Embedded newform 43.3.f.a.2.3

• Label: 43.3.f.a.2.3
• Level: $43$
• Weight: $3$
• Character: 43.2
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			2.3
Character	\(\chi\)	\(=\)	43.2
Dual form			43.3.f.a.22.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.686008 - 1.42451i) q^{2} +(0.358272 - 0.743960i) q^{3} +(0.935338 - 1.17288i) q^{4} +(-1.02237 - 0.233349i) q^{5} -1.30556 q^{6} -7.62808i q^{7} +(-8.47820 - 1.93509i) q^{8} +(5.18629 + 6.50340i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.686008	−	1.42451i	−0.343004	−	0.712255i	0.656096	−	0.754677i	\(-0.272207\pi\)
							−0.999100	+	0.0424228i	\(0.986492\pi\)
\(3\)	0.358272	−	0.743960i	0.119424	−	0.247987i	−0.832684	−	0.553749i	\(-0.813197\pi\)
							0.952108	+	0.305762i	\(0.0989112\pi\)
\(5\)	−1.02237	−	0.233349i	−0.204473	−	0.0466697i	0.119057	−	0.992887i	\(-0.462013\pi\)
							−0.323531	+	0.946218i	\(0.604870\pi\)
\(7\)		−	7.62808i		−	1.08973i	−0.838525	−	0.544863i	\(-0.816582\pi\)
							0.838525	−	0.544863i	\(-0.183418\pi\)
\(11\)	10.2815	+	12.8925i	0.934678	+	1.17205i	0.984868	+	0.173309i	\(0.0554459\pi\)
							−0.0501899	+	0.998740i	\(0.515983\pi\)
\(13\)	0.141055	−	0.618001i	0.0108504	−	0.0475385i	−0.969213	−	0.246224i	\(-0.920810\pi\)
							0.980063	+	0.198686i	\(0.0636672\pi\)
\(17\)	4.86398	+	21.3105i	0.286116	+	1.25356i	0.889806	+	0.456339i	\(0.150839\pi\)
							−0.603690	+	0.797219i	\(0.706303\pi\)
\(19\)	2.94459	+	2.34824i	0.154979	+	0.123591i	0.697909	−	0.716186i	\(-0.254114\pi\)
							−0.542931	+	0.839778i	\(0.682685\pi\)
\(23\)	−8.02279	−	10.0603i	−0.348817	−	0.437402i	0.576212	−	0.817301i	\(-0.304530\pi\)
							−0.925028	+	0.379898i	\(0.875959\pi\)
\(29\)	−9.06865	−	18.8313i	−0.312712	−	0.649354i	0.684078	−	0.729409i	\(-0.260205\pi\)
							−0.996790	+	0.0800551i	\(0.974490\pi\)
\(31\)	15.6275	−	7.52582i	0.504114	−	0.242768i	−0.164502	−	0.986377i	\(-0.552602\pi\)
							0.668615	+	0.743608i	\(0.266887\pi\)
\(37\)		−	6.21175i		−	0.167885i	−0.996471	−	0.0839426i	\(-0.973249\pi\)
							0.996471	−	0.0839426i	\(-0.0267512\pi\)
\(41\)	−52.1226	+	25.1009i	−1.27128	+	0.612217i	−0.943135	−	0.332409i	\(-0.892139\pi\)
							−0.328147	+	0.944627i	\(0.606424\pi\)
\(43\)	24.1220	+	35.5968i	0.560977	+	0.827832i
							\(47\)	−11.6182	+	14.5687i	−0.247196	+	0.309973i	−0.889913	−	0.456130i	\(-0.849235\pi\)
0.642718	+	0.766103i	\(0.277807\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.2.3	✓	42
3.2	odd	2		387.3.w.b.217.5		42
43.22	odd	14	inner	43.3.f.a.22.3	yes	42
129.65	even	14		387.3.w.b.280.5		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.2.3	✓	42	1.1	even	1	trivial
43.3.f.a.22.3	yes	42	43.22	odd	14	inner
387.3.w.b.217.5		42	3.2	odd	2
387.3.w.b.280.5		42	129.65	even	14