Embedded newform 348.3.p.a.199.18

• Label: 348.3.p.a.199.18
• Level: $348$
• Weight: $3$
• Character: 348.199
• 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			199.18
Character	\(\chi\)	\(=\)	348.199
Dual form			348.3.p.a.7.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34742 + 1.47799i) q^{2} +(-0.751509 - 1.56052i) q^{3} +(-0.368934 - 3.98295i) q^{4} +(1.46187 - 6.40485i) q^{5} +(3.31904 + 0.991951i) q^{6} +(-3.03117 - 6.29428i) q^{7} +(6.38388 + 4.82141i) q^{8} +(-1.87047 + 2.34549i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 4 q^{2} + 12 q^{4} + 8 q^{5} - 20 q^{8} + 180 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(175\)	\(205\)	\(233\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{7}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34742	+	1.47799i	−0.673709	+	0.738997i
							\(3\)	−0.751509	−	1.56052i	−0.250503	−	0.520175i
\(5\)	1.46187	−	6.40485i	0.292373	−	1.28097i								−0.588839	−	0.808250i	\(-0.700415\pi\)
							0.881212	−	0.472721i	\(-0.156728\pi\)
\(7\)	−3.03117	−	6.29428i	−0.433024	−	0.899184i	−0.997289	−	0.0735895i	\(-0.976555\pi\)
							0.564265	−	0.825594i	\(-0.309160\pi\)
\(11\)	13.6552	−	10.8897i	1.24138	−	0.989971i	0.241577	−	0.970382i	\(-0.422335\pi\)
							0.999808	−	0.0195895i	\(-0.00623594\pi\)
\(13\)	9.51612	+	11.9328i	0.732009	+	0.917910i	0.998951	−	0.0458023i	\(-0.0145844\pi\)
							−0.266941	+	0.963713i	\(0.586013\pi\)
\(17\)	−26.4350			−1.55500			−0.777501	−	0.628882i	\(-0.783513\pi\)
							−0.777501	+	0.628882i	\(0.783513\pi\)
\(19\)	15.0038	−	31.1557i	0.789674	−	1.63978i	0.0213010	−	0.999773i	\(-0.493219\pi\)
							0.768373	−	0.640002i	\(-0.221067\pi\)
\(23\)	−26.6899	+	6.09179i	−1.16043	+	0.264860i	−0.759038	−	0.651046i	\(-0.774330\pi\)
							−0.401391	+	0.915907i	\(0.631473\pi\)
\(29\)	5.22375	+	28.5256i	0.180129	+	0.983643i
							\(31\)	5.21661	+	1.19066i	0.168278	+	0.0384083i	0.305830	−	0.952086i	\(-0.401066\pi\)
−0.137552	+	0.990495i	\(0.543923\pi\)
\(37\)	17.2559	−	21.6382i	0.466376	−	0.584817i	−0.491903	−	0.870650i	\(-0.663699\pi\)
							0.958280	+	0.285832i	\(0.0922701\pi\)
\(41\)	−44.2348			−1.07890			−0.539449	−	0.842018i	\(-0.681367\pi\)
							−0.539449	+	0.842018i	\(0.681367\pi\)
\(43\)	−23.8077	+	5.43395i	−0.553667	+	0.126371i	−0.490195	−	0.871613i	\(-0.663074\pi\)
							−0.0634718	+	0.997984i	\(0.520217\pi\)
\(47\)	−39.7953	+	31.7357i	−0.846709	+	0.675228i	−0.947527	−	0.319677i	\(-0.896426\pi\)
							0.100818	+	0.994905i	\(0.467854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	348.3.p.a.199.18	yes	360
4.3	odd	2	inner	348.3.p.a.199.35	yes	360
29.7	even	7	inner	348.3.p.a.7.35	yes	360
116.7	odd	14	inner	348.3.p.a.7.18	✓	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
348.3.p.a.7.18	✓	360	116.7	odd	14	inner
348.3.p.a.7.35	yes	360	29.7	even	7	inner
348.3.p.a.199.18	yes	360	1.1	even	1	trivial
348.3.p.a.199.35	yes	360	4.3	odd	2	inner