Embedded newform 671.2.dc.a.112.5

• Label: 671.2.dc.a.112.5
• Level: $671$
• Weight: $2$
• Character: 671.112
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $960$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.dc (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(960\)
Relative dimension:	\(60\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			112.5
Character	\(\chi\)	\(=\)	671.112
Dual form			671.2.dc.a.6.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.48848 + 0.130416i) q^{2} +(-0.819826 - 0.266378i) q^{3} +(4.18648 - 0.440017i) q^{4} +(1.51849 - 0.876700i) q^{5} +(2.07486 + 0.555957i) q^{6} +(1.01726 - 0.0533122i) q^{7} +(-5.43816 + 0.861320i) q^{8} +(-1.82589 - 1.32659i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q - 20 q^{2} - 20 q^{3} - 8 q^{4} - 18 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 220 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)	\(551\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{53}{60}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48848	+	0.130416i	−1.75962	+	0.0922178i	−0.904570	−	0.426325i	\(-0.859808\pi\)
							−0.855051	+	0.518543i	\(0.826475\pi\)
\(3\)	−0.819826	−	0.266378i	−0.473327	−	0.153793i	0.0626332	−	0.998037i	\(-0.480050\pi\)
							−0.535960	+	0.844243i	\(0.680050\pi\)
\(5\)	1.51849	−	0.876700i	0.679089	−	0.392072i	−0.120423	−	0.992723i	\(-0.538425\pi\)
							0.799512	+	0.600651i	\(0.205092\pi\)
\(7\)	1.01726	−	0.0533122i	0.384487	−	0.0201501i	0.140887	−	0.990026i	\(-0.455005\pi\)
							0.243601	+	0.969876i	\(0.421671\pi\)
\(11\)	0.0714210	+	3.31586i	0.0215342	+	0.999768i
							\(13\)	−1.27745	−	6.00995i	−0.354302	−	1.66686i	−0.689173	−	0.724597i	\(-0.742026\pi\)
0.334871	−	0.942264i	\(-0.391307\pi\)
\(17\)	−0.644128	−	2.40392i	−0.156224	−	0.583036i	−0.998997	−	0.0447672i	\(-0.985745\pi\)
							0.842774	−	0.538268i	\(-0.180921\pi\)
\(19\)	−4.70249	−	5.22265i	−1.07883	−	1.19816i	−0.979150	−	0.203138i	\(-0.934886\pi\)
							−0.0996759	−	0.995020i	\(-0.531781\pi\)
\(23\)	2.66459	+	5.22955i	0.555605	+	1.09044i	0.982522	+	0.186146i	\(0.0595997\pi\)
							−0.426917	+	0.904291i	\(0.640400\pi\)
\(29\)	0.0266477	−	0.0694197i	0.00494836	−	0.0128909i	−0.931084	−	0.364805i	\(-0.881136\pi\)
							0.936032	+	0.351914i	\(0.114469\pi\)
\(31\)	−5.26272	+	6.49891i	−0.945212	+	1.16724i	0.0405646	+	0.999177i	\(0.487084\pi\)
							−0.985776	+	0.168062i	\(0.946249\pi\)
\(37\)	4.18453	−	8.21260i	0.687932	−	1.35014i	−0.237558	−	0.971373i	\(-0.576347\pi\)
							0.925491	−	0.378770i	\(-0.123653\pi\)
\(41\)	2.94722			0.460278			0.230139	−	0.973158i	\(-0.426082\pi\)
							0.230139	+	0.973158i	\(0.426082\pi\)
\(43\)	−0.0709303	−	1.35343i	−0.0108168	−	0.206396i	−0.998691	−	0.0511424i	\(-0.983714\pi\)
							0.987875	−	0.155254i	\(-0.0496196\pi\)
\(47\)	−0.707765	−	6.73393i	−0.103238	−	0.982245i	−0.916415	−	0.400229i	\(-0.868931\pi\)
							0.813177	−	0.582016i	\(-0.197736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.dc.a.112.5	yes	960
11.6	odd	10		671.2.cu.a.600.5	yes	960
61.6	odd	60		671.2.cu.a.189.5	✓	960
671.6	even	60	inner	671.2.dc.a.6.5	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.cu.a.189.5	✓	960	61.6	odd	60
671.2.cu.a.600.5	yes	960	11.6	odd	10
671.2.dc.a.6.5	yes	960	671.6	even	60	inner
671.2.dc.a.112.5	yes	960	1.1	even	1	trivial