Embedded newform 348.3.p.a.7.35

• Label: 348.3.p.a.7.35
• Level: $348$
• Weight: $3$
• Character: 348.7
• 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			7.35
Character	\(\chi\)	\(=\)	348.7
Dual form			348.3.p.a.199.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.572703 - 1.91625i) q^{2} +(0.751509 - 1.56052i) q^{3} +(-3.34402 - 2.19488i) q^{4} +(1.46187 + 6.40485i) q^{5} +(-2.55996 - 2.33379i) q^{6} +(3.03117 - 6.29428i) q^{7} +(-6.12108 + 5.15096i) q^{8} +(-1.87047 - 2.34549i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 4 q^{2} + 12 q^{4} + 8 q^{5} - 20 q^{8} + 180 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.572703	−	1.91625i	0.286352	−	0.958125i
							\(3\)	0.751509	−	1.56052i	0.250503	−	0.520175i
\(5\)	1.46187	+	6.40485i	0.292373	+	1.28097i								0.881212	+	0.472721i	\(0.156728\pi\)
							−0.588839	+	0.808250i	\(0.700415\pi\)
\(7\)	3.03117	−	6.29428i	0.433024	−	0.899184i	−0.564265	−	0.825594i	\(-0.690840\pi\)
							0.997289	−	0.0735895i	\(-0.0234455\pi\)
\(11\)	−13.6552	−	10.8897i	−1.24138	−	0.989971i	−0.999808	−	0.0195895i	\(-0.993764\pi\)
							−0.241577	−	0.970382i	\(-0.577665\pi\)
\(13\)	9.51612	−	11.9328i	0.732009	−	0.917910i	−0.266941	−	0.963713i	\(-0.586013\pi\)
							0.998951	+	0.0458023i	\(0.0145844\pi\)
\(17\)	−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	−0.768373	−	0.640002i	\(-0.778933\pi\)
							−0.0213010	−	0.999773i	\(-0.506781\pi\)
\(23\)	26.6899	+	6.09179i	1.16043	+	0.264860i	0.759038	−	0.651046i	\(-0.225670\pi\)
							0.401391	+	0.915907i	\(0.368527\pi\)
\(29\)	5.22375	−	28.5256i	0.180129	−	0.983643i
							\(31\)	−5.21661	+	1.19066i	−0.168278	+	0.0384083i	−0.305830	−	0.952086i	\(-0.598934\pi\)
0.137552	+	0.990495i	\(0.456077\pi\)
\(37\)	17.2559	+	21.6382i	0.466376	+	0.584817i	0.958280	−	0.285832i	\(-0.0922701\pi\)
							−0.491903	+	0.870650i	\(0.663699\pi\)
\(41\)	−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	\(-0.336926\pi\)
							0.0634718	+	0.997984i	\(0.479783\pi\)
\(47\)	39.7953	+	31.7357i	0.846709	+	0.675228i	0.947527	−	0.319677i	\(-0.103574\pi\)
							−0.100818	+	0.994905i	\(0.532146\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
348.3.p.a.7.18	✓	360	4.3	odd	2	inner
348.3.p.a.7.35	yes	360	1.1	even	1	trivial
348.3.p.a.199.18	yes	360	29.25	even	7	inner
348.3.p.a.199.35	yes	360	116.83	odd	14	inner