Embedded newform 349.2.h.a.17.4

• Label: 349.2.h.a.17.4
• Level: $349$
• Weight: $2$
• Character: 349.17
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $784$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(784\)
Relative dimension:	\(28\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			17.4
Character	\(\chi\)	\(=\)	349.17
Dual form			349.2.h.a.308.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53864 + 1.62432i) q^{2} +(-0.352185 - 0.414624i) q^{3} +(-0.162732 - 3.00141i) q^{4} +(-2.61649 + 1.98901i) q^{5} +(1.21537 + 0.0658952i) q^{6} +(-1.47986 + 2.45955i) q^{7} +(1.71517 + 1.45688i) q^{8} +(0.437467 - 2.66843i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 784 q - 29 q^{2} - 25 q^{3} + 3 q^{4} - 21 q^{5} - 29 q^{6} - 29 q^{7} + 58 q^{8} - 57 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{43}{58}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53864	+	1.62432i	−1.08798	+	1.14857i	−0.0998427	+	0.995003i	\(0.531834\pi\)
							−0.988139	+	0.153565i	\(0.950925\pi\)
\(3\)	−0.352185	−	0.414624i	−0.203334	−	0.239383i	0.651038	−	0.759045i	\(-0.274334\pi\)
							−0.854372	+	0.519662i	\(0.826058\pi\)
\(5\)	−2.61649	+	1.98901i	−1.17013	+	0.889510i	−0.995466	−	0.0951178i	\(-0.969677\pi\)
							−0.174665	+	0.984628i	\(0.555884\pi\)
\(7\)	−1.47986	+	2.45955i	−0.559335	+	0.929623i	0.440162	+	0.897918i	\(0.354921\pi\)
							−0.999497	+	0.0317042i	\(0.989907\pi\)
\(11\)	−0.998883	+	4.53797i	−0.301175	+	1.36825i	0.546378	+	0.837539i	\(0.316006\pi\)
							−0.847553	+	0.530711i	\(0.821925\pi\)
\(13\)	2.16897	−	0.602211i	0.601564	−	0.167023i	0.0466220	−	0.998913i	\(-0.485154\pi\)
							0.554942	+	0.831889i	\(0.312741\pi\)
\(17\)	−1.40500	−	1.65409i	−0.340763	−	0.401177i	0.564911	−	0.825152i	\(-0.308910\pi\)
							−0.905674	+	0.423975i	\(0.860635\pi\)
\(19\)	−1.62892	−	4.08827i	−0.373699	−	0.937914i	−0.988770	−	0.149443i	\(-0.952252\pi\)
							0.615071	−	0.788472i	\(-0.289127\pi\)
\(23\)	−2.51423	−	9.05543i	−0.524253	−	1.88819i	−0.454781	−	0.890603i	\(-0.650282\pi\)
							−0.0694716	−	0.997584i	\(-0.522131\pi\)
\(29\)	1.90763	+	1.80701i	0.354239	+	0.335553i	0.844039	−	0.536282i	\(-0.180172\pi\)
							−0.489800	+	0.871835i	\(0.662930\pi\)
\(31\)	4.07718	−	7.69037i	0.732282	−	1.38123i	−0.184898	−	0.982758i	\(-0.559195\pi\)
							0.917180	−	0.398473i	\(-0.130460\pi\)
\(37\)	−0.848582	−	3.05632i	−0.139506	−	0.502455i	0.860469	−	0.509502i	\(-0.170171\pi\)
							−0.999975	+	0.00704736i	\(0.997757\pi\)
\(41\)	−0.513328	+	0.604336i	−0.0801683	+	0.0943814i	−0.800784	−	0.598953i	\(-0.795583\pi\)
							0.720616	+	0.693335i	\(0.243859\pi\)
\(43\)	−2.12378	+	9.64841i	−0.323873	+	1.47137i	0.480369	+	0.877066i	\(0.340503\pi\)
							−0.804242	+	0.594302i	\(0.797428\pi\)
\(47\)	−2.93859	+	8.72143i	−0.428638	+	1.27215i	0.488144	+	0.872763i	\(0.337674\pi\)
							−0.916782	+	0.399388i	\(0.869223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.h.a.17.4	✓	784
349.308	even	58	inner	349.2.h.a.308.4	yes	784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.h.a.17.4	✓	784	1.1	even	1	trivial
349.2.h.a.308.4	yes	784	349.308	even	58	inner