Embedded newform 475.2.v.b.113.23

• Label: 475.2.v.b.113.23
• Level: $475$
• Weight: $2$
• Character: 475.113
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.v (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			113.23
Character	\(\chi\)	\(=\)	475.113
Dual form			475.2.v.b.227.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0403442 - 0.254723i) q^{2} +(1.86848 + 0.952039i) q^{3} +(1.83886 - 0.597481i) q^{4} +(-1.59545 + 1.56670i) q^{5} +(0.167124 - 0.514355i) q^{6} +(-0.170667 + 0.170667i) q^{7} +(-0.460546 - 0.903872i) q^{8} +(0.821490 + 1.13068i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 20 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0403442	−	0.254723i	−0.0285276	−	0.180116i	0.969310	−	0.245841i	\(-0.0790641\pi\)
							−0.997838	+	0.0657246i	\(0.979064\pi\)
\(3\)	1.86848	+	0.952039i	1.07877	+	0.549660i	0.900737	−	0.434365i	\(-0.143027\pi\)
							0.178031	+	0.984025i	\(0.443027\pi\)
\(5\)	−1.59545	+	1.56670i	−0.713506	+	0.700649i
							\(7\)	−0.170667	+	0.170667i	−0.0645062	+	0.0645062i	−0.738624	−	0.674118i	\(-0.764524\pi\)
0.674118	+	0.738624i	\(0.264524\pi\)
\(11\)	2.38128	+	1.73010i	0.717984	+	0.521646i	0.885740	−	0.464183i	\(-0.153652\pi\)
							−0.167755	+	0.985829i	\(0.553652\pi\)
\(13\)	0.220642	−	1.39308i	0.0611951	−	0.386371i	−0.938014	−	0.346598i	\(-0.887337\pi\)
							0.999209	−	0.0397723i	\(-0.0126633\pi\)
\(17\)	1.11117	+	2.18080i	0.269499	+	0.528921i	0.985604	−	0.169071i	\(-0.0540769\pi\)
							−0.716105	+	0.697993i	\(0.754077\pi\)
\(19\)	4.12257	+	1.41576i	0.945783	+	0.324798i
							\(23\)	0.469575	+	2.96478i	0.0979131	+	0.618199i	0.987032	+	0.160523i	\(0.0513181\pi\)
−0.889119	+	0.457676i	\(0.848682\pi\)
\(29\)	2.44731	+	7.53203i	0.454453	+	1.39866i	0.871776	+	0.489905i	\(0.162968\pi\)
							−0.417323	+	0.908758i	\(0.637032\pi\)
\(31\)	−6.65683	−	2.16294i	−1.19560	−	0.388475i	−0.357461	−	0.933928i	\(-0.616358\pi\)
							−0.838141	+	0.545453i	\(0.816358\pi\)
\(37\)	−10.7831	−	1.70787i	−1.77273	−	0.280773i	−0.817348	−	0.576145i	\(-0.804556\pi\)
							−0.955382	+	0.295372i	\(0.904556\pi\)
\(41\)	−5.75559	−	7.92188i	−0.898871	−	1.23719i	−0.970827	−	0.239783i	\(-0.922924\pi\)
							0.0719552	−	0.997408i	\(-0.477076\pi\)
\(43\)	−4.80592	−	4.80592i	−0.732895	−	0.732895i	0.238297	−	0.971192i	\(-0.423411\pi\)
							−0.971192	+	0.238297i	\(0.923411\pi\)
\(47\)	−0.347574	−	0.177098i	−0.0506989	−	0.0258324i	0.428457	−	0.903562i	\(-0.359057\pi\)
							−0.479156	+	0.877730i	\(0.659057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.v.b.113.23	✓	368
19.18	odd	2	inner	475.2.v.b.113.24	yes	368
25.2	odd	20	inner	475.2.v.b.227.24	yes	368
475.227	even	20	inner	475.2.v.b.227.23	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.v.b.113.23	✓	368	1.1	even	1	trivial
475.2.v.b.113.24	yes	368	19.18	odd	2	inner
475.2.v.b.227.23	yes	368	475.227	even	20	inner
475.2.v.b.227.24	yes	368	25.2	odd	20	inner