Embedded newform 349.2.k.a.3.3

• Label: 349.2.k.a.3.3
• Level: $349$
• Weight: $2$
• Character: 349.3
• Analytic conductor: $2.787$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.k (of order \(174\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.78677903054\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(29\) over \(\Q(\zeta_{174})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{174}]$

Embedding invariants

Embedding label			3.3
Character	\(\chi\)	\(=\)	349.3
Dual form			349.2.k.a.233.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.36455 - 0.565422i) q^{2} +(-2.19178 + 0.400086i) q^{3} +(3.48776 + 1.76918i) q^{4} +(-0.444524 - 3.49845i) q^{5} +(5.40880 + 0.293257i) q^{6} +(2.72845 + 0.0492679i) q^{7} +(-3.54070 - 3.00750i) q^{8} +(1.83731 - 0.693883i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 55 q^{2} - 58 q^{3} - 85 q^{4} - 60 q^{5} - 58 q^{6} - 58 q^{7} - 145 q^{8} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{174}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.36455	−	0.565422i	−1.67199	−	0.399814i	−0.717147	−	0.696921i	\(-0.754553\pi\)
							−0.954844	+	0.297108i	\(0.903978\pi\)
\(3\)	−2.19178	+	0.400086i	−1.26543	+	0.230990i	−0.770215	−	0.637784i	\(-0.779851\pi\)
							−0.495210	+	0.868773i	\(0.664909\pi\)
\(5\)	−0.444524	−	3.49845i	−0.198797	−	1.56455i	−0.709443	−	0.704763i	\(-0.751053\pi\)
							0.510646	−	0.859791i	\(-0.329406\pi\)
\(7\)	2.72845	+	0.0492679i	1.03126	+	0.0186215i	0.531097	−	0.847311i	\(-0.321780\pi\)
							0.500160	+	0.865933i	\(0.333274\pi\)
\(11\)	−1.15172	+	5.23232i	−0.347257	+	1.57760i	0.402202	+	0.915551i	\(0.368245\pi\)
							−0.749458	+	0.662051i	\(0.769686\pi\)
\(13\)	1.38154	+	1.35682i	0.383171	+	0.376315i	0.866545	−	0.499099i	\(-0.166336\pi\)
							−0.483373	+	0.875414i	\(0.660589\pi\)
\(17\)	−2.56322	−	3.01765i	−0.621671	−	0.731888i	0.357727	−	0.933826i	\(-0.383552\pi\)
							−0.979398	+	0.201938i	\(0.935276\pi\)
\(19\)	5.71261	+	0.830921i	1.31056	+	0.190626i	0.763141	−	0.646231i	\(-0.223656\pi\)
							0.547420	+	0.836858i	\(0.315610\pi\)
\(23\)	0.928070	−	0.944979i	0.193516	−	0.197042i	−0.609868	−	0.792503i	\(-0.708778\pi\)
							0.803384	+	0.595461i	\(0.203031\pi\)
\(29\)	9.35530	−	2.78035i	1.73723	−	0.516299i	0.748135	−	0.663546i	\(-0.230949\pi\)
							0.989100	+	0.147248i	\(0.0470414\pi\)
\(31\)	1.41509	−	2.66915i	0.254158	−	0.479393i	−0.723297	−	0.690537i	\(-0.757374\pi\)
							0.977455	+	0.211144i	\(0.0677190\pi\)
\(37\)	−1.63018	−	5.87138i	−0.268000	−	0.965249i	−0.967968	−	0.251075i	\(-0.919216\pi\)
							0.699968	−	0.714175i	\(-0.253198\pi\)
\(41\)	−5.28551	+	6.22258i	−0.825458	+	0.971804i	−0.999917	−	0.0128600i	\(-0.995906\pi\)
							0.174459	+	0.984664i	\(0.444182\pi\)
\(43\)	−6.79918	−	6.21150i	−1.03686	−	0.947245i	−0.0382188	−	0.999269i	\(-0.512168\pi\)
							−0.998646	+	0.0520241i	\(0.983433\pi\)
\(47\)	−0.125485	+	0.372427i	−0.0183039	+	0.0543241i	−0.956416	−	0.292008i	\(-0.905677\pi\)
							0.938112	+	0.346332i	\(0.112573\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.k.a.3.3	✓	1624
349.233	even	174	inner	349.2.k.a.233.3	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.k.a.3.3	✓	1624	1.1	even	1	trivial
349.2.k.a.233.3	yes	1624	349.233	even	174	inner