Embedded newform 845.2.t.e.657.1

• Label: 845.2.t.e.657.1
• 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			657.1
Root			\(-2.25081i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.657
Dual form			845.2.t.e.418.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94926 - 1.12540i) q^{2} +(0.514229 + 1.91913i) q^{3} +(1.53307 + 2.65535i) q^{4} +(-0.247944 + 2.22228i) q^{5} +(1.15743 - 4.31958i) q^{6} +(-0.638592 - 1.10607i) q^{7} -2.39966i q^{8} +(-0.820542 + 0.473740i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94926	−	1.12540i	−1.37833	−	0.795780i	−0.386373	−	0.922342i	\(-0.626272\pi\)
							−0.991959	+	0.126562i	\(0.959606\pi\)
\(3\)	0.514229	+	1.91913i	0.296890	+	1.10801i	0.939705	+	0.341987i	\(0.111100\pi\)
							−0.642815	+	0.766022i	\(0.722233\pi\)
\(5\)	−0.247944	+	2.22228i	−0.110884	+	0.993833i
							\(7\)	−0.638592	−	1.10607i	−0.241365	−	0.418057i	0.719738	−	0.694245i	\(-0.244262\pi\)
−0.961103	+	0.276189i	\(0.910928\pi\)
\(11\)	−1.41373	−	5.27612i	−0.426257	−	1.59081i	−0.761163	−	0.648560i	\(-0.775371\pi\)
							0.334907	−	0.942251i	\(-0.391295\pi\)
\(13\)		0			0
							\(17\)	−3.11179	−	0.833802i	−0.754721	−	0.202227i	−0.139110	−	0.990277i	\(-0.544424\pi\)
−0.615611	+	0.788050i	\(0.711091\pi\)
\(19\)	−1.17707	−	0.315395i	−0.270039	−	0.0723567i	0.121259	−	0.992621i	\(-0.461307\pi\)
							−0.391297	+	0.920264i	\(0.627974\pi\)
\(23\)	−0.160006	+	0.0428736i	−0.0333637	+	0.00893976i	−0.275462	−	0.961312i	\(-0.588831\pi\)
							0.242099	+	0.970252i	\(0.422164\pi\)
\(29\)	−8.41068	−	4.85591i	−1.56182	−	0.901719i	−0.997073	−	0.0764575i	\(-0.975639\pi\)
							−0.564751	−	0.825262i	\(-0.691028\pi\)
\(31\)	−0.233305	−	0.233305i	−0.0419027	−	0.0419027i	0.685845	−	0.727748i	\(-0.259433\pi\)
							−0.727748	+	0.685845i	\(0.759433\pi\)
\(37\)	0.660816	−	1.14457i	0.108638	−	0.188166i	−0.806581	−	0.591124i	\(-0.798685\pi\)
							0.915219	+	0.402958i	\(0.132018\pi\)
\(41\)	0.483595	−	0.129579i	0.0755249	−	0.0202368i	−0.220859	−	0.975306i	\(-0.570886\pi\)
							0.296384	+	0.955069i	\(0.404219\pi\)
\(43\)	1.72444	−	6.43569i	0.262974	−	0.981434i	−0.700504	−	0.713648i	\(-0.747041\pi\)
							0.963479	−	0.267786i	\(-0.0862919\pi\)
\(47\)	−3.20027			−0.466808			−0.233404	−	0.972380i	\(-0.574986\pi\)
							−0.233404	+	0.972380i	\(0.574986\pi\)