Embedded newform 354.2.g.a.23.4

• Label: 354.2.g.a.23.4
• Level: $354$
• Weight: $2$
• Character: 354.23
• Analytic conductor: $2.827$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	354.g (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.82670423155\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(10\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			23.4
Character	\(\chi\)	\(=\)	354.23
Dual form			354.2.g.a.77.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.725995 + 0.687699i) q^{2} +(-0.713094 + 1.57845i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(-1.84131 - 0.301868i) q^{5} +(-0.567794 - 1.63634i) q^{6} +(1.58049 - 2.98111i) q^{7} +(0.647386 + 0.762162i) q^{8} +(-1.98299 - 2.25116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{15}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.725995	+	0.687699i	−0.513356	+	0.486277i
							\(3\)	−0.713094	+	1.57845i	−0.411705	+	0.911317i
\(5\)	−1.84131	−	0.301868i	−0.823460	−	0.134999i								−0.264702	−	0.964330i	\(-0.585274\pi\)
							−0.558758	+	0.829331i	\(0.688722\pi\)
\(7\)	1.58049	−	2.98111i	0.597368	−	1.12675i	−0.381904	−	0.924202i	\(-0.624731\pi\)
							0.979271	−	0.202553i	\(-0.0649238\pi\)
\(11\)	1.74687	−	0.189983i	0.526700	−	0.0572821i	0.159093	−	0.987264i	\(-0.449143\pi\)
							0.367606	+	0.929981i	\(0.380177\pi\)
\(13\)	1.17888	−	2.54811i	0.326963	−	0.706718i	−0.672480	−	0.740116i	\(-0.734771\pi\)
							0.999442	+	0.0333976i	\(0.0106328\pi\)
\(17\)	1.40656	−	0.745708i	0.341140	−	0.180861i	−0.289026	−	0.957321i	\(-0.593332\pi\)
							0.630166	+	0.776460i	\(0.282987\pi\)
\(19\)	1.10300	−	0.663652i	0.253045	−	0.152252i	−0.383380	−	0.923591i	\(-0.625240\pi\)
							0.636425	+	0.771338i	\(0.280412\pi\)
\(23\)	1.11873	+	2.80779i	0.233270	+	0.585464i	0.998285	−	0.0585490i	\(-0.0186474\pi\)
							−0.765014	+	0.644013i	\(0.777268\pi\)
\(29\)	3.12024	−	3.29400i	0.579414	−	0.611680i	−0.368724	−	0.929539i	\(-0.620205\pi\)
							0.948138	+	0.317859i	\(0.102964\pi\)
\(31\)	0.351453	−	0.584120i	0.0631229	−	0.104911i	−0.823689	−	0.567042i	\(-0.808088\pi\)
							0.886811	+	0.462131i	\(0.152915\pi\)
\(37\)	7.43707	+	6.31710i	1.22265	+	1.03853i	0.997944	+	0.0640977i	\(0.0204169\pi\)
							0.224703	+	0.974427i	\(0.427859\pi\)
\(41\)	−2.41918	−	0.963890i	−0.377813	−	0.150534i	0.173498	−	0.984834i	\(-0.444493\pi\)
							−0.551311	+	0.834300i	\(0.685872\pi\)
\(43\)	0.988209	−	9.08644i	0.150701	−	1.38567i	−0.637595	−	0.770372i	\(-0.720071\pi\)
							0.788296	−	0.615297i	\(-0.210964\pi\)
\(47\)	−1.85325	−	11.3043i	−0.270324	−	1.64890i	−0.677478	−	0.735543i	\(-0.736927\pi\)
							0.407154	−	0.913360i	\(-0.366521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.23.4	✓	280
3.2	odd	2		354.2.g.b.23.3	yes	280
59.18	odd	58		354.2.g.b.77.3	yes	280
177.77	even	58	inner	354.2.g.a.77.4	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.23.4	✓	280	1.1	even	1	trivial
354.2.g.a.77.4	yes	280	177.77	even	58	inner
354.2.g.b.23.3	yes	280	3.2	odd	2
354.2.g.b.77.3	yes	280	59.18	odd	58