Embedded newform 400.2.bl.a.29.10

• Label: 400.2.bl.a.29.10
• Level: $400$
• Weight: $2$
• Character: 400.29
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bl (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.19401608085\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(58\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			29.10
Character	\(\chi\)	\(=\)	400.29
Dual form			400.2.bl.a.69.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26619 - 0.629895i) q^{2} +(2.62418 - 0.415629i) q^{3} +(1.20646 + 1.59513i) q^{4} +(-0.886089 - 2.05301i) q^{5} +(-3.58451 - 1.12669i) q^{6} +3.34726 q^{7} +(-0.522846 - 2.77968i) q^{8} +(3.86039 - 1.25432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 10 q^{2} - 10 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} - 40 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26619	−	0.629895i	−0.895330	−	0.445403i
							\(3\)	2.62418	−	0.415629i	1.51507	−	0.239963i	0.657155	−	0.753756i	\(-0.271760\pi\)
0.857915	+	0.513792i	\(0.171760\pi\)
\(5\)	−0.886089	−	2.05301i	−0.396271	−	0.918134i
							\(7\)	3.34726			1.26515			0.632573	−	0.774500i	\(-0.281999\pi\)
0.632573	+	0.774500i	\(0.281999\pi\)
\(11\)	2.00913	+	3.94314i	0.605776	+	1.18890i	0.966605	+	0.256270i	\(0.0824936\pi\)
							−0.360829	+	0.932632i	\(0.617506\pi\)
\(13\)	−2.70927	−	1.38044i	−0.751415	−	0.382865i	0.0359486	−	0.999354i	\(-0.488555\pi\)
							−0.787364	+	0.616488i	\(0.788555\pi\)
\(17\)	−2.55464	−	3.51617i	−0.619592	−	0.852796i	0.377731	−	0.925915i	\(-0.376704\pi\)
							−0.997323	+	0.0731198i	\(0.976704\pi\)
\(19\)	3.59167	+	0.568865i	0.823987	+	0.130507i	0.554168	−	0.832405i	\(-0.313036\pi\)
							0.269818	+	0.962911i	\(0.413036\pi\)
\(23\)	−0.537689	+	1.65484i	−0.112116	+	0.345057i	−0.991335	−	0.131361i	\(-0.958065\pi\)
							0.879219	+	0.476418i	\(0.158065\pi\)
\(29\)	−0.662057	−	4.18007i	−0.122941	−	0.776219i	−0.969710	−	0.244258i	\(-0.921455\pi\)
							0.846769	−	0.531960i	\(-0.178545\pi\)
\(31\)	−4.07439	+	2.96022i	−0.731783	+	0.531671i	−0.890127	−	0.455713i	\(-0.849384\pi\)
							0.158344	+	0.987384i	\(0.449384\pi\)
\(37\)	0.850161	−	1.66854i	0.139766	−	0.274306i	−0.810504	−	0.585732i	\(-0.800807\pi\)
							0.950270	+	0.311427i	\(0.100807\pi\)
\(41\)	−2.65492	+	0.862637i	−0.414629	+	0.134721i	−0.508901	−	0.860825i	\(-0.669948\pi\)
							0.0942712	+	0.995547i	\(0.469948\pi\)
\(43\)	8.45766	+	8.45766i	1.28978	+	1.28978i	0.934918	+	0.354863i	\(0.115473\pi\)
							0.354863	+	0.934918i	\(0.384527\pi\)
\(47\)	4.41317	−	6.07421i	0.643728	−	0.886015i	−0.355080	−	0.934836i	\(-0.615546\pi\)
							0.998808	+	0.0488208i	\(0.0155463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bl.a.29.10	✓	464
16.5	even	4	inner	400.2.bl.a.229.4	yes	464
25.19	even	10	inner	400.2.bl.a.269.4	yes	464
400.69	even	20	inner	400.2.bl.a.69.10	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bl.a.29.10	✓	464	1.1	even	1	trivial
400.2.bl.a.69.10	yes	464	400.69	even	20	inner
400.2.bl.a.229.4	yes	464	16.5	even	4	inner
400.2.bl.a.269.4	yes	464	25.19	even	10	inner