Embedded newform 43.9.h.a.3.7

• Label: 43.9.h.a.3.7
• Level: $43$
• Weight: $9$
• Character: 43.3
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.77760 + 16.1503i) q^{2} +(155.893 + 11.6826i) q^{3} +(-40.7292 - 51.0728i) q^{4} +(221.849 - 719.217i) q^{5} +(-1401.15 + 2426.87i) q^{6} +(890.825 - 514.318i) q^{7} +(-3332.26 + 760.567i) q^{8} +(17678.6 + 2664.61i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.77760	+	16.1503i	−0.486100	+	1.00940i	0.503291	+	0.864117i	\(0.332122\pi\)
							−0.989391	+	0.145279i	\(0.953592\pi\)
\(3\)	155.893	+	11.6826i	1.92461	+	0.144230i	0.981902	−	0.189389i	\(-0.0606507\pi\)
							0.942708	+	0.333619i	\(0.108270\pi\)
\(5\)	221.849	−	719.217i	0.354958	−	1.15075i	−0.585453	−	0.810707i	\(-0.699083\pi\)
							0.940411	−	0.340040i	\(-0.110441\pi\)
\(7\)	890.825	−	514.318i	0.371023	−	0.214210i	−0.302883	−	0.953028i	\(-0.597949\pi\)
							0.673905	+	0.738818i	\(0.264616\pi\)
\(11\)	−12909.4	+	16187.9i	−0.881728	+	1.10565i	0.111986	+	0.993710i	\(0.464279\pi\)
							−0.993715	+	0.111943i	\(0.964293\pi\)
\(13\)	34213.7	−	31745.7i	1.19792	−	1.11150i	0.206818	−	0.978379i	\(-0.433689\pi\)
							0.991099	−	0.133125i	\(-0.0425012\pi\)
\(17\)	67370.7	−	20781.1i	0.806631	−	0.248813i	0.136101	−	0.990695i	\(-0.456543\pi\)
							0.670530	+	0.741882i	\(0.266067\pi\)
\(19\)	11467.9	+	76084.3i	0.0879970	+	0.583822i	0.988760	+	0.149511i	\(0.0477700\pi\)
							−0.900763	+	0.434311i	\(0.856992\pi\)
\(23\)	105651.	+	269195.i	0.377541	+	0.961958i	0.985755	+	0.168185i	\(0.0537907\pi\)
							−0.608215	+	0.793773i	\(0.708114\pi\)
\(29\)	−634318.	+	47535.6i	−0.896840	+	0.0672089i	−0.515168	−	0.857089i	\(-0.672271\pi\)
							−0.381672	+	0.924298i	\(0.624652\pi\)
\(31\)	−15998.9	+	10907.9i	−0.0173238	+	0.0118112i	−0.571951	−	0.820288i	\(-0.693813\pi\)
							0.554627	+	0.832099i	\(0.312861\pi\)
\(37\)	−1.28721e6	−	743173.i	−0.686821	−	0.396536i	0.115599	−	0.993296i	\(-0.463121\pi\)
							−0.802420	+	0.596760i	\(0.796455\pi\)
\(41\)	661684.	+	318650.i	0.234161	+	0.112766i	0.547285	−	0.836946i	\(-0.315661\pi\)
							−0.313124	+	0.949712i	\(0.601376\pi\)
\(43\)	−3.34644e6	−	699686.i	−0.978833	−	0.204658i
							\(47\)	−3.14581e6	−	3.94472e6i	−0.644675	−	0.808397i	0.346904	−	0.937901i	\(-0.387233\pi\)
−0.991579	+	0.129503i	\(0.958662\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.7	✓	336
43.29	odd	42	inner	43.9.h.a.29.7	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.7	✓	336	1.1	even	1	trivial
43.9.h.a.29.7	yes	336	43.29	odd	42	inner