Newform orbit 51.4.i.a

• Label: 51.4.i.a
• Level: $51$
• Weight: $4$
• Character orbit: 51.i
• Analytic conductor: $3.009$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	51.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.00909741029\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−4.62770	+	1.91686i	1.47526	+	4.98233i	12.0844	−	12.0844i	20.2465	+	4.02728i	−16.3775	−	20.2289i	1.72626	+	8.67851i	−17.4241	+	42.0654i	−22.6472	+	14.7004i	−101.414	+	20.1726i
5.2	−4.29762	+	1.78013i	5.07708	−	1.10599i	9.64380	−	9.64380i	−13.2387	−	2.63334i	−19.8506	+	13.7910i	−2.60698	−	13.1062i	−10.0371	+	24.2317i	24.5536	−	11.2304i	61.5826	−	12.2495i
5.3	−4.28664	+	1.77558i	−2.74194	−	4.41381i	9.56570	−	9.56570i	−0.612146	−	0.121763i	19.5908	+	14.0519i	1.68601	+	8.47614i	−9.81533	+	23.6963i	−11.9635	+	24.2049i	2.84025	−	0.564960i
5.4	−3.00477	+	1.24462i	−4.72345	+	2.16542i	1.82273	−	1.82273i	−1.80227	−	0.358494i	11.4978	−	12.3855i	−0.238671	−	1.19988i	6.74866	−	16.2927i	17.6219	−	20.4565i	5.86160	−	1.16595i
5.5	−1.89600	+	0.785347i	1.88038	+	4.84398i	−2.67883	+	2.67883i	−8.49133	−	1.68903i	−7.36940	−	7.70742i	−2.04524	−	10.2821i	9.25801	−	22.3508i	−19.9283	+	18.2171i	17.4260	−	3.46625i
5.6	−1.54951	+	0.641828i	0.682978	−	5.15107i	−3.66781	+	3.66781i	11.5824	+	2.30388i	2.24782	+	8.42000i	−7.03184	−	35.3514i	8.46384	−	20.4335i	−26.0671	−	7.03614i	−19.4257	+	3.86401i
5.7	−1.39521	+	0.577915i	5.18290	−	0.370861i	−4.04423	+	4.04423i	6.67638	+	1.32802i	−7.01690	+	3.51270i	3.58738	+	18.0350i	7.92864	−	19.1414i	26.7249	−	3.84427i	−10.0824	+	2.00552i
5.8	−0.358536	+	0.148510i	0.276004	−	5.18882i	−5.55036	+	5.55036i	−18.8139	−	3.74231i	0.671636	+	1.90137i	4.84695	+	24.3673i	2.35380	−	5.68258i	−26.8476	−	2.86427i	7.30121	−	1.45230i
5.9	0.358536	−	0.148510i	−4.89946	−	1.73068i	−5.55036	+	5.55036i	18.8139	+	3.74231i	−2.01366	+	0.107111i	4.84695	+	24.3673i	−2.35380	+	5.68258i	21.0095	+	16.9588i	7.30121	−	1.45230i
5.10	1.39521	−	0.577915i	−2.32604	+	4.64645i	−4.04423	+	4.04423i	−6.67638	−	1.32802i	−0.560061	+	7.82703i	3.58738	+	18.0350i	−7.92864	+	19.1414i	−16.1791	−	21.6157i	−10.0824	+	2.00552i
5.11	1.54951	−	0.641828i	−5.02033	−	1.34024i	−3.66781	+	3.66781i	−11.5824	−	2.30388i	−8.63927	+	1.14548i	−7.03184	−	35.3514i	−8.46384	+	20.4335i	23.4075	+	13.4569i	−19.4257	+	3.86401i
5.12	1.89600	−	0.785347i	3.75567	+	3.59096i	−2.67883	+	2.67883i	8.49133	+	1.68903i	9.94087	+	3.85893i	−2.04524	−	10.2821i	−9.25801	+	22.3508i	1.21006	+	26.9729i	17.4260	−	3.46625i
5.13	3.00477	−	1.24462i	3.80817	−	3.53523i	1.82273	−	1.82273i	1.80227	+	0.358494i	7.04268	−	15.3623i	−0.238671	−	1.19988i	−6.74866	+	16.2927i	2.00435	−	26.9255i	5.86160	−	1.16595i
5.14	4.28664	−	1.77558i	−3.02854	−	4.22232i	9.56570	−	9.56570i	0.612146	+	0.121763i	−20.4793	−	12.7221i	1.68601	+	8.47614i	9.81533	−	23.6963i	−8.65595	+	25.5749i	2.84025	−	0.564960i
5.15	4.29762	−	1.78013i	−2.96472	+	4.26737i	9.64380	−	9.64380i	13.2387	+	2.63334i	−5.14475	+	23.6171i	−2.60698	−	13.1062i	10.0371	−	24.2317i	−9.42088	−	25.3031i	61.5826	−	12.2495i
5.16	4.62770	−	1.91686i	4.03852	+	3.26962i	12.0844	−	12.0844i	−20.2465	−	4.02728i	24.9564	+	7.38954i	1.72626	+	8.67851i	17.4241	−	42.0654i	5.61923	+	26.4088i	−101.414	+	20.1726i
11.1	−1.87197	+	4.51934i	5.11979	−	0.887546i	−11.2633	−	11.2633i	−7.31533	+	10.9482i	−5.57298	+	24.7995i	−22.3550	+	14.9371i	35.8324	−	14.8423i	25.4245	−	9.08810i	−35.7843	−	53.5551i
11.2	−1.77125	+	4.27617i	−4.82349	+	1.93234i	−9.49143	−	9.49143i	0.436777	−	0.653684i	0.280586	−	24.0487i	−5.84895	+	3.90814i	23.1892	−	9.60530i	19.5321	−	18.6412i	2.02162	+	3.02557i
11.3	−1.71142	+	4.13173i	−0.158947	−	5.19372i	−8.48536	−	8.48536i	5.85923	−	8.76896i	21.7311	+	8.23190i	20.1927	−	13.4923i	16.5274	−	6.84586i	−26.9495	+	1.65105i	26.2033	+	39.2161i
11.4	−1.25707	+	3.03485i	2.05971	+	4.77049i	−1.97320	−	1.97320i	−5.32451	+	7.96869i	−17.0669	+	0.254048i	22.1633	−	14.8090i	−15.8099	+	6.54869i	−18.5152	+	19.6517i	−17.4904	−	26.1763i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
17.e	odd	16	1	inner
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	51.4.i.a	✓	128
3.b	odd	2	1	inner	51.4.i.a	✓	128
17.e	odd	16	1	inner	51.4.i.a	✓	128
51.i	even	16	1	inner	51.4.i.a	✓	128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
51.4.i.a	✓	128	1.a	even	1	1	trivial
51.4.i.a	✓	128	3.b	odd	2	1	inner
51.4.i.a	✓	128	17.e	odd	16	1	inner
51.4.i.a	✓	128	51.i	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(51, [\chi])\).