Embedded newform 348.3.p.a.7.1

• Label: 348.3.p.a.7.1
• Level: $348$
• Weight: $3$
• Character: 348.7
• Analytic conductor: $9.482$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• 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			7.1
Character	\(\chi\)	\(=\)	348.7
Dual form			348.3.p.a.199.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99911 + 0.0596572i) q^{2} +(-0.751509 + 1.56052i) q^{3} +(3.99288 - 0.238523i) q^{4} +(-0.103650 - 0.454120i) q^{5} +(1.40925 - 3.16449i) q^{6} +(0.138351 - 0.287290i) q^{7} +(-7.96798 + 0.715037i) q^{8} +(-1.87047 - 2.34549i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 4 q^{2} + 12 q^{4} + 8 q^{5} - 20 q^{8} + 180 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(175\)	\(205\)	\(233\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{7}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99911	+	0.0596572i	−0.999555	+	0.0298286i
							\(3\)	−0.751509	+	1.56052i	−0.250503	+	0.520175i
\(5\)	−0.103650	−	0.454120i	−0.0207300	−	0.0908241i								0.963505	−	0.267691i	\(-0.0862608\pi\)
							−0.984235	+	0.176867i	\(0.943404\pi\)
\(7\)	0.138351	−	0.287290i	0.0197645	−	0.0410414i	−0.890853	−	0.454292i	\(-0.849892\pi\)
							0.910617	+	0.413250i	\(0.135607\pi\)
\(11\)	6.75012	+	5.38304i	0.613647	+	0.489367i	0.880295	−	0.474427i	\(-0.157345\pi\)
							−0.266648	+	0.963794i	\(0.585916\pi\)
\(13\)	8.10928	−	10.1687i	0.623791	−	0.782209i	−0.365082	−	0.930976i	\(-0.618959\pi\)
							0.988872	+	0.148767i	\(0.0475303\pi\)
\(17\)	−0.612846			−0.0360498			−0.0180249	−	0.999838i	\(-0.505738\pi\)
							−0.0180249	+	0.999838i	\(0.505738\pi\)
\(19\)	−2.15286	−	4.47047i	−0.113309	−	0.235288i	0.836601	−	0.547813i	\(-0.184539\pi\)
							−0.949909	+	0.312525i	\(0.898825\pi\)
\(23\)	−11.6639	−	2.66221i	−0.507126	−	0.115748i	−0.0387000	−	0.999251i	\(-0.512322\pi\)
							−0.468426	+	0.883503i	\(0.655179\pi\)
\(29\)	−12.6529	+	26.0941i	−0.436307	+	0.899798i
							\(31\)	26.0908	−	5.95504i	0.841637	−	0.192098i	0.220095	−	0.975479i	\(-0.429363\pi\)
0.621543	+	0.783380i	\(0.286506\pi\)
\(37\)	18.8526	+	23.6405i	0.509531	+	0.638932i	0.968349	−	0.249598i	\(-0.0802986\pi\)
							−0.458818	+	0.888530i	\(0.651727\pi\)
\(41\)	50.8037			1.23911			0.619557	−	0.784952i	\(-0.287312\pi\)
							0.619557	+	0.784952i	\(0.287312\pi\)
\(43\)	20.9188	+	4.77459i	0.486485	+	0.111037i	0.458726	−	0.888578i	\(-0.348306\pi\)
							0.0277585	+	0.999615i	\(0.491163\pi\)
\(47\)	43.8959	+	35.0058i	0.933956	+	0.744805i	0.967034	−	0.254648i	\(-0.0819595\pi\)
							−0.0330778	+	0.999453i	\(0.510531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	348.3.p.a.7.1	✓	360
4.3	odd	2	inner	348.3.p.a.7.51	yes	360
29.25	even	7	inner	348.3.p.a.199.51	yes	360
116.83	odd	14	inner	348.3.p.a.199.1	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
348.3.p.a.7.1	✓	360	1.1	even	1	trivial
348.3.p.a.7.51	yes	360	4.3	odd	2	inner
348.3.p.a.199.1	yes	360	116.83	odd	14	inner
348.3.p.a.199.51	yes	360	29.25	even	7	inner