Newform orbit 51.5.f.a

• Label: 51.5.f.a
• Level: $51$
• Weight: $5$
• Character orbit: 51.f
• Analytic conductor: $5.272$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	51.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.27186811728\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 6 q^{3} + 312 q^{4} - 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
38.1	−7.33659			8.49542	+	2.97118i	37.8256			−3.84842	−	3.84842i	−62.3274	−	21.7983i	−14.0710	−	14.0710i	−160.125			63.3442	+	50.4828i	28.2343	+	28.2343i
38.2	−7.23155			−8.14037	+	3.83853i	36.2953			6.69823	+	6.69823i	58.8675	−	27.7585i	40.9766	+	40.9766i	−146.767			51.5313	−	62.4942i	−48.4386	−	48.4386i
38.3	−6.32469			−0.488073	−	8.98676i	24.0018			17.5527	+	17.5527i	3.08692	+	56.8385i	2.97942	+	2.97942i	−50.6087			−80.5236	+	8.77239i	−111.015	−	111.015i
38.4	−5.57352			−7.66547	−	4.71599i	15.0641			−21.8128	−	21.8128i	42.7237	+	26.2847i	−59.0644	−	59.0644i	5.21604			36.5189	+	72.3006i	121.574	+	121.574i
38.5	−4.90419			6.18041	−	6.54236i	8.05107			−18.2674	−	18.2674i	−30.3099	+	32.0850i	14.1681	+	14.1681i	38.9831			−4.60506	−	80.8690i	89.5867	+	89.5867i
38.6	−4.56859			2.39660	+	8.67504i	4.87200			22.1843	+	22.1843i	−10.9491	−	39.6327i	−16.2413	−	16.2413i	50.8392			−69.5126	+	41.5813i	−101.351	−	101.351i
38.7	−4.30873			−0.332116	+	8.99387i	2.56516			−31.4431	−	31.4431i	1.43100	−	38.7522i	36.9552	+	36.9552i	57.8871			−80.7794	−	5.97402i	135.480	+	135.480i
38.8	−2.32615			8.28695	−	3.51090i	−10.5890			21.0845	+	21.0845i	−19.2767	+	8.16687i	5.62962	+	5.62962i	61.8500			56.3472	−	58.1893i	−49.0457	−	49.0457i
38.9	−2.25239			−8.31602	+	3.44149i	−10.9267			8.06459	+	8.06459i	18.7309	−	7.75159i	−15.1734	−	15.1734i	60.6496			57.3123	−	57.2390i	−18.1646	−	18.1646i
38.10	−1.46781			−5.97020	−	6.73474i	−13.8455			−5.76005	−	5.76005i	8.76309	+	9.88529i	56.4128	+	56.4128i	43.8075			−9.71340	+	80.4155i	8.45463	+	8.45463i
38.11	−0.828441			8.60834	+	2.62611i	−15.3137			−20.1408	−	20.1408i	−7.13150	−	2.17558i	−53.5715	−	53.5715i	25.9415			67.2071	+	45.2130i	16.6854	+	16.6854i
38.12	0.828441			−2.62611	−	8.60834i	−15.3137			20.1408	+	20.1408i	−2.17558	−	7.13150i	−53.5715	−	53.5715i	−25.9415			−67.2071	+	45.2130i	16.6854	+	16.6854i
38.13	1.46781			6.73474	+	5.97020i	−13.8455			5.76005	+	5.76005i	9.88529	+	8.76309i	56.4128	+	56.4128i	−43.8075			9.71340	+	80.4155i	8.45463	+	8.45463i
38.14	2.25239			−3.44149	+	8.31602i	−10.9267			−8.06459	−	8.06459i	−7.75159	+	18.7309i	−15.1734	−	15.1734i	−60.6496			−57.3123	−	57.2390i	−18.1646	−	18.1646i
38.15	2.32615			3.51090	−	8.28695i	−10.5890			−21.0845	−	21.0845i	8.16687	−	19.2767i	5.62962	+	5.62962i	−61.8500			−56.3472	−	58.1893i	−49.0457	−	49.0457i
38.16	4.30873			−8.99387	+	0.332116i	2.56516			31.4431	+	31.4431i	−38.7522	+	1.43100i	36.9552	+	36.9552i	−57.8871			80.7794	−	5.97402i	135.480	+	135.480i
38.17	4.56859			−8.67504	−	2.39660i	4.87200			−22.1843	−	22.1843i	−39.6327	−	10.9491i	−16.2413	−	16.2413i	−50.8392			69.5126	+	41.5813i	−101.351	−	101.351i
38.18	4.90419			6.54236	−	6.18041i	8.05107			18.2674	+	18.2674i	32.0850	−	30.3099i	14.1681	+	14.1681i	−38.9831			4.60506	−	80.8690i	89.5867	+	89.5867i
38.19	5.57352			4.71599	+	7.66547i	15.0641			21.8128	+	21.8128i	26.2847	+	42.7237i	−59.0644	−	59.0644i	−5.21604			−36.5189	+	72.3006i	121.574	+	121.574i
38.20	6.32469			8.98676	+	0.488073i	24.0018			−17.5527	−	17.5527i	56.8385	+	3.08692i	2.97942	+	2.97942i	50.6087			80.5236	+	8.77239i	−111.015	−	111.015i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
17.c	even	4	1	inner
51.f	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	51.5.f.a	✓	44
3.b	odd	2	1	inner	51.5.f.a	✓	44
17.c	even	4	1	inner	51.5.f.a	✓	44
51.f	odd	4	1	inner	51.5.f.a	✓	44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
51.5.f.a	✓	44	1.a	even	1	1	trivial
51.5.f.a	✓	44	3.b	odd	2	1	inner
51.5.f.a	✓	44	17.c	even	4	1	inner
51.5.f.a	✓	44	51.f	odd	4	1	inner

Hecke kernels

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