Embedded newform 43.7.h.a.3.4

• Label: 43.7.h.a.3.4
• Level: $43$
• Weight: $7$
• Character: 43.3
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.58142 + 9.51343i) q^{2} +(-35.2682 - 2.64299i) q^{3} +(-29.6125 - 37.1329i) q^{4} +(22.3499 - 72.4567i) q^{5} +(186.723 - 323.413i) q^{6} +(336.875 - 194.495i) q^{7} +(-169.911 + 38.7811i) q^{8} +(516.004 + 77.7751i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.58142	+	9.51343i	−0.572678	+	1.18918i	0.390572	+	0.920572i	\(0.372277\pi\)
							−0.963250	+	0.268606i	\(0.913437\pi\)
\(3\)	−35.2682	−	2.64299i	−1.30623	−	0.0978884i	−0.596643	−	0.802506i	\(-0.703499\pi\)
							−0.709587	+	0.704618i	\(0.751118\pi\)
\(5\)	22.3499	−	72.4567i	0.178800	−	0.579654i	−0.821160	−	0.570698i	\(-0.806673\pi\)
							0.999960	−	0.00895630i	\(-0.00285092\pi\)
\(7\)	336.875	−	194.495i	0.982144	−	0.567041i	0.0792272	−	0.996857i	\(-0.474755\pi\)
							0.902917	+	0.429815i	\(0.141421\pi\)
\(11\)	−313.422	+	393.019i	−0.235479	+	0.295281i	−0.885504	−	0.464631i	\(-0.846187\pi\)
							0.650025	+	0.759913i	\(0.274758\pi\)
\(13\)	−241.920	+	224.469i	−0.110114	+	0.102171i	−0.733309	−	0.679896i	\(-0.762025\pi\)
							0.623195	+	0.782067i	\(0.285834\pi\)
\(17\)	1982.76	−	611.602i	0.403575	−	0.124486i	−0.0863207	−	0.996267i	\(-0.527511\pi\)
							0.489896	+	0.871781i	\(0.337035\pi\)
\(19\)	1101.49	+	7307.89i	0.160590	+	1.06544i	0.913358	+	0.407158i	\(0.133480\pi\)
							−0.752768	+	0.658286i	\(0.771282\pi\)
\(23\)	2320.49	+	5912.50i	0.190720	+	0.485946i	0.993964	−	0.109704i	\(-0.0349901\pi\)
							−0.803245	+	0.595649i	\(0.796895\pi\)
\(29\)	10814.3	−	810.421i	0.443410	−	0.0332290i	0.148845	−	0.988861i	\(-0.452444\pi\)
							0.294565	+	0.955632i	\(0.404825\pi\)
\(31\)	31001.5	−	21136.5i	1.04063	−	0.709492i	0.0827273	−	0.996572i	\(-0.473637\pi\)
							0.957907	+	0.287080i	\(0.0926846\pi\)
\(37\)	34759.8	+	20068.6i	0.686233	+	0.396197i	0.802199	−	0.597056i	\(-0.203663\pi\)
							−0.115966	+	0.993253i	\(0.536996\pi\)
\(41\)	−56214.1	−	27071.3i	−0.815631	−	0.392787i	−0.0209244	−	0.999781i	\(-0.506661\pi\)
							−0.794706	+	0.606994i	\(0.792375\pi\)
\(43\)	79469.1	−	2455.33i	0.999523	−	0.0308819i
							\(47\)	−59508.1	−	74620.8i	−0.573169	−	0.718731i	0.407762	−	0.913088i	\(-0.366309\pi\)
−0.980931	+	0.194358i	\(0.937738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.4	✓	252
43.29	odd	42	inner	43.7.h.a.29.4	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.4	✓	252	1.1	even	1	trivial
43.7.h.a.29.4	yes	252	43.29	odd	42	inner