Embedded newform 840.2.dc.e.557.83

• Label: 840.2.dc.e.557.83
• 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			557.83
Character	\(\chi\)	\(=\)	840.557
Dual form			840.2.dc.e.653.83

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.163037 + 1.40478i) q^{2} +(-1.48718 + 0.887862i) q^{3} +(-1.94684 - 0.458064i) q^{4} +(1.12280 + 1.93373i) q^{5} +(-1.00479 - 2.23392i) q^{6} +(-2.64253 + 0.130595i) q^{7} +(0.960888 - 2.66021i) q^{8} +(1.42340 - 2.64082i) q^{9} +(-2.89953 + 1.26202i) q^{10} +(1.45101 - 2.51322i) q^{11} +(3.30199 - 1.04730i) q^{12} +(1.13395 + 1.13395i) q^{13} +(0.247372 - 3.73347i) q^{14} +(-3.38669 - 1.87891i) q^{15} +(3.58035 + 1.78355i) q^{16} +(-1.55483 - 5.80269i) q^{17} +(3.47771 + 2.43012i) q^{18} +(-4.22157 - 7.31197i) q^{19} +(-1.30014 - 4.27898i) q^{20} +(3.81396 - 2.54042i) q^{21} +(3.29397 + 2.44811i) q^{22} +(0.0569428 - 0.212513i) q^{23} +(0.932884 + 4.80934i) q^{24} +(-2.47864 + 4.34239i) q^{25} +(-1.77783 + 1.40808i) q^{26} +(0.227832 + 5.19116i) q^{27} +(5.20439 + 0.956199i) q^{28} +5.64627i q^{29} +(3.19162 - 4.45124i) q^{30} +(3.40485 - 5.89737i) q^{31} +(-3.08924 + 4.73884i) q^{32} +(0.0734846 + 5.02591i) q^{33} +(8.40502 - 1.23814i) q^{34} +(-3.21957 - 4.96330i) q^{35} +(-3.98080 + 4.48924i) q^{36} +(6.70395 + 1.79632i) q^{37} +(10.9600 - 4.73827i) q^{38} +(-2.69318 - 0.679596i) q^{39} +(6.22301 - 1.12878i) q^{40} -4.91810i q^{41} +(2.94692 + 5.77197i) q^{42} +(0.826434 + 0.826434i) q^{43} +(-3.97610 + 4.22818i) q^{44} +(6.70483 - 0.212637i) q^{45} +(0.289251 + 0.114640i) q^{46} +(-4.89758 - 1.31230i) q^{47} +(-6.90818 + 0.526400i) q^{48} +(6.96589 - 0.690202i) q^{49} +(-5.69601 - 4.18992i) q^{50} +(7.46429 + 7.24917i) q^{51} +(-1.68820 - 2.72704i) q^{52} +(-1.85188 - 6.91130i) q^{53} +(-7.32960 - 0.526297i) q^{54} +(6.48910 - 0.0159840i) q^{55} +(-2.19176 + 7.15515i) q^{56} +(12.7703 + 7.12604i) q^{57} +(-7.93179 - 0.920551i) q^{58} +(0.222735 + 0.128596i) q^{59} +(5.73268 + 5.20926i) q^{60} +(0.0940230 - 0.0542842i) q^{61} +(7.72942 + 5.74457i) q^{62} +(-3.41650 + 7.16432i) q^{63} +(-6.15339 - 5.11232i) q^{64} +(-0.919556 + 3.46596i) q^{65} +(-7.07230 - 0.716180i) q^{66} +(3.83501 + 14.3124i) q^{67} +(0.368991 + 12.0091i) q^{68} +(0.103998 + 0.366603i) q^{69} +(7.49728 - 3.71359i) q^{70} -12.8900i q^{71} +(-5.65739 - 6.32407i) q^{72} +(-0.0731140 - 0.272865i) q^{73} +(-3.61643 + 9.12474i) q^{74} +(-0.169267 - 8.65860i) q^{75} +(4.86936 + 16.1690i) q^{76} +(-3.50612 + 6.83075i) q^{77} +(1.39377 - 3.67254i) q^{78} +(0.639229 - 0.369059i) q^{79} +(0.571112 + 8.92602i) q^{80} +(-4.94786 - 7.51789i) q^{81} +(6.90886 + 0.801832i) q^{82} +(-2.53311 + 2.53311i) q^{83} +(-8.58883 + 3.19874i) q^{84} +(9.47509 - 9.52188i) q^{85} +(-1.29570 + 1.02622i) q^{86} +(-5.01311 - 8.39701i) q^{87} +(-5.29143 - 6.27491i) q^{88} +(0.894453 + 1.54924i) q^{89} +(-0.794427 + 9.45351i) q^{90} +(-3.14458 - 2.84841i) q^{91} +(-0.208203 + 0.387645i) q^{92} +(0.172434 + 11.7935i) q^{93} +(2.64199 - 6.66609i) q^{94} +(9.39942 - 16.3733i) q^{95} +(0.386810 - 9.79032i) q^{96} +(-7.84637 - 7.84637i) q^{97} +(-0.166114 + 9.89810i) q^{98} +(-4.57160 - 7.40918i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.163037	+	1.40478i	−0.115285	+	0.993332i
							\(3\)	−1.48718	+	0.887862i	−0.858623	+	0.512607i
\(5\)	1.12280	+	1.93373i	0.502132	+	0.864791i
							\(7\)	−2.64253	+	0.130595i	−0.998781	+	0.0493603i
\(11\)	1.45101	−	2.51322i	0.437496	−	0.757765i								−0.560000	−	0.828493i	\(-0.689199\pi\)
							0.997496	+	0.0707274i	\(0.0225320\pi\)
\(13\)	1.13395	+	1.13395i	0.314501	+	0.314501i	0.846651	−	0.532149i	\(-0.178615\pi\)
							−0.532149	+	0.846651i	\(0.678615\pi\)
\(17\)	−1.55483	−	5.80269i	−0.377101	−	1.40736i	−0.850251	−	0.526377i	\(-0.823550\pi\)
							0.473150	−	0.880982i	\(-0.343117\pi\)
\(19\)	−4.22157	−	7.31197i	−0.968495	−	1.67748i	−0.699917	−	0.714224i	\(-0.746780\pi\)
							−0.268577	−	0.963258i	\(-0.586553\pi\)
\(23\)	0.0569428	−	0.212513i	0.0118734	−	0.0443121i	−0.959735	−	0.280907i	\(-0.909365\pi\)
							0.971608	+	0.236595i	\(0.0760314\pi\)
\(29\)			5.64627i			1.04849i	0.851569	+	0.524243i	\(0.175652\pi\)
							−0.851569	+	0.524243i	\(0.824348\pi\)
\(31\)	3.40485	−	5.89737i	0.611529	−	1.05920i	−0.379454	−	0.925211i	\(-0.623888\pi\)
							0.990983	−	0.133988i	\(-0.0427785\pi\)
\(37\)	6.70395	+	1.79632i	1.10212	+	0.295313i	0.763629	−	0.645656i	\(-0.223416\pi\)
							0.338494	+	0.940969i	\(0.390082\pi\)
\(41\)		−	4.91810i		−	0.768078i	−0.923317	−	0.384039i	\(-0.874533\pi\)
							0.923317	−	0.384039i	\(-0.125467\pi\)
\(43\)	0.826434	+	0.826434i	0.126030	+	0.126030i	0.767308	−	0.641278i	\(-0.221596\pi\)
							−0.641278	+	0.767308i	\(0.721596\pi\)
\(47\)	−4.89758	−	1.31230i	−0.714386	−	0.191419i	−0.116721	−	0.993165i	\(-0.537238\pi\)
							−0.597665	+	0.801746i	\(0.703905\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.dc.e.557.83	yes	704
3.2	odd	2	inner	840.2.dc.e.557.94	yes	704
5.3	odd	4	inner	840.2.dc.e.53.171	yes	704
7.2	even	3	inner	840.2.dc.e.317.156	yes	704
8.5	even	2	inner	840.2.dc.e.557.109	yes	704
15.8	even	4	inner	840.2.dc.e.53.6	✓	704
21.2	odd	6	inner	840.2.dc.e.317.21	yes	704
24.5	odd	2	inner	840.2.dc.e.557.68	yes	704
35.23	odd	12	inner	840.2.dc.e.653.68	yes	704
40.13	odd	4	inner	840.2.dc.e.53.21	yes	704
56.37	even	6	inner	840.2.dc.e.317.6	yes	704
105.23	even	12	inner	840.2.dc.e.653.109	yes	704
120.53	even	4	inner	840.2.dc.e.53.156	yes	704
168.149	odd	6	inner	840.2.dc.e.317.171	yes	704
280.93	odd	12	inner	840.2.dc.e.653.94	yes	704
840.653	even	12	inner	840.2.dc.e.653.83	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.dc.e.53.6	✓	704	15.8	even	4	inner
840.2.dc.e.53.21	yes	704	40.13	odd	4	inner
840.2.dc.e.53.156	yes	704	120.53	even	4	inner
840.2.dc.e.53.171	yes	704	5.3	odd	4	inner
840.2.dc.e.317.6	yes	704	56.37	even	6	inner
840.2.dc.e.317.21	yes	704	21.2	odd	6	inner
840.2.dc.e.317.156	yes	704	7.2	even	3	inner
840.2.dc.e.317.171	yes	704	168.149	odd	6	inner
840.2.dc.e.557.68	yes	704	24.5	odd	2	inner
840.2.dc.e.557.83	yes	704	1.1	even	1	trivial
840.2.dc.e.557.94	yes	704	3.2	odd	2	inner
840.2.dc.e.557.109	yes	704	8.5	even	2	inner
840.2.dc.e.653.68	yes	704	35.23	odd	12	inner
840.2.dc.e.653.83	yes	704	840.653	even	12	inner
840.2.dc.e.653.94	yes	704	280.93	odd	12	inner
840.2.dc.e.653.109	yes	704	105.23	even	12	inner