Embedded newform 43.5.h.a.5.3

• Label: 43.5.h.a.5.3
• Level: $43$
• Weight: $5$
• Character: 43.5
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	43.5
Dual form			43.5.h.a.26.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.02577 - 1.37534i) q^{2} +(4.35150 - 14.1072i) q^{3} +(20.0029 + 9.63287i) q^{4} +(39.0145 - 15.3121i) q^{5} +(-45.6234 + 79.0220i) q^{6} +(34.4295 - 19.8779i) q^{7} +(-29.9675 - 23.8983i) q^{8} +(-113.152 - 77.1459i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.02577	−	1.37534i	−1.50644	−	0.343836i	−0.611942	−	0.790902i	\(-0.709612\pi\)
							−0.894501	+	0.447067i	\(0.852469\pi\)
\(3\)	4.35150	−	14.1072i	0.483499	−	1.56747i	−0.304951	−	0.952368i	\(-0.598640\pi\)
							0.788450	−	0.615099i	\(-0.210884\pi\)
\(5\)	39.0145	−	15.3121i	1.56058	−	0.612483i	0.581067	−	0.813856i	\(-0.302635\pi\)
							0.979515	+	0.201372i	\(0.0645402\pi\)
\(7\)	34.4295	−	19.8779i	0.702642	−	0.405671i	−0.105688	−	0.994399i	\(-0.533705\pi\)
							0.808331	+	0.588728i	\(0.200371\pi\)
\(11\)	81.4184	−	39.2090i	0.672879	−	0.324042i	−0.0660581	−	0.997816i	\(-0.521042\pi\)
							0.738937	+	0.673774i	\(0.235328\pi\)
\(13\)	−57.9087	−	8.72833i	−0.342655	−	0.0516469i	−0.0245394	−	0.999699i	\(-0.507812\pi\)
							−0.318116	+	0.948052i	\(0.603050\pi\)
\(17\)	−195.427	+	497.941i	−0.676219	+	1.72298i	0.0145408	+	0.999894i	\(0.495371\pi\)
							−0.690760	+	0.723084i	\(0.742724\pi\)
\(19\)	141.992	+	208.265i	0.393331	+	0.576910i	0.970916	−	0.239420i	\(-0.0769573\pi\)
							−0.577585	+	0.816330i	\(0.696005\pi\)
\(23\)	−43.4576	+	579.902i	−0.0821506	+	1.09622i	0.791985	+	0.610541i	\(0.209048\pi\)
							−0.874135	+	0.485682i	\(0.838571\pi\)
\(29\)	129.368	+	419.401i	0.153827	+	0.498693i	0.999380	−	0.0351967i	\(-0.0112058\pi\)
							−0.845554	+	0.533890i	\(0.820730\pi\)
\(31\)	−1133.81	−	1052.02i	−1.17982	−	1.09471i	−0.993687	−	0.112186i	\(-0.964215\pi\)
							−0.186131	−	0.982525i	\(-0.559595\pi\)
\(37\)	−115.344	−	66.5939i	−0.0842543	−	0.0486442i	0.457281	−	0.889322i	\(-0.348823\pi\)
							−0.541535	+	0.840678i	\(0.682157\pi\)
\(41\)	266.996	−	1169.78i	0.158831	−	0.695886i	−0.831309	−	0.555810i	\(-0.812408\pi\)
							0.990141	−	0.140076i	\(-0.0447347\pi\)
\(43\)	−790.071	+	1671.70i	−0.427296	+	0.904112i
							\(47\)	3040.94	+	1464.44i	1.37662	+	0.662943i	0.968275	−	0.249886i	\(-0.0803932\pi\)
0.408341	+	0.912830i	\(0.366108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.5.3	✓	168
43.26	odd	42	inner	43.5.h.a.26.3	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.5.3	✓	168	1.1	even	1	trivial
43.5.h.a.26.3	yes	168	43.26	odd	42	inner