Embedded newform 475.2.bc.a.111.14

• Label: 475.2.bc.a.111.14
• 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			111.14
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50426 - 0.105188i) q^{2} +(0.0527124 + 1.50949i) q^{3} +(0.271205 + 0.0381154i) q^{4} +(-2.06163 - 0.865851i) q^{5} +(0.0794869 - 2.27621i) q^{6} +(1.96137 + 3.39719i) q^{7} +(2.54601 + 0.541172i) q^{8} +(0.716924 - 0.0501322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50426	−	0.105188i	−1.06367	−	0.0743794i	−0.472821	−	0.881158i	\(-0.656764\pi\)
							−0.590853	+	0.806779i	\(0.701209\pi\)
\(3\)	0.0527124	+	1.50949i	0.0304335	+	0.871502i	0.914607	+	0.404344i	\(0.132500\pi\)
							−0.884174	+	0.467158i	\(0.845278\pi\)
\(5\)	−2.06163	−	0.865851i	−0.921987	−	0.387220i
							\(7\)	1.96137	+	3.39719i	0.741328	+	1.28402i	0.951891	+	0.306437i	\(0.0991370\pi\)
−0.210563	+	0.977580i	\(0.567530\pi\)
\(11\)	−0.272836	−	2.59586i	−0.0822630	−	0.782680i	−0.955422	−	0.295244i	\(-0.904599\pi\)
							0.873159	−	0.487436i	\(-0.162068\pi\)
\(13\)	1.35972	+	2.01587i	0.377118	+	0.559101i	0.969043	−	0.246892i	\(-0.0794094\pi\)
							−0.591925	+	0.805993i	\(0.701632\pi\)
\(17\)	0.613434	−	1.51830i	0.148780	−	0.368243i	−0.834329	−	0.551267i	\(-0.814144\pi\)
							0.983108	+	0.183025i	\(0.0585888\pi\)
\(19\)	−2.85943	+	3.28993i	−0.655999	+	0.754762i
							\(23\)	−1.15893	−	1.11917i	−0.241654	−	0.233362i	0.563962	−	0.825801i	\(-0.309276\pi\)
−0.805616	+	0.592439i	\(0.798165\pi\)
\(29\)	3.38100	+	8.36827i	0.627836	+	1.55395i	0.819863	+	0.572560i	\(0.194050\pi\)
							−0.192027	+	0.981390i	\(0.561506\pi\)
\(31\)	−3.66284	+	4.06800i	−0.657866	+	0.730634i	−0.976086	−	0.217383i	\(-0.930248\pi\)
							0.318220	+	0.948017i	\(0.396915\pi\)
\(37\)	−9.77963	−	7.10532i	−1.60776	−	1.16811i	−0.870008	−	0.493037i	\(-0.835887\pi\)
							−0.737753	−	0.675070i	\(-0.764113\pi\)
\(41\)	−7.15233	−	2.05090i	−1.11701	−	0.320296i	−0.334163	−	0.942515i	\(-0.608454\pi\)
							−0.782843	+	0.622219i	\(0.786231\pi\)
\(43\)	−0.970995	+	5.50679i	−0.148075	+	0.839777i	0.816771	+	0.576962i	\(0.195762\pi\)
							−0.964846	+	0.262815i	\(0.915349\pi\)
\(47\)	1.96447	+	4.86223i	0.286547	+	0.709228i	0.999964	+	0.00848879i	\(0.00270210\pi\)
							−0.713417	+	0.700740i	\(0.752853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.111.14	yes	1152
19.6	even	9	inner	475.2.bc.a.386.14	yes	1152
25.16	even	5	inner	475.2.bc.a.16.14	✓	1152
475.291	even	45	inner	475.2.bc.a.291.14	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.14	✓	1152	25.16	even	5	inner
475.2.bc.a.111.14	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.14	yes	1152	475.291	even	45	inner
475.2.bc.a.386.14	yes	1152	19.6	even	9	inner