Embedded newform 361.3.f.i.307.3

• Label: 361.3.f.i.307.3
• Level: $361$
• Weight: $3$
• Character: 361.307
• Analytic conductor: $9.837$
• Analytic rank: $0$
• Dimension: $18$
• CM: no
• Inner twists: $6$

Newspace parameters

Level:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	361.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.83653754341\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Relative dimension:	\(3\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial:	\( x^{18} - 19x^{15} + 722x^{12} + 6857x^{9} + 130340x^{6} - 361x^{3} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			307.3
Root			\(-0.0243863 - 0.138302i\) of defining polynomial
Character	\(\chi\)	\(=\)	361.307
Dual form			361.3.f.i.127.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.47714 - 1.76038i) q^{2} +(-1.46156 - 4.01559i) q^{3} +(-0.222421 - 1.26141i) q^{4} +(1.01408 - 5.75112i) q^{5} +(-9.22789 - 3.35868i) q^{6} +(-4.69093 - 8.12493i) q^{7} +(5.41145 + 3.12430i) q^{8} +(-7.09445 + 5.95295i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 63 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.47714	−	1.76038i	0.738568	−	0.880191i	−0.257725	−	0.966218i	\(-0.582973\pi\)
							0.996293	+	0.0860277i	\(0.0274173\pi\)
\(3\)	−1.46156	−	4.01559i	−0.487186	−	1.33853i	−0.903218	−	0.429182i	\(-0.858802\pi\)
							0.416032	−	0.909350i	\(-0.363420\pi\)
\(5\)	1.01408	−	5.75112i	0.202816	−	1.15022i	−0.698025	−	0.716074i	\(-0.745937\pi\)
							0.900840	−	0.434151i	\(-0.142951\pi\)
\(7\)	−4.69093	−	8.12493i	−0.670133	−	1.16070i	−0.977866	−	0.209232i	\(-0.932904\pi\)
							0.307733	−	0.951473i	\(-0.400430\pi\)
\(11\)	2.33137	−	4.03804i	0.211942	−	0.367095i	−0.740380	−	0.672189i	\(-0.765354\pi\)
							0.952322	+	0.305094i	\(0.0986877\pi\)
\(13\)	−1.95908	+	5.38252i	−0.150698	+	0.414040i	−0.991954	−	0.126597i	\(-0.959594\pi\)
							0.841256	+	0.540637i	\(0.181817\pi\)
\(17\)	14.5397	+	12.2003i	0.855279	+	0.717664i	0.960946	−	0.276738i	\(-0.0892533\pi\)
							−0.105667	+	0.994402i	\(0.533698\pi\)
\(19\)		0			0
							\(23\)	−2.22619	−	12.6254i	−0.0967909	−	0.548928i	−0.994184	−	0.107696i	\(-0.965653\pi\)
0.897393	−	0.441232i	\(-0.145458\pi\)
\(29\)	20.6099	+	24.5620i	0.710687	+	0.846964i	0.993691	−	0.112155i	\(-0.0357752\pi\)
							−0.283003	+	0.959119i	\(0.591331\pi\)
\(31\)	−23.3288	+	13.4689i	−0.752542	+	0.434480i	−0.826612	−	0.562773i	\(-0.809735\pi\)
							0.0740699	+	0.997253i	\(0.476401\pi\)
\(37\)		−	0.140762i		−	0.00380438i	−0.999998	−	0.00190219i	\(-0.999395\pi\)
							0.999998	−	0.00190219i	\(-0.000605487\pi\)
\(41\)	3.98749	+	10.9555i	0.0972558	+	0.267208i	0.978774	−	0.204942i	\(-0.0657004\pi\)
							−0.881518	+	0.472150i	\(0.843478\pi\)
\(43\)	5.69525	−	32.2994i	0.132448	−	0.751148i	−0.844156	−	0.536098i	\(-0.819898\pi\)
							0.976603	−	0.215050i	\(-0.0689913\pi\)
\(47\)	8.22826	−	6.90433i	0.175069	−	0.146901i	−0.551043	−	0.834477i	\(-0.685770\pi\)
							0.726112	+	0.687576i	\(0.241325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	361.3.f.i.307.3		18
19.2	odd	18		19.3.d.a.12.3	yes	6
19.3	odd	18		361.3.d.c.293.1		6
19.4	even	9		361.3.f.h.333.3		18
19.5	even	9		361.3.b.b.360.2		6
19.6	even	9		361.3.f.h.127.1		18
19.7	even	3	inner	361.3.f.i.299.3		18
19.8	odd	6		361.3.f.h.116.3		18
19.9	even	9		361.3.f.h.262.1		18
19.10	odd	18	inner	361.3.f.i.262.3		18
19.11	even	3	inner	361.3.f.i.116.1		18
19.12	odd	6		361.3.f.h.299.1		18
19.13	odd	18	inner	361.3.f.i.127.3		18
19.14	odd	18		361.3.b.b.360.5		6
19.15	odd	18	inner	361.3.f.i.333.1		18
19.16	even	9		19.3.d.a.8.3	✓	6
19.17	even	9		361.3.d.c.69.1		6
19.18	odd	2		361.3.f.h.307.1		18
57.2	even	18		171.3.p.d.145.1		6
57.35	odd	18		171.3.p.d.46.1		6
76.35	odd	18		304.3.r.b.65.3		6
76.59	even	18		304.3.r.b.145.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.d.a.8.3	✓	6	19.16	even	9
19.3.d.a.12.3	yes	6	19.2	odd	18
171.3.p.d.46.1		6	57.35	odd	18
171.3.p.d.145.1		6	57.2	even	18
304.3.r.b.65.3		6	76.35	odd	18
304.3.r.b.145.3		6	76.59	even	18
361.3.b.b.360.2		6	19.5	even	9
361.3.b.b.360.5		6	19.14	odd	18
361.3.d.c.69.1		6	19.17	even	9
361.3.d.c.293.1		6	19.3	odd	18
361.3.f.h.116.3		18	19.8	odd	6
361.3.f.h.127.1		18	19.6	even	9
361.3.f.h.262.1		18	19.9	even	9
361.3.f.h.299.1		18	19.12	odd	6
361.3.f.h.307.1		18	19.18	odd	2
361.3.f.h.333.3		18	19.4	even	9
361.3.f.i.116.1		18	19.11	even	3	inner
361.3.f.i.127.3		18	19.13	odd	18	inner
361.3.f.i.262.3		18	19.10	odd	18	inner
361.3.f.i.299.3		18	19.7	even	3	inner
361.3.f.i.307.3		18	1.1	even	1	trivial
361.3.f.i.333.1		18	19.15	odd	18	inner