Embedded newform 13.4.e.a.4.1

• Label: 13.4.e.a.4.1
• Level: $13$
• Weight: $4$
• Character: 13.4
• Analytic conductor: $0.767$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.767024830075\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	13.4
Dual form			13.4.e.a.10.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.00000 - 1.73205i) q^{2} +(3.50000 - 6.06218i) q^{3} +(2.00000 + 3.46410i) q^{4} +13.8564i q^{5} +(-21.0000 + 12.1244i) q^{6} +(19.5000 - 11.2583i) q^{7} +13.8564i q^{8} +(-11.0000 - 19.0526i) q^{9} +(24.0000 - 41.5692i) q^{10} +(-19.5000 - 11.2583i) q^{11} +28.0000 q^{12} +(-13.0000 + 45.0333i) q^{13} -78.0000 q^{14} +(84.0000 + 48.4974i) q^{15} +(40.0000 - 69.2820i) q^{16} +(-13.5000 - 23.3827i) q^{17} +76.2102i q^{18} +(-76.5000 + 44.1673i) q^{19} +(-48.0000 + 27.7128i) q^{20} -157.617i q^{21} +(39.0000 + 67.5500i) q^{22} +(-28.5000 + 49.3634i) q^{23} +(84.0000 + 48.4974i) q^{24} -67.0000 q^{25} +(117.000 - 112.583i) q^{26} +35.0000 q^{27} +(78.0000 + 45.0333i) q^{28} +(34.5000 - 59.7558i) q^{29} +(-168.000 - 290.985i) q^{30} -72.7461i q^{31} +(-144.000 + 83.1384i) q^{32} +(-136.500 + 78.8083i) q^{33} +93.5307i q^{34} +(156.000 + 270.200i) q^{35} +(44.0000 - 76.2102i) q^{36} +(-34.5000 - 19.9186i) q^{37} +306.000 q^{38} +(227.500 + 236.425i) q^{39} -192.000 q^{40} +(-340.500 - 196.588i) q^{41} +(-273.000 + 472.850i) q^{42} +(42.5000 + 73.6122i) q^{43} -90.0666i q^{44} +(264.000 - 152.420i) q^{45} +(171.000 - 98.7269i) q^{46} -342.946i q^{47} +(-280.000 - 484.974i) q^{48} +(82.0000 - 142.028i) q^{49} +(201.000 + 116.047i) q^{50} -189.000 q^{51} +(-182.000 + 45.0333i) q^{52} +426.000 q^{53} +(-105.000 - 60.6218i) q^{54} +(156.000 - 270.200i) q^{55} +(156.000 + 270.200i) q^{56} +618.342i q^{57} +(-207.000 + 119.512i) q^{58} +(-16.5000 + 9.52628i) q^{59} +387.979i q^{60} +(8.50000 + 14.7224i) q^{61} +(-126.000 + 218.238i) q^{62} +(-429.000 - 247.683i) q^{63} -64.0000 q^{64} +(-624.000 - 180.133i) q^{65} +546.000 q^{66} +(142.500 + 82.2724i) q^{67} +(54.0000 - 93.5307i) q^{68} +(199.500 + 345.544i) q^{69} -1080.80i q^{70} +(505.500 - 291.851i) q^{71} +(264.000 - 152.420i) q^{72} +1004.59i q^{73} +(69.0000 + 119.512i) q^{74} +(-234.500 + 406.166i) q^{75} +(-306.000 - 176.669i) q^{76} -507.000 q^{77} +(-273.000 - 1103.32i) q^{78} -1244.00 q^{79} +(960.000 + 554.256i) q^{80} +(419.500 - 726.595i) q^{81} +(681.000 + 1179.53i) q^{82} -426.084i q^{83} +(546.000 - 315.233i) q^{84} +(324.000 - 187.061i) q^{85} -294.449i q^{86} +(-241.500 - 418.290i) q^{87} +(156.000 - 270.200i) q^{88} +(265.500 + 153.286i) q^{89} -1056.00 q^{90} +(253.500 + 1024.51i) q^{91} -228.000 q^{92} +(-441.000 - 254.611i) q^{93} +(-594.000 + 1028.84i) q^{94} +(-612.000 - 1060.02i) q^{95} +1163.94i q^{96} +(1069.50 - 617.476i) q^{97} +(-492.000 + 284.056i) q^{98} +495.367i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 6 * q^2 + 7 * q^3 + 4 * q^4 - 42 * q^6 + 39 * q^7 - 22 * q^9 + 48 * q^10 - 39 * q^11 + 56 * q^12 - 26 * q^13 - 156 * q^14 + 168 * q^15 + 80 * q^16 - 27 * q^17 - 153 * q^19 - 96 * q^20 + 78 * q^22 - 57 * q^23 + 168 * q^24 - 134 * q^25 + 234 * q^26 + 70 * q^27 + 156 * q^28 + 69 * q^29 - 336 * q^30 - 288 * q^32 - 273 * q^33 + 312 * q^35 + 88 * q^36 - 69 * q^37 + 612 * q^38 + 455 * q^39 - 384 * q^40 - 681 * q^41 - 546 * q^42 + 85 * q^43 + 528 * q^45 + 342 * q^46 - 560 * q^48 + 164 * q^49 + 402 * q^50 - 378 * q^51 - 364 * q^52 + 852 * q^53 - 210 * q^54 + 312 * q^55 + 312 * q^56 - 414 * q^58 - 33 * q^59 + 17 * q^61 - 252 * q^62 - 858 * q^63 - 128 * q^64 - 1248 * q^65 + 1092 * q^66 + 285 * q^67 + 108 * q^68 + 399 * q^69 + 1011 * q^71 + 528 * q^72 + 138 * q^74 - 469 * q^75 - 612 * q^76 - 1014 * q^77 - 546 * q^78 - 2488 * q^79 + 1920 * q^80 + 839 * q^81 + 1362 * q^82 + 1092 * q^84 + 648 * q^85 - 483 * q^87 + 312 * q^88 + 531 * q^89 - 2112 * q^90 + 507 * q^91 - 456 * q^92 - 882 * q^93 - 1188 * q^94 - 1224 * q^95 + 2139 * q^97 - 984 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.00000	−	1.73205i	−1.06066	−	0.612372i	−0.135045	−	0.990839i	\(-0.543118\pi\)
							−0.925615	+	0.378467i	\(0.876451\pi\)
\(3\)	3.50000	−	6.06218i	0.673575	−	1.16667i	−0.303308	−	0.952893i	\(-0.598091\pi\)
							0.976883	−	0.213774i	\(-0.0685756\pi\)
\(5\)			13.8564i			1.23935i	0.784857	+	0.619677i	\(0.212737\pi\)
							−0.784857	+	0.619677i	\(0.787263\pi\)
\(7\)	19.5000	−	11.2583i	1.05290	−	0.607893i	0.129441	−	0.991587i	\(-0.458682\pi\)
							0.923460	+	0.383694i	\(0.125348\pi\)
\(11\)	−19.5000	−	11.2583i	−0.534497	−	0.308592i	0.208349	−	0.978055i	\(-0.433191\pi\)
							−0.742846	+	0.669462i	\(0.766525\pi\)
\(13\)	−13.0000	+	45.0333i	−0.277350	+	0.960769i
							\(17\)	−13.5000	−	23.3827i	−0.192602	−	0.333596i	0.753510	−	0.657437i	\(-0.228359\pi\)
−0.946112	+	0.323840i	\(0.895026\pi\)
\(19\)	−76.5000	+	44.1673i	−0.923700	+	0.533299i	−0.884814	−	0.465945i	\(-0.845714\pi\)
							−0.0388865	+	0.999244i	\(0.512381\pi\)
\(23\)	−28.5000	+	49.3634i	−0.258377	+	0.447521i	−0.965807	−	0.259261i	\(-0.916521\pi\)
							0.707431	+	0.706783i	\(0.249854\pi\)
\(29\)	34.5000	−	59.7558i	0.220913	−	0.382633i	−0.734172	−	0.678963i	\(-0.762430\pi\)
							0.955086	+	0.296330i	\(0.0957628\pi\)
\(31\)		−	72.7461i		−	0.421471i	−0.977543	−	0.210735i	\(-0.932414\pi\)
							0.977543	−	0.210735i	\(-0.0675858\pi\)
\(37\)	−34.5000	−	19.9186i	−0.153291	−	0.0885026i	0.421393	−	0.906878i	\(-0.361541\pi\)
							−0.574683	+	0.818376i	\(0.694875\pi\)
\(41\)	−340.500	−	196.588i	−1.29700	−	0.748826i	−0.317118	−	0.948386i	\(-0.602715\pi\)
							−0.979886	+	0.199560i	\(0.936049\pi\)
\(43\)	42.5000	+	73.6122i	0.150725	+	0.261064i	0.931494	−	0.363756i	\(-0.118506\pi\)
							−0.780769	+	0.624820i	\(0.785172\pi\)
\(47\)		−	342.946i		−	1.06434i	−0.846639	−	0.532168i	\(-0.821377\pi\)
							0.846639	−	0.532168i	\(-0.178623\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.4.e.a.4.1	✓	2
3.2	odd	2		117.4.q.c.82.1		2
4.3	odd	2		208.4.w.a.17.1		2
13.2	odd	12		169.4.c.i.146.1		4
13.3	even	3		169.4.e.b.23.1		2
13.4	even	6		169.4.b.b.168.1		2
13.5	odd	4		169.4.c.i.22.1		4
13.6	odd	12		169.4.a.h.1.2		2
13.7	odd	12		169.4.a.h.1.1		2
13.8	odd	4		169.4.c.i.22.2		4
13.9	even	3		169.4.b.b.168.2		2
13.10	even	6	inner	13.4.e.a.10.1	yes	2
13.11	odd	12		169.4.c.i.146.2		4
13.12	even	2		169.4.e.b.147.1		2
39.20	even	12		1521.4.a.q.1.2		2
39.23	odd	6		117.4.q.c.10.1		2
39.32	even	12		1521.4.a.q.1.1		2
52.23	odd	6		208.4.w.a.49.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.e.a.4.1	✓	2	1.1	even	1	trivial
13.4.e.a.10.1	yes	2	13.10	even	6	inner
117.4.q.c.10.1		2	39.23	odd	6
117.4.q.c.82.1		2	3.2	odd	2
169.4.a.h.1.1		2	13.7	odd	12
169.4.a.h.1.2		2	13.6	odd	12
169.4.b.b.168.1		2	13.4	even	6
169.4.b.b.168.2		2	13.9	even	3
169.4.c.i.22.1		4	13.5	odd	4
169.4.c.i.22.2		4	13.8	odd	4
169.4.c.i.146.1		4	13.2	odd	12
169.4.c.i.146.2		4	13.11	odd	12
169.4.e.b.23.1		2	13.3	even	3
169.4.e.b.147.1		2	13.12	even	2
208.4.w.a.17.1		2	4.3	odd	2
208.4.w.a.49.1		2	52.23	odd	6
1521.4.a.q.1.1		2	39.32	even	12
1521.4.a.q.1.2		2	39.20	even	12