Embedded newform 145.4.l.a.4.32

• Label: 145.4.l.a.4.32
• 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			4.32
Character	\(\chi\)	\(=\)	145.4
Dual form			145.4.l.a.109.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.607251 - 2.66054i) q^{2} +(6.07887 - 2.92743i) q^{3} +(0.498029 + 0.239838i) q^{4} +(-1.74951 + 11.0426i) q^{5} +(-4.09715 - 17.9508i) q^{6} +(12.3167 + 25.5759i) q^{7} +(14.5524 - 18.2481i) q^{8} +(11.5486 - 14.4815i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 186 q^{4} - 13 q^{5} + 26 q^{6} - 346 q^{9}+O(q^{10}) \)

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	\(-0.768331\pi\)
							0.961328	−	0.275406i	\(-0.0888123\pi\)
\(3\)	6.07887	−	2.92743i	1.16988	−	0.563384i	0.254933	−	0.966959i	\(-0.417947\pi\)
							0.914947	+	0.403574i	\(0.132232\pi\)
\(5\)	−1.74951	+	11.0426i	−0.156481	+	0.987681i
							\(7\)	12.3167	+	25.5759i	0.665039	+	1.38097i	0.911292	+	0.411762i	\(0.135086\pi\)
−0.246253	+	0.969206i	\(0.579199\pi\)
\(11\)	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	−0.100798	−	0.994907i	\(-0.532140\pi\)
							0.947533	+	0.319659i	\(0.103568\pi\)
\(17\)	−76.2857			−1.08835			−0.544176	−	0.838971i	\(-0.683158\pi\)
							−0.544176	+	0.838971i	\(0.683158\pi\)
\(19\)	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	−0.776907	−	0.629615i	\(-0.783213\pi\)
							−0.426790	+	0.904351i	\(0.640356\pi\)
\(29\)	−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	\(-0.569418\pi\)
							0.999980	−	0.00631717i	\(-0.00201083\pi\)
\(41\)		−	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	\(-0.907751\pi\)
							0.739406	−	0.673260i	\(-0.235106\pi\)
\(47\)	243.665	+	305.546i	0.756217	+	0.948266i	0.999766	−	0.0216228i	\(-0.00688328\pi\)
							−0.243550	+	0.969888i	\(0.578312\pi\)

(See \(a_n\) instead)

Twists

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

    

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