Embedded newform 141.2.g.a.11.5

• Label: 141.2.g.a.11.5
• Level: $141$
• Weight: $2$
• Character: 141.11
• Analytic conductor: $1.126$
• Analytic rank: $0$
• Dimension: $308$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 141 = 3 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	141.g (of order \(46\), degree \(22\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.12589066850\)
Analytic rank:	\(0\)
Dimension:	\(308\)
Relative dimension:	\(14\) over \(\Q(\zeta_{46})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{46}]$

Embedding invariants

Embedding label			11.5
Character	\(\chi\)	\(=\)	141.11
Dual form			141.2.g.a.77.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.840871 - 0.785319i) q^{2} +(1.70711 - 0.292854i) q^{3} +(-0.0461463 - 0.674635i) q^{4} +(1.24240 - 2.39773i) q^{5} +(-1.66545 - 1.09438i) q^{6} +(-2.61153 + 1.13435i) q^{7} +(-1.94321 + 2.38853i) q^{8} +(2.82847 - 0.999870i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 308 q - 19 q^{3} - 26 q^{4} - 15 q^{6} - 42 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/141\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(95\)
\(\chi(n)\)	\(e\left(\frac{7}{46}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.840871	−	0.785319i	−0.594585	−	0.555304i	0.326236	−	0.945288i	\(-0.394220\pi\)
							−0.920822	+	0.389984i	\(0.872480\pi\)
\(3\)	1.70711	−	0.292854i	0.985602	−	0.169079i
							\(5\)	1.24240	−	2.39773i	0.555619	−	1.07230i	−0.429923	−	0.902866i	\(-0.641459\pi\)
0.985542	−	0.169431i	\(-0.0541930\pi\)
\(7\)	−2.61153	+	1.13435i	−0.987064	+	0.428743i	−0.829342	−	0.558741i	\(-0.811285\pi\)
							−0.157722	+	0.987484i	\(0.550415\pi\)
\(11\)	−0.878821	+	4.22912i	−0.264975	+	1.27513i	0.610559	+	0.791971i	\(0.290945\pi\)
							−0.875534	+	0.483157i	\(0.839490\pi\)
\(13\)	2.76336	−	4.54415i	0.766417	−	1.26032i	−0.193793	−	0.981042i	\(-0.562079\pi\)
							0.960210	−	0.279277i	\(-0.0900949\pi\)
\(17\)	1.77058	−	0.367930i	0.429428	−	0.0892362i	0.0162108	−	0.999869i	\(-0.494840\pi\)
							0.413217	+	0.910632i	\(0.364405\pi\)
\(19\)	−3.66132	+	1.89714i	−0.839965	+	0.435235i	−0.825325	−	0.564658i	\(-0.809008\pi\)
							−0.0146400	+	0.999893i	\(0.504660\pi\)
\(23\)	2.37354	+	2.54144i	0.494917	+	0.529926i	0.928572	−	0.371152i	\(-0.121037\pi\)
							−0.433655	+	0.901079i	\(0.642776\pi\)
\(29\)	5.05842	−	3.07609i	0.939325	−	0.571216i	0.0356168	−	0.999366i	\(-0.488660\pi\)
							0.903708	+	0.428150i	\(0.140834\pi\)
\(31\)	0.384563	+	2.79790i	0.0690695	+	0.502518i	0.992829	+	0.119545i	\(0.0381434\pi\)
							−0.923759	+	0.382974i	\(0.874900\pi\)
\(37\)	1.56811	+	4.41224i	0.257796	+	0.725368i	0.998562	+	0.0536050i	\(0.0170712\pi\)
							−0.740767	+	0.671763i	\(0.765538\pi\)
\(41\)	−8.39294	+	6.82816i	−1.31076	+	1.06638i	−0.317768	+	0.948169i	\(0.602933\pi\)
							−0.992989	+	0.118211i	\(0.962284\pi\)
\(43\)	−4.35983	+	0.298220i	−0.664867	+	0.0454782i	−0.396776	−	0.917915i	\(-0.629871\pi\)
							−0.268091	+	0.963394i	\(0.586393\pi\)
\(47\)	−1.98369	−	6.56239i	−0.289351	−	0.957223i

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	141.2.g.a.11.5	✓	308
3.2	odd	2	inner	141.2.g.a.11.10	yes	308
47.30	odd	46	inner	141.2.g.a.77.10	yes	308
141.77	even	46	inner	141.2.g.a.77.5	yes	308

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
141.2.g.a.11.5	✓	308	1.1	even	1	trivial
141.2.g.a.11.10	yes	308	3.2	odd	2	inner
141.2.g.a.77.5	yes	308	141.77	even	46	inner
141.2.g.a.77.10	yes	308	47.30	odd	46	inner