Embedded newform 225.4.u.a.4.2

• Label: 225.4.u.a.4.2
• 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			4.2
Character	\(\chi\)	\(=\)	225.4
Dual form			225.4.u.a.169.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12585 - 5.29669i) q^{2} +(-5.17983 + 0.411540i) q^{3} +(-19.4791 + 8.67264i) q^{4} +(6.78358 + 8.88724i) q^{5} +(8.01150 + 26.9726i) q^{6} +(-21.8175 + 12.5963i) q^{7} +(42.4038 + 58.3638i) q^{8} +(26.6613 - 4.26341i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12585	−	5.29669i	−0.398047	−	1.87266i	−0.481964	−	0.876191i	\(-0.660076\pi\)
							0.0839169	−	0.996473i	\(-0.473257\pi\)
\(3\)	−5.17983	+	0.411540i	−0.996859	+	0.0792009i
							\(5\)	6.78358	+	8.88724i	0.606741	+	0.794899i
\(7\)	−21.8175	+	12.5963i	−1.17803	+	0.680137i								−0.955558	−	0.294802i	\(-0.904746\pi\)
							−0.222473	+	0.974939i	\(0.571413\pi\)
\(11\)	−65.1298	+	13.8438i	−1.78521	+	0.379459i	−0.977637	−	0.210302i	\(-0.932555\pi\)
							−0.807578	+	0.589761i	\(0.799222\pi\)
\(13\)	10.6682	−	50.1902i	0.227603	−	1.07079i	−0.704811	−	0.709396i	\(-0.748968\pi\)
							0.932414	−	0.361393i	\(-0.117699\pi\)
\(17\)	−10.5574	−	14.5310i	−0.150620	−	0.207310i	0.727039	−	0.686596i	\(-0.240896\pi\)
							−0.877659	+	0.479286i	\(0.840896\pi\)
\(19\)	−2.15095	+	1.56276i	−0.0259717	+	0.0188696i	−0.600695	−	0.799478i	\(-0.705110\pi\)
							0.574724	+	0.818348i	\(0.305110\pi\)
\(23\)	116.533	−	104.927i	1.05647	−	0.951253i	0.0575831	−	0.998341i	\(-0.481661\pi\)
							0.998891	+	0.0470874i	\(0.0149939\pi\)
\(29\)	−6.30331	+	59.9720i	−0.0403619	+	0.384018i	0.955630	+	0.294571i	\(0.0951767\pi\)
							−0.995992	+	0.0894472i	\(0.971490\pi\)
\(31\)	−1.75046	−	16.6545i	−0.0101417	−	0.0964917i	0.988280	−	0.152651i	\(-0.0487811\pi\)
							−0.998422	+	0.0561596i	\(0.982114\pi\)
\(37\)	161.427	−	52.4508i	0.717254	−	0.233050i	0.0724219	−	0.997374i	\(-0.476927\pi\)
							0.644832	+	0.764324i	\(0.276927\pi\)
\(41\)	173.259	+	36.8273i	0.659963	+	0.140279i	0.525705	−	0.850667i	\(-0.323802\pi\)
							0.134258	+	0.990946i	\(0.457135\pi\)
\(43\)	175.313	−	101.217i	0.621743	−	0.358963i	−0.155805	−	0.987788i	\(-0.549797\pi\)
							0.777547	+	0.628825i	\(0.216464\pi\)
\(47\)	179.454	+	18.8614i	0.556937	+	0.0585365i	0.378816	−	0.925472i	\(-0.376331\pi\)
							0.178121	+	0.984009i	\(0.442998\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.u.a.4.2	✓	704
9.7	even	3	inner	225.4.u.a.79.2	yes	704
25.19	even	10	inner	225.4.u.a.94.2	yes	704
225.169	even	30	inner	225.4.u.a.169.2	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.u.a.4.2	✓	704	1.1	even	1	trivial
225.4.u.a.79.2	yes	704	9.7	even	3	inner
225.4.u.a.94.2	yes	704	25.19	even	10	inner
225.4.u.a.169.2	yes	704	225.169	even	30	inner