Embedded newform 845.2.t.e.418.1

• Label: 845.2.t.e.418.1
• Level: $845$
• Weight: $2$
• Character: 845.418
• 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			418.1
Root			\(2.25081i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.418
Dual form			845.2.t.e.657.1

$q$-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 \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

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