		2.89286	−	0.794570i		0
97.2		0		−1.65691	+	0.504627i		0		1.76220	+	0.641388i		0		−0.498467	+	2.82694i		0		2.49070	−	1.67224i		0
97.3		0		−1.01321	−	1.40478i		0		−1.03120	−	0.375327i		0		−0.257635	+	1.46112i		0		−0.946819	+	2.84667i		0
97.4		0		−0.790581	−	1.54110i		0		3.49481	+	1.27201i		0		0.718530	−	4.07499i		0		−1.74996	+	2.43673i		0
97.5		0		−0.523054	+	1.65119i		0		−0.117882	−	0.0429055i		0		0.455772	−	2.58481i		0		−2.45283	−	1.72732i		0
97.6		0		0.523054	−	1.65119i		0		−0.117882	−	0.0429055i		0		−0.455772	+	2.58481i		0		−2.45283	−	1.72732i		0
97.7		0		0.790581	+	1.54110i		0		3.49481	+	1.27201i		0		−0.718530	+	4.07499i		0		−1.74996	+	2.43673i		0
97.8		0		1.01321	+	1.40478i		0		−1.03120	−	0.375327i		0		0.257635	−	1.46112i		0		−0.946819	+	2.84667i		0
97.9		0		1.65691	−	0.504627i		0		1.76220	+	0.641388i		0		0.498467	−	2.82694i		0		2.49070	−	1.67224i		0
97.10		0		1.71652	−	0.231448i		0		−3.16823	−	1.15314i		0		−0.327751	+	1.85877i		0		2.89286	−	0.794570i		0
193.1		0		−1.72168	−	0.189239i		0		0.679279	−	3.85238i		0		−2.91962	+	2.44985i		0		2.92838	+	0.651618i		0
193.2		0		−1.68152	+	0.415309i		0		−0.564773	+	3.20298i		0		−2.06064	+	1.72908i		0		2.65504	−	1.39670i		0
193.3		0		−1.28924	+	1.15666i		0		−0.110022	+	0.623966i		0		3.44796	−	2.89318i		0		0.324271	−	2.98242i		0
193.4		0		−0.998006	−	1.41562i		0		−0.476400	+	2.70180i		0		0.930891	−	0.781110i		0		−1.00797	+	2.82560i		0
193.5		0		−0.141380	+	1.72627i		0		0.298268	−	1.69156i		0		−0.0848018	+	0.0711572i		0		−2.96002	−	0.488119i		0
193.6		0		0.141380	−	1.72627i		0		0.298268	−	1.69156i		0		0.0848018	−	0.0711572i		0		−2.96002	−	0.488119i		0
193.7		0		0.998006	+	1.41562i		0		−0.476400	+	2.70180i		0		−0.930891	+	0.781110i		0		−1.00797	+	2.82560i		0
193.8		0		1.28924	−	1.15666i		0		−0.110022	+	0.623966i		0		−3.44796	+	2.89318i		0		0.324271	−	2.98242i		0
193.9		0		1.68152	−	0.415309i		0		−0.564773	+	3.20298i		0		2.06064	−	1.72908i		0		2.65504	−	1.39670i		0
193.10		0		1.72168	+	0.189239i		0		0.679279	−	3.85238i		0		2.91962	−	2.44985i		0		2.92838	+	0.651618i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
27.e	even	9	1	inner
108.j	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	864.2.y.d	✓	60
4.b	odd	2	1	inner	864.2.y.d	✓	60
27.e	even	9	1	inner	864.2.y.d	✓	60
108.j	odd	18	1	inner	864.2.y.d	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
864.2.y.d	✓	60	1.a	even	1	1	trivial
864.2.y.d	✓	60	4.b	odd	2	1	inner
864.2.y.d	✓	60	27.e	even	9	1	inner
864.2.y.d	✓	60	108.j	odd	18	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\):

T5^30 + 6*T5^28 - 19*T5^27 + 63*T5^26 - 675*T5^25 + 3456*T5^24 + 270*T5^23 + 12006*T5^22 + 32910*T5^21 - 110277*T5^20 - 36891*T5^19 + 5454057*T5^18 - 5670288*T5^17 + 38454480*T5^16 - 64453833*T5^15 + 6453216*T5^14 + 6415488*T5^13 - 46378302*T5^12 + 182526282*T5^11 + 648979668*T5^10 + 233741913*T5^9 - 141687648*T5^8 + 84954231*T5^7 + 290361645*T5^6 + 19060056*T5^5 + 142231644*T5^4 - 44959672*T5^3 + 5313888*T5^2 + 3861408*T5 + 341056

T7^60 - 21*T7^58 + 405*T7^56 + 4647*T7^54 - 122247*T7^52 + 3429648*T7^50 + 36003726*T7^48 - 665047908*T7^46 + 14409106767*T7^44 + 4135988142*T7^42 - 1031987293470*T7^40 + 20247963820056*T7^38 + 66493664344269*T7^36 - 1106804855944800*T7^34 + 9135389772541260*T7^32 + 55252118608258734*T7^30 - 325394889128996886*T7^28 + 2641688492668860483*T7^26 + 21565600611249156741*T7^24 - 81169025622683446728*T7^22 + 168171436900233937395*T7^20 + 4645449475766190655218*T7^18 + 7525600430759861469156*T7^16 - 14112603550681256806272*T7^14 + 25587942682483626677217*T7^12 + 183363582154487896665936*T7^10 + 194323612644299132283456*T7^8 + 320252144158550496143232*T7^6 + 702268816003058338845696*T7^4 - 2946339846016726901760*T7^2 + 105665856469516652544