Embedded newform 605.2.g.o.366.2

• Label: 605.2.g.o.366.2
• Level: $605$
• Weight: $2$
• Character: 605.366
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - x^11 + 6*x^10 - 12*x^9 + 43*x^8 + 72*x^7 + 155*x^6 + 162*x^5 + 541*x^4 + 114*x^3 + 24*x^2 + 5*x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			366.2
Root			\(0.511560 - 1.57442i\) of defining polynomial
Character	\(\chi\)	\(=\)	605.366
Dual form			605.2.g.o.81.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12933 + 0.820508i) q^{2} +(-0.511560 - 1.57442i) q^{3} +(-0.0158755 + 0.0488598i) q^{4} +(0.809017 + 0.587785i) q^{5} +(1.86955 + 1.35830i) q^{6} +(1.45449 - 4.47645i) q^{7} +(-0.884894 - 2.72343i) q^{8} +(0.209948 - 0.152536i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} - q^{3} - 9 q^{4} + 3 q^{5} - 5 q^{6} + q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12933	+	0.820508i	−0.798559	+	0.580187i	−0.910491	−	0.413529i	\(-0.864296\pi\)
							0.111932	+	0.993716i	\(0.464296\pi\)
\(3\)	−0.511560	−	1.57442i	−0.295349	−	0.908991i	−0.983104	−	0.183048i	\(-0.941404\pi\)
							0.687755	−	0.725943i	\(-0.258596\pi\)
\(5\)	0.809017	+	0.587785i	0.361803	+	0.262866i
							\(7\)	1.45449	−	4.47645i	0.549744	−	1.69194i	−0.159690	−	0.987167i	\(-0.551050\pi\)
0.709434	−	0.704771i	\(-0.248950\pi\)
\(11\)		0			0
							\(13\)	−4.08665	+	2.96912i	−1.13343	+	0.823487i	−0.986191	−	0.165614i	\(-0.947039\pi\)
−0.147241	+	0.989101i	\(0.547039\pi\)
\(17\)	−4.29660	−	3.12166i	−1.04208	−	0.757114i	−0.0713876	−	0.997449i	\(-0.522743\pi\)
							−0.970690	+	0.240335i	\(0.922743\pi\)
\(19\)	0.698227	+	2.14892i	0.160184	+	0.492997i	0.998649	−	0.0519590i	\(-0.0165465\pi\)
							−0.838465	+	0.544956i	\(0.816547\pi\)
\(23\)	1.05137			0.219227			0.109613	−	0.993974i	\(-0.465039\pi\)
							0.109613	+	0.993974i	\(0.465039\pi\)
\(29\)	0.862733	−	2.65522i	0.160206	−	0.493062i	−0.838446	−	0.544985i	\(-0.816535\pi\)
							0.998651	+	0.0519234i	\(0.0165352\pi\)
\(31\)	−3.02612	+	2.19860i	−0.543507	+	0.394881i	−0.825386	−	0.564569i	\(-0.809042\pi\)
							0.281879	+	0.959450i	\(0.409042\pi\)
\(37\)	−0.244699	+	0.753107i	−0.0402283	+	0.123810i	−0.969154	−	0.246457i	\(-0.920734\pi\)
							0.928925	+	0.370267i	\(0.120734\pi\)
\(41\)	−1.90173	−	5.85292i	−0.297000	−	0.914072i	−0.982542	−	0.186039i	\(-0.940435\pi\)
							0.685542	−	0.728033i	\(-0.259565\pi\)
\(43\)	−2.70682			−0.412786			−0.206393	−	0.978469i	\(-0.566172\pi\)
							−0.206393	+	0.978469i	\(0.566172\pi\)
\(47\)	−3.47309	−	10.6891i	−0.506603	−	1.55916i	−0.798059	−	0.602579i	\(-0.794140\pi\)
							0.291456	−	0.956584i	\(-0.405860\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.g.o.366.2		12
11.2	odd	10		605.2.a.g.1.2	✓	3
11.3	even	5	inner	605.2.g.o.511.2		12
11.4	even	5	inner	605.2.g.o.81.2		12
11.5	even	5	inner	605.2.g.o.251.2		12
11.6	odd	10		605.2.g.p.251.2		12
11.7	odd	10		605.2.g.p.81.2		12
11.8	odd	10		605.2.g.p.511.2		12
11.9	even	5		605.2.a.h.1.2	yes	3
11.10	odd	2		605.2.g.p.366.2		12
33.2	even	10		5445.2.a.bd.1.2		3
33.20	odd	10		5445.2.a.bb.1.2		3
44.31	odd	10		9680.2.a.cb.1.3		3
44.35	even	10		9680.2.a.bz.1.3		3
55.9	even	10		3025.2.a.p.1.2		3
55.24	odd	10		3025.2.a.u.1.2		3

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.g.1.2	✓	3	11.2	odd	10
605.2.a.h.1.2	yes	3	11.9	even	5
605.2.g.o.81.2		12	11.4	even	5	inner
605.2.g.o.251.2		12	11.5	even	5	inner
605.2.g.o.366.2		12	1.1	even	1	trivial
605.2.g.o.511.2		12	11.3	even	5	inner
605.2.g.p.81.2		12	11.7	odd	10
605.2.g.p.251.2		12	11.6	odd	10
605.2.g.p.366.2		12	11.10	odd	2
605.2.g.p.511.2		12	11.8	odd	10
3025.2.a.p.1.2		3	55.9	even	10
3025.2.a.u.1.2		3	55.24	odd	10
5445.2.a.bb.1.2		3	33.20	odd	10
5445.2.a.bd.1.2		3	33.2	even	10
9680.2.a.bz.1.3		3	44.35	even	10
9680.2.a.cb.1.3		3	44.31	odd	10