Embedded newform 354.3.h.a.71.6

• Label: 354.3.h.a.71.6
• Level: $354$
• Weight: $3$
• Character: 354.71
• Analytic conductor: $9.646$
• Analytic rank: $0$
• Dimension: $1120$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.64580135835\)
Analytic rank:	\(0\)
Dimension:	\(1120\)
Relative dimension:	\(40\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			71.6
Character	\(\chi\)	\(=\)	354.71
Dual form			354.3.h.a.5.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 - 1.34018i) q^{2} +(-1.79506 + 2.40370i) q^{3} +(-1.59219 + 1.21035i) q^{4} +(1.34145 - 0.534483i) q^{5} +(4.03197 + 1.32030i) q^{6} +(-2.67432 - 1.23727i) q^{7} +(2.34106 + 1.58728i) q^{8} +(-2.55552 - 8.62956i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.451561	−	1.34018i	−0.225780	−	0.670092i
							\(3\)	−1.79506	+	2.40370i	−0.598353	+	0.801232i
\(5\)	1.34145	−	0.534483i	0.268290	−	0.106897i								−0.232114	−	0.972689i	\(-0.574564\pi\)
							0.500404	+	0.865792i	\(0.333185\pi\)
\(7\)	−2.67432	−	1.23727i	−0.382046	−	0.176753i	0.219460	−	0.975621i	\(-0.429570\pi\)
							−0.601506	+	0.798868i	\(0.705432\pi\)
\(11\)	6.19440	−	1.71987i	0.563127	−	0.156351i	0.0257120	−	0.999669i	\(-0.491815\pi\)
							0.537415	+	0.843318i	\(0.319401\pi\)
\(13\)	−2.72893	−	5.14730i	−0.209917	−	0.395946i	0.756087	−	0.654471i	\(-0.227109\pi\)
							−0.966005	+	0.258525i	\(0.916764\pi\)
\(17\)	8.27961	+	17.8961i	0.487036	+	1.05271i	0.982871	+	0.184294i	\(0.0590000\pi\)
							−0.495835	+	0.868417i	\(0.665138\pi\)
\(19\)	26.8608	+	5.91251i	1.41373	+	0.311185i	0.855198	−	0.518301i	\(-0.173435\pi\)
							0.558528	+	0.829486i	\(0.311366\pi\)
\(23\)	−25.2869	+	4.14557i	−1.09943	+	0.180242i	−0.684071	−	0.729415i	\(-0.739792\pi\)
							−0.415359	+	0.909658i	\(0.636344\pi\)
\(29\)	−17.0250	+	50.5284i	−0.587069	+	1.74236i	0.0794170	+	0.996841i	\(0.474694\pi\)
							−0.666486	+	0.745517i	\(0.732202\pi\)
\(31\)	55.8016	−	12.2829i	1.80005	−	0.396221i	0.816573	−	0.577243i	\(-0.195871\pi\)
							0.983479	+	0.181022i	\(0.0579404\pi\)
\(37\)	36.8291	+	54.3189i	0.995381	+	1.46808i	0.880125	+	0.474741i	\(0.157458\pi\)
							0.115256	+	0.993336i	\(0.463231\pi\)
\(41\)	54.0965	+	8.86867i	1.31943	+	0.216309i	0.780029	−	0.625743i	\(-0.215204\pi\)
							0.539397	+	0.842052i	\(0.318652\pi\)
\(43\)	8.91419	−	32.1060i	0.207307	−	0.746651i	−0.784635	−	0.619958i	\(-0.787150\pi\)
							0.991942	−	0.126694i	\(-0.0404365\pi\)
\(47\)	−13.7791	−	5.49008i	−0.293172	−	0.116810i	0.218915	−	0.975744i	\(-0.429748\pi\)
							−0.512087	+	0.858934i	\(0.671127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.71.6	yes	1120
3.2	odd	2	inner	354.3.h.a.71.21	yes	1120
59.5	even	29	inner	354.3.h.a.5.21	yes	1120
177.5	odd	58	inner	354.3.h.a.5.6	✓	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.6	✓	1120	177.5	odd	58	inner
354.3.h.a.5.21	yes	1120	59.5	even	29	inner
354.3.h.a.71.6	yes	1120	1.1	even	1	trivial
354.3.h.a.71.21	yes	1120	3.2	odd	2	inner