Embedded newform 67.2.e.b.22.1

• Label: 67.2.e.b.22.1
• Level: $67$
• Weight: $2$
• Character: 67.22
• Analytic conductor: $0.535$
• Analytic rank: $0$
• Dimension: $10$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	67.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.534997693543\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			22.1
Root			\(0.142315 + 0.989821i\) of defining polynomial
Character	\(\chi\)	\(=\)	67.22
Dual form			67.2.e.b.64.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.459493 + 0.134919i) q^{2} +(0.260554 + 1.81219i) q^{3} +(-1.48958 + 0.957293i) q^{4} +(-0.345139 + 0.755750i) q^{5} +(-0.364223 - 0.797537i) q^{6} +(1.04408 - 0.306569i) q^{7} +(1.18251 - 1.36469i) q^{8} +(-0.337683 + 0.0991526i) q^{9} +(0.0566239 - 0.393828i) q^{10} +(1.19505 - 2.61680i) q^{11} +(-2.12292 - 2.44998i) q^{12} +(2.87491 + 3.31782i) q^{13} +(-0.438384 + 0.281733i) q^{14} +(-1.45949 - 0.428546i) q^{15} +(1.11189 - 2.43470i) q^{16} +(-2.76321 - 1.77580i) q^{17} +(0.141785 - 0.0911198i) q^{18} +(-3.40746 - 1.00052i) q^{19} +(-0.209362 - 1.45615i) q^{20} +(0.827602 + 1.81219i) q^{21} +(-0.196061 + 1.36364i) q^{22} +(-0.912860 - 6.34908i) q^{23} +(2.78118 + 1.78736i) q^{24} +(2.82227 + 3.25707i) q^{25} +(-1.76864 - 1.13663i) q^{26} +(2.01399 + 4.41003i) q^{27} +(-1.26176 + 1.45615i) q^{28} -1.97270 q^{29} +0.728446 q^{30} +(-1.76031 + 2.03151i) q^{31} +(-0.696384 + 4.84346i) q^{32} +(5.05353 + 1.48385i) q^{33} +(1.50926 + 0.443160i) q^{34} +(-0.128663 + 0.894870i) q^{35} +(0.408086 - 0.470956i) q^{36} +8.07686 q^{37} +1.70069 q^{38} +(-5.26347 + 6.07437i) q^{39} +(0.623231 + 1.36469i) q^{40} +(-2.28074 - 1.46575i) q^{41} +(-0.624777 - 0.721031i) q^{42} +(-6.51473 - 4.18676i) q^{43} +(0.724922 + 5.04194i) q^{44} +(0.0416130 - 0.289425i) q^{45} +(1.27607 + 2.79420i) q^{46} +(-1.56014 - 10.8510i) q^{47} +(4.70185 + 1.38059i) q^{48} +(-4.89266 + 3.14432i) q^{49} +(-1.73625 - 1.11582i) q^{50} +(2.49814 - 5.47016i) q^{51} +(-7.45852 - 2.19002i) q^{52} +(6.30856 - 4.05427i) q^{53} +(-1.52041 - 1.75465i) q^{54} +(1.56519 + 1.80632i) q^{55} +(0.816259 - 1.78736i) q^{56} +(0.925309 - 6.43566i) q^{57} +(0.906440 - 0.266155i) q^{58} +(-6.97706 + 8.05195i) q^{59} +(2.58427 - 0.758810i) q^{60} +(-3.84302 - 8.41503i) q^{61} +(0.534760 - 1.17096i) q^{62} +(-0.322170 + 0.207046i) q^{63} +(0.428340 + 2.97917i) q^{64} +(-3.49969 + 1.02760i) q^{65} -2.52226 q^{66} +(7.85223 + 2.31139i) q^{67} +5.81597 q^{68} +(11.2679 - 3.30856i) q^{69} +(-0.0616156 - 0.428546i) q^{70} +(-1.45889 + 0.937571i) q^{71} +(-0.264000 + 0.578079i) q^{72} +(-0.608158 - 1.33168i) q^{73} +(-3.71126 + 1.08972i) q^{74} +(-5.16709 + 5.96314i) q^{75} +(6.03346 - 1.77158i) q^{76} +(0.445499 - 3.09851i) q^{77} +(1.59898 - 3.50127i) q^{78} +(-10.1410 - 11.7033i) q^{79} +(1.45626 + 1.68062i) q^{80} +(-8.35529 + 5.36962i) q^{81} +(1.24574 + 0.365783i) q^{82} +(-3.35960 + 7.35650i) q^{83} +(-2.96758 - 1.90715i) q^{84} +(2.29575 - 1.47539i) q^{85} +(3.55835 + 1.04483i) q^{86} +(-0.513994 - 3.57491i) q^{87} +(-2.15795 - 4.72526i) q^{88} +(-2.19746 + 15.2836i) q^{89} +(0.0199281 + 0.138603i) q^{90} +(4.01877 + 2.58271i) q^{91} +(7.43771 + 8.58357i) q^{92} +(-4.14014 - 2.66071i) q^{93} +(2.18089 + 4.77548i) q^{94} +(1.93219 - 2.22987i) q^{95} -8.95874 q^{96} +1.03304 q^{97} +(1.82391 - 2.10491i) q^{98} +(-0.144086 + 1.00214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	10 * q + 4 * q^2 + 3 * q^3 - 14 * q^4 - 9 * q^5 + 10 * q^6 + 7 * q^7 - 7 * q^8 - 6 * q^9 - 8 * q^10 - 12 * q^11 - 13 * q^12 + 15 * q^13 + 5 * q^14 - 6 * q^15 + 12 * q^16 + q^17 + 24 * q^18 - 13 * q^19 + 6 * q^20 + q^21 - 7 * q^22 - 7 * q^23 - 23 * q^24 + 12 * q^25 + 28 * q^26 + 9 * q^27 + 10 * q^28 - 24 * q^29 - 20 * q^30 - q^31 - q^32 + 3 * q^33 - 15 * q^34 - 3 * q^35 + 37 * q^36 + 2 * q^37 - 36 * q^38 - 23 * q^39 + 25 * q^40 - 7 * q^41 + 7 * q^42 - 2 * q^43 + 41 * q^44 + 12 * q^45 + 6 * q^46 + 33 * q^47 + 8 * q^48 + 2 * q^49 - 4 * q^50 + 30 * q^51 - 21 * q^52 + 21 * q^53 - 14 * q^54 + 13 * q^55 - 17 * q^56 + 28 * q^57 - 3 * q^58 - 38 * q^59 + 4 * q^60 - 50 * q^61 + 4 * q^62 - 13 * q^63 - 31 * q^64 - 8 * q^65 - 78 * q^66 + 32 * q^67 - 30 * q^68 + 21 * q^69 - 10 * q^70 - 16 * q^71 - 53 * q^72 + 3 * q^73 - 8 * q^74 - 14 * q^75 + 5 * q^76 + 7 * q^77 + 70 * q^78 - 19 * q^79 + 9 * q^80 - 9 * q^81 - 16 * q^82 + 5 * q^83 + 25 * q^84 - 13 * q^85 + 19 * q^86 + 6 * q^87 + 48 * q^88 + 7 * q^89 - 15 * q^90 - 6 * q^91 + 45 * q^92 - 19 * q^93 + 22 * q^94 + 15 * q^95 + 14 * q^96 + 54 * q^97 + 47 * q^98 + 16 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{10}{11}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.459493	+	0.134919i	−0.324911	+	0.0954024i	−0.440120	−	0.897939i	\(-0.645064\pi\)
							0.115210	+	0.993341i	\(0.463246\pi\)
\(3\)	0.260554	+	1.81219i	0.150431	+	1.04627i	0.915499	+	0.402321i	\(0.131796\pi\)
							−0.765068	+	0.643950i	\(0.777294\pi\)
\(5\)	−0.345139	+	0.755750i	−0.154351	+	0.337981i	−0.970972	−	0.239193i	\(-0.923117\pi\)
							0.816621	+	0.577174i	\(0.195845\pi\)
\(7\)	1.04408	−	0.306569i	0.394624	−	0.115872i	−0.0783990	−	0.996922i	\(-0.524981\pi\)
							0.473023	+	0.881050i	\(0.343163\pi\)
\(11\)	1.19505	−	2.61680i	0.360322	−	0.788995i	−0.639474	−	0.768812i	\(-0.720848\pi\)
							0.999796	−	0.0201827i	\(-0.00642479\pi\)
\(13\)	2.87491	+	3.31782i	0.797356	+	0.920198i	0.998233	−	0.0594185i	\(-0.0189247\pi\)
							−0.200877	+	0.979616i	\(0.564379\pi\)
\(17\)	−2.76321	−	1.77580i	−0.670176	−	0.430696i	0.160813	−	0.986985i	\(-0.448588\pi\)
							−0.830989	+	0.556289i	\(0.812225\pi\)
\(19\)	−3.40746	−	1.00052i	−0.781724	−	0.229535i	−0.133565	−	0.991040i	\(-0.542643\pi\)
							−0.648159	+	0.761505i	\(0.724461\pi\)
\(23\)	−0.912860	−	6.34908i	−0.190345	−	1.32388i	−0.831097	−	0.556128i	\(-0.812286\pi\)
							0.640752	−	0.767748i	\(-0.278623\pi\)
\(29\)	−1.97270			−0.366320			−0.183160	−	0.983083i	\(-0.558633\pi\)
							−0.183160	+	0.983083i	\(0.558633\pi\)
\(31\)	−1.76031	+	2.03151i	−0.316161	+	0.364869i	−0.891480	−	0.453060i	\(-0.850332\pi\)
							0.575319	+	0.817929i	\(0.304878\pi\)
\(37\)	8.07686			1.32783			0.663914	−	0.747809i	\(-0.268894\pi\)
							0.663914	+	0.747809i	\(0.268894\pi\)
\(41\)	−2.28074	−	1.46575i	−0.356192	−	0.228911i	0.350290	−	0.936641i	\(-0.386083\pi\)
							−0.706483	+	0.707730i	\(0.749719\pi\)
\(43\)	−6.51473	−	4.18676i	−0.993487	−	0.638476i	−0.0604184	−	0.998173i	\(-0.519244\pi\)
							−0.933069	+	0.359698i	\(0.882880\pi\)
\(47\)	−1.56014	−	10.8510i	−0.227570	−	1.58278i	−0.708297	−	0.705914i	\(-0.750536\pi\)
							0.480727	−	0.876870i	\(-0.340373\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.e.b.22.1	✓	10
3.2	odd	2		603.2.u.a.424.1		10
67.8	odd	22		4489.2.a.h.1.3		5
67.59	even	11		4489.2.a.i.1.3		5
67.64	even	11	inner	67.2.e.b.64.1	yes	10
201.131	odd	22		603.2.u.a.64.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.e.b.22.1	✓	10	1.1	even	1	trivial
67.2.e.b.64.1	yes	10	67.64	even	11	inner
603.2.u.a.64.1		10	201.131	odd	22
603.2.u.a.424.1		10	3.2	odd	2
4489.2.a.h.1.3		5	67.8	odd	22
4489.2.a.i.1.3		5	67.59	even	11