Embedded newform 43.7.h.a.29.20

• Label: 43.7.h.a.29.20
• Level: $43$
• Weight: $7$
• Character: 43.29
• 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			29.20
Character	\(\chi\)	\(=\)	43.29
Dual form			43.7.h.a.3.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.93639 + 12.3270i) q^{2} +(-47.2449 + 3.54052i) q^{3} +(-76.8118 + 96.3189i) q^{4} +(8.18721 + 26.5423i) q^{5} +(-324.108 - 561.372i) q^{6} +(-116.163 - 67.0667i) q^{7} +(-789.619 - 180.225i) q^{8} +(1498.69 - 225.891i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.93639	+	12.3270i	0.742049	+	1.54088i	0.838108	+	0.545504i	\(0.183662\pi\)
							−0.0960596	+	0.995376i	\(0.530624\pi\)
\(3\)	−47.2449	+	3.54052i	−1.74981	+	0.131130i	−0.910764	−	0.412928i	\(-0.864506\pi\)
							−0.839048	+	0.544058i	\(0.816887\pi\)
\(5\)	8.18721	+	26.5423i	0.0654977	+	0.212338i	0.982405	−	0.186761i	\(-0.0597991\pi\)
							−0.916908	+	0.399099i	\(0.869323\pi\)
\(7\)	−116.163	−	67.0667i	−0.338667	−	0.195530i	0.321015	−	0.947074i	\(-0.395976\pi\)
							−0.659683	+	0.751544i	\(0.729309\pi\)
\(11\)	1029.21	+	1290.59i	0.773262	+	0.969640i	0.999991	−	0.00431837i	\(-0.00137458\pi\)
							−0.226729	+	0.973958i	\(0.572803\pi\)
\(13\)	−1882.34	−	1746.56i	−0.856777	−	0.794973i	0.123553	−	0.992338i	\(-0.460571\pi\)
							−0.980330	+	0.197365i	\(0.936762\pi\)
\(17\)	−5984.32	−	1845.92i	−1.21806	−	0.375721i	−0.381897	−	0.924205i	\(-0.624729\pi\)
							−0.836161	+	0.548484i	\(0.815205\pi\)
\(19\)	1459.69	−	9684.38i	0.212813	−	1.41192i	−0.585984	−	0.810323i	\(-0.699292\pi\)
							0.798797	−	0.601601i	\(-0.205470\pi\)
\(23\)	−3680.64	+	9378.11i	−0.302510	+	0.770782i	0.696094	+	0.717951i	\(0.254920\pi\)
							−0.998603	+	0.0528314i	\(0.983175\pi\)
\(29\)	−40361.4	−	3024.67i	−1.65490	−	0.124018i	−0.785744	−	0.618552i	\(-0.787720\pi\)
							−0.869158	+	0.494534i	\(0.835339\pi\)
\(31\)	−2922.85	−	1992.77i	−0.0981119	−	0.0668916i	0.513270	−	0.858227i	\(-0.328434\pi\)
							−0.611382	+	0.791335i	\(0.709386\pi\)
\(37\)	5219.63	−	3013.56i	0.103047	−	0.0594941i	−0.447591	−	0.894238i	\(-0.647718\pi\)
							0.550638	+	0.834744i	\(0.314385\pi\)
\(41\)	−82505.0	+	39732.3i	−1.19710	+	0.576491i	−0.922846	−	0.385168i	\(-0.874143\pi\)
							−0.274249	+	0.961659i	\(0.588429\pi\)
\(43\)	75826.2	−	23911.2i	0.953705	−	0.300743i
							\(47\)	−39357.9	+	49353.2i	−0.379086	+	0.475359i	−0.934371	−	0.356301i	\(-0.884038\pi\)
0.555285	+	0.831660i	\(0.312609\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.29.20	yes	252
43.3	odd	42	inner	43.7.h.a.3.20	✓	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.20	✓	252	43.3	odd	42	inner
43.7.h.a.29.20	yes	252	1.1	even	1	trivial