Embedded newform 833.2.bn.a.3.5

• Label: 833.2.bn.a.3.5
• Level: $833$
• Weight: $2$
• Character: 833.3
• Analytic conductor: $6.652$
• Analytic rank: $0$
• Dimension: $7872$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 833 = 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	833.bn (of order \(336\), degree \(96\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.65153848837\)
Analytic rank:	\(0\)
Dimension:	\(7872\)
Relative dimension:	\(82\) over \(\Q(\zeta_{336})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{336}]$

Embedding invariants

Embedding label			3.5
Character	\(\chi\)	\(=\)	833.3
Dual form			833.2.bn.a.278.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0487603 - 2.60721i) q^{2} +(2.46220 + 0.684483i) q^{3} +(-4.79655 + 0.179474i) q^{4} +(-3.36530 - 0.0314664i) q^{5} +(1.66453 - 6.45285i) q^{6} +(0.536053 + 2.59088i) q^{7} +(0.409382 + 7.28972i) q^{8} +(3.02436 + 1.82236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7872 q - 104 q^{2} - 88 q^{3} - 104 q^{4} - 88 q^{5} - 112 q^{6} - 96 q^{7} - 80 q^{8} - 104 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{1}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0487603	−	2.60721i	−0.0344788	−	1.84357i	−0.404485	−	0.914544i	\(-0.632549\pi\)
							0.370007	−	0.929029i	\(-0.379355\pi\)
\(3\)	2.46220	+	0.684483i	1.42155	+	0.395186i	0.891594	−	0.452836i	\(-0.149588\pi\)
							0.529961	+	0.848022i	\(0.322207\pi\)
\(5\)	−3.36530	−	0.0314664i	−1.50501	−	0.0140722i	−0.746347	−	0.665557i	\(-0.768194\pi\)
							−0.758662	+	0.651485i	\(0.774146\pi\)
\(7\)	0.536053	+	2.59088i	0.202609	+	0.979260i
							\(11\)	1.16788	+	3.24010i	0.352128	+	0.976926i	0.980068	+	0.198664i	\(0.0636603\pi\)
−0.627940	+	0.778262i	\(0.716102\pi\)
\(13\)	0.765273	+	0.0862256i	0.212249	+	0.0239147i	0.217448	−	0.976072i	\(-0.430227\pi\)
							−0.00519982	+	0.999986i	\(0.501655\pi\)
\(17\)	−0.729886	+	4.05799i	−0.177023	+	0.984207i
							\(19\)	2.19182	−	2.85644i	0.502838	−	0.655312i	−0.471112	−	0.882073i	\(-0.656147\pi\)
0.973950	+	0.226762i	\(0.0728138\pi\)
\(23\)	2.99005	−	1.61623i	0.623469	−	0.337008i	−0.132923	−	0.991126i	\(-0.542436\pi\)
							0.756392	+	0.654118i	\(0.226960\pi\)
\(29\)	7.44389	+	7.03758i	1.38230	+	1.30685i	0.903288	+	0.429034i	\(0.141146\pi\)
							0.479007	+	0.877811i	\(0.340997\pi\)
\(31\)	−2.24276	+	2.55738i	−0.402812	+	0.459319i	−0.917311	−	0.398172i	\(-0.869645\pi\)
							0.514498	+	0.857491i	\(0.327978\pi\)
\(37\)	−6.56382	+	4.03921i	−1.07908	+	0.664043i	−0.945047	−	0.326933i	\(-0.893985\pi\)
							−0.134037	+	0.990976i	\(0.542794\pi\)
\(41\)	4.50451	+	0.635932i	0.703487	+	0.0993159i	0.482416	−	0.875942i	\(-0.339759\pi\)
							0.221071	+	0.975258i	\(0.429045\pi\)
\(43\)	−4.59010	−	5.13632i	−0.699984	−	0.783282i	0.284516	−	0.958671i	\(-0.408167\pi\)
							−0.984499	+	0.175389i	\(0.943882\pi\)
\(47\)	3.27941	+	7.51649i	0.478351	+	1.09639i	0.973728	+	0.227715i	\(0.0731254\pi\)
							−0.495376	+	0.868678i	\(0.664970\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	833.2.bn.a.3.5	✓	7872
17.6	odd	16	inner	833.2.bn.a.346.78	yes	7872
49.33	odd	42	inner	833.2.bn.a.768.78	yes	7872
833.278	even	336	inner	833.2.bn.a.278.5	yes	7872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
833.2.bn.a.3.5	✓	7872	1.1	even	1	trivial
833.2.bn.a.278.5	yes	7872	833.278	even	336	inner
833.2.bn.a.346.78	yes	7872	17.6	odd	16	inner
833.2.bn.a.768.78	yes	7872	49.33	odd	42	inner