Embedded newform 475.2.bc.a.6.4

• Label: 475.2.bc.a.6.4
• 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			6.4
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.08209 - 1.30103i) q^{2} +(0.278378 - 0.0798235i) q^{3} +(1.76566 + 3.62014i) q^{4} +(1.47314 - 1.68222i) q^{5} +(-0.683459 - 0.195979i) q^{6} +(-1.61228 + 2.79255i) q^{7} +(0.520393 - 4.95121i) q^{8} +(-2.47302 + 1.54532i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.08209	−	1.30103i	−1.47226	−	0.919969i	−0.999243	−	0.0389145i	\(-0.987610\pi\)
							−0.473015	−	0.881054i	\(-0.656834\pi\)
\(3\)	0.278378	−	0.0798235i	0.160721	−	0.0460861i	−0.194312	−	0.980940i	\(-0.562247\pi\)
							0.355033	+	0.934854i	\(0.384470\pi\)
\(5\)	1.47314	−	1.68222i	0.658806	−	0.752313i
							\(7\)	−1.61228	+	2.79255i	−0.609385	+	1.05549i	0.381957	+	0.924180i	\(0.375250\pi\)
−0.991342	+	0.131306i	\(0.958083\pi\)
\(11\)	2.52936	−	2.80913i	0.762630	−	0.846986i	−0.229355	−	0.973343i	\(-0.573662\pi\)
							0.991985	+	0.126357i	\(0.0403284\pi\)
\(13\)	0.204827	+	5.86549i	0.0568089	+	1.62679i	0.609065	+	0.793120i	\(0.291545\pi\)
							−0.552256	+	0.833674i	\(0.686233\pi\)
\(17\)	−6.62525	+	0.463283i	−1.60686	+	0.112363i	−0.844777	−	0.535118i	\(-0.820267\pi\)
							−0.762082	+	0.647481i	\(0.775823\pi\)
\(19\)	−4.32330	+	0.555925i	−0.991834	+	0.127538i
							\(23\)	−6.71132	+	0.943214i	−1.39941	+	0.196674i	−0.798187	−	0.602410i	\(-0.794207\pi\)
−0.601220	+	0.799084i	\(0.705318\pi\)
\(29\)	−0.672162	−	0.0470021i	−0.124817	−	0.00872808i	0.00721115	−	0.999974i	\(-0.497705\pi\)
							−0.132029	+	0.991246i	\(0.542149\pi\)
\(31\)	−0.780417	+	0.347464i	−0.140167	+	0.0624064i	−0.475622	−	0.879650i	\(-0.657777\pi\)
							0.335454	+	0.942056i	\(0.391110\pi\)
\(37\)	−0.363154	+	1.11767i	−0.0597021	+	0.183744i	−0.976460	−	0.215700i	\(-0.930797\pi\)
							0.916758	+	0.399444i	\(0.130797\pi\)
\(41\)	−4.51396	+	5.77760i	−0.704962	+	0.902310i	−0.998561	−	0.0536188i	\(-0.982924\pi\)
							0.293600	+	0.955928i	\(0.405147\pi\)
\(43\)	0.0292560	+	0.165919i	0.00446150	+	0.0253024i	0.986957	−	0.160983i	\(-0.0514663\pi\)
							−0.982496	+	0.186285i	\(0.940355\pi\)
\(47\)	11.0436	+	0.772242i	1.61087	+	0.112643i	0.846591	−	0.532243i	\(-0.178651\pi\)
							0.764279	+	0.644886i	\(0.223095\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.4	✓	1152
19.16	even	9	inner	475.2.bc.a.206.45	yes	1152
25.21	even	5	inner	475.2.bc.a.196.45	yes	1152
475.396	even	45	inner	475.2.bc.a.396.4	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.4	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.45	yes	1152	25.21	even	5	inner
475.2.bc.a.206.45	yes	1152	19.16	even	9	inner
475.2.bc.a.396.4	yes	1152	475.396	even	45	inner