Embedded newform 400.2.bl.a.389.9

• Label: 400.2.bl.a.389.9
• Level: $400$
• Weight: $2$
• Character: 400.389
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• 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			389.9
Character	\(\chi\)	\(=\)	400.389
Dual form			400.2.bl.a.109.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27001 - 0.622160i) q^{2} +(1.92954 - 0.983149i) q^{3} +(1.22583 + 1.58029i) q^{4} +(1.90024 + 1.17859i) q^{5} +(-3.06220 + 0.0481238i) q^{6} -0.337437 q^{7} +(-0.573620 - 2.76965i) q^{8} +(0.993180 - 1.36700i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 10 q^{2} - 10 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} - 40 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27001	−	0.622160i	−0.898030	−	0.439934i
							\(3\)	1.92954	−	0.983149i	1.11402	−	0.567621i	0.202667	−	0.979248i	\(-0.435039\pi\)
0.911353	+	0.411627i	\(0.135039\pi\)
\(5\)	1.90024	+	1.17859i	0.849815	+	0.527081i
							\(7\)	−0.337437			−0.127539			−0.0637697	−	0.997965i	\(-0.520312\pi\)
−0.0637697	+	0.997965i	\(0.520312\pi\)
\(11\)	1.06644	+	0.168908i	0.321545	+	0.0509277i	0.315121	−	0.949052i	\(-0.397955\pi\)
							0.00642437	+	0.999979i	\(0.497955\pi\)
\(13\)	−0.676137	−	4.26896i	−0.187527	−	1.18400i	−0.884375	−	0.466778i	\(-0.845415\pi\)
							0.696848	−	0.717219i	\(-0.254585\pi\)
\(17\)	1.29208	−	0.419821i	0.313375	−	0.101822i	−0.148107	−	0.988971i	\(-0.547318\pi\)
							0.461482	+	0.887150i	\(0.347318\pi\)
\(19\)	4.71610	+	2.40298i	1.08195	+	0.551280i	0.901709	−	0.432342i	\(-0.142313\pi\)
							0.180239	+	0.983623i	\(0.442313\pi\)
\(23\)	−0.226540	+	0.164591i	−0.0472369	+	0.0343196i	−0.611153	−	0.791512i	\(-0.709294\pi\)
							0.563916	+	0.825832i	\(0.309294\pi\)
\(29\)	−3.52071	−	6.90978i	−0.653779	−	1.28311i	−0.945192	−	0.326516i	\(-0.894125\pi\)
							0.291413	−	0.956597i	\(-0.405875\pi\)
\(31\)	1.69748	+	5.22431i	0.304877	+	0.938314i	0.979723	+	0.200356i	\(0.0642099\pi\)
							−0.674846	+	0.737958i	\(0.735790\pi\)
\(37\)	−0.823099	+	0.130366i	−0.135317	+	0.0214320i	−0.223725	−	0.974652i	\(-0.571822\pi\)
							0.0884087	+	0.996084i	\(0.471822\pi\)
\(41\)	−1.79636	+	2.47248i	−0.280544	+	0.386136i	−0.925914	−	0.377734i	\(-0.876703\pi\)
							0.645370	+	0.763870i	\(0.276703\pi\)
\(43\)	4.82814	−	4.82814i	0.736285	−	0.736285i	−0.235572	−	0.971857i	\(-0.575696\pi\)
							0.971857	+	0.235572i	\(0.0756963\pi\)
\(47\)	−2.30680	−	0.749523i	−0.336481	−	0.109329i	0.135903	−	0.990722i	\(-0.456606\pi\)
							−0.472384	+	0.881393i	\(0.656606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bl.a.389.9	yes	464
16.13	even	4	inner	400.2.bl.a.189.27	yes	464
25.9	even	10	inner	400.2.bl.a.309.27	yes	464
400.109	even	20	inner	400.2.bl.a.109.9	✓	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bl.a.109.9	✓	464	400.109	even	20	inner
400.2.bl.a.189.27	yes	464	16.13	even	4	inner
400.2.bl.a.309.27	yes	464	25.9	even	10	inner
400.2.bl.a.389.9	yes	464	1.1	even	1	trivial