Embedded newform 43.7.h.a.12.9

• Label: 43.7.h.a.12.9
• 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			12.9
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60125 - 3.32504i) q^{2} +(-14.5501 - 21.3411i) q^{3} +(31.4115 - 39.3888i) q^{4} +(42.5700 - 45.8796i) q^{5} +(-47.6616 + 82.5522i) q^{6} +(440.964 - 254.590i) q^{7} +(-411.538 - 93.9308i) q^{8} +(22.5967 - 57.5754i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60125	−	3.32504i	−0.200157	−	0.415630i	0.776596	−	0.629999i	\(-0.216945\pi\)
							−0.976753	+	0.214369i	\(0.931230\pi\)
\(3\)	−14.5501	−	21.3411i	−0.538893	−	0.790412i	0.456145	−	0.889906i	\(-0.349230\pi\)
							−0.995038	+	0.0994941i	\(0.968278\pi\)
\(5\)	42.5700	−	45.8796i	0.340560	−	0.367037i	−0.539428	−	0.842032i	\(-0.681360\pi\)
							0.879989	+	0.474995i	\(0.157550\pi\)
\(7\)	440.964	−	254.590i	1.28561	−	0.742246i	0.307740	−	0.951470i	\(-0.400427\pi\)
							0.977868	+	0.209224i	\(0.0670938\pi\)
\(11\)	409.031	+	512.909i	0.307311	+	0.385356i	0.911373	−	0.411581i	\(-0.135023\pi\)
							−0.604062	+	0.796938i	\(0.706452\pi\)
\(13\)	−2196.86	+	677.643i	−0.999938	+	0.308440i	−0.751160	−	0.660121i	\(-0.770505\pi\)
							−0.248778	+	0.968561i	\(0.580029\pi\)
\(17\)	2211.57	−	2052.04i	0.450147	−	0.417676i	−0.422245	−	0.906482i	\(-0.638758\pi\)
							0.872393	+	0.488806i	\(0.162567\pi\)
\(19\)	−9954.54	+	3906.87i	−1.45131	+	0.569597i	−0.954487	−	0.298254i	\(-0.903596\pi\)
							−0.496824	+	0.867851i	\(0.665500\pi\)
\(23\)	12451.4	−	1876.74i	1.02337	−	0.154248i	0.384150	−	0.923271i	\(-0.374495\pi\)
							0.639222	+	0.769022i	\(0.279256\pi\)
\(29\)	17776.1	−	26072.8i	0.728859	−	1.06904i	−0.265618	−	0.964078i	\(-0.585576\pi\)
							0.994476	−	0.104961i	\(-0.0334717\pi\)
\(31\)	−1368.13	+	18256.5i	−0.0459244	+	0.612819i	0.926000	+	0.377524i	\(0.123225\pi\)
							−0.971924	+	0.235294i	\(0.924395\pi\)
\(37\)	54459.7	+	31442.3i	1.07515	+	0.620739i	0.929584	−	0.368609i	\(-0.120166\pi\)
							0.145567	+	0.989348i	\(0.453499\pi\)
\(41\)	37976.6	−	18288.5i	0.551016	−	0.265355i	−0.137592	−	0.990489i	\(-0.543936\pi\)
							0.688608	+	0.725134i	\(0.258222\pi\)
\(43\)	69505.6	−	38604.8i	0.874208	−	0.485552i
							\(47\)	−31196.6	+	39119.3i	−0.300479	+	0.376789i	−0.909033	−	0.416724i	\(-0.863178\pi\)
0.608554	+	0.793512i	\(0.291750\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.9	✓	252
43.18	odd	42	inner	43.7.h.a.18.9	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.9	✓	252	1.1	even	1	trivial
43.7.h.a.18.9	yes	252	43.18	odd	42	inner