Embedded newform 354.2.g.b.23.3

• Label: 354.2.g.b.23.3
• 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			23.3
Character	\(\chi\)	\(=\)	354.23
Dual form			354.2.g.b.77.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.725995 - 0.687699i) q^{2} +(-1.30996 - 1.13314i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(1.84131 + 0.301868i) q^{5} +(-1.73028 + 0.0782045i) q^{6} +(1.58049 - 2.98111i) q^{7} +(-0.647386 - 0.762162i) q^{8} +(0.431990 + 2.96873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.30996	−	1.13314i	−0.756306	−	0.654218i
\(5\)	1.84131	+	0.301868i	0.823460	+	0.134999i								0.558758	−	0.829331i	\(-0.311278\pi\)
							0.264702	+	0.964330i	\(0.414726\pi\)
\(7\)	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	−0.367606	−	0.929981i	\(-0.619823\pi\)
							−0.159093	+	0.987264i	\(0.550857\pi\)
\(13\)	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	−0.630166	−	0.776460i	\(-0.717013\pi\)
							0.289026	+	0.957321i	\(0.406668\pi\)
\(19\)	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	0.765014	−	0.644013i	\(-0.222732\pi\)
							−0.998285	+	0.0585490i	\(0.981353\pi\)
\(29\)	−3.12024	+	3.29400i	−0.579414	+	0.611680i	−0.948138	−	0.317859i	\(-0.897036\pi\)
							0.368724	+	0.929539i	\(0.379795\pi\)
\(31\)	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	0.551311	−	0.834300i	\(-0.314128\pi\)
							−0.173498	+	0.984834i	\(0.555507\pi\)
\(43\)	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	\(0.263073\pi\)
							−0.407154	+	0.913360i	\(0.633479\pi\)

(See \(a_n\) instead)

Twists

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

    

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