Embedded newform 349.2.g.a.31.20

• Label: 349.2.g.a.31.20
• 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			31.20
Character	\(\chi\)	\(=\)	349.31
Dual form			349.2.g.a.304.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.226756 + 1.38315i) q^{2} +(-1.17622 + 2.21859i) q^{3} +(0.0336242 - 0.0113293i) q^{4} +(-0.238431 + 0.225854i) q^{5} +(-3.33535 - 1.12381i) q^{6} +(-4.90800 - 0.533778i) q^{7} +(1.33635 + 2.52062i) q^{8} +(-1.85508 - 2.73603i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 756 q - 26 q^{2} - 25 q^{3} - 46 q^{4} - 19 q^{5} - 9 q^{6} - 19 q^{7} - 95 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.226756	+	1.38315i	0.160341	+	0.978034i	0.937310	+	0.348496i	\(0.113307\pi\)
							−0.776970	+	0.629538i	\(0.783244\pi\)
\(3\)	−1.17622	+	2.21859i	−0.679092	+	1.28090i	0.269072	+	0.963120i	\(0.413283\pi\)
							−0.948164	+	0.317782i	\(0.897062\pi\)
\(5\)	−0.238431	+	0.225854i	−0.106630	+	0.101005i	−0.739157	−	0.673533i	\(-0.764776\pi\)
							0.632528	+	0.774538i	\(0.282017\pi\)
\(7\)	−4.90800	−	0.533778i	−1.85505	−	0.201749i	−0.888614	−	0.458655i	\(-0.848331\pi\)
							−0.966436	+	0.256906i	\(0.917297\pi\)
\(11\)	−1.29340	−	4.65841i	−0.389975	−	1.40456i	−0.854706	−	0.519112i	\(-0.826263\pi\)
							0.464731	−	0.885452i	\(-0.346151\pi\)
\(13\)	0.0844088	+	1.55683i	0.0234108	+	0.431787i	0.986620	+	0.163035i	\(0.0521283\pi\)
							−0.963210	+	0.268752i	\(0.913389\pi\)
\(17\)	−2.65412	+	5.00620i	−0.643719	+	1.21418i	0.320039	+	0.947404i	\(0.396304\pi\)
							−0.963758	+	0.266778i	\(0.914041\pi\)
\(19\)	1.73163	+	2.03863i	0.397263	+	0.467695i	0.924033	−	0.382314i	\(-0.124873\pi\)
							−0.526769	+	0.850008i	\(0.676597\pi\)
\(23\)	−0.0856525	−	1.57977i	−0.0178598	−	0.329404i	−0.993786	−	0.111305i	\(-0.964497\pi\)
							0.975927	−	0.218099i	\(-0.0699857\pi\)
\(29\)	0.159083	+	0.970361i	0.0295409	+	0.180192i	0.997476	−	0.0710105i	\(-0.0226224\pi\)
							−0.967935	+	0.251202i	\(0.919174\pi\)
\(31\)	5.39050	−	1.18654i	0.968163	−	0.213109i	0.297377	−	0.954760i	\(-0.403888\pi\)
							0.670786	+	0.741651i	\(0.265957\pi\)
\(37\)	0.179991	+	3.31973i	0.0295903	+	0.545761i	0.975319	+	0.220802i	\(0.0708676\pi\)
							−0.945728	+	0.324959i	\(0.894650\pi\)
\(41\)	−3.39428	−	6.40230i	−0.530098	−	0.999871i	−0.993512	−	0.113729i	\(-0.963720\pi\)
							0.463414	−	0.886142i	\(-0.346624\pi\)
\(43\)	−1.33975	−	4.82534i	−0.204310	−	0.735857i	−0.992692	−	0.120672i	\(-0.961495\pi\)
							0.788383	−	0.615185i	\(-0.210919\pi\)
\(47\)	−1.35549	+	3.40201i	−0.197718	+	0.496235i	−0.994088	−	0.108575i	\(-0.965371\pi\)
							0.796370	+	0.604809i	\(0.206751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.g.a.31.20	✓	756
349.304	even	29	inner	349.2.g.a.304.20	yes	756

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.g.a.31.20	✓	756	1.1	even	1	trivial
349.2.g.a.304.20	yes	756	349.304	even	29	inner