Embedded newform 700.2.bj.a.323.85

• Label: 700.2.bj.a.323.85
• Level: $700$
• Weight: $2$
• Character: 700.323
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $720$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.bj (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(720\)
Relative dimension:	\(90\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			323.85
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.85

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40266 - 0.180426i) q^{2} +(0.474137 + 0.930546i) q^{3} +(1.93489 - 0.506153i) q^{4} +(-2.11953 + 0.712450i) q^{5} +(0.832946 + 1.21969i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(2.62267 - 1.05906i) q^{8} +(1.12225 - 1.54464i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40266	−	0.180426i	0.991828	−	0.127581i
							\(3\)	0.474137	+	0.930546i	0.273743	+	0.537251i	0.986420	−	0.164241i	\(-0.0525175\pi\)
−0.712677	+	0.701492i	\(0.752518\pi\)
\(5\)	−2.11953	+	0.712450i	−0.947883	+	0.318617i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	2.90711	+	4.00129i	0.876527	+	1.20644i								0.977371	+	0.211533i	\(0.0678456\pi\)
							−0.100844	+	0.994902i	\(0.532154\pi\)
\(13\)	0.496662	+	3.13580i	0.137749	+	0.869714i	0.955682	+	0.294401i	\(0.0951200\pi\)
							−0.817933	+	0.575314i	\(0.804880\pi\)
\(17\)	4.14186	+	2.11038i	1.00455	+	0.511843i	0.877256	−	0.480023i	\(-0.159372\pi\)
							0.127291	+	0.991865i	\(0.459372\pi\)
\(19\)	0.918951	+	2.82824i	0.210822	+	0.648843i	0.999424	+	0.0339390i	\(0.0108052\pi\)
							−0.788602	+	0.614904i	\(0.789195\pi\)
\(23\)	0.437135	−	2.75996i	0.0911489	−	0.575492i	−0.899270	−	0.437394i	\(-0.855901\pi\)
							0.990419	−	0.138097i	\(-0.0440986\pi\)
\(29\)	−10.1850	−	3.30931i	−1.89131	−	0.614524i	−0.978541	−	0.206051i	\(-0.933939\pi\)
							−0.912770	−	0.408474i	\(-0.866061\pi\)
\(31\)	0.809846	−	0.263135i	0.145453	−	0.0472604i	−0.235386	−	0.971902i	\(-0.575635\pi\)
							0.380838	+	0.924642i	\(0.375635\pi\)
\(37\)	−6.51368	+	1.03167i	−1.07084	+	0.169605i	−0.666876	−	0.745169i	\(-0.732369\pi\)
							−0.403967	+	0.914774i	\(0.632369\pi\)
\(41\)	−6.30605	−	4.58162i	−0.984840	−	0.715528i	−0.0260550	−	0.999661i	\(-0.508294\pi\)
							−0.958785	+	0.284132i	\(0.908294\pi\)
\(43\)	3.69854	−	3.69854i	0.564021	−	0.564021i	−0.366426	−	0.930447i	\(-0.619419\pi\)
							0.930447	+	0.366426i	\(0.119419\pi\)
\(47\)	5.32712	−	2.71430i	0.777040	−	0.395922i	−0.0200372	−	0.999799i	\(-0.506378\pi\)
							0.797077	+	0.603878i	\(0.206378\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.85	yes	720
4.3	odd	2	inner	700.2.bj.a.323.14	✓	720
25.12	odd	20	inner	700.2.bj.a.687.14	yes	720
100.87	even	20	inner	700.2.bj.a.687.85	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.14	✓	720	4.3	odd	2	inner
700.2.bj.a.323.85	yes	720	1.1	even	1	trivial
700.2.bj.a.687.14	yes	720	25.12	odd	20	inner
700.2.bj.a.687.85	yes	720	100.87	even	20	inner