Embedded newform 348.3.p.a.103.42

• Label: 348.3.p.a.103.42
• Level: $348$
• Weight: $3$
• Character: 348.103
• Analytic conductor: $9.482$
• Analytic rank: $0$
• Dimension: $360$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	348.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.48231319974\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(60\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			103.42
Character	\(\chi\)	\(=\)	348.103
Dual form			348.3.p.a.223.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.34918 + 1.47638i) q^{2} +(1.35417 + 1.07992i) q^{3} +(-0.359412 + 3.98382i) q^{4} +(5.34462 + 2.57383i) q^{5} +(0.232656 + 3.45628i) q^{6} +(-7.67602 - 6.12142i) q^{7} +(-6.36656 + 4.84427i) q^{8} +(0.667563 + 2.92478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 4 q^{2} + 12 q^{4} + 8 q^{5} - 20 q^{8} + 180 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/348\mathbb{Z}\right)^\times\).

\(n\)	\(175\)	\(205\)	\(233\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{7}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.34918	+	1.47638i	0.674591	+	0.738191i
							\(3\)	1.35417	+	1.07992i	0.451391	+	0.359972i
\(5\)	5.34462	+	2.57383i	1.06892	+	0.514767i								0.883760	−	0.467941i	\(-0.155004\pi\)
							0.185165	+	0.982708i	\(0.440718\pi\)
\(7\)	−7.67602	−	6.12142i	−1.09657	−	0.874489i	−0.103813	−	0.994597i	\(-0.533104\pi\)
							−0.992761	+	0.120108i	\(0.961676\pi\)
\(11\)	20.1223	+	4.59277i	1.82930	+	0.417525i	0.991679	−	0.128733i	\(-0.0410911\pi\)
							0.837617	+	0.546258i	\(0.183948\pi\)
\(13\)	−3.27967	+	14.3692i	−0.252282	+	1.10532i	0.677010	+	0.735974i	\(0.263276\pi\)
							−0.929292	+	0.369346i	\(0.879582\pi\)
\(17\)	−2.90209			−0.170711			−0.0853556	−	0.996351i	\(-0.527203\pi\)
							−0.0853556	+	0.996351i	\(0.527203\pi\)
\(19\)	−13.6004	+	10.8459i	−0.715810	+	0.570839i	−0.912229	−	0.409681i	\(-0.865640\pi\)
							0.196419	+	0.980520i	\(0.437069\pi\)
\(23\)	2.20470	+	4.57810i	0.0958565	+	0.199048i	0.943383	−	0.331704i	\(-0.107624\pi\)
							−0.847527	+	0.530752i	\(0.821909\pi\)
\(29\)	20.8750	−	20.1304i	0.719828	−	0.694152i
							\(31\)	5.95771	−	12.3713i	0.192184	−	0.399075i	−0.782503	−	0.622646i	\(-0.786057\pi\)
0.974687	+	0.223572i	\(0.0717718\pi\)
\(37\)	−12.8894	−	56.4722i	−0.348363	−	1.52628i	−0.780898	−	0.624659i	\(-0.785238\pi\)
							0.432535	−	0.901617i	\(-0.357619\pi\)
\(41\)	65.2367			1.59114			0.795569	−	0.605862i	\(-0.207172\pi\)
							0.795569	+	0.605862i	\(0.207172\pi\)
\(43\)	−9.17012	−	19.0419i	−0.213259	−	0.442836i	0.766709	−	0.641995i	\(-0.221893\pi\)
							−0.979967	+	0.199159i	\(0.936179\pi\)
\(47\)	29.1356	+	6.65001i	0.619906	+	0.141490i	0.520929	−	0.853600i	\(-0.325586\pi\)
							0.0989775	+	0.995090i	\(0.468443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	348.3.p.a.103.42	✓	360
4.3	odd	2	inner	348.3.p.a.103.59	yes	360
29.20	even	7	inner	348.3.p.a.223.59	yes	360
116.107	odd	14	inner	348.3.p.a.223.42	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
348.3.p.a.103.42	✓	360	1.1	even	1	trivial
348.3.p.a.103.59	yes	360	4.3	odd	2	inner
348.3.p.a.223.42	yes	360	116.107	odd	14	inner
348.3.p.a.223.59	yes	360	29.20	even	7	inner