Embedded newform 169.2.e.b.147.2

• Label: 169.2.e.b.147.2
• Level: $169$
• Weight: $2$
• Character: 169.147
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.e (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	12.0.17213603549184.1
Defining polynomial:	\( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \)
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			147.2
Root			\(1.56052 - 0.900969i\) of defining polynomial
Character	\(\chi\)	\(=\)	169.147
Dual form			169.2.e.b.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.694498 - 0.400969i) q^{2} +(1.12349 - 1.94594i) q^{3} +(-0.678448 - 1.17511i) q^{4} +0.246980i q^{5} +(-1.56052 + 0.900969i) q^{6} +(2.04113 - 1.17845i) q^{7} +2.69202i q^{8} +(-1.02446 - 1.77441i) q^{9} +(0.0990311 - 0.171527i) q^{10} +(-3.67799 - 2.12349i) q^{11} -3.04892 q^{12} -1.89008 q^{14} +(0.480608 + 0.277479i) q^{15} +(-0.277479 + 0.480608i) q^{16} +(1.07942 + 1.86960i) q^{17} +1.64310i q^{18} +(-0.0763367 + 0.0440730i) q^{19} +(0.290227 - 0.167563i) q^{20} -5.29590i q^{21} +(1.70291 + 2.94952i) q^{22} +(0.746980 - 1.29381i) q^{23} +(5.23852 + 3.02446i) q^{24} +4.93900 q^{25} +2.13706 q^{27} +(-2.76960 - 1.59903i) q^{28} +(-2.31551 + 4.01058i) q^{29} +(-0.222521 - 0.385418i) q^{30} -6.63102i q^{31} +(5.04814 - 2.91454i) q^{32} +(-8.26437 + 4.77144i) q^{33} -1.73125i q^{34} +(0.291053 + 0.504118i) q^{35} +(-1.39008 + 2.40770i) q^{36} +(4.92944 + 2.84601i) q^{37} +0.0706876 q^{38} -0.664874 q^{40} +(10.0388 + 5.79590i) q^{41} +(-2.12349 + 3.67799i) q^{42} +(-0.147948 - 0.256254i) q^{43} +5.76271i q^{44} +(0.438244 - 0.253020i) q^{45} +(-1.03755 + 0.599031i) q^{46} +7.35690i q^{47} +(0.623490 + 1.07992i) q^{48} +(-0.722521 + 1.25144i) q^{49} +(-3.43013 - 1.98039i) q^{50} +4.85086 q^{51} -10.3937 q^{53} +(-1.48419 - 0.856896i) q^{54} +(0.524459 - 0.908389i) q^{55} +(3.17241 + 5.49477i) q^{56} +0.198062i q^{57} +(3.21624 - 1.85690i) q^{58} +(5.87180 - 3.39008i) q^{59} -0.753020i q^{60} +(-1.73609 - 3.00700i) q^{61} +(-2.65883 + 4.60523i) q^{62} +(-4.18211 - 2.41454i) q^{63} -3.56465 q^{64} +7.65279 q^{66} +(-6.65102 - 3.83997i) q^{67} +(1.46466 - 2.53686i) q^{68} +(-1.67845 - 2.90716i) q^{69} -0.466812i q^{70} +(-7.50400 + 4.33244i) q^{71} +(4.77676 - 2.75786i) q^{72} -6.73556i q^{73} +(-2.28232 - 3.95310i) q^{74} +(5.54892 - 9.61101i) q^{75} +(0.103581 + 0.0598025i) q^{76} -10.0097 q^{77} +9.97046 q^{79} +(-0.118700 - 0.0685317i) q^{80} +(5.47434 - 9.48184i) q^{81} +(-4.64795 - 8.05048i) q^{82} +1.60925i q^{83} +(-6.22324 + 3.59299i) q^{84} +(-0.461754 + 0.266594i) q^{85} +0.237291i q^{86} +(5.20291 + 9.01170i) q^{87} +(5.71648 - 9.90123i) q^{88} +(-2.49823 - 1.44235i) q^{89} -0.405813 q^{90} -2.02715 q^{92} +(-12.9036 - 7.44989i) q^{93} +(2.94989 - 5.10935i) q^{94} +(-0.0108851 - 0.0188536i) q^{95} -13.0978i q^{96} +(-6.97896 + 4.02930i) q^{97} +(1.00358 - 0.579417i) q^{98} +8.70171i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	12 * q + 4 * q^3 + 6 * q^9 + 10 * q^10 - 20 * q^14 - 4 * q^16 - 4 * q^17 - 6 * q^22 - 10 * q^23 + 20 * q^25 + 4 * q^27 + 2 * q^29 - 2 * q^30 - 8 * q^35 - 14 * q^36 - 48 * q^38 - 12 * q^40 - 16 * q^42 + 26 * q^43 - 2 * q^48 - 8 * q^49 + 4 * q^51 + 4 * q^53 - 12 * q^55 - 8 * q^56 - 8 * q^61 + 2 * q^62 + 44 * q^64 + 20 * q^66 + 42 * q^68 - 12 * q^69 + 16 * q^74 + 30 * q^75 - 32 * q^77 - 20 * q^79 + 2 * q^81 - 28 * q^82 + 36 * q^87 + 30 * q^88 + 48 * q^90 - 10 * q^94 + 6 * q^95

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

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

Coefficient data

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

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.694498	−	0.400969i	−0.491085	−	0.283528i	0.233940	−	0.972251i	\(-0.424838\pi\)
							−0.725024	+	0.688723i	\(0.758171\pi\)
\(3\)	1.12349	−	1.94594i	0.648647	−	1.12349i	−0.334799	−	0.942290i	\(-0.608668\pi\)
							0.983446	−	0.181200i	\(-0.0579982\pi\)
\(5\)			0.246980i			0.110453i	0.998474	+	0.0552263i	\(0.0175880\pi\)
							−0.998474	+	0.0552263i	\(0.982412\pi\)
\(7\)	2.04113	−	1.17845i	0.771475	−	0.445411i	−0.0619254	−	0.998081i	\(-0.519724\pi\)
							0.833401	+	0.552669i	\(0.186391\pi\)
\(11\)	−3.67799	−	2.12349i	−1.10896	−	0.640256i	−0.170397	−	0.985375i	\(-0.554505\pi\)
							−0.938559	+	0.345119i	\(0.887839\pi\)
\(13\)		0			0
							\(17\)	1.07942	+	1.86960i	0.261797	+	0.453446i	0.966720	−	0.255839i	\(-0.0823516\pi\)
−0.704922	+	0.709284i	\(0.749018\pi\)
\(19\)	−0.0763367	+	0.0440730i	−0.0175128	+	0.0101110i	−0.508731	−	0.860926i	\(-0.669885\pi\)
							0.491218	+	0.871037i	\(0.336552\pi\)
\(23\)	0.746980	−	1.29381i	0.155756	−	0.269777i	−0.777578	−	0.628786i	\(-0.783552\pi\)
							0.933334	+	0.359009i	\(0.116885\pi\)
\(29\)	−2.31551	+	4.01058i	−0.429980	+	0.744747i	−0.996871	−	0.0790460i	\(-0.974813\pi\)
							0.566891	+	0.823793i	\(0.308146\pi\)
\(31\)		−	6.63102i		−	1.19097i	−0.803368	−	0.595483i	\(-0.796961\pi\)
							0.803368	−	0.595483i	\(-0.203039\pi\)
\(37\)	4.92944	+	2.84601i	0.810394	+	0.467881i	0.847093	−	0.531445i	\(-0.178351\pi\)
							−0.0366986	+	0.999326i	\(0.511684\pi\)
\(41\)	10.0388	+	5.79590i	1.56780	+	0.905167i	0.996426	+	0.0844742i	\(0.0269210\pi\)
							0.571370	+	0.820693i	\(0.306412\pi\)
\(43\)	−0.147948	−	0.256254i	−0.0225619	−	0.0390784i	0.854524	−	0.519412i	\(-0.173849\pi\)
							−0.877086	+	0.480334i	\(0.840516\pi\)
\(47\)			7.35690i			1.07311i	0.843864	+	0.536557i	\(0.180275\pi\)
							−0.843864	+	0.536557i	\(0.819725\pi\)
\(53\)	−10.3937			−1.42769			−0.713844	−	0.700304i	\(-0.753048\pi\)
							−0.713844	+	0.700304i	\(0.753048\pi\)
\(59\)	5.87180	−	3.39008i	0.764443	−	0.441351i	−0.0664458	−	0.997790i	\(-0.521166\pi\)
							0.830889	+	0.556439i	\(0.187833\pi\)
\(61\)	−1.73609	−	3.00700i	−0.222284	−	0.385007i	0.733217	−	0.679995i	\(-0.238018\pi\)
							−0.955501	+	0.294987i	\(0.904685\pi\)
\(67\)	−6.65102	−	3.83997i	−0.812552	−	0.469127i	0.0352895	−	0.999377i	\(-0.488765\pi\)
							−0.847841	+	0.530250i	\(0.822098\pi\)
\(71\)	−7.50400	+	4.33244i	−0.890561	+	0.514166i	−0.874126	−	0.485699i	\(-0.838565\pi\)
							−0.0164351	+	0.999865i	\(0.505232\pi\)
\(73\)		−	6.73556i		−	0.788338i	−0.919038	−	0.394169i	\(-0.871032\pi\)
							0.919038	−	0.394169i	\(-0.128968\pi\)
\(79\)	9.97046			1.12176			0.560882	−	0.827896i	\(-0.310462\pi\)
							0.560882	+	0.827896i	\(0.310462\pi\)
\(83\)			1.60925i			0.176638i	0.996092	+	0.0883192i	\(0.0281495\pi\)
							−0.996092	+	0.0883192i	\(0.971850\pi\)
\(89\)	−2.49823	−	1.44235i	−0.264812	−	0.152889i	0.361716	−	0.932288i	\(-0.382191\pi\)
							−0.626528	+	0.779399i	\(0.715524\pi\)
\(97\)	−6.97896	+	4.02930i	−0.708606	+	0.409114i	−0.810545	−	0.585677i	\(-0.800829\pi\)
							0.101939	+	0.994791i	\(0.467495\pi\)

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	169.2.e.b.147.2		12
13.2	odd	12		169.2.c.c.146.1		6
13.3	even	3	inner	169.2.e.b.23.5		12
13.4	even	6		169.2.b.b.168.2		6
13.5	odd	4		169.2.c.c.22.1		6
13.6	odd	12		169.2.a.b.1.3	✓	3
13.7	odd	12		169.2.a.c.1.1	yes	3
13.8	odd	4		169.2.c.b.22.3		6
13.9	even	3		169.2.b.b.168.5		6
13.10	even	6	inner	169.2.e.b.23.2		12
13.11	odd	12		169.2.c.b.146.3		6
13.12	even	2	inner	169.2.e.b.147.5		12
39.17	odd	6		1521.2.b.l.1351.5		6
39.20	even	12		1521.2.a.o.1.3		3
39.32	even	12		1521.2.a.r.1.1		3
39.35	odd	6		1521.2.b.l.1351.2		6
52.7	even	12		2704.2.a.ba.1.3		3
52.19	even	12		2704.2.a.z.1.3		3
52.35	odd	6		2704.2.f.o.337.6		6
52.43	odd	6		2704.2.f.o.337.5		6
65.19	odd	12		4225.2.a.bg.1.1		3
65.59	odd	12		4225.2.a.bb.1.3		3
91.6	even	12		8281.2.a.bf.1.3		3
91.20	even	12		8281.2.a.bj.1.1		3

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.3	✓	3	13.6	odd	12
169.2.a.c.1.1	yes	3	13.7	odd	12
169.2.b.b.168.2		6	13.4	even	6
169.2.b.b.168.5		6	13.9	even	3
169.2.c.b.22.3		6	13.8	odd	4
169.2.c.b.146.3		6	13.11	odd	12
169.2.c.c.22.1		6	13.5	odd	4
169.2.c.c.146.1		6	13.2	odd	12
169.2.e.b.23.2		12	13.10	even	6	inner
169.2.e.b.23.5		12	13.3	even	3	inner
169.2.e.b.147.2		12	1.1	even	1	trivial
169.2.e.b.147.5		12	13.12	even	2	inner
1521.2.a.o.1.3		3	39.20	even	12
1521.2.a.r.1.1		3	39.32	even	12
1521.2.b.l.1351.2		6	39.35	odd	6
1521.2.b.l.1351.5		6	39.17	odd	6
2704.2.a.z.1.3		3	52.19	even	12
2704.2.a.ba.1.3		3	52.7	even	12
2704.2.f.o.337.5		6	52.43	odd	6
2704.2.f.o.337.6		6	52.35	odd	6
4225.2.a.bb.1.3		3	65.59	odd	12
4225.2.a.bg.1.1		3	65.19	odd	12
8281.2.a.bf.1.3		3	91.6	even	12
8281.2.a.bj.1.1		3	91.20	even	12