Embedded newform 70.3.l.c.67.1

• Label: 70.3.l.c.67.1
• Level: $70$
• Weight: $3$
• Character: 70.67
• Analytic conductor: $1.907$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	70.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.90736185052\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 2*x^15 + 2*x^14 - 8*x^13 - 722*x^12 + 1354*x^11 - 1232*x^10 + 9306*x^9 + 505731*x^8 - 1157418*x^7 + 1256080*x^6 - 9824090*x^5 + 5294974*x^4 + 30978760*x^3 + 12152450*x^2 + 35619250*x + 52200625
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			67.1
Root			\(4.90868 - 1.31528i\) of defining polynomial
Character	\(\chi\)	\(=\)	70.67
Dual form			70.3.l.c.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 - 1.36603i) q^{2} +(-1.31528 - 4.90868i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(2.11728 + 4.52958i) q^{5} -7.18681 q^{6} +(-0.145113 - 6.99850i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(-14.5710 + 8.41254i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{2} + 2 q^{3} - 2 q^{5} - 8 q^{6} + 12 q^{7} - 32 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	−	1.36603i	0.183013	−	0.683013i
							\(3\)	−1.31528	−	4.90868i	−0.438426	−	1.63623i	−0.732733	−	0.680516i	\(-0.761756\pi\)
0.294308	−	0.955711i	\(-0.404911\pi\)
\(5\)	2.11728	+	4.52958i	0.423457	+	0.905916i
							\(7\)	−0.145113	−	6.99850i	−0.0207304	−	0.999785i
\(11\)	0.474367	−	0.821627i	0.0431242	−	0.0746934i								−0.843658	−	0.536882i	\(-0.819602\pi\)
							0.886782	+	0.462188i	\(0.152936\pi\)
\(13\)	0.862338	−	0.862338i	0.0663337	−	0.0663337i	−0.673162	−	0.739495i	\(-0.735064\pi\)
							0.739495	+	0.673162i	\(0.235064\pi\)
\(17\)	29.3872	−	7.87429i	1.72866	−	0.463193i	0.748787	−	0.662810i	\(-0.230636\pi\)
							0.979874	+	0.199617i	\(0.0639698\pi\)
\(19\)	20.9123	−	12.0737i	1.10065	−	0.635459i	0.164256	−	0.986418i	\(-0.447478\pi\)
							0.936391	+	0.350959i	\(0.114144\pi\)
\(23\)	−5.47452	−	1.46689i	−0.238023	−	0.0637780i	0.137836	−	0.990455i	\(-0.455985\pi\)
							−0.375858	+	0.926677i	\(0.622652\pi\)
\(29\)		−	7.33636i		−	0.252978i	−0.991968	−	0.126489i	\(-0.959629\pi\)
							0.991968	−	0.126489i	\(-0.0403708\pi\)
\(31\)	−23.5316	+	40.7579i	−0.759083	+	1.31477i	0.184235	+	0.982882i	\(0.441019\pi\)
							−0.943319	+	0.331889i	\(0.892314\pi\)
\(37\)	−2.13074	+	7.95202i	−0.0575875	+	0.214919i	−0.988723	−	0.149753i	\(-0.952152\pi\)
							0.931136	+	0.364672i	\(0.118819\pi\)
\(41\)	53.3601			1.30147			0.650733	−	0.759306i	\(-0.274462\pi\)
							0.650733	+	0.759306i	\(0.274462\pi\)
\(43\)	−33.0776	+	33.0776i	−0.769247	+	0.769247i	−0.977974	−	0.208727i	\(-0.933068\pi\)
							0.208727	+	0.977974i	\(0.433068\pi\)
\(47\)	−7.87429	+	29.3872i	−0.167538	+	0.625260i	0.830165	+	0.557518i	\(0.188246\pi\)
							−0.997703	+	0.0677424i	\(0.978420\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	70.3.l.c.67.1	yes	16
5.2	odd	4		350.3.p.e.193.1		16
5.3	odd	4	inner	70.3.l.c.53.4	yes	16
5.4	even	2		350.3.p.e.207.4		16
7.2	even	3	inner	70.3.l.c.37.4	yes	16
7.3	odd	6		490.3.f.p.197.4		8
7.4	even	3		490.3.f.o.197.1		8
35.2	odd	12		350.3.p.e.93.4		16
35.3	even	12		490.3.f.p.393.4		8
35.9	even	6		350.3.p.e.107.1		16
35.18	odd	12		490.3.f.o.393.1		8
35.23	odd	12	inner	70.3.l.c.23.1	✓	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.3.l.c.23.1	✓	16	35.23	odd	12	inner
70.3.l.c.37.4	yes	16	7.2	even	3	inner
70.3.l.c.53.4	yes	16	5.3	odd	4	inner
70.3.l.c.67.1	yes	16	1.1	even	1	trivial
350.3.p.e.93.4		16	35.2	odd	12
350.3.p.e.107.1		16	35.9	even	6
350.3.p.e.193.1		16	5.2	odd	4
350.3.p.e.207.4		16	5.4	even	2
490.3.f.o.197.1		8	7.4	even	3
490.3.f.o.393.1		8	35.18	odd	12
490.3.f.p.197.4		8	7.3	odd	6
490.3.f.p.393.4		8	35.3	even	12