Embedded newform 392.2.bc.a.299.34

• Label: 392.2.bc.a.299.34
• Level: $392$
• Weight: $2$
• Character: 392.299
• 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.bc (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			299.34
Character	\(\chi\)	\(=\)	392.299
Dual form			392.2.bc.a.59.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.471848 - 1.33318i) q^{2} +(0.423331 - 0.620912i) q^{3} +(-1.55472 - 1.25811i) q^{4} +(-0.325145 + 4.33876i) q^{5} +(-0.628037 - 0.857350i) q^{6} +(1.83367 + 1.90726i) q^{7} +(-2.41088 + 1.47907i) q^{8} +(0.889700 + 2.26692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 22 q^{3} - 13 q^{4} - 14 q^{6} - 4 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.471848	−	1.33318i	0.333647	−	0.942698i
							\(3\)	0.423331	−	0.620912i	0.244410	−	0.358484i	−0.684357	−	0.729147i	\(-0.739917\pi\)
0.928767	+	0.370663i	\(0.120870\pi\)
\(5\)	−0.325145	+	4.33876i	−0.145409	+	1.94035i	0.157888	+	0.987457i	\(0.449531\pi\)
							−0.303298	+	0.952896i	\(0.598088\pi\)
\(7\)	1.83367	+	1.90726i	0.693064	+	0.720876i
							\(11\)	2.18432	−	5.56556i	0.658597	−	1.67808i	−0.0738622	−	0.997268i	\(-0.523532\pi\)
0.732460	−	0.680810i	\(-0.238372\pi\)
\(13\)	−0.323677	−	0.405879i	−0.0897719	−	0.112570i	0.734917	−	0.678157i	\(-0.237221\pi\)
							−0.824689	+	0.565587i	\(0.808650\pi\)
\(17\)	2.75214	+	2.96610i	0.667493	+	0.719386i	0.972484	−	0.232969i	\(-0.0748441\pi\)
							−0.304992	+	0.952355i	\(0.598654\pi\)
\(19\)	1.66175	+	0.959411i	0.381231	+	0.220104i	0.678354	−	0.734735i	\(-0.262694\pi\)
							−0.297123	+	0.954839i	\(0.596027\pi\)
\(23\)	−2.06160	+	2.22188i	−0.429874	+	0.463294i	−0.910426	−	0.413673i	\(-0.864246\pi\)
							0.480552	+	0.876966i	\(0.340436\pi\)
\(29\)	1.73734	−	0.396537i	0.322616	−	0.0736351i	−0.0581458	−	0.998308i	\(-0.518519\pi\)
							0.380762	+	0.924673i	\(0.375662\pi\)
\(31\)	−1.89172	−	3.27656i	−0.339763	−	0.588488i	0.644625	−	0.764499i	\(-0.277014\pi\)
							−0.984388	+	0.176012i	\(0.943680\pi\)
\(37\)	2.10829	−	6.83491i	0.346601	−	1.12365i	−0.599662	−	0.800253i	\(-0.704698\pi\)
							0.946263	−	0.323399i	\(-0.104826\pi\)
\(41\)	1.51789	+	3.15193i	0.237054	+	0.492249i	0.985227	−	0.171256i	\(-0.0547825\pi\)
							−0.748172	+	0.663505i	\(0.769068\pi\)
\(43\)	−4.68192	−	2.25469i	−0.713986	−	0.343837i	0.0413777	−	0.999144i	\(-0.486825\pi\)
							−0.755363	+	0.655306i	\(0.772540\pi\)
\(47\)	−4.17337	+	0.629035i	−0.608749	+	0.0917541i	−0.446180	−	0.894943i	\(-0.647216\pi\)
							−0.162569	+	0.986697i	\(0.551978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.bc.a.299.34	yes	648
8.3	odd	2	inner	392.2.bc.a.299.39	yes	648
49.10	odd	42	inner	392.2.bc.a.59.39	yes	648
392.59	even	42	inner	392.2.bc.a.59.34	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.bc.a.59.34	✓	648	392.59	even	42	inner
392.2.bc.a.59.39	yes	648	49.10	odd	42	inner
392.2.bc.a.299.34	yes	648	1.1	even	1	trivial
392.2.bc.a.299.39	yes	648	8.3	odd	2	inner