Embedded newform 310.3.l.a.29.11

• Label: 310.3.l.a.29.11
• Level: $310$
• Weight: $3$
• Character: 310.29
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			29.11
Character	\(\chi\)	\(=\)	310.29
Dual form			310.3.l.a.139.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.831254 + 1.14412i) q^{2} +(1.44587 - 1.05048i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(-4.76702 - 1.50849i) q^{5} +2.52747i q^{6} +(-2.45402 + 0.797360i) q^{7} +(2.68999 + 0.874032i) q^{8} +(-1.79414 + 5.52178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 64 q^{4} - 12 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.831254	+	1.14412i	−0.415627	+	0.572061i
							\(3\)	1.44587	−	1.05048i	0.481956	−	0.350162i	−0.320126	−	0.947375i	\(-0.603725\pi\)
0.802083	+	0.597213i	\(0.203725\pi\)
\(5\)	−4.76702	−	1.50849i	−0.953404	−	0.301698i
							\(7\)	−2.45402	+	0.797360i	−0.350574	+	0.113909i	−0.479010	−	0.877809i	\(-0.659004\pi\)
0.128436	+	0.991718i	\(0.459004\pi\)
\(11\)	16.4861	−	5.35665i	1.49873	−	0.486968i	0.559085	−	0.829110i	\(-0.311153\pi\)
							0.939648	+	0.342142i	\(0.111153\pi\)
\(13\)	−10.6449	+	7.73397i	−0.818838	+	0.594921i	−0.916380	−	0.400311i	\(-0.868902\pi\)
							0.0975411	+	0.995231i	\(0.468902\pi\)
\(17\)	−6.69759	+	20.6131i	−0.393976	+	1.21253i	0.535781	+	0.844357i	\(0.320017\pi\)
							−0.929756	+	0.368175i	\(0.879983\pi\)
\(19\)	15.9772	+	11.6081i	0.840903	+	0.610952i	0.922623	−	0.385703i	\(-0.126041\pi\)
							−0.0817195	+	0.996655i	\(0.526041\pi\)
\(23\)	−1.97027	+	6.06387i	−0.0856640	+	0.263647i	−0.984708	−	0.174211i	\(-0.944263\pi\)
							0.899044	+	0.437857i	\(0.144263\pi\)
\(29\)	−19.6669	+	27.0691i	−0.678169	+	0.933419i	−0.999910	−	0.0134086i	\(-0.995732\pi\)
							0.321742	+	0.946828i	\(0.395732\pi\)
\(31\)	29.9697	−	7.92579i	0.966764	−	0.255671i
							\(37\)	−14.2895			−0.386203			−0.193102	−	0.981179i	\(-0.561855\pi\)
−0.193102	+	0.981179i	\(0.561855\pi\)
\(41\)	24.7536	+	17.9846i	0.603747	+	0.438648i	0.847207	−	0.531263i	\(-0.178283\pi\)
							−0.243460	+	0.969911i	\(0.578283\pi\)
\(43\)	−15.6852	−	11.3960i	−0.364772	−	0.265023i	0.390267	−	0.920702i	\(-0.372383\pi\)
							−0.755040	+	0.655679i	\(0.772383\pi\)
\(47\)	−8.95052	−	12.3193i	−0.190437	−	0.262113i	0.703113	−	0.711078i	\(-0.251793\pi\)
							−0.893549	+	0.448965i	\(0.851793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.l.a.29.11	✓	128
5.4	even	2	inner	310.3.l.a.29.22	yes	128
31.15	odd	10	inner	310.3.l.a.139.22	yes	128
155.139	odd	10	inner	310.3.l.a.139.11	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.l.a.29.11	✓	128	1.1	even	1	trivial
310.3.l.a.29.22	yes	128	5.4	even	2	inner
310.3.l.a.139.11	yes	128	155.139	odd	10	inner
310.3.l.a.139.22	yes	128	31.15	odd	10	inner