Embedded newform 666.2.bj.c.469.2

• Label: 666.2.bj.c.469.2
• Level: $666$
• Weight: $2$
• Character: 666.469
• Analytic conductor: $5.318$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.bj (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.31803677462\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\Q(\zeta_{36})\)
Defining polynomial:	\( x^{12} - x^{6} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			469.2
Root			\(0.642788 + 0.766044i\) of defining polynomial
Character	\(\chi\)	\(=\)	666.469
Dual form			666.2.bj.c.595.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642788 + 0.766044i) q^{2} +(-0.173648 + 0.984808i) q^{4} +(-0.273629 - 0.751790i) q^{5} +(-0.138449 + 0.0503913i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(0.400019 - 0.692853i) q^{10} +(2.40570 + 4.16679i) q^{11} +(1.91858 + 0.338298i) q^{13} +(-0.127595 - 0.0736672i) q^{14} +(-0.939693 - 0.342020i) q^{16} +(3.43779 - 0.606175i) q^{17} +(-4.33625 + 5.16775i) q^{19} +(0.787884 - 0.138925i) q^{20} +(-1.64560 + 4.52124i) q^{22} +(3.61610 + 2.08776i) q^{23} +(3.33991 - 2.80251i) q^{25} +(0.974090 + 1.68717i) q^{26} +(-0.0255844 - 0.145096i) q^{28} +(-2.63134 + 1.51921i) q^{29} +8.13740i q^{31} +(-0.342020 - 0.939693i) q^{32} +(2.67412 + 2.24386i) q^{34} +(0.0757674 + 0.0902961i) q^{35} +(6.08227 + 0.0772535i) q^{37} -6.74601 q^{38} +(0.612865 + 0.514255i) q^{40} +(-0.676822 + 3.83845i) q^{41} -8.10852i q^{43} +(-4.52124 + 1.64560i) q^{44} +(0.725070 + 4.11208i) q^{46} +(4.16911 - 7.22111i) q^{47} +(-5.34568 + 4.48556i) q^{49} +(4.29370 + 0.757095i) q^{50} +(-0.666317 + 1.83069i) q^{52} +(-10.2327 - 3.72440i) q^{53} +(2.47428 - 2.94874i) q^{55} +(0.0947047 - 0.112865i) q^{56} +(-2.85517 - 1.03920i) q^{58} +(2.96569 - 8.14816i) q^{59} +(-0.346344 - 0.0610698i) q^{61} +(-6.23361 + 5.23062i) q^{62} +(0.500000 - 0.866025i) q^{64} +(-0.270651 - 1.53494i) q^{65} +(-2.25471 + 0.820647i) q^{67} +3.49082i q^{68} +(-0.0204685 + 0.116082i) q^{70} +(-5.34953 - 4.48879i) q^{71} +1.13399 q^{73} +(3.85043 + 4.70895i) q^{74} +(-4.33625 - 5.16775i) q^{76} +(-0.543037 - 0.455662i) q^{77} +(0.646510 + 1.77627i) q^{79} +0.800038i q^{80} +(-3.37547 + 1.94883i) q^{82} +(1.71997 + 9.75442i) q^{83} +(-1.39639 - 2.41863i) q^{85} +(6.21149 - 5.21206i) q^{86} +(-4.16679 - 2.40570i) q^{88} +(3.45924 - 9.50418i) q^{89} +(-0.282673 + 0.0498429i) q^{91} +(-2.68397 + 3.19863i) q^{92} +(8.21154 - 1.44792i) q^{94} +(5.07158 + 1.84591i) q^{95} +(5.08812 + 2.93763i) q^{97} +(-6.87228 - 1.21177i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	12 * q - 12 * q^7 + 6 * q^10 + 6 * q^11 + 6 * q^13 + 18 * q^14 - 18 * q^19 - 18 * q^25 - 12 * q^26 - 6 * q^28 - 18 * q^29 + 12 * q^34 - 18 * q^35 + 30 * q^37 + 24 * q^38 + 12 * q^40 - 24 * q^41 - 6 * q^44 + 30 * q^46 - 6 * q^47 + 12 * q^49 + 36 * q^50 - 12 * q^52 + 12 * q^53 - 18 * q^55 + 6 * q^58 - 36 * q^61 + 6 * q^64 - 36 * q^65 - 30 * q^67 - 12 * q^70 - 12 * q^71 + 48 * q^74 - 18 * q^76 - 12 * q^77 + 6 * q^79 + 48 * q^83 + 18 * q^85 + 36 * q^86 - 36 * q^88 + 18 * q^89 - 6 * q^91 - 18 * q^92 + 36 * q^97 - 36 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\)	\(371\)	\(631\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{18}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.642788	+	0.766044i	0.454519	+	0.541675i
							\(3\)		0			0
\(5\)	−0.273629	−	0.751790i	−0.122371	−	0.336211i								0.863349	−	0.504608i	\(-0.168363\pi\)
							−0.985719	+	0.168397i	\(0.946141\pi\)
\(7\)	−0.138449	+	0.0503913i	−0.0523288	+	0.0190461i	−0.368052	−	0.929805i	\(-0.619975\pi\)
							0.315723	+	0.948851i	\(0.397753\pi\)
\(11\)	2.40570	+	4.16679i	0.725346	+	1.25634i	0.958832	+	0.283975i	\(0.0916533\pi\)
							−0.233486	+	0.972360i	\(0.575013\pi\)
\(13\)	1.91858	+	0.338298i	0.532119	+	0.0938270i	0.433251	−	0.901274i	\(-0.357367\pi\)
							0.0988686	+	0.995100i	\(0.468478\pi\)
\(17\)	3.43779	−	0.606175i	0.833786	−	0.147019i	0.259572	−	0.965724i	\(-0.416419\pi\)
							0.574214	+	0.818705i	\(0.305308\pi\)
\(19\)	−4.33625	+	5.16775i	−0.994805	+	1.18556i	−0.0121861	+	0.999926i	\(0.503879\pi\)
							−0.982619	+	0.185636i	\(0.940565\pi\)
\(23\)	3.61610	+	2.08776i	0.754009	+	0.435327i	0.827141	−	0.561995i	\(-0.189966\pi\)
							−0.0731316	+	0.997322i	\(0.523299\pi\)
\(29\)	−2.63134	+	1.51921i	−0.488628	+	0.282109i	−0.724005	−	0.689795i	\(-0.757701\pi\)
							0.235377	+	0.971904i	\(0.424367\pi\)
\(31\)			8.13740i			1.46152i	0.682634	+	0.730760i	\(0.260834\pi\)
							−0.682634	+	0.730760i	\(0.739166\pi\)
\(37\)	6.08227	+	0.0772535i	0.999919	+	0.0127004i
							\(41\)	−0.676822	+	3.83845i	−0.105702	+	0.599465i	0.885236	+	0.465142i	\(0.153997\pi\)
−0.990938	+	0.134322i	\(0.957114\pi\)
\(43\)		−	8.10852i		−	1.23654i	−0.785966	−	0.618269i	\(-0.787834\pi\)
							0.785966	−	0.618269i	\(-0.212166\pi\)
\(47\)	4.16911	−	7.22111i	0.608127	−	1.05331i	−0.383422	−	0.923573i	\(-0.625254\pi\)
							0.991549	−	0.129734i	\(-0.0414122\pi\)
\(53\)	−10.2327	−	3.72440i	−1.40557	−	0.511586i	−0.475744	−	0.879584i	\(-0.657821\pi\)
							−0.929827	+	0.367998i	\(0.880043\pi\)
\(59\)	2.96569	−	8.14816i	0.386100	−	1.06080i	−0.582642	−	0.812729i	\(-0.697981\pi\)
							0.968742	−	0.248072i	\(-0.0797968\pi\)
\(61\)	−0.346344	−	0.0610698i	−0.0443448	−	0.00781919i	0.151432	−	0.988468i	\(-0.451612\pi\)
							−0.195777	+	0.980649i	\(0.562723\pi\)
\(67\)	−2.25471	+	0.820647i	−0.275457	+	0.100258i	−0.476055	−	0.879416i	\(-0.657934\pi\)
							0.200598	+	0.979674i	\(0.435711\pi\)
\(71\)	−5.34953	−	4.48879i	−0.634873	−	0.532721i	0.267566	−	0.963539i	\(-0.413781\pi\)
							−0.902439	+	0.430818i	\(0.858225\pi\)
\(73\)	1.13399			0.132724			0.0663618	−	0.997796i	\(-0.478861\pi\)
							0.0663618	+	0.997796i	\(0.478861\pi\)
\(79\)	0.646510	+	1.77627i	0.0727380	+	0.199846i	0.970734	−	0.240158i	\(-0.0771992\pi\)
							−0.897996	+	0.440004i	\(0.854977\pi\)
\(83\)	1.71997	+	9.75442i	0.188791	+	1.07069i	0.920987	+	0.389593i	\(0.127384\pi\)
							−0.732196	+	0.681094i	\(0.761505\pi\)
\(89\)	3.45924	−	9.50418i	0.366679	−	1.00744i	−0.609937	−	0.792450i	\(-0.708805\pi\)
							0.976616	−	0.214992i	\(-0.0689724\pi\)
\(97\)	5.08812	+	2.93763i	0.516620	+	0.298271i	0.735551	−	0.677470i	\(-0.236923\pi\)
							−0.218930	+	0.975740i	\(0.570257\pi\)

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	666.2.bj.c.469.2		12
3.2	odd	2		74.2.h.a.25.1	yes	12
12.11	even	2		592.2.bq.b.321.2		12
37.3	even	18	inner	666.2.bj.c.595.2		12
111.59	even	36		2738.2.a.r.1.1		6
111.77	odd	18		74.2.h.a.3.1	✓	12
111.89	even	36		2738.2.a.s.1.2		6
444.299	even	18		592.2.bq.b.225.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.h.a.3.1	✓	12	111.77	odd	18
74.2.h.a.25.1	yes	12	3.2	odd	2
592.2.bq.b.225.2		12	444.299	even	18
592.2.bq.b.321.2		12	12.11	even	2
666.2.bj.c.469.2		12	1.1	even	1	trivial
666.2.bj.c.595.2		12	37.3	even	18	inner
2738.2.a.r.1.1		6	111.59	even	36
2738.2.a.s.1.2		6	111.89	even	36