Embedded newform 310.3.o.d.67.13

• Label: 310.3.o.d.67.13
• Level: $310$
• Weight: $3$
• Character: 310.67
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(15\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			67.13
Character	\(\chi\)	\(=\)	310.67
Dual form			310.3.o.d.273.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +(1.05621 + 3.94184i) q^{3} +2.00000i q^{4} +(-4.52312 + 2.13106i) q^{5} +(-2.88563 + 4.99806i) q^{6} +(-1.21778 + 0.326302i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(-6.62831 + 3.82686i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 60 q^{2} - 2 q^{3} - 4 q^{6} - 12 q^{7} - 120 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	+	1.00000i	0.500000	+	0.500000i
							\(3\)	1.05621	+	3.94184i	0.352071	+	1.31395i	0.884130	+	0.467241i	\(0.154752\pi\)
−0.532059	+	0.846707i	\(0.678582\pi\)
\(5\)	−4.52312	+	2.13106i	−0.904623	+	0.426212i
							\(7\)	−1.21778	+	0.326302i	−0.173968	+	0.0466146i	−0.344752	−	0.938694i	\(-0.612037\pi\)
0.170783	+	0.985309i	\(0.445370\pi\)
\(11\)	1.03221	+	1.78785i	0.0938376	+	0.162531i	0.909123	−	0.416528i	\(-0.136753\pi\)
							−0.815285	+	0.579059i	\(0.803420\pi\)
\(13\)	−9.64361	−	2.58400i	−0.741816	−	0.198769i	−0.131931	−	0.991259i	\(-0.542118\pi\)
							−0.609885	+	0.792490i	\(0.708784\pi\)
\(17\)	13.8075	−	3.69970i	0.812203	−	0.217629i	0.171268	−	0.985224i	\(-0.445213\pi\)
							0.640935	+	0.767595i	\(0.278547\pi\)
\(19\)	6.81206	+	3.93294i	0.358529	+	0.206997i	0.668435	−	0.743770i	\(-0.266964\pi\)
							−0.309906	+	0.950767i	\(0.600298\pi\)
\(23\)	−24.3680	+	24.3680i	−1.05948	+	1.05948i	−0.0613643	+	0.998115i	\(0.519545\pi\)
							−0.998115	+	0.0613643i	\(0.980455\pi\)
\(29\)			2.53259i			0.0873306i	0.999046	+	0.0436653i	\(0.0139035\pi\)
							−0.999046	+	0.0436653i	\(0.986096\pi\)
\(31\)	−30.9498	−	1.76392i	−0.998380	−	0.0569005i
							\(37\)	40.2991	−	10.7981i	1.08916	−	0.291841i	0.330819	−	0.943694i	\(-0.392675\pi\)
0.758345	+	0.651854i	\(0.226008\pi\)
\(41\)	7.08271	+	12.2676i	0.172749	+	0.299210i	0.939380	−	0.342878i	\(-0.111402\pi\)
							−0.766631	+	0.642088i	\(0.778068\pi\)
\(43\)	9.76386	+	36.4392i	0.227066	+	0.847423i	0.981566	+	0.191123i	\(0.0612129\pi\)
							−0.754500	+	0.656300i	\(0.772120\pi\)
\(47\)	28.7340	+	28.7340i	0.611363	+	0.611363i	0.943301	−	0.331938i	\(-0.107703\pi\)
							−0.331938	+	0.943301i	\(0.607703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.o.d.67.13	✓	60
5.3	odd	4	inner	310.3.o.d.253.3	yes	60
31.25	even	3	inner	310.3.o.d.87.3	yes	60
155.118	odd	12	inner	310.3.o.d.273.13	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.o.d.67.13	✓	60	1.1	even	1	trivial
310.3.o.d.87.3	yes	60	31.25	even	3	inner
310.3.o.d.253.3	yes	60	5.3	odd	4	inner
310.3.o.d.273.13	yes	60	155.118	odd	12	inner