Embedded newform 980.2.v.b.717.1

• Label: 980.2.v.b.717.1
• Level: $980$
• Weight: $2$
• Character: 980.717
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.v (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	16.0.878943153267362859319296.1
Defining polynomial:	\( x^{16} - 45x^{12} + 2021x^{8} - 180x^{4} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			717.1
Root			\(-2.50053 + 0.670015i\) of defining polynomial
Character	\(\chi\)	\(=\)	980.717
Dual form			980.2.v.b.313.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.50053 + 0.670015i) q^{3} +(-2.02742 - 0.943162i) q^{5} +(3.20565 - 1.85078i) q^{9} +(2.35078 - 4.07167i) q^{11} +(1.83051 + 1.83051i) q^{13} +(5.70156 + 1.00000i) q^{15} +(-0.270099 - 1.00802i) q^{17} +(2.37681 + 4.11675i) q^{19} +(-5.05643 - 1.35487i) q^{23} +(3.22089 + 3.82438i) q^{25} +(-1.28422 + 1.28422i) q^{27} +0.701562i q^{29} +(-7.61921 - 4.39895i) q^{31} +(-3.15012 + 11.7564i) q^{33} +(-1.35487 + 5.05643i) q^{37} +(-5.80372 - 3.35078i) q^{39} -4.75362i q^{41} +(-5.00000 + 5.00000i) q^{43} +(-8.24479 + 0.728873i) q^{45} +(-11.0101 - 2.95016i) q^{47} +(1.35078 + 2.33962i) q^{51} +(1.83013 + 6.83013i) q^{53} +(-8.60627 + 6.03784i) q^{55} +(-8.70156 - 8.70156i) q^{57} +(-2.37681 + 4.11675i) q^{59} +(8.23351 - 4.75362i) q^{61} +(-1.98476 - 5.43770i) q^{65} +(-6.83013 + 1.83013i) q^{67} +13.5515 q^{69} -5.40312 q^{71} +(-4.25480 + 1.14007i) q^{73} +(-10.6163 - 7.40492i) q^{75} +(-5.80372 + 3.35078i) q^{79} +(-3.20156 + 5.54527i) q^{81} +(7.86835 + 7.86835i) q^{83} +(-0.403124 + 2.29844i) q^{85} +(-0.470057 - 1.75428i) q^{87} +(6.77576 + 11.7360i) q^{89} +(21.9994 + 5.89472i) q^{93} +(-0.936032 - 10.5881i) q^{95} +(-5.49154 + 5.49154i) q^{97} -17.4031i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{11} + 40 q^{15} - 4 q^{23} + 12 q^{25} + 4 q^{37} - 80 q^{43} - 4 q^{51} - 40 q^{53} - 88 q^{57} - 20 q^{65} - 40 q^{67} + 16 q^{71} + 96 q^{85} + 52 q^{93} + 44 q^{95}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)	−2.50053	+	0.670015i	−1.44368	+	0.386833i	−0.893821	−	0.448424i	\(-0.851986\pi\)
−0.549860	+	0.835257i	\(0.685319\pi\)
\(5\)	−2.02742	−	0.943162i	−0.906691	−	0.421795i
							\(7\)		0			0
\(11\)	2.35078	−	4.07167i	0.708787	−	1.22766i								−0.256520	−	0.966539i	\(-0.582576\pi\)
							0.965307	−	0.261116i	\(-0.0840907\pi\)
\(13\)	1.83051	+	1.83051i	0.507693	+	0.507693i	0.913818	−	0.406125i	\(-0.133120\pi\)
							−0.406125	+	0.913818i	\(0.633120\pi\)
\(17\)	−0.270099	−	1.00802i	−0.0655087	−	0.244482i	0.925405	−	0.378979i	\(-0.123725\pi\)
							−0.990914	+	0.134497i	\(0.957058\pi\)
\(19\)	2.37681	+	4.11675i	0.545277	+	0.944448i	0.998589	+	0.0530961i	\(0.0169090\pi\)
							−0.453312	+	0.891352i	\(0.649758\pi\)
\(23\)	−5.05643	−	1.35487i	−1.05434	−	0.282509i	−0.310295	−	0.950640i	\(-0.600428\pi\)
							−0.744043	+	0.668131i	\(0.767094\pi\)
\(29\)			0.701562i			0.130277i	0.997876	+	0.0651384i	\(0.0207489\pi\)
							−0.997876	+	0.0651384i	\(0.979251\pi\)
\(31\)	−7.61921	−	4.39895i	−1.36845	−	0.790075i	−0.377720	−	0.925920i	\(-0.623292\pi\)
							−0.990730	+	0.135845i	\(0.956625\pi\)
\(37\)	−1.35487	+	5.05643i	−0.222739	+	0.831272i	0.760559	+	0.649268i	\(0.224925\pi\)
							−0.983298	+	0.182003i	\(0.941742\pi\)
\(41\)		−	4.75362i		−	0.742390i	−0.928555	−	0.371195i	\(-0.878948\pi\)
							0.928555	−	0.371195i	\(-0.121052\pi\)
\(43\)	−5.00000	+	5.00000i	−0.762493	+	0.762493i	−0.976772	−	0.214280i	\(-0.931260\pi\)
							0.214280	+	0.976772i	\(0.431260\pi\)
\(47\)	−11.0101	−	2.95016i	−1.60599	−	0.430325i	−0.659148	−	0.752013i	\(-0.729083\pi\)
							−0.946846	+	0.321689i	\(0.895750\pi\)
\(53\)	1.83013	+	6.83013i	0.251387	+	0.938190i	0.970065	+	0.242846i	\(0.0780811\pi\)
							−0.718677	+	0.695344i	\(0.755252\pi\)
\(59\)	−2.37681	+	4.11675i	−0.309434	+	0.535956i	−0.978239	−	0.207483i	\(-0.933473\pi\)
							0.668805	+	0.743438i	\(0.266806\pi\)
\(61\)	8.23351	−	4.75362i	1.05419	−	0.608638i	0.130372	−	0.991465i	\(-0.458383\pi\)
							0.923820	+	0.382827i	\(0.125049\pi\)
\(67\)	−6.83013	+	1.83013i	−0.834433	+	0.223586i	−0.650647	−	0.759381i	\(-0.725502\pi\)
							−0.183786	+	0.982966i	\(0.558835\pi\)
\(71\)	−5.40312			−0.641233			−0.320616	−	0.947209i	\(-0.603890\pi\)
							−0.320616	+	0.947209i	\(0.603890\pi\)
\(73\)	−4.25480	+	1.14007i	−0.497987	+	0.133435i	−0.499066	−	0.866564i	\(-0.666323\pi\)
							0.00107850	+	0.999999i	\(0.499657\pi\)
\(79\)	−5.80372	+	3.35078i	−0.652970	+	0.376992i	−0.789593	−	0.613631i	\(-0.789708\pi\)
							0.136623	+	0.990623i	\(0.456375\pi\)
\(83\)	7.86835	+	7.86835i	0.863664	+	0.863664i	0.991762	−	0.128098i	\(-0.0408872\pi\)
							−0.128098	+	0.991762i	\(0.540887\pi\)
\(89\)	6.77576	+	11.7360i	0.718229	+	1.24401i	0.961701	+	0.274101i	\(0.0883804\pi\)
							−0.243472	+	0.969908i	\(0.578286\pi\)
\(97\)	−5.49154	+	5.49154i	−0.557582	+	0.557582i	−0.928618	−	0.371037i	\(-0.879002\pi\)
							0.371037	+	0.928618i	\(0.379002\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.v.b.717.1		16
5.3	odd	4	inner	980.2.v.b.913.1		16
7.2	even	3	inner	980.2.v.b.117.4		16
7.3	odd	6		140.2.m.a.97.1	yes	8
7.4	even	3		140.2.m.a.97.4	yes	8
7.5	odd	6	inner	980.2.v.b.117.1		16
7.6	odd	2	inner	980.2.v.b.717.4		16
21.11	odd	6		1260.2.ba.a.937.1		8
21.17	even	6		1260.2.ba.a.937.4		8
28.3	even	6		560.2.bj.b.97.4		8
28.11	odd	6		560.2.bj.b.97.1		8
35.3	even	12		140.2.m.a.13.4	yes	8
35.4	even	6		700.2.m.c.657.1		8
35.13	even	4	inner	980.2.v.b.913.4		16
35.17	even	12		700.2.m.c.293.1		8
35.18	odd	12		140.2.m.a.13.1	✓	8
35.23	odd	12	inner	980.2.v.b.313.4		16
35.24	odd	6		700.2.m.c.657.4		8
35.32	odd	12		700.2.m.c.293.4		8
35.33	even	12	inner	980.2.v.b.313.1		16
105.38	odd	12		1260.2.ba.a.433.1		8
105.53	even	12		1260.2.ba.a.433.4		8
140.3	odd	12		560.2.bj.b.433.1		8
140.123	even	12		560.2.bj.b.433.4		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.m.a.13.1	✓	8	35.18	odd	12
140.2.m.a.13.4	yes	8	35.3	even	12
140.2.m.a.97.1	yes	8	7.3	odd	6
140.2.m.a.97.4	yes	8	7.4	even	3
560.2.bj.b.97.1		8	28.11	odd	6
560.2.bj.b.97.4		8	28.3	even	6
560.2.bj.b.433.1		8	140.3	odd	12
560.2.bj.b.433.4		8	140.123	even	12
700.2.m.c.293.1		8	35.17	even	12
700.2.m.c.293.4		8	35.32	odd	12
700.2.m.c.657.1		8	35.4	even	6
700.2.m.c.657.4		8	35.24	odd	6
980.2.v.b.117.1		16	7.5	odd	6	inner
980.2.v.b.117.4		16	7.2	even	3	inner
980.2.v.b.313.1		16	35.33	even	12	inner
980.2.v.b.313.4		16	35.23	odd	12	inner
980.2.v.b.717.1		16	1.1	even	1	trivial
980.2.v.b.717.4		16	7.6	odd	2	inner
980.2.v.b.913.1		16	5.3	odd	4	inner
980.2.v.b.913.4		16	35.13	even	4	inner
1260.2.ba.a.433.1		8	105.38	odd	12
1260.2.ba.a.433.4		8	105.53	even	12
1260.2.ba.a.937.1		8	21.11	odd	6
1260.2.ba.a.937.4		8	21.17	even	6