Embedded newform 1680.2.cz.d.97.8

• Label: 1680.2.cz.d.97.8
• Level: $1680$
• Weight: $2$
• Character: 1680.97
• Analytic conductor: $13.415$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.cz (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4148675396\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			97.8
Root			\(-0.944649 + 1.05244i\) of defining polynomial
Character	\(\chi\)	\(=\)	1680.97
Dual form			1680.2.cz.d.433.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{3} +(1.28999 + 1.82645i) q^{5} +(1.75993 + 1.97552i) q^{7} +1.00000i q^{9} +2.67187 q^{11} +(-1.22714 - 1.22714i) q^{13} +(-0.379340 + 2.20366i) q^{15} +(4.74624 - 4.74624i) q^{17} +6.01729 q^{19} +(-0.152445 + 2.64136i) q^{21} +(0.175684 - 0.175684i) q^{23} +(-1.67187 + 4.71220i) q^{25} +(-0.707107 + 0.707107i) q^{27} -0.304889i q^{29} -7.25379i q^{31} +(1.88930 + 1.88930i) q^{33} +(-1.33791 + 5.76281i) q^{35} +(-0.735441 - 0.735441i) q^{37} -1.73544i q^{39} +7.05736i q^{41} +(-0.304889 + 0.304889i) q^{43} +(-1.82645 + 1.28999i) q^{45} +(0.556866 - 0.556866i) q^{47} +(-0.805321 + 6.95352i) q^{49} +6.71220 q^{51} +(-4.99031 + 4.99031i) q^{53} +(3.44668 + 4.88005i) q^{55} +(4.25487 + 4.25487i) q^{57} -7.98837 q^{59} -5.53409i q^{61} +(-1.97552 + 1.75993i) q^{63} +(0.658323 - 3.82432i) q^{65} +(3.43055 + 3.43055i) q^{67} +0.248455 q^{69} -15.3087 q^{71} +(-10.0208 - 10.0208i) q^{73} +(-4.51422 + 2.14984i) q^{75} +(4.70230 + 5.27832i) q^{77} +11.2973i q^{79} -1.00000 q^{81} +(-4.88941 - 4.88941i) q^{83} +(14.7914 + 2.54621i) q^{85} +(0.215589 - 0.215589i) q^{87} -6.91251 q^{89} +(0.264559 - 4.58392i) q^{91} +(5.12921 - 5.12921i) q^{93} +(7.76222 + 10.9903i) q^{95} +(8.84137 - 8.84137i) q^{97} +2.67187i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q + 8 * q^7 + 16 * q^11 - 8 * q^15 + 8 * q^21 + 40 * q^23 + 8 * q^35 + 32 * q^37 + 16 * q^43 + 16 * q^51 + 24 * q^53 + 8 * q^57 - 8 * q^63 + 40 * q^65 + 32 * q^67 - 64 * q^71 - 24 * q^77 - 16 * q^81 + 48 * q^85 + 48 * q^91 + 24 * q^93 + 72 * q^95

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(1\)	\(1\)

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
							\(3\)	0.707107	+	0.707107i	0.408248	+	0.408248i
\(5\)	1.28999	+	1.82645i	0.576899	+	0.816815i
							\(7\)	1.75993	+	1.97552i	0.665189	+	0.746675i
\(11\)	2.67187			0.805600										0.402800	−	0.915288i	\(-0.368037\pi\)
							0.402800	+	0.915288i	\(0.368037\pi\)
\(13\)	−1.22714	−	1.22714i	−0.340348	−	0.340348i	0.516150	−	0.856498i	\(-0.327365\pi\)
							−0.856498	+	0.516150i	\(0.827365\pi\)
\(17\)	4.74624	−	4.74624i	1.15113	−	1.15113i	0.164807	−	0.986326i	\(-0.447300\pi\)
							0.986326	−	0.164807i	\(-0.0527002\pi\)
\(19\)	6.01729			1.38046			0.690231	−	0.723589i	\(-0.257509\pi\)
							0.690231	+	0.723589i	\(0.257509\pi\)
\(23\)	0.175684	−	0.175684i	0.0366327	−	0.0366327i	−0.688553	−	0.725186i	\(-0.741754\pi\)
							0.725186	+	0.688553i	\(0.241754\pi\)
\(29\)		−	0.304889i		−	0.0566165i	−0.999599	−	0.0283083i	\(-0.990988\pi\)
							0.999599	−	0.0283083i	\(-0.00901200\pi\)
\(31\)		−	7.25379i		−	1.30282i	−0.758726	−	0.651410i	\(-0.774178\pi\)
							0.758726	−	0.651410i	\(-0.225822\pi\)
\(37\)	−0.735441	−	0.735441i	−0.120906	−	0.120906i	0.644065	−	0.764971i	\(-0.277247\pi\)
							−0.764971	+	0.644065i	\(0.777247\pi\)
\(41\)			7.05736i			1.10217i	0.834447	+	0.551087i	\(0.185787\pi\)
							−0.834447	+	0.551087i	\(0.814213\pi\)
\(43\)	−0.304889	+	0.304889i	−0.0464952	+	0.0464952i	−0.729972	−	0.683477i	\(-0.760467\pi\)
							0.683477	+	0.729972i	\(0.260467\pi\)
\(47\)	0.556866	−	0.556866i	0.0812273	−	0.0812273i	−0.665326	−	0.746553i	\(-0.731707\pi\)
							0.746553	+	0.665326i	\(0.231707\pi\)
\(53\)	−4.99031	+	4.99031i	−0.685472	+	0.685472i	−0.961228	−	0.275756i	\(-0.911072\pi\)
							0.275756	+	0.961228i	\(0.411072\pi\)
\(59\)	−7.98837			−1.04000			−0.519999	−	0.854167i	\(-0.674068\pi\)
							−0.519999	+	0.854167i	\(0.674068\pi\)
\(61\)		−	5.53409i		−	0.708567i	−0.935138	−	0.354284i	\(-0.884725\pi\)
							0.935138	−	0.354284i	\(-0.115275\pi\)
\(67\)	3.43055	+	3.43055i	0.419109	+	0.419109i	0.884896	−	0.465788i	\(-0.154229\pi\)
							−0.465788	+	0.884896i	\(0.654229\pi\)
\(71\)	−15.3087			−1.81681			−0.908407	−	0.418087i	\(-0.862701\pi\)
							−0.908407	+	0.418087i	\(0.862701\pi\)
\(73\)	−10.0208	−	10.0208i	−1.17285	−	1.17285i	−0.981527	−	0.191323i	\(-0.938722\pi\)
							−0.191323	−	0.981527i	\(-0.561278\pi\)
\(79\)			11.2973i			1.27104i	0.772084	+	0.635521i	\(0.219215\pi\)
							−0.772084	+	0.635521i	\(0.780785\pi\)
\(83\)	−4.88941	−	4.88941i	−0.536682	−	0.536682i	0.385871	−	0.922553i	\(-0.373901\pi\)
							−0.922553	+	0.385871i	\(0.873901\pi\)
\(89\)	−6.91251			−0.732725			−0.366363	−	0.930472i	\(-0.619397\pi\)
							−0.366363	+	0.930472i	\(0.619397\pi\)
\(97\)	8.84137	−	8.84137i	0.897705	−	0.897705i	−0.0975276	−	0.995233i	\(-0.531093\pi\)
							0.995233	+	0.0975276i	\(0.0310934\pi\)

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	1680.2.cz.d.97.8		16
4.3	odd	2		105.2.m.a.97.7	yes	16
5.3	odd	4	inner	1680.2.cz.d.433.1		16
7.6	odd	2	inner	1680.2.cz.d.97.1		16
12.11	even	2		315.2.p.e.307.1		16
20.3	even	4		105.2.m.a.13.8	yes	16
20.7	even	4		525.2.m.b.118.1		16
20.19	odd	2		525.2.m.b.307.2		16
28.3	even	6		735.2.v.a.607.2		32
28.11	odd	6		735.2.v.a.607.1		32
28.19	even	6		735.2.v.a.472.7		32
28.23	odd	6		735.2.v.a.472.8		32
28.27	even	2		105.2.m.a.97.8	yes	16
35.13	even	4	inner	1680.2.cz.d.433.8		16
60.23	odd	4		315.2.p.e.118.2		16
84.83	odd	2		315.2.p.e.307.2		16
140.3	odd	12		735.2.v.a.313.8		32
140.23	even	12		735.2.v.a.178.2		32
140.27	odd	4		525.2.m.b.118.2		16
140.83	odd	4		105.2.m.a.13.7	✓	16
140.103	odd	12		735.2.v.a.178.1		32
140.123	even	12		735.2.v.a.313.7		32
140.139	even	2		525.2.m.b.307.1		16
420.83	even	4		315.2.p.e.118.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.m.a.13.7	✓	16	140.83	odd	4
105.2.m.a.13.8	yes	16	20.3	even	4
105.2.m.a.97.7	yes	16	4.3	odd	2
105.2.m.a.97.8	yes	16	28.27	even	2
315.2.p.e.118.1		16	420.83	even	4
315.2.p.e.118.2		16	60.23	odd	4
315.2.p.e.307.1		16	12.11	even	2
315.2.p.e.307.2		16	84.83	odd	2
525.2.m.b.118.1		16	20.7	even	4
525.2.m.b.118.2		16	140.27	odd	4
525.2.m.b.307.1		16	140.139	even	2
525.2.m.b.307.2		16	20.19	odd	2
735.2.v.a.178.1		32	140.103	odd	12
735.2.v.a.178.2		32	140.23	even	12
735.2.v.a.313.7		32	140.123	even	12
735.2.v.a.313.8		32	140.3	odd	12
735.2.v.a.472.7		32	28.19	even	6
735.2.v.a.472.8		32	28.23	odd	6
735.2.v.a.607.1		32	28.11	odd	6
735.2.v.a.607.2		32	28.3	even	6
1680.2.cz.d.97.1		16	7.6	odd	2	inner
1680.2.cz.d.97.8		16	1.1	even	1	trivial
1680.2.cz.d.433.1		16	5.3	odd	4	inner
1680.2.cz.d.433.8		16	35.13	even	4	inner