Newform orbit 56.5.k.a

• Label: 56.5.k.a
• Level: $56$
• Weight: $5$
• Character orbit: 56.k
• Analytic conductor: $5.789$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	56.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.78871793270\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{2} - 2 q^{3} - 4 q^{4} + 28 q^{6} - 4 q^{8} - 704 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−3.99270	+	0.241514i	2.90447	−	5.03069i	15.8833	−	1.92859i	−35.7453	+	20.6376i	−10.3817	+	20.7875i	33.5978	−	35.6678i	−62.9517	+	11.5363i	23.6281	+	40.9250i	137.736	−	91.0326i
11.2	−3.91997	−	0.796124i	4.42688	−	7.66758i	14.7324	+	6.24157i	19.7872	−	11.4241i	−23.4576	+	26.5324i	24.1747	+	42.6214i	−52.7814	−	36.1956i	1.30543	+	2.26108i	−86.6602	+	29.0292i
11.3	−3.73587	−	1.42944i	−2.68606	+	4.65239i	11.9134	+	10.6804i	15.6318	−	9.02500i	16.6851	−	13.5412i	−44.4242	−	20.6759i	−29.2399	−	56.9301i	26.0702	+	45.1548i	−71.2989	+	11.3715i
11.4	−3.72154	+	1.46634i	−5.61505	+	9.72556i	11.6997	−	10.9141i	29.2132	−	16.8663i	6.63571	−	44.4276i	48.9331	+	2.56001i	−27.5373	+	57.7728i	−22.5576	−	39.0709i	−83.9866	+	105.605i
11.5	−3.66880	+	1.59370i	−4.03038	+	6.98083i	10.9202	−	11.6939i	−15.7223	+	9.07726i	3.66135	−	32.0345i	−48.8060	+	4.35649i	−21.4276	+	60.3064i	8.01201	+	13.8772i	43.2155	−	58.3593i
11.6	−3.49436	+	1.94665i	6.97904	−	12.0881i	8.42110	−	13.6046i	12.6763	−	7.31864i	−0.856068	+	55.8258i	−37.6711	−	31.3351i	−2.94298	+	63.9323i	−56.9141	−	98.5781i	−30.0486	+	50.2502i
11.7	−3.42567	−	2.06513i	−8.62600	+	14.9407i	7.47046	+	14.1489i	−27.4474	+	15.8467i	60.4043	−	33.3680i	24.0616	+	42.6853i	3.62804	−	63.8971i	−108.316	−	187.609i	126.751	+	2.39667i
11.8	−2.66196	−	2.98562i	6.96678	−	12.0668i	−1.82790	+	15.8952i	−33.4946	+	19.3381i	−54.5723	+	11.3212i	−35.7007	+	33.5627i	52.3230	−	36.8552i	−56.5720	−	97.9856i	146.898	+	48.5249i
11.9	−2.66151	−	2.98603i	−0.962732	+	1.66750i	−1.83276	+	15.8947i	−3.89042	+	2.24614i	7.54153	−	1.56332i	31.0044	−	37.9437i	52.3399	−	36.8312i	38.6463	+	66.9373i	17.0614	+	5.63881i
11.10	−2.31118	+	3.26473i	2.16063	−	3.74231i	−5.31692	−	15.0907i	−4.23878	+	2.44726i	7.22405	+	15.7030i	7.11583	+	48.4806i	61.5555	+	17.5191i	31.1634	+	53.9766i	1.80693	−	19.4945i
11.11	−1.61448	+	3.65971i	−5.08078	+	8.80018i	−10.7869	−	11.8170i	−25.7945	+	14.8925i	−24.0033	−	32.8019i	24.6862	−	42.3272i	60.6622	−	20.3986i	−11.1287	−	19.2755i	−12.8574	−	118.444i
11.12	−1.25522	−	3.79795i	−0.962732	+	1.66750i	−12.8488	+	9.53456i	3.89042	−	2.24614i	7.54153	+	1.56332i	−31.0044	+	37.9437i	52.3399	+	36.8312i	38.6463	+	66.9373i	−13.4141	−	11.9562i
11.13	−1.25464	−	3.79814i	6.96678	−	12.0668i	−12.8517	+	9.53063i	33.4946	−	19.3381i	−54.5723	−	11.3212i	35.7007	−	33.5627i	52.3230	+	36.8552i	−56.5720	−	97.9856i	−115.473	−	102.955i
11.14	−0.907680	+	3.89565i	−0.285840	+	0.495090i	−14.3522	−	7.07202i	40.2341	−	23.2292i	−1.66925	−	1.56292i	−22.8899	−	43.3249i	40.5774	−	49.4922i	40.3366	+	69.8650i	53.9731	+	177.823i
11.15	−0.229998	+	3.99338i	7.74459	−	13.4140i	−15.8942	−	1.83694i	−13.5276	+	7.81018i	51.7861	+	34.0123i	43.5544	−	22.4503i	10.9912	−	63.0491i	−79.4574	−	137.624i	−28.0777	−	55.8173i
11.16	−0.0756202	−	3.99929i	−8.62600	+	14.9407i	−15.9886	+	0.604854i	27.4474	−	15.8467i	60.4043	+	33.3680i	−24.0616	−	42.6853i	3.62804	+	63.8971i	−108.316	−	187.609i	−65.4512	−	108.571i
11.17	0.630001	−	3.95008i	−2.68606	+	4.65239i	−15.2062	−	4.97710i	−15.6318	+	9.02500i	16.6851	+	13.5412i	44.4242	+	20.6759i	−29.2399	+	56.9301i	26.0702	+	45.1548i	25.8014	+	67.4324i
11.18	0.635272	+	3.94923i	−6.43733	+	11.1498i	−15.1929	+	5.01767i	8.48321	−	4.89778i	−48.1225	−	18.3394i	−4.25339	+	48.8150i	−29.4675	−	56.8125i	−42.3785	−	73.4017i	24.7316	+	30.3907i
11.19	1.27052	−	3.79286i	4.42688	−	7.66758i	−12.7715	−	9.63782i	−19.7872	+	11.4241i	−23.4576	−	26.5324i	−24.1747	−	42.6214i	−52.7814	+	36.1956i	1.30543	+	2.26108i	18.1900	+	89.5645i
11.20	1.42092	+	3.73911i	2.04179	−	3.53649i	−11.9620	+	10.6260i	−20.9617	+	12.1023i	16.1246	+	2.60942i	−48.9779	−	1.47008i	−56.7288	−	29.6284i	32.1621	+	55.7065i	−75.0368	−	61.1819i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
8.d	odd	2	1	inner
56.k	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	56.5.k.a	✓	60
4.b	odd	2	1		224.5.o.a		60
7.c	even	3	1	inner	56.5.k.a	✓	60
8.b	even	2	1		224.5.o.a		60
8.d	odd	2	1	inner	56.5.k.a	✓	60
28.g	odd	6	1		224.5.o.a		60
56.k	odd	6	1	inner	56.5.k.a	✓	60
56.p	even	6	1		224.5.o.a		60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.5.k.a	✓	60	1.a	even	1	1	trivial
56.5.k.a	✓	60	7.c	even	3	1	inner
56.5.k.a	✓	60	8.d	odd	2	1	inner
56.5.k.a	✓	60	56.k	odd	6	1	inner
224.5.o.a		60	4.b	odd	2	1
224.5.o.a		60	8.b	even	2	1
224.5.o.a		60	28.g	odd	6	1
224.5.o.a		60	56.p	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(56, [\chi])\).