Embedded newform 280.2.ba.b.59.5

• Label: 280.2.ba.b.59.5
• Level: $280$
• Weight: $2$
• Character: 280.59
• Analytic conductor: $2.236$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	280.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.23581125660\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.5
Character	\(\chi\)	\(=\)	280.59
Dual form			280.2.ba.b.19.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32026 + 0.506870i) q^{2} +(0.360276 - 0.624015i) q^{3} +(1.48617 - 1.33840i) q^{4} +(-1.56432 - 1.59778i) q^{5} +(-0.159362 + 1.00647i) q^{6} +(1.35637 + 2.27162i) q^{7} +(-1.28373 + 2.52033i) q^{8} +(1.24040 + 2.14844i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{4} - 52 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\)	\(57\)	\(71\)	\(141\)	\(241\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32026	+	0.506870i	−0.933564	+	0.358411i
							\(3\)	0.360276	−	0.624015i	0.208005	−	0.360276i	−0.743081	−	0.669202i	\(-0.766636\pi\)
0.951086	+	0.308926i	\(0.0999696\pi\)
\(5\)	−1.56432	−	1.59778i	−0.699587	−	0.714548i
							\(7\)	1.35637	+	2.27162i	0.512658	+	0.858593i
\(11\)	1.95654	−	3.38883i	0.589920	−	1.02177i								−0.404322	−	0.914616i	\(-0.632493\pi\)
							0.994242	−	0.107155i	\(-0.0341740\pi\)
\(13\)		−	2.55354i		−	0.708224i	−0.935203	−	0.354112i	\(-0.884783\pi\)
							0.935203	−	0.354112i	\(-0.115217\pi\)
\(17\)	2.55984	−	4.43377i	0.620852	−	1.07535i	−0.368476	−	0.929637i	\(-0.620120\pi\)
							0.989327	−	0.145709i	\(-0.0465464\pi\)
\(19\)	3.25858	−	1.88134i	0.747569	−	0.431609i	−0.0772457	−	0.997012i	\(-0.524613\pi\)
							0.824815	+	0.565403i	\(0.191279\pi\)
\(23\)	−3.50209	−	6.06580i	−0.730237	−	1.26481i	−0.956782	−	0.290807i	\(-0.906076\pi\)
							0.226545	−	0.974001i	\(-0.427257\pi\)
\(29\)			3.39188i			0.629856i	0.949116	+	0.314928i	\(0.101980\pi\)
							−0.949116	+	0.314928i	\(0.898020\pi\)
\(31\)	1.59267	−	2.75858i	0.286052	−	0.495456i	−0.686812	−	0.726835i	\(-0.740990\pi\)
							0.972864	+	0.231379i	\(0.0743237\pi\)
\(37\)	0.850773	+	1.47358i	0.139866	+	0.242256i	0.927446	−	0.373957i	\(-0.121999\pi\)
							−0.787580	+	0.616213i	\(0.788666\pi\)
\(41\)			3.97345i			0.620548i	0.950647	+	0.310274i	\(0.100421\pi\)
							−0.950647	+	0.310274i	\(0.899579\pi\)
\(43\)			0.898763i			0.137060i	0.997649	+	0.0685300i	\(0.0218309\pi\)
							−0.997649	+	0.0685300i	\(0.978169\pi\)
\(47\)	5.64460	−	3.25891i	0.823350	−	0.475361i	−0.0282202	−	0.999602i	\(-0.508984\pi\)
							0.851570	+	0.524240i	\(0.175651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.ba.b.59.5	yes	80
4.3	odd	2		1120.2.bq.b.1039.17		80
5.4	even	2	inner	280.2.ba.b.59.36	yes	80
7.5	odd	6	inner	280.2.ba.b.19.31	yes	80
8.3	odd	2	inner	280.2.ba.b.59.10	yes	80
8.5	even	2		1120.2.bq.b.1039.18		80
20.19	odd	2		1120.2.bq.b.1039.23		80
28.19	even	6		1120.2.bq.b.719.24		80
35.19	odd	6	inner	280.2.ba.b.19.10	yes	80
40.19	odd	2	inner	280.2.ba.b.59.31	yes	80
40.29	even	2		1120.2.bq.b.1039.24		80
56.5	odd	6		1120.2.bq.b.719.23		80
56.19	even	6	inner	280.2.ba.b.19.36	yes	80
140.19	even	6		1120.2.bq.b.719.18		80
280.19	even	6	inner	280.2.ba.b.19.5	✓	80
280.229	odd	6		1120.2.bq.b.719.17		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.ba.b.19.5	✓	80	280.19	even	6	inner
280.2.ba.b.19.10	yes	80	35.19	odd	6	inner
280.2.ba.b.19.31	yes	80	7.5	odd	6	inner
280.2.ba.b.19.36	yes	80	56.19	even	6	inner
280.2.ba.b.59.5	yes	80	1.1	even	1	trivial
280.2.ba.b.59.10	yes	80	8.3	odd	2	inner
280.2.ba.b.59.31	yes	80	40.19	odd	2	inner
280.2.ba.b.59.36	yes	80	5.4	even	2	inner
1120.2.bq.b.719.17		80	280.229	odd	6
1120.2.bq.b.719.18		80	140.19	even	6
1120.2.bq.b.719.23		80	56.5	odd	6
1120.2.bq.b.719.24		80	28.19	even	6
1120.2.bq.b.1039.17		80	4.3	odd	2
1120.2.bq.b.1039.18		80	8.5	even	2
1120.2.bq.b.1039.23		80	20.19	odd	2
1120.2.bq.b.1039.24		80	40.29	even	2