Embedded newform 825.2.ct.c.707.4

• Label: 825.2.ct.c.707.4
• Level: $825$
• Weight: $2$
• Character: 825.707
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.ct (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(32\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			707.4
Character	\(\chi\)	\(=\)	825.707
Dual form			825.2.ct.c.818.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.372338 + 2.35085i) q^{2} +(1.70152 + 0.323798i) q^{3} +(-3.48576 - 1.13259i) q^{4} +(-1.39474 + 3.87945i) q^{6} +(-3.57328 + 1.82068i) q^{7} +(1.79930 - 3.53132i) q^{8} +(2.79031 + 1.10189i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\)	\(376\)	\(551\)	\(727\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.372338	+	2.35085i	−0.263283	+	1.66230i	0.401945	+	0.915664i	\(0.368334\pi\)
							−0.665228	+	0.746640i	\(0.731666\pi\)
\(3\)	1.70152	+	0.323798i	0.982370	+	0.186945i
							\(5\)		0			0
\(7\)	−3.57328	+	1.82068i	−1.35057	+	0.688151i								−0.971461	−	0.237200i	\(-0.923770\pi\)
							−0.379111	+	0.925351i	\(0.623770\pi\)
\(11\)	2.27830	−	2.41026i	0.686934	−	0.726720i
							\(13\)	−0.780179	+	4.92586i	−0.216383	+	1.36619i	0.605189	+	0.796082i	\(0.293097\pi\)
−0.821572	+	0.570105i	\(0.806903\pi\)
\(17\)	−5.68785	+	0.900868i	−1.37951	+	0.218493i	−0.801722	−	0.597697i	\(-0.796083\pi\)
							−0.577785	+	0.816189i	\(0.696083\pi\)
\(19\)	−2.98920	+	0.971249i	−0.685769	+	0.222820i	−0.631119	−	0.775686i	\(-0.717404\pi\)
							−0.0546496	+	0.998506i	\(0.517404\pi\)
\(23\)	−2.49175	−	2.49175i	−0.519566	−	0.519566i	0.397874	−	0.917440i	\(-0.369748\pi\)
							−0.917440	+	0.397874i	\(0.869748\pi\)
\(29\)	−1.66998	+	5.13968i	−0.310108	+	0.954414i	0.667614	+	0.744508i	\(0.267316\pi\)
							−0.977722	+	0.209906i	\(0.932684\pi\)
\(31\)	−1.67071	+	1.21384i	−0.300068	+	0.218012i	−0.727623	−	0.685977i	\(-0.759375\pi\)
							0.427555	+	0.903989i	\(0.359375\pi\)
\(37\)	0.387104	−	0.197239i	0.0636395	−	0.0324260i	−0.421881	−	0.906651i	\(-0.638630\pi\)
							0.485521	+	0.874225i	\(0.338630\pi\)
\(41\)	6.07055	−	1.97244i	0.948061	−	0.308044i	0.206133	−	0.978524i	\(-0.433912\pi\)
							0.741927	+	0.670480i	\(0.233912\pi\)
\(43\)	−1.94060	+	1.94060i	−0.295938	+	0.295938i	−0.839421	−	0.543482i	\(-0.817105\pi\)
							0.543482	+	0.839421i	\(0.317105\pi\)
\(47\)	−4.23353	−	2.15709i	−0.617523	−	0.314644i	0.117099	−	0.993120i	\(-0.462641\pi\)
							−0.734622	+	0.678476i	\(0.762641\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.ct.c.707.4	yes	256
3.2	odd	2	inner	825.2.ct.c.707.30	yes	256
5.2	odd	4	inner	825.2.ct.c.443.4	yes	256
5.3	odd	4	inner	825.2.ct.c.443.29	yes	256
5.4	even	2	inner	825.2.ct.c.707.29	yes	256
11.4	even	5	inner	825.2.ct.c.257.3	✓	256
15.2	even	4	inner	825.2.ct.c.443.30	yes	256
15.8	even	4	inner	825.2.ct.c.443.3	yes	256
15.14	odd	2	inner	825.2.ct.c.707.3	yes	256
33.26	odd	10	inner	825.2.ct.c.257.29	yes	256
55.4	even	10	inner	825.2.ct.c.257.30	yes	256
55.37	odd	20	inner	825.2.ct.c.818.3	yes	256
55.48	odd	20	inner	825.2.ct.c.818.30	yes	256
165.59	odd	10	inner	825.2.ct.c.257.4	yes	256
165.92	even	20	inner	825.2.ct.c.818.29	yes	256
165.158	even	20	inner	825.2.ct.c.818.4	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.ct.c.257.3	✓	256	11.4	even	5	inner
825.2.ct.c.257.4	yes	256	165.59	odd	10	inner
825.2.ct.c.257.29	yes	256	33.26	odd	10	inner
825.2.ct.c.257.30	yes	256	55.4	even	10	inner
825.2.ct.c.443.3	yes	256	15.8	even	4	inner
825.2.ct.c.443.4	yes	256	5.2	odd	4	inner
825.2.ct.c.443.29	yes	256	5.3	odd	4	inner
825.2.ct.c.443.30	yes	256	15.2	even	4	inner
825.2.ct.c.707.3	yes	256	15.14	odd	2	inner
825.2.ct.c.707.4	yes	256	1.1	even	1	trivial
825.2.ct.c.707.29	yes	256	5.4	even	2	inner
825.2.ct.c.707.30	yes	256	3.2	odd	2	inner
825.2.ct.c.818.3	yes	256	55.37	odd	20	inner
825.2.ct.c.818.4	yes	256	165.158	even	20	inner
825.2.ct.c.818.29	yes	256	165.92	even	20	inner
825.2.ct.c.818.30	yes	256	55.48	odd	20	inner