Embedded newform 840.2.dc.e.317.171

• Label: 840.2.dc.e.317.171
• Level: $840$
• Weight: $2$
• Character: 840.317
• 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			317.171
Character	\(\chi\)	\(=\)	840.317
Dual form			840.2.dc.e.53.171

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40478 - 0.163037i) q^{2} +(-0.887862 + 1.48718i) q^{3} +(1.94684 - 0.458064i) q^{4} +(1.11326 - 1.93924i) q^{5} +(-1.00479 + 2.23392i) q^{6} +(0.130595 - 2.64253i) q^{7} +(2.66021 - 0.960888i) q^{8} +(-1.42340 - 2.64082i) q^{9} +(1.24772 - 2.90572i) q^{10} +(1.45101 + 2.51322i) q^{11} +(-1.04730 + 3.30199i) q^{12} +(-1.13395 - 1.13395i) q^{13} +(-0.247372 - 3.73347i) q^{14} +(1.89557 + 3.37740i) q^{15} +(3.58035 - 1.78355i) q^{16} +(-5.80269 - 1.55483i) q^{17} +(-2.43012 - 3.47771i) q^{18} +(4.22157 - 7.31197i) q^{19} +(1.27904 - 4.28533i) q^{20} +(3.81396 + 2.54042i) q^{21} +(2.44811 + 3.29397i) q^{22} +(0.212513 - 0.0569428i) q^{23} +(-0.932884 + 4.80934i) q^{24} +(-2.52130 - 4.31776i) q^{25} +(-1.77783 - 1.40808i) q^{26} +(5.19116 + 0.227832i) q^{27} +(-0.956199 - 5.20439i) q^{28} +5.64627i q^{29} +(3.21351 + 4.43546i) q^{30} +(3.40485 + 5.89737i) q^{31} +(4.73884 - 3.08924i) q^{32} +(-5.02591 - 0.0734846i) q^{33} +(-8.40502 - 1.23814i) q^{34} +(-4.97911 - 3.19508i) q^{35} +(-3.98080 - 4.48924i) q^{36} +(1.79632 + 6.70395i) q^{37} +(4.73827 - 10.9600i) q^{38} +(2.69318 - 0.679596i) q^{39} +(1.09811 - 6.22850i) q^{40} +4.91810i q^{41} +(5.77197 + 2.94692i) q^{42} +(-0.826434 - 0.826434i) q^{43} +(3.97610 + 4.22818i) q^{44} +(-6.70580 - 0.179604i) q^{45} +(0.289251 - 0.114640i) q^{46} +(-1.31230 - 4.89758i) q^{47} +(-0.526400 + 6.90818i) q^{48} +(-6.96589 - 0.690202i) q^{49} +(-4.24584 - 5.65445i) q^{50} +(7.46429 - 7.24917i) q^{51} +(-2.72704 - 1.68820i) q^{52} +(6.91130 + 1.85188i) q^{53} +(7.32960 - 0.526297i) q^{54} +(6.48910 - 0.0159840i) q^{55} +(-2.19176 - 7.15515i) q^{56} +(7.12604 + 12.7703i) q^{57} +(0.920551 + 7.93179i) q^{58} +(-0.222735 + 0.128596i) q^{59} +(5.23744 + 5.70695i) q^{60} +(0.0940230 + 0.0542842i) q^{61} +(5.74457 + 7.72942i) q^{62} +(-7.16432 + 3.41650i) q^{63} +(6.15339 - 5.11232i) q^{64} +(-3.46139 + 0.936620i) q^{65} +(-7.07230 + 0.716180i) q^{66} +(14.3124 + 3.83501i) q^{67} +(-12.0091 - 0.368991i) q^{68} +(-0.103998 + 0.366603i) q^{69} +(-7.51549 - 3.67661i) q^{70} +12.8900i q^{71} +(-6.32407 - 5.65739i) q^{72} +(0.272865 + 0.0731140i) q^{73} +(3.61643 + 9.12474i) q^{74} +(8.65985 + 0.0839485i) q^{75} +(4.86936 - 16.1690i) q^{76} +(6.83075 - 3.50612i) q^{77} +(3.67254 - 1.39377i) q^{78} +(-0.639229 - 0.369059i) q^{79} +(0.527132 - 8.92873i) q^{80} +(-4.94786 + 7.51789i) q^{81} +(0.801832 + 6.90886i) 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} +(-8.39701 - 5.01311i) q^{87} +(6.27491 + 5.29143i) q^{88} +(-0.894453 + 1.54924i) q^{89} +(-9.44948 + 0.840989i) q^{90} +(-3.14458 + 2.84841i) q^{91} +(0.387645 - 0.208203i) q^{92} +(-11.7935 - 0.172434i) q^{93} +(-2.64199 - 6.66609i) q^{94} +(-9.47996 - 16.3268i) q^{95} +(0.386810 + 9.79032i) q^{96} +(-7.84637 - 7.84637i) q^{97} +(-9.89810 + 0.166114i) 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 \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40478	−	0.163037i	0.993332	−	0.115285i
							\(3\)	−0.887862	+	1.48718i	−0.512607	+	0.858623i
\(5\)	1.11326	−	1.93924i	0.497865	−	0.867254i
							\(7\)	0.130595	−	2.64253i	0.0493603	−	0.998781i
\(11\)	1.45101	+	2.51322i	0.437496	+	0.757765i								0.997496	−	0.0707274i	\(-0.0225320\pi\)
							−0.560000	+	0.828493i	\(0.689199\pi\)
\(13\)	−1.13395	−	1.13395i	−0.314501	−	0.314501i	0.532149	−	0.846651i	\(-0.321385\pi\)
							−0.846651	+	0.532149i	\(0.821385\pi\)
\(17\)	−5.80269	−	1.55483i	−1.40736	−	0.377101i	−0.526377	−	0.850251i	\(-0.676450\pi\)
							−0.880982	+	0.473150i	\(0.843117\pi\)
\(19\)	4.22157	−	7.31197i	0.968495	−	1.67748i	0.268577	−	0.963258i	\(-0.413447\pi\)
							0.699917	−	0.714224i	\(-0.253220\pi\)
\(23\)	0.212513	−	0.0569428i	0.0443121	−	0.0118734i	−0.236595	−	0.971608i	\(-0.576031\pi\)
							0.280907	+	0.959735i	\(0.409365\pi\)
\(29\)			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	0.990983	+	0.133988i	\(0.0427785\pi\)
							−0.379454	+	0.925211i	\(0.623888\pi\)
\(37\)	1.79632	+	6.70395i	0.295313	+	1.10212i	0.940969	+	0.338494i	\(0.109918\pi\)
							−0.645656	+	0.763629i	\(0.723416\pi\)
\(41\)			4.91810i			0.768078i	0.923317	+	0.384039i	\(0.125467\pi\)
							−0.923317	+	0.384039i	\(0.874533\pi\)
\(43\)	−0.826434	−	0.826434i	−0.126030	−	0.126030i	0.641278	−	0.767308i	\(-0.278404\pi\)
							−0.767308	+	0.641278i	\(0.778404\pi\)
\(47\)	−1.31230	−	4.89758i	−0.191419	−	0.714386i	−0.993165	−	0.116721i	\(-0.962762\pi\)
							0.801746	−	0.597665i	\(-0.203905\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.dc.e.317.171	yes	704
3.2	odd	2	inner	840.2.dc.e.317.6	yes	704
5.3	odd	4	inner	840.2.dc.e.653.83	yes	704
7.4	even	3	inner	840.2.dc.e.557.68	yes	704
8.5	even	2	inner	840.2.dc.e.317.21	yes	704
15.8	even	4	inner	840.2.dc.e.653.94	yes	704
21.11	odd	6	inner	840.2.dc.e.557.109	yes	704
24.5	odd	2	inner	840.2.dc.e.317.156	yes	704
35.18	odd	12	inner	840.2.dc.e.53.156	yes	704
40.13	odd	4	inner	840.2.dc.e.653.109	yes	704
56.53	even	6	inner	840.2.dc.e.557.94	yes	704
105.53	even	12	inner	840.2.dc.e.53.21	yes	704
120.53	even	4	inner	840.2.dc.e.653.68	yes	704
168.53	odd	6	inner	840.2.dc.e.557.83	yes	704
280.53	odd	12	inner	840.2.dc.e.53.6	✓	704
840.53	even	12	inner	840.2.dc.e.53.171	yes	704

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.dc.e.53.6	✓	704	280.53	odd	12	inner
840.2.dc.e.53.21	yes	704	105.53	even	12	inner
840.2.dc.e.53.156	yes	704	35.18	odd	12	inner
840.2.dc.e.53.171	yes	704	840.53	even	12	inner
840.2.dc.e.317.6	yes	704	3.2	odd	2	inner
840.2.dc.e.317.21	yes	704	8.5	even	2	inner
840.2.dc.e.317.156	yes	704	24.5	odd	2	inner
840.2.dc.e.317.171	yes	704	1.1	even	1	trivial
840.2.dc.e.557.68	yes	704	7.4	even	3	inner
840.2.dc.e.557.83	yes	704	168.53	odd	6	inner
840.2.dc.e.557.94	yes	704	56.53	even	6	inner
840.2.dc.e.557.109	yes	704	21.11	odd	6	inner
840.2.dc.e.653.68	yes	704	120.53	even	4	inner
840.2.dc.e.653.83	yes	704	5.3	odd	4	inner
840.2.dc.e.653.94	yes	704	15.8	even	4	inner
840.2.dc.e.653.109	yes	704	40.13	odd	4	inner