Embedded newform 349.2.h.a.17.9

• Label: 349.2.h.a.17.9
• 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			17.9
Character	\(\chi\)	\(=\)	349.17
Dual form			349.2.h.a.308.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.998387 + 1.05398i) q^{2} +(1.55038 + 1.82525i) q^{3} +(-0.00582860 - 0.107502i) q^{4} +(2.23112 - 1.69606i) q^{5} +(-3.47167 - 0.188228i) q^{6} +(-0.345966 + 0.575000i) q^{7} +(-2.09386 - 1.77854i) q^{8} +(-0.442511 + 2.69920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 784 q - 29 q^{2} - 25 q^{3} + 3 q^{4} - 21 q^{5} - 29 q^{6} - 29 q^{7} + 58 q^{8} - 57 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.998387	+	1.05398i	−0.705966	+	0.745280i	−0.976047	−	0.217558i	\(-0.930191\pi\)
							0.270081	+	0.962838i	\(0.412949\pi\)
\(3\)	1.55038	+	1.82525i	0.895114	+	1.05381i	0.998250	+	0.0591423i	\(0.0188366\pi\)
							−0.103136	+	0.994667i	\(0.532888\pi\)
\(5\)	2.23112	−	1.69606i	0.997789	−	0.758499i	0.0272954	−	0.999627i	\(-0.491311\pi\)
							0.970493	+	0.241128i	\(0.0775174\pi\)
\(7\)	−0.345966	+	0.575000i	−0.130763	+	0.217329i	−0.915320	−	0.402727i	\(-0.868063\pi\)
							0.784558	+	0.620056i	\(0.212890\pi\)
\(11\)	−1.12024	+	5.08932i	−0.337766	+	1.53449i	0.435372	+	0.900251i	\(0.356617\pi\)
							−0.773138	+	0.634238i	\(0.781314\pi\)
\(13\)	−0.608689	+	0.169002i	−0.168820	+	0.0468726i	−0.350910	−	0.936409i	\(-0.614128\pi\)
							0.182090	+	0.983282i	\(0.441714\pi\)
\(17\)	−0.248318	−	0.292342i	−0.0602259	−	0.0709034i	0.731224	−	0.682137i	\(-0.238949\pi\)
							−0.791450	+	0.611234i	\(0.790673\pi\)
\(19\)	1.90490	+	4.78094i	0.437014	+	1.09682i	0.968335	+	0.249653i	\(0.0803164\pi\)
							−0.531321	+	0.847170i	\(0.678304\pi\)
\(23\)	0.404779	+	1.45788i	0.0844023	+	0.303990i	0.993687	−	0.112184i	\(-0.0357846\pi\)
							−0.909285	+	0.416174i	\(0.863371\pi\)
\(29\)	−4.11114	−	3.89428i	−0.763420	−	0.723150i	0.203264	−	0.979124i	\(-0.434845\pi\)
							−0.966684	+	0.255974i	\(0.917604\pi\)
\(31\)	3.41620	−	6.44364i	0.613568	−	1.15731i	−0.360862	−	0.932619i	\(-0.617518\pi\)
							0.974430	−	0.224692i	\(-0.0721376\pi\)
\(37\)	0.452807	+	1.63086i	0.0744409	+	0.268112i	0.991399	−	0.130870i	\(-0.0417772\pi\)
							−0.916959	+	0.398983i	\(0.869363\pi\)
\(41\)	−0.302676	+	0.356338i	−0.0472700	+	0.0556506i	−0.785289	−	0.619129i	\(-0.787486\pi\)
							0.738019	+	0.674780i	\(0.235761\pi\)
\(43\)	−0.0633769	+	0.287924i	−0.00966488	+	0.0439080i	−0.981206	−	0.192962i	\(-0.938191\pi\)
							0.971541	+	0.236870i	\(0.0761216\pi\)
\(47\)	2.48816	−	7.38459i	0.362935	−	1.07715i	−0.598265	−	0.801299i	\(-0.704143\pi\)
							0.961200	−	0.275854i	\(-0.0889606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.h.a.17.9	✓	784
349.308	even	58	inner	349.2.h.a.308.9	yes	784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.h.a.17.9	✓	784	1.1	even	1	trivial
349.2.h.a.308.9	yes	784	349.308	even	58	inner