Embedded newform 1680.2.cz.d.433.4

• Label: 1680.2.cz.d.433.4
• Level: $1680$
• Weight: $2$
• Character: 1680.433
• 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			433.4
Root			\(-1.40927 - 0.118126i\) of defining polynomial
Character	\(\chi\)	\(=\)	1680.433
Dual form			1680.2.cz.d.97.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +(2.23450 - 0.0836010i) q^{5} +(2.64501 - 0.0627175i) q^{7} -1.00000i q^{9} -3.98602 q^{11} +(0.500437 - 0.500437i) q^{13} +(-1.52092 + 1.63915i) q^{15} +(-1.67840 - 1.67840i) q^{17} +7.21850 q^{19} +(-1.82596 + 1.91465i) q^{21} +(5.16007 + 5.16007i) q^{23} +(4.98602 - 0.373614i) q^{25} +(0.707107 + 0.707107i) q^{27} +3.65191i q^{29} -4.93821i q^{31} +(2.81854 - 2.81854i) q^{33} +(5.90504 - 0.361268i) q^{35} +(0.292275 - 0.292275i) q^{37} +0.707725i q^{39} -7.63184i q^{41} +(-3.65191 - 3.65191i) q^{43} +(-0.0836010 - 2.23450i) q^{45} +(0.305303 + 0.305303i) q^{47} +(6.99213 - 0.331777i) q^{49} +2.37361 q^{51} +(5.39653 + 5.39653i) q^{53} +(-8.90678 + 0.333235i) q^{55} +(-5.10425 + 5.10425i) q^{57} -6.10959 q^{59} +7.11047i q^{61} +(-0.0627175 - 2.64501i) q^{63} +(1.07639 - 1.16007i) q^{65} +(-0.944185 + 0.944185i) q^{67} -7.29744 q^{69} -1.19297 q^{71} +(1.38298 - 1.38298i) q^{73} +(-3.26147 + 3.78983i) q^{75} +(-10.5431 + 0.249993i) q^{77} +8.64027i q^{79} -1.00000 q^{81} +(11.9895 - 11.9895i) q^{83} +(-3.89070 - 3.61007i) q^{85} +(-2.58229 - 2.58229i) q^{87} +7.82581 q^{89} +(1.29227 - 1.35505i) q^{91} +(3.49184 + 3.49184i) q^{93} +(16.1298 - 0.603474i) q^{95} +(7.43671 + 7.43671i) q^{97} +3.98602i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(1\)	\(1\)

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
							\(7\)	2.64501	−	0.0627175i	0.999719	−	0.0237050i
\(11\)	−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	\(-0.718709\pi\)
							0.773091	+	0.634295i	\(0.218709\pi\)
\(17\)	−1.67840	−	1.67840i	−0.407071	−	0.407071i	0.473645	−	0.880716i	\(-0.342938\pi\)
							−0.880716	+	0.473645i	\(0.842938\pi\)
\(19\)	7.21850			1.65604			0.828019	−	0.560700i	\(-0.189468\pi\)
							0.828019	+	0.560700i	\(0.189468\pi\)
\(23\)	5.16007	+	5.16007i	1.07595	+	1.07595i	0.996868	+	0.0790800i	\(0.0251983\pi\)
							0.0790800	+	0.996868i	\(0.474802\pi\)
\(29\)			3.65191i			0.678143i	0.940761	+	0.339071i	\(0.110113\pi\)
							−0.940761	+	0.339071i	\(0.889887\pi\)
\(31\)		−	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	−0.682674	−	0.730723i	\(-0.739183\pi\)
							0.730723	+	0.682674i	\(0.239183\pi\)
\(41\)		−	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	0.371516	−	0.928427i	\(-0.378838\pi\)
							−0.928427	+	0.371516i	\(0.878838\pi\)
\(47\)	0.305303	+	0.305303i	0.0445331	+	0.0445331i	0.729023	−	0.684490i	\(-0.239975\pi\)
							−0.684490	+	0.729023i	\(0.739975\pi\)
\(53\)	5.39653	+	5.39653i	0.741270	+	0.741270i	0.972822	−	0.231553i	\(-0.0743805\pi\)
							−0.231553	+	0.972822i	\(0.574381\pi\)
\(59\)	−6.10959			−0.795401			−0.397701	−	0.917515i	\(-0.630192\pi\)
							−0.397701	+	0.917515i	\(0.630192\pi\)
\(61\)			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	−0.762426	−	0.647075i	\(-0.775992\pi\)
							0.647075	+	0.762426i	\(0.275992\pi\)
\(71\)	−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	\(-0.713487\pi\)
							0.783393	+	0.621527i	\(0.213487\pi\)
\(79\)			8.64027i			0.972106i	0.873929	+	0.486053i	\(0.161564\pi\)
							−0.873929	+	0.486053i	\(0.838436\pi\)
\(83\)	11.9895	−	11.9895i	1.31602	−	1.31602i	0.399122	−	0.916898i	\(-0.369315\pi\)
							0.916898	−	0.399122i	\(-0.130685\pi\)
\(89\)	7.82581			0.829534			0.414767	−	0.909928i	\(-0.363863\pi\)
							0.414767	+	0.909928i	\(0.363863\pi\)
\(97\)	7.43671	+	7.43671i	0.755083	+	0.755083i	0.975423	−	0.220340i	\(-0.0707167\pi\)
							−0.220340	+	0.975423i	\(0.570717\pi\)

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.433.4		16
4.3	odd	2		105.2.m.a.13.4	yes	16
5.2	odd	4	inner	1680.2.cz.d.97.5		16
7.6	odd	2	inner	1680.2.cz.d.433.5		16
12.11	even	2		315.2.p.e.118.5		16
20.3	even	4		525.2.m.b.307.6		16
20.7	even	4		105.2.m.a.97.3	yes	16
20.19	odd	2		525.2.m.b.118.5		16
28.3	even	6		735.2.v.a.313.4		32
28.11	odd	6		735.2.v.a.313.3		32
28.19	even	6		735.2.v.a.178.5		32
28.23	odd	6		735.2.v.a.178.6		32
28.27	even	2		105.2.m.a.13.3	✓	16
35.27	even	4	inner	1680.2.cz.d.97.4		16
60.47	odd	4		315.2.p.e.307.6		16
84.83	odd	2		315.2.p.e.118.6		16
140.27	odd	4		105.2.m.a.97.4	yes	16
140.47	odd	12		735.2.v.a.472.3		32
140.67	even	12		735.2.v.a.607.5		32
140.83	odd	4		525.2.m.b.307.5		16
140.87	odd	12		735.2.v.a.607.6		32
140.107	even	12		735.2.v.a.472.4		32
140.139	even	2		525.2.m.b.118.6		16
420.167	even	4		315.2.p.e.307.5		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.m.a.13.3	✓	16	28.27	even	2
105.2.m.a.13.4	yes	16	4.3	odd	2
105.2.m.a.97.3	yes	16	20.7	even	4
105.2.m.a.97.4	yes	16	140.27	odd	4
315.2.p.e.118.5		16	12.11	even	2
315.2.p.e.118.6		16	84.83	odd	2
315.2.p.e.307.5		16	420.167	even	4
315.2.p.e.307.6		16	60.47	odd	4
525.2.m.b.118.5		16	20.19	odd	2
525.2.m.b.118.6		16	140.139	even	2
525.2.m.b.307.5		16	140.83	odd	4
525.2.m.b.307.6		16	20.3	even	4
735.2.v.a.178.5		32	28.19	even	6
735.2.v.a.178.6		32	28.23	odd	6
735.2.v.a.313.3		32	28.11	odd	6
735.2.v.a.313.4		32	28.3	even	6
735.2.v.a.472.3		32	140.47	odd	12
735.2.v.a.472.4		32	140.107	even	12
735.2.v.a.607.5		32	140.67	even	12
735.2.v.a.607.6		32	140.87	odd	12
1680.2.cz.d.97.4		16	35.27	even	4	inner
1680.2.cz.d.97.5		16	5.2	odd	4	inner
1680.2.cz.d.433.4		16	1.1	even	1	trivial
1680.2.cz.d.433.5		16	7.6	odd	2	inner