Embedded newform 667.2.s.a.16.4

• Label: 667.2.s.a.16.4
• Level: $667$
• Weight: $2$
• Character: 667.16
• Analytic conductor: $5.326$
• Analytic rank: $0$
• Dimension: $3480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	667.s (of order \(77\), degree \(60\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.32602181482\)
Analytic rank:	\(0\)
Dimension:	\(3480\)
Relative dimension:	\(58\) over \(\Q(\zeta_{77})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{77}]$

Embedding invariants

Embedding label			16.4
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29489 + 0.991977i) q^{2} +(-0.162496 + 1.58757i) q^{3} +(2.91223 - 3.09614i) q^{4} +(0.0749307 - 3.67257i) q^{5} +(-1.20193 - 3.80450i) q^{6} +(0.954585 - 2.33823i) q^{7} +(-1.91243 + 5.29164i) q^{8} +(0.443809 + 0.0918140i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/667\mathbb{Z}\right)^\times\).

\(n\)	\(465\)	\(553\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29489	+	0.991977i	−1.62273	+	0.701434i	−0.996912	−	0.0785320i	\(-0.974977\pi\)
							−0.625822	+	0.779966i	\(0.715236\pi\)
\(3\)	−0.162496	+	1.58757i	−0.0938171	+	0.916586i	0.836041	+	0.548667i	\(0.184865\pi\)
							−0.929858	+	0.367919i	\(0.880071\pi\)
\(5\)	0.0749307	−	3.67257i	0.0335100	−	1.64242i	−0.553504	−	0.832847i	\(-0.686710\pi\)
							0.587014	−	0.809577i	\(-0.300303\pi\)
\(7\)	0.954585	−	2.33823i	0.360799	−	0.883768i	−0.633160	−	0.774021i	\(-0.718243\pi\)
							0.993960	−	0.109747i	\(-0.0350042\pi\)
\(11\)	−3.77444	+	2.01130i	−1.13804	+	0.606429i	−0.928486	−	0.371368i	\(-0.878889\pi\)
							−0.209552	+	0.977798i	\(0.567200\pi\)
\(13\)	2.12392	−	3.79708i	0.589071	−	1.05312i	−0.401466	−	0.915874i	\(-0.631499\pi\)
							0.990537	−	0.137247i	\(-0.0438253\pi\)
\(17\)	0.177431	−	1.23406i	0.0430333	−	0.299303i	−0.956926	−	0.290330i	\(-0.906235\pi\)
							0.999960	−	0.00897267i	\(-0.00285613\pi\)
\(19\)	1.16687	−	1.24056i	0.267698	−	0.284604i	−0.580525	−	0.814242i	\(-0.697153\pi\)
							0.848224	+	0.529638i	\(0.177672\pi\)
\(23\)	−2.31366	+	4.20083i	−0.482431	+	0.875934i
							\(29\)	−3.55261	−	4.04709i	−0.659703	−	0.751527i
\(31\)	−4.64787	−	7.56731i	−0.834783	−	1.35913i								−0.931472	−	0.363814i	\(-0.881475\pi\)
							0.0966888	−	0.995315i	\(-0.469175\pi\)
\(37\)	−5.55103	−	1.14838i	−0.912584	−	0.188793i	−0.277041	−	0.960858i	\(-0.589354\pi\)
							−0.635543	+	0.772065i	\(0.719224\pi\)
\(41\)	1.02981	+	2.25498i	0.160830	+	0.352168i	0.972841	−	0.231473i	\(-0.0743546\pi\)
							−0.812011	+	0.583642i	\(0.801627\pi\)
\(43\)	−4.70025	+	7.65257i	−0.716781	+	1.16701i	0.262585	+	0.964909i	\(0.415425\pi\)
							−0.979365	+	0.202097i	\(0.935224\pi\)
\(47\)	2.17181	−	9.51531i	0.316791	−	1.38795i	−0.526355	−	0.850265i	\(-0.676442\pi\)
							0.843145	−	0.537686i	\(-0.180701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.4	✓	3480
23.13	even	11	inner	667.2.s.a.335.4	yes	3480
29.20	even	7	inner	667.2.s.a.223.4	yes	3480
667.542	even	77	inner	667.2.s.a.542.4	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.4	✓	3480	1.1	even	1	trivial
667.2.s.a.223.4	yes	3480	29.20	even	7	inner
667.2.s.a.335.4	yes	3480	23.13	even	11	inner
667.2.s.a.542.4	yes	3480	667.542	even	77	inner