Embedded newform 145.4.l.a.4.13

• Label: 145.4.l.a.4.13
• Level: $145$
• Weight: $4$
• Character: 145.4
• Analytic conductor: $8.555$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 145 = 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	145.l (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.55527695083\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(44\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	145.4
Dual form			145.4.l.a.109.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.607251 + 2.66054i) q^{2} +(-6.07887 + 2.92743i) q^{3} +(0.498029 + 0.239838i) q^{4} +(-3.21496 + 10.7081i) q^{5} +(-4.09715 - 17.9508i) q^{6} +(-12.3167 - 25.5759i) q^{7} +(-14.5524 + 18.2481i) q^{8} +(11.5486 - 14.4815i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 186 q^{4} - 13 q^{5} + 26 q^{6} - 346 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(117\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.607251	+	2.66054i	−0.214696	+	0.940643i	0.746632	+	0.665237i	\(0.231669\pi\)
							−0.961328	+	0.275406i	\(0.911188\pi\)
\(3\)	−6.07887	+	2.92743i	−1.16988	+	0.563384i	−0.914947	−	0.403574i	\(-0.867768\pi\)
							−0.254933	+	0.966959i	\(0.582053\pi\)
\(5\)	−3.21496	+	10.7081i	−0.287554	+	0.957764i
							\(7\)	−12.3167	−	25.5759i	−0.665039	−	1.38097i	−0.911292	−	0.411762i	\(-0.864914\pi\)
0.246253	−	0.969206i	\(-0.420801\pi\)
\(11\)	6.40645	−	5.10897i	0.175601	−	0.140038i	−0.531745	−	0.846905i	\(-0.678464\pi\)
							0.707346	+	0.706867i	\(0.249892\pi\)
\(13\)	−39.6883	+	31.6503i	−0.846734	+	0.675248i	−0.947533	−	0.319659i	\(-0.896432\pi\)
							0.100798	+	0.994907i	\(0.467860\pi\)
\(17\)	76.2857			1.08835			0.544176	−	0.838971i	\(-0.316842\pi\)
							0.544176	+	0.838971i	\(0.316842\pi\)
\(19\)	14.4441	−	29.9936i	0.174406	−	0.362158i	−0.795381	−	0.606109i	\(-0.792729\pi\)
							0.969787	+	0.243951i	\(0.0784437\pi\)
\(23\)	132.773	−	30.3045i	1.20370	−	0.274736i	0.426790	−	0.904351i	\(-0.359644\pi\)
							0.776907	+	0.629615i	\(0.216787\pi\)
\(29\)	−133.035	−	81.7969i	−0.851860	−	0.523769i
							\(31\)	43.7711	+	9.99047i	0.253598	+	0.0578820i	0.347430	−	0.937706i	\(-0.387055\pi\)
−0.0938321	+	0.995588i	\(0.529912\pi\)
\(37\)	−176.364	+	221.153i	−0.783622	+	0.982631i	0.216358	+	0.976314i	\(0.430582\pi\)
							−0.999980	+	0.00631717i	\(0.997989\pi\)
\(41\)		−	180.101i		−	0.686025i	−0.939331	−	0.343012i	\(-0.888553\pi\)
							0.939331	−	0.343012i	\(-0.111447\pi\)
\(43\)	61.7210	+	270.417i	0.218892	+	0.959030i	0.958298	+	0.285770i	\(0.0922493\pi\)
							−0.739406	+	0.673260i	\(0.764894\pi\)
\(47\)	−243.665	−	305.546i	−0.756217	−	0.948266i	0.243550	−	0.969888i	\(-0.421688\pi\)
							−0.999766	+	0.0216228i	\(0.993117\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	145.4.l.a.4.13	✓	264
5.4	even	2	inner	145.4.l.a.4.32	yes	264
29.22	even	14	inner	145.4.l.a.109.32	yes	264
145.109	even	14	inner	145.4.l.a.109.13	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.4.l.a.4.13	✓	264	1.1	even	1	trivial
145.4.l.a.4.32	yes	264	5.4	even	2	inner
145.4.l.a.109.13	yes	264	145.109	even	14	inner
145.4.l.a.109.32	yes	264	29.22	even	14	inner