Newform orbit 504.2.bk.c

• Label: 504.2.bk.c
• Level: $504$
• Weight: $2$
• Character orbit: 504.bk
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 168)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 2 q^{4} - 16 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.41405	−	0.0213058i		0		1.99909	+	0.0602550i	1.14053	−	1.97545i		0		1.95181	+	1.78618i	−2.82554	−	0.127796i		0		−1.65485	+	2.76909i
19.2	−1.34646	−	0.432485i		0		1.62591	+	1.16465i	0.155280	−	0.268953i		0		−2.58581	−	0.560001i	−1.68554	−	2.27133i		0		−0.325396	+	0.294978i
19.3	−1.34597	+	0.434022i		0		1.62325	−	1.16836i	−1.44142	+	2.49662i		0		−2.63862	+	0.194181i	−1.67775	+	2.27710i		0		0.856518	−	3.98597i
19.4	−1.09919	+	0.889821i		0		0.416438	−	1.95616i	0.225540	−	0.390646i		0		0.458196	+	2.60577i	1.28289	+	2.52075i		0		0.0996941	+	0.630084i
19.5	−0.647235	−	1.25741i		0		−1.16217	+	1.62768i	2.08776	−	3.61611i		0		2.39694	−	1.12013i	2.79887	+	0.407838i		0		−5.89822	−	0.284706i
19.6	−0.582416	−	1.28872i		0		−1.32158	+	1.50114i	0.128707	−	0.222928i		0		−0.623918	+	2.57113i	2.70425	+	0.828860i		0		−0.362252	−	0.0360307i
19.7	−0.221012	+	1.39684i		0		−1.90231	−	0.617436i	−0.225540	+	0.390646i		0		−0.458196	−	2.60577i	1.28289	−	2.52075i		0		−0.495822	−	0.401380i
19.8	0.297109	+	1.38265i		0		−1.82345	+	0.821596i	1.44142	−	2.49662i		0		2.63862	−	0.194181i	−1.67775	−	2.27710i		0		3.88021	+	1.25122i
19.9	0.321935	−	1.37708i		0		−1.79272	−	0.886663i	−1.25150	+	2.16767i		0		1.36321	−	2.26752i	−1.79815	+	2.18327i		0		2.58215	+	2.42127i
19.10	0.647418	−	1.25732i		0		−1.16170	−	1.62802i	1.61398	−	2.79550i		0		−1.82725	−	1.91341i	−2.79905	+	0.406616i		0		−2.46991	−	3.83914i
19.11	0.725478	+	1.21395i		0		−0.947364	+	1.76139i	−1.14053	+	1.97545i		0		−1.95181	−	1.78618i	−2.82554	+	0.127796i		0		−3.22553	+	0.0485996i
19.12	0.765161	−	1.18934i		0		−0.829058	−	1.82007i	−1.61398	+	2.79550i		0		1.82725	+	1.91341i	−2.79905	−	0.406616i		0		2.08984	+	4.05858i
19.13	1.03162	−	0.967346i		0		0.128485	−	1.99587i	1.25150	−	2.16767i		0		−1.36321	+	2.26752i	−1.79815	−	2.18327i		0		−0.805806	−	3.44685i
19.14	1.04777	+	0.949827i		0		0.195657	+	1.99041i	−0.155280	+	0.268953i		0		2.58581	+	0.560001i	−1.68554	+	2.27133i		0		−0.418156	+	0.134312i
19.15	1.40727	−	0.139971i		0		1.96082	−	0.393955i	−0.128707	+	0.222928i		0		0.623918	−	2.57113i	2.70425	−	0.828860i		0		−0.149923	+	0.331735i
19.16	1.41257	−	0.0681843i		0		1.99070	−	0.192630i	−2.08776	+	3.61611i		0		−2.39694	+	1.12013i	2.79887	−	0.407838i		0		−2.70255	+	5.25036i
451.1	−1.41405	+	0.0213058i		0		1.99909	−	0.0602550i	1.14053	+	1.97545i		0		1.95181	−	1.78618i	−2.82554	+	0.127796i		0		−1.65485	−	2.76909i
451.2	−1.34646	+	0.432485i		0		1.62591	−	1.16465i	0.155280	+	0.268953i		0		−2.58581	+	0.560001i	−1.68554	+	2.27133i		0		−0.325396	−	0.294978i
451.3	−1.34597	−	0.434022i		0		1.62325	+	1.16836i	−1.44142	−	2.49662i		0		−2.63862	−	0.194181i	−1.67775	−	2.27710i		0		0.856518	+	3.98597i
451.4	−1.09919	−	0.889821i		0		0.416438	+	1.95616i	0.225540	+	0.390646i		0		0.458196	−	2.60577i	1.28289	−	2.52075i		0		0.0996941	−	0.630084i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
8.d	odd	2	1	inner
56.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	504.2.bk.c		32
3.b	odd	2	1		168.2.t.a	✓	32
4.b	odd	2	1		2016.2.bs.c		32
7.d	odd	6	1	inner	504.2.bk.c		32
8.b	even	2	1		2016.2.bs.c		32
8.d	odd	2	1	inner	504.2.bk.c		32
12.b	even	2	1		672.2.bb.a		32
21.g	even	6	1		168.2.t.a	✓	32
21.g	even	6	1		1176.2.p.a		32
21.h	odd	6	1		1176.2.p.a		32
24.f	even	2	1		168.2.t.a	✓	32
24.h	odd	2	1		672.2.bb.a		32
28.f	even	6	1		2016.2.bs.c		32
56.j	odd	6	1		2016.2.bs.c		32
56.m	even	6	1	inner	504.2.bk.c		32
84.j	odd	6	1		672.2.bb.a		32
84.j	odd	6	1		4704.2.p.a		32
84.n	even	6	1		4704.2.p.a		32
168.s	odd	6	1		4704.2.p.a		32
168.v	even	6	1		1176.2.p.a		32
168.ba	even	6	1		672.2.bb.a		32
168.ba	even	6	1		4704.2.p.a		32
168.be	odd	6	1		168.2.t.a	✓	32
168.be	odd	6	1		1176.2.p.a		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.2.t.a	✓	32	3.b	odd	2	1
168.2.t.a	✓	32	21.g	even	6	1
168.2.t.a	✓	32	24.f	even	2	1
168.2.t.a	✓	32	168.be	odd	6	1
504.2.bk.c		32	1.a	even	1	1	trivial
504.2.bk.c		32	7.d	odd	6	1	inner
504.2.bk.c		32	8.d	odd	2	1	inner
504.2.bk.c		32	56.m	even	6	1	inner
672.2.bb.a		32	12.b	even	2	1
672.2.bb.a		32	24.h	odd	2	1
672.2.bb.a		32	84.j	odd	6	1
672.2.bb.a		32	168.ba	even	6	1
1176.2.p.a		32	21.g	even	6	1
1176.2.p.a		32	21.h	odd	6	1
1176.2.p.a		32	168.v	even	6	1
1176.2.p.a		32	168.be	odd	6	1
2016.2.bs.c		32	4.b	odd	2	1
2016.2.bs.c		32	8.b	even	2	1
2016.2.bs.c		32	28.f	even	6	1
2016.2.bs.c		32	56.j	odd	6	1
4704.2.p.a		32	84.j	odd	6	1
4704.2.p.a		32	84.n	even	6	1
4704.2.p.a		32	168.s	odd	6	1
4704.2.p.a		32	168.ba	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^32 + 48*T5^30 + 1426*T5^28 + 26656*T5^26 + 365635*T5^24 + 3640464*T5^22 + 27644914*T5^20 + 153899712*T5^18 + 637309729*T5^16 + 1742145008*T5^14 + 3033922768*T5^12 + 1037391232*T5^10 + 247743680*T5^8 + 30407680*T5^6 + 2704384*T5^4 + 126976*T5^2 + 4096