Embedded newform 344.2.z.a.101.9

• Label: 344.2.z.a.101.9
• Level: $344$
• Weight: $2$
• Character: 344.101
• Analytic conductor: $2.747$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 344 = 2^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	344.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.74685382953\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(42\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			101.9
Character	\(\chi\)	\(=\)	344.101
Dual form			344.2.z.a.109.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16997 + 0.794468i) q^{2} +(-0.623819 - 0.244831i) q^{3} +(0.737642 - 1.85900i) q^{4} +(2.60221 - 0.195009i) q^{5} +(0.924357 - 0.209160i) q^{6} +(-0.0646287 + 0.111940i) q^{7} +(0.613900 + 2.76100i) q^{8} +(-1.86995 - 1.73506i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 8 q^{2} - 12 q^{4} - 8 q^{6} - 36 q^{7} - 14 q^{8} - 64 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/344\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(89\)	\(173\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{21}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16997	+	0.794468i	−0.827291	+	0.561773i
							\(3\)	−0.623819	−	0.244831i	−0.360162	−	0.141353i	0.178359	−	0.983966i	\(-0.442921\pi\)
−0.538521	+	0.842612i	\(0.681017\pi\)
\(5\)	2.60221	−	0.195009i	1.16374	−	0.0872105i	0.521183	−	0.853445i	\(-0.325491\pi\)
							0.642561	+	0.766235i	\(0.277872\pi\)
\(7\)	−0.0646287	+	0.111940i	−0.0244274	+	0.0423094i	−0.877981	−	0.478696i	\(-0.841110\pi\)
							0.853553	+	0.521006i	\(0.174443\pi\)
\(11\)	3.12788	+	0.713919i	0.943092	+	0.215255i	0.666307	−	0.745677i	\(-0.267874\pi\)
							0.276785	+	0.960932i	\(0.410731\pi\)
\(13\)	−1.00880	−	1.47964i	−0.279792	−	0.410379i	0.660459	−	0.750862i	\(-0.270362\pi\)
							−0.940251	+	0.340483i	\(0.889409\pi\)
\(17\)	0.593443	−	7.91895i	0.143931	−	1.92063i	−0.196761	−	0.980452i	\(-0.563042\pi\)
							0.340692	−	0.940175i	\(-0.389339\pi\)
\(19\)	2.92327	+	3.15054i	0.670644	+	0.722783i	0.973101	−	0.230381i	\(-0.0739971\pi\)
							−0.302456	+	0.953163i	\(0.597807\pi\)
\(23\)	−1.75633	+	0.541757i	−0.366221	+	0.112964i	−0.472403	−	0.881383i	\(-0.656613\pi\)
							0.106183	+	0.994347i	\(0.466137\pi\)
\(29\)	4.95266	−	1.94378i	0.919686	−	0.360950i	0.142191	−	0.989839i	\(-0.454585\pi\)
							0.777495	+	0.628889i	\(0.216490\pi\)
\(31\)	7.84424	+	1.18233i	1.40887	+	0.212352i	0.809029	−	0.587769i	\(-0.199994\pi\)
							0.599838	+	0.800122i	\(0.295232\pi\)
\(37\)	9.30036	−	5.36957i	1.52897	−	0.882751i	0.529565	−	0.848270i	\(-0.322355\pi\)
							0.999405	−	0.0344817i	\(-0.0109780\pi\)
\(41\)	−2.85521	+	3.58032i	−0.445909	+	0.559152i	−0.953090	−	0.302687i	\(-0.902116\pi\)
							0.507181	+	0.861840i	\(0.330688\pi\)
\(43\)	−5.87172	+	2.91939i	−0.895429	+	0.445204i
							\(47\)	−1.81013	−	7.93069i	−0.264034	−	1.15681i	−0.916831	−	0.399276i	\(-0.869261\pi\)
0.652796	−	0.757533i	\(-0.273596\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	344.2.z.a.101.9	✓	504
8.5	even	2	inner	344.2.z.a.101.16	yes	504
43.23	even	21	inner	344.2.z.a.109.16	yes	504
344.109	even	42	inner	344.2.z.a.109.9	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
344.2.z.a.101.9	✓	504	1.1	even	1	trivial
344.2.z.a.101.16	yes	504	8.5	even	2	inner
344.2.z.a.109.9	yes	504	344.109	even	42	inner
344.2.z.a.109.16	yes	504	43.23	even	21	inner