Embedded newform 320.5.p.p.193.1

• Label: 320.5.p.p.193.1
• Level: $320$
• Weight: $5$
• Character: 320.193
• 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			193.1
Root			\(-4.49029i\) of defining polynomial
Character	\(\chi\)	\(=\)	320.193
Dual form			320.5.p.p.257.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.28874 - 8.28874i) q^{3} +(10.2887 - 22.7847i) q^{5} +(-3.20721 + 3.20721i) q^{7} +56.4063i q^{9} +19.5694 q^{11} +(-219.840 - 219.840i) q^{13} +(-274.137 + 103.576i) q^{15} +(281.352 - 281.352i) q^{17} +305.648i q^{19} +53.1675 q^{21} +(-633.243 - 633.243i) q^{23} +(-413.284 - 468.851i) q^{25} +(-203.851 + 203.851i) q^{27} -885.210i q^{29} +1383.32 q^{31} +(-162.205 - 162.205i) q^{33} +(40.0772 + 106.073i) q^{35} +(-453.987 + 453.987i) q^{37} +3644.39i q^{39} -193.003 q^{41} +(1016.66 + 1016.66i) q^{43} +(1285.20 + 580.350i) q^{45} +(-1203.53 + 1203.53i) q^{47} +2380.43i q^{49} -4664.11 q^{51} +(530.174 + 530.174i) q^{53} +(201.344 - 445.882i) q^{55} +(2533.43 - 2533.43i) q^{57} -934.801i q^{59} +1122.32 q^{61} +(-180.907 - 180.907i) q^{63} +(-7270.86 + 2747.11i) q^{65} +(-4760.40 + 4760.40i) q^{67} +10497.6i q^{69} -8831.10 q^{71} +(3008.63 + 3008.63i) q^{73} +(-460.584 + 7311.79i) q^{75} +(-62.7631 + 62.7631i) q^{77} +3349.09i q^{79} +7948.24 q^{81} +(3166.08 + 3166.08i) q^{83} +(-3515.76 - 9305.28i) q^{85} +(-7337.27 + 7337.27i) q^{87} +1732.33i q^{89} +1410.15 q^{91} +(-11466.0 - 11466.0i) q^{93} +(6964.09 + 3144.73i) q^{95} +(1659.43 - 1659.43i) q^{97} +1103.84i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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
							\(3\)	−8.28874	−	8.28874i	−0.920971	−	0.920971i	0.0761273	−	0.997098i	\(-0.475744\pi\)
−0.997098	+	0.0761273i	\(0.975744\pi\)
\(5\)	10.2887	−	22.7847i	0.411549	−	0.911387i
							\(7\)	−3.20721	+	3.20721i	−0.0654533	+	0.0654533i	−0.739076	−	0.673622i	\(-0.764737\pi\)
0.673622	+	0.739076i	\(0.264737\pi\)
\(11\)	19.5694			0.161730			0.0808652	−	0.996725i	\(-0.474232\pi\)
							0.0808652	+	0.996725i	\(0.474232\pi\)
\(13\)	−219.840	−	219.840i	−1.30083	−	1.30083i	−0.927833	−	0.372996i	\(-0.878331\pi\)
							−0.372996	−	0.927833i	\(-0.621669\pi\)
\(17\)	281.352	−	281.352i	0.973537	−	0.973537i	−0.0261217	−	0.999659i	\(-0.508316\pi\)
							0.999659	+	0.0261217i	\(0.00831574\pi\)
\(19\)			305.648i			0.846670i	0.905973	+	0.423335i	\(0.139141\pi\)
							−0.905973	+	0.423335i	\(0.860859\pi\)
\(23\)	−633.243	−	633.243i	−1.19706	−	1.19706i	−0.975044	−	0.222014i	\(-0.928737\pi\)
							−0.222014	−	0.975044i	\(-0.571263\pi\)
\(29\)		−	885.210i		−	1.05257i	−0.850309	−	0.526284i	\(-0.823585\pi\)
							0.850309	−	0.526284i	\(-0.176415\pi\)
\(31\)	1383.32			1.43946			0.719730	−	0.694254i	\(-0.244266\pi\)
							0.719730	+	0.694254i	\(0.244266\pi\)
\(37\)	−453.987	+	453.987i	−0.331619	+	0.331619i	−0.853201	−	0.521582i	\(-0.825342\pi\)
							0.521582	+	0.853201i	\(0.325342\pi\)
\(41\)	−193.003			−0.114814			−0.0574071	−	0.998351i	\(-0.518283\pi\)
							−0.0574071	+	0.998351i	\(0.518283\pi\)
\(43\)	1016.66	+	1016.66i	0.549846	+	0.549846i	0.926396	−	0.376550i	\(-0.122890\pi\)
							−0.376550	+	0.926396i	\(0.622890\pi\)
\(47\)	−1203.53	+	1203.53i	−0.544828	+	0.544828i	−0.924940	−	0.380112i	\(-0.875885\pi\)
							0.380112	+	0.924940i	\(0.375885\pi\)
\(53\)	530.174	+	530.174i	0.188741	+	0.188741i	0.795152	−	0.606410i	\(-0.207391\pi\)
							−0.606410	+	0.795152i	\(0.707391\pi\)
\(59\)		−	934.801i		−	0.268544i	−0.990945	−	0.134272i	\(-0.957130\pi\)
							0.990945	−	0.134272i	\(-0.0428695\pi\)
\(61\)	1122.32			0.301617			0.150809	−	0.988563i	\(-0.451812\pi\)
							0.150809	+	0.988563i	\(0.451812\pi\)
\(67\)	−4760.40	+	4760.40i	−1.06046	+	1.06046i	−0.0624093	+	0.998051i	\(0.519878\pi\)
							−0.998051	+	0.0624093i	\(0.980122\pi\)
\(71\)	−8831.10			−1.75185			−0.875927	−	0.482443i	\(-0.839750\pi\)
							−0.875927	+	0.482443i	\(0.839750\pi\)
\(73\)	3008.63	+	3008.63i	0.564578	+	0.564578i	0.930604	−	0.366027i	\(-0.119282\pi\)
							−0.366027	+	0.930604i	\(0.619282\pi\)
\(79\)			3349.09i			0.536627i	0.963332	+	0.268313i	\(0.0864663\pi\)
							−0.963332	+	0.268313i	\(0.913534\pi\)
\(83\)	3166.08	+	3166.08i	0.459586	+	0.459586i	0.898519	−	0.438934i	\(-0.144644\pi\)
							−0.438934	+	0.898519i	\(0.644644\pi\)
\(89\)			1732.33i			0.218700i	0.994003	+	0.109350i	\(0.0348770\pi\)
							−0.994003	+	0.109350i	\(0.965123\pi\)
\(97\)	1659.43	−	1659.43i	0.176366	−	0.176366i	−0.613404	−	0.789770i	\(-0.710200\pi\)
							0.789770	+	0.613404i	\(0.210200\pi\)

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.193.1		6
4.3	odd	2		320.5.p.o.193.3		6
5.2	odd	4	inner	320.5.p.p.257.1		6
8.3	odd	2		80.5.p.g.33.1		6
8.5	even	2		40.5.l.c.33.3	yes	6
20.7	even	4		320.5.p.o.257.3		6
24.5	odd	2		360.5.v.c.73.3		6
40.3	even	4		400.5.p.o.257.3		6
40.13	odd	4		200.5.l.e.57.1		6
40.19	odd	2		400.5.p.o.193.3		6
40.27	even	4		80.5.p.g.17.1		6
40.29	even	2		200.5.l.e.193.1		6
40.37	odd	4		40.5.l.c.17.3	✓	6
120.77	even	4		360.5.v.c.217.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.l.c.17.3	✓	6	40.37	odd	4
40.5.l.c.33.3	yes	6	8.5	even	2
80.5.p.g.17.1		6	40.27	even	4
80.5.p.g.33.1		6	8.3	odd	2
200.5.l.e.57.1		6	40.13	odd	4
200.5.l.e.193.1		6	40.29	even	2
320.5.p.o.193.3		6	4.3	odd	2
320.5.p.o.257.3		6	20.7	even	4
320.5.p.p.193.1		6	1.1	even	1	trivial
320.5.p.p.257.1		6	5.2	odd	4	inner
360.5.v.c.73.3		6	24.5	odd	2
360.5.v.c.217.3		6	120.77	even	4
400.5.p.o.193.3		6	40.19	odd	2
400.5.p.o.257.3		6	40.3	even	4