Embedded newform 845.2.t.e.188.2

• Label: 845.2.t.e.188.2
• Level: $845$
• Weight: $2$
• Character: 845.188
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $20$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.t (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 26*x^18 + 279*x^16 + 1604*x^14 + 5353*x^12 + 10466*x^10 + 11441*x^8 + 6176*x^6 + 1263*x^4 + 78*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			188.2
Root			\(-1.83163i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.188
Dual form			845.2.t.e.427.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58624 - 0.915816i) q^{2} +(1.91432 - 0.512942i) q^{3} +(0.677439 + 1.17336i) q^{4} +(1.69810 + 1.45480i) q^{5} +(-3.50634 - 0.939520i) q^{6} +(1.76945 + 3.06478i) q^{7} +1.18163i q^{8} +(0.803451 - 0.463873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58624	−	0.915816i	−1.12164	−	0.647580i	−0.179822	−	0.983699i	\(-0.557552\pi\)
							−0.941819	+	0.336119i	\(0.890885\pi\)
\(3\)	1.91432	−	0.512942i	1.10524	−	0.296147i	0.340341	−	0.940302i	\(-0.389458\pi\)
							0.764895	+	0.644155i	\(0.222791\pi\)
\(5\)	1.69810	+	1.45480i	0.759414	+	0.650608i
							\(7\)	1.76945	+	3.06478i	0.668790	+	1.15838i	0.978243	+	0.207464i	\(0.0665209\pi\)
−0.309453	+	0.950915i	\(0.600146\pi\)
\(11\)	−3.74209	+	1.00269i	−1.12828	+	0.302323i	−0.774231	−	0.632903i	\(-0.781863\pi\)
							−0.354053	+	0.935226i	\(0.615197\pi\)
\(13\)		0			0
							\(17\)	−0.524334	+	1.95684i	−0.127170	+	0.474603i	−0.999908	−	0.0135853i	\(-0.995676\pi\)
0.872738	+	0.488189i	\(0.162342\pi\)
\(19\)	0.139057	−	0.518968i	0.0319019	−	0.119059i	−0.948139	−	0.317857i	\(-0.897037\pi\)
							0.980041	+	0.198797i	\(0.0637036\pi\)
\(23\)	−0.0788026	−	0.294095i	−0.0164315	−	0.0613231i	0.957223	−	0.289350i	\(-0.0934392\pi\)
							−0.973655	+	0.228027i	\(0.926773\pi\)
\(29\)	1.71273	+	0.988843i	0.318045	+	0.183624i	0.650521	−	0.759488i	\(-0.274551\pi\)
							−0.332476	+	0.943112i	\(0.607884\pi\)
\(31\)	−4.13563	+	4.13563i	−0.742781	+	0.742781i	−0.973112	−	0.230331i	\(-0.926019\pi\)
							0.230331	+	0.973112i	\(0.426019\pi\)
\(37\)	2.70887	−	4.69189i	0.445335	−	0.771342i	−0.552741	−	0.833353i	\(-0.686418\pi\)
							0.998075	+	0.0620109i	\(0.0197514\pi\)
\(41\)	0.174136	+	0.649884i	0.0271955	+	0.101495i	0.978190	−	0.207714i	\(-0.0666024\pi\)
							−0.950994	+	0.309209i	\(0.899936\pi\)
\(43\)	8.51164	+	2.28069i	1.29801	+	0.347802i	0.840698	−	0.541504i	\(-0.182145\pi\)
							0.457314	+	0.889305i	\(0.348811\pi\)
\(47\)	−9.75201			−1.42248			−0.711238	−	0.702951i	\(-0.751865\pi\)
							−0.711238	+	0.702951i	\(0.751865\pi\)