Embedded newform 605.2.g.i.81.2

• Label: 605.2.g.i.81.2
• Level: $605$
• Weight: $2$
• Character: 605.81
• 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			81.2
Root			\(-0.535233 + 1.64728i\) of defining polynomial
Character	\(\chi\)	\(=\)	605.81
Dual form			605.2.g.i.366.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40126 + 1.01807i) q^{2} +(-0.309017 + 0.951057i) q^{3} +(0.309017 + 0.951057i) q^{4} +(0.809017 - 0.587785i) q^{5} +(-1.40126 + 1.01807i) q^{6} +(0.535233 + 1.64728i) q^{7} +(0.535233 - 1.64728i) q^{8} +(1.61803 + 1.17557i) q^{9} +1.73205 q^{10} -1.00000 q^{12} +(2.80252 + 2.03615i) q^{13} +(-0.927051 + 2.85317i) q^{14} +(0.309017 + 0.951057i) q^{15} +(4.04508 - 2.93893i) q^{16} +(-5.60503 + 4.07230i) q^{17} +(1.07047 + 3.29456i) q^{18} +(1.07047 - 3.29456i) q^{19} +(0.809017 + 0.587785i) q^{20} -1.73205 q^{21} +(1.40126 + 1.01807i) q^{24} +(0.309017 - 0.951057i) q^{25} +(1.85410 + 5.70634i) q^{26} +(-4.04508 + 2.93893i) q^{27} +(-1.40126 + 1.01807i) q^{28} +(-0.535233 + 1.64728i) q^{30} +(6.47214 + 4.70228i) q^{31} +5.19615 q^{32} -12.0000 q^{34} +(1.40126 + 1.01807i) q^{35} +(-0.618034 + 1.90211i) q^{36} +(-2.47214 - 7.60845i) q^{37} +(4.85410 - 3.52671i) q^{38} +(-2.80252 + 2.03615i) q^{39} +(-0.535233 - 1.64728i) q^{40} +(-3.74663 + 11.5309i) q^{41} +(-2.42705 - 1.76336i) q^{42} -8.66025 q^{43} +2.00000 q^{45} +(2.78115 - 8.55951i) q^{47} +(1.54508 + 4.75528i) q^{48} +(3.23607 - 2.35114i) q^{49} +(1.40126 - 1.01807i) q^{50} +(-2.14093 - 6.58911i) q^{51} +(-1.07047 + 3.29456i) q^{52} +(-4.85410 - 3.52671i) q^{53} -8.66025 q^{54} +3.00000 q^{56} +(2.80252 + 2.03615i) q^{57} +(-3.70820 - 11.4127i) q^{59} +(-0.809017 + 0.587785i) q^{60} +(7.00629 - 5.09037i) q^{61} +(4.28187 + 13.1782i) q^{62} +(-1.07047 + 3.29456i) q^{63} +(-0.809017 - 0.587785i) q^{64} +3.46410 q^{65} -5.00000 q^{67} +(-5.60503 - 4.07230i) q^{68} +(0.927051 + 2.85317i) q^{70} +(9.70820 - 7.05342i) q^{71} +(2.80252 - 2.03615i) q^{72} +(4.28187 - 13.1782i) q^{74} +(0.809017 + 0.587785i) q^{75} +3.46410 q^{76} -6.00000 q^{78} +(-8.40755 - 6.10844i) q^{79} +(1.54508 - 4.75528i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-16.9894 + 12.3435i) q^{82} +(2.80252 - 2.03615i) q^{83} +(-0.535233 - 1.64728i) q^{84} +(-2.14093 + 6.58911i) q^{85} +(-12.1353 - 8.81678i) q^{86} +3.00000 q^{89} +(2.80252 + 2.03615i) q^{90} +(-1.85410 + 5.70634i) q^{91} +(-6.47214 + 4.70228i) q^{93} +(12.6113 - 9.16267i) q^{94} +(-1.07047 - 3.29456i) q^{95} +(-1.60570 + 4.94183i) q^{96} +(8.09017 + 5.87785i) 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 \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

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

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40126	+	1.01807i	0.990839	+	0.719887i	0.960105	−	0.279641i	\(-0.0902153\pi\)
							0.0307347	+	0.999528i	\(0.490215\pi\)
\(3\)	−0.309017	+	0.951057i	−0.178411	+	0.549093i	−0.999773	−	0.0213149i	\(-0.993215\pi\)
							0.821362	+	0.570408i	\(0.193215\pi\)
\(5\)	0.809017	−	0.587785i	0.361803	−	0.262866i
							\(7\)	0.535233	+	1.64728i	0.202299	+	0.622613i	0.999813	+	0.0193127i	\(0.00614781\pi\)
−0.797514	+	0.603300i	\(0.793852\pi\)
\(11\)		0			0
							\(13\)	2.80252	+	2.03615i	0.777278	+	0.564726i	0.904161	−	0.427192i	\(-0.140497\pi\)
−0.126883	+	0.991918i	\(0.540497\pi\)
\(17\)	−5.60503	+	4.07230i	−1.35942	+	0.987677i	−0.360939	+	0.932589i	\(0.617544\pi\)
							−0.998482	+	0.0550873i	\(0.982456\pi\)
\(19\)	1.07047	−	3.29456i	0.245582	−	0.755823i	−0.749958	−	0.661485i	\(-0.769927\pi\)
							0.995540	−	0.0943381i	\(-0.0300735\pi\)
\(23\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(29\)		0			0		0.951057	−	0.309017i	\(-0.100000\pi\)
							−0.951057	+	0.309017i	\(0.900000\pi\)
\(31\)	6.47214	+	4.70228i	1.16243	+	0.844555i	0.990083	−	0.140482i	\(-0.0448651\pi\)
							0.172347	+	0.985036i	\(0.444865\pi\)
\(37\)	−2.47214	−	7.60845i	−0.406417	−	1.25082i	−0.919707	−	0.392607i	\(-0.871573\pi\)
							0.513290	−	0.858215i	\(-0.328427\pi\)
\(41\)	−3.74663	+	11.5309i	−0.585126	+	1.80083i	0.0136398	+	0.999907i	\(0.495658\pi\)
							−0.598765	+	0.800924i	\(0.704342\pi\)
\(43\)	−8.66025			−1.32068			−0.660338	−	0.750968i	\(-0.729587\pi\)
							−0.660338	+	0.750968i	\(0.729587\pi\)
\(47\)	2.78115	−	8.55951i	0.405673	−	1.24853i	−0.514659	−	0.857395i	\(-0.672082\pi\)
							0.920332	−	0.391138i	\(-0.127918\pi\)
\(53\)	−4.85410	−	3.52671i	−0.666762	−	0.484431i	0.202178	−	0.979349i	\(-0.435198\pi\)
							−0.868940	+	0.494918i	\(0.835198\pi\)
\(59\)	−3.70820	−	11.4127i	−0.482767	−	1.48580i	−0.835189	−	0.549963i	\(-0.814642\pi\)
							0.352422	−	0.935841i	\(-0.385358\pi\)
\(61\)	7.00629	−	5.09037i	0.897064	−	0.651755i	−0.0406463	−	0.999174i	\(-0.512942\pi\)
							0.937710	+	0.347419i	\(0.112942\pi\)
\(67\)	−5.00000			−0.610847			−0.305424	−	0.952217i	\(-0.598798\pi\)
							−0.305424	+	0.952217i	\(0.598798\pi\)
\(71\)	9.70820	−	7.05342i	1.15215	−	0.837087i	0.163386	−	0.986562i	\(-0.447758\pi\)
							0.988766	+	0.149475i	\(0.0477583\pi\)
\(73\)		0			0		0.951057	−	0.309017i	\(-0.100000\pi\)
							−0.951057	+	0.309017i	\(0.900000\pi\)
\(79\)	−8.40755	−	6.10844i	−0.945923	−	0.687254i	0.00391577	−	0.999992i	\(-0.498754\pi\)
							−0.949839	+	0.312739i	\(0.898754\pi\)
\(83\)	2.80252	−	2.03615i	0.307616	−	0.223496i	−0.423257	−	0.906010i	\(-0.639113\pi\)
							0.730873	+	0.682514i	\(0.239113\pi\)
\(89\)	3.00000			0.317999			0.159000	−	0.987279i	\(-0.449173\pi\)
							0.159000	+	0.987279i	\(0.449173\pi\)
\(97\)	8.09017	+	5.87785i	0.821432	+	0.596806i	0.917122	−	0.398606i	\(-0.130506\pi\)
							−0.0956901	+	0.995411i	\(0.530506\pi\)

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.81.2		8
11.2	odd	10	inner	605.2.g.i.511.2		8
11.3	even	5	inner	605.2.g.i.366.2		8
11.4	even	5	inner	605.2.g.i.251.1		8
11.5	even	5		605.2.a.e.1.1	✓	2
11.6	odd	10		605.2.a.e.1.2	yes	2
11.7	odd	10	inner	605.2.g.i.251.2		8
11.8	odd	10	inner	605.2.g.i.366.1		8
11.9	even	5	inner	605.2.g.i.511.1		8
11.10	odd	2	inner	605.2.g.i.81.1		8
33.5	odd	10		5445.2.a.u.1.2		2
33.17	even	10		5445.2.a.u.1.1		2
44.27	odd	10		9680.2.a.bu.1.1		2
44.39	even	10		9680.2.a.bu.1.2		2
55.39	odd	10		3025.2.a.l.1.1		2
55.49	even	10		3025.2.a.l.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.e.1.1	✓	2	11.5	even	5
605.2.a.e.1.2	yes	2	11.6	odd	10
605.2.g.i.81.1		8	11.10	odd	2	inner
605.2.g.i.81.2		8	1.1	even	1	trivial
605.2.g.i.251.1		8	11.4	even	5	inner
605.2.g.i.251.2		8	11.7	odd	10	inner
605.2.g.i.366.1		8	11.8	odd	10	inner
605.2.g.i.366.2		8	11.3	even	5	inner
605.2.g.i.511.1		8	11.9	even	5	inner
605.2.g.i.511.2		8	11.2	odd	10	inner
3025.2.a.l.1.1		2	55.39	odd	10
3025.2.a.l.1.2		2	55.49	even	10
5445.2.a.u.1.1		2	33.17	even	10
5445.2.a.u.1.2		2	33.5	odd	10
9680.2.a.bu.1.1		2	44.27	odd	10
9680.2.a.bu.1.2		2	44.39	even	10