Embedded newform 43.7.h.a.3.17

• Label: 43.7.h.a.3.17
• 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			3.17
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.47573 - 9.29395i) q^{2} +(30.6638 + 2.29794i) q^{3} +(-26.4420 - 33.1572i) q^{4} +(-67.0797 + 217.467i) q^{5} +(158.600 - 274.703i) q^{6} +(521.839 - 301.284i) q^{7} +(217.132 - 49.5589i) q^{8} +(214.131 + 32.2751i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.47573	−	9.29395i	0.559466	−	1.16174i	−0.408985	−	0.912541i	\(-0.634117\pi\)
							0.968451	−	0.249203i	\(-0.0801685\pi\)
\(3\)	30.6638	+	2.29794i	1.13570	+	0.0851087i	0.629279	−	0.777180i	\(-0.283350\pi\)
							0.506418	+	0.862288i	\(0.330969\pi\)
\(5\)	−67.0797	+	217.467i	−0.536638	+	1.73974i	0.127217	+	0.991875i	\(0.459396\pi\)
							−0.663855	+	0.747862i	\(0.731081\pi\)
\(7\)	521.839	−	301.284i	1.52140	−	0.878378i	0.521715	−	0.853120i	\(-0.325293\pi\)
							0.999681	−	0.0252582i	\(-0.00804078\pi\)
\(11\)	1090.36	−	1367.27i	0.819203	−	1.02725i	−0.179848	−	0.983694i	\(-0.557561\pi\)
							0.999051	−	0.0435536i	\(-0.0138679\pi\)
\(13\)	−2017.10	+	1871.60i	−0.918117	+	0.851888i	−0.989348	−	0.145572i	\(-0.953498\pi\)
							0.0712307	+	0.997460i	\(0.477307\pi\)
\(17\)	−1266.53	+	390.672i	−0.257791	+	0.0795181i	−0.420957	−	0.907081i	\(-0.638306\pi\)
							0.163166	+	0.986599i	\(0.447829\pi\)
\(19\)	897.955	+	5957.54i	0.130916	+	0.868573i	0.953817	+	0.300388i	\(0.0971162\pi\)
							−0.822901	+	0.568185i	\(0.807646\pi\)
\(23\)	−478.704	−	1219.72i	−0.0393445	−	0.100248i	0.909851	−	0.414935i	\(-0.136196\pi\)
							−0.949196	+	0.314687i	\(0.898100\pi\)
\(29\)	8931.60	−	669.331i	0.366214	−	0.0274440i	0.109648	−	0.993971i	\(-0.465028\pi\)
							0.256567	+	0.966527i	\(0.417409\pi\)
\(31\)	−6189.16	+	4219.70i	−0.207753	+	0.141643i	−0.662727	−	0.748861i	\(-0.730601\pi\)
							0.454974	+	0.890505i	\(0.349649\pi\)
\(37\)	−7270.58	−	4197.67i	−0.143537	−	0.0828711i	0.426512	−	0.904482i	\(-0.359742\pi\)
							−0.570049	+	0.821611i	\(0.693076\pi\)
\(41\)	−33757.3	−	16256.7i	−0.489798	−	0.235874i	0.172649	−	0.984983i	\(-0.444767\pi\)
							−0.662446	+	0.749109i	\(0.730482\pi\)
\(43\)	−42463.6	+	67217.6i	−0.534087	+	0.845430i
							\(47\)	−74746.9	−	93729.6i	−0.719945	−	0.902783i	0.278389	−	0.960468i	\(-0.410199\pi\)
−0.998335	+	0.0576852i	\(0.981628\pi\)