Embedded newform 845.2.v.a.389.46

• Label: 845.2.v.a.389.46
• Level: $845$
• Weight: $2$
• Character: 845.389
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $1056$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(1056\)
Relative dimension:	\(88\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			389.46
Character	\(\chi\)	\(=\)	845.389
Dual form			845.2.v.a.454.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0655812 + 0.0580999i) q^{2} +(0.0139291 + 0.00528261i) q^{3} +(-0.240148 - 1.97780i) q^{4} +(1.39180 - 1.75011i) q^{5} +(0.000606567 + 0.00115572i) q^{6} +(-2.84284 + 4.11857i) q^{7} +(0.198703 - 0.287871i) q^{8} +(-2.24537 - 1.98922i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1056 q - 106 q^{4} - 13 q^{5} - 26 q^{6} + 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{19}{26}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0655812	+	0.0580999i	0.0463729	+	0.0410828i	0.685991	−	0.727610i	\(-0.259369\pi\)
							−0.639618	+	0.768693i	\(0.720907\pi\)
\(3\)	0.0139291	+	0.00528261i	0.00804196	+	0.00304991i	0.358623	−	0.933483i	\(-0.383247\pi\)
							−0.350581	+	0.936533i	\(0.614016\pi\)
\(5\)	1.39180	−	1.75011i	0.622433	−	0.782673i
							\(7\)	−2.84284	+	4.11857i	−1.07449	+	1.55667i	−0.269861	+	0.962899i	\(0.586978\pi\)
−0.804632	+	0.593773i	\(0.797638\pi\)
\(11\)	−0.433313	−	0.489109i	−0.130649	−	0.147472i	0.679546	−	0.733633i	\(-0.262177\pi\)
							−0.810194	+	0.586161i	\(0.800638\pi\)
\(13\)	−2.77555	+	2.30137i	−0.769800	+	0.638285i
							\(17\)	−2.82236	−	1.94813i	−0.684523	−	0.472492i	0.174348	−	0.984684i	\(-0.444218\pi\)
−0.858871	+	0.512192i	\(0.828834\pi\)
\(19\)			5.04860i			1.15823i	0.815247	+	0.579114i	\(0.196601\pi\)
							−0.815247	+	0.579114i	\(0.803399\pi\)
\(23\)		−	4.31530i		−	0.899802i	−0.893078	−	0.449901i	\(-0.851459\pi\)
							0.893078	−	0.449901i	\(-0.148541\pi\)
\(29\)	−3.06966	−	2.71949i	−0.570022	−	0.504996i	0.328158	−	0.944623i	\(-0.393572\pi\)
							−0.898180	+	0.439627i	\(0.855111\pi\)
\(31\)	2.55610	+	4.87024i	0.459089	+	0.874721i	0.999446	+	0.0332848i	\(0.0105969\pi\)
							−0.540357	+	0.841436i	\(0.681711\pi\)
\(37\)	−5.59787	+	2.93799i	−0.920285	+	0.483003i	−0.857187	−	0.515005i	\(-0.827790\pi\)
							−0.0630974	+	0.998007i	\(0.520098\pi\)
\(41\)	6.62418	+	2.51222i	1.03452	+	0.392343i	0.812662	−	0.582736i	\(-0.198018\pi\)
							0.221861	+	0.975078i	\(0.428787\pi\)
\(43\)	−3.51711	+	6.70129i	−0.536354	+	1.02194i	0.454971	+	0.890506i	\(0.349650\pi\)
							−0.991325	+	0.131431i	\(0.958043\pi\)
\(47\)	0.796867	−	6.56279i	0.116235	−	0.957281i	−0.811580	−	0.584241i	\(-0.801392\pi\)
							0.927815	−	0.373040i	\(-0.121685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.v.a.389.46	yes	1056
5.4	even	2	inner	845.2.v.a.389.43	✓	1056
169.116	even	26	inner	845.2.v.a.454.43	yes	1056
845.454	even	26	inner	845.2.v.a.454.46	yes	1056

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.v.a.389.43	✓	1056	5.4	even	2	inner
845.2.v.a.389.46	yes	1056	1.1	even	1	trivial
845.2.v.a.454.43	yes	1056	169.116	even	26	inner
845.2.v.a.454.46	yes	1056	845.454	even	26	inner