Embedded newform 400.2.be.a.221.5

• Label: 400.2.be.a.221.5
• Level: $400$
• Weight: $2$
• Character: 400.221
• 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.be (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			221.5
Character	\(\chi\)	\(=\)	400.221
Dual form			400.2.be.a.181.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37273 - 0.340022i) q^{2} +(0.176699 + 1.11563i) q^{3} +(1.76877 + 0.933517i) q^{4} +(0.507439 + 2.17773i) q^{5} +(0.136780 - 1.59154i) q^{6} -4.59337i q^{7} +(-2.11063 - 1.88289i) q^{8} +(1.63976 - 0.532790i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37273	−	0.340022i	−0.970666	−	0.240432i
							\(3\)	0.176699	+	1.11563i	0.102017	+	0.644110i	0.984716	+	0.174170i	\(0.0557244\pi\)
−0.882699	+	0.469940i	\(0.844276\pi\)
\(5\)	0.507439	+	2.17773i	0.226934	+	0.973910i
							\(7\)		−	4.59337i		−	1.73613i	−0.496448	−	0.868066i	\(-0.665363\pi\)
0.496448	−	0.868066i	\(-0.334637\pi\)
\(11\)	−2.56559	−	5.03525i	−0.773554	−	1.51818i	−0.853330	−	0.521372i	\(-0.825420\pi\)
							0.0797758	−	0.996813i	\(-0.474580\pi\)
\(13\)	1.56683	−	3.07508i	0.434561	−	0.852873i	−0.565052	−	0.825055i	\(-0.691144\pi\)
							0.999613	−	0.0278181i	\(-0.00885593\pi\)
\(17\)	0.981966	−	0.713440i	0.238162	−	0.173035i	−0.462302	−	0.886722i	\(-0.652976\pi\)
							0.700464	+	0.713688i	\(0.252976\pi\)
\(19\)	6.70459	+	1.06190i	1.53814	+	0.243617i	0.867225	−	0.497917i	\(-0.165902\pi\)
							0.670915	+	0.741535i	\(0.265902\pi\)
\(23\)	−1.41375	−	0.459354i	−0.294786	−	0.0957819i	0.157890	−	0.987457i	\(-0.449531\pi\)
							−0.452677	+	0.891675i	\(0.649531\pi\)
\(29\)	1.08272	+	6.83601i	0.201056	+	1.26942i	0.857280	+	0.514850i	\(0.172152\pi\)
							−0.656225	+	0.754566i	\(0.727848\pi\)
\(31\)	1.40171	−	1.01840i	0.251754	−	0.182910i	−0.454750	−	0.890619i	\(-0.650271\pi\)
							0.706504	+	0.707709i	\(0.250271\pi\)
\(37\)	−4.04911	−	2.06313i	−0.665670	−	0.339176i	0.0882914	−	0.996095i	\(-0.471859\pi\)
							−0.753961	+	0.656919i	\(0.771859\pi\)
\(41\)	−1.56322	+	0.507919i	−0.244133	+	0.0793237i	−0.428528	−	0.903529i	\(-0.640968\pi\)
							0.184395	+	0.982852i	\(0.440968\pi\)
\(43\)	1.39763	−	1.39763i	0.213137	−	0.213137i	−0.592462	−	0.805598i	\(-0.701844\pi\)
							0.805598	+	0.592462i	\(0.201844\pi\)
\(47\)	−6.49783	−	4.72095i	−0.947806	−	0.688621i	0.00248086	−	0.999997i	\(-0.499210\pi\)
							−0.950287	+	0.311375i	\(0.899210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.be.a.221.5	yes	464
16.5	even	4	inner	400.2.be.a.21.51	✓	464
25.6	even	5	inner	400.2.be.a.381.51	yes	464
400.181	even	20	inner	400.2.be.a.181.5	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.51	✓	464	16.5	even	4	inner
400.2.be.a.181.5	yes	464	400.181	even	20	inner
400.2.be.a.221.5	yes	464	1.1	even	1	trivial
400.2.be.a.381.51	yes	464	25.6	even	5	inner