Embedded newform 225.4.u.a.4.17

• Label: 225.4.u.a.4.17
• 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			4.17
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.801643 - 3.77143i) q^{2} +(5.10602 - 0.963611i) q^{3} +(-6.27271 + 2.79279i) q^{4} +(-2.58409 - 10.8776i) q^{5} +(-7.72740 - 18.4845i) q^{6} +(11.3837 - 6.57239i) q^{7} +(-2.56924 - 3.53626i) q^{8} +(25.1429 - 9.84043i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.801643	−	3.77143i	−0.283424	−	1.33340i	−0.857451	−	0.514566i	\(-0.827953\pi\)
							0.574027	−	0.818836i	\(-0.305380\pi\)
\(3\)	5.10602	−	0.963611i	0.982654	−	0.185447i
							\(5\)	−2.58409	−	10.8776i	−0.231128	−	0.972923i
\(7\)	11.3837	−	6.57239i	0.614663	−	0.354876i								−0.160125	−	0.987097i	\(-0.551190\pi\)
							0.774788	+	0.632221i	\(0.217857\pi\)
\(11\)	48.9308	−	10.4006i	1.34120	−	0.285081i	0.519241	−	0.854628i	\(-0.326215\pi\)
							0.821959	+	0.569547i	\(0.192881\pi\)
\(13\)	8.01697	−	37.7169i	0.171039	−	0.804675i	−0.806051	−	0.591847i	\(-0.798399\pi\)
							0.977090	−	0.212828i	\(-0.0682675\pi\)
\(17\)	58.1473	+	80.0329i	0.829576	+	1.14181i	0.988002	+	0.154442i	\(0.0493580\pi\)
							−0.158426	+	0.987371i	\(0.550642\pi\)
\(19\)	−57.9269	+	42.0864i	−0.699439	+	0.508172i	−0.879749	−	0.475438i	\(-0.842290\pi\)
							0.180310	+	0.983610i	\(0.442290\pi\)
\(23\)	−88.3239	+	79.5272i	−0.800731	+	0.720981i	−0.964087	−	0.265586i	\(-0.914435\pi\)
							0.163356	+	0.986567i	\(0.447768\pi\)
\(29\)	29.1601	−	277.440i	0.186720	−	1.77653i	−0.353923	−	0.935275i	\(-0.615152\pi\)
							0.540643	−	0.841252i	\(-0.318181\pi\)
\(31\)	−5.75958	−	54.7988i	−0.0333694	−	0.317489i	−0.998456	−	0.0555561i	\(-0.982307\pi\)
							0.965086	−	0.261933i	\(-0.0843598\pi\)
\(37\)	−262.122	+	85.1687i	−1.16467	+	0.378423i	−0.826649	−	0.562718i	\(-0.809756\pi\)
							−0.338016	+	0.941140i	\(0.609756\pi\)
\(41\)	223.886	+	47.5884i	0.852808	+	0.181270i	0.613520	−	0.789679i	\(-0.289753\pi\)
							0.239287	+	0.970949i	\(0.423086\pi\)
\(43\)	−452.097	+	261.018i	−1.60335	+	0.925696i	−0.612542	+	0.790438i	\(0.709853\pi\)
							−0.990810	+	0.135258i	\(0.956814\pi\)
\(47\)	221.112	+	23.2398i	0.686223	+	0.0721249i	0.441222	−	0.897398i	\(-0.354545\pi\)
							0.245001	+	0.969523i	\(0.421212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.17	✓	704
9.7	even	3	inner	225.4.u.a.79.17	yes	704
25.19	even	10	inner	225.4.u.a.94.17	yes	704
225.169	even	30	inner	225.4.u.a.169.17	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.17	✓	704	1.1	even	1	trivial
225.4.u.a.79.17	yes	704	9.7	even	3	inner
225.4.u.a.94.17	yes	704	25.19	even	10	inner
225.4.u.a.169.17	yes	704	225.169	even	30	inner