Embedded newform 140.3.t.a.51.26

• Label: 140.3.t.a.51.26
• Level: $140$
• Weight: $3$
• Character: 140.51
• Analytic conductor: $3.815$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	140.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.81472370104\)
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			51.26
Character	\(\chi\)	\(=\)	140.51
Dual form			140.3.t.a.11.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.68963 - 1.07012i) q^{2} +(-4.93310 + 2.84813i) q^{3} +(1.70967 - 3.61622i) q^{4} +(1.11803 - 1.93649i) q^{5} +(-5.28724 + 10.0913i) q^{6} +(6.48796 - 2.62800i) q^{7} +(-0.981105 - 7.93961i) q^{8} +(11.7237 - 20.3060i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 2 q^{2} + 6 q^{4} + 20 q^{8} + 96 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.68963	−	1.07012i	0.844813	−	0.535062i
							\(3\)	−4.93310	+	2.84813i	−1.64437	+	0.949376i	−0.665113	+	0.746743i	\(0.731616\pi\)
−0.979255	+	0.202633i	\(0.935050\pi\)
\(5\)	1.11803	−	1.93649i	0.223607	−	0.387298i
							\(7\)	6.48796	−	2.62800i	0.926852	−	0.375428i
\(11\)	5.26172	−	3.03786i	0.478339	−	0.276169i								−0.241385	−	0.970429i	\(-0.577602\pi\)
							0.719724	+	0.694260i	\(0.244268\pi\)
\(13\)	11.5457			0.888134			0.444067	−	0.895993i	\(-0.353535\pi\)
							0.444067	+	0.895993i	\(0.353535\pi\)
\(17\)	2.84067	+	4.92019i	0.167098	+	0.289423i	0.937398	−	0.348259i	\(-0.113227\pi\)
							−0.770300	+	0.637682i	\(0.779894\pi\)
\(19\)	−29.1569	−	16.8338i	−1.53458	−	0.885988i	−0.999142	−	0.0414103i	\(-0.986815\pi\)
							−0.535434	−	0.844577i	\(-0.679852\pi\)
\(23\)	17.6580	+	10.1948i	0.767738	+	0.443254i	0.832067	−	0.554675i	\(-0.187157\pi\)
							−0.0643288	+	0.997929i	\(0.520491\pi\)
\(29\)	−10.7706			−0.371399			−0.185699	−	0.982607i	\(-0.559455\pi\)
							−0.185699	+	0.982607i	\(0.559455\pi\)
\(31\)	14.2872	−	8.24874i	0.460879	−	0.266088i	−0.251535	−	0.967848i	\(-0.580935\pi\)
							0.712414	+	0.701760i	\(0.247602\pi\)
\(37\)	−20.3664	+	35.2756i	−0.550443	+	0.953396i	0.447799	+	0.894134i	\(0.352208\pi\)
							−0.998242	+	0.0592617i	\(0.981125\pi\)
\(41\)	41.1307			1.00319			0.501594	−	0.865103i	\(-0.332747\pi\)
							0.501594	+	0.865103i	\(0.332747\pi\)
\(43\)			6.10425i			0.141959i	0.997478	+	0.0709796i	\(0.0226125\pi\)
							−0.997478	+	0.0709796i	\(0.977387\pi\)
\(47\)	−12.6485	−	7.30261i	−0.269117	−	0.155375i	0.359369	−	0.933195i	\(-0.382992\pi\)
							−0.628486	+	0.777821i	\(0.716325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.3.t.a.51.26	yes	64
4.3	odd	2	inner	140.3.t.a.51.16	yes	64
7.4	even	3	inner	140.3.t.a.11.16	✓	64
28.11	odd	6	inner	140.3.t.a.11.26	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.3.t.a.11.16	✓	64	7.4	even	3	inner
140.3.t.a.11.26	yes	64	28.11	odd	6	inner
140.3.t.a.51.16	yes	64	4.3	odd	2	inner
140.3.t.a.51.26	yes	64	1.1	even	1	trivial