Newform orbit 816.2.cj.f

• Label: 816.2.cj.f
• Level: $816$
• Weight: $2$
• Character orbit: 816.cj
• Analytic conductor: $6.516$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.cj (of order \(16\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.51579280494\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) over \(\Q(\zeta_{16})\)
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
65.1		0		−1.70679	−	0.294712i		0		1.03780	+	1.55318i		0		−1.66829	−	1.11471i		0		2.82629	+	1.00602i		0
65.2		0		−1.55059	+	0.771806i		0		−1.28076	−	1.91679i		0		0.176284	+	0.117789i		0		1.80863	−	2.39350i		0
65.3		0		−1.21751	−	1.23194i		0		−0.254485	−	0.380864i		0		1.87193	+	1.25078i		0		−0.0353621	+	2.99979i		0
65.4		0		0.0580152	+	1.73108i		0		1.04136	+	1.55850i		0		2.88539	+	1.92796i		0		−2.99327	+	0.200858i		0
65.5		0		0.197491	−	1.72075i		0		1.60395	+	2.40048i		0		−2.36431	−	1.57978i		0		−2.92199	−	0.679668i		0
65.6		0		0.430496	+	1.67770i		0		0.503287	+	0.753222i		0		−4.13347	−	2.76189i		0		−2.62935	+	1.44449i		0
65.7		0		1.33848	+	1.09930i		0		−1.81973	−	2.72341i		0		1.68007	+	1.12259i		0		0.583074	+	2.94279i		0
65.8		0		1.47754	−	0.903815i		0		−1.04129	−	1.55841i		0		−1.50251	−	1.00395i		0		1.36624	−	2.67084i		0
65.9		0		1.67997	−	0.421554i		0		2.05763	+	3.07946i		0		3.05491	+	2.04122i		0		2.64458	−	1.41640i		0
113.1		0		−1.70679	+	0.294712i		0		1.03780	−	1.55318i		0		−1.66829	+	1.11471i		0		2.82629	−	1.00602i		0
113.2		0		−1.55059	−	0.771806i		0		−1.28076	+	1.91679i		0		0.176284	−	0.117789i		0		1.80863	+	2.39350i		0
113.3		0		−1.21751	+	1.23194i		0		−0.254485	+	0.380864i		0		1.87193	−	1.25078i		0		−0.0353621	−	2.99979i		0
113.4		0		0.0580152	−	1.73108i		0		1.04136	−	1.55850i		0		2.88539	−	1.92796i		0		−2.99327	−	0.200858i		0
113.5		0		0.197491	+	1.72075i		0		1.60395	−	2.40048i		0		−2.36431	+	1.57978i		0		−2.92199	+	0.679668i		0
113.6		0		0.430496	−	1.67770i		0		0.503287	−	0.753222i		0		−4.13347	+	2.76189i		0		−2.62935	−	1.44449i		0
113.7		0		1.33848	−	1.09930i		0		−1.81973	+	2.72341i		0		1.68007	−	1.12259i		0		0.583074	−	2.94279i		0
113.8		0		1.47754	+	0.903815i		0		−1.04129	+	1.55841i		0		−1.50251	+	1.00395i		0		1.36624	+	2.67084i		0
113.9		0		1.67997	+	0.421554i		0		2.05763	−	3.07946i		0		3.05491	−	2.04122i		0		2.64458	+	1.41640i		0
209.1		0		−1.71825	−	0.218217i		0		3.19088	+	0.634705i		0		0.130307	+	0.655095i		0		2.90476	+	0.749904i		0
209.2		0		−1.42432	−	0.985552i		0		−3.81385	−	0.758621i		0		−0.838845	−	4.21716i		0		1.05737	+	2.80748i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
51.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	816.2.cj.f		72
3.b	odd	2	1		816.2.cj.e		72
4.b	odd	2	1		408.2.bh.a	✓	72
12.b	even	2	1		408.2.bh.b	yes	72
17.e	odd	16	1		816.2.cj.e		72
51.i	even	16	1	inner	816.2.cj.f		72
68.i	even	16	1		408.2.bh.b	yes	72
204.t	odd	16	1		408.2.bh.a	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
408.2.bh.a	✓	72	4.b	odd	2	1
408.2.bh.a	✓	72	204.t	odd	16	1
408.2.bh.b	yes	72	12.b	even	2	1
408.2.bh.b	yes	72	68.i	even	16	1
816.2.cj.e		72	3.b	odd	2	1
816.2.cj.e		72	17.e	odd	16	1
816.2.cj.f		72	1.a	even	1	1	trivial
816.2.cj.f		72	51.i	even	16	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^72 - 8*T5^70 - 16*T5^69 - 60*T5^68 + 1088*T5^67 + 2304*T5^66 - 13088*T5^65 - 27128*T5^64 + 19328*T5^63 + 125912*T5^62 - 768544*T5^61 - 5506256*T5^60 - 4763360*T5^59 + 162787944*T5^58 + 453350272*T5^57 + 318415373*T5^56 - 2205435040*T5^55 - 4546577648*T5^54 - 16733530384*T5^53 - 131258781444*T5^52 - 90697389600*T5^51 + 457956089608*T5^50 - 10029361152*T5^49 + 5396577284488*T5^48 + 24697512813440*T5^47 + 48894152579280*T5^46 - 10629430315616*T5^45 + 92945636579512*T5^44 - 1148728144835264*T5^43 + 2746609396680176*T5^42 - 2447754759516224*T5^41 + 9761908419286972*T5^40 - 27160475476513536*T5^39 + 27207875141871168*T5^38 - 56716993978016896*T5^37 - 260604010330985312*T5^36 + 318412135663815424*T5^35 + 1611846664337798208*T5^34 + 786609688433994240*T5^33 + 2681062266276633792*T5^32 + 25425474887259622400*T5^31 + 53575224400584331776*T5^30 + 28901140014659756800*T5^29 + 40620964333842216128*T5^28 + 257868341620882037760*T5^27 + 357649785267369621504*T5^26 + 19991950102959558144*T5^25 + 79949360503676083824*T5^24 + 949018694611112805888*T5^23 + 1169271228704565391104*T5^22 + 371097483869654011648*T5^21 - 32487474114683383360*T5^20 - 420245483487914684928*T5^19 - 1519537006094999314304*T5^18 - 1668732289484851011584*T5^17 - 404823025765259309952*T5^16 + 488532173609324369920*T5^15 + 815111023056350220544*T5^14 + 1474119300835603311104*T5^13 + 2049776580473235481472*T5^12 + 1924067178056318542848*T5^11 + 1342152046337038433024*T5^10 + 743424805856776330240*T5^9 + 343365341113478398016*T5^8 + 133703803268485619712*T5^7 + 43948535503092360192*T5^6 + 12208640822910373888*T5^5 + 2874206847720740864*T5^4 + 555736656209821696*T5^3 + 84185334763667456*T5^2 + 9508594165415936*T5 + 616397059162112