Newform orbit 245.4.x.a

• Label: 245.4.x.a
• Level: $245$
• Weight: $4$
• Character orbit: 245.x
• Analytic conductor: $14.455$
• Analytic rank: $0$
• Dimension: $1968$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.4554679514\)
Analytic rank:	\(0\)
Dimension:	\(1968\)
Relative dimension:	\(82\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1968 q - 26 q^{2} - 22 q^{3} + 2 q^{5} - 112 q^{6} - 32 q^{7} + 96 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.19555	−	5.03225i	−0.568398	−	0.660490i	−15.0618	+	16.2327i	5.01771	−	9.99113i	−2.07581	+	4.31046i	−4.13261	−	18.0533i	73.2980	+	25.6481i	3.91097	−	25.9476i	−61.2945	−	3.31436i
3.2	−2.17912	−	4.99460i	5.64085	+	6.55479i	−14.7560	+	15.9032i	−1.73058	+	11.0456i	20.4464	−	42.4574i	18.0098	−	4.31803i	70.4377	+	24.6472i	−7.12189	+	47.2507i	58.9394	−	15.4261i
3.3	−2.09561	−	4.80320i	1.02986	+	1.19672i	−13.2377	+	14.2669i	7.88484	+	7.92649i	3.58988	−	7.45447i	−16.7675	+	7.86463i	56.6967	+	19.8390i	3.65262	−	24.2335i	21.5489	−	54.4833i
3.4	−2.09171	−	4.79426i	−6.09036	−	7.07713i	−13.1682	+	14.1920i	8.40956	+	7.36744i	−21.1903	+	44.0021i	9.65855	−	15.8023i	56.0869	+	19.6257i	−8.96911	+	59.5061i	17.7310	−	55.7282i
3.5	−2.08690	−	4.78323i	−0.587086	−	0.682206i	−13.0828	+	14.0999i	−6.27246	−	9.25507i	−2.03796	+	4.23187i	16.6026	+	8.20701i	55.3390	+	19.3640i	3.90341	−	25.8974i	−31.1791	+	49.3171i
3.6	−2.04360	−	4.68398i	−3.40425	−	3.95581i	−12.3220	+	13.2799i	−6.36544	+	9.19137i	−11.5720	+	24.0295i	5.41555	+	17.7108i	48.7951	+	17.0742i	−0.0353796	+	0.234728i	56.0605	+	11.0321i
3.7	−1.95041	−	4.47039i	3.41618	+	3.96967i	−10.7389	+	11.5738i	−11.0761	+	1.52300i	11.0830	−	23.0141i	−18.1138	−	3.85857i	35.8555	+	12.5464i	−0.0638664	+	0.423726i	28.4114	+	46.5441i
3.8	−1.93956	−	4.44551i	−5.26970	−	6.12351i	−10.5593	+	11.3802i	−9.06725	−	6.54102i	−17.0012	+	35.3034i	−18.3842	+	2.24102i	34.4471	+	12.0536i	−5.70342	+	37.8397i	−11.4917	+	52.9953i
3.9	−1.90213	−	4.35973i	5.01688	+	5.82972i	−9.94775	+	10.7211i	−3.94143	−	10.4626i	15.8732	−	32.9611i	9.88579	+	15.6611i	29.7456	+	10.4084i	−4.79241	+	31.7956i	−38.1168	+	37.0847i
3.10	−1.82881	−	4.19168i	3.80858	+	4.42565i	−8.78422	+	9.46714i	9.36486	−	6.10733i	11.5857	−	24.0580i	−5.36903	+	17.7249i	21.2149	+	7.42340i	−1.05696	+	7.01250i	−42.7265	−	28.0853i
3.11	−1.81625	−	4.16289i	−4.59264	−	5.33674i	−8.58951	+	9.25729i	8.19553	−	7.60482i	−13.8749	+	28.8115i	−8.80150	+	16.2952i	19.8419	+	6.94299i	−3.36436	+	22.3211i	−46.5432	−	20.3048i
3.12	−1.71313	−	3.92653i	−0.292494	−	0.339884i	−7.04144	+	7.58886i	−11.1664	−	0.557960i	−0.833485	+	1.73075i	14.0478	−	12.0689i	9.51219	+	3.32846i	3.99417	−	26.4996i	16.9386	+	44.8011i
3.13	−1.70260	−	3.90239i	−0.484436	−	0.562925i	−6.88845	+	7.42398i	−3.62983	+	10.5747i	−1.37195	+	2.84889i	6.44764	−	17.3617i	8.54980	+	2.99170i	3.94194	−	26.1530i	47.4468	−	3.83945i
3.14	−1.70199	−	3.90100i	4.18810	+	4.86666i	−6.87964	+	7.41448i	11.1264	−	1.09676i	11.8567	−	24.6208i	7.82092	−	16.7879i	8.49471	+	2.97243i	−2.12007	+	14.0657i	−23.2155	−	41.5374i
3.15	−1.55680	−	3.56822i	−2.92642	−	3.40056i	−4.86721	+	5.24561i	10.5899	−	3.58523i	−7.57812	+	15.7361i	17.5712	+	5.85251i	−3.10190	−	1.08540i	1.02425	−	6.79549i	−29.2793	−	32.2057i
3.16	−1.53310	−	3.51390i	−2.62570	−	3.05111i	−4.55569	+	4.90987i	7.65438	+	8.14926i	−6.69585	+	13.9041i	−15.5727	−	10.0245i	−4.71203	−	1.64881i	1.60912	−	10.6758i	16.9007	−	39.3903i
3.17	−1.50738	−	3.45496i	−5.55290	−	6.45259i	−4.22317	+	4.55149i	−6.21496	−	9.29377i	−13.9231	+	28.9116i	11.9422	−	14.1557i	−6.37242	−	2.22981i	−6.77704	+	44.9627i	−22.7413	+	35.4817i
3.18	−1.37225	−	3.14522i	1.78033	+	2.06878i	−2.56798	+	2.76762i	5.39526	+	9.79240i	4.06373	−	8.43843i	14.4695	+	11.5599i	−13.6831	−	4.78793i	2.91386	−	19.3322i	23.3957	−	30.4069i
3.19	−1.36642	−	3.13188i	5.17361	+	6.01184i	−2.50015	+	2.69452i	−3.53538	−	10.6067i	11.7590	−	24.4178i	−12.2482	−	13.8918i	−13.9467	−	4.88016i	−5.35189	+	35.5075i	−28.3879	+	25.5656i
3.20	−1.28491	−	2.94504i	−0.870046	−	1.01101i	−1.58089	+	1.70379i	2.35912	−	10.9286i	−1.85954	+	3.86138i	−13.3272	−	12.8602i	−17.2136	−	6.02329i	3.75898	−	24.9392i	−35.2164	+	7.09457i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
49.h	odd	42	1	inner
245.x	even	84	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	245.4.x.a	✓	1968
5.c	odd	4	1	inner	245.4.x.a	✓	1968
49.h	odd	42	1	inner	245.4.x.a	✓	1968
245.x	even	84	1	inner	245.4.x.a	✓	1968

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
245.4.x.a	✓	1968	1.a	even	1	1	trivial
245.4.x.a	✓	1968	5.c	odd	4	1	inner
245.4.x.a	✓	1968	49.h	odd	42	1	inner
245.4.x.a	✓	1968	245.x	even	84	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(245, [\chi])\).