Embedded newform 320.5.p.i.257.1

• Label: 320.5.p.i.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:	\(\Z[a_1, \ldots, a_{5}]\)
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.i.193.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.00000 - 9.00000i) q^{3} +(15.0000 - 20.0000i) q^{5} +(-29.0000 - 29.0000i) q^{7} -81.0000i q^{9} -118.000 q^{11} +(-69.0000 + 69.0000i) q^{13} +(-45.0000 - 315.000i) q^{15} +(-271.000 - 271.000i) q^{17} +280.000i q^{19} -522.000 q^{21} +(-269.000 + 269.000i) q^{23} +(-175.000 - 600.000i) q^{25} -680.000i q^{29} -202.000 q^{31} +(-1062.00 + 1062.00i) q^{33} +(-1015.00 + 145.000i) q^{35} +(651.000 + 651.000i) q^{37} +1242.00i q^{39} +1682.00 q^{41} +(1089.00 - 1089.00i) q^{43} +(-1620.00 - 1215.00i) q^{45} +(-1269.00 - 1269.00i) q^{47} -719.000i q^{49} -4878.00 q^{51} +(611.000 - 611.000i) q^{53} +(-1770.00 + 2360.00i) q^{55} +(2520.00 + 2520.00i) q^{57} +1160.00i q^{59} +5598.00 q^{61} +(-2349.00 + 2349.00i) q^{63} +(345.000 + 2415.00i) q^{65} +(-751.000 - 751.000i) q^{67} +4842.00i q^{69} -6442.00 q^{71} +(-2951.00 + 2951.00i) q^{73} +(-6975.00 - 3825.00i) q^{75} +(3422.00 + 3422.00i) q^{77} -10560.0i q^{79} +6561.00 q^{81} +(-6231.00 + 6231.00i) q^{83} +(-9485.00 + 1355.00i) q^{85} +(-6120.00 - 6120.00i) q^{87} -14480.0i q^{89} +4002.00 q^{91} +(-1818.00 + 1818.00i) q^{93} +(5600.00 + 4200.00i) q^{95} +(-7311.00 - 7311.00i) q^{97} +9558.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 18 * q^3 + 30 * q^5 - 58 * q^7 - 236 * q^11 - 138 * q^13 - 90 * q^15 - 542 * q^17 - 1044 * q^21 - 538 * q^23 - 350 * q^25 - 404 * q^31 - 2124 * q^33 - 2030 * q^35 + 1302 * q^37 + 3364 * q^41 + 2178 * q^43 - 3240 * q^45 - 2538 * q^47 - 9756 * q^51 + 1222 * q^53 - 3540 * q^55 + 5040 * q^57 + 11196 * q^61 - 4698 * q^63 + 690 * q^65 - 1502 * q^67 - 12884 * q^71 - 5902 * q^73 - 13950 * q^75 + 6844 * q^77 + 13122 * q^81 - 12462 * q^83 - 18970 * q^85 - 12240 * q^87 + 8004 * q^91 - 3636 * q^93 + 11200 * q^95 - 14622 * 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
							\(3\)	9.00000	−	9.00000i	1.00000	−	1.00000i		−	1.00000i	\(-0.5\pi\)
1.00000			\(0\)
\(5\)	15.0000	−	20.0000i	0.600000	−	0.800000i
							\(7\)	−29.0000	−	29.0000i	−0.591837	−	0.591837i	0.346291	−	0.938127i	\(-0.387441\pi\)
−0.938127	+	0.346291i	\(0.887441\pi\)
\(11\)	−118.000			−0.975207			−0.487603	−	0.873065i	\(-0.662129\pi\)
							−0.487603	+	0.873065i	\(0.662129\pi\)
\(13\)	−69.0000	+	69.0000i	−0.408284	+	0.408284i	−0.881140	−	0.472856i	\(-0.843223\pi\)
							0.472856	+	0.881140i	\(0.343223\pi\)
\(17\)	−271.000	−	271.000i	−0.937716	−	0.937716i	0.0604547	−	0.998171i	\(-0.480745\pi\)
							−0.998171	+	0.0604547i	\(0.980745\pi\)
\(19\)			280.000i			0.775623i	0.921739	+	0.387812i	\(0.126769\pi\)
							−0.921739	+	0.387812i	\(0.873231\pi\)
\(23\)	−269.000	+	269.000i	−0.508507	+	0.508507i	−0.914068	−	0.405561i	\(-0.867076\pi\)
							0.405561	+	0.914068i	\(0.367076\pi\)
\(29\)		−	680.000i		−	0.808561i	−0.914635	−	0.404281i	\(-0.867522\pi\)
							0.914635	−	0.404281i	\(-0.132478\pi\)
\(31\)	−202.000			−0.210198			−0.105099	−	0.994462i	\(-0.533516\pi\)
							−0.105099	+	0.994462i	\(0.533516\pi\)
\(37\)	651.000	+	651.000i	0.475530	+	0.475530i	0.903699	−	0.428169i	\(-0.140841\pi\)
							−0.428169	+	0.903699i	\(0.640841\pi\)
\(41\)	1682.00			1.00059			0.500297	−	0.865854i	\(-0.333224\pi\)
							0.500297	+	0.865854i	\(0.333224\pi\)
\(43\)	1089.00	−	1089.00i	0.588967	−	0.588967i	−0.348385	−	0.937352i	\(-0.613270\pi\)
							0.937352	+	0.348385i	\(0.113270\pi\)
\(47\)	−1269.00	−	1269.00i	−0.574468	−	0.574468i	0.358906	−	0.933374i	\(-0.383150\pi\)
							−0.933374	+	0.358906i	\(0.883150\pi\)
\(53\)	611.000	−	611.000i	0.217515	−	0.217515i	−0.589935	−	0.807450i	\(-0.700847\pi\)
							0.807450	+	0.589935i	\(0.200847\pi\)
\(59\)			1160.00i			0.333238i	0.986021	+	0.166619i	\(0.0532849\pi\)
							−0.986021	+	0.166619i	\(0.946715\pi\)
\(61\)	5598.00			1.50443			0.752217	−	0.658915i	\(-0.228984\pi\)
							0.752217	+	0.658915i	\(0.228984\pi\)
\(67\)	−751.000	−	751.000i	−0.167298	−	0.167298i	0.618493	−	0.785791i	\(-0.287744\pi\)
							−0.785791	+	0.618493i	\(0.787744\pi\)
\(71\)	−6442.00			−1.27792			−0.638961	−	0.769240i	\(-0.720635\pi\)
							−0.638961	+	0.769240i	\(0.720635\pi\)
\(73\)	−2951.00	+	2951.00i	−0.553762	+	0.553762i	−0.927525	−	0.373762i	\(-0.878068\pi\)
							0.373762	+	0.927525i	\(0.378068\pi\)
\(79\)		−	10560.0i		−	1.69204i	−0.533154	−	0.846018i	\(-0.678993\pi\)
							0.533154	−	0.846018i	\(-0.321007\pi\)
\(83\)	−6231.00	+	6231.00i	−0.904485	+	0.904485i	−0.995820	−	0.0913348i	\(-0.970887\pi\)
							0.0913348	+	0.995820i	\(0.470887\pi\)
\(89\)		−	14480.0i		−	1.82805i	−0.405656	−	0.914026i	\(-0.632957\pi\)
							0.405656	−	0.914026i	\(-0.367043\pi\)
\(97\)	−7311.00	−	7311.00i	−0.777022	−	0.777022i	0.202301	−	0.979323i	\(-0.435158\pi\)
							−0.979323	+	0.202301i	\(0.935158\pi\)

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.5.c.a.3.1	✓	2	40.3	even	4
10.5.c.a.7.1	yes	2	8.3	odd	2
50.5.c.b.7.1		2	40.19	odd	2
50.5.c.b.43.1		2	40.27	even	4
80.5.p.b.17.1		2	8.5	even	2
80.5.p.b.33.1		2	40.13	odd	4
90.5.g.b.37.1		2	24.11	even	2
90.5.g.b.73.1		2	120.83	odd	4
320.5.p.b.193.1		2	20.3	even	4
320.5.p.b.257.1		2	4.3	odd	2
320.5.p.i.193.1		2	5.3	odd	4	inner
320.5.p.i.257.1		2	1.1	even	1	trivial
400.5.p.c.193.1		2	40.37	odd	4
400.5.p.c.257.1		2	40.29	even	2
450.5.g.a.307.1		2	120.59	even	2
450.5.g.a.343.1		2	120.107	odd	4