Newform orbit 51.5.g.a

• Label: 51.5.g.a
• Level: $51$
• Weight: $5$
• Character orbit: 51.g
• Analytic conductor: $5.272$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	51.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.27186811728\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 4 q^{3} - 4 q^{6} - 8 q^{7} + 332 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−5.39150	+	5.39150i	4.97338	−	7.50103i		−	42.1365i	0.0713453	−	0.0295522i	13.6279	+	67.2558i	−32.5040	+	78.4716i	140.915	+	140.915i	−31.5310	−	74.6109i	−0.225328	+	0.543989i
2.2	−5.07027	+	5.07027i	−5.87219	+	6.82036i		−	35.4153i	36.9735	−	15.3149i	−4.80748	−	64.3547i	10.2311	−	24.7001i	98.4408	+	98.4408i	−12.0347	−	80.1010i	−109.815	+	265.116i
2.3	−4.55719	+	4.55719i	−8.40530	−	3.21729i		−	25.5359i	−31.9841	+	13.2482i	52.9663	−	23.6428i	15.1639	−	36.6089i	43.4570	+	43.4570i	60.2982	+	54.0845i	85.3827	−	206.132i
2.4	−4.30507	+	4.30507i	4.82260	+	7.59885i		−	21.0672i	−20.8403	+	8.63235i	−53.4752	−	11.9519i	−9.24442	+	22.3180i	21.8147	+	21.8147i	−34.4850	+	73.2925i	52.5562	−	126.882i
2.5	−3.86343	+	3.86343i	8.99848	−	0.165289i		−	13.8521i	22.7328	−	9.41624i	−34.1264	+	35.4036i	31.6199	−	76.3371i	−8.29814	−	8.29814i	80.9454	−	2.97470i	−51.4476	+	124.206i
2.6	−2.74558	+	2.74558i	−6.01726	−	6.69273i			0.923601i	30.6275	−	12.6863i	34.8963	+	1.85454i	−14.0916	+	34.0201i	−46.4651	−	46.4651i	−8.58515	+	80.5437i	−49.2588	+	118.921i
2.7	−2.70562	+	2.70562i	2.06082	−	8.76088i			1.35927i	−6.20074	+	2.56843i	18.1278	+	29.2794i	12.1646	−	29.3678i	−46.9675	−	46.9675i	−72.5060	−	36.1092i	9.82764	−	23.7260i
2.8	−2.48560	+	2.48560i	−8.01487	+	4.09413i			3.64362i	−3.43814	+	1.42413i	9.74537	−	30.0981i	−23.6240	+	57.0333i	−48.8261	−	48.8261i	47.4762	−	65.6278i	5.00603	−	12.0856i
2.9	−1.03357	+	1.03357i	8.97316	−	0.694614i			13.8635i	−27.9436	+	11.5746i	−8.55646	+	9.99232i	−11.6176	+	28.0474i	−30.8660	−	30.8660i	80.0350	−	12.4658i	16.9185	−	40.8448i
2.10	−1.02767	+	1.02767i	3.33247	+	8.36030i			13.8878i	28.9057	−	11.9731i	−12.0164	−	5.16698i	−8.06628	+	19.4737i	−30.7149	−	30.7149i	−58.7893	+	55.7209i	−17.4012	+	42.0101i
2.11	−0.886043	+	0.886043i	−4.29839	+	7.90720i			14.4299i	−20.0897	+	8.32143i	−3.19757	−	10.8147i	29.6755	−	71.6430i	−26.9622	−	26.9622i	−44.0477	−	67.9765i	10.4272	−	25.1735i
2.12	0.886043	−	0.886043i	−8.63066	−	2.55182i			14.4299i	20.0897	−	8.32143i	−9.90816	+	5.38612i	29.6755	−	71.6430i	26.9622	+	26.9622i	67.9765	+	44.0477i	10.4272	−	25.1735i
2.13	1.02767	−	1.02767i	−3.55522	−	8.26804i			13.8878i	−28.9057	+	11.9731i	−12.1505	−	4.84325i	−8.06628	+	19.4737i	30.7149	+	30.7149i	−55.7209	+	58.7893i	−17.4012	+	42.0101i
2.14	1.03357	−	1.03357i	6.83615	−	5.85381i			13.8635i	27.9436	−	11.5746i	1.01531	−	13.1160i	−11.6176	+	28.0474i	30.8660	+	30.8660i	12.4658	−	80.0350i	16.9185	−	40.8448i
2.15	2.48560	−	2.48560i	−8.56235	+	2.77238i			3.64362i	3.43814	−	1.42413i	−14.3915	+	28.1736i	−23.6240	+	57.0333i	48.8261	+	48.8261i	65.6278	−	47.4762i	5.00603	−	12.0856i
2.16	2.70562	−	2.70562i	7.65210	+	4.73766i			1.35927i	6.20074	−	2.56843i	33.5219	−	7.88536i	12.1646	−	29.3678i	46.9675	+	46.9675i	36.1092	+	72.5060i	9.82764	−	23.7260i
2.17	2.74558	−	2.74558i	0.477626	+	8.98732i			0.923601i	−30.6275	+	12.6863i	25.9867	+	23.3640i	−14.0916	+	34.0201i	46.4651	+	46.4651i	−80.5437	+	8.58515i	−49.2588	+	118.921i
2.18	3.86343	−	3.86343i	6.47976	−	6.24601i		−	13.8521i	−22.7328	+	9.41624i	0.903092	−	49.1651i	31.6199	−	76.3371i	8.29814	+	8.29814i	2.97470	−	80.9454i	−51.4476	+	124.206i
2.19	4.30507	−	4.30507i	−1.96310	−	8.78329i		−	21.0672i	20.8403	−	8.63235i	−46.2640	−	29.3614i	−9.24442	+	22.3180i	−21.8147	−	21.8147i	−73.2925	+	34.4850i	52.5562	−	126.882i
2.20	4.55719	−	4.55719i	−3.66848	+	8.21841i		−	25.5359i	31.9841	−	13.2482i	20.7349	+	54.1708i	15.1639	−	36.6089i	−43.4570	−	43.4570i	−54.0845	−	60.2982i	85.3827	−	206.132i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
17.d	even	8	1	inner
51.g	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	51.5.g.a	✓	88
3.b	odd	2	1	inner	51.5.g.a	✓	88
17.d	even	8	1	inner	51.5.g.a	✓	88
51.g	odd	8	1	inner	51.5.g.a	✓	88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
51.5.g.a	✓	88	1.a	even	1	1	trivial
51.5.g.a	✓	88	3.b	odd	2	1	inner
51.5.g.a	✓	88	17.d	even	8	1	inner
51.5.g.a	✓	88	51.g	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(51, [\chi])\).