Newform orbit 322.3.l.a

• Label: 322.3.l.a
• Level: $322$
• Weight: $3$
• Character orbit: 322.l
• Analytic conductor: $8.774$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	322.l (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.77386451240\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(32\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 64 q^{4} + 12 q^{7} + 128 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−0.926113	−	1.06879i	−2.81320	+	4.37742i	−0.284630	+	1.97964i	7.04097	+	3.21550i	7.28388	−	1.04726i	−1.65855	−	6.80068i	2.37942	−	1.52916i	−7.50898	−	16.4424i	−3.08403	−	10.5032i
13.2	−0.926113	−	1.06879i	−2.70071	+	4.20238i	−0.284630	+	1.97964i	−6.28405	−	2.86983i	6.99263	−	1.00539i	5.44264	−	4.40201i	2.37942	−	1.52916i	−6.62746	−	14.5121i	2.75249	+	9.37413i
13.3	−0.926113	−	1.06879i	−2.64804	+	4.12043i	−0.284630	+	1.97964i	0.485699	+	0.221811i	6.85626	−	0.985782i	−6.16367	+	3.31801i	2.37942	−	1.52916i	−6.22710	−	13.6354i	−0.212742	−	0.724533i
13.4	−0.926113	−	1.06879i	−1.79979	+	2.80052i	−0.284630	+	1.97964i	3.38518	+	1.54596i	4.65998	−	0.670004i	2.52388	+	6.52917i	2.37942	−	1.52916i	−0.864958	−	1.89399i	−1.48275	−	5.04978i
13.5	−0.926113	−	1.06879i	−1.18077	+	1.83732i	−0.284630	+	1.97964i	−4.05655	−	1.85256i	3.05723	−	0.439564i	2.12029	+	6.67116i	2.37942	−	1.52916i	1.75723	+	3.84779i	1.77682	+	6.05129i
13.6	−0.926113	−	1.06879i	−0.911745	+	1.41870i	−0.284630	+	1.97964i	7.01357	+	3.20299i	2.36068	−	0.339414i	6.99734	−	0.192921i	2.37942	−	1.52916i	2.55730	+	5.59970i	−3.07203	−	10.4624i
13.7	−0.926113	−	1.06879i	−0.863388	+	1.34346i	−0.284630	+	1.97964i	−1.17095	−	0.534754i	2.23547	−	0.321412i	−3.40112	−	6.11820i	2.37942	−	1.52916i	2.67930	+	5.86684i	0.512889	+	1.74674i
13.8	−0.926113	−	1.06879i	−0.0351598	+	0.0547097i	−0.284630	+	1.97964i	−6.29651	−	2.87552i	0.0910352	−	0.0130889i	−5.20875	−	4.67642i	2.37942	−	1.52916i	3.73698	+	8.18284i	2.75795	+	9.39271i
13.9	−0.926113	−	1.06879i	0.0351598	−	0.0547097i	−0.284630	+	1.97964i	6.29651	+	2.87552i	−0.0910352	+	0.0130889i	−6.91014	+	1.11799i	2.37942	−	1.52916i	3.73698	+	8.18284i	−2.75795	−	9.39271i
13.10	−0.926113	−	1.06879i	0.863388	−	1.34346i	−0.284630	+	1.97964i	1.17095	+	0.534754i	−2.23547	+	0.321412i	−6.16896	+	3.30817i	2.37942	−	1.52916i	2.67930	+	5.86684i	−0.512889	−	1.74674i
13.11	−0.926113	−	1.06879i	0.911745	−	1.41870i	−0.284630	+	1.97964i	−7.01357	−	3.20299i	−2.36068	+	0.339414i	5.78224	+	3.94534i	2.37942	−	1.52916i	2.55730	+	5.59970i	3.07203	+	10.4624i
13.12	−0.926113	−	1.06879i	1.18077	−	1.83732i	−0.284630	+	1.97964i	4.05655	+	1.85256i	−3.05723	+	0.439564i	5.39040	−	4.46582i	2.37942	−	1.52916i	1.75723	+	3.84779i	−1.77682	−	6.05129i
13.13	−0.926113	−	1.06879i	1.79979	−	2.80052i	−0.284630	+	1.97964i	−3.38518	−	1.54596i	−4.65998	+	0.670004i	5.65316	−	4.12817i	2.37942	−	1.52916i	−0.864958	−	1.89399i	1.48275	+	5.04978i
13.14	−0.926113	−	1.06879i	2.64804	−	4.12043i	−0.284630	+	1.97964i	−0.485699	−	0.221811i	−6.85626	+	0.985782i	−3.39136	−	6.12362i	2.37942	−	1.52916i	−6.22710	−	13.6354i	0.212742	+	0.724533i
13.15	−0.926113	−	1.06879i	2.70071	−	4.20238i	−0.284630	+	1.97964i	6.28405	+	2.86983i	−6.99263	+	1.00539i	2.19873	+	6.64572i	2.37942	−	1.52916i	−6.62746	−	14.5121i	−2.75249	−	9.37413i
13.16	−0.926113	−	1.06879i	2.81320	−	4.37742i	−0.284630	+	1.97964i	−7.04097	−	3.21550i	−7.28388	+	1.04726i	−5.07199	+	4.82441i	2.37942	−	1.52916i	−7.50898	−	16.4424i	3.08403	+	10.5032i
13.17	0.926113	+	1.06879i	−2.90066	+	4.51352i	−0.284630	+	1.97964i	−1.05554	−	0.482050i	−7.51035	+	1.07983i	−6.74552	−	1.87028i	−2.37942	+	1.52916i	−8.21927	−	17.9977i	−0.462340	−	1.57459i
13.18	0.926113	+	1.06879i	−2.74480	+	4.27100i	−0.284630	+	1.97964i	−7.25647	−	3.31392i	−7.10680	+	1.02180i	4.69765	+	5.18962i	−2.37942	+	1.52916i	−6.96874	−	15.2594i	−3.17842	−	10.8247i
13.19	0.926113	+	1.06879i	−2.25417	+	3.50756i	−0.284630	+	1.97964i	4.91620	+	2.24515i	−5.83647	+	0.839158i	2.34357	+	6.59603i	−2.37942	+	1.52916i	−3.48296	−	7.62662i	2.15336	+	7.33366i
13.20	0.926113	+	1.06879i	−1.50484	+	2.34158i	−0.284630	+	1.97964i	5.82966	+	2.66232i	−3.89632	+	0.560206i	−6.60219	−	2.32617i	−2.37942	+	1.52916i	0.520283	+	1.13926i	2.55346	+	8.69630i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
23.c	even	11	1	inner
161.l	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	322.3.l.a	✓	320
7.b	odd	2	1	inner	322.3.l.a	✓	320
23.c	even	11	1	inner	322.3.l.a	✓	320
161.l	odd	22	1	inner	322.3.l.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
322.3.l.a	✓	320	1.a	even	1	1	trivial
322.3.l.a	✓	320	7.b	odd	2	1	inner
322.3.l.a	✓	320	23.c	even	11	1	inner
322.3.l.a	✓	320	161.l	odd	22	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).