Embedded newform 840.2.dc.e.557.84

• Label: 840.2.dc.e.557.84
• Level: $840$
• Weight: $2$
• Character: 840.557
• Analytic conductor: $6.707$
• Analytic rank: $0$
• Dimension: $704$
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.dc (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.70743376979\)
Analytic rank:	\(0\)
Dimension:	\(704\)
Relative dimension:	\(176\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.84
Character	\(\chi\)	\(=\)	840.557
Dual form			840.2.dc.e.653.84

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.126222 - 1.40857i) q^{2} +(-1.32941 - 1.11026i) q^{3} +(-1.96814 + 0.355585i) q^{4} +(-2.14683 + 0.625407i) q^{5} +(-1.39607 + 2.01271i) q^{6} +(0.163401 - 2.64070i) q^{7} +(0.749289 + 2.72737i) q^{8} +(0.534667 + 2.95197i) q^{9} +(1.15191 + 2.94501i) q^{10} +(-2.26343 + 3.92037i) q^{11} +(3.01125 + 1.71241i) q^{12} +(-0.229579 - 0.229579i) q^{13} +(-3.74024 + 0.103154i) q^{14} +(3.54838 + 1.55210i) q^{15} +(3.74712 - 1.39968i) q^{16} +(-1.62167 - 6.05216i) q^{17} +(4.09057 - 1.12572i) q^{18} +(3.28122 + 5.68324i) q^{19} +(4.00286 - 1.99427i) q^{20} +(-3.14908 + 3.32916i) q^{21} +(5.80781 + 2.69336i) q^{22} +(-1.48752 + 5.55149i) q^{23} +(2.03197 - 4.45770i) q^{24} +(4.21773 - 2.68528i) q^{25} +(-0.294400 + 0.352356i) q^{26} +(2.56665 - 4.51800i) q^{27} +(0.617400 + 5.25536i) q^{28} +0.429607i q^{29} +(1.73836 - 5.19405i) q^{30} +(2.50770 - 4.34346i) q^{31} +(-2.44452 - 5.10141i) q^{32} +(7.36164 - 2.69880i) q^{33} +(-8.32019 + 3.04815i) q^{34} +(1.30072 + 5.77132i) q^{35} +(-2.10197 - 5.61976i) q^{36} +(-1.00618 - 0.269605i) q^{37} +(7.59108 - 5.33918i) q^{38} +(0.0503136 + 0.560097i) q^{39} +(-3.31431 - 5.38659i) q^{40} +2.07585i q^{41} +(5.08684 + 4.01548i) q^{42} +(1.23989 + 1.23989i) q^{43} +(3.06070 - 8.52066i) q^{44} +(-2.99402 - 6.00299i) q^{45} +(8.00742 + 1.39455i) q^{46} +(7.33221 + 1.96466i) q^{47} +(-6.53546 - 2.29951i) q^{48} +(-6.94660 - 0.862984i) q^{49} +(-4.31478 - 5.60202i) q^{50} +(-4.56357 + 9.84627i) q^{51} +(0.533478 + 0.370208i) q^{52} +(-3.00932 - 11.2309i) q^{53} +(-6.68788 - 3.04503i) q^{54} +(2.40736 - 9.83192i) q^{55} +(7.32461 - 1.53299i) q^{56} +(1.94776 - 11.1984i) q^{57} +(0.605131 - 0.0542259i) q^{58} +(1.56340 + 0.902632i) q^{59} +(-7.53559 - 1.79300i) q^{60} +(12.1211 - 6.99814i) q^{61} +(-6.43459 - 2.98403i) q^{62} +(7.88264 - 0.929542i) q^{63} +(-6.87713 + 4.08718i) q^{64} +(0.636448 + 0.349286i) q^{65} +(-4.73065 - 10.0287i) q^{66} +(0.386465 + 1.44231i) q^{67} +(5.34373 + 11.3348i) q^{68} +(8.14109 - 5.72869i) q^{69} +(7.96512 - 2.56062i) q^{70} +15.8264i q^{71} +(-7.65051 + 3.67012i) q^{72} +(3.68214 + 13.7419i) q^{73} +(-0.252755 + 1.45130i) q^{74} +(-8.58845 - 1.11291i) q^{75} +(-8.47877 - 10.0186i) q^{76} +(9.98268 + 6.61762i) q^{77} +(0.782584 - 0.141567i) q^{78} +(4.85675 - 2.80405i) q^{79} +(-7.16904 + 5.34835i) q^{80} +(-8.42826 + 3.15664i) q^{81} +(2.92398 - 0.262019i) q^{82} +(-4.47966 + 4.47966i) q^{83} +(5.01401 - 7.67201i) q^{84} +(7.26651 + 11.9787i) q^{85} +(1.58998 - 1.90298i) q^{86} +(0.476973 - 0.571124i) q^{87} +(-12.3883 - 3.23572i) q^{88} +(4.03760 + 6.99333i) q^{89} +(-8.07771 + 4.97500i) q^{90} +(-0.643763 + 0.568737i) q^{91} +(0.953607 - 11.4550i) q^{92} +(-8.15611 + 2.99006i) q^{93} +(1.84187 - 10.5759i) q^{94} +(-10.5986 - 10.1488i) q^{95} +(-2.41409 + 9.49590i) q^{96} +(5.18418 + 5.18418i) q^{97} +(-0.338757 + 9.89370i) q^{98} +(-12.7830 - 4.58548i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	704 * q - 16 * q^6 - 24 * q^7 + 12 * q^10 + 10 * q^12 + 32 * q^15 - 72 * q^16 + 6 * q^18 + 64 * q^22 - 8 * q^25 + 24 * q^28 + 8 * q^30 - 16 * q^31 + 44 * q^33 - 240 * q^36 + 16 * q^40 - 78 * q^42 - 8 * q^46 - 20 * q^48 - 60 * q^52 + 272 * q^55 + 8 * q^57 - 20 * q^58 - 14 * q^60 + 12 * q^63 - 84 * q^66 + 112 * q^70 - 38 * q^72 + 104 * q^73 - 64 * q^76 - 16 * q^78 - 168 * q^81 - 72 * q^82 - 64 * q^87 + 52 * q^88 - 44 * q^90 + 40 * q^96 - 80 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(281\)	\(337\)	\(421\)	\(631\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.126222	−	1.40857i	−0.0892526	−	0.996009i
							\(3\)	−1.32941	−	1.11026i	−0.767536	−	0.641006i
\(5\)	−2.14683	+	0.625407i	−0.960090	+	0.279691i
							\(7\)	0.163401	−	2.64070i	0.0617596	−	0.998091i
\(11\)	−2.26343	+	3.92037i	−0.682449	+	1.18204i								0.291782	+	0.956485i	\(0.405752\pi\)
							−0.974231	+	0.225551i	\(0.927582\pi\)
\(13\)	−0.229579	−	0.229579i	−0.0636738	−	0.0636738i	0.674553	−	0.738227i	\(-0.264336\pi\)
							−0.738227	+	0.674553i	\(0.764336\pi\)
\(17\)	−1.62167	−	6.05216i	−0.393313	−	1.46786i	−0.824635	−	0.565665i	\(-0.808619\pi\)
							0.431322	−	0.902198i	\(-0.358047\pi\)
\(19\)	3.28122	+	5.68324i	0.752764	+	1.30383i	0.946478	+	0.322767i	\(0.104613\pi\)
							−0.193714	+	0.981058i	\(0.562053\pi\)
\(23\)	−1.48752	+	5.55149i	−0.310169	+	1.15757i	0.618235	+	0.785993i	\(0.287848\pi\)
							−0.928404	+	0.371572i	\(0.878819\pi\)
\(29\)			0.429607i			0.0797760i	0.999204	+	0.0398880i	\(0.0127001\pi\)
							−0.999204	+	0.0398880i	\(0.987300\pi\)
\(31\)	2.50770	−	4.34346i	0.450396	−	0.780109i	−0.548015	−	0.836469i	\(-0.684616\pi\)
							0.998410	+	0.0563602i	\(0.0179495\pi\)
\(37\)	−1.00618	−	0.269605i	−0.165415	−	0.0443228i	0.175161	−	0.984540i	\(-0.443955\pi\)
							−0.340576	+	0.940217i	\(0.610622\pi\)
\(41\)			2.07585i			0.324194i	0.986775	+	0.162097i	\(0.0518257\pi\)
							−0.986775	+	0.162097i	\(0.948174\pi\)
\(43\)	1.23989	+	1.23989i	0.189082	+	0.189082i	0.795299	−	0.606217i	\(-0.207314\pi\)
							−0.606217	+	0.795299i	\(0.707314\pi\)
\(47\)	7.33221	+	1.96466i	1.06951	+	0.286575i	0.750293	−	0.661105i	\(-0.229912\pi\)
							0.319220	+	0.947681i	\(0.396579\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.dc.e.557.84	yes	704
3.2	odd	2	inner	840.2.dc.e.557.93	yes	704
5.3	odd	4	inner	840.2.dc.e.53.5	✓	704
7.2	even	3	inner	840.2.dc.e.317.31	yes	704
8.5	even	2	inner	840.2.dc.e.557.58	yes	704
15.8	even	4	inner	840.2.dc.e.53.172	yes	704
21.2	odd	6	inner	840.2.dc.e.317.146	yes	704
24.5	odd	2	inner	840.2.dc.e.557.119	yes	704
35.23	odd	12	inner	840.2.dc.e.653.119	yes	704
40.13	odd	4	inner	840.2.dc.e.53.146	yes	704
56.37	even	6	inner	840.2.dc.e.317.172	yes	704
105.23	even	12	inner	840.2.dc.e.653.58	yes	704
120.53	even	4	inner	840.2.dc.e.53.31	yes	704
168.149	odd	6	inner	840.2.dc.e.317.5	yes	704
280.93	odd	12	inner	840.2.dc.e.653.93	yes	704
840.653	even	12	inner	840.2.dc.e.653.84	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.dc.e.53.5	✓	704	5.3	odd	4	inner
840.2.dc.e.53.31	yes	704	120.53	even	4	inner
840.2.dc.e.53.146	yes	704	40.13	odd	4	inner
840.2.dc.e.53.172	yes	704	15.8	even	4	inner
840.2.dc.e.317.5	yes	704	168.149	odd	6	inner
840.2.dc.e.317.31	yes	704	7.2	even	3	inner
840.2.dc.e.317.146	yes	704	21.2	odd	6	inner
840.2.dc.e.317.172	yes	704	56.37	even	6	inner
840.2.dc.e.557.58	yes	704	8.5	even	2	inner
840.2.dc.e.557.84	yes	704	1.1	even	1	trivial
840.2.dc.e.557.93	yes	704	3.2	odd	2	inner
840.2.dc.e.557.119	yes	704	24.5	odd	2	inner
840.2.dc.e.653.58	yes	704	105.23	even	12	inner
840.2.dc.e.653.84	yes	704	840.653	even	12	inner
840.2.dc.e.653.93	yes	704	280.93	odd	12	inner
840.2.dc.e.653.119	yes	704	35.23	odd	12	inner