Embedded newform 168.2.v.a.11.17

• Label: 168.2.v.a.11.17
• 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			11.17
Character	\(\chi\)	\(=\)	168.11
Dual form			168.2.v.a.107.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.389211 + 1.35960i) q^{2} +(-0.987527 - 1.42295i) q^{3} +(-1.69703 + 1.05834i) q^{4} +(-1.77783 - 3.07929i) q^{5} +(1.55029 - 1.89647i) q^{6} +(-0.793456 - 2.52397i) q^{7} +(-2.09943 - 1.89537i) q^{8} +(-1.04958 + 2.81041i) q^{9} +(3.49466 - 3.61563i) q^{10} +(-0.396703 - 0.229036i) q^{11} +(3.18183 + 1.36965i) q^{12} +0.799063i q^{13} +(3.12277 - 2.06114i) q^{14} +(-2.62603 + 5.57065i) q^{15} +(1.75982 - 3.59208i) q^{16} +(5.48456 + 3.16651i) q^{17} +(-4.22954 - 0.333169i) q^{18} +(-2.61696 - 4.53272i) q^{19} +(6.27598 + 3.34410i) q^{20} +(-2.80793 + 3.62154i) q^{21} +(0.156997 - 0.628501i) q^{22} +(-1.55053 - 2.68559i) q^{23} +(-0.623772 + 4.85911i) q^{24} +(-3.82137 + 6.61880i) q^{25} +(-1.08641 + 0.311004i) q^{26} +(5.03556 - 1.28185i) q^{27} +(4.01774 + 3.44351i) q^{28} +2.60340 q^{29} +(-8.59594 - 1.40219i) q^{30} +(-4.42520 - 2.55489i) q^{31} +(5.56873 + 0.994581i) q^{32} +(0.0658472 + 0.790668i) q^{33} +(-2.17055 + 8.68926i) q^{34} +(-6.36142 + 6.93048i) q^{35} +(-1.19320 - 5.88016i) q^{36} +(2.89424 - 1.67099i) q^{37} +(5.14413 - 5.32221i) q^{38} +(1.13703 - 0.789096i) q^{39} +(-2.10396 + 9.83439i) q^{40} -10.1452i q^{41} +(-6.01672 - 2.40812i) q^{42} +3.56359 q^{43} +(0.915615 - 0.0311657i) q^{44} +(10.5200 - 1.76446i) q^{45} +(3.04785 - 3.15336i) q^{46} +(2.15923 + 3.73989i) q^{47} +(-6.84922 + 1.04314i) q^{48} +(-5.74086 + 4.00532i) q^{49} +(-10.4862 - 2.61943i) q^{50} +(-0.910362 - 10.9313i) q^{51} +(-0.845682 - 1.35603i) q^{52} +(-1.16913 + 2.02499i) q^{53} +(3.70270 + 6.34744i) q^{54} +1.62875i q^{55} +(-3.11805 + 6.80278i) q^{56} +(-3.86551 + 8.19999i) q^{57} +(1.01327 + 3.53959i) q^{58} +(-1.49741 - 0.864528i) q^{59} +(-1.43921 - 12.2328i) q^{60} +(-4.60841 + 2.66067i) q^{61} +(1.75130 - 7.01090i) q^{62} +(7.92618 + 0.419175i) q^{63} +(0.815178 + 7.95836i) q^{64} +(2.46055 - 1.42060i) q^{65} +(-1.04936 + 0.397262i) q^{66} +(0.00979145 - 0.0169593i) q^{67} +(-12.6587 + 0.430877i) q^{68} +(-2.29028 + 4.85842i) q^{69} +(-11.8986 - 5.95157i) q^{70} +9.04022 q^{71} +(7.53026 - 3.91090i) q^{72} +(4.15674 - 7.19969i) q^{73} +(3.39835 + 3.28465i) q^{74} +(13.1919 - 1.09863i) q^{75} +(9.23823 + 4.92251i) q^{76} +(-0.263315 + 1.18300i) q^{77} +(1.51540 + 1.23878i) q^{78} +(10.8825 - 6.28304i) q^{79} +(-14.1897 + 0.967099i) q^{80} +(-6.79677 - 5.89949i) q^{81} +(13.7934 - 3.94861i) q^{82} -0.694004i q^{83} +(0.932309 - 9.11761i) q^{84} -22.5181i q^{85} +(1.38699 + 4.84506i) q^{86} +(-2.57093 - 3.70452i) q^{87} +(0.398740 + 1.23274i) q^{88} +(7.02189 - 4.05409i) q^{89} +(6.49348 + 13.6163i) q^{90} +(2.01681 - 0.634021i) q^{91} +(5.47357 + 2.91654i) q^{92} +(0.734523 + 8.81987i) q^{93} +(-4.24437 + 4.39129i) q^{94} +(-9.30504 + 16.1168i) q^{95} +(-4.08404 - 8.90621i) q^{96} +7.05333 q^{97} +(-7.68004 - 6.24636i) q^{98} +(1.06006 - 0.874504i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.389211	+	1.35960i	0.275214	+	0.961383i
							\(3\)	−0.987527	−	1.42295i	−0.570149	−	0.821541i
\(5\)	−1.77783	−	3.07929i	−0.795070	−	1.37710i								−0.922795	−	0.385292i	\(-0.874101\pi\)
							0.127724	−	0.991810i	\(-0.459233\pi\)
\(7\)	−0.793456	−	2.52397i	−0.299898	−	0.953971i
							\(11\)	−0.396703	−	0.229036i	−0.119610	−	0.0690571i	0.439001	−	0.898486i	\(-0.355332\pi\)
−0.558612	+	0.829429i	\(0.688666\pi\)
\(13\)			0.799063i			0.221620i	0.993842	+	0.110810i	\(0.0353445\pi\)
							−0.993842	+	0.110810i	\(0.964655\pi\)
\(17\)	5.48456	+	3.16651i	1.33020	+	0.767993i	0.985330	−	0.170657i	\(-0.0545891\pi\)
							0.344872	+	0.938650i	\(0.387922\pi\)
\(19\)	−2.61696	−	4.53272i	−0.600373	−	1.03988i	−0.992764	−	0.120078i	\(-0.961686\pi\)
							0.392392	−	0.919798i	\(-0.371648\pi\)
\(23\)	−1.55053	−	2.68559i	−0.323307	−	0.559985i	0.657861	−	0.753139i	\(-0.271461\pi\)
							−0.981168	+	0.193155i	\(0.938128\pi\)
\(29\)	2.60340			0.483440			0.241720	−	0.970346i	\(-0.422288\pi\)
							0.241720	+	0.970346i	\(0.422288\pi\)
\(31\)	−4.42520	−	2.55489i	−0.794790	−	0.458872i	0.0468563	−	0.998902i	\(-0.485080\pi\)
							−0.841646	+	0.540030i	\(0.818413\pi\)
\(37\)	2.89424	−	1.67099i	0.475810	−	0.274709i	−0.242859	−	0.970062i	\(-0.578085\pi\)
							0.718669	+	0.695352i	\(0.244752\pi\)
\(41\)		−	10.1452i		−	1.58441i	−0.610256	−	0.792205i	\(-0.708933\pi\)
							0.610256	−	0.792205i	\(-0.291067\pi\)
\(43\)	3.56359			0.543442			0.271721	−	0.962376i	\(-0.412407\pi\)
							0.271721	+	0.962376i	\(0.412407\pi\)
\(47\)	2.15923	+	3.73989i	0.314956	+	0.545519i	0.979428	−	0.201794i	\(-0.0646770\pi\)
							−0.664472	+	0.747313i	\(0.731344\pi\)
\(53\)	−1.16913	+	2.02499i	−0.160592	+	0.278154i	−0.935081	−	0.354433i	\(-0.884674\pi\)
							0.774489	+	0.632587i	\(0.218007\pi\)
\(59\)	−1.49741	−	0.864528i	−0.194946	−	0.112552i	0.399350	−	0.916798i	\(-0.369236\pi\)
							−0.594296	+	0.804247i	\(0.702569\pi\)
\(61\)	−4.60841	+	2.66067i	−0.590047	+	0.340664i	−0.765116	−	0.643892i	\(-0.777318\pi\)
							0.175069	+	0.984556i	\(0.443985\pi\)
\(67\)	0.00979145	−	0.0169593i	0.00119622	−	0.00207191i	−0.865427	−	0.501036i	\(-0.832953\pi\)
							0.866623	+	0.498964i	\(0.166286\pi\)
\(71\)	9.04022			1.07288			0.536439	−	0.843939i	\(-0.319769\pi\)
							0.536439	+	0.843939i	\(0.319769\pi\)
\(73\)	4.15674	−	7.19969i	0.486510	−	0.842660i	−0.513370	−	0.858167i	\(-0.671603\pi\)
							0.999880	+	0.0155078i	\(0.00493650\pi\)
\(79\)	10.8825	−	6.28304i	1.22438	−	0.706897i	0.258533	−	0.966003i	\(-0.416761\pi\)
							0.965849	+	0.259105i	\(0.0834277\pi\)
\(83\)		−	0.694004i		−	0.0761768i	−0.999274	−	0.0380884i	\(-0.987873\pi\)
							0.999274	−	0.0380884i	\(-0.0121268\pi\)
\(89\)	7.02189	−	4.05409i	0.744319	−	0.429733i	−0.0793185	−	0.996849i	\(-0.525274\pi\)
							0.823638	+	0.567116i	\(0.191941\pi\)
\(97\)	7.05333			0.716157			0.358078	−	0.933692i	\(-0.383432\pi\)
							0.358078	+	0.933692i	\(0.383432\pi\)

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.11.17	yes	56
3.2	odd	2	inner	168.2.v.a.11.12	yes	56
4.3	odd	2		672.2.bd.a.431.19		56
7.2	even	3	inner	168.2.v.a.107.22	yes	56
8.3	odd	2	inner	168.2.v.a.11.7	✓	56
8.5	even	2		672.2.bd.a.431.20		56
12.11	even	2		672.2.bd.a.431.2		56
21.2	odd	6	inner	168.2.v.a.107.7	yes	56
24.5	odd	2		672.2.bd.a.431.1		56
24.11	even	2	inner	168.2.v.a.11.22	yes	56
28.23	odd	6		672.2.bd.a.527.1		56
56.37	even	6		672.2.bd.a.527.2		56
56.51	odd	6	inner	168.2.v.a.107.12	yes	56
84.23	even	6		672.2.bd.a.527.20		56
168.107	even	6	inner	168.2.v.a.107.17	yes	56
168.149	odd	6		672.2.bd.a.527.19		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.7	✓	56	8.3	odd	2	inner
168.2.v.a.11.12	yes	56	3.2	odd	2	inner
168.2.v.a.11.17	yes	56	1.1	even	1	trivial
168.2.v.a.11.22	yes	56	24.11	even	2	inner
168.2.v.a.107.7	yes	56	21.2	odd	6	inner
168.2.v.a.107.12	yes	56	56.51	odd	6	inner
168.2.v.a.107.17	yes	56	168.107	even	6	inner
168.2.v.a.107.22	yes	56	7.2	even	3	inner
672.2.bd.a.431.1		56	24.5	odd	2
672.2.bd.a.431.2		56	12.11	even	2
672.2.bd.a.431.19		56	4.3	odd	2
672.2.bd.a.431.20		56	8.5	even	2
672.2.bd.a.527.1		56	28.23	odd	6
672.2.bd.a.527.2		56	56.37	even	6
672.2.bd.a.527.19		56	168.149	odd	6
672.2.bd.a.527.20		56	84.23	even	6