Embedded newform 315.2.u.a.59.10

• Label: 315.2.u.a.59.10
• 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			59.10
Character	\(\chi\)	\(=\)	315.59
Dual form			315.2.u.a.299.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.851246 + 1.47440i) q^{2} +(1.72149 - 0.191018i) q^{3} +(-0.449240 - 0.778107i) q^{4} +(2.23006 + 0.163754i) q^{5} +(-1.18377 + 2.70076i) q^{6} +(2.57388 + 0.612504i) q^{7} -1.87533 q^{8} +(2.92702 - 0.657668i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.851246	+	1.47440i	−0.601922	+	1.04256i	0.390608	+	0.920557i	\(0.372265\pi\)
							−0.992530	+	0.122002i	\(0.961069\pi\)
\(3\)	1.72149	−	0.191018i	0.993900	−	0.110284i
							\(5\)	2.23006	+	0.163754i	0.997315	+	0.0732330i
\(7\)	2.57388	+	0.612504i	0.972834	+	0.231505i
							\(11\)		−	5.58381i		−	1.68358i	−0.539803	−	0.841792i	\(-0.681501\pi\)
0.539803	−	0.841792i	\(-0.318499\pi\)
\(13\)	−1.03818	+	1.79817i	−0.287938	+	0.498724i	−0.973318	−	0.229462i	\(-0.926303\pi\)
							0.685379	+	0.728186i	\(0.259637\pi\)
\(17\)	−2.46562	−	1.42353i	−0.598001	−	0.345256i	0.170254	−	0.985400i	\(-0.445541\pi\)
							−0.768255	+	0.640144i	\(0.778875\pi\)
\(19\)	−2.21883	+	1.28104i	−0.509034	+	0.293891i	−0.732437	−	0.680835i	\(-0.761617\pi\)
							0.223402	+	0.974726i	\(0.428284\pi\)
\(23\)	−2.27154			−0.473649			−0.236824	−	0.971552i	\(-0.576107\pi\)
							−0.236824	+	0.971552i	\(0.576107\pi\)
\(29\)	−8.84546	+	5.10693i	−1.64256	+	0.948333i	−0.662643	+	0.748936i	\(0.730565\pi\)
							−0.979919	+	0.199397i	\(0.936102\pi\)
\(31\)	−2.16128	+	1.24782i	−0.388177	+	0.224114i	−0.681370	−	0.731939i	\(-0.738616\pi\)
							0.293193	+	0.956053i	\(0.405282\pi\)
\(37\)	−5.94937	+	3.43487i	−0.978071	+	0.564689i	−0.901687	−	0.432389i	\(-0.857671\pi\)
							−0.0763837	+	0.997078i	\(0.524337\pi\)
\(41\)	−2.24887	+	3.89516i	−0.351215	+	0.608322i	−0.986463	−	0.163986i	\(-0.947565\pi\)
							0.635248	+	0.772308i	\(0.280898\pi\)
\(43\)	−3.55092	+	2.05013i	−0.541511	+	0.312641i	−0.745691	−	0.666292i	\(-0.767880\pi\)
							0.204180	+	0.978933i	\(0.434547\pi\)
\(47\)	−6.32348	−	3.65087i	−0.922375	−	0.532533i	−0.0379830	−	0.999278i	\(-0.512093\pi\)
							−0.884392	+	0.466745i	\(0.845427\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.u.a.59.10	✓	88
3.2	odd	2		945.2.u.a.584.35		88
5.4	even	2	inner	315.2.u.a.59.35	yes	88
7.5	odd	6		315.2.bq.a.194.35	yes	88
9.2	odd	6		315.2.bq.a.164.10	yes	88
9.7	even	3		945.2.bq.a.899.35		88
15.14	odd	2		945.2.u.a.584.10		88
21.5	even	6		945.2.bq.a.719.10		88
35.19	odd	6		315.2.bq.a.194.10	yes	88
45.29	odd	6		315.2.bq.a.164.35	yes	88
45.34	even	6		945.2.bq.a.899.10		88
63.47	even	6	inner	315.2.u.a.299.35	yes	88
63.61	odd	6		945.2.u.a.89.10		88
105.89	even	6		945.2.bq.a.719.35		88
315.124	odd	6		945.2.u.a.89.35		88
315.299	even	6	inner	315.2.u.a.299.10	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.10	✓	88	1.1	even	1	trivial
315.2.u.a.59.35	yes	88	5.4	even	2	inner
315.2.u.a.299.10	yes	88	315.299	even	6	inner
315.2.u.a.299.35	yes	88	63.47	even	6	inner
315.2.bq.a.164.10	yes	88	9.2	odd	6
315.2.bq.a.164.35	yes	88	45.29	odd	6
315.2.bq.a.194.10	yes	88	35.19	odd	6
315.2.bq.a.194.35	yes	88	7.5	odd	6
945.2.u.a.89.10		88	63.61	odd	6
945.2.u.a.89.35		88	315.124	odd	6
945.2.u.a.584.10		88	15.14	odd	2
945.2.u.a.584.35		88	3.2	odd	2
945.2.bq.a.719.10		88	21.5	even	6
945.2.bq.a.719.35		88	105.89	even	6
945.2.bq.a.899.10		88	45.34	even	6
945.2.bq.a.899.35		88	9.7	even	3