Embedded newform 349.2.g.a.31.10

• Label: 349.2.g.a.31.10
• 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			31.10
Character	\(\chi\)	\(=\)	349.31
Dual form			349.2.g.a.304.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.168433 - 1.02740i) q^{2} +(0.294114 - 0.554758i) q^{3} +(0.868128 - 0.292507i) q^{4} +(1.98905 - 1.88412i) q^{5} +(-0.619496 - 0.208733i) q^{6} +(1.60551 + 0.174610i) q^{7} +(-1.42208 - 2.68232i) q^{8} +(1.46231 + 2.15674i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 26 q^{2} - 25 q^{3} - 46 q^{4} - 19 q^{5} - 9 q^{6} - 19 q^{7} - 95 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.168433	−	1.02740i	−0.119100	−	0.726480i	−0.977092	−	0.212817i	\(-0.931736\pi\)
							0.857992	−	0.513664i	\(-0.171712\pi\)
\(3\)	0.294114	−	0.554758i	0.169807	−	0.320290i	−0.783974	−	0.620794i	\(-0.786810\pi\)
							0.953780	+	0.300505i	\(0.0971551\pi\)
\(5\)	1.98905	−	1.88412i	0.889528	−	0.842606i	−0.0987744	−	0.995110i	\(-0.531492\pi\)
							0.988303	+	0.152504i	\(0.0487336\pi\)
\(7\)	1.60551	+	0.174610i	0.606827	+	0.0659965i	0.406374	−	0.913707i	\(-0.366793\pi\)
							0.200453	+	0.979703i	\(0.435758\pi\)
\(11\)	0.848532	+	3.05614i	0.255842	+	0.921459i	0.974181	+	0.225767i	\(0.0724887\pi\)
							−0.718339	+	0.695693i	\(0.755098\pi\)
\(13\)	−0.0125749	−	0.231930i	−0.00348764	−	0.0643257i	0.996271	−	0.0862737i	\(-0.0274959\pi\)
							−0.999759	+	0.0219480i	\(0.993013\pi\)
\(17\)	−3.76974	+	7.11048i	−0.914296	+	1.72454i	−0.264586	+	0.964362i	\(0.585235\pi\)
							−0.649710	+	0.760182i	\(0.725110\pi\)
\(19\)	−4.39208	−	5.17075i	−1.00761	−	1.18625i	−0.982442	−	0.186569i	\(-0.940263\pi\)
							−0.0251699	−	0.999683i	\(-0.508013\pi\)
\(23\)	0.212676	+	3.92258i	0.0443460	+	0.817915i	0.933119	+	0.359567i	\(0.117076\pi\)
							−0.888773	+	0.458347i	\(0.848442\pi\)
\(29\)	−0.234421	−	1.42990i	−0.0435309	−	0.265527i	0.956090	−	0.293074i	\(-0.0946784\pi\)
							−0.999620	+	0.0275479i	\(0.991230\pi\)
\(31\)	−8.99875	+	1.98077i	−1.61622	+	0.355757i	−0.928823	−	0.370523i	\(-0.879179\pi\)
							−0.687399	+	0.726280i	\(0.741248\pi\)
\(37\)	−0.388774	−	7.17052i	−0.0639141	−	1.17883i	−0.837973	−	0.545712i	\(-0.816259\pi\)
							0.774059	−	0.633114i	\(-0.218224\pi\)
\(41\)	1.23214	+	2.32406i	0.192428	+	0.362958i	0.960887	−	0.276942i	\(-0.0893209\pi\)
							−0.768459	+	0.639899i	\(0.778976\pi\)
\(43\)	−3.08112	−	11.0972i	−0.469867	−	1.69231i	−0.695859	−	0.718179i	\(-0.744976\pi\)
							0.225992	−	0.974129i	\(-0.427438\pi\)
\(47\)	−1.79348	+	4.50129i	−0.261606	+	0.656581i	−0.999839	−	0.0179649i	\(-0.994281\pi\)
							0.738233	+	0.674546i	\(0.235661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.g.a.31.10	✓	756
349.304	even	29	inner	349.2.g.a.304.10	yes	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.g.a.31.10	✓	756	1.1	even	1	trivial
349.2.g.a.304.10	yes	756	349.304	even	29	inner