Embedded newform 43.7.h.a.18.21

• Label: 43.7.h.a.18.21
• Level: $43$
• Weight: $7$
• Character: 43.18
• 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			18.21
Character	\(\chi\)	\(=\)	43.18
Dual form			43.7.h.a.12.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.66612 - 13.8423i) q^{2} +(-20.8194 + 30.5364i) q^{3} +(-107.270 - 134.512i) q^{4} +(161.163 + 173.692i) q^{5} +(283.911 + 491.748i) q^{6} +(325.973 + 188.201i) q^{7} +(-1618.40 + 369.389i) q^{8} +(-232.693 - 592.892i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.66612	−	13.8423i	0.833265	−	1.73029i	0.165102	−	0.986277i	\(-0.447205\pi\)
							0.668163	−	0.744015i	\(-0.267081\pi\)
\(3\)	−20.8194	+	30.5364i	−0.771088	+	1.13098i	0.216862	+	0.976202i	\(0.430418\pi\)
							−0.987950	+	0.154776i	\(0.950534\pi\)
\(5\)	161.163	+	173.692i	1.28930	+	1.38954i	0.878449	+	0.477836i	\(0.158579\pi\)
							0.410853	+	0.911702i	\(0.365231\pi\)
\(7\)	325.973	+	188.201i	0.950359	+	0.548690i	0.893192	−	0.449675i	\(-0.148460\pi\)
							0.0571666	+	0.998365i	\(0.481793\pi\)
\(11\)	54.5870	−	68.4499i	0.0410120	−	0.0514274i	−0.760903	−	0.648866i	\(-0.775243\pi\)
							0.801915	+	0.597439i	\(0.203815\pi\)
\(13\)	−439.732	−	135.639i	−0.200151	−	0.0617384i	0.193059	−	0.981187i	\(-0.438159\pi\)
							−0.393210	+	0.919449i	\(0.628635\pi\)
\(17\)	2671.95	+	2479.20i	0.543852	+	0.504621i	0.903570	−	0.428440i	\(-0.140937\pi\)
							−0.359718	+	0.933061i	\(0.617127\pi\)
\(19\)	6892.89	+	2705.26i	1.00494	+	0.394410i	0.810016	−	0.586408i	\(-0.199458\pi\)
							0.194925	+	0.980818i	\(0.437554\pi\)
\(23\)	9898.49	+	1491.96i	0.813552	+	0.122623i	0.542623	−	0.839976i	\(-0.317431\pi\)
							0.270929	+	0.962599i	\(0.412669\pi\)
\(29\)	−8394.41	−	12312.3i	−0.344188	−	0.504831i	0.614644	−	0.788805i	\(-0.289300\pi\)
							−0.958832	+	0.283973i	\(0.908347\pi\)
\(31\)	−832.764	−	11112.5i	−0.0279535	−	0.373014i	−0.993584	−	0.113096i	\(-0.963923\pi\)
							0.965631	−	0.259918i	\(-0.0836957\pi\)
\(37\)	−9433.01	+	5446.15i	−0.186228	+	0.107519i	−0.590216	−	0.807246i	\(-0.700957\pi\)
							0.403987	+	0.914765i	\(0.367624\pi\)
\(41\)	9308.02	+	4482.50i	0.135053	+	0.0650383i	0.500190	−	0.865916i	\(-0.333263\pi\)
							−0.365137	+	0.930954i	\(0.618978\pi\)
\(43\)	−70999.7	−	35782.7i	−0.893000	−	0.450057i
							\(47\)	−23733.6	−	29761.0i	−0.228597	−	0.286652i	0.654283	−	0.756249i	\(-0.272970\pi\)
−0.882880	+	0.469598i	\(0.844399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.18.21	yes	252
43.12	odd	42	inner	43.7.h.a.12.21	✓	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.21	✓	252	43.12	odd	42	inner
43.7.h.a.18.21	yes	252	1.1	even	1	trivial