Newform orbit 468.2.x.a

• Label: 468.2.x.a
• Level: $468$
• Weight: $2$
• Character orbit: 468.x
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73699881460\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{4} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
155.1	−1.41165	+	0.0851639i	1.62419	+	0.601677i	1.98549	−	0.240443i	−0.0539519	−	0.0934475i	−2.34402	−	0.711034i	0.701778	−	1.21551i	−2.78234	+	0.508513i	2.27597	+	1.95447i	0.0841194	+	0.127320i
155.2	−1.40836	+	0.128492i	−0.146395	+	1.72585i	1.96698	−	0.361926i	−1.37233	−	2.37694i	−0.0155803	−	2.44944i	0.132323	−	0.229190i	−2.72372	+	0.762463i	−2.95714	−	0.505312i	2.23815	+	3.17126i
155.3	−1.40259	−	0.180938i	−1.48903	−	0.884756i	1.93452	+	0.507565i	−0.813740	−	1.40944i	1.92841	+	1.51037i	2.52751	−	4.37778i	−2.62151	−	1.06193i	1.43441	+	2.63485i	0.886323	+	2.12410i
155.4	−1.39742	−	0.217295i	−0.607102	−	1.62217i	1.90557	+	0.607304i	1.04978	+	1.81828i	0.495888	+	2.39877i	−1.06921	+	1.85192i	−2.53091	−	1.26273i	−2.26285	+	1.96964i	−1.07189	−	2.76901i
155.5	−1.39651	+	0.223087i	1.40970	−	1.00636i	1.90046	−	0.623086i	2.09220	+	3.62379i	−1.74415	+	1.71987i	1.58137	−	2.73901i	−2.51501	+	1.29411i	0.974487	−	2.83732i	−3.73019	−	4.59391i
155.6	−1.39043	+	0.258258i	−1.48505	+	0.891412i	1.86661	−	0.718180i	0.471306	+	0.816326i	1.83465	−	1.62298i	0.990611	−	1.71579i	−2.40991	+	1.48065i	1.41077	−	2.64759i	−0.866142	−	1.01333i
155.7	−1.35137	−	0.416895i	−1.60923	+	0.640608i	1.65240	+	1.12676i	−1.71364	−	2.96811i	2.44173	−	0.194818i	−1.73612	+	3.00705i	−1.76326	−	2.21154i	2.17924	−	2.06177i	1.07837	+	4.72542i
155.8	−1.33378	+	0.470144i	0.402900	−	1.68454i	1.55793	−	1.25414i	−1.91288	−	3.31320i	0.254596	+	2.43622i	−0.429256	+	0.743494i	−1.48831	+	2.40519i	−2.67534	−	1.35740i	4.10903	+	3.51975i
155.9	−1.33377	−	0.470158i	1.02286	+	1.39777i	1.55790	+	1.25417i	0.948662	+	1.64313i	−0.707094	−	2.34521i	−1.03247	+	1.78829i	−1.48823	−	2.40524i	−0.907507	+	2.85945i	−0.492768	−	2.63759i
155.10	−1.31964	−	0.508471i	−1.71011	+	0.274811i	1.48291	+	1.34200i	1.53885	+	2.66536i	2.39647	+	0.506889i	−0.853117	+	1.47764i	−1.27455	−	2.52498i	2.84896	−	0.939914i	−0.675470	−	4.29979i
155.11	−1.31061	−	0.531327i	0.980761	−	1.42762i	1.43538	+	1.39272i	−0.603056	−	1.04452i	−2.04393	+	1.34995i	1.29487	−	2.24278i	−1.14123	−	2.58797i	−1.07622	−	2.80031i	0.235386	+	1.68938i
155.12	−1.30407	+	0.547185i	−0.427954	+	1.67835i	1.40118	−	1.42713i	0.966845	+	1.67462i	−0.360286	−	2.42285i	−2.32962	+	4.03501i	−1.04633	+	2.62778i	−2.63371	−	1.43651i	−2.17716	−	1.65478i
155.13	−1.30066	+	0.555233i	−0.599821	−	1.62487i	1.38343	−	1.44434i	0.649096	+	1.12427i	1.68235	+	1.78037i	−0.0466486	+	0.0807978i	−0.997430	+	2.64672i	−2.28043	+	1.94927i	−1.46848	−	1.10189i
155.14	−1.16970	+	0.794861i	0.749074	+	1.56169i	0.736391	−	1.85950i	−0.751028	−	1.30082i	−2.11752	−	1.23130i	1.92659	−	3.33696i	0.616686	+	2.76038i	−1.87778	+	2.33965i	1.91245	+	0.924603i
155.15	−1.15843	+	0.811191i	−1.68925	−	0.382661i	0.683938	−	1.87942i	−0.876150	−	1.51754i	2.26730	−	0.927017i	−1.81800	+	3.14887i	0.732273	+	2.73199i	2.70714	+	1.29282i	2.24597	+	1.04724i
155.16	−1.11545	−	0.869356i	−0.980761	+	1.42762i	0.488441	+	1.93944i	0.603056	+	1.04452i	2.33510	−	0.739807i	1.29487	−	2.24278i	1.14123	−	2.58797i	−1.07622	−	2.80031i	0.235386	−	1.68938i
155.17	−1.10017	−	0.888609i	1.71011	−	0.274811i	0.420749	+	1.95524i	−1.53885	−	2.66536i	−2.12561	−	1.21728i	−0.853117	+	1.47764i	1.27455	−	2.52498i	2.84896	−	0.939914i	−0.675470	+	4.29979i
155.18	−1.07643	+	0.917225i	1.70089	−	0.327082i	0.317396	−	1.97465i	−0.208307	−	0.360798i	−1.53088	+	1.91218i	−0.416471	+	0.721348i	1.46955	+	2.41670i	2.78603	−	1.11266i	0.555160	+	0.197309i
155.19	−1.07406	−	0.920002i	−1.02286	−	1.39777i	0.307192	+	1.97627i	−0.948662	−	1.64313i	−0.187338	+	2.44232i	−1.03247	+	1.78829i	1.48823	−	2.40524i	−0.907507	+	2.85945i	−0.492768	+	2.63759i
155.20	−1.03673	−	0.961872i	1.60923	−	0.640608i	0.149603	+	1.99440i	1.71364	+	2.96811i	−2.28451	−	0.883739i	−1.73612	+	3.00705i	1.76326	−	2.21154i	2.17924	−	2.06177i	1.07837	−	4.72542i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
13.b	even	2	1	inner
36.h	even	6	1	inner
52.b	odd	2	1	inner
117.n	odd	6	1	inner
468.x	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.x.a	✓	160
4.b	odd	2	1	inner	468.2.x.a	✓	160
9.d	odd	6	1	inner	468.2.x.a	✓	160
13.b	even	2	1	inner	468.2.x.a	✓	160
36.h	even	6	1	inner	468.2.x.a	✓	160
52.b	odd	2	1	inner	468.2.x.a	✓	160
117.n	odd	6	1	inner	468.2.x.a	✓	160
468.x	even	6	1	inner	468.2.x.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.x.a	✓	160	1.a	even	1	1	trivial
468.2.x.a	✓	160	4.b	odd	2	1	inner
468.2.x.a	✓	160	9.d	odd	6	1	inner
468.2.x.a	✓	160	13.b	even	2	1	inner
468.2.x.a	✓	160	36.h	even	6	1	inner
468.2.x.a	✓	160	52.b	odd	2	1	inner
468.2.x.a	✓	160	117.n	odd	6	1	inner
468.2.x.a	✓	160	468.x	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(468, [\chi])\).