Embedded newform 43.7.d.a.7.12

• Label: 43.7.d.a.7.12
• Level: $43$
• Weight: $7$
• Character: 43.7
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.12
Character	\(\chi\)	\(=\)	43.7
Dual form			43.7.d.a.37.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.95704i q^{2} +(10.4424 - 6.02890i) q^{3} +60.1700 q^{4} +(-193.167 + 111.525i) q^{5} +(11.7988 + 20.4361i) q^{6} +(-30.2778 - 17.4809i) q^{7} +243.006i q^{8} +(-291.805 + 505.421i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{3} - 1194 q^{4} - 3 q^{5} + 111 q^{6} - 147 q^{7} + 3852 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.95704i			0.244630i	0.992491	+	0.122315i	\(0.0390319\pi\)
							−0.992491	+	0.122315i	\(0.960968\pi\)
\(3\)	10.4424	−	6.02890i	0.386754	−	0.223293i	−0.293999	−	0.955806i	\(-0.594986\pi\)
							0.680753	+	0.732513i	\(0.261653\pi\)
\(5\)	−193.167	+	111.525i	−1.54534	+	0.892200i	−0.546847	+	0.837233i	\(0.684172\pi\)
							−0.998488	+	0.0549670i	\(0.982495\pi\)
\(7\)	−30.2778	−	17.4809i	−0.0882733	−	0.0509646i	0.455214	−	0.890382i	\(-0.349563\pi\)
							−0.543487	+	0.839418i	\(0.682896\pi\)
\(11\)	688.201			0.517056			0.258528	−	0.966004i	\(-0.416763\pi\)
							0.258528	+	0.966004i	\(0.416763\pi\)
\(13\)	−591.384	+	1024.31i	−0.269178	+	0.466230i	−0.968650	−	0.248430i	\(-0.920085\pi\)
							0.699472	+	0.714660i	\(0.253419\pi\)
\(17\)	−2373.46	+	4110.95i	−0.483097	+	0.836749i	−0.999812	−	0.0194089i	\(-0.993822\pi\)
							0.516714	+	0.856158i	\(0.327155\pi\)
\(19\)	−8832.15	+	5099.24i	−1.28767	+	0.743438i	−0.978239	−	0.207483i	\(-0.933473\pi\)
							−0.309434	+	0.950921i	\(0.600140\pi\)
\(23\)	1298.84	+	2249.66i	0.106751	+	0.184899i	0.914452	−	0.404694i	\(-0.132622\pi\)
							−0.807701	+	0.589592i	\(0.799288\pi\)
\(29\)	34988.7	+	20200.7i	1.43461	+	0.828273i	0.997468	−	0.0711195i	\(-0.0226572\pi\)
							0.437143	+	0.899392i	\(0.355991\pi\)
\(31\)	−27966.9	−	48440.1i	−0.938771	−	1.62600i	−0.767768	−	0.640728i	\(-0.778633\pi\)
							−0.171003	−	0.985271i	\(-0.554701\pi\)
\(37\)	40553.8	−	23413.8i	0.800620	−	0.462238i	−0.0430676	−	0.999072i	\(-0.513713\pi\)
							0.843688	+	0.536834i	\(0.180380\pi\)
\(41\)	46510.8			0.674843			0.337421	−	0.941354i	\(-0.390445\pi\)
							0.337421	+	0.941354i	\(0.390445\pi\)
\(43\)	71663.4	−	34434.3i	0.901347	−	0.433098i
							\(47\)	−54839.3			−0.528200			−0.264100	−	0.964495i	\(-0.585075\pi\)
−0.264100	+	0.964495i	\(0.585075\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.d.a.7.12	✓	42
43.37	odd	6	inner	43.7.d.a.37.10	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.d.a.7.12	✓	42	1.1	even	1	trivial
43.7.d.a.37.10	yes	42	43.37	odd	6	inner