Embedded newform 225.4.u.a.169.17

• Label: 225.4.u.a.169.17
• Level: $225$
• Weight: $4$
• Character: 225.169
• 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			169.17
Character	\(\chi\)	\(=\)	225.169
Dual form			225.4.u.a.4.17

$q$-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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.17	✓	704	225.4	even	30	inner
225.4.u.a.79.17	yes	704	25.4	even	10	inner
225.4.u.a.94.17	yes	704	9.4	even	3	inner
225.4.u.a.169.17	yes	704	1.1	even	1	trivial