Newform orbit 504.2.ch.b

• Label: 504.2.ch.b
• Level: $504$
• Weight: $2$
• Character orbit: 504.ch
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{4} - 20 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
269.1	−1.41410	+	0.0182389i		0		1.99933	−	0.0515832i	−0.785247	+	0.453362i		0		−2.47043	−	0.947077i	−2.82631	+	0.109409i		0		1.10215	−	0.655420i
269.2	−1.40843	−	0.127777i		0		1.96735	+	0.359931i	1.87230	−	1.08097i		0		2.55958	+	0.669737i	−2.72488	−	0.758320i		0		−2.77512	+	1.28324i
269.3	−1.39540	+	0.229930i		0		1.89426	−	0.641688i	−3.16007	+	1.82447i		0		−1.64838	+	2.06951i	−2.49571	+	1.33096i		0		3.99006	−	3.27246i
269.4	−1.30255	−	0.550794i		0		1.39325	+	1.43487i	−1.00441	+	0.579896i		0		1.24394	+	2.33508i	−1.02446	−	2.63638i		0		1.62769	−	0.202118i
269.5	−1.13135	−	0.848557i		0		0.559903	+	1.92003i	−1.53798	+	0.887954i		0		0.843933	−	2.50754i	0.995807	−	2.64733i		0		2.49347	+	0.300478i
269.6	−1.11071	−	0.875401i		0		0.467346	+	1.94463i	3.18706	−	1.84005i		0		−0.998380	−	2.45015i	1.18325	−	2.56903i		0		−5.15068	−	0.746198i
269.7	−0.937852	−	1.05851i		0		−0.240867	+	1.98544i	−0.331990	+	0.191675i		0		−2.03027	+	1.69647i	2.32750	−	1.60709i		0		0.514246	+	0.171651i
269.8	−0.896824	+	1.09348i		0		−0.391414	−	1.96132i	3.16007	−	1.82447i		0		−1.64838	+	2.06951i	2.49571	+	1.33096i		0		−0.839002	+	5.09172i
269.9	−0.722843	+	1.21552i		0		−0.954995	−	1.75727i	0.785247	−	0.453362i		0		−2.47043	−	0.947077i	2.82631	+	0.109409i		0		−0.0165377	+	1.28220i
269.10	−0.593556	+	1.28362i		0		−1.29538	−	1.52381i	−1.87230	+	1.08097i		0		2.55958	+	0.669737i	2.72488	−	0.758320i		0		−0.276248	−	3.04495i
269.11	−0.447766	−	1.34146i		0		−1.59901	+	1.20132i	−0.331990	+	0.191675i		0		−2.03027	+	1.69647i	2.32750	+	1.60709i		0		0.405777	+	0.359525i
269.12	−0.202765	−	1.39960i		0		−1.91777	+	0.567582i	3.18706	−	1.84005i		0		−0.998380	−	2.45015i	1.18325	+	2.56903i		0		−3.22157	−	4.08752i
269.13	−0.174271	+	1.40343i		0		−1.93926	−	0.489157i	1.00441	−	0.579896i		0		1.24394	+	2.33508i	1.02446	−	2.63638i		0		0.638806	+	1.51068i
269.14	−0.169197	−	1.40406i		0		−1.94274	+	0.475124i	−1.53798	+	0.887954i		0		0.843933	−	2.50754i	0.995807	+	2.64733i		0		1.50696	+	2.00917i
269.15	0.169197	+	1.40406i		0		−1.94274	+	0.475124i	1.53798	−	0.887954i		0		0.843933	−	2.50754i	−0.995807	−	2.64733i		0		1.50696	+	2.00917i
269.16	0.174271	−	1.40343i		0		−1.93926	−	0.489157i	−1.00441	+	0.579896i		0		1.24394	+	2.33508i	−1.02446	+	2.63638i		0		0.638806	+	1.51068i
269.17	0.202765	+	1.39960i		0		−1.91777	+	0.567582i	−3.18706	+	1.84005i		0		−0.998380	−	2.45015i	−1.18325	−	2.56903i		0		−3.22157	−	4.08752i
269.18	0.447766	+	1.34146i		0		−1.59901	+	1.20132i	0.331990	−	0.191675i		0		−2.03027	+	1.69647i	−2.32750	−	1.60709i		0		0.405777	+	0.359525i
269.19	0.593556	−	1.28362i		0		−1.29538	−	1.52381i	1.87230	−	1.08097i		0		2.55958	+	0.669737i	−2.72488	+	0.758320i		0		−0.276248	−	3.04495i
269.20	0.722843	−	1.21552i		0		−0.954995	−	1.75727i	−0.785247	+	0.453362i		0		−2.47043	−	0.947077i	−2.82631	−	0.109409i		0		−0.0165377	+	1.28220i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
8.b	even	2	1	inner
21.g	even	6	1	inner
24.h	odd	2	1	inner
56.j	odd	6	1	inner
168.ba	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.ch.b	✓	56
3.b	odd	2	1	inner	504.2.ch.b	✓	56
4.b	odd	2	1		2016.2.cp.b		56
7.d	odd	6	1	inner	504.2.ch.b	✓	56
8.b	even	2	1	inner	504.2.ch.b	✓	56
8.d	odd	2	1		2016.2.cp.b		56
12.b	even	2	1		2016.2.cp.b		56
21.g	even	6	1	inner	504.2.ch.b	✓	56
24.f	even	2	1		2016.2.cp.b		56
24.h	odd	2	1	inner	504.2.ch.b	✓	56
28.f	even	6	1		2016.2.cp.b		56
56.j	odd	6	1	inner	504.2.ch.b	✓	56
56.m	even	6	1		2016.2.cp.b		56
84.j	odd	6	1		2016.2.cp.b		56
168.ba	even	6	1	inner	504.2.ch.b	✓	56
168.be	odd	6	1		2016.2.cp.b		56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.ch.b	✓	56	1.a	even	1	1	trivial
504.2.ch.b	✓	56	3.b	odd	2	1	inner
504.2.ch.b	✓	56	7.d	odd	6	1	inner
504.2.ch.b	✓	56	8.b	even	2	1	inner
504.2.ch.b	✓	56	21.g	even	6	1	inner
504.2.ch.b	✓	56	24.h	odd	2	1	inner
504.2.ch.b	✓	56	56.j	odd	6	1	inner
504.2.ch.b	✓	56	168.ba	even	6	1	inner
2016.2.cp.b		56	4.b	odd	2	1
2016.2.cp.b		56	8.d	odd	2	1
2016.2.cp.b		56	12.b	even	2	1
2016.2.cp.b		56	24.f	even	2	1
2016.2.cp.b		56	28.f	even	6	1
2016.2.cp.b		56	56.m	even	6	1
2016.2.cp.b		56	84.j	odd	6	1
2016.2.cp.b		56	168.be	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^28 - 37*T5^26 + 882*T5^24 - 12429*T5^22 + 126330*T5^20 - 820581*T5^18 + 3879393*T5^16 - 12338064*T5^14 + 28587144*T5^12 - 42674992*T5^10 + 45801904*T5^8 - 29457024*T5^6 + 12848256*T5^4 - 1762560*T5^2 + 186624