Embedded newform 43.3.d.a.7.2

• Label: 43.3.d.a.7.2
• Level: $43$
• Weight: $3$
• Character: 43.7
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 37x^{10} + 483x^{8} + 2718x^{6} + 6923x^{4} + 7253x^{2} + 1849 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.2
Root			\(-1.61947i\) of defining polynomial
Character	\(\chi\)	\(=\)	43.7
Dual form			43.3.d.a.37.5

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.61947i q^{2} +(2.75697 - 1.59174i) q^{3} +1.37732 q^{4} +(-6.40200 + 3.69620i) q^{5} +(-2.57777 - 4.46483i) q^{6} +(1.18578 + 0.684612i) q^{7} -8.70840i q^{8} +(0.567259 - 0.982521i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 26 q^{4} - 3 q^{5} - 9 q^{6} + 9 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.61947i		−	0.809735i	−0.914375	−	0.404868i	\(-0.867318\pi\)
							0.914375	−	0.404868i	\(-0.132682\pi\)
\(3\)	2.75697	−	1.59174i	0.918990	−	0.530579i	0.0356775	−	0.999363i	\(-0.488641\pi\)
							0.883313	+	0.468784i	\(0.155308\pi\)
\(5\)	−6.40200	+	3.69620i	−1.28040	+	0.739239i	−0.976921	−	0.213599i	\(-0.931481\pi\)
							−0.303479	+	0.952838i	\(0.598148\pi\)
\(7\)	1.18578	+	0.684612i	0.169398	+	0.0978017i	0.582302	−	0.812973i	\(-0.302152\pi\)
							−0.412904	+	0.910774i	\(0.635486\pi\)
\(11\)	2.61047			0.237315			0.118658	−	0.992935i	\(-0.462141\pi\)
							0.118658	+	0.992935i	\(0.462141\pi\)
\(13\)	−7.87371	+	13.6377i	−0.605670	+	1.04905i	0.386275	+	0.922384i	\(0.373762\pi\)
							−0.991945	+	0.126668i	\(0.959572\pi\)
\(17\)	−5.97417	+	10.3476i	−0.351422	+	0.608680i	−0.986499	−	0.163769i	\(-0.947635\pi\)
							0.635077	+	0.772449i	\(0.280968\pi\)
\(19\)	29.6812	−	17.1365i	1.56217	−	0.901919i	0.565132	−	0.825000i	\(-0.308825\pi\)
							0.997037	−	0.0769189i	\(-0.0245083\pi\)
\(23\)	−6.65538	−	11.5275i	−0.289365	−	0.501194i	0.684294	−	0.729207i	\(-0.260111\pi\)
							−0.973658	+	0.228012i	\(0.926777\pi\)
\(29\)	−43.7338	−	25.2497i	−1.50806	−	0.870679i	−0.999956	−	0.00938436i	\(-0.997013\pi\)
							−0.508105	−	0.861295i	\(-0.669654\pi\)
\(31\)	−12.2265	−	21.1770i	−0.394404	−	0.683128i	0.598621	−	0.801033i	\(-0.295716\pi\)
							−0.993025	+	0.117904i	\(0.962382\pi\)
\(37\)	7.17954	−	4.14511i	0.194042	−	0.112030i	−0.399832	−	0.916589i	\(-0.630931\pi\)
							0.593873	+	0.804559i	\(0.297598\pi\)
\(41\)	68.7184			1.67606			0.838030	−	0.545625i	\(-0.183708\pi\)
							0.838030	+	0.545625i	\(0.183708\pi\)
\(43\)	−39.9435	+	15.9223i	−0.928918	+	0.370285i
							\(47\)	−18.2745			−0.388819			−0.194410	−	0.980920i	\(-0.562279\pi\)
−0.194410	+	0.980920i	\(0.562279\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.d.a.7.2	✓	12
3.2	odd	2		387.3.j.c.136.5		12
4.3	odd	2		688.3.t.c.609.2		12
43.37	odd	6	inner	43.3.d.a.37.5	yes	12
129.80	even	6		387.3.j.c.37.2		12
172.123	even	6		688.3.t.c.209.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.d.a.7.2	✓	12	1.1	even	1	trivial
43.3.d.a.37.5	yes	12	43.37	odd	6	inner
387.3.j.c.37.2		12	129.80	even	6
387.3.j.c.136.5		12	3.2	odd	2
688.3.t.c.209.2		12	172.123	even	6
688.3.t.c.609.2		12	4.3	odd	2