Embedded newform 225.4.u.a.4.15

• Label: 225.4.u.a.4.15
• Level: $225$
• Weight: $4$
• Character: 225.4
• Analytic conductor: $13.275$
• Analytic rank: $0$
• Dimension: $704$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.u (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2754297513\)
Analytic rank:	\(0\)
Dimension:	\(704\)
Relative dimension:	\(88\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.15
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.822434 - 3.86925i) q^{2} +(-4.35066 - 2.84108i) q^{3} +(-6.98632 + 3.11051i) q^{4} +(9.64418 + 5.65595i) q^{5} +(-7.41470 + 19.1704i) q^{6} +(13.6923 - 7.90527i) q^{7} +(-0.819642 - 1.12814i) q^{8} +(10.8566 + 24.7211i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.822434	−	3.86925i	−0.290774	−	1.36799i	−0.844620	−	0.535366i	\(-0.820174\pi\)
							0.553846	−	0.832619i	\(-0.313160\pi\)
\(3\)	−4.35066	−	2.84108i	−0.837286	−	0.546766i
							\(5\)	9.64418	+	5.65595i	0.862602	+	0.505884i
\(7\)	13.6923	−	7.90527i	0.739316	−	0.426844i								−0.0825046	−	0.996591i	\(-0.526292\pi\)
							0.821821	+	0.569746i	\(0.192959\pi\)
\(11\)	47.4538	−	10.0866i	1.30071	−	0.276475i	0.495058	−	0.868860i	\(-0.335147\pi\)
							0.805655	+	0.592384i	\(0.201813\pi\)
\(13\)	10.8165	−	50.8875i	0.230765	−	1.08567i	−0.698316	−	0.715790i	\(-0.746067\pi\)
							0.929081	−	0.369876i	\(-0.120600\pi\)
\(17\)	−21.5701	−	29.6887i	−0.307736	−	0.423562i	0.626937	−	0.779070i	\(-0.284308\pi\)
							−0.934674	+	0.355507i	\(0.884308\pi\)
\(19\)	71.0880	−	51.6485i	0.858353	−	0.623630i	−0.0690835	−	0.997611i	\(-0.522007\pi\)
							0.927436	+	0.373981i	\(0.122007\pi\)
\(23\)	−12.6839	+	11.4207i	−0.114990	+	0.103538i	−0.724620	−	0.689148i	\(-0.757985\pi\)
							0.609630	+	0.792686i	\(0.291318\pi\)
\(29\)	14.8253	−	141.053i	0.0949307	−	0.903205i	−0.838610	−	0.544733i	\(-0.816631\pi\)
							0.933540	−	0.358473i	\(-0.116702\pi\)
\(31\)	29.2015	+	277.834i	0.169185	+	1.60969i	0.668800	+	0.743443i	\(0.266808\pi\)
							−0.499614	+	0.866248i	\(0.666525\pi\)
\(37\)	195.316	−	63.4622i	0.867833	−	0.281976i	0.158937	−	0.987289i	\(-0.449193\pi\)
							0.708896	+	0.705313i	\(0.249193\pi\)
\(41\)	−487.739	−	103.672i	−1.85786	−	0.394899i	−0.863806	−	0.503825i	\(-0.831926\pi\)
							−0.994050	+	0.108925i	\(0.965259\pi\)
\(43\)	−67.2915	+	38.8508i	−0.238648	+	0.137783i	−0.614555	−	0.788874i	\(-0.710664\pi\)
							0.375907	+	0.926657i	\(0.377331\pi\)
\(47\)	−488.449	−	51.3380i	−1.51591	−	0.159328i	−0.690278	−	0.723544i	\(-0.742512\pi\)
							−0.825627	+	0.564216i	\(0.809179\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.15	✓	704
9.7	even	3	inner	225.4.u.a.79.15	yes	704
25.19	even	10	inner	225.4.u.a.94.15	yes	704
225.169	even	30	inner	225.4.u.a.169.15	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.15	✓	704	1.1	even	1	trivial
225.4.u.a.79.15	yes	704	9.7	even	3	inner
225.4.u.a.94.15	yes	704	25.19	even	10	inner
225.4.u.a.169.15	yes	704	225.169	even	30	inner