Embedded newform 845.2.w.a.116.6

• Label: 845.2.w.a.116.6
• Level: $845$
• Weight: $2$
• Character: 845.116
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $744$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.w (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(744\)
Relative dimension:	\(62\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			116.6
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.63215 - 1.84231i) q^{2} +(0.881350 + 2.32393i) q^{3} +(-0.489142 + 4.02845i) q^{4} +(0.239316 + 0.970942i) q^{5} +(2.84291 - 5.41672i) q^{6} +(3.13894 - 2.16665i) q^{7} +(4.16879 - 2.87751i) q^{8} +(-2.37833 + 2.10702i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{7}{26}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.63215	−	1.84231i	−1.15410	−	1.30271i	−0.945837	−	0.324641i	\(-0.894756\pi\)
							−0.208266	−	0.978072i	\(-0.566782\pi\)
\(3\)	0.881350	+	2.32393i	0.508848	+	1.34172i	0.905731	+	0.423852i	\(0.139322\pi\)
							−0.396884	+	0.917869i	\(0.629908\pi\)
\(5\)	0.239316	+	0.970942i	0.107025	+	0.434218i
							\(7\)	3.13894	−	2.16665i	1.18641	−	0.818918i	0.199451	−	0.979908i	\(-0.436084\pi\)
0.986956	+	0.160990i	\(0.0514687\pi\)
\(11\)	1.99996	−	2.25749i	0.603010	−	0.680658i	−0.365162	−	0.930944i	\(-0.618986\pi\)
							0.968172	+	0.250286i	\(0.0805248\pi\)
\(13\)	−0.902837	−	3.49069i	−0.250402	−	0.968142i
							\(17\)	−4.16193	−	6.02960i	−1.00942	−	1.46239i	−0.882405	−	0.470491i	\(-0.844077\pi\)
−0.127012	−	0.991901i	\(-0.540539\pi\)
\(19\)		−	7.01272i		−	1.60883i	−0.594069	−	0.804414i	\(-0.702479\pi\)
							0.594069	−	0.804414i	\(-0.297521\pi\)
\(23\)	1.99943			0.416911			0.208455	−	0.978032i	\(-0.433156\pi\)
							0.208455	+	0.978032i	\(0.433156\pi\)
\(29\)	−2.20666	+	1.95493i	−0.409767	+	0.363022i	−0.842681	−	0.538413i	\(-0.819024\pi\)
							0.432914	+	0.901435i	\(0.357485\pi\)
\(31\)	2.11227	−	4.02459i	0.379374	−	0.722837i	−0.618650	−	0.785667i	\(-0.712320\pi\)
							0.998024	+	0.0628294i	\(0.0200124\pi\)
\(37\)	−0.352611	+	0.671845i	−0.0579689	+	0.110451i	−0.912717	−	0.408593i	\(-0.866020\pi\)
							0.854748	+	0.519044i	\(0.173712\pi\)
\(41\)	−7.52829	+	2.85510i	−1.17572	+	0.445892i	−0.863613	−	0.504155i	\(-0.831804\pi\)
							−0.312108	+	0.950047i	\(0.601035\pi\)
\(43\)	3.18720	−	1.67277i	0.486043	−	0.255095i	−0.203856	−	0.979001i	\(-0.565347\pi\)
							0.689899	+	0.723906i	\(0.257655\pi\)
\(47\)	7.07689	−	0.859290i	1.03227	−	0.125340i	0.413172	−	0.910653i	\(-0.364421\pi\)
							0.619099	+	0.785313i	\(0.287498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.116.6	yes	744
169.51	even	26	inner	845.2.w.a.51.6	✓	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.6	✓	744	169.51	even	26	inner
845.2.w.a.116.6	yes	744	1.1	even	1	trivial