Embedded newform 310.3.i.a.99.19

• Label: 310.3.i.a.99.19
• 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			99.19
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-2.02117 - 3.50078i) q^{3} -2.00000 q^{4} +(-4.84404 - 1.23907i) q^{5} +(4.95084 - 2.85837i) q^{6} +(-5.44792 + 3.14536i) q^{7} -2.82843i q^{8} +(-3.67029 + 6.35712i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.02117	−	3.50078i	−0.673725	−	1.16693i	−0.976840	−	0.213971i	\(-0.931360\pi\)
0.303115	−	0.952954i	\(-0.401973\pi\)
\(5\)	−4.84404	−	1.23907i	−0.968808	−	0.247814i
							\(7\)	−5.44792	+	3.14536i	−0.778274	+	0.449337i	−0.835818	−	0.549006i	\(-0.815006\pi\)
0.0575440	+	0.998343i	\(0.481673\pi\)
\(11\)	5.08962	+	2.93849i	0.462693	+	0.267136i	0.713176	−	0.700985i	\(-0.247256\pi\)
							−0.250483	+	0.968121i	\(0.580589\pi\)
\(13\)	1.81240	−	3.13917i	0.139416	−	0.241475i	−0.787860	−	0.615854i	\(-0.788811\pi\)
							0.927276	+	0.374380i	\(0.122144\pi\)
\(17\)	3.83698	+	6.64585i	0.225705	+	0.390932i	0.956531	−	0.291632i	\(-0.0941982\pi\)
							−0.730826	+	0.682564i	\(0.760865\pi\)
\(19\)	2.59008	+	4.48614i	0.136320	+	0.236113i	0.926101	−	0.377276i	\(-0.123139\pi\)
							−0.789781	+	0.613389i	\(0.789806\pi\)
\(23\)	21.4690			0.933437			0.466718	−	0.884406i	\(-0.345436\pi\)
							0.466718	+	0.884406i	\(0.345436\pi\)
\(29\)			23.6911i			0.816936i	0.912773	+	0.408468i	\(0.133937\pi\)
							−0.912773	+	0.408468i	\(0.866063\pi\)
\(31\)	28.7371	+	11.6268i	0.927002	+	0.375057i
							\(37\)	11.4239	+	19.7867i	0.308753	+	0.534777i	0.978090	−	0.208183i	\(-0.0667549\pi\)
−0.669337	+	0.742959i	\(0.733422\pi\)
\(41\)	−2.10215	+	3.64104i	−0.0512721	+	0.0888058i	−0.890522	−	0.454939i	\(-0.849661\pi\)
							0.839250	+	0.543745i	\(0.182994\pi\)
\(43\)	−1.26250	−	2.18671i	−0.0293605	−	0.0508538i	0.850972	−	0.525211i	\(-0.176014\pi\)
							−0.880332	+	0.474358i	\(0.842680\pi\)
\(47\)			9.95716i			0.211854i	0.994374	+	0.105927i	\(0.0337810\pi\)
							−0.994374	+	0.105927i	\(0.966219\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.19	yes	64
5.4	even	2	inner	310.3.i.a.99.14	✓	64
31.26	odd	6	inner	310.3.i.a.119.30	yes	64
155.119	odd	6	inner	310.3.i.a.119.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.14	✓	64	5.4	even	2	inner
310.3.i.a.99.19	yes	64	1.1	even	1	trivial
310.3.i.a.119.3	yes	64	155.119	odd	6	inner
310.3.i.a.119.30	yes	64	31.26	odd	6	inner