Embedded newform 280.2.q.e.121.3

• Label: 280.2.q.e.121.3
• Level: $280$
• Weight: $2$
• Character: 280.121
• Analytic conductor: $2.236$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	280.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.23581125660\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.11337408.1
Defining polynomial:	\( x^{6} + 18x^{4} + 81x^{2} + 12 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.3
Root			\(2.78499i\) of defining polynomial
Character	\(\chi\)	\(=\)	280.121
Dual form			280.2.q.e.81.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.64497 + 2.84918i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(2.64497 + 0.0641892i) q^{7} +(-3.91187 + 6.77556i) q^{9} +(-2.91187 - 5.04351i) q^{11} +2.75615 q^{13} -3.28995 q^{15} +(-1.00000 - 1.73205i) q^{17} +(0.378076 - 0.654846i) q^{19} +(4.16802 + 7.64158i) q^{21} +(0.266897 - 0.462279i) q^{23} +(-0.500000 - 0.866025i) q^{25} -15.8698 q^{27} -0.823739 q^{29} +(1.28995 + 2.23425i) q^{31} +(9.57989 - 16.5929i) q^{33} +(-1.37808 + 2.25852i) q^{35} +(-2.37808 + 4.11895i) q^{37} +(4.53379 + 7.85276i) q^{39} +6.06759 q^{41} +0.710055 q^{43} +(-3.91187 - 6.77556i) q^{45} +(6.44566 - 11.1642i) q^{47} +(6.99176 + 0.339557i) q^{49} +(3.28995 - 5.69835i) q^{51} +(4.20181 + 7.27776i) q^{53} +5.82374 q^{55} +2.48770 q^{57} +(-4.00000 - 6.92820i) q^{59} +(-4.70181 + 8.14378i) q^{61} +(-10.7817 + 17.6701i) q^{63} +(-1.37808 + 2.38690i) q^{65} +(-5.93492 - 10.2796i) q^{67} +1.75615 q^{69} +(-1.75615 - 3.04174i) q^{73} +(1.64497 - 2.84918i) q^{75} +(-7.37808 - 13.5268i) q^{77} +(4.75615 - 8.23790i) q^{79} +(-14.3698 - 24.8893i) q^{81} -6.71005 q^{83} +2.00000 q^{85} +(-1.35503 - 2.34698i) q^{87} +(-0.878076 + 1.52087i) q^{89} +(7.28995 + 0.176915i) q^{91} +(-4.24385 + 7.35056i) q^{93} +(0.378076 + 0.654846i) q^{95} -2.00000 q^{97} +45.5634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 3 * q^5 + 6 * q^7 - 9 * q^9 - 3 * q^11 + 6 * q^13 - 6 * q^17 - 3 * q^19 - 3 * q^23 - 3 * q^25 - 36 * q^27 + 24 * q^29 - 12 * q^31 + 18 * q^33 - 3 * q^35 - 9 * q^37 + 18 * q^39 + 18 * q^41 + 24 * q^43 - 9 * q^45 + 15 * q^47 - 12 * q^49 - 9 * q^53 + 6 * q^55 + 36 * q^57 - 24 * q^59 + 6 * q^61 + 9 * q^63 - 3 * q^65 - 6 * q^67 - 39 * q^77 + 18 * q^79 - 27 * q^81 - 60 * q^83 + 12 * q^85 - 18 * q^87 + 24 * q^91 - 36 * q^93 - 3 * q^95 - 12 * q^97 + 126 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\)	\(57\)	\(71\)	\(141\)	\(241\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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			0
							\(3\)	1.64497	+	2.84918i	0.949725	+	1.64497i	0.746002	+	0.665944i	\(0.231971\pi\)
0.203724	+	0.979028i	\(0.434696\pi\)
\(5\)	−0.500000	+	0.866025i	−0.223607	+	0.387298i
							\(7\)	2.64497	+	0.0641892i	0.999706	+	0.0242612i
\(11\)	−2.91187	−	5.04351i	−0.877962	−	1.52067i								−0.853574	−	0.520972i	\(-0.825570\pi\)
							−0.0243876	−	0.999703i	\(-0.507764\pi\)
\(13\)	2.75615			0.764419			0.382209	−	0.924076i	\(-0.375163\pi\)
							0.382209	+	0.924076i	\(0.375163\pi\)
\(17\)	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	\(-0.244646\pi\)
							−0.961436	+	0.275029i	\(0.911312\pi\)
\(19\)	0.378076	−	0.654846i	0.0867365	−	0.150232i	−0.819393	−	0.573232i	\(-0.805690\pi\)
							0.906130	+	0.423000i	\(0.139023\pi\)
\(23\)	0.266897	−	0.462279i	0.0556518	−	0.0963918i	−0.836857	−	0.547421i	\(-0.815610\pi\)
							0.892509	+	0.451029i	\(0.148943\pi\)
\(29\)	−0.823739			−0.152964			−0.0764822	−	0.997071i	\(-0.524369\pi\)
							−0.0764822	+	0.997071i	\(0.524369\pi\)
\(31\)	1.28995	+	2.23425i	0.231681	+	0.401283i	0.958303	−	0.285754i	\(-0.0922441\pi\)
							−0.726622	+	0.687038i	\(0.758911\pi\)
\(37\)	−2.37808	+	4.11895i	−0.390953	+	0.677151i	−0.992576	−	0.121630i	\(-0.961188\pi\)
							0.601622	+	0.798781i	\(0.294521\pi\)
\(41\)	6.06759			0.947598			0.473799	−	0.880633i	\(-0.342882\pi\)
							0.473799	+	0.880633i	\(0.342882\pi\)
\(43\)	0.710055			0.108282			0.0541412	−	0.998533i	\(-0.482758\pi\)
							0.0541412	+	0.998533i	\(0.482758\pi\)
\(47\)	6.44566	−	11.1642i	0.940197	−	1.62847i	0.175102	−	0.984550i	\(-0.443974\pi\)
							0.765095	−	0.643918i	\(-0.222692\pi\)
\(53\)	4.20181	+	7.27776i	0.577164	+	0.999677i	0.995803	+	0.0915241i	\(0.0291738\pi\)
							−0.418639	+	0.908153i	\(0.637493\pi\)
\(59\)	−4.00000	−	6.92820i	−0.520756	−	0.901975i	−0.999709	−	0.0241347i	\(-0.992317\pi\)
							0.478953	−	0.877841i	\(-0.341016\pi\)
\(61\)	−4.70181	+	8.14378i	−0.602006	+	1.04270i	0.390511	+	0.920598i	\(0.372298\pi\)
							−0.992517	+	0.122106i	\(0.961035\pi\)
\(67\)	−5.93492	−	10.2796i	−0.725066	−	1.25585i	−0.958947	−	0.283585i	\(-0.908476\pi\)
							0.233881	−	0.972265i	\(-0.424857\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	−1.75615	−	3.04174i	−0.205542	−	0.356009i	0.744763	−	0.667329i	\(-0.232562\pi\)
							−0.950305	+	0.311320i	\(0.899229\pi\)
\(79\)	4.75615	−	8.23790i	0.535109	−	0.926836i	−0.464049	−	0.885809i	\(-0.653604\pi\)
							0.999158	−	0.0410263i	\(-0.0130627\pi\)
\(83\)	−6.71005			−0.736524			−0.368262	−	0.929722i	\(-0.620047\pi\)
							−0.368262	+	0.929722i	\(0.620047\pi\)
\(89\)	−0.878076	+	1.52087i	−0.0930758	+	0.161212i	−0.908804	−	0.417223i	\(-0.863003\pi\)
							0.815728	+	0.578436i	\(0.196337\pi\)
\(97\)	−2.00000			−0.203069			−0.101535	−	0.994832i	\(-0.532375\pi\)
							−0.101535	+	0.994832i	\(0.532375\pi\)

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	280.2.q.e.121.3	yes	6
3.2	odd	2		2520.2.bi.q.1801.3		6
4.3	odd	2		560.2.q.l.401.1		6
5.2	odd	4		1400.2.bh.i.849.6		12
5.3	odd	4		1400.2.bh.i.849.1		12
5.4	even	2		1400.2.q.j.401.1		6
7.2	even	3		1960.2.a.w.1.1		3
7.3	odd	6		1960.2.q.w.361.1		6
7.4	even	3	inner	280.2.q.e.81.3	✓	6
7.5	odd	6		1960.2.a.v.1.3		3
7.6	odd	2		1960.2.q.w.961.1		6
21.11	odd	6		2520.2.bi.q.361.3		6
28.11	odd	6		560.2.q.l.81.1		6
28.19	even	6		3920.2.a.cb.1.1		3
28.23	odd	6		3920.2.a.cc.1.3		3
35.4	even	6		1400.2.q.j.1201.1		6
35.9	even	6		9800.2.a.ce.1.3		3
35.18	odd	12		1400.2.bh.i.249.6		12
35.19	odd	6		9800.2.a.cf.1.1		3
35.32	odd	12		1400.2.bh.i.249.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.e.81.3	✓	6	7.4	even	3	inner
280.2.q.e.121.3	yes	6	1.1	even	1	trivial
560.2.q.l.81.1		6	28.11	odd	6
560.2.q.l.401.1		6	4.3	odd	2
1400.2.q.j.401.1		6	5.4	even	2
1400.2.q.j.1201.1		6	35.4	even	6
1400.2.bh.i.249.1		12	35.32	odd	12
1400.2.bh.i.249.6		12	35.18	odd	12
1400.2.bh.i.849.1		12	5.3	odd	4
1400.2.bh.i.849.6		12	5.2	odd	4
1960.2.a.v.1.3		3	7.5	odd	6
1960.2.a.w.1.1		3	7.2	even	3
1960.2.q.w.361.1		6	7.3	odd	6
1960.2.q.w.961.1		6	7.6	odd	2
2520.2.bi.q.361.3		6	21.11	odd	6
2520.2.bi.q.1801.3		6	3.2	odd	2
3920.2.a.cb.1.1		3	28.19	even	6
3920.2.a.cc.1.3		3	28.23	odd	6
9800.2.a.ce.1.3		3	35.9	even	6
9800.2.a.cf.1.1		3	35.19	odd	6