Newform orbit 345.3.o.a

• Label: 345.3.o.a
• Level: $345$
• Weight: $3$
• Character orbit: 345.o
• Analytic conductor: $9.401$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 345 = 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	345.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.40056912043\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 8 q^{2} - 92 q^{4} + 12 q^{6} + 8 q^{8} - 96 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
61.1	−0.546428	+	3.80049i	0.719520	+	1.57553i	−10.3072	−	3.02646i	1.68991	−	1.46431i	−6.38095	+	1.87362i	6.33380	−	9.85559i	10.7541	−	23.5482i	−1.96458	+	2.26725i	4.64170	+	7.22262i
61.2	−0.544086	+	3.78420i	−0.719520	−	1.57553i	−10.1862	−	2.99094i	−1.68991	+	1.46431i	6.35360	−	1.86559i	2.47822	−	3.85618i	10.5078	−	23.0088i	−1.96458	+	2.26725i	−4.62180	−	7.19167i
61.3	−0.503373	+	3.50104i	−0.719520	−	1.57553i	−8.16591	−	2.39773i	1.68991	−	1.46431i	5.87817	−	1.72599i	−6.46335	+	10.0572i	6.62768	−	14.5126i	−1.96458	+	2.26725i	4.27596	+	6.65353i
61.4	−0.474841	+	3.30259i	0.719520	+	1.57553i	−6.84366	−	2.00948i	−1.68991	+	1.46431i	−5.54499	+	1.62815i	−0.0381659	+	0.0593874i	4.34193	−	9.50750i	−1.96458	+	2.26725i	−4.03359	−	6.27639i
61.5	−0.459573	+	3.19640i	−0.719520	−	1.57553i	−6.16781	−	1.81103i	1.68991	−	1.46431i	5.36670	−	1.57580i	3.50913	−	5.46032i	3.25739	−	7.13269i	−1.96458	+	2.26725i	3.90390	+	6.07458i
61.6	−0.416496	+	2.89679i	0.719520	+	1.57553i	−4.37998	−	1.28608i	1.68991	−	1.46431i	−4.86366	+	1.42810i	−1.79021	+	2.78562i	0.686757	−	1.50379i	−1.96458	+	2.26725i	3.53797	+	5.50519i
61.7	−0.401566	+	2.79296i	−0.719520	−	1.57553i	−3.80137	−	1.11618i	−1.68991	+	1.46431i	4.68932	−	1.37691i	2.23034	−	3.47048i	−0.0447117	+	0.0979049i	−1.96458	+	2.26725i	−3.41115	−	5.30785i
61.8	−0.267033	+	1.85726i	0.719520	+	1.57553i	0.459881	+	0.135033i	−1.68991	+	1.46431i	−3.11829	+	0.915614i	−6.17188	+	9.60364i	−3.49146	+	7.64522i	−1.96458	+	2.26725i	−2.26834	−	3.52961i
61.9	−0.260994	+	1.81525i	0.719520	+	1.57553i	0.610942	+	0.179389i	1.68991	−	1.46431i	−3.04778	+	0.894908i	3.92381	−	6.10557i	−3.53244	+	7.73497i	−1.96458	+	2.26725i	2.21704	+	3.44979i
61.10	−0.239867	+	1.66831i	−0.719520	−	1.57553i	1.11224	+	0.326582i	1.68991	−	1.46431i	2.80107	−	0.822467i	4.95786	−	7.71458i	−3.61231	+	7.90985i	−1.96458	+	2.26725i	2.03758	+	3.17054i
61.11	−0.223239	+	1.55266i	0.719520	+	1.57553i	1.47705	+	0.433701i	−1.68991	+	1.46431i	−2.60689	+	0.765451i	3.03691	−	4.72552i	−3.60965	+	7.90404i	−1.96458	+	2.26725i	−1.89633	−	2.95075i
61.12	−0.149776	+	1.04171i	−0.719520	−	1.57553i	2.77524	+	0.814884i	1.68991	−	1.46431i	1.74902	−	0.513557i	−4.54855	+	7.07768i	−3.01331	+	6.59823i	−1.96458	+	2.26725i	1.27229	+	1.97972i
61.13	−0.0727214	+	0.505788i	−0.719520	−	1.57553i	3.58744	+	1.05337i	−1.68991	+	1.46431i	0.849208	−	0.249350i	0.975433	−	1.51780i	−1.64275	+	3.59713i	−1.96458	+	2.26725i	−0.617740	−	0.961222i
61.14	−0.0536364	+	0.373049i	−0.719520	−	1.57553i	3.70168	+	1.08691i	−1.68991	+	1.46431i	0.626342	−	0.183911i	−1.87772	+	2.92179i	−1.23027	+	2.69392i	−1.96458	+	2.26725i	−0.455620	−	0.708959i
61.15	−0.0291100	+	0.202465i	0.719520	+	1.57553i	3.79783	+	1.11514i	1.68991	−	1.46431i	−0.339934	+	0.0998136i	−5.53545	+	8.61333i	−0.676218	+	1.48071i	−1.96458	+	2.26725i	0.247278	+	0.384772i
61.16	0.0666286	−	0.463412i	0.719520	+	1.57553i	3.62766	+	1.06518i	1.68991	−	1.46431i	0.778060	−	0.228459i	0.136651	−	0.212633i	1.51327	−	3.31361i	−1.96458	+	2.26725i	−0.565984	−	0.880689i
61.17	0.0899494	−	0.625612i	−0.719520	−	1.57553i	3.45467	+	1.01438i	1.68991	−	1.46431i	−1.05039	+	0.308422i	−4.46997	+	6.95541i	1.99560	−	4.36976i	−1.96458	+	2.26725i	−0.764086	−	1.18894i
61.18	0.108513	−	0.754721i	0.719520	+	1.57553i	3.28014	+	0.963137i	−1.68991	+	1.46431i	1.26716	−	0.372072i	−4.17798	+	6.50107i	2.34982	−	5.14539i	−1.96458	+	2.26725i	0.921772	+	1.43431i
61.19	0.187935	−	1.30712i	0.719520	+	1.57553i	2.16474	+	0.635624i	−1.68991	+	1.46431i	2.19462	−	0.644400i	7.40363	−	11.5203i	3.43198	−	7.51500i	−1.96458	+	2.26725i	1.59644	+	2.48410i
61.20	0.217051	−	1.50962i	−0.719520	−	1.57553i	1.60612	+	0.471599i	−1.68991	+	1.46431i	−2.53463	+	0.744234i	1.37842	−	2.14486i	3.59482	−	7.87157i	−1.96458	+	2.26725i	1.84377	+	2.86896i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.d	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	345.3.o.a	✓	320
23.d	odd	22	1	inner	345.3.o.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
345.3.o.a	✓	320	1.a	even	1	1	trivial
345.3.o.a	✓	320	23.d	odd	22	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(345, [\chi])\).