Embedded newform 671.2.h.b.70.17

• Label: 671.2.h.b.70.17
• Level: $671$
• Weight: $2$
• Character: 671.70
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $236$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(236\)
Relative dimension:	\(59\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			70.17
Character	\(\chi\)	\(=\)	671.70
Dual form			671.2.h.b.278.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.428672 - 1.31932i) q^{2} +(-0.369969 + 0.268798i) q^{3} +(0.0612004 - 0.0444647i) q^{4} +(2.08561 + 1.51529i) q^{5} +(0.513225 + 0.372880i) q^{6} +(-3.54951 - 2.57887i) q^{7} +(-2.32945 - 1.69245i) q^{8} +(-0.862426 + 2.65428i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 236 q + 2 q^{2} - 2 q^{3} - 62 q^{4} - 6 q^{5} + 9 q^{6} + q^{7} + 12 q^{8} - 59 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/671\mathbb{Z}\right)^\times\).

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.428672	−	1.31932i	−0.303117	−	0.932897i	−0.980373	−	0.197150i	\(-0.936831\pi\)
							0.677257	−	0.735747i	\(-0.263169\pi\)
\(3\)	−0.369969	+	0.268798i	−0.213602	+	0.155191i	−0.689442	−	0.724341i	\(-0.742144\pi\)
							0.475840	+	0.879532i	\(0.342144\pi\)
\(5\)	2.08561	+	1.51529i	0.932714	+	0.677657i	0.946656	−	0.322246i	\(-0.104438\pi\)
							−0.0139417	+	0.999903i	\(0.504438\pi\)
\(7\)	−3.54951	−	2.57887i	−1.34159	−	0.974720i	−0.999384	−	0.0350944i	\(-0.988827\pi\)
							−0.342203	−	0.939626i	\(-0.611173\pi\)
\(11\)	−3.29411	−	0.385804i	−0.993211	−	0.116324i
							\(13\)	−1.09448	−	0.795186i	−0.303554	−	0.220545i	0.425572	−	0.904925i	\(-0.360073\pi\)
−0.729126	+	0.684380i	\(0.760073\pi\)
\(17\)	1.81953	+	1.32197i	0.441301	+	0.320624i	0.786152	−	0.618034i	\(-0.212070\pi\)
							−0.344851	+	0.938658i	\(0.612070\pi\)
\(19\)	−4.24691	+	3.08556i	−0.974308	+	0.707877i	−0.956429	−	0.291964i	\(-0.905691\pi\)
							−0.0178790	+	0.999840i	\(0.505691\pi\)
\(23\)	−1.91585	−	5.89638i	−0.399482	−	1.22948i	−0.925415	−	0.378954i	\(-0.876284\pi\)
							0.525933	−	0.850526i	\(-0.323716\pi\)
\(29\)	−1.64667	−	5.06794i	−0.305780	−	0.941093i	−0.979385	−	0.202002i	\(-0.935255\pi\)
							0.673606	−	0.739091i	\(-0.264745\pi\)
\(31\)	−7.10831			−1.27669			−0.638345	−	0.769750i	\(-0.720381\pi\)
							−0.638345	+	0.769750i	\(0.720381\pi\)
\(37\)	2.10356	+	1.52832i	0.345822	+	0.251255i	0.747114	−	0.664695i	\(-0.231439\pi\)
							−0.401292	+	0.915950i	\(0.631439\pi\)
\(41\)	0.843079	+	2.59473i	0.131667	+	0.405229i	0.995057	−	0.0993086i	\(-0.0316631\pi\)
							−0.863390	+	0.504537i	\(0.831663\pi\)
\(43\)	0.683764	−	2.10441i	0.104273	−	0.320919i	−0.885286	−	0.465047i	\(-0.846038\pi\)
							0.989559	+	0.144127i	\(0.0460375\pi\)
\(47\)	0.599024	−	1.84361i	0.0873766	−	0.268918i	−0.897815	−	0.440372i	\(-0.854847\pi\)
							0.985192	+	0.171454i	\(0.0548466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.h.b.70.17	✓	236
11.3	even	5		671.2.m.b.619.17	yes	236
61.34	even	5		671.2.m.b.400.17	yes	236
671.278	even	5	inner	671.2.h.b.278.17	yes	236

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.h.b.70.17	✓	236	1.1	even	1	trivial
671.2.h.b.278.17	yes	236	671.278	even	5	inner
671.2.m.b.400.17	yes	236	61.34	even	5
671.2.m.b.619.17	yes	236	11.3	even	5