Embedded newform 349.2.h.a.17.8

• Label: 349.2.h.a.17.8
• 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			17.8
Character	\(\chi\)	\(=\)	349.17
Dual form			349.2.h.a.308.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12530 + 1.18796i) q^{2} +(1.22054 + 1.43693i) q^{3} +(-0.0366819 - 0.676559i) q^{4} +(-0.841182 + 0.639450i) q^{5} +(-3.08050 - 0.167020i) q^{6} +(-2.16228 + 3.59373i) q^{7} +(-1.64928 - 1.40091i) q^{8} +(-0.0897083 + 0.547197i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 784 q - 29 q^{2} - 25 q^{3} + 3 q^{4} - 21 q^{5} - 29 q^{6} - 29 q^{7} + 58 q^{8} - 57 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12530	+	1.18796i	−0.795706	+	0.840017i	−0.990021	−	0.140923i	\(-0.954993\pi\)
							0.194315	+	0.980939i	\(0.437752\pi\)
\(3\)	1.22054	+	1.43693i	0.704680	+	0.829614i	0.991735	−	0.128300i	\(-0.0409521\pi\)
							−0.287055	+	0.957914i	\(0.592676\pi\)
\(5\)	−0.841182	+	0.639450i	−0.376188	+	0.285971i	−0.776134	−	0.630568i	\(-0.782822\pi\)
							0.399946	+	0.916539i	\(0.369029\pi\)
\(7\)	−2.16228	+	3.59373i	−0.817264	+	1.35830i	0.113822	+	0.993501i	\(0.463691\pi\)
							−0.931085	+	0.364802i	\(0.881137\pi\)
\(11\)	0.659731	−	2.99719i	0.198916	−	0.903686i	−0.766351	−	0.642422i	\(-0.777930\pi\)
							0.965267	−	0.261264i	\(-0.0841393\pi\)
\(13\)	−3.90863	+	1.08522i	−1.08406	+	0.300987i	−0.763214	−	0.646145i	\(-0.776380\pi\)
							−0.320843	+	0.947132i	\(0.603966\pi\)
\(17\)	3.67633	+	4.32811i	0.891641	+	1.04972i	0.998462	+	0.0554427i	\(0.0176570\pi\)
							−0.106821	+	0.994278i	\(0.534067\pi\)
\(19\)	−1.99221	−	5.00007i	−0.457044	−	1.14709i	−0.959303	−	0.282378i	\(-0.908877\pi\)
							0.502259	−	0.864717i	\(-0.332502\pi\)
\(23\)	1.04128	+	3.75035i	0.217122	+	0.782002i	0.989212	+	0.146490i	\(0.0467975\pi\)
							−0.772090	+	0.635513i	\(0.780789\pi\)
\(29\)	1.69861	+	1.60901i	0.315423	+	0.298785i	0.828988	−	0.559266i	\(-0.188917\pi\)
							−0.513565	+	0.858051i	\(0.671675\pi\)
\(31\)	−4.53853	+	8.56057i	−0.815144	+	1.53752i	0.0278567	+	0.999612i	\(0.491132\pi\)
							−0.843001	+	0.537913i	\(0.819213\pi\)
\(37\)	−1.32775	−	4.78212i	−0.218281	−	0.786176i	−0.988862	−	0.148834i	\(-0.952448\pi\)
							0.770581	−	0.637342i	\(-0.219966\pi\)
\(41\)	2.75000	−	3.23755i	0.429478	−	0.505621i	−0.504273	−	0.863544i	\(-0.668239\pi\)
							0.933751	+	0.357924i	\(0.116515\pi\)
\(43\)	0.830776	−	3.77425i	0.126692	−	0.575568i	−0.869911	−	0.493210i	\(-0.835824\pi\)
							0.996603	−	0.0823589i	\(-0.0262454\pi\)
\(47\)	−1.78073	+	5.28501i	−0.259746	+	0.770898i	0.735850	+	0.677145i	\(0.236783\pi\)
							−0.995595	+	0.0937529i	\(0.970114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.h.a.17.8	✓	784
349.308	even	58	inner	349.2.h.a.308.8	yes	784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.h.a.17.8	✓	784	1.1	even	1	trivial
349.2.h.a.308.8	yes	784	349.308	even	58	inner