Embedded newform 43.7.h.a.3.16

• Label: 43.7.h.a.3.16
• 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			3.16
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.92333 - 8.14687i) q^{2} +(7.09953 + 0.532036i) q^{3} +(-11.0757 - 13.8885i) q^{4} +(43.9242 - 142.399i) q^{5} +(32.1882 - 55.7516i) q^{6} +(-72.9514 + 42.1185i) q^{7} +(407.599 - 93.0319i) q^{8} +(-670.737 - 101.097i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.92333	−	8.14687i	0.490416	−	1.01836i	−0.498083	−	0.867130i	\(-0.665962\pi\)
							0.988499	−	0.151230i	\(-0.0483233\pi\)
\(3\)	7.09953	+	0.532036i	0.262946	+	0.0197050i	0.205551	−	0.978646i	\(-0.434101\pi\)
							0.0573950	+	0.998352i	\(0.481721\pi\)
\(5\)	43.9242	−	142.399i	0.351394	−	1.13919i	−0.591559	−	0.806262i	\(-0.701487\pi\)
							0.942952	−	0.332928i	\(-0.108037\pi\)
\(7\)	−72.9514	+	42.1185i	−0.212686	+	0.122794i	−0.602559	−	0.798074i	\(-0.705852\pi\)
							0.389873	+	0.920869i	\(0.372519\pi\)
\(11\)	1140.28	−	1429.87i	0.856710	−	1.07428i	−0.139747	−	0.990187i	\(-0.544629\pi\)
							0.996458	−	0.0840938i	\(-0.0267995\pi\)
\(13\)	798.493	−	740.893i	0.363447	−	0.337229i	−0.477218	−	0.878785i	\(-0.658355\pi\)
							0.840665	+	0.541555i	\(0.182164\pi\)
\(17\)	−9192.36	+	2835.47i	−1.87103	+	0.577136i	−0.875437	+	0.483333i	\(0.839426\pi\)
							−0.995591	+	0.0938025i	\(0.970098\pi\)
\(19\)	−423.918	−	2812.51i	−0.0618046	−	0.410047i	−0.998245	−	0.0592200i	\(-0.981139\pi\)
							0.936440	−	0.350827i	\(-0.114099\pi\)
\(23\)	7075.99	+	18029.3i	0.581572	+	1.48182i	0.854326	+	0.519737i	\(0.173970\pi\)
							−0.272754	+	0.962084i	\(0.587935\pi\)
\(29\)	17389.7	−	1303.18i	0.713013	−	0.0534329i	0.286715	−	0.958016i	\(-0.407437\pi\)
							0.426298	+	0.904583i	\(0.359818\pi\)
\(31\)	44993.2	−	30675.8i	1.51029	−	1.02970i	0.526546	−	0.850147i	\(-0.323487\pi\)
							0.983748	−	0.179554i	\(-0.0574655\pi\)
\(37\)	84503.8	+	48788.3i	1.66829	+	0.963187i	0.968561	+	0.248778i	\(0.0800289\pi\)
							0.699728	+	0.714409i	\(0.253304\pi\)
\(41\)	2649.15	+	1275.76i	0.0384374	+	0.0185105i	0.453004	−	0.891509i	\(-0.350352\pi\)
							−0.414566	+	0.910019i	\(0.636067\pi\)
\(43\)	−74858.4	−	26787.8i	−0.941532	−	0.336924i
							\(47\)	53629.5	+	67249.2i	0.516547	+	0.647730i	0.969872	−	0.243616i	\(-0.0783336\pi\)
−0.453325	+	0.891345i	\(0.649762\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.16	✓	252
43.29	odd	42	inner	43.7.h.a.29.16	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.16	✓	252	1.1	even	1	trivial
43.7.h.a.29.16	yes	252	43.29	odd	42	inner