Embedded newform 392.2.z.a.205.8

• Label: 392.2.z.a.205.8
• Level: $392$
• Weight: $2$
• Character: 392.205
• 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			205.8
Character	\(\chi\)	\(=\)	392.205
Dual form			392.2.z.a.109.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29596 + 0.566107i) q^{2} +(0.502307 - 3.33259i) q^{3} +(1.35904 - 1.46731i) q^{4} +(-0.588711 + 0.231052i) q^{5} +(1.23563 + 4.60328i) q^{6} +(-1.76263 + 1.97310i) q^{7} +(-0.930618 + 2.67095i) q^{8} +(-7.98712 - 2.46370i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29596	+	0.566107i	−0.916385	+	0.400298i
							\(3\)	0.502307	−	3.33259i	0.290007	−	1.92407i	−0.0836307	−	0.996497i	\(-0.526652\pi\)
0.373638	−	0.927575i	\(-0.378110\pi\)
\(5\)	−0.588711	+	0.231052i	−0.263279	+	0.103330i	−0.493309	−	0.869854i	\(-0.664213\pi\)
							0.230030	+	0.973184i	\(0.426118\pi\)
\(7\)	−1.76263	+	1.97310i	−0.666213	+	0.745762i
							\(11\)	−0.526052	−	1.70542i	−0.158611	−	0.514203i	0.841031	−	0.540988i	\(-0.181949\pi\)
−0.999641	+	0.0267846i	\(0.991473\pi\)
\(13\)	−0.264361	−	0.0603387i	−0.0733206	−	0.0167350i	0.185703	−	0.982606i	\(-0.440544\pi\)
							−0.259024	+	0.965871i	\(0.583401\pi\)
\(17\)	−4.13715	−	2.82066i	−1.00341	−	0.684111i	−0.0541235	−	0.998534i	\(-0.517236\pi\)
							−0.949283	+	0.314423i	\(0.898189\pi\)
\(19\)	1.56736	+	0.904917i	0.359577	+	0.207602i	0.668895	−	0.743357i	\(-0.266767\pi\)
							−0.309318	+	0.950959i	\(0.600101\pi\)
\(23\)	−1.95361	+	1.33195i	−0.407356	+	0.277731i	−0.749614	−	0.661875i	\(-0.769761\pi\)
							0.342258	+	0.939606i	\(0.388808\pi\)
\(29\)	−4.18049	−	8.68089i	−0.776298	−	1.61200i	−0.790776	−	0.612105i	\(-0.790323\pi\)
							0.0144781	−	0.999895i	\(-0.495391\pi\)
\(31\)	2.51248	+	4.35174i	0.451254	+	0.781595i	0.998464	−	0.0554000i	\(-0.0176434\pi\)
							−0.547210	+	0.836995i	\(0.684310\pi\)
\(37\)	0.463012	+	0.0346979i	0.0761187	+	0.00570431i	0.112735	−	0.993625i	\(-0.464039\pi\)
							−0.0366163	+	0.999329i	\(0.511658\pi\)
\(41\)	3.40699	−	4.27223i	0.532083	−	0.667211i	−0.441043	−	0.897486i	\(-0.645391\pi\)
							0.973125	+	0.230276i	\(0.0739627\pi\)
\(43\)	6.05352	−	4.82752i	0.923153	−	0.736190i	−0.0416586	−	0.999132i	\(-0.513264\pi\)
							0.964812	+	0.262942i	\(0.0846928\pi\)
\(47\)	6.04782	+	5.61156i	0.882165	+	0.818529i	0.984348	−	0.176236i	\(-0.0563922\pi\)
							−0.102183	+	0.994766i	\(0.532583\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.205.8	yes	648
8.5	even	2	inner	392.2.z.a.205.34	yes	648
49.11	even	21	inner	392.2.z.a.109.34	yes	648
392.109	even	42	inner	392.2.z.a.109.8	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.8	✓	648	392.109	even	42	inner
392.2.z.a.109.34	yes	648	49.11	even	21	inner
392.2.z.a.205.8	yes	648	1.1	even	1	trivial
392.2.z.a.205.34	yes	648	8.5	even	2	inner