Embedded newform 392.2.z.a.109.43

• Label: 392.2.z.a.109.43
• Level: $392$
• Weight: $2$
• Character: 392.109
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			109.43
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.43

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04049 - 0.957798i) q^{2} +(-0.167383 - 1.11051i) q^{3} +(0.165244 - 1.99316i) q^{4} +(2.65001 + 1.04005i) q^{5} +(-1.23781 - 0.995159i) q^{6} +(2.56499 + 0.648729i) q^{7} +(-1.73711 - 2.23214i) q^{8} +(1.66150 - 0.512504i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04049	−	0.957798i	0.735738	−	0.677266i
							\(3\)	−0.167383	−	1.11051i	−0.0966385	−	0.641154i	−0.983742	−	0.179585i	\(-0.942525\pi\)
0.887104	−	0.461570i	\(-0.152714\pi\)
\(5\)	2.65001	+	1.04005i	1.18512	+	0.465126i	0.874320	−	0.485349i	\(-0.161308\pi\)
							0.310801	+	0.950475i	\(0.399403\pi\)
\(7\)	2.56499	+	0.648729i	0.969473	+	0.245196i
							\(11\)	−1.29629	+	4.20248i	−0.390848	+	1.26710i	0.519933	+	0.854207i	\(0.325957\pi\)
−0.910781	+	0.412890i	\(0.864519\pi\)
\(13\)	−1.42183	+	0.324524i	−0.394346	+	0.0900069i	−0.415096	−	0.909778i	\(-0.636252\pi\)
							0.0207497	+	0.999785i	\(0.493395\pi\)
\(17\)	−0.726083	+	0.495035i	−0.176101	+	0.120064i	−0.648163	−	0.761502i	\(-0.724462\pi\)
							0.472062	+	0.881566i	\(0.343510\pi\)
\(19\)	−4.43002	+	2.55768i	−1.01632	+	0.586771i	−0.913035	−	0.407881i	\(-0.866268\pi\)
							−0.103282	+	0.994652i	\(0.532934\pi\)
\(23\)	−4.97250	−	3.39019i	−1.03684	−	0.706904i	−0.0797832	−	0.996812i	\(-0.525423\pi\)
							−0.957054	+	0.289908i	\(0.906375\pi\)
\(29\)	2.61300	−	5.42596i	0.485223	−	1.00758i	−0.504344	−	0.863503i	\(-0.668266\pi\)
							0.989567	−	0.144073i	\(-0.0460200\pi\)
\(31\)	−3.42577	+	5.93360i	−0.615286	+	1.06571i	0.375049	+	0.927005i	\(0.377626\pi\)
							−0.990334	+	0.138701i	\(0.955707\pi\)
\(37\)	−3.38341	+	0.253551i	−0.556229	+	0.0416836i	−0.349879	−	0.936795i	\(-0.613777\pi\)
							−0.206349	+	0.978478i	\(0.566158\pi\)
\(41\)	2.18561	+	2.74067i	0.341335	+	0.428020i	0.922638	−	0.385666i	\(-0.126028\pi\)
							−0.581303	+	0.813687i	\(0.697457\pi\)
\(43\)	−6.60466	−	5.26704i	−1.00720	−	0.803216i	−0.0266823	−	0.999644i	\(-0.508494\pi\)
							−0.980518	+	0.196428i	\(0.937066\pi\)
\(47\)	−1.71704	+	1.59318i	−0.250456	+	0.232389i	−0.795402	−	0.606082i	\(-0.792740\pi\)
							0.544947	+	0.838471i	\(0.316550\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.43	yes	648
8.5	even	2	inner	392.2.z.a.109.42	✓	648
49.9	even	21	inner	392.2.z.a.205.42	yes	648
392.205	even	42	inner	392.2.z.a.205.43	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.42	✓	648	8.5	even	2	inner
392.2.z.a.109.43	yes	648	1.1	even	1	trivial
392.2.z.a.205.42	yes	648	49.9	even	21	inner
392.2.z.a.205.43	yes	648	392.205	even	42	inner