Embedded newform 43.3.h.a.5.4

• Label: 43.3.h.a.5.4
• Level: $43$
• Weight: $3$
• Character: 43.5
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(6\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	43.5
Dual form			43.3.h.a.26.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.187226 - 0.0427331i) q^{2} +(0.614362 - 1.99171i) q^{3} +(-3.57065 - 1.71953i) q^{4} +(8.48040 - 3.32831i) q^{5} +(-0.200136 + 0.346646i) q^{6} +(-4.12743 + 2.38297i) q^{7} +(1.19561 + 0.953468i) q^{8} +(3.84667 + 2.62262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.187226	−	0.0427331i	−0.0936130	−	0.0213665i	0.175458	−	0.984487i	\(-0.443859\pi\)
							−0.269071	+	0.963120i	\(0.586717\pi\)
\(3\)	0.614362	−	1.99171i	0.204787	−	0.663904i	−0.793669	−	0.608350i	\(-0.791832\pi\)
							0.998456	−	0.0555533i	\(-0.0176923\pi\)
\(5\)	8.48040	−	3.32831i	1.69608	−	0.665662i	0.697383	−	0.716698i	\(-0.254347\pi\)
							0.998697	+	0.0510358i	\(0.0162523\pi\)
\(7\)	−4.12743	+	2.38297i	−0.589632	+	0.340424i	−0.764952	−	0.644087i	\(-0.777237\pi\)
							0.175320	+	0.984512i	\(0.443904\pi\)
\(11\)	−10.1026	+	4.86514i	−0.918416	+	0.442286i	−0.832505	−	0.554017i	\(-0.813094\pi\)
							−0.0859104	+	0.996303i	\(0.527380\pi\)
\(13\)	2.51824	+	0.379563i	0.193710	+	0.0291972i	0.245181	−	0.969477i	\(-0.421153\pi\)
							−0.0514703	+	0.998675i	\(0.516391\pi\)
\(17\)	−4.50010	+	11.4661i	−0.264712	+	0.674475i	−0.999995	−	0.00328554i	\(-0.998954\pi\)
							0.735283	+	0.677761i	\(0.237049\pi\)
\(19\)	1.22281	+	1.79353i	0.0643583	+	0.0943963i	0.857069	−	0.515202i	\(-0.172283\pi\)
							−0.792710	+	0.609598i	\(0.791331\pi\)
\(23\)	−1.47228	+	19.6462i	−0.0640122	+	0.854184i	0.869109	+	0.494621i	\(0.164693\pi\)
							−0.933121	+	0.359563i	\(0.882926\pi\)
\(29\)	−15.5909	−	50.5445i	−0.537617	−	1.74291i	−0.660833	−	0.750533i	\(-0.729797\pi\)
							0.123216	−	0.992380i	\(-0.460679\pi\)
\(31\)	20.4898	+	19.0117i	0.660960	+	0.613282i	0.937610	−	0.347689i	\(-0.113034\pi\)
							−0.276649	+	0.960971i	\(0.589224\pi\)
\(37\)	−37.2387	−	21.4998i	−1.00645	−	0.581075i	−0.0963002	−	0.995352i	\(-0.530701\pi\)
							−0.910151	+	0.414278i	\(0.864034\pi\)
\(41\)	−14.1140	+	61.8373i	−0.344243	+	1.50823i	0.445777	+	0.895144i	\(0.352928\pi\)
							−0.790019	+	0.613082i	\(0.789930\pi\)
\(43\)	−34.0741	−	26.2289i	−0.792420	−	0.609975i
							\(47\)	−35.3746	−	17.0355i	−0.752652	−	0.362458i	0.0178964	−	0.999840i	\(-0.494303\pi\)
−0.770548	+	0.637382i	\(0.780017\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.5.4	✓	72
3.2	odd	2		387.3.bn.b.91.3		72
43.26	odd	42	inner	43.3.h.a.26.4	yes	72
129.26	even	42		387.3.bn.b.370.3		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.5.4	✓	72	1.1	even	1	trivial
43.3.h.a.26.4	yes	72	43.26	odd	42	inner
387.3.bn.b.91.3		72	3.2	odd	2
387.3.bn.b.370.3		72	129.26	even	42