Newform orbit 162.3.h.a

• Label: 162.3.h.a
• Level: $162$
• Weight: $3$
• Character orbit: 162.h
• Analytic conductor: $4.414$
• Analytic rank: $0$
• Dimension: $324$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	162.h (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.41418028264\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(18\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 324 q+O(q^{10}) \)

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	−0.326140	−	1.37609i	−2.76034	+	1.17495i	−1.78727	+	0.897598i	6.70564	−	4.99216i	2.51710	+	3.41529i	−6.54938	+	4.30759i	1.81808	+	2.16670i	6.23898	−	6.48654i	−9.05666	−	7.59944i
5.2	−0.326140	−	1.37609i	−2.75282	−	1.19248i	−1.78727	+	0.897598i	−1.51997	+	1.13158i	−0.743160	+	4.17705i	−2.05983	+	1.35477i	1.81808	+	2.16670i	6.15598	+	6.56535i	2.05288	+	1.72257i
5.3	−0.326140	−	1.37609i	−2.45330	+	1.72665i	−1.78727	+	0.897598i	−4.47588	+	3.33217i	3.17615	+	2.81284i	7.95611	−	5.23282i	1.81808	+	2.16670i	3.03734	−	8.47199i	6.04514	+	5.07248i
5.4	−0.326140	−	1.37609i	−1.15412	−	2.76912i	−1.78727	+	0.897598i	5.90063	−	4.39286i	−3.43416	+	2.49129i	11.1239	−	7.31631i	1.81808	+	2.16670i	−6.33603	+	6.39177i	−7.96942	−	6.68713i
5.5	−0.326140	−	1.37609i	−0.663139	+	2.92579i	−1.78727	+	0.897598i	0.605521	−	0.450794i	4.24244	−	0.0416761i	1.59852	−	1.05136i	1.81808	+	2.16670i	−8.12049	−	3.88041i	−0.817819	−	0.686231i
5.6	−0.326140	−	1.37609i	0.243053	−	2.99014i	−1.78727	+	0.897598i	−2.26822	+	1.68862i	−4.19398	+	0.640740i	−8.30694	+	5.46356i	1.81808	+	2.16670i	−8.88185	−	1.45353i	3.06346	+	2.57055i
5.7	−0.326140	−	1.37609i	2.26424	+	1.96805i	−1.78727	+	0.897598i	5.11004	−	3.80428i	1.96976	−	3.75766i	1.09207	−	0.718268i	1.81808	+	2.16670i	1.25355	+	8.91227i	−6.90163	−	5.79116i
5.8	−0.326140	−	1.37609i	2.56706	−	1.55248i	−1.78727	+	0.897598i	−0.00733898	+	0.00546367i	−2.97357	−	3.02620i	2.72886	−	1.79480i	1.81808	+	2.16670i	4.17964	−	7.97061i	0.00991205	+	0.00831720i
5.9	−0.326140	−	1.37609i	2.89210	+	0.797352i	−1.78727	+	0.897598i	−7.40578	+	5.51340i	0.154002	−	4.23984i	−1.07376	+	0.706223i	1.81808	+	2.16670i	7.72846	+	4.61204i	10.0023	+	8.39290i
5.10	0.326140	+	1.37609i	−2.96504	−	0.456664i	−1.78727	+	0.897598i	−1.81547	+	1.35157i	−0.338606	−	4.22911i	−1.49360	+	0.982359i	−1.81808	−	2.16670i	8.58292	+	2.70805i	−2.45198	−	2.05746i
5.11	0.326140	+	1.37609i	−2.78435	+	1.11687i	−1.78727	+	0.897598i	5.92375	−	4.41007i	−2.44501	−	3.46727i	6.83506	−	4.49549i	−1.81808	−	2.16670i	6.50519	−	6.21953i	8.00064	+	6.71334i
5.12	0.326140	+	1.37609i	−2.21854	−	2.01943i	−1.78727	+	0.897598i	2.97271	−	2.21310i	2.05537	−	3.71153i	−8.93104	+	5.87404i	−1.81808	−	2.16670i	0.843823	+	8.96036i	4.01495	+	3.36895i
5.13	0.326140	+	1.37609i	−1.18720	+	2.75509i	−1.78727	+	0.897598i	−3.75904	+	2.79850i	−4.17846	−	0.735156i	1.03380	−	0.679940i	−1.81808	−	2.16670i	−6.18109	−	6.54172i	−5.07698	−	4.26009i
5.14	0.326140	+	1.37609i	−0.0281193	−	2.99987i	−1.78727	+	0.897598i	1.73126	−	1.28887i	4.11893	−	1.01707i	3.67681	−	2.41828i	−1.81808	−	2.16670i	−8.99842	+	0.168708i	2.33824	+	1.96202i
5.15	0.326140	+	1.37609i	1.73974	−	2.44404i	−1.78727	+	0.897598i	−6.68826	+	4.97922i	3.93062	+	1.59694i	−10.2458	+	6.73874i	−1.81808	−	2.16670i	−2.94663	−	8.50396i	−9.03318	−	7.57974i
5.16	0.326140	+	1.37609i	2.18959	+	2.05078i	−1.78727	+	0.897598i	−1.03753	+	0.772415i	−2.10794	+	3.68193i	−4.52918	+	2.97889i	−1.81808	−	2.16670i	0.588642	+	8.98073i	−1.40130	−	1.17583i
5.17	0.326140	+	1.37609i	2.81870	−	1.02711i	−1.78727	+	0.897598i	−1.67736	+	1.24875i	2.33268	+	3.54381i	11.2134	−	7.37517i	−1.81808	−	2.16670i	6.89011	−	5.79020i	−2.26544	−	1.90093i
5.18	0.326140	+	1.37609i	2.88671	−	0.816639i	−1.78727	+	0.897598i	6.99459	−	5.20728i	2.06524	+	3.70604i	−4.06903	+	2.67625i	−1.81808	−	2.16670i	7.66620	−	4.71480i	9.44691	+	7.92690i
11.1	−1.02866	−	0.970492i	−2.92008	−	0.687859i	0.116290	+	1.99662i	0.533779	−	4.56677i	2.33621	+	3.54149i	7.85421	−	3.94453i	1.81808	−	2.16670i	8.05370	+	4.01720i	−4.98110	+	4.17964i
11.2	−1.02866	−	0.970492i	−2.72678	+	1.25087i	0.116290	+	1.99662i	0.295075	−	2.52452i	4.01889	+	1.35959i	−6.23888	+	3.13328i	1.81808	−	2.16670i	5.87063	−	6.82170i	−2.75356	+	2.31051i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
81.h	odd	54	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.3.h.a	✓	324
81.h	odd	54	1	inner	162.3.h.a	✓	324

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
162.3.h.a	✓	324	1.a	even	1	1	trivial
162.3.h.a	✓	324	81.h	odd	54	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(162, [\chi])\).