Embedded newform 343.2.k.a.100.17

• Label: 343.2.k.a.100.17
• Level: $343$
• Weight: $2$
• Character: 343.100
• Analytic conductor: $2.739$
• Analytic rank: $0$
• Dimension: $2688$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 343 = 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	343.k (of order \(147\), degree \(84\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.73886878933\)
Analytic rank:	\(0\)
Dimension:	\(2688\)
Relative dimension:	\(32\) over \(\Q(\zeta_{147})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{147}]$

Embedding invariants

Embedding label			100.17
Character	\(\chi\)	\(=\)	343.100
Dual form			343.2.k.a.319.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0868690 - 0.119029i) q^{2} +(-0.0888885 + 1.66211i) q^{3} +(0.603277 - 1.88406i) q^{4} +(-2.91488 + 0.0623044i) q^{5} +(0.205561 - 0.133805i) q^{6} +(1.96523 - 1.77141i) q^{7} +(-0.556364 + 0.184726i) q^{8} +(0.228186 + 0.0244765i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2688 q - 84 q^{2} - 84 q^{3} - 84 q^{4} - 84 q^{5} - 98 q^{6} - 84 q^{7} - 84 q^{8} - 98 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/343\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{76}{147}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0868690	−	0.119029i	−0.0614256	−	0.0841663i	0.775949	−	0.630795i	\(-0.217271\pi\)
							−0.837375	+	0.546629i	\(0.815911\pi\)
\(3\)	−0.0888885	+	1.66211i	−0.0513198	+	0.959619i	0.850086	+	0.526644i	\(0.176550\pi\)
							−0.901406	+	0.432975i	\(0.857464\pi\)
\(5\)	−2.91488	+	0.0623044i	−1.30357	+	0.0278634i	−0.667990	−	0.744170i	\(-0.732845\pi\)
							−0.635582	+	0.772033i	\(0.719240\pi\)
\(7\)	1.96523	−	1.77141i	0.742785	−	0.669530i
							\(11\)	0.605549	−	5.12800i	0.182580	−	1.54615i	−0.532090	−	0.846688i	\(-0.678593\pi\)
0.714670	−	0.699462i	\(-0.246577\pi\)
\(13\)	2.17052	+	4.16048i	0.601994	+	1.15391i	0.974185	+	0.225753i	\(0.0724842\pi\)
							−0.372191	+	0.928156i	\(0.621393\pi\)
\(17\)	2.67565	−	0.640856i	0.648942	−	0.155430i	0.104892	−	0.994484i	\(-0.466550\pi\)
							0.544049	+	0.839053i	\(0.316890\pi\)
\(19\)	4.19761	−	7.27048i	0.962998	−	1.66796i	0.248099	−	0.968735i	\(-0.420194\pi\)
							0.714899	−	0.699228i	\(-0.246473\pi\)
\(23\)	0.159950	−	0.124783i	0.0333518	−	0.0260191i	−0.598286	−	0.801282i	\(-0.704152\pi\)
							0.631638	+	0.775263i	\(0.282383\pi\)
\(29\)	−3.37976	+	1.36831i	−0.627605	+	0.254089i	−0.666910	−	0.745138i	\(-0.732384\pi\)
							0.0393050	+	0.999227i	\(0.487486\pi\)
\(31\)	1.55135	−	1.43945i	0.278631	−	0.258532i	−0.528461	−	0.848958i	\(-0.677231\pi\)
							0.807092	+	0.590426i	\(0.201040\pi\)
\(37\)	5.07870	+	5.35756i	0.834932	+	0.880777i	0.994319	−	0.106439i	\(-0.0339449\pi\)
							−0.159387	+	0.987216i	\(0.550952\pi\)
\(41\)	1.00571	−	2.73284i	0.157065	−	0.426797i	−0.835351	−	0.549716i	\(-0.814736\pi\)
							0.992417	+	0.122919i	\(0.0392256\pi\)
\(43\)	−2.59153	−	0.860448i	−0.395205	−	0.131217i	0.110600	−	0.993865i	\(-0.464723\pi\)
							−0.505805	+	0.862648i	\(0.668804\pi\)
\(47\)	0.165374	+	0.0885364i	0.0241223	+	0.0129144i	0.484003	−	0.875067i	\(-0.339183\pi\)
							−0.459880	+	0.887981i	\(0.652108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	343.2.k.a.100.17	✓	2688
343.319	even	147	inner	343.2.k.a.319.17	yes	2688

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.k.a.100.17	✓	2688	1.1	even	1	trivial
343.2.k.a.319.17	yes	2688	343.319	even	147	inner