Embedded newform 475.2.bc.a.6.10

• Label: 475.2.bc.a.6.10
• 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			6.10
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68487 - 1.05282i) q^{2} +(1.58487 - 0.454455i) q^{3} +(0.853608 + 1.75016i) q^{4} +(1.47769 + 1.67822i) q^{5} +(-3.14876 - 0.902893i) q^{6} +(-0.289478 + 0.501391i) q^{7} +(-0.0109594 + 0.104272i) q^{8} +(-0.238856 + 0.149254i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68487	−	1.05282i	−1.19138	−	0.744459i	−0.218519	−	0.975833i	\(-0.570122\pi\)
							−0.972865	+	0.231374i	\(0.925678\pi\)
\(3\)	1.58487	−	0.454455i	0.915026	−	0.262379i	0.215098	−	0.976593i	\(-0.430993\pi\)
							0.699928	+	0.714213i	\(0.253215\pi\)
\(5\)	1.47769	+	1.67822i	0.660844	+	0.750523i
							\(7\)	−0.289478	+	0.501391i	−0.109412	+	0.189508i	−0.915532	−	0.402244i	\(-0.868230\pi\)
0.806120	+	0.591752i	\(0.201564\pi\)
\(11\)	1.10659	−	1.22899i	0.333649	−	0.370555i	−0.552853	−	0.833279i	\(-0.686461\pi\)
							0.886503	+	0.462723i	\(0.153128\pi\)
\(13\)	0.0337473	+	0.966396i	0.00935982	+	0.268030i	0.995044	+	0.0994308i	\(0.0317022\pi\)
							−0.985685	+	0.168599i	\(0.946076\pi\)
\(17\)	3.19928	−	0.223715i	0.775939	−	0.0542589i	0.323705	−	0.946158i	\(-0.395072\pi\)
							0.452234	+	0.891899i	\(0.350627\pi\)
\(19\)	1.18483	+	4.19478i	0.271818	+	0.962349i
							\(23\)	0.915838	−	0.128713i	0.190965	−	0.0268384i	−0.0430419	−	0.999073i	\(-0.513705\pi\)
0.234007	+	0.972235i	\(0.424816\pi\)
\(29\)	6.15398	+	0.430328i	1.14277	+	0.0799100i	0.628563	−	0.777759i	\(-0.283643\pi\)
							0.514203	+	0.857669i	\(0.328088\pi\)
\(31\)	4.22024	−	1.87897i	0.757978	−	0.337474i	0.00890436	−	0.999960i	\(-0.497166\pi\)
							0.749074	+	0.662487i	\(0.230499\pi\)
\(37\)	−1.83371	+	5.64358i	−0.301460	+	0.927798i	0.679515	+	0.733662i	\(0.262190\pi\)
							−0.980975	+	0.194136i	\(0.937810\pi\)
\(41\)	4.79459	−	6.13680i	0.748789	−	0.958407i	−0.251165	−	0.967944i	\(-0.580814\pi\)
							0.999955	+	0.00953758i	\(0.00303595\pi\)
\(43\)	0.250231	+	1.41913i	0.0381599	+	0.216415i	0.997925	−	0.0643891i	\(-0.0205099\pi\)
							−0.959765	+	0.280804i	\(0.909399\pi\)
\(47\)	−2.85246	−	0.199463i	−0.416074	−	0.0290947i	−0.139814	−	0.990178i	\(-0.544650\pi\)
							−0.276259	+	0.961083i	\(0.589095\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.10	✓	1152
19.16	even	9	inner	475.2.bc.a.206.39	yes	1152
25.21	even	5	inner	475.2.bc.a.196.39	yes	1152
475.396	even	45	inner	475.2.bc.a.396.10	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.10	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.39	yes	1152	25.21	even	5	inner
475.2.bc.a.206.39	yes	1152	19.16	even	9	inner
475.2.bc.a.396.10	yes	1152	475.396	even	45	inner