Embedded newform 310.3.i.a.99.2

• Label: 310.3.i.a.99.2
• Level: $310$
• Weight: $3$
• Character: 310.99
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			99.2
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.18

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-2.49294 - 4.31790i) q^{3} -2.00000 q^{4} +(-3.07623 + 3.94168i) q^{5} +(-6.10643 + 3.52555i) q^{6} +(1.56148 - 0.901524i) q^{7} +2.82843i q^{8} +(-7.92949 + 13.7343i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.41421i		−	0.707107i
							\(3\)	−2.49294	−	4.31790i	−0.830980	−	1.43930i	−0.897262	−	0.441498i	\(-0.854447\pi\)
0.0662825	−	0.997801i	\(-0.478886\pi\)
\(5\)	−3.07623	+	3.94168i	−0.615245	+	0.788336i
							\(7\)	1.56148	−	0.901524i	0.223069	−	0.128789i	−0.384301	−	0.923208i	\(-0.625558\pi\)
0.607371	+	0.794419i	\(0.292224\pi\)
\(11\)	2.02224	+	1.16754i	0.183840	+	0.106140i	0.589096	−	0.808063i	\(-0.299484\pi\)
							−0.405255	+	0.914203i	\(0.632817\pi\)
\(13\)	−4.03854	+	6.99495i	−0.310657	+	0.538073i	−0.978505	−	0.206224i	\(-0.933882\pi\)
							0.667848	+	0.744298i	\(0.267216\pi\)
\(17\)	1.46230	+	2.53278i	0.0860176	+	0.148987i	0.905824	−	0.423653i	\(-0.139253\pi\)
							−0.819807	+	0.572640i	\(0.805919\pi\)
\(19\)	4.30284	+	7.45274i	0.226465	+	0.392249i	0.956758	−	0.290885i	\(-0.0939497\pi\)
							−0.730293	+	0.683134i	\(0.760616\pi\)
\(23\)	−5.76962			−0.250853			−0.125427	−	0.992103i	\(-0.540030\pi\)
							−0.125427	+	0.992103i	\(0.540030\pi\)
\(29\)			36.1695i			1.24723i	0.781733	+	0.623613i	\(0.214336\pi\)
							−0.781733	+	0.623613i	\(0.785664\pi\)
\(31\)	26.5852	−	15.9445i	0.857587	−	0.514339i
							\(37\)	−10.0782	−	17.4560i	−0.272384	−	0.471782i	0.697088	−	0.716986i	\(-0.254479\pi\)
−0.969472	+	0.245203i	\(0.921145\pi\)
\(41\)	1.92351	−	3.33161i	0.0469148	−	0.0812588i	−0.841614	−	0.540079i	\(-0.818394\pi\)
							0.888529	+	0.458820i	\(0.151728\pi\)
\(43\)	20.6346	+	35.7402i	0.479874	+	0.831167i	0.999734	−	0.0230852i	\(-0.00734889\pi\)
							−0.519859	+	0.854252i	\(0.674016\pi\)
\(47\)		−	8.08780i		−	0.172081i	−0.996292	−	0.0860404i	\(-0.972579\pi\)
							0.996292	−	0.0860404i	\(-0.0274214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.2	✓	64
5.4	even	2	inner	310.3.i.a.99.31	yes	64
31.26	odd	6	inner	310.3.i.a.119.15	yes	64
155.119	odd	6	inner	310.3.i.a.119.18	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.2	✓	64	1.1	even	1	trivial
310.3.i.a.99.31	yes	64	5.4	even	2	inner
310.3.i.a.119.15	yes	64	31.26	odd	6	inner
310.3.i.a.119.18	yes	64	155.119	odd	6	inner