Embedded newform 1680.2.cz.d.97.5

• Label: 1680.2.cz.d.97.5
• 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			97.5
Root			\(1.40927 - 0.118126i\) of defining polynomial
Character	\(\chi\)	\(=\)	1680.97
Dual form			1680.2.cz.d.433.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{3} +(-2.23450 - 0.0836010i) q^{5} +(0.0627175 + 2.64501i) q^{7} +1.00000i q^{9} -3.98602 q^{11} +(-0.500437 - 0.500437i) q^{13} +(-1.52092 - 1.63915i) q^{15} +(1.67840 - 1.67840i) q^{17} -7.21850 q^{19} +(-1.82596 + 1.91465i) q^{21} +(5.16007 - 5.16007i) q^{23} +(4.98602 + 0.373614i) q^{25} +(-0.707107 + 0.707107i) q^{27} -3.65191i q^{29} -4.93821i q^{31} +(-2.81854 - 2.81854i) q^{33} +(0.0809828 - 5.91553i) q^{35} +(0.292275 + 0.292275i) q^{37} -0.707725i q^{39} -7.63184i q^{41} +(-3.65191 + 3.65191i) q^{43} +(0.0836010 - 2.23450i) q^{45} +(-0.305303 + 0.305303i) q^{47} +(-6.99213 + 0.331777i) q^{49} +2.37361 q^{51} +(5.39653 - 5.39653i) q^{53} +(8.90678 + 0.333235i) q^{55} +(-5.10425 - 5.10425i) q^{57} +6.10959 q^{59} +7.11047i q^{61} +(-2.64501 + 0.0627175i) q^{63} +(1.07639 + 1.16007i) q^{65} +(-0.944185 - 0.944185i) q^{67} +7.29744 q^{69} -1.19297 q^{71} +(-1.38298 - 1.38298i) q^{73} +(3.26147 + 3.78983i) q^{75} +(-0.249993 - 10.5431i) q^{77} -8.64027i q^{79} -1.00000 q^{81} +(-11.9895 - 11.9895i) q^{83} +(-3.89070 + 3.61007i) q^{85} +(2.58229 - 2.58229i) q^{87} -7.82581 q^{89} +(1.29227 - 1.35505i) q^{91} +(3.49184 - 3.49184i) q^{93} +(16.1298 + 0.603474i) q^{95} +(-7.43671 + 7.43671i) q^{97} -3.98602i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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
\(5\)	−2.23450	−	0.0836010i	−0.999301	−	0.0373875i
							\(7\)	0.0627175	+	2.64501i	0.0237050	+	0.999719i
\(11\)	−3.98602			−1.20183										−0.600915	−	0.799313i	\(-0.705197\pi\)
							−0.600915	+	0.799313i	\(0.705197\pi\)
\(13\)	−0.500437	−	0.500437i	−0.138796	−	0.138796i	0.634295	−	0.773091i	\(-0.281291\pi\)
							−0.773091	+	0.634295i	\(0.781291\pi\)
\(17\)	1.67840	−	1.67840i	0.407071	−	0.407071i	−0.473645	−	0.880716i	\(-0.657062\pi\)
							0.880716	+	0.473645i	\(0.157062\pi\)
\(19\)	−7.21850			−1.65604			−0.828019	−	0.560700i	\(-0.810532\pi\)
							−0.828019	+	0.560700i	\(0.810532\pi\)
\(23\)	5.16007	−	5.16007i	1.07595	−	1.07595i	0.0790800	−	0.996868i	\(-0.474802\pi\)
							0.996868	−	0.0790800i	\(-0.0251983\pi\)
\(29\)		−	3.65191i		−	0.678143i	−0.940761	−	0.339071i	\(-0.889887\pi\)
							0.940761	−	0.339071i	\(-0.110113\pi\)
\(31\)		−	4.93821i		−	0.886929i	−0.896292	−	0.443465i	\(-0.853749\pi\)
							0.896292	−	0.443465i	\(-0.146251\pi\)
\(37\)	0.292275	+	0.292275i	0.0480497	+	0.0480497i	0.730723	−	0.682674i	\(-0.239183\pi\)
							−0.682674	+	0.730723i	\(0.739183\pi\)
\(41\)		−	7.63184i		−	1.19189i	−0.803024	−	0.595947i	\(-0.796777\pi\)
							0.803024	−	0.595947i	\(-0.203223\pi\)
\(43\)	−3.65191	+	3.65191i	−0.556911	+	0.556911i	−0.928427	−	0.371516i	\(-0.878838\pi\)
							0.371516	+	0.928427i	\(0.378838\pi\)
\(47\)	−0.305303	+	0.305303i	−0.0445331	+	0.0445331i	−0.729023	−	0.684490i	\(-0.760025\pi\)
							0.684490	+	0.729023i	\(0.260025\pi\)
\(53\)	5.39653	−	5.39653i	0.741270	−	0.741270i	−0.231553	−	0.972822i	\(-0.574381\pi\)
							0.972822	+	0.231553i	\(0.0743805\pi\)
\(59\)	6.10959			0.795401			0.397701	−	0.917515i	\(-0.369808\pi\)
							0.397701	+	0.917515i	\(0.369808\pi\)
\(61\)			7.11047i			0.910402i	0.890389	+	0.455201i	\(0.150433\pi\)
							−0.890389	+	0.455201i	\(0.849567\pi\)
\(67\)	−0.944185	−	0.944185i	−0.115351	−	0.115351i	0.647075	−	0.762426i	\(-0.275992\pi\)
							−0.762426	+	0.647075i	\(0.775992\pi\)
\(71\)	−1.19297			−0.141579			−0.0707897	−	0.997491i	\(-0.522552\pi\)
							−0.0707897	+	0.997491i	\(0.522552\pi\)
\(73\)	−1.38298	−	1.38298i	−0.161865	−	0.161865i	0.621527	−	0.783393i	\(-0.286513\pi\)
							−0.783393	+	0.621527i	\(0.786513\pi\)
\(79\)		−	8.64027i		−	0.972106i	−0.873929	−	0.486053i	\(-0.838436\pi\)
							0.873929	−	0.486053i	\(-0.161564\pi\)
\(83\)	−11.9895	−	11.9895i	−1.31602	−	1.31602i	−0.916898	−	0.399122i	\(-0.869315\pi\)
							−0.399122	−	0.916898i	\(-0.630685\pi\)
\(89\)	−7.82581			−0.829534			−0.414767	−	0.909928i	\(-0.636137\pi\)
							−0.414767	+	0.909928i	\(0.636137\pi\)
\(97\)	−7.43671	+	7.43671i	−0.755083	+	0.755083i	−0.975423	−	0.220340i	\(-0.929283\pi\)
							0.220340	+	0.975423i	\(0.429283\pi\)

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.97.5		16
4.3	odd	2		105.2.m.a.97.3	yes	16
5.3	odd	4	inner	1680.2.cz.d.433.4		16
7.6	odd	2	inner	1680.2.cz.d.97.4		16
12.11	even	2		315.2.p.e.307.6		16
20.3	even	4		105.2.m.a.13.4	yes	16
20.7	even	4		525.2.m.b.118.5		16
20.19	odd	2		525.2.m.b.307.6		16
28.3	even	6		735.2.v.a.607.6		32
28.11	odd	6		735.2.v.a.607.5		32
28.19	even	6		735.2.v.a.472.3		32
28.23	odd	6		735.2.v.a.472.4		32
28.27	even	2		105.2.m.a.97.4	yes	16
35.13	even	4	inner	1680.2.cz.d.433.5		16
60.23	odd	4		315.2.p.e.118.5		16
84.83	odd	2		315.2.p.e.307.5		16
140.3	odd	12		735.2.v.a.313.4		32
140.23	even	12		735.2.v.a.178.6		32
140.27	odd	4		525.2.m.b.118.6		16
140.83	odd	4		105.2.m.a.13.3	✓	16
140.103	odd	12		735.2.v.a.178.5		32
140.123	even	12		735.2.v.a.313.3		32
140.139	even	2		525.2.m.b.307.5		16
420.83	even	4		315.2.p.e.118.6		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.m.a.13.3	✓	16	140.83	odd	4
105.2.m.a.13.4	yes	16	20.3	even	4
105.2.m.a.97.3	yes	16	4.3	odd	2
105.2.m.a.97.4	yes	16	28.27	even	2
315.2.p.e.118.5		16	60.23	odd	4
315.2.p.e.118.6		16	420.83	even	4
315.2.p.e.307.5		16	84.83	odd	2
315.2.p.e.307.6		16	12.11	even	2
525.2.m.b.118.5		16	20.7	even	4
525.2.m.b.118.6		16	140.27	odd	4
525.2.m.b.307.5		16	140.139	even	2
525.2.m.b.307.6		16	20.19	odd	2
735.2.v.a.178.5		32	140.103	odd	12
735.2.v.a.178.6		32	140.23	even	12
735.2.v.a.313.3		32	140.123	even	12
735.2.v.a.313.4		32	140.3	odd	12
735.2.v.a.472.3		32	28.19	even	6
735.2.v.a.472.4		32	28.23	odd	6
735.2.v.a.607.5		32	28.11	odd	6
735.2.v.a.607.6		32	28.3	even	6
1680.2.cz.d.97.4		16	7.6	odd	2	inner
1680.2.cz.d.97.5		16	1.1	even	1	trivial
1680.2.cz.d.433.4		16	5.3	odd	4	inner
1680.2.cz.d.433.5		16	35.13	even	4	inner