Embedded newform 349.2.i.a.9.12

• Label: 349.2.i.a.9.12
• Level: $349$
• Weight: $2$
• Character: 349.9
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $1568$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.i (of order \(87\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(1568\)
Relative dimension:	\(28\) over \(\Q(\zeta_{87})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{87}]$

Embedding invariants

Embedding label			9.12
Character	\(\chi\)	\(=\)	349.9
Dual form			349.2.i.a.194.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.736757 - 0.373722i) q^{2} +(2.21410 - 0.836181i) q^{3} +(-0.778263 - 1.06309i) q^{4} +(0.343155 - 0.0886359i) q^{5} +(-1.94375 - 0.211396i) q^{6} +(5.07513 + 0.183344i) q^{7} +(0.443393 + 2.70458i) q^{8} +(1.95198 - 1.71965i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{87}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.736757	−	0.373722i	−0.520966	−	0.264262i	0.172172	−	0.985067i	\(-0.444922\pi\)
							−0.693138	+	0.720805i	\(0.743772\pi\)
\(3\)	2.21410	−	0.836181i	1.27831	−	0.482770i	0.381166	−	0.924507i	\(-0.375523\pi\)
							0.897145	+	0.441737i	\(0.145638\pi\)
\(5\)	0.343155	−	0.0886359i	0.153464	−	0.0396392i	−0.172570	−	0.984997i	\(-0.555207\pi\)
							0.326034	+	0.945358i	\(0.394288\pi\)
\(7\)	5.07513	+	0.183344i	1.91822	+	0.0692975i	0.969251	−	0.246075i	\(-0.0791409\pi\)
							0.948968	+	0.315372i	\(0.102129\pi\)
\(11\)	0.382558	+	0.176990i	0.115346	+	0.0533646i	0.476713	−	0.879059i	\(-0.341828\pi\)
							−0.361368	+	0.932423i	\(0.617690\pi\)
\(13\)	0.0973284	+	5.39004i	0.0269940	+	1.49493i	0.677554	+	0.735473i	\(0.263040\pi\)
							−0.650560	+	0.759455i	\(0.725466\pi\)
\(17\)	0.900628	−	5.49359i	0.218434	−	1.33239i	−0.618801	−	0.785548i	\(-0.712381\pi\)
							0.837235	−	0.546843i	\(-0.184171\pi\)
\(19\)	−4.99723	−	1.48515i	−1.14644	−	0.340718i	−0.344890	−	0.938643i	\(-0.612084\pi\)
							−0.801552	+	0.597925i	\(0.795992\pi\)
\(23\)	−0.130895	−	7.24894i	−0.0272934	−	1.51151i	−0.668254	−	0.743934i	\(-0.732958\pi\)
							0.640960	−	0.767574i	\(-0.278536\pi\)
\(29\)	−3.90254	+	2.54437i	−0.724684	+	0.472478i	−0.854758	−	0.519026i	\(-0.826295\pi\)
							0.130074	+	0.991504i	\(0.458479\pi\)
\(31\)	2.15042	+	3.17163i	0.386226	+	0.569641i	0.970253	−	0.242093i	\(-0.0778340\pi\)
							−0.584027	+	0.811734i	\(0.698524\pi\)
\(37\)	−6.31440	+	3.79925i	−1.03808	+	0.624593i	−0.929244	−	0.369467i	\(-0.879540\pi\)
							−0.108837	+	0.994060i	\(0.534713\pi\)
\(41\)	−0.462340	−	2.82015i	−0.0722054	−	0.440434i	−0.998044	−	0.0625146i	\(-0.980088\pi\)
							0.925839	−	0.377919i	\(-0.123360\pi\)
\(43\)	−0.201667	−	2.22783i	−0.0307539	−	0.339740i	−0.996711	−	0.0810381i	\(-0.974176\pi\)
							0.965957	−	0.258702i	\(-0.0832948\pi\)
\(47\)	−8.32332	−	6.32722i	−1.21408	−	0.922920i	−0.215517	−	0.976500i	\(-0.569144\pi\)
							−0.998564	+	0.0535798i	\(0.982937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.i.a.9.12	✓	1568
349.194	even	87	inner	349.2.i.a.194.12	yes	1568

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.i.a.9.12	✓	1568	1.1	even	1	trivial
349.2.i.a.194.12	yes	1568	349.194	even	87	inner