Embedded newform 4000.2.c.f.1249.9

• Label: 4000.2.c.f.1249.9
• Level: $4000$
• Weight: $2$
• Character: 4000.1249
• Analytic conductor: $31.940$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(31.9401608085\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 22x^{10} + 179x^{8} + 646x^{6} + 929x^{4} + 252x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.9
Root			\(1.81030i\) of defining polynomial
Character	\(\chi\)	\(=\)	4000.1249
Dual form			4000.2.c.f.1249.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.81030i q^{3} -4.37760i q^{7} -0.277175 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\)	\(1377\)	\(2501\)	\(2751\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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	0
			\(3\)	1.81030i	1.04518i	0.852586	+	0.522588i	\(0.175033\pi\)
−0.852586	+	0.522588i				\(0.824967\pi\)
\(5\)	0	0
			\(7\)	− 4.37760i	− 1.65458i	−0.561777	−	0.827289i	\(-0.689882\pi\)
0.561777	−	0.827289i				\(-0.310118\pi\)
\(11\)	−5.82433	−1.75610	−0.878051	−	0.478567i	\(-0.841156\pi\)
			−0.878051	+	0.478567i	\(0.841156\pi\)
\(13\)	− 5.34625i	− 1.48278i	−0.671073	−	0.741391i	\(-0.734166\pi\)
			0.671073	−	0.741391i	\(-0.265834\pi\)
\(17\)	5.41151i	1.31248i	0.754550	+	0.656242i	\(0.227855\pi\)
			−0.754550	+	0.656242i	\(0.772145\pi\)
\(19\)	−1.59789	−0.366580	−0.183290	−	0.983059i	\(-0.558675\pi\)
			−0.183290	+	0.983059i	\(0.558675\pi\)
\(23\)	− 0.340859i	− 0.0710740i	−0.999368	−	0.0355370i	\(-0.988686\pi\)
			0.999368	−	0.0355370i	\(-0.0113142\pi\)
\(29\)	4.96428	0.921844	0.460922	−	0.887441i	\(-0.347519\pi\)
			0.460922	+	0.887441i	\(0.347519\pi\)
\(31\)	4.63796	0.833002	0.416501	−	0.909135i	\(-0.363256\pi\)
			0.416501	+	0.909135i	\(0.363256\pi\)
\(37\)	8.40977i	1.38256i	0.722588	+	0.691278i	\(0.242952\pi\)
			−0.722588	+	0.691278i	\(0.757048\pi\)
\(41\)	−2.34244	−0.365828	−0.182914	−	0.983129i	\(-0.558553\pi\)
			−0.182914	+	0.983129i	\(0.558553\pi\)
\(43\)	− 7.10407i	− 1.08336i	−0.840585	−	0.541680i	\(-0.817788\pi\)
			0.840585	−	0.541680i	\(-0.182212\pi\)
\(47\)	− 0.879427i	− 0.128278i	−0.997941	−	0.0641388i	\(-0.979570\pi\)
			0.997941	−	0.0641388i	\(-0.0204300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.c.f.1249.9		12
4.3	odd	2		4000.2.c.g.1249.4		12
5.2	odd	4		4000.2.a.m.1.5	yes	6
5.3	odd	4		4000.2.a.k.1.2	✓	6
5.4	even	2	inner	4000.2.c.f.1249.4		12
20.3	even	4		4000.2.a.n.1.5	yes	6
20.7	even	4		4000.2.a.l.1.2	yes	6
20.19	odd	2		4000.2.c.g.1249.9		12
40.3	even	4		8000.2.a.bw.1.2		6
40.13	odd	4		8000.2.a.bv.1.5		6
40.27	even	4		8000.2.a.bu.1.5		6
40.37	odd	4		8000.2.a.bx.1.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.k.1.2	✓	6	5.3	odd	4
4000.2.a.l.1.2	yes	6	20.7	even	4
4000.2.a.m.1.5	yes	6	5.2	odd	4
4000.2.a.n.1.5	yes	6	20.3	even	4
4000.2.c.f.1249.4		12	5.4	even	2	inner
4000.2.c.f.1249.9		12	1.1	even	1	trivial
4000.2.c.g.1249.4		12	4.3	odd	2
4000.2.c.g.1249.9		12	20.19	odd	2
8000.2.a.bu.1.5		6	40.27	even	4
8000.2.a.bv.1.5		6	40.13	odd	4
8000.2.a.bw.1.2		6	40.3	even	4
8000.2.a.bx.1.2		6	40.37	odd	4