Embedded newform 475.2.bc.a.396.20

• Label: 475.2.bc.a.396.20
• 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			396.20
Character	\(\chi\)	\(=\)	475.396
Dual form			475.2.bc.a.6.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.695663 + 0.434699i) q^{2} +(-2.62473 - 0.752630i) q^{3} +(-0.581758 + 1.19278i) q^{4} +(-1.90436 + 1.17193i) q^{5} +(2.15310 - 0.617391i) q^{6} +(2.44900 + 4.24179i) q^{7} +(-0.285284 - 2.71430i) q^{8} +(3.77862 + 2.36115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.695663	+	0.434699i	−0.491908	+	0.307378i	−0.753097	−	0.657909i	\(-0.771441\pi\)
							0.261189	+	0.965288i	\(0.415885\pi\)
\(3\)	−2.62473	−	0.752630i	−1.51539	−	0.434531i	−0.588073	−	0.808808i	\(-0.700113\pi\)
							−0.927317	+	0.374277i	\(0.877891\pi\)
\(5\)	−1.90436	+	1.17193i	−0.851654	+	0.524105i
							\(7\)	2.44900	+	4.24179i	0.925634	+	1.60325i	0.790538	+	0.612413i	\(0.209801\pi\)
0.135096	+	0.990832i	\(0.456866\pi\)
\(11\)	1.65000	+	1.83251i	0.497494	+	0.552523i	0.938634	−	0.344915i	\(-0.112092\pi\)
							−0.441140	+	0.897439i	\(0.645426\pi\)
\(13\)	−0.143033	+	4.09593i	−0.0396702	+	1.13601i	0.802451	+	0.596718i	\(0.203529\pi\)
							−0.842121	+	0.539288i	\(0.818693\pi\)
\(17\)	−6.33655	−	0.443095i	−1.53684	−	0.107466i	−0.723936	−	0.689867i	\(-0.757669\pi\)
							−0.812902	+	0.582401i	\(0.802114\pi\)
\(19\)	2.92849	+	3.22861i	0.671842	+	0.740695i
							\(23\)	−5.68316	−	0.798717i	−1.18502	−	0.166544i	−0.480999	−	0.876721i	\(-0.659726\pi\)
−0.704023	+	0.710177i	\(0.748615\pi\)
\(29\)	3.11933	−	0.218125i	0.579245	−	0.0405048i	0.222871	−	0.974848i	\(-0.428457\pi\)
							0.356375	+	0.934343i	\(0.384013\pi\)
\(31\)	−5.14065	−	2.28876i	−0.923288	−	0.411074i	−0.110661	−	0.993858i	\(-0.535297\pi\)
							−0.812627	+	0.582784i	\(0.801963\pi\)
\(37\)	0.109436	+	0.336809i	0.0179911	+	0.0553710i	0.959649	−	0.281201i	\(-0.0907326\pi\)
							−0.941658	+	0.336572i	\(0.890733\pi\)
\(41\)	2.63592	+	3.37382i	0.411661	+	0.526902i	0.948498	−	0.316784i	\(-0.102603\pi\)
							−0.536837	+	0.843686i	\(0.680381\pi\)
\(43\)	1.16013	−	6.57941i	0.176918	−	1.00335i	−0.758989	−	0.651104i	\(-0.774306\pi\)
							0.935907	−	0.352248i	\(-0.114583\pi\)
\(47\)	−8.52292	+	0.595980i	−1.24320	+	0.0869327i	−0.676156	−	0.736758i	\(-0.736356\pi\)
							−0.567039	+	0.823691i	\(0.691911\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.396.20	yes	1152
19.6	even	9	inner	475.2.bc.a.196.29	yes	1152
25.6	even	5	inner	475.2.bc.a.206.29	yes	1152
475.6	even	45	inner	475.2.bc.a.6.20	✓	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.20	✓	1152	475.6	even	45	inner
475.2.bc.a.196.29	yes	1152	19.6	even	9	inner
475.2.bc.a.206.29	yes	1152	25.6	even	5	inner
475.2.bc.a.396.20	yes	1152	1.1	even	1	trivial