Embedded newform 845.2.t.e.188.1

• Label: 845.2.t.e.188.1
• 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			188.1
Root			\(-2.64975i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.188
Dual form			845.2.t.e.427.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29475 - 1.32488i) q^{2} +(-1.25278 + 0.335680i) q^{3} +(2.51060 + 4.34849i) q^{4} +(-1.81654 - 1.30391i) q^{5} +(3.31955 + 0.889471i) q^{6} +(-0.0561740 - 0.0972962i) q^{7} -8.00544i q^{8} +(-1.14131 + 0.658935i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29475	−	1.32488i	−1.62264	−	0.936830i	−0.986211	−	0.165491i	\(-0.947079\pi\)
							−0.636425	−	0.771338i	\(-0.719588\pi\)
\(3\)	−1.25278	+	0.335680i	−0.723291	+	0.193805i	−0.601640	−	0.798768i	\(-0.705486\pi\)
							−0.121651	+	0.992573i	\(0.538819\pi\)
\(5\)	−1.81654	−	1.30391i	−0.812382	−	0.583126i
							\(7\)	−0.0561740	−	0.0972962i	−0.0212318	−	0.0367745i	0.855214	−	0.518275i	\(-0.173425\pi\)
−0.876446	+	0.481500i	\(0.840092\pi\)
\(11\)	1.78976	−	0.479564i	0.539632	−	0.144594i	0.0212994	−	0.999773i	\(-0.493220\pi\)
							0.518332	+	0.855179i	\(0.326553\pi\)
\(13\)		0			0
							\(17\)	0.706500	−	2.63669i	0.171351	−	0.639492i	−0.825793	−	0.563973i	\(-0.809272\pi\)
0.997144	−	0.0755186i	\(-0.0240612\pi\)
\(19\)	−1.80138	+	6.72284i	−0.413265	+	1.54233i	0.375021	+	0.927016i	\(0.377636\pi\)
							−0.788286	+	0.615309i	\(0.789031\pi\)
\(23\)	0.831519	+	3.10327i	0.173384	+	0.647077i	0.996821	+	0.0796701i	\(0.0253867\pi\)
							−0.823438	+	0.567407i	\(0.807947\pi\)
\(29\)	4.03134	+	2.32749i	0.748601	+	0.432205i	0.825188	−	0.564858i	\(-0.191069\pi\)
							−0.0765874	+	0.997063i	\(0.524402\pi\)
\(31\)	0.624367	−	0.624367i	0.112140	−	0.112140i	−0.648810	−	0.760950i	\(-0.724733\pi\)
							0.760950	+	0.648810i	\(0.224733\pi\)
\(37\)	0.737435	−	1.27728i	0.121234	−	0.209983i	−0.799021	−	0.601303i	\(-0.794648\pi\)
							0.920254	+	0.391321i	\(0.127982\pi\)
\(41\)	−1.40424	−	5.24069i	−0.219305	−	0.818458i	−0.984606	−	0.174786i	\(-0.944077\pi\)
							0.765301	−	0.643672i	\(-0.222590\pi\)
\(43\)	−3.76415	−	1.00860i	−0.574027	−	0.153810i	−0.0398840	−	0.999204i	\(-0.512699\pi\)
							−0.534143	+	0.845394i	\(0.679366\pi\)
\(47\)	0.345095			0.0503372			0.0251686	−	0.999683i	\(-0.491988\pi\)
							0.0251686	+	0.999683i	\(0.491988\pi\)