Embedded newform 475.2.bc.a.396.18

• Label: 475.2.bc.a.396.18
• Level: $475$
• Weight: $2$
• Character: 475.396
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $1152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.bc (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(1152\)
Relative dimension:	\(48\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

Embedding invariants

Embedding label			396.18
Character	\(\chi\)	\(=\)	475.396
Dual form			475.2.bc.a.6.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.764890 + 0.477956i) q^{2} +(-1.97717 - 0.566943i) q^{3} +(-0.520128 + 1.06642i) q^{4} +(0.973170 - 2.01319i) q^{5} +(1.78329 - 0.511350i) q^{6} +(-2.15040 - 3.72461i) q^{7} +(-0.300419 - 2.85829i) q^{8} +(1.04362 + 0.652125i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.764890	+	0.477956i	−0.540859	+	0.337966i	−0.772683	−	0.634792i	\(-0.781086\pi\)
							0.231825	+	0.972758i	\(0.425530\pi\)
\(3\)	−1.97717	−	0.566943i	−1.14152	−	0.327325i	−0.348969	−	0.937134i	\(-0.613468\pi\)
							−0.792548	+	0.609809i	\(0.791246\pi\)
\(5\)	0.973170	−	2.01319i	0.435215	−	0.900327i
							\(7\)	−2.15040	−	3.72461i	−0.812776	−	1.40777i	−0.910914	−	0.412597i	\(-0.864622\pi\)
0.0981372	−	0.995173i	\(-0.468712\pi\)
\(11\)	3.98869	+	4.42989i	1.20263	+	1.33566i	0.927303	+	0.374311i	\(0.122121\pi\)
							0.275331	+	0.961349i	\(0.411212\pi\)
\(13\)	−0.0991372	+	2.83892i	−0.0274957	+	0.787374i	0.904834	+	0.425765i	\(0.139995\pi\)
							−0.932330	+	0.361609i	\(0.882227\pi\)
\(17\)	1.20931	+	0.0845630i	0.293300	+	0.0205095i	0.215649	−	0.976471i	\(-0.430813\pi\)
							0.0776514	+	0.996981i	\(0.475258\pi\)
\(19\)	−4.35319	−	0.223086i	−0.998689	−	0.0511795i
							\(23\)	−7.63695	−	1.07330i	−1.59242	−	0.223799i	−0.713457	−	0.700699i	\(-0.752872\pi\)
−0.878958	+	0.476899i	\(0.841761\pi\)
\(29\)	−3.53587	+	0.247252i	−0.656595	+	0.0459136i	−0.394168	−	0.919038i	\(-0.628967\pi\)
							−0.262427	+	0.964952i	\(0.584523\pi\)
\(31\)	−2.74225	−	1.22093i	−0.492523	−	0.219285i	0.145418	−	0.989370i	\(-0.453547\pi\)
							−0.637942	+	0.770085i	\(0.720214\pi\)
\(37\)	3.43449	+	10.5703i	0.564627	+	1.73774i	0.669056	+	0.743212i	\(0.266698\pi\)
							−0.104429	+	0.994532i	\(0.533302\pi\)
\(41\)	0.198281	+	0.253789i	0.0309663	+	0.0396351i	0.803245	−	0.595649i	\(-0.203105\pi\)
							−0.772278	+	0.635284i	\(0.780883\pi\)
\(43\)	0.329176	−	1.86685i	0.0501988	−	0.284692i	−0.949367	−	0.314170i	\(-0.898274\pi\)
							0.999565	+	0.0294789i	\(0.00938477\pi\)
\(47\)	−7.15568	+	0.500374i	−1.04376	+	0.0729870i	−0.581331	−	0.813667i	\(-0.697468\pi\)
							−0.462432	+	0.886655i	\(0.653023\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.396.18	yes	1152
19.6	even	9	inner	475.2.bc.a.196.31	yes	1152
25.6	even	5	inner	475.2.bc.a.206.31	yes	1152
475.6	even	45	inner	475.2.bc.a.6.18	✓	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.18	✓	1152	475.6	even	45	inner
475.2.bc.a.196.31	yes	1152	19.6	even	9	inner
475.2.bc.a.206.31	yes	1152	25.6	even	5	inner
475.2.bc.a.396.18	yes	1152	1.1	even	1	trivial