Embedded newform 349.2.g.a.31.3

• Label: 349.2.g.a.31.3
• Level: $349$
• Weight: $2$
• Character: 349.31
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $756$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.g (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(756\)
Relative dimension:	\(27\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			31.3
Character	\(\chi\)	\(=\)	349.31
Dual form			349.2.g.a.304.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.397225 - 2.42297i) q^{2} +(-0.263553 + 0.497114i) q^{3} +(-3.81768 + 1.28632i) q^{4} +(1.72972 - 1.63847i) q^{5} +(1.30918 + 0.441115i) q^{6} +(3.06696 + 0.333552i) q^{7} +(2.33302 + 4.40055i) q^{8} +(1.50590 + 2.22103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 26 q^{2} - 25 q^{3} - 46 q^{4} - 19 q^{5} - 9 q^{6} - 19 q^{7} - 95 q^{8} - 44 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{29}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.397225	−	2.42297i	−0.280881	−	1.71330i	−0.630318	−	0.776337i	\(-0.717076\pi\)
							0.349438	−	0.936960i	\(-0.386373\pi\)
\(3\)	−0.263553	+	0.497114i	−0.152163	+	0.287009i	−0.947861	−	0.318684i	\(-0.896759\pi\)
							0.795698	+	0.605693i	\(0.207104\pi\)
\(5\)	1.72972	−	1.63847i	0.773552	−	0.732748i	−0.195204	−	0.980763i	\(-0.562537\pi\)
							0.968756	+	0.248015i	\(0.0797783\pi\)
\(7\)	3.06696	+	0.333552i	1.15920	+	0.126071i	0.667443	−	0.744661i	\(-0.267389\pi\)
							0.491759	+	0.870732i	\(0.336354\pi\)
\(11\)	−1.56432	−	5.63419i	−0.471662	−	1.69877i	−0.690773	−	0.723072i	\(-0.742730\pi\)
							0.219111	−	0.975700i	\(-0.429684\pi\)
\(13\)	−0.189421	−	3.49366i	−0.0525359	−	0.968968i	−0.899391	−	0.437145i	\(-0.855990\pi\)
							0.846855	−	0.531823i	\(-0.178493\pi\)
\(17\)	−0.280696	+	0.529448i	−0.0680787	+	0.128410i	−0.915215	−	0.402965i	\(-0.867980\pi\)
							0.847137	+	0.531375i	\(0.178324\pi\)
\(19\)	2.03327	+	2.39374i	0.466463	+	0.549163i	0.944214	−	0.329333i	\(-0.106824\pi\)
							−0.477751	+	0.878495i	\(0.658548\pi\)
\(23\)	0.00480440	+	0.0886120i	0.00100179	+	0.0184769i	0.998992	−	0.0448981i	\(-0.0142963\pi\)
							−0.997990	+	0.0633750i	\(0.979814\pi\)
\(29\)	−1.06741	−	6.51091i	−0.198213	−	1.20905i	−0.879093	−	0.476651i	\(-0.841851\pi\)
							0.680880	−	0.732395i	\(-0.261598\pi\)
\(31\)	−4.04405	+	0.890163i	−0.726333	+	0.159878i	−0.562721	−	0.826647i	\(-0.690245\pi\)
							−0.163612	+	0.986525i	\(0.552314\pi\)
\(37\)	0.363615	+	6.70649i	0.0597779	+	1.10254i	0.862458	+	0.506129i	\(0.168924\pi\)
							−0.802680	+	0.596410i	\(0.796593\pi\)
\(41\)	0.136311	+	0.257109i	0.0212881	+	0.0401537i	0.893928	−	0.448211i	\(-0.147939\pi\)
							−0.872640	+	0.488364i	\(0.837594\pi\)
\(43\)	−1.37199	−	4.94146i	−0.209227	−	0.753566i	−0.991441	−	0.130556i	\(-0.958324\pi\)
							0.782214	−	0.623009i	\(-0.214090\pi\)
\(47\)	−0.254001	+	0.637494i	−0.0370499	+	0.0929881i	−0.946337	−	0.323180i	\(-0.895248\pi\)
							0.909288	+	0.416168i	\(0.136627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.g.a.31.3	✓	756
349.304	even	29	inner	349.2.g.a.304.3	yes	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.g.a.31.3	✓	756	1.1	even	1	trivial
349.2.g.a.304.3	yes	756	349.304	even	29	inner