Embedded newform 315.2.u.a.59.19

• Label: 315.2.u.a.59.19
• Level: $315$
• Weight: $2$
• Character: 315.59
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.19
Character	\(\chi\)	\(=\)	315.59
Dual form			315.2.u.a.299.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.252360 + 0.437100i) q^{2} +(1.57415 - 0.722528i) q^{3} +(0.872629 + 1.51144i) q^{4} +(-1.06742 + 1.96485i) q^{5} +(-0.0814357 + 0.870398i) q^{6} +(1.93769 - 1.80149i) q^{7} -1.89030 q^{8} +(1.95591 - 2.27474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.252360	+	0.437100i	−0.178445	+	0.309076i	−0.941348	−	0.337437i	\(-0.890440\pi\)
							0.762903	+	0.646513i	\(0.223773\pi\)
\(3\)	1.57415	−	0.722528i	0.908837	−	0.417152i
							\(5\)	−1.06742	+	1.96485i	−0.477363	+	0.878706i
\(7\)	1.93769	−	1.80149i	0.732379	−	0.680898i
							\(11\)			4.33915i			1.30830i	0.756364	+	0.654151i	\(0.226974\pi\)
−0.756364	+	0.654151i	\(0.773026\pi\)
\(13\)	1.79624	−	3.11118i	0.498188	−	0.862887i	−0.501810	−	0.864978i	\(-0.667332\pi\)
							0.999998	+	0.00209129i	\(0.000665678\pi\)
\(17\)	0.0823636	+	0.0475527i	0.0199761	+	0.0115332i	0.509955	−	0.860201i	\(-0.329662\pi\)
							−0.489979	+	0.871734i	\(0.662995\pi\)
\(19\)	−5.66909	+	3.27305i	−1.30058	+	0.750889i	−0.980503	−	0.196504i	\(-0.937041\pi\)
							−0.320074	+	0.947392i	\(0.603708\pi\)
\(23\)	3.53299			0.736679			0.368339	−	0.929691i	\(-0.379926\pi\)
							0.368339	+	0.929691i	\(0.379926\pi\)
\(29\)	2.04197	−	1.17893i	0.379184	−	0.218922i	−0.298279	−	0.954479i	\(-0.596413\pi\)
							0.677463	+	0.735557i	\(0.263079\pi\)
\(31\)	5.71238	−	3.29804i	1.02597	−	0.592346i	0.110145	−	0.993916i	\(-0.464869\pi\)
							0.915828	+	0.401570i	\(0.131535\pi\)
\(37\)	−6.99799	+	4.04029i	−1.15046	+	0.664219i	−0.949000	−	0.315277i	\(-0.897903\pi\)
							−0.201462	+	0.979496i	\(0.564569\pi\)
\(41\)	1.07650	−	1.86456i	0.168122	−	0.291195i	−0.769638	−	0.638481i	\(-0.779563\pi\)
							0.937759	+	0.347285i	\(0.112897\pi\)
\(43\)	4.58395	−	2.64654i	0.699045	−	0.403594i	−0.107946	−	0.994157i	\(-0.534427\pi\)
							0.806992	+	0.590563i	\(0.201094\pi\)
\(47\)	−10.1606	−	5.86621i	−1.48207	−	0.855675i	−0.482280	−	0.876017i	\(-0.660191\pi\)
							−0.999793	+	0.0203422i	\(0.993524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.u.a.59.19	✓	88
3.2	odd	2		945.2.u.a.584.26		88
5.4	even	2	inner	315.2.u.a.59.26	yes	88
7.5	odd	6		315.2.bq.a.194.26	yes	88
9.2	odd	6		315.2.bq.a.164.19	yes	88
9.7	even	3		945.2.bq.a.899.26		88
15.14	odd	2		945.2.u.a.584.19		88
21.5	even	6		945.2.bq.a.719.19		88
35.19	odd	6		315.2.bq.a.194.19	yes	88
45.29	odd	6		315.2.bq.a.164.26	yes	88
45.34	even	6		945.2.bq.a.899.19		88
63.47	even	6	inner	315.2.u.a.299.26	yes	88
63.61	odd	6		945.2.u.a.89.19		88
105.89	even	6		945.2.bq.a.719.26		88
315.124	odd	6		945.2.u.a.89.26		88
315.299	even	6	inner	315.2.u.a.299.19	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.19	✓	88	1.1	even	1	trivial
315.2.u.a.59.26	yes	88	5.4	even	2	inner
315.2.u.a.299.19	yes	88	315.299	even	6	inner
315.2.u.a.299.26	yes	88	63.47	even	6	inner
315.2.bq.a.164.19	yes	88	9.2	odd	6
315.2.bq.a.164.26	yes	88	45.29	odd	6
315.2.bq.a.194.19	yes	88	35.19	odd	6
315.2.bq.a.194.26	yes	88	7.5	odd	6
945.2.u.a.89.19		88	63.61	odd	6
945.2.u.a.89.26		88	315.124	odd	6
945.2.u.a.584.19		88	15.14	odd	2
945.2.u.a.584.26		88	3.2	odd	2
945.2.bq.a.719.19		88	21.5	even	6
945.2.bq.a.719.26		88	105.89	even	6
945.2.bq.a.899.19		88	45.34	even	6
945.2.bq.a.899.26		88	9.7	even	3