Embedded newform 475.2.bc.a.111.40

• Label: 475.2.bc.a.111.40
• Level: $475$
• Weight: $2$
• Character: 475.111
• 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			111.40
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.95901 + 0.136987i) q^{2} +(-0.0107840 - 0.308812i) q^{3} +(1.83842 + 0.258373i) q^{4} +(-0.891650 + 2.05060i) q^{5} +(0.0211775 - 0.606443i) q^{6} +(1.49611 + 2.59133i) q^{7} +(-0.275672 - 0.0585958i) q^{8} +(2.89744 - 0.202609i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.95901	+	0.136987i	1.38523	+	0.0968647i	0.742817	−	0.669494i	\(-0.233489\pi\)
							0.642412	+	0.766359i	\(0.277934\pi\)
\(3\)	−0.0107840	−	0.308812i	−0.00622612	−	0.178293i	−0.998520	−	0.0543860i	\(-0.982680\pi\)
							0.992294	−	0.123907i	\(-0.0395424\pi\)
\(5\)	−0.891650	+	2.05060i	−0.398758	+	0.917056i
							\(7\)	1.49611	+	2.59133i	0.565475	+	0.979431i	0.997005	+	0.0773328i	\(0.0246404\pi\)
−0.431531	+	0.902098i	\(0.642026\pi\)
\(11\)	−0.0475609	−	0.452512i	−0.0143402	−	0.136438i	0.985009	−	0.172501i	\(-0.0551848\pi\)
							−0.999350	+	0.0360635i	\(0.988518\pi\)
\(13\)	0.885529	+	1.31285i	0.245602	+	0.364119i	0.931601	−	0.363483i	\(-0.118413\pi\)
							−0.685999	+	0.727602i	\(0.740635\pi\)
\(17\)	0.654475	−	1.61988i	0.158734	−	0.392879i	−0.826769	−	0.562542i	\(-0.809823\pi\)
							0.985502	+	0.169663i	\(0.0542678\pi\)
\(19\)	3.63849	+	2.40029i	0.834727	+	0.550664i
							\(23\)	−2.34965	−	2.26903i	−0.489935	−	0.473125i	0.408170	−	0.912906i	\(-0.366167\pi\)
−0.898105	+	0.439781i	\(0.855056\pi\)
\(29\)	0.323811	+	0.801460i	0.0601302	+	0.148827i	0.954258	−	0.298984i	\(-0.0966479\pi\)
							−0.894128	+	0.447812i	\(0.852203\pi\)
\(31\)	2.41613	−	2.68338i	0.433949	−	0.481950i	−0.486016	−	0.873950i	\(-0.661550\pi\)
							0.919965	+	0.392001i	\(0.128217\pi\)
\(37\)	−1.72310	−	1.25190i	−0.283275	−	0.205811i	0.437070	−	0.899428i	\(-0.356016\pi\)
							−0.720345	+	0.693616i	\(0.756016\pi\)
\(41\)	1.75746	+	0.503944i	0.274469	+	0.0787028i	0.410048	−	0.912064i	\(-0.365512\pi\)
							−0.135579	+	0.990766i	\(0.543290\pi\)
\(43\)	0.372864	−	2.11462i	0.0568612	−	0.322476i	−0.943088	−	0.332543i	\(-0.892093\pi\)
							0.999949	+	0.0100670i	\(0.00320448\pi\)
\(47\)	−3.93738	−	9.74536i	−0.574326	−	1.42151i	−0.882562	−	0.470197i	\(-0.844183\pi\)
							0.308236	−	0.951310i	\(-0.400262\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.111.40	yes	1152
19.6	even	9	inner	475.2.bc.a.386.40	yes	1152
25.16	even	5	inner	475.2.bc.a.16.40	✓	1152
475.291	even	45	inner	475.2.bc.a.291.40	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.40	✓	1152	25.16	even	5	inner
475.2.bc.a.111.40	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.40	yes	1152	475.291	even	45	inner
475.2.bc.a.386.40	yes	1152	19.6	even	9	inner