Embedded newform 845.2.t.e.657.2

• Label: 845.2.t.e.657.2
• Level: $845$
• Weight: $2$
• Character: 845.657
• 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			657.2
Root			\(-1.02262i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.657
Dual form			845.2.t.e.418.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.885613 - 0.511309i) q^{2} +(-0.721300 - 2.69193i) q^{3} +(-0.477126 - 0.826407i) q^{4} +(1.45744 + 1.69584i) q^{5} +(-0.737614 + 2.75281i) q^{6} +(0.481787 + 0.834479i) q^{7} +3.02107i q^{8} +(-4.12812 + 2.38337i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.885613	−	0.511309i	−0.626223	−	0.361550i	0.153065	−	0.988216i	\(-0.451086\pi\)
							−0.779288	+	0.626666i	\(0.784419\pi\)
\(3\)	−0.721300	−	2.69193i	−0.416443	−	1.55418i	−0.781929	−	0.623368i	\(-0.785764\pi\)
							0.365486	−	0.930817i	\(-0.380903\pi\)
\(5\)	1.45744	+	1.69584i	0.651785	+	0.758404i
							\(7\)	0.481787	+	0.834479i	0.182098	+	0.315404i	0.942595	−	0.333938i	\(-0.108378\pi\)
−0.760497	+	0.649342i	\(0.775044\pi\)
\(11\)	0.430490	+	1.60661i	0.129797	+	0.484411i	0.999965	−	0.00834492i	\(-0.00265630\pi\)
							−0.870168	+	0.492756i	\(0.835990\pi\)
\(13\)		0			0
							\(17\)	7.00342	+	1.87656i	1.69858	+	0.455133i	0.972581	−	0.232564i	\(-0.0747114\pi\)
0.725998	+	0.687697i	\(0.241378\pi\)
\(19\)	2.64041	+	0.707496i	0.605752	+	0.162311i	0.548642	−	0.836057i	\(-0.315145\pi\)
							0.0571095	+	0.998368i	\(0.481812\pi\)
\(23\)	3.72214	−	0.997344i	0.776120	−	0.207961i	0.151046	−	0.988527i	\(-0.451736\pi\)
							0.625073	+	0.780566i	\(0.285069\pi\)
\(29\)	0.253107	+	0.146132i	0.0470008	+	0.0271360i	0.523316	−	0.852139i	\(-0.324695\pi\)
							−0.476315	+	0.879274i	\(0.658028\pi\)
\(31\)	0.125649	+	0.125649i	0.0225673	+	0.0225673i	0.718300	−	0.695733i	\(-0.244920\pi\)
							−0.695733	+	0.718300i	\(0.744920\pi\)
\(37\)	−2.04061	+	3.53443i	−0.335474	+	0.581057i	−0.983576	−	0.180496i	\(-0.942230\pi\)
							0.648102	+	0.761553i	\(0.275563\pi\)
\(41\)	−6.69071	+	1.79277i	−1.04491	+	0.279984i	−0.740148	−	0.672444i	\(-0.765245\pi\)
							−0.304765	+	0.952427i	\(0.598578\pi\)
\(43\)	−2.05706	+	7.67707i	−0.313699	+	1.17074i	0.611495	+	0.791248i	\(0.290568\pi\)
							−0.925194	+	0.379494i	\(0.876098\pi\)
\(47\)	7.84582			1.14443			0.572215	−	0.820103i	\(-0.306084\pi\)
							0.572215	+	0.820103i	\(0.306084\pi\)