Embedded newform 344.2.z.a.109.26

• Label: 344.2.z.a.109.26
• Level: $344$
• Weight: $2$
• Character: 344.109
• Analytic conductor: $2.747$
• Analytic rank: $0$
• Dimension: $504$
• 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			109.26
Character	\(\chi\)	\(=\)	344.109
Dual form			344.2.z.a.101.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.580064 - 1.28978i) q^{2} +(-2.24905 + 0.882686i) q^{3} +(-1.32705 - 1.49631i) q^{4} +(0.933127 + 0.0699282i) q^{5} +(-0.166122 + 3.41278i) q^{6} +(-0.726414 - 1.25819i) q^{7} +(-2.69968 + 0.843650i) q^{8} +(2.07992 - 1.92988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 8 q^{2} - 12 q^{4} - 8 q^{6} - 36 q^{7} - 14 q^{8} - 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(89\)	\(173\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{8}{21}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.580064	−	1.28978i	0.410167	−	0.912010i
							\(3\)	−2.24905	+	0.882686i	−1.29849	+	0.509619i	−0.911057	−	0.412280i	\(-0.864733\pi\)
−0.387430	+	0.921899i	\(0.626637\pi\)
\(5\)	0.933127	+	0.0699282i	0.417307	+	0.0312729i	0.281730	−	0.959494i	\(-0.409092\pi\)
							0.135578	+	0.990767i	\(0.456711\pi\)
\(7\)	−0.726414	−	1.25819i	−0.274559	−	0.475549i	0.695465	−	0.718560i	\(-0.255198\pi\)
							−0.970024	+	0.243011i	\(0.921865\pi\)
\(11\)	−3.74960	+	0.855821i	−1.13055	+	0.258040i	−0.746572	−	0.665304i	\(-0.768302\pi\)
							−0.383974	+	0.923344i	\(0.625445\pi\)
\(13\)	−1.53323	+	2.24883i	−0.425241	+	0.623714i	−0.977745	−	0.209796i	\(-0.932720\pi\)
							0.552504	+	0.833510i	\(0.313672\pi\)
\(17\)	0.126328	+	1.68573i	0.0306391	+	0.408851i	0.991345	+	0.131282i	\(0.0419094\pi\)
							−0.960706	+	0.277568i	\(0.910472\pi\)
\(19\)	−3.85320	+	4.15276i	−0.883984	+	0.952708i	−0.999174	−	0.0406354i	\(-0.987062\pi\)
							0.115190	+	0.993343i	\(0.463252\pi\)
\(23\)	−1.74069	−	0.536930i	−0.362958	−	0.111958i	0.107912	−	0.994160i	\(-0.465584\pi\)
							−0.470870	+	0.882203i	\(0.656060\pi\)
\(29\)	−0.580992	−	0.228023i	−0.107888	−	0.0423427i	0.310783	−	0.950481i	\(-0.399409\pi\)
							−0.418671	+	0.908138i	\(0.637504\pi\)
\(31\)	0.768235	−	0.115793i	0.137979	−	0.0207970i	−0.0796896	−	0.996820i	\(-0.525393\pi\)
							0.217669	+	0.976023i	\(0.430155\pi\)
\(37\)	2.11904	+	1.22343i	0.348368	+	0.201130i	0.663966	−	0.747763i	\(-0.268872\pi\)
							−0.315598	+	0.948893i	\(0.602205\pi\)
\(41\)	−0.330272	−	0.414148i	−0.0515798	−	0.0646791i	0.755372	−	0.655296i	\(-0.227456\pi\)
							−0.806952	+	0.590617i	\(0.798885\pi\)
\(43\)	−4.96207	−	4.28694i	−0.756708	−	0.653753i
							\(47\)	−0.534974	+	2.34388i	−0.0780340	+	0.341889i	−0.998841	−	0.0481289i	\(-0.984674\pi\)
0.920807	+	0.390018i	\(0.127531\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
344.2.z.a.101.4	✓	504	43.15	even	21	inner
344.2.z.a.101.26	yes	504	344.101	even	42	inner
344.2.z.a.109.4	yes	504	8.5	even	2	inner
344.2.z.a.109.26	yes	504	1.1	even	1	trivial