Embedded newform 225.4.u.a.4.11

• Label: 225.4.u.a.4.11
• 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			4.11
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.939504 - 4.42002i) q^{2} +(-1.03009 + 5.09303i) q^{3} +(-11.3455 + 5.05136i) q^{4} +(9.01568 + 6.61193i) q^{5} +(23.4791 - 0.231910i) q^{6} +(13.1856 - 7.61268i) q^{7} +(11.7378 + 16.1557i) q^{8} +(-24.8778 - 10.4925i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.939504	−	4.42002i	−0.332165	−	1.56271i	−0.754517	−	0.656281i	\(-0.772129\pi\)
							0.422352	−	0.906432i	\(-0.361205\pi\)
\(3\)	−1.03009	+	5.09303i	−0.198241	+	0.980153i
							\(5\)	9.01568	+	6.61193i	0.806387	+	0.591389i
\(7\)	13.1856	−	7.61268i	0.711953	−	0.411046i								−0.0998308	−	0.995004i	\(-0.531830\pi\)
							0.811784	+	0.583958i	\(0.198497\pi\)
\(11\)	−21.4888	+	4.56759i	−0.589011	+	0.125198i	−0.492769	−	0.870160i	\(-0.664015\pi\)
							−0.0962413	+	0.995358i	\(0.530682\pi\)
\(13\)	−13.5454	+	63.7259i	−0.288985	+	1.35957i	0.558833	+	0.829280i	\(0.311249\pi\)
							−0.847818	+	0.530287i	\(0.822084\pi\)
\(17\)	1.96104	+	2.69913i	0.0279777	+	0.0385080i	0.822777	−	0.568365i	\(-0.192424\pi\)
							−0.794799	+	0.606873i	\(0.792424\pi\)
\(19\)	3.72704	−	2.70786i	0.0450022	−	0.0326960i	−0.565057	−	0.825052i	\(-0.691146\pi\)
							0.610059	+	0.792356i	\(0.291146\pi\)
\(23\)	−27.9452	+	25.1620i	−0.253347	+	0.228114i	−0.786011	−	0.618212i	\(-0.787857\pi\)
							0.532665	+	0.846326i	\(0.321191\pi\)
\(29\)	−6.82316	+	64.9180i	−0.0436907	+	0.415689i	0.950715	+	0.310065i	\(0.100351\pi\)
							−0.994406	+	0.105624i	\(0.966316\pi\)
\(31\)	18.6032	+	176.997i	0.107782	+	1.02547i	0.906050	+	0.423171i	\(0.139083\pi\)
							−0.798268	+	0.602302i	\(0.794250\pi\)
\(37\)	−337.350	+	109.612i	−1.49892	+	0.487028i	−0.939702	−	0.341994i	\(-0.888898\pi\)
							−0.559216	+	0.829022i	\(0.688898\pi\)
\(41\)	197.967	+	42.0792i	0.754080	+	0.160285i	0.568885	−	0.822417i	\(-0.307375\pi\)
							0.185195	+	0.982702i	\(0.440708\pi\)
\(43\)	−90.8647	+	52.4608i	−0.322250	+	0.186051i	−0.652395	−	0.757879i	\(-0.726236\pi\)
							0.330145	+	0.943930i	\(0.392902\pi\)
\(47\)	−18.3570	−	1.92940i	−0.0569713	−	0.00598792i	0.0759999	−	0.997108i	\(-0.475785\pi\)
							−0.132971	+	0.991120i	\(0.542452\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.11	✓	704
9.7	even	3	inner	225.4.u.a.79.11	yes	704
25.19	even	10	inner	225.4.u.a.94.11	yes	704
225.169	even	30	inner	225.4.u.a.169.11	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.11	✓	704	1.1	even	1	trivial
225.4.u.a.79.11	yes	704	9.7	even	3	inner
225.4.u.a.94.11	yes	704	25.19	even	10	inner
225.4.u.a.169.11	yes	704	225.169	even	30	inner