Embedded newform 43.6.e.a.4.6

• Label: 43.6.e.a.4.6
• Level: $43$
• Weight: $6$
• Character: 43.4
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.6
Character	\(\chi\)	\(=\)	43.4
Dual form			43.6.e.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.83850 + 4.81333i) q^{2} +(-10.8788 - 13.6416i) q^{3} +(-1.31337 - 5.75427i) q^{4} +(77.8725 + 37.5014i) q^{5} +107.420 q^{6} -36.5256 q^{7} +(-144.759 - 69.7122i) q^{8} +(-13.6719 + 59.9004i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 3 q^{2} - 9 q^{3} - 331 q^{4} + 67 q^{5} - 20 q^{6} - 624 q^{7} + 589 q^{8} - 579 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.83850	+	4.81333i	−0.678558	+	0.850885i	−0.995220	−	0.0976536i	\(-0.968866\pi\)
							0.316663	+	0.948538i	\(0.397438\pi\)
\(3\)	−10.8788	−	13.6416i	−0.697875	−	0.875108i	0.298988	−	0.954257i	\(-0.403351\pi\)
							−0.996863	+	0.0791493i	\(0.974780\pi\)
\(5\)	77.8725	+	37.5014i	1.39303	+	0.670846i	0.971734	−	0.236079i	\(-0.0758623\pi\)
							0.421292	+	0.906925i	\(0.361577\pi\)
\(7\)	−36.5256			−0.281742			−0.140871	−	0.990028i	\(-0.544990\pi\)
							−0.140871	+	0.990028i	\(0.544990\pi\)
\(11\)	−163.208	+	715.061i	−0.406686	+	1.78181i	0.192602	+	0.981277i	\(0.438307\pi\)
							−0.599289	+	0.800533i	\(0.704550\pi\)
\(13\)	−808.880	−	389.536i	−1.32747	−	0.639277i	−0.370331	−	0.928900i	\(-0.620756\pi\)
							−0.957141	+	0.289623i	\(0.906470\pi\)
\(17\)	−1337.47	+	644.092i	−1.12244	+	0.540538i	−0.900645	−	0.434556i	\(-0.856905\pi\)
							−0.221793	+	0.975094i	\(0.571191\pi\)
\(19\)	129.348	+	566.711i	0.0822008	+	0.360145i	0.999254	−	0.0386197i	\(-0.0122961\pi\)
							−0.917053	+	0.398765i	\(0.869439\pi\)
\(23\)	−392.863	+	1721.24i	−0.154854	+	0.678458i	0.836580	+	0.547845i	\(0.184552\pi\)
							−0.991433	+	0.130613i	\(0.958306\pi\)
\(29\)	2234.95	−	2802.53i	0.493483	−	0.618808i	−0.471262	−	0.881993i	\(-0.656202\pi\)
							0.964745	+	0.263185i	\(0.0847730\pi\)
\(31\)	−4712.45	+	5909.23i	−0.880730	+	1.10440i	0.113111	+	0.993582i	\(0.463918\pi\)
							−0.993841	+	0.110818i	\(0.964653\pi\)
\(37\)	−8027.40			−0.963986			−0.481993	−	0.876175i	\(-0.660087\pi\)
							−0.481993	+	0.876175i	\(0.660087\pi\)
\(41\)	11133.0	−	13960.4i	1.03432	−	1.29699i	0.0804525	−	0.996758i	\(-0.474363\pi\)
							0.953865	−	0.300235i	\(-0.0970651\pi\)
\(43\)	11627.5	+	3436.61i	0.958990	+	0.283439i
							\(47\)	1179.52	+	5167.82i	0.0778863	+	0.341242i	0.998825	−	0.0484656i	\(-0.0154331\pi\)
−0.920939	+	0.389708i	\(0.872576\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.e.a.4.6	✓	108
43.11	even	7	inner	43.6.e.a.11.6	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.e.a.4.6	✓	108	1.1	even	1	trivial
43.6.e.a.11.6	yes	108	43.11	even	7	inner