Embedded newform 392.2.z.a.109.42

• Label: 392.2.z.a.109.42
• 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			109.42
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03288 - 0.966005i) q^{2} +(0.167383 + 1.11051i) q^{3} +(0.133667 - 1.99553i) q^{4} +(-2.65001 - 1.04005i) q^{5} +(1.24565 + 0.985329i) q^{6} +(2.56499 + 0.648729i) q^{7} +(-1.78963 - 2.19026i) q^{8} +(1.66150 - 0.512504i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03288	−	0.966005i	0.730354	−	0.683069i
							\(3\)	0.167383	+	1.11051i	0.0966385	+	0.641154i	0.983742	+	0.179585i	\(0.0574754\pi\)
−0.887104	+	0.461570i	\(0.847286\pi\)
\(5\)	−2.65001	−	1.04005i	−1.18512	−	0.465126i	−0.310801	−	0.950475i	\(-0.600597\pi\)
							−0.874320	+	0.485349i	\(0.838692\pi\)
\(7\)	2.56499	+	0.648729i	0.969473	+	0.245196i
							\(11\)	1.29629	−	4.20248i	0.390848	−	1.26710i	−0.519933	−	0.854207i	\(-0.674043\pi\)
0.910781	−	0.412890i	\(-0.135481\pi\)
\(13\)	1.42183	−	0.324524i	0.394346	−	0.0900069i	−0.0207497	−	0.999785i	\(-0.506605\pi\)
							0.415096	+	0.909778i	\(0.363748\pi\)
\(17\)	−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	0.103282	−	0.994652i	\(-0.467066\pi\)
							0.913035	+	0.407881i	\(0.133732\pi\)
\(23\)	−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	\(0.331734\pi\)
							−0.989567	+	0.144073i	\(0.953980\pi\)
\(31\)	−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	0.206349	−	0.978478i	\(-0.433842\pi\)
							0.349879	+	0.936795i	\(0.386223\pi\)
\(41\)	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	0.980518	−	0.196428i	\(-0.0629343\pi\)
							0.0266823	+	0.999644i	\(0.491506\pi\)
\(47\)	−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.109.42	✓	648
8.5	even	2	inner	392.2.z.a.109.43	yes	648
49.9	even	21	inner	392.2.z.a.205.43	yes	648
392.205	even	42	inner	392.2.z.a.205.42	yes	648

    

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