Embedded newform 845.2.t.e.427.3

• Label: 845.2.t.e.427.3
• Level: $845$
• Weight: $2$
• Character: 845.427
• 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			427.3
Root			\(0.493902i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.427
Dual form			845.2.t.e.188.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.427732 + 0.246951i) q^{2} +(0.908353 + 0.243392i) q^{3} +(-0.878030 + 1.52079i) q^{4} +(0.284413 + 2.21791i) q^{5} +(-0.448637 + 0.120212i) q^{6} +(-1.83775 + 3.18307i) q^{7} -1.85513i q^{8} +(-1.83221 - 1.05783i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.427732	+	0.246951i	−0.302452	+	0.174621i	−0.643544	−	0.765409i	\(-0.722537\pi\)
							0.341092	+	0.940030i	\(0.389203\pi\)
\(3\)	0.908353	+	0.243392i	0.524438	+	0.140523i	0.511318	−	0.859391i	\(-0.329157\pi\)
							0.0131191	+	0.999914i	\(0.495824\pi\)
\(5\)	0.284413	+	2.21791i	0.127193	+	0.991878i
							\(7\)	−1.83775	+	3.18307i	−0.694603	+	1.20309i	0.275712	+	0.961240i	\(0.411086\pi\)
−0.970314	+	0.241847i	\(0.922247\pi\)
\(11\)	−0.664257	−	0.177987i	−0.200281	−	0.0536651i	0.157284	−	0.987553i	\(-0.449726\pi\)
							−0.357565	+	0.933888i	\(0.616393\pi\)
\(13\)		0			0
							\(17\)	−0.614565	−	2.29359i	−0.149054	−	0.556277i	−0.999541	−	0.0302815i	\(-0.990360\pi\)
0.850487	−	0.525995i	\(-0.176307\pi\)
\(19\)	1.41763	+	5.29067i	0.325227	+	1.21376i	0.914084	+	0.405526i	\(0.132912\pi\)
							−0.588857	+	0.808238i	\(0.700422\pi\)
\(23\)	−0.350507	+	1.30811i	−0.0730858	+	0.272760i	−0.992792	−	0.119846i	\(-0.961760\pi\)
							0.919707	+	0.392606i	\(0.128426\pi\)
\(29\)	8.24134	−	4.75814i	1.53038	−	0.883564i	0.531034	−	0.847350i	\(-0.321803\pi\)
							0.999344	−	0.0362142i	\(-0.0115299\pi\)
\(31\)	−4.81595	−	4.81595i	−0.864970	−	0.864970i	0.126940	−	0.991910i	\(-0.459484\pi\)
							−0.991910	+	0.126940i	\(0.959484\pi\)
\(37\)	0.917615	+	1.58936i	0.150855	+	0.261289i	0.931542	−	0.363634i	\(-0.118464\pi\)
							−0.780687	+	0.624922i	\(0.785131\pi\)
\(41\)	−0.143350	+	0.534988i	−0.0223874	+	0.0835510i	−0.976216	−	0.216801i	\(-0.930438\pi\)
							0.953828	+	0.300352i	\(0.0971043\pi\)
\(43\)	2.09285	−	0.560778i	0.319157	−	0.0855178i	−0.0956841	−	0.995412i	\(-0.530504\pi\)
							0.414841	+	0.909894i	\(0.363837\pi\)
\(47\)	3.80918			0.555626			0.277813	−	0.960635i	\(-0.410390\pi\)
							0.277813	+	0.960635i	\(0.410390\pi\)