Embedded newform 315.2.u.a.59.5

• Label: 315.2.u.a.59.5
• 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			59.5
Character	\(\chi\)	\(=\)	315.59
Dual form			315.2.u.a.299.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.14897 + 1.99008i) q^{2} +(-1.60485 + 0.651500i) q^{3} +(-1.64028 - 2.84105i) q^{4} +(1.90726 - 1.16720i) q^{5} +(0.547395 - 3.94234i) q^{6} +(2.44187 + 1.01847i) q^{7} +2.94267 q^{8} +(2.15110 - 2.09112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.14897	+	1.99008i	−0.812447	+	1.40720i	0.0986993	+	0.995117i	\(0.468532\pi\)
							−0.911147	+	0.412083i	\(0.864802\pi\)
\(3\)	−1.60485	+	0.651500i	−0.926561	+	0.376144i
							\(5\)	1.90726	−	1.16720i	0.852954	−	0.521986i
\(7\)	2.44187	+	1.01847i	0.922940	+	0.384945i
							\(11\)			5.03602i			1.51842i	0.650848	+	0.759208i	\(0.274413\pi\)
−0.650848	+	0.759208i	\(0.725587\pi\)
\(13\)	1.81474	−	3.14323i	0.503320	−	0.871775i	−0.496673	−	0.867938i	\(-0.665445\pi\)
							0.999993	−	0.00383730i	\(-0.00122145\pi\)
\(17\)	5.20593	+	3.00564i	1.26262	+	0.728976i	0.973581	−	0.228340i	\(-0.0733299\pi\)
							0.289042	+	0.957316i	\(0.406663\pi\)
\(19\)	−2.26155	+	1.30571i	−0.518836	+	0.299550i	−0.736458	−	0.676483i	\(-0.763503\pi\)
							0.217622	+	0.976033i	\(0.430170\pi\)
\(23\)	−2.67708			−0.558210			−0.279105	−	0.960261i	\(-0.590038\pi\)
							−0.279105	+	0.960261i	\(0.590038\pi\)
\(29\)	−1.65943	+	0.958073i	−0.308149	+	0.177910i	−0.646098	−	0.763255i	\(-0.723600\pi\)
							0.337949	+	0.941164i	\(0.390267\pi\)
\(31\)	−1.26456	+	0.730091i	−0.227121	+	0.131128i	−0.609243	−	0.792983i	\(-0.708527\pi\)
							0.382122	+	0.924112i	\(0.375193\pi\)
\(37\)	−3.39771	+	1.96167i	−0.558579	+	0.322496i	−0.752575	−	0.658506i	\(-0.771189\pi\)
							0.193996	+	0.981002i	\(0.437855\pi\)
\(41\)	0.294240	−	0.509638i	0.0459525	−	0.0795921i	−0.842134	−	0.539268i	\(-0.818701\pi\)
							0.888087	+	0.459676i	\(0.152034\pi\)
\(43\)	4.96969	−	2.86925i	0.757871	−	0.437557i	−0.0706596	−	0.997500i	\(-0.522510\pi\)
							0.828531	+	0.559943i	\(0.189177\pi\)
\(47\)	3.99224	+	2.30492i	0.582328	+	0.336207i	0.762058	−	0.647509i	\(-0.224189\pi\)
							−0.179730	+	0.983716i	\(0.557522\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.u.a.59.5	✓	88
3.2	odd	2		945.2.u.a.584.40		88
5.4	even	2	inner	315.2.u.a.59.40	yes	88
7.5	odd	6		315.2.bq.a.194.40	yes	88
9.2	odd	6		315.2.bq.a.164.5	yes	88
9.7	even	3		945.2.bq.a.899.40		88
15.14	odd	2		945.2.u.a.584.5		88
21.5	even	6		945.2.bq.a.719.5		88
35.19	odd	6		315.2.bq.a.194.5	yes	88
45.29	odd	6		315.2.bq.a.164.40	yes	88
45.34	even	6		945.2.bq.a.899.5		88
63.47	even	6	inner	315.2.u.a.299.40	yes	88
63.61	odd	6		945.2.u.a.89.5		88
105.89	even	6		945.2.bq.a.719.40		88
315.124	odd	6		945.2.u.a.89.40		88
315.299	even	6	inner	315.2.u.a.299.5	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.5	✓	88	1.1	even	1	trivial
315.2.u.a.59.40	yes	88	5.4	even	2	inner
315.2.u.a.299.5	yes	88	315.299	even	6	inner
315.2.u.a.299.40	yes	88	63.47	even	6	inner
315.2.bq.a.164.5	yes	88	9.2	odd	6
315.2.bq.a.164.40	yes	88	45.29	odd	6
315.2.bq.a.194.5	yes	88	35.19	odd	6
315.2.bq.a.194.40	yes	88	7.5	odd	6
945.2.u.a.89.5		88	63.61	odd	6
945.2.u.a.89.40		88	315.124	odd	6
945.2.u.a.584.5		88	15.14	odd	2
945.2.u.a.584.40		88	3.2	odd	2
945.2.bq.a.719.5		88	21.5	even	6
945.2.bq.a.719.40		88	105.89	even	6
945.2.bq.a.899.5		88	45.34	even	6
945.2.bq.a.899.40		88	9.7	even	3