Newform orbit 168.3.be.a

• Label: 168.3.be.a
• Level: $168$
• Weight: $3$
• Character orbit: 168.be
• Analytic conductor: $4.578$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	168.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.57766844125\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(60\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{3} - 2 q^{4} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.97785	+	0.296805i	−1.08792	+	2.79579i	3.82381	−	1.17408i	3.91482	−	2.26023i	1.32193	−	5.85256i	−0.799710	−	6.95417i	−7.21447	+	3.45708i	−6.63288	−	6.08316i	−7.07210	+	5.63234i
59.2	−1.96931	−	0.349002i	−1.67664	−	2.48774i	3.75639	+	1.37459i	−2.04175	+	1.17880i	2.43361	+	5.48430i	0.799694	−	6.95417i	−6.91779	−	4.01799i	−3.37774	+	8.34211i	4.43224	−	1.60886i
59.3	−1.96912	+	0.350078i	2.05788	+	2.18292i	3.75489	−	1.37869i	−1.35250	+	0.780864i	−4.81641	−	3.57802i	−6.09469	+	3.44307i	−6.91119	+	4.02932i	−0.530263	+	8.98437i	2.38987	−	2.01109i
59.4	−1.96463	−	0.374469i	−2.61901	−	1.46315i	3.71955	+	1.47139i	7.45533	−	4.30434i	4.59748	+	3.85528i	−3.31214	+	6.16682i	−6.75654	−	4.28359i	4.71841	+	7.66398i	−16.2588	+	5.66464i
59.5	−1.95967	−	0.399601i	2.83538	−	0.980100i	3.68064	+	1.56618i	−3.58905	+	2.07214i	−5.94808	+	0.787653i	5.63217	−	4.15676i	−6.58700	−	4.53998i	7.07881	−	5.55792i	7.86140	−	2.62653i
59.6	−1.92340	+	0.548206i	−2.96784	+	0.438066i	3.39894	−	2.10884i	−1.76255	+	1.01761i	5.46820	−	2.46957i	6.88289	+	1.27509i	−5.38145	+	5.91946i	8.61620	−	2.60022i	2.83223	−	2.92351i
59.7	−1.90324	+	0.614542i	2.84017	−	0.966154i	3.24468	−	2.33925i	6.95650	−	4.01634i	−4.81179	+	3.58423i	5.23931	+	4.64215i	−4.73785	+	6.44614i	7.13309	−	5.48808i	−10.7717	+	11.9191i
59.8	−1.84821	−	0.764271i	−2.48908	+	1.67466i	2.83178	+	2.82507i	−7.78745	+	4.49609i	5.88024	−	1.19280i	−6.95319	+	0.808201i	−3.07461	−	7.38558i	3.39101	−	8.33673i	17.8291	−	2.35800i
59.9	−1.84309	+	0.776552i	0.0152337	−	2.99996i	2.79393	−	2.86251i	−4.49938	+	2.59772i	2.30155	+	5.54102i	1.91166	+	6.73391i	−2.92657	+	7.44548i	−8.99954	−	0.0914007i	6.27549	−	8.28183i
59.10	−1.75207	−	0.964487i	−0.327520	+	2.98207i	2.13953	+	3.37971i	1.96566	−	1.13487i	3.45001	−	4.90892i	5.01182	+	4.88688i	−0.488933	−	7.98505i	−8.78546	−	1.95337i	−4.53855	+	0.0925307i
59.11	−1.70834	−	1.03999i	1.25749	−	2.72373i	1.83685	+	3.55331i	−1.40984	+	0.813973i	−4.98088	+	3.34527i	−4.74607	+	5.14537i	0.557443	−	7.98055i	−5.83741	−	6.85015i	3.25501	+	0.0756784i
59.12	−1.64427	−	1.13859i	2.81135	+	1.04706i	1.40723	+	3.74429i	5.55759	−	3.20868i	−3.43043	−	4.92262i	−4.78906	−	5.10538i	1.94936	−	7.75887i	6.80733	+	5.88729i	−12.7915	−	1.05190i
59.13	−1.59406	+	1.20788i	0.0152337	−	2.99996i	1.08204	−	3.85087i	4.49938	−	2.59772i	3.59932	+	4.80051i	−1.91166	−	6.73391i	2.92657	+	7.44548i	−8.99954	−	0.0914007i	−4.03453	+	9.57564i
59.14	−1.48383	+	1.34099i	2.84017	−	0.966154i	0.403507	−	3.97960i	−6.95650	+	4.01634i	−2.91873	+	5.24224i	−5.23931	−	4.64215i	4.73785	+	6.44614i	7.13309	−	5.48808i	4.93641	−	15.2881i
59.15	−1.43646	+	1.39161i	−2.96784	+	0.438066i	0.126837	−	3.99799i	1.76255	−	1.01761i	3.65357	−	4.75935i	−6.88289	−	1.27509i	5.38145	+	5.91946i	8.61620	−	2.60022i	−1.11572	+	3.91454i
59.16	−1.28774	+	1.53027i	2.05788	+	2.18292i	−0.683463	−	3.94118i	1.35250	−	0.780864i	−5.99047	+	0.338089i	6.09469	−	3.44307i	6.91119	+	4.02932i	−0.530263	+	8.98437i	−0.546726	+	3.07523i
59.17	−1.24597	+	1.56447i	−1.08792	+	2.79579i	−0.895127	−	3.89856i	−3.91482	+	2.26023i	−3.01842	−	5.18547i	0.799710	+	6.95417i	7.21447	+	3.45708i	−6.63288	−	6.08316i	1.34169	−	8.94079i
59.18	−1.24027	−	1.56899i	0.693678	−	2.91870i	−0.923485	+	3.89194i	5.61499	−	3.24182i	−5.43977	+	2.53158i	6.83130	−	1.52755i	7.25179	−	3.37809i	−8.03762	−	4.04928i	−12.0505	−	4.78917i
59.19	−1.14156	−	1.64220i	−2.95161	−	0.536678i	−1.39367	+	3.74936i	−1.83075	+	1.05698i	2.48811	+	5.45979i	6.30723	+	3.03625i	7.74817	−	1.99143i	8.42395	+	3.16812i	3.82570	+	1.79985i
59.20	−1.13960	−	1.64356i	1.34360	+	2.68230i	−1.40260	+	3.74603i	−5.28741	+	3.05269i	2.87737	−	5.26505i	1.01576	−	6.92591i	7.75524	−	1.96372i	−5.38950	+	7.20787i	11.0428	+	5.21134i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
8.d	odd	2	1	inner
21.g	even	6	1	inner
24.f	even	2	1	inner
56.m	even	6	1	inner
168.be	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	168.3.be.a	✓	120
3.b	odd	2	1	inner	168.3.be.a	✓	120
7.d	odd	6	1	inner	168.3.be.a	✓	120
8.d	odd	2	1	inner	168.3.be.a	✓	120
21.g	even	6	1	inner	168.3.be.a	✓	120
24.f	even	2	1	inner	168.3.be.a	✓	120
56.m	even	6	1	inner	168.3.be.a	✓	120
168.be	odd	6	1	inner	168.3.be.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.3.be.a	✓	120	1.a	even	1	1	trivial
168.3.be.a	✓	120	3.b	odd	2	1	inner
168.3.be.a	✓	120	7.d	odd	6	1	inner
168.3.be.a	✓	120	8.d	odd	2	1	inner
168.3.be.a	✓	120	21.g	even	6	1	inner
168.3.be.a	✓	120	24.f	even	2	1	inner
168.3.be.a	✓	120	56.m	even	6	1	inner
168.3.be.a	✓	120	168.be	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(168, [\chi])\).