Embedded newform 168.2.v.a.11.8

• Label: 168.2.v.a.11.8
• Level: $168$
• Weight: $2$
• Character: 168.11
• Analytic conductor: $1.341$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34148675396\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.8
Character	\(\chi\)	\(=\)	168.11
Dual form			168.2.v.a.107.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.935706 - 1.06040i) q^{2} +(-1.45135 + 0.945295i) q^{3} +(-0.248908 + 1.98445i) q^{4} +(-0.187787 - 0.325257i) q^{5} +(2.36043 + 0.654497i) q^{6} +(0.198565 - 2.63829i) q^{7} +(2.33722 - 1.59292i) q^{8} +(1.21283 - 2.74391i) q^{9} +(-0.169190 + 0.503475i) q^{10} +(-4.18624 - 2.41693i) q^{11} +(-1.51464 - 3.11542i) q^{12} -4.40701i q^{13} +(-2.98345 + 2.25810i) q^{14} +(0.580009 + 0.294548i) q^{15} +(-3.87609 - 0.987893i) q^{16} +(2.39410 + 1.38223i) q^{17} +(-4.04450 + 1.28140i) q^{18} +(1.83569 + 3.17951i) q^{19} +(0.692199 - 0.291695i) q^{20} +(2.20578 + 4.01678i) q^{21} +(1.35417 + 6.70064i) q^{22} +(-3.60943 - 6.25172i) q^{23} +(-1.88635 + 4.52125i) q^{24} +(2.42947 - 4.20797i) q^{25} +(-4.67320 + 4.12366i) q^{26} +(0.833557 + 5.12886i) q^{27} +(5.18613 + 1.05073i) q^{28} -0.968468 q^{29} +(-0.230379 - 0.890653i) q^{30} +(-2.78588 - 1.60843i) q^{31} +(2.57932 + 5.03459i) q^{32} +(8.36041 - 0.449426i) q^{33} +(-0.774447 - 3.83207i) q^{34} +(-0.895411 + 0.430853i) q^{35} +(5.14327 + 3.08979i) q^{36} +(0.459519 - 0.265303i) q^{37} +(1.65390 - 4.92166i) q^{38} +(4.16592 + 6.39611i) q^{39} +(-0.957009 - 0.461068i) q^{40} +3.88604i q^{41} +(2.19545 - 6.09754i) q^{42} +0.747189 q^{43} +(5.83827 - 7.70580i) q^{44} +(-1.12023 + 0.120788i) q^{45} +(-3.25197 + 9.67722i) q^{46} +(5.20322 + 9.01223i) q^{47} +(6.55941 - 2.23027i) q^{48} +(-6.92114 - 1.04774i) q^{49} +(-6.73541 + 1.36120i) q^{50} +(-4.78129 + 0.257025i) q^{51} +(8.74549 + 1.09694i) q^{52} +(1.40600 - 2.43526i) q^{53} +(4.65869 - 5.68301i) q^{54} +1.81547i q^{55} +(-3.73849 - 6.48257i) q^{56} +(-5.66981 - 2.87932i) q^{57} +(0.906201 + 1.02697i) q^{58} +(1.65184 + 0.953689i) q^{59} +(-0.728884 + 1.07768i) q^{60} +(-8.43102 + 4.86765i) q^{61} +(0.901184 + 4.45918i) q^{62} +(-6.99840 - 3.74465i) q^{63} +(2.92522 - 7.44601i) q^{64} +(-1.43341 + 0.827580i) q^{65} +(-8.29946 - 8.44488i) q^{66} +(3.85681 - 6.68019i) q^{67} +(-3.33888 + 4.40691i) q^{68} +(11.1483 + 5.66145i) q^{69} +(1.29472 + 0.546344i) q^{70} +5.55936 q^{71} +(-1.53616 - 8.34507i) q^{72} +(-0.445737 + 0.772039i) q^{73} +(-0.711303 - 0.239029i) q^{74} +(0.451759 + 8.40380i) q^{75} +(-6.76651 + 2.85143i) q^{76} +(-7.20780 + 10.5646i) q^{77} +(2.88438 - 10.4024i) q^{78} +(-11.3282 + 6.54032i) q^{79} +(0.406561 + 1.44624i) q^{80} +(-6.05807 - 6.65581i) q^{81} +(4.12077 - 3.63619i) q^{82} -10.1716i q^{83} +(-8.52015 + 3.37744i) q^{84} -1.03826i q^{85} +(-0.699150 - 0.792322i) q^{86} +(1.40559 - 0.915488i) q^{87} +(-13.6342 + 1.01945i) q^{88} +(-4.70650 + 2.71730i) q^{89} +(1.17629 + 1.07487i) q^{90} +(-11.6270 - 0.875077i) q^{91} +(13.3046 - 5.60663i) q^{92} +(5.56373 - 0.299087i) q^{93} +(4.68792 - 13.9503i) q^{94} +(0.689440 - 1.19414i) q^{95} +(-8.50267 - 4.86874i) q^{96} +5.31623 q^{97} +(5.36513 + 8.31958i) q^{98} +(-11.7090 + 8.55533i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	56 * q - 2 * q^3 - 2 * q^4 - 8 * q^6 - 2 * q^9 + 6 * q^10 + 10 * q^12 - 10 * q^16 - 10 * q^18 - 4 * q^19 - 20 * q^22 - 8 * q^24 - 16 * q^25 - 8 * q^27 - 22 * q^28 - 12 * q^30 - 14 * q^33 - 56 * q^34 + 4 * q^36 + 14 * q^40 - 8 * q^42 - 16 * q^43 - 12 * q^46 + 64 * q^48 - 16 * q^49 - 34 * q^51 - 8 * q^52 + 32 * q^54 + 4 * q^57 - 18 * q^58 + 40 * q^60 + 4 * q^64 - 20 * q^66 - 36 * q^67 - 42 * q^70 + 22 * q^72 + 4 * q^73 + 104 * q^76 + 28 * q^78 - 10 * q^81 + 52 * q^82 + 4 * q^84 + 46 * q^88 + 140 * q^90 + 72 * q^91 - 46 * q^96 - 32 * q^97 - 44 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\)	\(73\)	\(85\)	\(113\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.935706	−	1.06040i	−0.661644	−	0.749818i
							\(3\)	−1.45135	+	0.945295i	−0.837937	+	0.545766i
\(5\)	−0.187787	−	0.325257i	−0.0839810	−	0.145459i								0.820976	−	0.570963i	\(-0.193430\pi\)
							−0.904957	+	0.425504i	\(0.860097\pi\)
\(7\)	0.198565	−	2.63829i	0.0750504	−	0.997180i
							\(11\)	−4.18624	−	2.41693i	−1.26220	−	0.728731i	−0.288700	−	0.957420i	\(-0.593223\pi\)
−0.973500	+	0.228688i	\(0.926556\pi\)
\(13\)		−	4.40701i		−	1.22228i	−0.791521	−	0.611142i	\(-0.790710\pi\)
							0.791521	−	0.611142i	\(-0.209290\pi\)
\(17\)	2.39410	+	1.38223i	0.580653	+	0.335240i	0.761393	−	0.648291i	\(-0.224516\pi\)
							−0.180740	+	0.983531i	\(0.557849\pi\)
\(19\)	1.83569	+	3.17951i	0.421137	+	0.729431i	0.996051	−	0.0887833i	\(-0.0282979\pi\)
							−0.574914	+	0.818214i	\(0.694965\pi\)
\(23\)	−3.60943	−	6.25172i	−0.752618	−	1.30357i	−0.946550	−	0.322558i	\(-0.895457\pi\)
							0.193932	−	0.981015i	\(-0.437876\pi\)
\(29\)	−0.968468			−0.179840			−0.0899200	−	0.995949i	\(-0.528661\pi\)
							−0.0899200	+	0.995949i	\(0.528661\pi\)
\(31\)	−2.78588	−	1.60843i	−0.500359	−	0.288883i	0.228503	−	0.973543i	\(-0.426617\pi\)
							−0.728862	+	0.684661i	\(0.759950\pi\)
\(37\)	0.459519	−	0.265303i	0.0755445	−	0.0436156i	−0.461752	−	0.887009i	\(-0.652779\pi\)
							0.537296	+	0.843393i	\(0.319446\pi\)
\(41\)			3.88604i			0.606897i	0.952848	+	0.303449i	\(0.0981381\pi\)
							−0.952848	+	0.303449i	\(0.901862\pi\)
\(43\)	0.747189			0.113945			0.0569726	−	0.998376i	\(-0.481855\pi\)
							0.0569726	+	0.998376i	\(0.481855\pi\)
\(47\)	5.20322	+	9.01223i	0.758967	+	1.31457i	0.943378	+	0.331720i	\(0.107629\pi\)
							−0.184411	+	0.982849i	\(0.559038\pi\)
\(53\)	1.40600	−	2.43526i	0.193129	−	0.334509i	−0.753157	−	0.657841i	\(-0.771470\pi\)
							0.946285	+	0.323332i	\(0.104803\pi\)
\(59\)	1.65184	+	0.953689i	0.215051	+	0.124160i	0.603657	−	0.797244i	\(-0.293710\pi\)
							−0.388606	+	0.921404i	\(0.627043\pi\)
\(61\)	−8.43102	+	4.86765i	−1.07948	+	0.623239i	−0.930756	−	0.365640i	\(-0.880850\pi\)
							−0.148725	+	0.988879i	\(0.547517\pi\)
\(67\)	3.85681	−	6.68019i	0.471184	−	0.816115i	−0.528273	−	0.849075i	\(-0.677160\pi\)
							0.999457	+	0.0329601i	\(0.0104934\pi\)
\(71\)	5.55936			0.659774			0.329887	−	0.944020i	\(-0.392989\pi\)
							0.329887	+	0.944020i	\(0.392989\pi\)
\(73\)	−0.445737	+	0.772039i	−0.0521696	+	0.0903603i	−0.890931	−	0.454139i	\(-0.849947\pi\)
							0.838761	+	0.544499i	\(0.183280\pi\)
\(79\)	−11.3282	+	6.54032i	−1.27452	+	0.735844i	−0.975835	−	0.218509i	\(-0.929881\pi\)
							−0.298684	+	0.954352i	\(0.596547\pi\)
\(83\)		−	10.1716i		−	1.11648i	−0.829680	−	0.558239i	\(-0.811477\pi\)
							0.829680	−	0.558239i	\(-0.188523\pi\)
\(89\)	−4.70650	+	2.71730i	−0.498888	+	0.288033i	−0.728254	−	0.685307i	\(-0.759668\pi\)
							0.229366	+	0.973340i	\(0.426335\pi\)
\(97\)	5.31623			0.539781			0.269891	−	0.962891i	\(-0.413012\pi\)
							0.269891	+	0.962891i	\(0.413012\pi\)

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	168.2.v.a.11.8	✓	56
3.2	odd	2	inner	168.2.v.a.11.21	yes	56
4.3	odd	2		672.2.bd.a.431.23		56
7.2	even	3	inner	168.2.v.a.107.11	yes	56
8.3	odd	2	inner	168.2.v.a.11.18	yes	56
8.5	even	2		672.2.bd.a.431.24		56
12.11	even	2		672.2.bd.a.431.14		56
21.2	odd	6	inner	168.2.v.a.107.18	yes	56
24.5	odd	2		672.2.bd.a.431.13		56
24.11	even	2	inner	168.2.v.a.11.11	yes	56
28.23	odd	6		672.2.bd.a.527.13		56
56.37	even	6		672.2.bd.a.527.14		56
56.51	odd	6	inner	168.2.v.a.107.21	yes	56
84.23	even	6		672.2.bd.a.527.24		56
168.107	even	6	inner	168.2.v.a.107.8	yes	56
168.149	odd	6		672.2.bd.a.527.23		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.8	✓	56	1.1	even	1	trivial
168.2.v.a.11.11	yes	56	24.11	even	2	inner
168.2.v.a.11.18	yes	56	8.3	odd	2	inner
168.2.v.a.11.21	yes	56	3.2	odd	2	inner
168.2.v.a.107.8	yes	56	168.107	even	6	inner
168.2.v.a.107.11	yes	56	7.2	even	3	inner
168.2.v.a.107.18	yes	56	21.2	odd	6	inner
168.2.v.a.107.21	yes	56	56.51	odd	6	inner
672.2.bd.a.431.13		56	24.5	odd	2
672.2.bd.a.431.14		56	12.11	even	2
672.2.bd.a.431.23		56	4.3	odd	2
672.2.bd.a.431.24		56	8.5	even	2
672.2.bd.a.527.13		56	28.23	odd	6
672.2.bd.a.527.14		56	56.37	even	6
672.2.bd.a.527.23		56	168.149	odd	6
672.2.bd.a.527.24		56	84.23	even	6