Embedded newform 605.2.g.i.511.1

• Label: 605.2.g.i.511.1
• Level: $605$
• Weight: $2$
• Character: 605.511
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.324000000.3
Defining polynomial:	\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			511.1
Root			\(1.40126 + 1.01807i\) of defining polynomial
Character	\(\chi\)	\(=\)	605.511
Dual form			605.2.g.i.251.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.535233 - 1.64728i) q^{2} +(0.809017 + 0.587785i) q^{3} +(-0.809017 + 0.587785i) q^{4} +(-0.309017 + 0.951057i) q^{5} +(0.535233 - 1.64728i) q^{6} +(-1.40126 + 1.01807i) q^{7} +(-1.40126 - 1.01807i) q^{8} +(-0.618034 - 1.90211i) q^{9} +1.73205 q^{10} -1.00000 q^{12} +(-1.07047 - 3.29456i) q^{13} +(2.42705 + 1.76336i) q^{14} +(-0.809017 + 0.587785i) q^{15} +(-1.54508 + 4.75528i) q^{16} +(2.14093 - 6.58911i) q^{17} +(-2.80252 + 2.03615i) q^{18} +(-2.80252 - 2.03615i) q^{19} +(-0.309017 - 0.951057i) q^{20} -1.73205 q^{21} +(-0.535233 - 1.64728i) q^{24} +(-0.809017 - 0.587785i) q^{25} +(-4.85410 + 3.52671i) q^{26} +(1.54508 - 4.75528i) q^{27} +(0.535233 - 1.64728i) q^{28} +(1.40126 + 1.01807i) q^{30} +(-2.47214 - 7.60845i) q^{31} +5.19615 q^{32} -12.0000 q^{34} +(-0.535233 - 1.64728i) q^{35} +(1.61803 + 1.17557i) q^{36} +(6.47214 - 4.70228i) q^{37} +(-1.85410 + 5.70634i) q^{38} +(1.07047 - 3.29456i) q^{39} +(1.40126 - 1.01807i) q^{40} +(9.80881 + 7.12652i) q^{41} +(0.927051 + 2.85317i) q^{42} -8.66025 q^{43} +2.00000 q^{45} +(-7.28115 - 5.29007i) q^{47} +(-4.04508 + 2.93893i) q^{48} +(-1.23607 + 3.80423i) q^{49} +(-0.535233 + 1.64728i) q^{50} +(5.60503 - 4.07230i) q^{51} +(2.80252 + 2.03615i) q^{52} +(1.85410 + 5.70634i) q^{53} -8.66025 q^{54} +3.00000 q^{56} +(-1.07047 - 3.29456i) q^{57} +(9.70820 - 7.05342i) q^{59} +(0.309017 - 0.951057i) q^{60} +(-2.67617 + 8.23639i) q^{61} +(-11.2101 + 8.14459i) q^{62} +(2.80252 + 2.03615i) q^{63} +(0.309017 + 0.951057i) q^{64} +3.46410 q^{65} -5.00000 q^{67} +(2.14093 + 6.58911i) q^{68} +(-2.42705 + 1.76336i) q^{70} +(-3.70820 + 11.4127i) q^{71} +(-1.07047 + 3.29456i) q^{72} +(-11.2101 - 8.14459i) q^{74} +(-0.309017 - 0.951057i) q^{75} +3.46410 q^{76} -6.00000 q^{78} +(3.21140 + 9.88367i) q^{79} +(-4.04508 - 2.93893i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(6.48936 - 19.9722i) q^{82} +(-1.07047 + 3.29456i) q^{83} +(1.40126 - 1.01807i) q^{84} +(5.60503 + 4.07230i) q^{85} +(4.63525 + 14.2658i) q^{86} +3.00000 q^{89} +(-1.07047 - 3.29456i) q^{90} +(4.85410 + 3.52671i) q^{91} +(2.47214 - 7.60845i) q^{93} +(-4.81710 + 14.8255i) q^{94} +(2.80252 - 2.03615i) q^{95} +(4.20378 + 3.05422i) q^{96} +(-3.09017 - 9.51057i) q^{97} +6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 2 * q^3 - 2 * q^4 + 2 * q^5 + 4 * q^9 - 8 * q^12 + 6 * q^14 - 2 * q^15 + 10 * q^16 + 2 * q^20 - 2 * q^25 - 12 * q^26 - 10 * q^27 + 16 * q^31 - 96 * q^34 + 4 * q^36 + 16 * q^37 + 12 * q^38 - 6 * q^42 + 16 * q^45 - 18 * q^47 - 10 * q^48 + 8 * q^49 - 12 * q^53 + 24 * q^56 + 24 * q^59 - 2 * q^60 - 2 * q^64 - 40 * q^67 - 6 * q^70 + 24 * q^71 + 2 * q^75 - 48 * q^78 - 10 * q^80 - 2 * q^81 - 42 * q^82 - 30 * q^86 + 24 * q^89 + 12 * q^91 - 16 * q^93 + 20 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.535233	−	1.64728i	−0.378467	−	1.16480i	−0.941110	−	0.338101i	\(-0.890215\pi\)
							0.562643	−	0.826700i	\(-0.309785\pi\)
\(3\)	0.809017	+	0.587785i	0.467086	+	0.339358i	0.796305	−	0.604896i	\(-0.206785\pi\)
							−0.329218	+	0.944254i	\(0.606785\pi\)
\(5\)	−0.309017	+	0.951057i	−0.138197	+	0.425325i
							\(7\)	−1.40126	+	1.01807i	−0.529626	+	0.384796i	−0.820218	−	0.572051i	\(-0.806148\pi\)
0.290592	+	0.956847i	\(0.406148\pi\)
\(11\)		0			0
							\(13\)	−1.07047	−	3.29456i	−0.296894	−	0.913746i	−0.982579	−	0.185847i	\(-0.940497\pi\)
0.685685	−	0.727899i	\(-0.259503\pi\)
\(17\)	2.14093	−	6.58911i	0.519252	−	1.59809i	−0.256157	−	0.966635i	\(-0.582456\pi\)
							0.775409	−	0.631459i	\(-0.217544\pi\)
\(19\)	−2.80252	−	2.03615i	−0.642942	−	0.467124i	0.217918	−	0.975967i	\(-0.430073\pi\)
							−0.860859	+	0.508843i	\(0.830073\pi\)
\(23\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(29\)		0			0		−0.587785	−	0.809017i	\(-0.700000\pi\)
							0.587785	+	0.809017i	\(0.300000\pi\)
\(31\)	−2.47214	−	7.60845i	−0.444009	−	1.36652i	−0.883567	−	0.468304i	\(-0.844865\pi\)
							0.439558	−	0.898214i	\(-0.355135\pi\)
\(37\)	6.47214	−	4.70228i	1.06401	−	0.773050i	0.0891861	−	0.996015i	\(-0.471573\pi\)
							0.974827	+	0.222965i	\(0.0715734\pi\)
\(41\)	9.80881	+	7.12652i	1.53188	+	1.11298i	0.955183	+	0.296016i	\(0.0956582\pi\)
							0.576696	+	0.816959i	\(0.304342\pi\)
\(43\)	−8.66025			−1.32068			−0.660338	−	0.750968i	\(-0.729587\pi\)
							−0.660338	+	0.750968i	\(0.729587\pi\)
\(47\)	−7.28115	−	5.29007i	−1.06207	−	0.771636i	−0.0875959	−	0.996156i	\(-0.527918\pi\)
							−0.974469	+	0.224520i	\(0.927918\pi\)
\(53\)	1.85410	+	5.70634i	0.254680	+	0.783826i	0.993892	+	0.110353i	\(0.0351982\pi\)
							−0.739212	+	0.673473i	\(0.764802\pi\)
\(59\)	9.70820	−	7.05342i	1.26390	−	0.918277i	0.264958	−	0.964260i	\(-0.414642\pi\)
							0.998942	+	0.0459824i	\(0.0146418\pi\)
\(61\)	−2.67617	+	8.23639i	−0.342648	+	1.05456i	0.620183	+	0.784457i	\(0.287058\pi\)
							−0.962831	+	0.270105i	\(0.912942\pi\)
\(67\)	−5.00000			−0.610847			−0.305424	−	0.952217i	\(-0.598798\pi\)
							−0.305424	+	0.952217i	\(0.598798\pi\)
\(71\)	−3.70820	+	11.4127i	−0.440083	+	1.35444i	0.447704	+	0.894182i	\(0.352242\pi\)
							−0.887787	+	0.460254i	\(0.847758\pi\)
\(73\)		0			0		−0.587785	−	0.809017i	\(-0.700000\pi\)
							0.587785	+	0.809017i	\(0.300000\pi\)
\(79\)	3.21140	+	9.88367i	0.361311	+	1.11200i	0.952259	+	0.305291i	\(0.0987536\pi\)
							−0.590949	+	0.806709i	\(0.701246\pi\)
\(83\)	−1.07047	+	3.29456i	−0.117499	+	0.361625i	−0.992460	−	0.122569i	\(-0.960887\pi\)
							0.874961	+	0.484193i	\(0.160887\pi\)
\(89\)	3.00000			0.317999			0.159000	−	0.987279i	\(-0.449173\pi\)
							0.159000	+	0.987279i	\(0.449173\pi\)
\(97\)	−3.09017	−	9.51057i	−0.313759	−	0.965652i	−0.976262	−	0.216592i	\(-0.930506\pi\)
							0.662503	−	0.749059i	\(-0.269494\pi\)

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	605.2.g.i.511.1		8
11.2	odd	10	inner	605.2.g.i.251.2		8
11.3	even	5		605.2.a.e.1.1	✓	2
11.4	even	5	inner	605.2.g.i.366.2		8
11.5	even	5	inner	605.2.g.i.81.2		8
11.6	odd	10	inner	605.2.g.i.81.1		8
11.7	odd	10	inner	605.2.g.i.366.1		8
11.8	odd	10		605.2.a.e.1.2	yes	2
11.9	even	5	inner	605.2.g.i.251.1		8
11.10	odd	2	inner	605.2.g.i.511.2		8
33.8	even	10		5445.2.a.u.1.1		2
33.14	odd	10		5445.2.a.u.1.2		2
44.3	odd	10		9680.2.a.bu.1.1		2
44.19	even	10		9680.2.a.bu.1.2		2
55.14	even	10		3025.2.a.l.1.2		2
55.19	odd	10		3025.2.a.l.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.e.1.1	✓	2	11.3	even	5
605.2.a.e.1.2	yes	2	11.8	odd	10
605.2.g.i.81.1		8	11.6	odd	10	inner
605.2.g.i.81.2		8	11.5	even	5	inner
605.2.g.i.251.1		8	11.9	even	5	inner
605.2.g.i.251.2		8	11.2	odd	10	inner
605.2.g.i.366.1		8	11.7	odd	10	inner
605.2.g.i.366.2		8	11.4	even	5	inner
605.2.g.i.511.1		8	1.1	even	1	trivial
605.2.g.i.511.2		8	11.10	odd	2	inner
3025.2.a.l.1.1		2	55.19	odd	10
3025.2.a.l.1.2		2	55.14	even	10
5445.2.a.u.1.1		2	33.8	even	10
5445.2.a.u.1.2		2	33.14	odd	10
9680.2.a.bu.1.1		2	44.3	odd	10
9680.2.a.bu.1.2		2	44.19	even	10