Embedded newform 845.2.v.a.389.52

• Label: 845.2.v.a.389.52
• 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			389.52
Character	\(\chi\)	\(=\)	845.389
Dual form			845.2.v.a.454.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.379492 + 0.336200i) q^{2} +(0.878070 + 0.333008i) q^{3} +(-0.210090 - 1.73025i) q^{4} +(-0.877226 - 2.05681i) q^{5} +(0.221263 + 0.421581i) q^{6} +(0.412702 - 0.597902i) q^{7} +(1.07799 - 1.56175i) q^{8} +(-1.58542 - 1.40456i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1056 q - 106 q^{4} - 13 q^{5} - 26 q^{6} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.379492	+	0.336200i	0.268341	+	0.237730i	0.786553	−	0.617523i	\(-0.211864\pi\)
							−0.518212	+	0.855252i	\(0.673402\pi\)
\(3\)	0.878070	+	0.333008i	0.506954	+	0.192262i	0.594803	−	0.803872i	\(-0.297230\pi\)
							−0.0878492	+	0.996134i	\(0.527999\pi\)
\(5\)	−0.877226	−	2.05681i	−0.392307	−	0.919834i
							\(7\)	0.412702	−	0.597902i	0.155987	−	0.225986i	−0.737197	−	0.675678i	\(-0.763851\pi\)
0.893184	+	0.449692i	\(0.148466\pi\)
\(11\)	1.39585	+	1.57559i	0.420864	+	0.475058i	0.920255	−	0.391319i	\(-0.127981\pi\)
							−0.499391	+	0.866377i	\(0.666443\pi\)
\(13\)	−2.00144	+	2.99904i	−0.555099	+	0.831784i
							\(17\)	2.77349	+	1.91440i	0.672671	+	0.464311i	0.854796	−	0.518964i	\(-0.173682\pi\)
−0.182125	+	0.983275i	\(0.558298\pi\)
\(19\)		−	8.01083i		−	1.83781i	−0.394478	−	0.918905i	\(-0.629075\pi\)
							0.394478	−	0.918905i	\(-0.370925\pi\)
\(23\)		−	4.41327i		−	0.920231i	−0.887859	−	0.460115i	\(-0.847808\pi\)
							0.887859	−	0.460115i	\(-0.152192\pi\)
\(29\)	−2.46242	−	2.18151i	−0.457259	−	0.405096i	0.402802	−	0.915287i	\(-0.368036\pi\)
							−0.860061	+	0.510191i	\(0.829575\pi\)
\(31\)	1.34164	+	2.55629i	0.240966	+	0.459123i	0.975664	−	0.219272i	\(-0.0703682\pi\)
							−0.734698	+	0.678395i	\(0.762676\pi\)
\(37\)	1.47192	−	0.772524i	0.241982	−	0.127002i	−0.339373	−	0.940652i	\(-0.610215\pi\)
							0.581355	+	0.813650i	\(0.302523\pi\)
\(41\)	−11.1917	−	4.24444i	−1.74784	−	0.662870i	−0.747844	−	0.663875i	\(-0.768911\pi\)
							−0.999999	+	0.00100509i	\(0.999680\pi\)
\(43\)	2.74251	−	5.22542i	0.418229	−	0.796869i	−0.581633	−	0.813451i	\(-0.697586\pi\)
							0.999863	+	0.0165821i	\(0.00527849\pi\)
\(47\)	0.556201	−	4.58072i	0.0811302	−	0.668167i	−0.894239	−	0.447589i	\(-0.852283\pi\)
							0.975369	−	0.220578i	\(-0.0707943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.v.a.389.52	yes	1056
5.4	even	2	inner	845.2.v.a.389.37	✓	1056
169.116	even	26	inner	845.2.v.a.454.37	yes	1056
845.454	even	26	inner	845.2.v.a.454.52	yes	1056

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.v.a.389.37	✓	1056	5.4	even	2	inner
845.2.v.a.389.52	yes	1056	1.1	even	1	trivial
845.2.v.a.454.37	yes	1056	169.116	even	26	inner
845.2.v.a.454.52	yes	1056	845.454	even	26	inner