Embedded newform 230.3.i.a.19.11

• Label: 230.3.i.a.19.11
• Level: $230$
• Weight: $3$
• Character: 230.19
• 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			19.11
Character	\(\chi\)	\(=\)	230.19
Dual form			230.3.i.a.109.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 + 0.587486i) q^{2} +(1.23261 + 4.19789i) q^{3} +(1.30972 - 1.51150i) q^{4} +(-2.21082 + 4.48467i) q^{5} +(-4.05185 - 4.67609i) q^{6} +(0.828200 + 5.76026i) q^{7} +(-0.796860 + 2.71386i) q^{8} +(-8.53169 + 5.48298i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{15}{22}\right)\)

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	0.862031	+	0.506855i	\(0.169192\pi\)
−0.451161	+	0.892443i	\(0.648990\pi\)
\(5\)	−2.21082	+	4.48467i	−0.442164	+	0.896934i
							\(7\)	0.828200	+	5.76026i	0.118314	+	0.822894i	0.959411	+	0.282010i	\(0.0910011\pi\)
−0.841097	+	0.540884i	\(0.818090\pi\)
\(11\)	−9.75571	−	4.45528i	−0.886883	−	0.405026i	−0.0807319	−	0.996736i	\(-0.525726\pi\)
							−0.806151	+	0.591710i	\(0.798453\pi\)
\(13\)	7.82709	+	1.12537i	0.602084	+	0.0865666i	0.436615	−	0.899648i	\(-0.356177\pi\)
							0.165469	+	0.986215i	\(0.447086\pi\)
\(17\)	6.38126	+	7.36437i	0.375369	+	0.433198i	0.911730	−	0.410790i	\(-0.134747\pi\)
							−0.536361	+	0.843988i	\(0.680202\pi\)
\(19\)	10.2733	+	8.90189i	0.540702	+	0.468521i	0.881877	−	0.471479i	\(-0.156280\pi\)
							−0.341176	+	0.940000i	\(0.610825\pi\)
\(23\)	−7.56911	−	21.7189i	−0.329092	−	0.944298i
							\(29\)	−21.9342	−	25.3134i	−0.756352	−	0.872876i	0.238816	−	0.971065i	\(-0.423241\pi\)
−0.995168	+	0.0981884i	\(0.968695\pi\)
\(31\)	27.3144	+	8.02023i	0.881109	+	0.258717i	0.690834	−	0.723014i	\(-0.257244\pi\)
							0.190276	+	0.981731i	\(0.439062\pi\)
\(37\)	−31.7003	+	20.3725i	−0.856764	+	0.550609i	−0.893678	−	0.448709i	\(-0.851884\pi\)
							0.0369133	+	0.999318i	\(0.488247\pi\)
\(41\)	−26.8368	−	17.2470i	−0.654557	−	0.420658i	0.170772	−	0.985311i	\(-0.445374\pi\)
							−0.825329	+	0.564652i	\(0.809010\pi\)
\(43\)	−25.0414	+	7.35281i	−0.582357	+	0.170996i	−0.559625	−	0.828746i	\(-0.689055\pi\)
							−0.0227324	+	0.999742i	\(0.507237\pi\)
\(47\)			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.19.11	✓	240
5.4	even	2	inner	230.3.i.a.19.14	yes	240
23.17	odd	22	inner	230.3.i.a.109.14	yes	240
115.109	odd	22	inner	230.3.i.a.109.11	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.11	✓	240	1.1	even	1	trivial
230.3.i.a.19.14	yes	240	5.4	even	2	inner
230.3.i.a.109.11	yes	240	115.109	odd	22	inner
230.3.i.a.109.14	yes	240	23.17	odd	22	inner