Embedded newform 475.2.bc.a.6.11

• Label: 475.2.bc.a.6.11
• Level: $475$
• Weight: $2$
• Character: 475.6
• 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			6.11
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48496 - 0.927909i) q^{2} +(-0.380214 + 0.109025i) q^{3} +(0.467363 + 0.958236i) q^{4} +(0.439600 + 2.19243i) q^{5} +(0.665769 + 0.190906i) q^{6} +(1.66674 - 2.88687i) q^{7} +(-0.170928 + 1.62627i) q^{8} +(-2.41147 + 1.50685i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48496	−	0.927909i	−1.05003	−	0.656131i	−0.108856	−	0.994057i	\(-0.534719\pi\)
							−0.941172	+	0.337927i	\(0.890274\pi\)
\(3\)	−0.380214	+	0.109025i	−0.219517	+	0.0629454i	−0.383593	−	0.923502i	\(-0.625313\pi\)
							0.164076	+	0.986448i	\(0.447536\pi\)
\(5\)	0.439600	+	2.19243i	0.196595	+	0.980485i
							\(7\)	1.66674	−	2.88687i	0.629967	−	1.09113i	−0.357591	−	0.933878i	\(-0.616402\pi\)
0.987558	−	0.157256i	\(-0.0502648\pi\)
\(11\)	−1.99119	+	2.21144i	−0.600366	+	0.666774i	−0.964350	−	0.264632i	\(-0.914750\pi\)
							0.363984	+	0.931405i	\(0.381416\pi\)
\(13\)	−0.0343934	−	0.984899i	−0.00953902	−	0.273162i	−0.994783	−	0.102009i	\(-0.967473\pi\)
							0.985244	−	0.171153i	\(-0.0547492\pi\)
\(17\)	−2.57630	+	0.180153i	−0.624845	+	0.0436934i	−0.378670	−	0.925532i	\(-0.623618\pi\)
							−0.246175	+	0.969225i	\(0.579174\pi\)
\(19\)	−2.43428	−	3.61584i	−0.558461	−	0.829531i
							\(23\)	0.353760	−	0.0497178i	0.0737641	−	0.0103669i	−0.102194	−	0.994764i	\(-0.532586\pi\)
0.175959	+	0.984398i	\(0.443698\pi\)
\(29\)	−7.93211	−	0.554667i	−1.47296	−	0.102999i	−0.689428	−	0.724355i	\(-0.742138\pi\)
							−0.783529	+	0.621355i	\(0.786582\pi\)
\(31\)	−6.99116	+	3.11267i	−1.25565	+	0.559052i	−0.923291	−	0.384101i	\(-0.874511\pi\)
							−0.332359	+	0.943153i	\(0.607845\pi\)
\(37\)	−1.80821	+	5.56510i	−0.297268	+	0.914897i	0.685182	+	0.728372i	\(0.259723\pi\)
							−0.982450	+	0.186525i	\(0.940277\pi\)
\(41\)	−2.96385	+	3.79356i	−0.462876	+	0.592454i	−0.961651	−	0.274278i	\(-0.911561\pi\)
							0.498775	+	0.866732i	\(0.333783\pi\)
\(43\)	1.80177	+	10.2183i	0.274767	+	1.55828i	0.739700	+	0.672936i	\(0.234967\pi\)
							−0.464933	+	0.885346i	\(0.653922\pi\)
\(47\)	9.96106	+	0.696545i	1.45297	+	0.101602i	0.774289	−	0.632832i	\(-0.218107\pi\)
							0.678681	+	0.734433i	\(0.262552\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.11	✓	1152
19.16	even	9	inner	475.2.bc.a.206.38	yes	1152
25.21	even	5	inner	475.2.bc.a.196.38	yes	1152
475.396	even	45	inner	475.2.bc.a.396.11	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.11	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.38	yes	1152	25.21	even	5	inner
475.2.bc.a.206.38	yes	1152	19.16	even	9	inner
475.2.bc.a.396.11	yes	1152	475.396	even	45	inner