Embedded newform 43.7.h.a.12.6

• Label: 43.7.h.a.12.6
• Level: $43$
• Weight: $7$
• Character: 43.12
• 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			12.6
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.52301 - 7.31560i) q^{2} +(12.1765 + 17.8597i) q^{3} +(-1.20308 + 1.50861i) q^{4} +(-66.5759 + 71.7517i) q^{5} +(87.7562 - 151.998i) q^{6} +(-41.7082 + 24.0802i) q^{7} +(-491.357 - 112.149i) q^{8} +(95.6334 - 243.670i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.52301	−	7.31560i	−0.440376	−	0.914450i	−0.996519	−	0.0833636i	\(-0.973434\pi\)
							0.556143	−	0.831086i	\(-0.312281\pi\)
\(3\)	12.1765	+	17.8597i	0.450982	+	0.661469i	0.982647	−	0.185486i	\(-0.0593858\pi\)
							−0.531665	+	0.846955i	\(0.678433\pi\)
\(5\)	−66.5759	+	71.7517i	−0.532607	+	0.574014i	−0.940835	−	0.338866i	\(-0.889957\pi\)
							0.408227	+	0.912880i	\(0.366147\pi\)
\(7\)	−41.7082	+	24.0802i	−0.121598	+	0.0702047i	−0.559565	−	0.828786i	\(-0.689032\pi\)
							0.437967	+	0.898991i	\(0.355699\pi\)
\(11\)	−1420.66	−	1781.45i	−1.06736	−	1.33843i	−0.937923	−	0.346844i	\(-0.887253\pi\)
							−0.129440	−	0.991587i	\(-0.541318\pi\)
\(13\)	−1281.33	+	395.239i	−0.583220	+	0.179899i	−0.572305	−	0.820041i	\(-0.693951\pi\)
							−0.0109148	+	0.999940i	\(0.503474\pi\)
\(17\)	−768.939	+	713.471i	−0.156511	+	0.145221i	−0.754542	−	0.656252i	\(-0.772141\pi\)
							0.598030	+	0.801473i	\(0.295950\pi\)
\(19\)	−5897.43	+	2314.57i	−0.859809	+	0.337450i	−0.753951	−	0.656931i	\(-0.771854\pi\)
							−0.105858	+	0.994381i	\(0.533759\pi\)
\(23\)	−16098.9	+	2426.52i	−1.32316	+	0.199435i	−0.772352	−	0.635195i	\(-0.780920\pi\)
							−0.550812	+	0.834630i	\(0.685682\pi\)
\(29\)	12371.4	−	18145.5i	0.507254	−	0.744004i	−0.484130	−	0.874996i	\(-0.660864\pi\)
							0.991383	+	0.130992i	\(0.0418162\pi\)
\(31\)	2416.11	−	32240.8i	0.0811021	−	1.08223i	−0.797070	−	0.603887i	\(-0.793618\pi\)
							0.878172	−	0.478345i	\(-0.158763\pi\)
\(37\)	−40757.4	−	23531.3i	−0.804640	−	0.464559i	0.0404510	−	0.999182i	\(-0.487121\pi\)
							−0.845091	+	0.534622i	\(0.820454\pi\)
\(41\)	56516.7	−	27217.0i	0.820022	−	0.394902i	0.0236585	−	0.999720i	\(-0.492469\pi\)
							0.796363	+	0.604818i	\(0.206754\pi\)
\(43\)	79234.5	+	6576.89i	0.996573	+	0.0827209i
							\(47\)	−57707.2	+	72362.6i	−0.555823	+	0.696980i	−0.977780	−	0.209636i	\(-0.932772\pi\)
0.421956	+	0.906616i	\(0.361344\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.6	✓	252
43.18	odd	42	inner	43.7.h.a.18.6	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.6	✓	252	1.1	even	1	trivial
43.7.h.a.18.6	yes	252	43.18	odd	42	inner