Embedded newform 168.2.v.a.11.10

• Label: 168.2.v.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	168.11
Dual form			168.2.v.a.107.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.819527 + 1.15255i) q^{2} +(-1.71959 - 0.207430i) q^{3} +(-0.656749 - 1.88910i) q^{4} +(-0.692094 - 1.19874i) q^{5} +(1.64832 - 1.81192i) q^{6} +(2.08863 + 1.62408i) q^{7} +(2.71550 + 0.791228i) q^{8} +(2.91395 + 0.713387i) q^{9} +(1.94880 + 0.184728i) q^{10} +(3.82316 + 2.20731i) q^{11} +(0.737481 + 3.38469i) q^{12} -6.43609i q^{13} +(-3.58352 + 1.07627i) q^{14} +(0.941460 + 2.20490i) q^{15} +(-3.13736 + 2.48132i) q^{16} +(2.52866 + 1.45992i) q^{17} +(-3.21027 + 2.77383i) q^{18} +(1.58928 + 2.75271i) q^{19} +(-1.81001 + 2.09470i) q^{20} +(-3.25469 - 3.22599i) q^{21} +(-5.67722 + 2.59745i) q^{22} +(-1.84696 - 3.19902i) q^{23} +(-4.50541 - 1.92386i) q^{24} +(1.54201 - 2.67084i) q^{25} +(7.41792 + 5.27455i) q^{26} +(-4.86280 - 1.83117i) q^{27} +(1.69634 - 5.01223i) q^{28} +6.67884 q^{29} +(-3.31281 - 0.721896i) q^{30} +(2.17580 + 1.25620i) q^{31} +(-0.288700 - 5.64948i) q^{32} +(-6.11640 - 4.58869i) q^{33} +(-3.75494 + 1.71796i) q^{34} +(0.501330 - 3.62774i) q^{35} +(-0.566076 - 5.97324i) q^{36} +(-5.00149 + 2.88761i) q^{37} +(-4.47510 - 0.424197i) q^{38} +(-1.33504 + 11.0674i) q^{39} +(-0.930905 - 3.80279i) q^{40} -0.497427i q^{41} +(6.38542 - 1.10741i) q^{42} +0.865898 q^{43} +(1.65895 - 8.67197i) q^{44} +(-1.16156 - 3.98680i) q^{45} +(5.20067 + 0.492975i) q^{46} +(-1.59001 - 2.75398i) q^{47} +(5.90966 - 3.61607i) q^{48} +(1.72472 + 6.78420i) q^{49} +(1.81456 + 3.96608i) q^{50} +(-4.04541 - 3.03498i) q^{51} +(-12.1584 + 4.22690i) q^{52} +(-4.12906 + 7.15175i) q^{53} +(6.09571 - 4.10393i) q^{54} -6.11065i q^{55} +(4.38665 + 6.06278i) q^{56} +(-2.16191 - 5.06319i) q^{57} +(-5.47350 + 7.69771i) q^{58} +(-6.62279 - 3.82367i) q^{59} +(3.54696 - 3.22657i) q^{60} +(2.99321 - 1.72813i) q^{61} +(-3.23096 + 1.47823i) q^{62} +(4.92754 + 6.22248i) q^{63} +(6.74792 + 4.29716i) q^{64} +(-7.71521 + 4.45438i) q^{65} +(10.3013 - 3.28890i) q^{66} +(-3.36806 + 5.83364i) q^{67} +(1.09724 - 5.73567i) q^{68} +(2.51243 + 5.88411i) q^{69} +(3.77030 + 3.55084i) q^{70} -1.90437 q^{71} +(7.34838 + 4.24280i) q^{72} +(3.23515 - 5.60345i) q^{73} +(0.770738 - 8.13095i) q^{74} +(-3.20563 + 4.27288i) q^{75} +(4.15638 - 4.81014i) q^{76} +(4.40032 + 10.8194i) q^{77} +(-11.6616 - 10.6087i) q^{78} +(1.65073 - 0.953048i) q^{79} +(5.14582 + 2.04358i) q^{80} +(7.98216 + 4.15754i) q^{81} +(0.573310 + 0.407655i) q^{82} +7.00064i q^{83} +(-3.95669 + 8.26708i) q^{84} -4.04161i q^{85} +(-0.709627 + 0.997992i) q^{86} +(-11.4848 - 1.38539i) q^{87} +(8.63533 + 9.01894i) q^{88} +(-8.22776 + 4.75030i) q^{89} +(5.54692 + 1.92854i) q^{90} +(10.4527 - 13.4426i) q^{91} +(-4.83027 + 5.59004i) q^{92} +(-3.48090 - 2.61147i) q^{93} +(4.47717 + 0.424393i) q^{94} +(2.19986 - 3.81027i) q^{95} +(-0.675427 + 9.77465i) q^{96} +1.19214 q^{97} +(-9.23259 - 3.57201i) q^{98} +(9.56583 + 9.15936i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.819527	+	1.15255i	−0.579493	+	0.814977i
							\(3\)	−1.71959	−	0.207430i	−0.992803	−	0.119760i
\(5\)	−0.692094	−	1.19874i	−0.309514	−	0.536094i								0.668742	−	0.743494i	\(-0.266833\pi\)
							−0.978256	+	0.207401i	\(0.933500\pi\)
\(7\)	2.08863	+	1.62408i	0.789426	+	0.613845i
							\(11\)	3.82316	+	2.20731i	1.15273	+	0.665528i	0.949550	−	0.313614i	\(-0.101540\pi\)
0.203177	+	0.979142i	\(0.434873\pi\)
\(13\)		−	6.43609i		−	1.78505i	−0.450999	−	0.892525i	\(-0.648932\pi\)
							0.450999	−	0.892525i	\(-0.351068\pi\)
\(17\)	2.52866	+	1.45992i	0.613289	+	0.354083i	0.774252	−	0.632878i	\(-0.218126\pi\)
							−0.160962	+	0.986961i	\(0.551460\pi\)
\(19\)	1.58928	+	2.75271i	0.364606	+	0.631515i	0.988713	−	0.149823i	\(-0.0478705\pi\)
							−0.624107	+	0.781339i	\(0.714537\pi\)
\(23\)	−1.84696	−	3.19902i	−0.385117	−	0.667043i	0.606668	−	0.794955i	\(-0.292506\pi\)
							−0.991785	+	0.127912i	\(0.959172\pi\)
\(29\)	6.67884			1.24023			0.620115	−	0.784511i	\(-0.287086\pi\)
							0.620115	+	0.784511i	\(0.287086\pi\)
\(31\)	2.17580	+	1.25620i	0.390786	+	0.225620i	0.682500	−	0.730885i	\(-0.260893\pi\)
							−0.291715	+	0.956505i	\(0.594226\pi\)
\(37\)	−5.00149	+	2.88761i	−0.822240	+	0.474720i	−0.851188	−	0.524861i	\(-0.824117\pi\)
							0.0289485	+	0.999581i	\(0.490784\pi\)
\(41\)		−	0.497427i		−	0.0776850i	−0.999245	−	0.0388425i	\(-0.987633\pi\)
							0.999245	−	0.0388425i	\(-0.0123671\pi\)
\(43\)	0.865898			0.132048			0.0660241	−	0.997818i	\(-0.478969\pi\)
							0.0660241	+	0.997818i	\(0.478969\pi\)
\(47\)	−1.59001	−	2.75398i	−0.231927	−	0.401710i	0.726448	−	0.687221i	\(-0.241170\pi\)
							−0.958375	+	0.285512i	\(0.907836\pi\)
\(53\)	−4.12906	+	7.15175i	−0.567171	+	0.982368i	0.429673	+	0.902984i	\(0.358629\pi\)
							−0.996844	+	0.0793841i	\(0.974705\pi\)
\(59\)	−6.62279	−	3.82367i	−0.862214	−	0.497799i	0.00253919	−	0.999997i	\(-0.499192\pi\)
							−0.864753	+	0.502197i	\(0.832525\pi\)
\(61\)	2.99321	−	1.72813i	0.383241	−	0.221265i	−0.295986	−	0.955192i	\(-0.595648\pi\)
							0.679228	+	0.733928i	\(0.262315\pi\)
\(67\)	−3.36806	+	5.83364i	−0.411473	+	0.712693i	−0.995051	−	0.0993644i	\(-0.968319\pi\)
							0.583578	+	0.812057i	\(0.301652\pi\)
\(71\)	−1.90437			−0.226007			−0.113004	−	0.993595i	\(-0.536047\pi\)
							−0.113004	+	0.993595i	\(0.536047\pi\)
\(73\)	3.23515	−	5.60345i	0.378646	−	0.655834i	−0.612219	−	0.790688i	\(-0.709723\pi\)
							0.990866	+	0.134854i	\(0.0430564\pi\)
\(79\)	1.65073	−	0.953048i	0.185721	−	0.107226i	−0.404257	−	0.914646i	\(-0.632470\pi\)
							0.589978	+	0.807419i	\(0.299136\pi\)
\(83\)			7.00064i			0.768420i	0.923246	+	0.384210i	\(0.125526\pi\)
							−0.923246	+	0.384210i	\(0.874474\pi\)
\(89\)	−8.22776	+	4.75030i	−0.872141	+	0.503531i	−0.868059	−	0.496461i	\(-0.834633\pi\)
							−0.00408165	+	0.999992i	\(0.501299\pi\)
\(97\)	1.19214			0.121044			0.0605218	−	0.998167i	\(-0.480724\pi\)
							0.0605218	+	0.998167i	\(0.480724\pi\)

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.11.10	yes	56
3.2	odd	2	inner	168.2.v.a.11.19	yes	56
4.3	odd	2		672.2.bd.a.431.27		56
7.2	even	3	inner	168.2.v.a.107.28	yes	56
8.3	odd	2	inner	168.2.v.a.11.1	✓	56
8.5	even	2		672.2.bd.a.431.28		56
12.11	even	2		672.2.bd.a.431.8		56
21.2	odd	6	inner	168.2.v.a.107.1	yes	56
24.5	odd	2		672.2.bd.a.431.7		56
24.11	even	2	inner	168.2.v.a.11.28	yes	56
28.23	odd	6		672.2.bd.a.527.7		56
56.37	even	6		672.2.bd.a.527.8		56
56.51	odd	6	inner	168.2.v.a.107.19	yes	56
84.23	even	6		672.2.bd.a.527.28		56
168.107	even	6	inner	168.2.v.a.107.10	yes	56
168.149	odd	6		672.2.bd.a.527.27		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.1	✓	56	8.3	odd	2	inner
168.2.v.a.11.10	yes	56	1.1	even	1	trivial
168.2.v.a.11.19	yes	56	3.2	odd	2	inner
168.2.v.a.11.28	yes	56	24.11	even	2	inner
168.2.v.a.107.1	yes	56	21.2	odd	6	inner
168.2.v.a.107.10	yes	56	168.107	even	6	inner
168.2.v.a.107.19	yes	56	56.51	odd	6	inner
168.2.v.a.107.28	yes	56	7.2	even	3	inner
672.2.bd.a.431.7		56	24.5	odd	2
672.2.bd.a.431.8		56	12.11	even	2
672.2.bd.a.431.27		56	4.3	odd	2
672.2.bd.a.431.28		56	8.5	even	2
672.2.bd.a.527.7		56	28.23	odd	6
672.2.bd.a.527.8		56	56.37	even	6
672.2.bd.a.527.27		56	168.149	odd	6
672.2.bd.a.527.28		56	84.23	even	6