Embedded newform 43.7.h.a.12.14

• Label: 43.7.h.a.12.14
• 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			12.14
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.71537 + 5.63853i) q^{2} +(26.7438 + 39.2259i) q^{3} +(15.4836 - 19.4158i) q^{4} +(-88.5928 + 95.4804i) q^{5} +(-148.557 + 257.309i) q^{6} +(-56.9713 + 32.8924i) q^{7} +(542.009 + 123.710i) q^{8} +(-557.109 + 1419.49i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.71537	+	5.63853i	0.339422	+	0.704817i	0.998899	−	0.0469078i	\(-0.0149367\pi\)
							−0.659477	+	0.751724i	\(0.729222\pi\)
\(3\)	26.7438	+	39.2259i	0.990510	+	1.45281i	0.888962	+	0.457981i	\(0.151427\pi\)
							0.101548	+	0.994831i	\(0.467620\pi\)
\(5\)	−88.5928	+	95.4804i	−0.708743	+	0.763843i	−0.980072	−	0.198644i	\(-0.936346\pi\)
							0.271329	+	0.962487i	\(0.412537\pi\)
\(7\)	−56.9713	+	32.8924i	−0.166097	+	0.0958962i	−0.580744	−	0.814086i	\(-0.697238\pi\)
							0.414647	+	0.909982i	\(0.363905\pi\)
\(11\)	−698.868	−	876.353i	−0.525070	−	0.658417i	0.446607	−	0.894730i	\(-0.352632\pi\)
							−0.971677	+	0.236313i	\(0.924061\pi\)
\(13\)	2664.69	−	821.948i	1.21288	−	0.374123i	0.378648	−	0.925541i	\(-0.376389\pi\)
							0.834229	+	0.551418i	\(0.185913\pi\)
\(17\)	573.969	−	532.566i	0.116827	−	0.108399i	−0.619597	−	0.784920i	\(-0.712704\pi\)
							0.736423	+	0.676521i	\(0.236513\pi\)
\(19\)	−8774.55	+	3443.76i	−1.27928	+	0.502078i	−0.905068	−	0.425267i	\(-0.860180\pi\)
							−0.374207	+	0.927345i	\(0.622085\pi\)
\(23\)	13600.0	−	2049.88i	1.11778	−	0.168478i	0.435938	−	0.899977i	\(-0.356417\pi\)
							0.681843	+	0.731498i	\(0.261179\pi\)
\(29\)	2791.31	−	4094.11i	0.114450	−	0.167867i	−0.764740	−	0.644339i	\(-0.777132\pi\)
							0.879189	+	0.476472i	\(0.158085\pi\)
\(31\)	149.682	−	1997.37i	0.00502441	−	0.0670461i	−0.994128	−	0.108211i	\(-0.965488\pi\)
							0.999152	+	0.0411648i	\(0.0131069\pi\)
\(37\)	6756.33	+	3900.77i	0.133385	+	0.0770096i	0.565207	−	0.824949i	\(-0.308796\pi\)
							−0.431823	+	0.901958i	\(0.642129\pi\)
\(41\)	83404.6	−	40165.5i	1.21015	−	0.582776i	0.283596	−	0.958944i	\(-0.408473\pi\)
							0.926552	+	0.376168i	\(0.122758\pi\)
\(43\)	71359.7	+	35059.4i	0.897527	+	0.440960i
							\(47\)	47685.4	−	59795.6i	0.459295	−	0.575938i	−0.497219	−	0.867625i	\(-0.665645\pi\)
0.956514	+	0.291688i	\(0.0942167\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.14	✓	252
43.18	odd	42	inner	43.7.h.a.18.14	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.14	✓	252	1.1	even	1	trivial
43.7.h.a.18.14	yes	252	43.18	odd	42	inner