Newform orbit 51.6.i.a

• Label: 51.6.i.a
• Level: $51$
• Weight: $6$
• Character orbit: 51.i
• Analytic conductor: $8.180$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	51.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.17957481046\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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	−10.1948	+	4.22283i	8.40425	−	13.1289i	63.4746	−	63.4746i	57.8056	+	11.4982i	−30.2386	+	169.337i	−29.7361	−	149.494i	−243.939	+	588.921i	−101.737	−	220.677i	−637.872	+	126.881i
5.2	−9.70306	+	4.01914i	−15.5727	−	0.701681i	55.3685	−	55.3685i	0.439511	+	0.0874241i	153.923	−	55.7802i	44.1496	+	221.955i	−186.098	+	449.280i	242.015	+	21.8541i	−4.61597	+	0.918174i
5.3	−8.83173	+	3.65822i	12.8084	+	8.88513i	41.9895	−	41.9895i	−35.5962	−	7.08053i	−145.624	−	31.6152i	17.8418	+	89.6969i	−100.170	+	241.832i	85.1091	+	227.608i	340.279	−	67.6856i
5.4	−7.92950	+	3.28451i	−5.36347	+	14.6367i	29.4616	−	29.4616i	−2.23570	−	0.444709i	−5.54469	−	133.678i	−20.9709	−	105.428i	−31.7446	+	76.6381i	−185.466	−	157.007i	19.1886	−	3.81686i
5.5	−7.58347	+	3.14118i	−9.32894	−	12.4888i	25.0147	−	25.0147i	−90.0003	−	17.9022i	109.975	+	65.4048i	−41.2907	−	207.583i	−10.6048	+	25.6024i	−68.9416	+	233.015i	738.749	−	146.946i
5.6	−6.31772	+	2.61688i	10.7883	−	11.2522i	10.4380	−	10.4380i	−54.9326	−	10.9268i	−38.7118	+	99.3201i	25.9411	+	130.415i	45.1108	−	108.907i	−10.2245	−	242.785i	375.642	−	74.7199i
5.7	−5.84929	+	2.42285i	−15.5553	−	1.01612i	5.71651	−	5.71651i	84.9848	+	16.9045i	93.4493	−	31.7446i	−20.8169	−	104.653i	57.9441	−	139.889i	240.935	+	31.6123i	−538.057	+	107.026i
5.8	−5.68898	+	2.35645i	−2.58585	−	15.3725i	4.18423	−	4.18423i	43.5813	+	8.66887i	50.9354	+	81.3604i	20.9119	+	105.131i	61.4625	−	148.383i	−229.627	+	79.5020i	−268.361	+	53.3804i
5.9	−5.62132	+	2.32843i	15.2141	+	3.39581i	3.55027	−	3.55027i	74.9632	+	14.9111i	−93.4302	+	16.3360i	−8.49010	−	42.6826i	62.8190	−	151.659i	219.937	+	103.328i	−456.112	+	90.7263i
5.10	−3.18567	+	1.31955i	−14.2273	+	6.37056i	−14.2201	+	14.2201i	−93.5871	−	18.6156i	36.9172	−	39.0681i	21.5377	+	108.277i	68.7620	−	166.006i	161.832	−	181.272i	322.702	−	64.1894i
5.11	−2.65877	+	1.10130i	−1.85557	+	15.4776i	−16.7712	+	16.7712i	50.0065	+	9.94691i	−12.1120	−	43.1950i	41.2495	+	207.375i	61.3622	−	148.142i	−236.114	−	57.4396i	−143.910	+	28.6255i
5.12	−2.62172	+	1.08595i	8.68145	+	12.9473i	−16.9333	+	16.9333i	−63.9319	−	12.7168i	−36.8205	−	24.5166i	−20.6863	−	103.997i	60.7561	−	146.678i	−92.2648	+	224.803i	181.422	−	36.0870i
5.13	−1.54493	+	0.639929i	13.4575	−	7.86745i	−20.6501	+	20.6501i	−22.0950	−	4.39496i	−15.7562	+	20.7664i	−29.7883	−	149.756i	39.1660	−	94.5551i	119.207	−	211.752i	36.9475	−	7.34932i
5.14	−0.975563	+	0.404092i	−13.5327	−	7.73731i	−21.8390	+	21.8390i	−3.85231	−	0.766273i	16.3286	+	2.07978i	0.0715928	+	0.359922i	25.4113	−	61.3483i	123.268	+	209.413i	4.06782	−	0.809140i
5.15	0.975563	−	0.404092i	−1.96960	−	15.4635i	−21.8390	+	21.8390i	3.85231	+	0.766273i	−8.17015	−	14.2898i	0.0715928	+	0.359922i	−25.4113	+	61.3483i	−235.241	+	60.9139i	4.06782	−	0.809140i
5.16	1.54493	−	0.639929i	−12.4185	+	9.42233i	−20.6501	+	20.6501i	22.0950	+	4.39496i	−13.1561	+	22.5038i	−29.7883	−	149.756i	−39.1660	+	94.5551i	65.4393	−	234.023i	36.9475	−	7.34932i
5.17	2.62172	−	1.08595i	8.63949	+	12.9753i	−16.9333	+	16.9333i	63.9319	+	12.7168i	36.7409	+	24.6356i	−20.6863	−	103.997i	−60.7561	+	146.678i	−93.7184	+	224.200i	181.422	−	36.0870i
5.18	2.65877	−	1.10130i	15.0096	+	4.20871i	−16.7712	+	16.7712i	−50.0065	−	9.94691i	44.5420	−	5.34002i	41.2495	+	207.375i	−61.3622	+	148.142i	207.574	+	126.342i	−143.910	+	28.6255i
5.19	3.18567	−	1.31955i	11.3302	−	10.7064i	−14.2201	+	14.2201i	93.5871	+	18.6156i	21.9666	−	49.0578i	21.5377	+	108.277i	−68.7620	+	166.006i	13.7460	−	242.611i	322.702	−	64.1894i
5.20	5.62132	−	2.32843i	−2.68486	+	15.3555i	3.55027	−	3.55027i	−74.9632	−	14.9111i	20.6617	+	92.5697i	−8.49010	−	42.6826i	−62.8190	+	151.659i	−228.583	−	82.4547i	−456.112	+	90.7263i

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	51.6.i.a	✓	224
3.b	odd	2	1	inner	51.6.i.a	✓	224
17.e	odd	16	1	inner	51.6.i.a	✓	224
51.i	even	16	1	inner	51.6.i.a	✓	224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
51.6.i.a	✓	224	1.a	even	1	1	trivial
51.6.i.a	✓	224	3.b	odd	2	1	inner
51.6.i.a	✓	224	17.e	odd	16	1	inner
51.6.i.a	✓	224	51.i	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(51, [\chi])\).