Embedded newform 320.5.p.p.257.3

• Label: 320.5.p.p.257.3
• Level: $320$
• Weight: $5$
• Character: 320.257
• Analytic conductor: $33.078$
• Analytic rank: $0$
• Dimension: $6$
• 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:	\(6\)
Relative dimension:	\(3\) over \(\Q(i)\)
Coefficient field:	6.0.313431616.3
Defining polynomial:	\( x^{6} + 27x^{4} + 145x^{2} + 144 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{5}\cdot 5^{2} \)
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			257.3
Root			\(-2.35597i\) of defining polynomial
Character	\(\chi\)	\(=\)	320.257
Dual form			320.5.p.p.193.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.3304 - 11.3304i) q^{3} +(-9.33042 + 23.1936i) q^{5} +(-42.8544 - 42.8544i) q^{7} -175.757i q^{9} +20.3872 q^{11} +(151.697 - 151.697i) q^{13} +(157.076 + 368.511i) q^{15} +(-150.269 - 150.269i) q^{17} +277.202i q^{19} -971.118 q^{21} +(-363.891 + 363.891i) q^{23} +(-450.886 - 432.812i) q^{25} +(-1073.64 - 1073.64i) q^{27} -413.744i q^{29} -659.526 q^{31} +(230.996 - 230.996i) q^{33} +(1393.80 - 594.099i) q^{35} +(-1058.50 - 1058.50i) q^{37} -3437.59i q^{39} -801.206 q^{41} +(-2008.78 + 2008.78i) q^{43} +(4076.44 + 1639.89i) q^{45} +(-1471.26 - 1471.26i) q^{47} +1272.01i q^{49} -3405.23 q^{51} +(1321.08 - 1321.08i) q^{53} +(-190.221 + 472.853i) q^{55} +(3140.82 + 3140.82i) q^{57} +937.237i q^{59} -2543.20 q^{61} +(-7531.97 + 7531.97i) q^{63} +(2103.01 + 4933.80i) q^{65} +(-140.942 - 140.942i) q^{67} +8246.08i q^{69} +2532.44 q^{71} +(5900.02 - 5900.02i) q^{73} +(-10012.7 + 204.788i) q^{75} +(-873.682 - 873.682i) q^{77} -196.937i q^{79} -10093.2 q^{81} +(9130.34 - 9130.34i) q^{83} +(4887.36 - 2083.21i) q^{85} +(-4687.89 - 4687.89i) q^{87} +10678.4i q^{89} -13001.8 q^{91} +(-7472.70 + 7472.70i) q^{93} +(-6429.32 - 2586.41i) q^{95} +(-11052.2 - 11052.2i) q^{97} -3583.19i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q + 8 * q^3 + 4 * q^5 - 40 * q^7 - 72 * q^11 + 62 * q^13 - 280 * q^15 + 418 * q^17 - 1736 * q^21 - 1760 * q^23 - 2974 * q^25 - 2248 * q^27 + 2864 * q^31 - 8 * q^33 + 4224 * q^35 - 5366 * q^37 + 1656 * q^41 + 2760 * q^43 + 14678 * q^45 - 432 * q^47 - 15840 * q^51 - 534 * q^53 - 136 * q^55 + 12576 * q^57 - 216 * q^61 - 19552 * q^63 - 15082 * q^65 + 1336 * q^67 + 2512 * q^71 + 20230 * q^73 - 22240 * q^75 - 5832 * q^77 - 16526 * q^81 + 22736 * q^83 + 6798 * q^85 - 22608 * q^87 - 12848 * q^91 - 36520 * q^93 + 33064 * q^95 - 9418 * 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
							\(3\)	11.3304	−	11.3304i	1.25894	−	1.25894i	0.307334	−	0.951602i	\(-0.400563\pi\)
0.951602	−	0.307334i	\(-0.0994369\pi\)
\(5\)	−9.33042	+	23.1936i	−0.373217	+	0.927744i
							\(7\)	−42.8544	−	42.8544i	−0.874581	−	0.874581i	0.118387	−	0.992968i	\(-0.462228\pi\)
−0.992968	+	0.118387i	\(0.962228\pi\)
\(11\)	20.3872			0.168489			0.0842447	−	0.996445i	\(-0.473152\pi\)
							0.0842447	+	0.996445i	\(0.473152\pi\)
\(13\)	151.697	−	151.697i	0.897617	−	0.897617i	−0.0976082	−	0.995225i	\(-0.531119\pi\)
							0.995225	+	0.0976082i	\(0.0311192\pi\)
\(17\)	−150.269	−	150.269i	−0.519963	−	0.519963i	0.397597	−	0.917560i	\(-0.369844\pi\)
							−0.917560	+	0.397597i	\(0.869844\pi\)
\(19\)			277.202i			0.767873i	0.923359	+	0.383937i	\(0.125432\pi\)
							−0.923359	+	0.383937i	\(0.874568\pi\)
\(23\)	−363.891	+	363.891i	−0.687885	+	0.687885i	−0.961764	−	0.273879i	\(-0.911693\pi\)
							0.273879	+	0.961764i	\(0.411693\pi\)
\(29\)		−	413.744i		−	0.491967i	−0.969274	−	0.245983i	\(-0.920889\pi\)
							0.969274	−	0.245983i	\(-0.0791108\pi\)
\(31\)	−659.526			−0.686291			−0.343145	−	0.939282i	\(-0.611492\pi\)
							−0.343145	+	0.939282i	\(0.611492\pi\)
\(37\)	−1058.50	−	1058.50i	−0.773193	−	0.773193i	0.205471	−	0.978663i	\(-0.434127\pi\)
							−0.978663	+	0.205471i	\(0.934127\pi\)
\(41\)	−801.206			−0.476624			−0.238312	−	0.971189i	\(-0.576594\pi\)
							−0.238312	+	0.971189i	\(0.576594\pi\)
\(43\)	−2008.78	+	2008.78i	−1.08642	+	1.08642i	−0.0905217	+	0.995894i	\(0.528853\pi\)
							−0.995894	+	0.0905217i	\(0.971147\pi\)
\(47\)	−1471.26	−	1471.26i	−0.666030	−	0.666030i	0.290765	−	0.956795i	\(-0.406090\pi\)
							−0.956795	+	0.290765i	\(0.906090\pi\)
\(53\)	1321.08	−	1321.08i	0.470304	−	0.470304i	−0.431709	−	0.902013i	\(-0.642089\pi\)
							0.902013	+	0.431709i	\(0.142089\pi\)
\(59\)			937.237i			0.269243i	0.990897	+	0.134622i	\(0.0429819\pi\)
							−0.990897	+	0.134622i	\(0.957018\pi\)
\(61\)	−2543.20			−0.683473			−0.341736	−	0.939796i	\(-0.611015\pi\)
							−0.341736	+	0.939796i	\(0.611015\pi\)
\(67\)	−140.942	−	140.942i	−0.0313971	−	0.0313971i	0.691234	−	0.722631i	\(-0.257067\pi\)
							−0.722631	+	0.691234i	\(0.757067\pi\)
\(71\)	2532.44			0.502368			0.251184	−	0.967939i	\(-0.419180\pi\)
							0.251184	+	0.967939i	\(0.419180\pi\)
\(73\)	5900.02	−	5900.02i	1.10715	−	1.10715i	0.113630	−	0.993523i	\(-0.463752\pi\)
							0.993523	−	0.113630i	\(-0.0362479\pi\)
\(79\)		−	196.937i		−	0.0315554i	−0.999876	−	0.0157777i	\(-0.994978\pi\)
							0.999876	−	0.0157777i	\(-0.00502240\pi\)
\(83\)	9130.34	−	9130.34i	1.32535	−	1.32535i	0.415973	−	0.909377i	\(-0.363441\pi\)
							0.909377	−	0.415973i	\(-0.136559\pi\)
\(89\)			10678.4i			1.34811i	0.738680	+	0.674056i	\(0.235449\pi\)
							−0.738680	+	0.674056i	\(0.764551\pi\)
\(97\)	−11052.2	−	11052.2i	−1.17464	−	1.17464i	−0.981090	−	0.193550i	\(-0.938000\pi\)
							−0.193550	−	0.981090i	\(-0.562000\pi\)

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.p.257.3		6
4.3	odd	2		320.5.p.o.257.1		6
5.3	odd	4	inner	320.5.p.p.193.3		6
8.3	odd	2		80.5.p.g.17.3		6
8.5	even	2		40.5.l.c.17.1	✓	6
20.3	even	4		320.5.p.o.193.1		6
24.5	odd	2		360.5.v.c.217.1		6
40.3	even	4		80.5.p.g.33.3		6
40.13	odd	4		40.5.l.c.33.1	yes	6
40.19	odd	2		400.5.p.o.257.1		6
40.27	even	4		400.5.p.o.193.1		6
40.29	even	2		200.5.l.e.57.3		6
40.37	odd	4		200.5.l.e.193.3		6
120.53	even	4		360.5.v.c.73.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.l.c.17.1	✓	6	8.5	even	2
40.5.l.c.33.1	yes	6	40.13	odd	4
80.5.p.g.17.3		6	8.3	odd	2
80.5.p.g.33.3		6	40.3	even	4
200.5.l.e.57.3		6	40.29	even	2
200.5.l.e.193.3		6	40.37	odd	4
320.5.p.o.193.1		6	20.3	even	4
320.5.p.o.257.1		6	4.3	odd	2
320.5.p.p.193.3		6	5.3	odd	4	inner
320.5.p.p.257.3		6	1.1	even	1	trivial
360.5.v.c.73.1		6	120.53	even	4
360.5.v.c.217.1		6	24.5	odd	2
400.5.p.o.193.1		6	40.27	even	4
400.5.p.o.257.1		6	40.19	odd	2