Embedded newform 320.5.p.d.257.1

• Label: 320.5.p.d.257.1
• Level: $320$
• Weight: $5$
• Character: 320.257
• Analytic conductor: $33.078$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	320.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(33.0783881868\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			257.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	320.257
Dual form			320.5.p.d.193.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{3} +(15.0000 + 20.0000i) q^{5} +(-19.0000 - 19.0000i) q^{7} +79.0000i q^{9} -202.000 q^{11} +(99.0000 - 99.0000i) q^{13} +(-35.0000 - 5.00000i) q^{15} +(-239.000 - 239.000i) q^{17} -40.0000i q^{19} +38.0000 q^{21} +(541.000 - 541.000i) q^{23} +(-175.000 + 600.000i) q^{25} +(-160.000 - 160.000i) q^{27} +200.000i q^{29} -758.000 q^{31} +(202.000 - 202.000i) q^{33} +(95.0000 - 665.000i) q^{35} +(-141.000 - 141.000i) q^{37} +198.000i q^{39} +1042.00 q^{41} +(759.000 - 759.000i) q^{43} +(-1580.00 + 1185.00i) q^{45} +(-459.000 - 459.000i) q^{47} -1679.00i q^{49} +478.000 q^{51} +(1819.00 - 1819.00i) q^{53} +(-3030.00 - 4040.00i) q^{55} +(40.0000 + 40.0000i) q^{57} -4600.00i q^{59} -2082.00 q^{61} +(1501.00 - 1501.00i) q^{63} +(3465.00 + 495.000i) q^{65} +(-5081.00 - 5081.00i) q^{67} +1082.00i q^{69} -3478.00 q^{71} +(-3479.00 + 3479.00i) q^{73} +(-425.000 - 775.000i) q^{75} +(3838.00 + 3838.00i) q^{77} -7680.00i q^{79} -6079.00 q^{81} +(-6081.00 + 6081.00i) q^{83} +(1195.00 - 8365.00i) q^{85} +(-200.000 - 200.000i) q^{87} -5680.00i q^{89} -3762.00 q^{91} +(758.000 - 758.000i) q^{93} +(800.000 - 600.000i) q^{95} +(561.000 + 561.000i) q^{97} -15958.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^3 + 30 * q^5 - 38 * q^7 - 404 * q^11 + 198 * q^13 - 70 * q^15 - 478 * q^17 + 76 * q^21 + 1082 * q^23 - 350 * q^25 - 320 * q^27 - 1516 * q^31 + 404 * q^33 + 190 * q^35 - 282 * q^37 + 2084 * q^41 + 1518 * q^43 - 3160 * q^45 - 918 * q^47 + 956 * q^51 + 3638 * q^53 - 6060 * q^55 + 80 * q^57 - 4164 * q^61 + 3002 * q^63 + 6930 * q^65 - 10162 * q^67 - 6956 * q^71 - 6958 * q^73 - 850 * q^75 + 7676 * q^77 - 12158 * q^81 - 12162 * q^83 + 2390 * q^85 - 400 * q^87 - 7524 * q^91 + 1516 * q^93 + 1600 * q^95 + 1122 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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
							\(3\)	−1.00000	+	1.00000i	−0.111111	+	0.111111i	−0.760477	−	0.649365i	\(-0.775035\pi\)
0.649365	+	0.760477i	\(0.275035\pi\)
\(5\)	15.0000	+	20.0000i	0.600000	+	0.800000i
							\(7\)	−19.0000	−	19.0000i	−0.387755	−	0.387755i	0.486131	−	0.873886i	\(-0.338408\pi\)
−0.873886	+	0.486131i	\(0.838408\pi\)
\(11\)	−202.000			−1.66942			−0.834711	−	0.550689i	\(-0.814365\pi\)
							−0.834711	+	0.550689i	\(0.814365\pi\)
\(13\)	99.0000	−	99.0000i	0.585799	−	0.585799i	−0.350692	−	0.936491i	\(-0.614054\pi\)
							0.936491	+	0.350692i	\(0.114054\pi\)
\(17\)	−239.000	−	239.000i	−0.826990	−	0.826990i	0.160110	−	0.987099i	\(-0.448815\pi\)
							−0.987099	+	0.160110i	\(0.948815\pi\)
\(19\)		−	40.0000i		−	0.110803i	−0.998464	−	0.0554017i	\(-0.982356\pi\)
							0.998464	−	0.0554017i	\(-0.0176439\pi\)
\(23\)	541.000	−	541.000i	1.02268	−	1.02268i	0.0229476	−	0.999737i	\(-0.492695\pi\)
							0.999737	−	0.0229476i	\(-0.00730510\pi\)
\(29\)			200.000i			0.237812i	0.992906	+	0.118906i	\(0.0379387\pi\)
							−0.992906	+	0.118906i	\(0.962061\pi\)
\(31\)	−758.000			−0.788762			−0.394381	−	0.918947i	\(-0.629041\pi\)
							−0.394381	+	0.918947i	\(0.629041\pi\)
\(37\)	−141.000	−	141.000i	−0.102995	−	0.102995i	0.653732	−	0.756726i	\(-0.273203\pi\)
							−0.756726	+	0.653732i	\(0.773203\pi\)
\(41\)	1042.00			0.619869			0.309935	−	0.950758i	\(-0.399693\pi\)
							0.309935	+	0.950758i	\(0.399693\pi\)
\(43\)	759.000	−	759.000i	0.410492	−	0.410492i	−0.471418	−	0.881910i	\(-0.656258\pi\)
							0.881910	+	0.471418i	\(0.156258\pi\)
\(47\)	−459.000	−	459.000i	−0.207786	−	0.207786i	0.595540	−	0.803326i	\(-0.296938\pi\)
							−0.803326	+	0.595540i	\(0.796938\pi\)
\(53\)	1819.00	−	1819.00i	0.647561	−	0.647561i	−0.304842	−	0.952403i	\(-0.598604\pi\)
							0.952403	+	0.304842i	\(0.0986035\pi\)
\(59\)		−	4600.00i		−	1.32146i	−0.750624	−	0.660730i	\(-0.770247\pi\)
							0.750624	−	0.660730i	\(-0.229753\pi\)
\(61\)	−2082.00			−0.559527			−0.279764	−	0.960069i	\(-0.590256\pi\)
							−0.279764	+	0.960069i	\(0.590256\pi\)
\(67\)	−5081.00	−	5081.00i	−1.13188	−	1.13188i	−0.989865	−	0.142013i	\(-0.954642\pi\)
							−0.142013	−	0.989865i	\(-0.545358\pi\)
\(71\)	−3478.00			−0.689942			−0.344971	−	0.938613i	\(-0.612111\pi\)
							−0.344971	+	0.938613i	\(0.612111\pi\)
\(73\)	−3479.00	+	3479.00i	−0.652843	+	0.652843i	−0.953677	−	0.300834i	\(-0.902735\pi\)
							0.300834	+	0.953677i	\(0.402735\pi\)
\(79\)		−	7680.00i		−	1.23057i	−0.788304	−	0.615286i	\(-0.789041\pi\)
							0.788304	−	0.615286i	\(-0.210959\pi\)
\(83\)	−6081.00	+	6081.00i	−0.882712	+	0.882712i	−0.993809	−	0.111098i	\(-0.964563\pi\)
							0.111098	+	0.993809i	\(0.464563\pi\)
\(89\)		−	5680.00i		−	0.717081i	−0.933514	−	0.358541i	\(-0.883274\pi\)
							0.933514	−	0.358541i	\(-0.116726\pi\)
\(97\)	561.000	+	561.000i	0.0596238	+	0.0596238i	0.736290	−	0.676666i	\(-0.236576\pi\)
							−0.676666	+	0.736290i	\(0.736576\pi\)

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	320.5.p.d.257.1		2
4.3	odd	2		320.5.p.g.257.1		2
5.3	odd	4	inner	320.5.p.d.193.1		2
8.3	odd	2		80.5.p.c.17.1		2
8.5	even	2		10.5.c.b.7.1	yes	2
20.3	even	4		320.5.p.g.193.1		2
24.5	odd	2		90.5.g.a.37.1		2
40.3	even	4		80.5.p.c.33.1		2
40.13	odd	4		10.5.c.b.3.1	✓	2
40.19	odd	2		400.5.p.b.257.1		2
40.27	even	4		400.5.p.b.193.1		2
40.29	even	2		50.5.c.a.7.1		2
40.37	odd	4		50.5.c.a.43.1		2
120.29	odd	2		450.5.g.b.307.1		2
120.53	even	4		90.5.g.a.73.1		2
120.77	even	4		450.5.g.b.343.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.5.c.b.3.1	✓	2	40.13	odd	4
10.5.c.b.7.1	yes	2	8.5	even	2
50.5.c.a.7.1		2	40.29	even	2
50.5.c.a.43.1		2	40.37	odd	4
80.5.p.c.17.1		2	8.3	odd	2
80.5.p.c.33.1		2	40.3	even	4
90.5.g.a.37.1		2	24.5	odd	2
90.5.g.a.73.1		2	120.53	even	4
320.5.p.d.193.1		2	5.3	odd	4	inner
320.5.p.d.257.1		2	1.1	even	1	trivial
320.5.p.g.193.1		2	20.3	even	4
320.5.p.g.257.1		2	4.3	odd	2
400.5.p.b.193.1		2	40.27	even	4
400.5.p.b.257.1		2	40.19	odd	2
450.5.g.b.307.1		2	120.29	odd	2
450.5.g.b.343.1		2	120.77	even	4