Embedded newform 67.2.e.b.24.1

• Label: 67.2.e.b.24.1
• Level: $67$
• Weight: $2$
• Character: 67.24
• 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			24.1
Root			\(-0.841254 + 0.540641i\) of defining polynomial
Character	\(\chi\)	\(=\)	67.24
Dual form			67.2.e.b.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.915415 - 2.00448i) q^{2} +(-0.601808 + 0.386758i) q^{3} +(-1.87023 - 2.15836i) q^{4} +(-0.0405070 + 0.281733i) q^{5} +(0.224345 + 1.56036i) q^{6} +(0.226900 - 0.496841i) q^{7} +(-1.80972 + 0.531382i) q^{8} +(-1.03365 + 2.26339i) q^{9} +(0.527646 + 0.339098i) q^{10} +(-0.508975 + 3.54000i) q^{11} +(1.96028 + 0.575591i) q^{12} +(0.942270 + 0.276675i) q^{13} +(-0.788201 - 0.909632i) q^{14} +(-0.0845850 - 0.185215i) q^{15} +(0.221378 - 1.53972i) q^{16} +(0.862774 - 0.995695i) q^{17} +(3.59069 + 4.14388i) q^{18} +(-3.05062 - 6.67992i) q^{19} +(0.683838 - 0.439476i) q^{20} +(0.0556075 + 0.386758i) q^{21} +(6.62993 + 4.26080i) q^{22} +(-6.22839 + 4.00274i) q^{23} +(0.883588 - 1.01971i) q^{24} +(4.71973 + 1.38584i) q^{25} +(1.41716 - 1.63549i) q^{26} +(-0.558746 - 3.88617i) q^{27} +(-1.49672 + 0.439476i) q^{28} -3.42094 q^{29} -0.448691 q^{30} +(-5.85621 + 1.71954i) q^{31} +(-6.05710 - 3.89266i) q^{32} +(-1.06282 - 2.32725i) q^{33} +(-1.20605 - 2.64089i) q^{34} +(0.130785 + 0.0840506i) q^{35} +(6.81838 - 2.00206i) q^{36} +5.01428 q^{37} -16.1823 q^{38} +(-0.674072 + 0.197925i) q^{39} +(-0.0764012 - 0.531382i) q^{40} +(0.877875 - 1.01312i) q^{41} +(0.826153 + 0.242580i) q^{42} +(5.80088 - 6.69457i) q^{43} +(8.59250 - 5.52206i) q^{44} +(-0.595800 - 0.382897i) q^{45} +(2.32186 + 16.1489i) q^{46} +(4.50791 - 2.89705i) q^{47} +(0.462273 + 1.01224i) q^{48} +(4.38866 + 5.06478i) q^{49} +(7.09840 - 8.19199i) q^{50} +(-0.134131 + 0.932902i) q^{51} +(-1.16510 - 2.55121i) q^{52} +(4.58316 + 5.28925i) q^{53} +(-8.30122 - 2.43746i) q^{54} +(-0.976716 - 0.286790i) q^{55} +(-0.146613 + 1.01971i) q^{56} +(4.41940 + 2.84017i) q^{57} +(-3.13158 + 6.85720i) q^{58} +(-0.837324 + 0.245860i) q^{59} +(-0.241568 + 0.528960i) q^{60} +(-1.86953 - 13.0029i) q^{61} +(-1.91408 + 13.3127i) q^{62} +(0.890008 + 1.02712i) q^{63} +(-10.7303 + 6.89594i) q^{64} +(-0.116117 + 0.254261i) q^{65} -5.63784 q^{66} +(-1.18356 + 8.09933i) q^{67} -3.76266 q^{68} +(2.20020 - 4.81777i) q^{69} +(0.288201 - 0.185215i) q^{70} +(-2.89602 - 3.34218i) q^{71} +(0.667903 - 4.64537i) q^{72} +(0.738601 + 5.13708i) q^{73} +(4.59015 - 10.0510i) q^{74} +(-3.37636 + 0.991388i) q^{75} +(-8.71232 + 19.0773i) q^{76} +(1.64333 + 1.05610i) q^{77} +(-0.220318 + 1.53235i) q^{78} +(7.48653 + 2.19824i) q^{79} +(0.424822 + 0.124739i) q^{80} +(-3.04910 - 3.51885i) q^{81} +(-1.22716 - 2.68711i) q^{82} +(-0.929807 + 6.46695i) q^{83} +(0.730766 - 0.843349i) q^{84} +(0.245571 + 0.283404i) q^{85} +(-8.10892 - 17.7561i) q^{86} +(2.05875 - 1.32308i) q^{87} +(-0.959989 - 6.67687i) q^{88} +(12.8365 + 8.24949i) q^{89} +(-1.31291 + 0.843758i) q^{90} +(0.351265 - 0.405381i) q^{91} +(20.2879 + 5.95707i) q^{92} +(2.85927 - 3.29977i) q^{93} +(-1.68048 - 11.6880i) q^{94} +(2.00552 - 0.588874i) q^{95} +5.15073 q^{96} -5.39459 q^{97} +(14.1697 - 4.16060i) q^{98} +(-7.48628 - 4.81114i) 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 \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.915415	−	2.00448i	0.647296	−	1.41738i	−0.246605	−	0.969116i	\(-0.579315\pi\)
							0.893901	−	0.448265i	\(-0.147958\pi\)
\(3\)	−0.601808	+	0.386758i	−0.347454	+	0.223295i	−0.702714	−	0.711472i	\(-0.748029\pi\)
							0.355260	+	0.934767i	\(0.384392\pi\)
\(5\)	−0.0405070	+	0.281733i	−0.0181153	+	0.125995i	−0.996872	−	0.0790290i	\(-0.974818\pi\)
							0.978757	+	0.205024i	\(0.0657271\pi\)
\(7\)	0.226900	−	0.496841i	0.0857601	−	0.187788i	−0.861893	−	0.507090i	\(-0.830721\pi\)
							0.947653	+	0.319302i	\(0.103448\pi\)
\(11\)	−0.508975	+	3.54000i	−0.153462	+	1.06735i	0.756898	+	0.653533i	\(0.226714\pi\)
							−0.910360	+	0.413817i	\(0.864195\pi\)
\(13\)	0.942270	+	0.276675i	0.261339	+	0.0767360i	0.409777	−	0.912186i	\(-0.365606\pi\)
							−0.148438	+	0.988922i	\(0.547425\pi\)
\(17\)	0.862774	−	0.995695i	0.209254	−	0.241491i	−0.641415	−	0.767194i	\(-0.721652\pi\)
							0.850668	+	0.525703i	\(0.176198\pi\)
\(19\)	−3.05062	−	6.67992i	−0.699859	−	1.53248i	−0.840142	−	0.542367i	\(-0.817528\pi\)
							0.140283	−	0.990111i	\(-0.455199\pi\)
\(23\)	−6.22839	+	4.00274i	−1.29871	+	0.834630i	−0.993071	−	0.117518i	\(-0.962506\pi\)
							−0.305639	+	0.952148i	\(0.598870\pi\)
\(29\)	−3.42094			−0.635252			−0.317626	−	0.948216i	\(-0.602886\pi\)
							−0.317626	+	0.948216i	\(0.602886\pi\)
\(31\)	−5.85621	+	1.71954i	−1.05181	+	0.308838i	−0.761547	−	0.648110i	\(-0.775560\pi\)
							−0.290260	+	0.956948i	\(0.593742\pi\)
\(37\)	5.01428			0.824343			0.412172	−	0.911106i	\(-0.364770\pi\)
							0.412172	+	0.911106i	\(0.364770\pi\)
\(41\)	0.877875	−	1.01312i	0.137101	−	0.158223i	−0.683047	−	0.730374i	\(-0.739346\pi\)
							0.820148	+	0.572151i	\(0.193891\pi\)
\(43\)	5.80088	−	6.69457i	0.884626	−	1.02091i	−0.114995	−	0.993366i	\(-0.536685\pi\)
							0.999621	−	0.0275467i	\(-0.00876948\pi\)
\(47\)	4.50791	−	2.89705i	0.657546	−	0.422579i	−0.168871	−	0.985638i	\(-0.554012\pi\)
							0.826416	+	0.563059i	\(0.190376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.e.b.24.1	yes	10
3.2	odd	2		603.2.u.a.91.1		10
67.9	even	11		4489.2.a.i.1.5		5
67.14	even	11	inner	67.2.e.b.14.1	✓	10
67.58	odd	22		4489.2.a.h.1.1		5
201.14	odd	22		603.2.u.a.550.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.e.b.14.1	✓	10	67.14	even	11	inner
67.2.e.b.24.1	yes	10	1.1	even	1	trivial
603.2.u.a.91.1		10	3.2	odd	2
603.2.u.a.550.1		10	201.14	odd	22
4489.2.a.h.1.1		5	67.58	odd	22
4489.2.a.i.1.5		5	67.9	even	11