Embedded newform 1960.2.q.w.961.3

• Label: 1960.2.q.w.961.3
• Level: $1960$
• Weight: $2$
• Character: 1960.961
• Analytic conductor: $15.651$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.6506787962\)
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:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.3
Root			\(-0.391571i\) of defining polynomial
Character	\(\chi\)	\(=\)	1960.961
Dual form			1960.2.q.w.361.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29211 + 2.23800i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-1.83911 + 3.18543i) q^{9} +(-0.839111 - 1.45338i) q^{11} +4.84667 q^{13} +2.58423 q^{15} +(1.00000 + 1.73205i) q^{17} +(3.42334 - 5.92939i) q^{19} +(1.13122 - 1.95934i) q^{23} +(-0.500000 - 0.866025i) q^{25} -1.75268 q^{27} +3.32178 q^{29} +(4.58423 + 7.94011i) q^{31} +(2.16845 - 3.75587i) q^{33} +(1.42334 - 2.46529i) q^{37} +(6.26245 + 10.8469i) q^{39} -9.52489 q^{41} +6.58423 q^{43} +(1.83911 + 3.18543i) q^{45} +(-6.10156 + 10.5682i) q^{47} +(-2.58423 + 4.47601i) q^{51} +(-3.74511 - 6.48673i) q^{53} -1.67822 q^{55} +17.6933 q^{57} +(4.00000 + 6.92820i) q^{59} +(-3.24511 + 5.62070i) q^{61} +(2.42334 - 4.19734i) q^{65} +(2.87634 + 4.98196i) q^{67} +5.84667 q^{69} +(-5.84667 - 10.1267i) q^{73} +(1.29211 - 2.23800i) q^{75} +(-2.84667 + 4.93058i) q^{79} +(3.25268 + 5.63380i) q^{81} +12.5842 q^{83} +2.00000 q^{85} +(4.29211 + 7.43416i) q^{87} +(-2.92334 + 5.06337i) q^{89} +(-11.8467 + 20.5190i) q^{93} +(-3.42334 - 5.92939i) q^{95} +2.00000 q^{97} +6.17287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q + 3 * q^5 - 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 - 9 * q^37 + 18 * q^39 - 18 * q^41 + 24 * q^43 + 9 * q^45 - 15 * q^47 - 9 * q^53 - 6 * q^55 + 36 * q^57 + 24 * q^59 - 6 * q^61 - 3 * q^65 - 6 * q^67 + 18 * q^79 - 27 * q^81 + 60 * q^83 + 12 * q^85 + 18 * q^87 - 36 * q^93 - 3 * q^95 + 12 * q^97 + 126 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\)	\(981\)	\(1081\)	\(1177\)	\(1471\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
							\(3\)	1.29211	+	2.23800i	0.746002	+	1.29211i	0.949725	+	0.313084i	\(0.101362\pi\)
−0.203724	+	0.979028i	\(0.565304\pi\)
\(5\)	0.500000	−	0.866025i	0.223607	−	0.387298i
							\(7\)		0			0
\(11\)	−0.839111	−	1.45338i	−0.253001	−	0.438211i								0.711349	−	0.702839i	\(-0.248084\pi\)
							−0.964351	+	0.264627i	\(0.914751\pi\)
\(13\)	4.84667			1.34422			0.672112	−	0.740449i	\(-0.265387\pi\)
							0.672112	+	0.740449i	\(0.265387\pi\)
\(17\)	1.00000	+	1.73205i	0.242536	+	0.420084i	0.961436	−	0.275029i	\(-0.0886875\pi\)
							−0.718900	+	0.695113i	\(0.755354\pi\)
\(19\)	3.42334	−	5.92939i	0.785367	−	1.36030i	−0.143412	−	0.989663i	\(-0.545808\pi\)
							0.928779	−	0.370633i	\(-0.120859\pi\)
\(23\)	1.13122	−	1.95934i	0.235876	−	0.408550i	−0.723651	−	0.690166i	\(-0.757537\pi\)
							0.959527	+	0.281617i	\(0.0908706\pi\)
\(29\)	3.32178			0.616839			0.308419	−	0.951250i	\(-0.400200\pi\)
							0.308419	+	0.951250i	\(0.400200\pi\)
\(31\)	4.58423	+	7.94011i	0.823351	+	1.42609i	0.903173	+	0.429277i	\(0.141232\pi\)
							−0.0798217	+	0.996809i	\(0.525435\pi\)
\(37\)	1.42334	−	2.46529i	0.233995	−	0.405291i	−0.724985	−	0.688765i	\(-0.758153\pi\)
							0.958980	+	0.283473i	\(0.0914867\pi\)
\(41\)	−9.52489			−1.48754			−0.743769	−	0.668437i	\(-0.766964\pi\)
							−0.743769	+	0.668437i	\(0.766964\pi\)
\(43\)	6.58423			1.00408			0.502042	−	0.864843i	\(-0.332582\pi\)
							0.502042	+	0.864843i	\(0.332582\pi\)
\(47\)	−6.10156	+	10.5682i	−0.890004	+	1.54153i	−0.0501344	+	0.998742i	\(0.515965\pi\)
							−0.839869	+	0.542789i	\(0.817368\pi\)
\(53\)	−3.74511	−	6.48673i	−0.514431	−	0.891021i	−0.999860	−	0.0167445i	\(-0.994670\pi\)
							0.485429	−	0.874276i	\(-0.338664\pi\)
\(59\)	4.00000	+	6.92820i	0.520756	+	0.901975i	0.999709	+	0.0241347i	\(0.00768307\pi\)
							−0.478953	+	0.877841i	\(0.658984\pi\)
\(61\)	−3.24511	+	5.62070i	−0.415494	+	0.719657i	−0.995480	−	0.0949692i	\(-0.969725\pi\)
							0.579986	+	0.814627i	\(0.303058\pi\)
\(67\)	2.87634	+	4.98196i	0.351401	+	0.608644i	0.986495	−	0.163791i	\(-0.0523722\pi\)
							−0.635094	+	0.772434i	\(0.719039\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	−5.84667	−	10.1267i	−0.684301	−	1.18524i	−0.973656	−	0.228022i	\(-0.926774\pi\)
							0.289355	−	0.957222i	\(-0.406559\pi\)
\(79\)	−2.84667	+	4.93058i	−0.320276	+	0.554734i	−0.980545	−	0.196295i	\(-0.937109\pi\)
							0.660269	+	0.751029i	\(0.270442\pi\)
\(83\)	12.5842			1.38130			0.690649	−	0.723190i	\(-0.257325\pi\)
							0.690649	+	0.723190i	\(0.257325\pi\)
\(89\)	−2.92334	+	5.06337i	−0.309873	+	0.536716i	−0.978334	−	0.207031i	\(-0.933620\pi\)
							0.668461	+	0.743747i	\(0.266953\pi\)
\(97\)	2.00000			0.203069			0.101535	−	0.994832i	\(-0.467625\pi\)
							0.101535	+	0.994832i	\(0.467625\pi\)

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	1960.2.q.w.961.3		6
7.2	even	3		1960.2.a.v.1.1		3
7.3	odd	6		280.2.q.e.81.1	✓	6
7.4	even	3	inner	1960.2.q.w.361.3		6
7.5	odd	6		1960.2.a.w.1.3		3
7.6	odd	2		280.2.q.e.121.1	yes	6
21.17	even	6		2520.2.bi.q.361.1		6
21.20	even	2		2520.2.bi.q.1801.1		6
28.3	even	6		560.2.q.l.81.3		6
28.19	even	6		3920.2.a.cc.1.1		3
28.23	odd	6		3920.2.a.cb.1.3		3
28.27	even	2		560.2.q.l.401.3		6
35.3	even	12		1400.2.bh.i.249.2		12
35.9	even	6		9800.2.a.cf.1.3		3
35.13	even	4		1400.2.bh.i.849.5		12
35.17	even	12		1400.2.bh.i.249.5		12
35.19	odd	6		9800.2.a.ce.1.1		3
35.24	odd	6		1400.2.q.j.1201.3		6
35.27	even	4		1400.2.bh.i.849.2		12
35.34	odd	2		1400.2.q.j.401.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.e.81.1	✓	6	7.3	odd	6
280.2.q.e.121.1	yes	6	7.6	odd	2
560.2.q.l.81.3		6	28.3	even	6
560.2.q.l.401.3		6	28.27	even	2
1400.2.q.j.401.3		6	35.34	odd	2
1400.2.q.j.1201.3		6	35.24	odd	6
1400.2.bh.i.249.2		12	35.3	even	12
1400.2.bh.i.249.5		12	35.17	even	12
1400.2.bh.i.849.2		12	35.27	even	4
1400.2.bh.i.849.5		12	35.13	even	4
1960.2.a.v.1.1		3	7.2	even	3
1960.2.a.w.1.3		3	7.5	odd	6
1960.2.q.w.361.3		6	7.4	even	3	inner
1960.2.q.w.961.3		6	1.1	even	1	trivial
2520.2.bi.q.361.1		6	21.17	even	6
2520.2.bi.q.1801.1		6	21.20	even	2
3920.2.a.cb.1.3		3	28.23	odd	6
3920.2.a.cc.1.1		3	28.19	even	6
9800.2.a.ce.1.1		3	35.19	odd	6
9800.2.a.cf.1.3		3	35.9	even	6