Embedded newform 344.2.z.a.101.4

• Label: 344.2.z.a.101.4
• 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			101.4
Character	\(\chi\)	\(=\)	344.101
Dual form			344.2.z.a.109.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38652 - 0.278518i) q^{2} +(2.24905 + 0.882686i) q^{3} +(1.84486 + 0.772339i) q^{4} +(-0.933127 + 0.0699282i) q^{5} +(-2.87250 - 1.85026i) q^{6} +(-0.726414 + 1.25819i) q^{7} +(-2.34281 - 1.58469i) q^{8} +(2.07992 + 1.92988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 8 q^{2} - 12 q^{4} - 8 q^{6} - 36 q^{7} - 14 q^{8} - 64 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38652	−	0.278518i	−0.980415	−	0.196942i
							\(3\)	2.24905	+	0.882686i	1.29849	+	0.509619i	0.911057	−	0.412280i	\(-0.135267\pi\)
0.387430	+	0.921899i	\(0.373363\pi\)
\(5\)	−0.933127	+	0.0699282i	−0.417307	+	0.0312729i	−0.281730	−	0.959494i	\(-0.590908\pi\)
							−0.135578	+	0.990767i	\(0.543289\pi\)
\(7\)	−0.726414	+	1.25819i	−0.274559	+	0.475549i	−0.970024	−	0.243011i	\(-0.921865\pi\)
							0.695465	+	0.718560i	\(0.255198\pi\)
\(11\)	3.74960	+	0.855821i	1.13055	+	0.258040i	0.746572	−	0.665304i	\(-0.231698\pi\)
							0.383974	+	0.923344i	\(0.374555\pi\)
\(13\)	1.53323	+	2.24883i	0.425241	+	0.623714i	0.977745	−	0.209796i	\(-0.0672800\pi\)
							−0.552504	+	0.833510i	\(0.686328\pi\)
\(17\)	0.126328	−	1.68573i	0.0306391	−	0.408851i	−0.960706	−	0.277568i	\(-0.910472\pi\)
							0.991345	−	0.131282i	\(-0.0419094\pi\)
\(19\)	3.85320	+	4.15276i	0.883984	+	0.952708i	0.999174	−	0.0406354i	\(-0.0129382\pi\)
							−0.115190	+	0.993343i	\(0.536748\pi\)
\(23\)	−1.74069	+	0.536930i	−0.362958	+	0.111958i	−0.470870	−	0.882203i	\(-0.656060\pi\)
							0.107912	+	0.994160i	\(0.465584\pi\)
\(29\)	0.580992	−	0.228023i	0.107888	−	0.0423427i	−0.310783	−	0.950481i	\(-0.600591\pi\)
							0.418671	+	0.908138i	\(0.362496\pi\)
\(31\)	0.768235	+	0.115793i	0.137979	+	0.0207970i	0.217669	−	0.976023i	\(-0.430155\pi\)
							−0.0796896	+	0.996820i	\(0.525393\pi\)
\(37\)	−2.11904	+	1.22343i	−0.348368	+	0.201130i	−0.663966	−	0.747763i	\(-0.731128\pi\)
							0.315598	+	0.948893i	\(0.397795\pi\)
\(41\)	−0.330272	+	0.414148i	−0.0515798	+	0.0646791i	−0.806952	−	0.590617i	\(-0.798885\pi\)
							0.755372	+	0.655296i	\(0.227456\pi\)
\(43\)	4.96207	−	4.28694i	0.756708	−	0.653753i
							\(47\)	−0.534974	−	2.34388i	−0.0780340	−	0.341889i	0.920807	−	0.390018i	\(-0.127531\pi\)
−0.998841	+	0.0481289i	\(0.984674\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	344.2.z.a.101.4	✓	504
8.5	even	2	inner	344.2.z.a.101.26	yes	504
43.23	even	21	inner	344.2.z.a.109.26	yes	504
344.109	even	42	inner	344.2.z.a.109.4	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
344.2.z.a.101.4	✓	504	1.1	even	1	trivial
344.2.z.a.101.26	yes	504	8.5	even	2	inner
344.2.z.a.109.4	yes	504	344.109	even	42	inner
344.2.z.a.109.26	yes	504	43.23	even	21	inner