Embedded newform 230.2.l.a.217.2

• Label: 230.2.l.a.217.2
• Level: $230$
• Weight: $2$
• Character: 230.217
• Analytic conductor: $1.837$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	230.l (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.83655924649\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			217.2
Character	\(\chi\)	\(=\)	230.217
Dual form			230.2.l.a.53.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0713392 + 0.997452i) q^{2} +(-1.71120 + 0.372248i) q^{3} +(-0.989821 - 0.142315i) q^{4} +(-1.89205 - 1.19169i) q^{5} +(-0.249224 - 1.73339i) q^{6} +(3.16947 - 1.73066i) q^{7} +(0.212565 - 0.977147i) q^{8} +(0.0607343 - 0.0277364i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{3} - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0713392	+	0.997452i	−0.0504444	+	0.705305i
							\(3\)	−1.71120	+	0.372248i	−0.987961	+	0.214918i	−0.677383	−	0.735630i	\(-0.736886\pi\)
−0.310578	+	0.950548i	\(0.600522\pi\)
\(5\)	−1.89205	−	1.19169i	−0.846152	−	0.532941i
							\(7\)	3.16947	−	1.73066i	1.19795	−	0.654129i	0.248676	−	0.968587i	\(-0.420005\pi\)
0.949271	+	0.314458i	\(0.101823\pi\)
\(11\)	−1.69797	−	1.47130i	−0.511958	−	0.443614i	0.360174	−	0.932885i	\(-0.382717\pi\)
							−0.872132	+	0.489271i	\(0.837263\pi\)
\(13\)	2.52851	−	4.63061i	0.701282	−	1.28430i	−0.247659	−	0.968847i	\(-0.579661\pi\)
							0.948941	−	0.315454i	\(-0.102157\pi\)
\(17\)	2.07167	+	1.55083i	0.502453	+	0.376131i	0.820230	−	0.572034i	\(-0.193845\pi\)
							−0.317777	+	0.948165i	\(0.602936\pi\)
\(19\)	0.731070	−	5.08470i	0.167719	−	1.16651i	−0.715865	−	0.698239i	\(-0.753967\pi\)
							0.883584	−	0.468273i	\(-0.155123\pi\)
\(23\)	−2.42036	−	4.14027i	−0.504680	−	0.863307i
							\(29\)	−4.62333	+	0.664735i	−0.858531	+	0.123438i	−0.557501	−	0.830176i	\(-0.688240\pi\)
−0.301031	+	0.953615i	\(0.597331\pi\)
\(31\)	−7.65941	+	4.92241i	−1.37567	+	0.884090i	−0.999105	−	0.0422994i	\(-0.986532\pi\)
							−0.376566	+	0.926390i	\(0.622895\pi\)
\(37\)	−2.86759	−	1.06956i	−0.471429	−	0.175834i	0.102510	−	0.994732i	\(-0.467313\pi\)
							−0.573939	+	0.818898i	\(0.694585\pi\)
\(41\)	1.83808	−	4.02483i	0.287060	−	0.628574i	−0.710082	−	0.704119i	\(-0.751342\pi\)
							0.997142	+	0.0755450i	\(0.0240696\pi\)
\(43\)	−2.56386	−	11.7859i	−0.390985	−	1.79733i	−0.579184	−	0.815197i	\(-0.696629\pi\)
							0.188200	−	0.982131i	\(-0.439735\pi\)
\(47\)	−0.0966659	−	0.0966659i	−0.0141002	−	0.0141002i	0.700022	−	0.714122i	\(-0.253174\pi\)
							−0.714122	+	0.700022i	\(0.753174\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.2.l.a.217.2	yes	240
5.3	odd	4	inner	230.2.l.a.33.8	yes	240
23.7	odd	22	inner	230.2.l.a.7.8	✓	240
115.53	even	44	inner	230.2.l.a.53.2	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.l.a.7.8	✓	240	23.7	odd	22	inner
230.2.l.a.33.8	yes	240	5.3	odd	4	inner
230.2.l.a.53.2	yes	240	115.53	even	44	inner
230.2.l.a.217.2	yes	240	1.1	even	1	trivial