Embedded newform 230.3.i.a.109.14

• Label: 230.3.i.a.109.14
• Level: $230$
• Weight: $3$
• Character: 230.109
• Analytic conductor: $6.267$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	230.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			109.14
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(-1.23261 + 4.19789i) q^{3} +(1.30972 + 1.51150i) q^{4} +(-3.16099 - 3.87403i) q^{5} +(-4.05185 + 4.67609i) q^{6} +(-0.828200 + 5.76026i) q^{7} +(0.796860 + 2.71386i) q^{8} +(-8.53169 - 5.48298i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{22}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28641	+	0.587486i	0.643207	+	0.293743i
							\(3\)	−1.23261	+	4.19789i	−0.410871	+	1.39930i	0.451161	+	0.892443i	\(0.351010\pi\)
−0.862031	+	0.506855i	\(0.830808\pi\)
\(5\)	−3.16099	−	3.87403i	−0.632198	−	0.774806i
							\(7\)	−0.828200	+	5.76026i	−0.118314	+	0.822894i	0.841097	+	0.540884i	\(0.181910\pi\)
−0.959411	+	0.282010i	\(0.908999\pi\)
\(11\)	−9.75571	+	4.45528i	−0.886883	+	0.405026i	−0.806151	−	0.591710i	\(-0.798453\pi\)
							−0.0807319	+	0.996736i	\(0.525726\pi\)
\(13\)	−7.82709	+	1.12537i	−0.602084	+	0.0865666i	−0.436615	−	0.899648i	\(-0.643823\pi\)
							−0.165469	+	0.986215i	\(0.552914\pi\)
\(17\)	−6.38126	+	7.36437i	−0.375369	+	0.433198i	−0.911730	−	0.410790i	\(-0.865253\pi\)
							0.536361	+	0.843988i	\(0.319798\pi\)
\(19\)	10.2733	−	8.90189i	0.540702	−	0.468521i	−0.341176	−	0.940000i	\(-0.610825\pi\)
							0.881877	+	0.471479i	\(0.156280\pi\)
\(23\)	7.56911	−	21.7189i	0.329092	−	0.944298i
							\(29\)	−21.9342	+	25.3134i	−0.756352	+	0.872876i	−0.995168	−	0.0981884i	\(-0.968695\pi\)
0.238816	+	0.971065i	\(0.423241\pi\)
\(31\)	27.3144	−	8.02023i	0.881109	−	0.258717i	0.190276	−	0.981731i	\(-0.439062\pi\)
							0.690834	+	0.723014i	\(0.257244\pi\)
\(37\)	31.7003	+	20.3725i	0.856764	+	0.550609i	0.893678	−	0.448709i	\(-0.148116\pi\)
							−0.0369133	+	0.999318i	\(0.511753\pi\)
\(41\)	−26.8368	+	17.2470i	−0.654557	+	0.420658i	−0.825329	−	0.564652i	\(-0.809010\pi\)
							0.170772	+	0.985311i	\(0.445374\pi\)
\(43\)	25.0414	+	7.35281i	0.582357	+	0.170996i	0.559625	−	0.828746i	\(-0.310945\pi\)
							0.0227324	+	0.999742i	\(0.492763\pi\)
\(47\)			75.4595i			1.60552i	0.596302	+	0.802760i	\(0.296636\pi\)
							−0.596302	+	0.802760i	\(0.703364\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.109.14	yes	240
5.4	even	2	inner	230.3.i.a.109.11	yes	240
23.19	odd	22	inner	230.3.i.a.19.11	✓	240
115.19	odd	22	inner	230.3.i.a.19.14	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.11	✓	240	23.19	odd	22	inner
230.3.i.a.19.14	yes	240	115.19	odd	22	inner
230.3.i.a.109.11	yes	240	5.4	even	2	inner
230.3.i.a.109.14	yes	240	1.1	even	1	trivial