Embedded newform 400.2.be.a.21.51

• Label: 400.2.be.a.21.51
• Level: $400$
• Weight: $2$
• Character: 400.21
• 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			21.51
Character	\(\chi\)	\(=\)	400.21
Dual form			400.2.be.a.381.51

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31042 + 0.531786i) q^{2} +(1.11563 - 0.176699i) q^{3} +(1.43441 + 1.39373i) q^{4} +(-2.17773 + 0.507439i) q^{5} +(1.55591 + 0.361728i) q^{6} +4.59337i q^{7} +(1.13851 + 2.58917i) q^{8} +(-1.63976 + 0.532790i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.31042	+	0.531786i	0.926608	+	0.376030i
							\(3\)	1.11563	−	0.176699i	0.644110	−	0.102017i	0.174170	−	0.984716i	\(-0.444276\pi\)
0.469940	+	0.882699i	\(0.344276\pi\)
\(5\)	−2.17773	+	0.507439i	−0.973910	+	0.226934i
							\(7\)			4.59337i			1.73613i	0.496448	+	0.868066i	\(0.334637\pi\)
−0.496448	+	0.868066i	\(0.665363\pi\)
\(11\)	5.03525	−	2.56559i	1.51818	−	0.773554i	0.521372	−	0.853330i	\(-0.325420\pi\)
							0.996813	+	0.0797758i	\(0.0254205\pi\)
\(13\)	−3.07508	−	1.56683i	−0.852873	−	0.434561i	−0.0278181	−	0.999613i	\(-0.508856\pi\)
							−0.825055	+	0.565052i	\(0.808856\pi\)
\(17\)	0.981966	−	0.713440i	0.238162	−	0.173035i	−0.462302	−	0.886722i	\(-0.652976\pi\)
							0.700464	+	0.713688i	\(0.252976\pi\)
\(19\)	1.06190	−	6.70459i	0.243617	−	1.53814i	−0.497917	−	0.867225i	\(-0.665902\pi\)
							0.741535	−	0.670915i	\(-0.234098\pi\)
\(23\)	1.41375	+	0.459354i	0.294786	+	0.0957819i	0.452677	−	0.891675i	\(-0.350469\pi\)
							−0.157890	+	0.987457i	\(0.550469\pi\)
\(29\)	6.83601	−	1.08272i	1.26942	−	0.201056i	0.514850	−	0.857280i	\(-0.327848\pi\)
							0.754566	+	0.656225i	\(0.227848\pi\)
\(31\)	1.40171	−	1.01840i	0.251754	−	0.182910i	−0.454750	−	0.890619i	\(-0.650271\pi\)
							0.706504	+	0.707709i	\(0.250271\pi\)
\(37\)	2.06313	−	4.04911i	0.339176	−	0.665670i	−0.656919	−	0.753961i	\(-0.728141\pi\)
							0.996095	+	0.0882914i	\(0.0281407\pi\)
\(41\)	1.56322	−	0.507919i	0.244133	−	0.0793237i	−0.184395	−	0.982852i	\(-0.559032\pi\)
							0.428528	+	0.903529i	\(0.359032\pi\)
\(43\)	1.39763	+	1.39763i	0.213137	+	0.213137i	0.805598	−	0.592462i	\(-0.201844\pi\)
							−0.592462	+	0.805598i	\(0.701844\pi\)
\(47\)	−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.21.51	✓	464
16.13	even	4	inner	400.2.be.a.221.5	yes	464
25.6	even	5	inner	400.2.be.a.181.5	yes	464
400.381	even	20	inner	400.2.be.a.381.51	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.be.a.21.51	✓	464	1.1	even	1	trivial
400.2.be.a.181.5	yes	464	25.6	even	5	inner
400.2.be.a.221.5	yes	464	16.13	even	4	inner
400.2.be.a.381.51	yes	464	400.381	even	20	inner