Embedded newform 225.3.x.a.52.2

• Label: 225.3.x.a.52.2
• Level: $225$
• Weight: $3$
• Character: 225.52
• Analytic conductor: $6.131$
• Analytic rank: $0$
• Dimension: $928$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13080594811\)
Analytic rank:	\(0\)
Dimension:	\(928\)
Relative dimension:	\(58\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			52.2
Character	\(\chi\)	\(=\)	225.52
Dual form			225.3.x.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.39531 + 2.95797i) q^{2} +(2.99721 + 0.129320i) q^{3} +(-2.18039 - 10.2579i) q^{4} +(0.996446 + 4.89970i) q^{5} +(-7.56179 + 8.55589i) q^{6} +(-6.10048 + 1.63462i) q^{7} +(21.9999 + 11.2095i) q^{8} +(8.96655 + 0.775200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 928 q - 8 q^{2} - 14 q^{3} - 10 q^{4} - 8 q^{5} - 12 q^{6} - 8 q^{7} - 16 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.39531	+	2.95797i	−1.19766	+	1.47898i	−0.359855	+	0.933008i	\(0.617174\pi\)
							−0.837802	+	0.545975i	\(0.816159\pi\)
\(3\)	2.99721	+	0.129320i	0.999070	+	0.0431068i
							\(5\)	0.996446	+	4.89970i	0.199289	+	0.979941i
\(7\)	−6.10048	+	1.63462i	−0.871497	+	0.233517i								−0.666735	−	0.745295i	\(-0.732309\pi\)
							−0.204762	+	0.978812i	\(0.565642\pi\)
\(11\)	−0.194216	−	1.84784i	−0.0176560	−	0.167985i	0.982141	−	0.188148i	\(-0.0602484\pi\)
							−0.999797	+	0.0201623i	\(0.993582\pi\)
\(13\)	8.33222	+	10.2894i	0.640940	+	0.791495i	0.989305	−	0.145864i	\(-0.0465962\pi\)
							−0.348365	+	0.937359i	\(0.613263\pi\)
\(17\)	−9.21873	+	18.0928i	−0.542278	+	1.06428i	0.443507	+	0.896271i	\(0.353734\pi\)
							−0.985785	+	0.168010i	\(0.946266\pi\)
\(19\)	−29.6645	+	9.63858i	−1.56129	+	0.507294i	−0.957152	−	0.289587i	\(-0.906482\pi\)
							−0.604137	+	0.796880i	\(0.706482\pi\)
\(23\)	7.93508	−	3.04599i	0.345004	−	0.132434i	−0.179695	−	0.983722i	\(-0.557511\pi\)
							0.524699	+	0.851288i	\(0.324178\pi\)
\(29\)	−31.2337	+	28.1230i	−1.07702	+	0.969757i	−0.999629	−	0.0272433i	\(-0.991327\pi\)
							−0.0773956	+	0.997000i	\(0.524660\pi\)
\(31\)	30.6413	−	34.0306i	0.988428	−	1.09776i	−0.00677940	−	0.999977i	\(-0.502158\pi\)
							0.995208	−	0.0977838i	\(-0.0311754\pi\)
\(37\)	−5.08518	−	32.1066i	−0.137437	−	0.867745i	−0.956008	−	0.293339i	\(-0.905233\pi\)
							0.818571	−	0.574405i	\(-0.194767\pi\)
\(41\)	1.42560	−	13.5637i	0.0347707	−	0.330821i	−0.963284	−	0.268483i	\(-0.913478\pi\)
							0.998055	−	0.0623379i	\(-0.0198556\pi\)
\(43\)	6.34971	+	23.6975i	0.147668	+	0.551104i	0.999622	+	0.0274876i	\(0.00875068\pi\)
							−0.851954	+	0.523616i	\(0.824583\pi\)
\(47\)	0.756141	−	14.4280i	0.0160881	−	0.306979i	−0.978719	−	0.205204i	\(-0.934214\pi\)
							0.994807	−	0.101776i	\(-0.0324524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.x.a.52.2	yes	928
9.4	even	3	inner	225.3.x.a.202.2	yes	928
25.13	odd	20	inner	225.3.x.a.88.2	yes	928
225.13	odd	60	inner	225.3.x.a.13.2	✓	928

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.3.x.a.13.2	✓	928	225.13	odd	60	inner
225.3.x.a.52.2	yes	928	1.1	even	1	trivial
225.3.x.a.88.2	yes	928	25.13	odd	20	inner
225.3.x.a.202.2	yes	928	9.4	even	3	inner